http://2010.igem.org/wiki/index.php?title=Special:Contributions/Ally_m&feed=atom&limit=50&target=Ally_m&year=&month=2010.igem.org - User contributions [en]2020-04-06T12:42:48ZFrom 2010.igem.orgMediaWiki 1.16.5http://2010.igem.org/Team:St_AndrewsTeam:St Andrews2010-10-28T03:02:11Z<p>Ally m: </p>
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<html><h1>University of St Andrews 2010 iGEM Team</h1><br />
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Welcome to the the University of St Andrews <a href="http://2010.igem.org/Main_Page"> iGEM 2010</a> Team Website. We are the first University of St Andrews iGEM Team. Our work is mainly based on Quorum Sensing and trying to investigate possible applications in Synthetic Biology.<br />
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<td width="260"> Comprised of 9 undergraduate <a href="http://2010.igem.org/Team:St Andrews/team/members">students</a> and guided by 5 <a href="http://2010.igem.org/Team:St Andrews/advisors">advisors</a>, our team stems from a variety of different scientific fields: Medicine, Biological Sciences, Chemistry, Computer Science and Physics.</td><br />
<td>Cholera is a bacterial disease that infects approximately 5 million people worldwide each year, approximately 100,000 of which are fatal. Symptoms of an acute cholera infection including diarrhoea and severe dehydration that can kill within hours if left untreated. <br/><br />
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Provision of safe water is critical to preventing cholera outbreaks however in many cases this is not feasible, particularly in areas recently hit by natural disaster. Our project involves investigating the basic science behind a potential means of preventing the disease through the application of synthetic biology. Lately Cholera outbreaks occurred in Pakistan and Haiti. In Haiti around 3500 confirmed causes have been reported <a href="http://www.elpais.com/articulo/internacional/Haiti/lucha/reloj/frenar/epidemia/colera/elpepiint/20101025elpepiint_5/Tes">news in spanish</a> and <a href="http://www.bbc.co.uk/news/world-latin-america-11632738">news in english</a>. Read more about our <a href="http://2010.igem.org/Team:St Andrews/project/objectives"> project</a>. </td><br />
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<td width="260"> We would like to thank our <a href="http://2010.igem.org/Team:St Andrews/team/sponsors">sponsors </a>: multiple faculties of the University of St Andrews, biotech companies, etc., whose generosity has made this all possible.</td><br />
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Part of our project is also to help other iGEM teams (both current and future) by fine-tunning the control over protein expression by designing new ribosome binding sites. <br />
We collaborated in iGEM 2010 with multiple teams in multiple contexts (wiki development (<a href="http://2010.igem.org/Team:TU_Delft">Delft</a> team), human practices data collection, exhange of DNA (<a href="http://2010.igem.org/Team:Sheffield">Sheffield</a> team)...).<br />
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In order to complete our project several biobricks must be constructed. Through use of standard protocols and procedures we plan to construct a bistable switch based on the Lux quorum sensing system and a CAI-1 sender using the cqsa gene from <i>Vibrio cholerae</i>. Follow our progress in the<a href="http://2010.igem.org/Team:St Andrews/project/laboratory"> laboratory. </a><br />
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<a href="http://2010.igem.org/Team:St Andrews/project/ethics">Human Practices</a> shapes the future and the very being of all science. Human Practices includes (but is not limited to) the purpose, effects and impact of science on society. A realm where ethics, economics and <i>E.coli</i> all intertwine.<br />
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To better understand the inner workings of the <i>Vibrio cholera</i> quorum sensing system we produced a series of computational models to simulate the operation of <i>Vibrio cholerae</i>. Based upon differential equations and solved computationally via the Fourth Order Runge-Kutta method our models provide a comprehensive view of <i>Vibrio cholerae</i> quorum sensing and bi-stable switching behaviour. To find out how we designed our models from the blackboard to the CPU check out our <a href="http://2010.igem.org/Team:St Andrews/project/modelling">modelling </a>. This work is potentially a flexible framework for future quorum sensing modelling. Possible future extensions can be other diffusion models for quorum sensing, other quorum sensing systems, etc. Different models are proposed as an example.</td><br />
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We proposed a new solution for the fine-tuning the translation of proteins designing Ribosome Binding Site by the usage of an RBS calculator. Check out more about this <a href="http://2010.igem.org/Team:St_Andrews/project/RBS">RBS</a>. <br />
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* [[Team:St_Andrews/FAQ | Frequently Asked Questions ]]<br />
* [http://igem.org/Main_Page iGEM 2010 ]</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_4Team:St Andrews/project/modelling/model 42010-10-28T02:58:32Z<p>Ally m: /* Results */</p>
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<h1> Model 4: Pseudo Multi Cell Two Dimensions </h1><br />
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=Theory & assumptions=<br />
<br />
Although our previous work presented many insights into the LuxR quorum sensing system, a new model of further complexity was deemed necessary to investigate some of the subtleties of bistable behaviour. This was accomplished by changing focus from the interactions in one cell, to the population of bacteria as a whole. In doing this, not only were we able to reproduce the results from previous models on a population-wide scale, but our results would be empirically verifiable in the lab.<br />
<br />
The primary development from previous models - the change of focus from one cell to the whole culture - was implemented by making many small adjustments to the previous differential equations we used. In addition to the biochemical reactions changing chemical concentrations, we also had a volumetric term describing the increase of volume (and subsequent decrease in concentration) as new cells were added to the colony. This can be understood by taking the time derivative of the HSL concentration: <br />
<br />
[[Image:4line1.gif|369px]]<br />
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[[Image:4line2.gif|488px]]<br />
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Previous models only included the first term of this equation - changes in the concentration due to biochemical processes inside the cell. In this form the concentration was also diluted by the second term as more bacteria were added to the system (when cells undergo binary fusion we assume the cell contents are equally distributed to both parent and daughter).<br />
<br />
Similar to model 3, the cell density was initially increased by a cell growth function, where the number of bacteria doubled every 20 minutes. Having ran to a realistic maximum cell density, there was a stationary phase in the cell growth to allow steady state concentrations of chemicals to be realised. The cell death function was then called, and the cell density was ran back to it's starting value.<br />
<br />
=Results=<br />
<br />
Our results show that in accordance with our single cell models, there is clear switching behaviour in the LuxR system. One point of interest is how the cell density at which the system becomes "up regulated" has changed from 10^6 CFU/ml to 10^9 CFU/ml. Tests have been carried out to confirm that this is related to the "diffusion constant" which describes the extent to which the cross-membrane concentration gradient of the signalling molecule (HSL) effects the rate of flow of HSL out of the cell. This shifted activation point can be seen below:<br />
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[[Image:Number_of_GFP_Molecules.JPG|800px]]<br />
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'''Figure 1: Number of GFP molecues in system against cell density'''<br />
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This graph shows the output of the model having been run through only the growth phase. As discussed above, the point at which the system becomes up-regulated in this setting has changed quite drastically from previous models. All other protein levels are in line with our previous calculations.<br />
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[[Image:GFPConcentration.JPG|821px]]<br />
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'''Figure 2: GFP concentration against cell density (Up and Down)'''<br />
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[[Image:GFP_fixed.JPG|821px]]<br />
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'''Figure 3: Number of GFP molecues in system against cell density (Up and Down)'''<br />
<br />
Here we have the GFP profile for the full cell growth and death cycle. Bistability is evident - indeed the system does not appear to turn "off" for the range of cell densities we have run through. Once again this anomaly can be analysed by considering the diffusion of HSL out of the cell.<br />
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[[Image:HSLINOUTONOFF.JPG|821px]]<br />
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'''Figure 4: HSL concentration against cell density '''<br />
<br />
Analysing this graphic tells us why the system doesn't revert back to it's down regulated state at any point in our simulation. As expected, when in the growth phase (the blue line) the concentration of HSL inside and outside the cells is the same. However, when the cells number is decreased (the red and green lines) this is not the case. It appears that the concentration outside is reduced but the concentration inside is the same at 10^10 CFU/ml as it is at 10000 CFU/ml. This explains why the GFP concentration does not decrease on the down phase of the simulation - ''' although the number of HSL molecules has diminished for the whole system, there is an uneven distribution in favour of the cell interior '''.<br />
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= Conclusions=<br />
While initial results from this model looked promising, the development of it was incomplete and further work would be required to realise its full potential.</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-28T02:51:07Z<p>Ally m: /* Conclusions */</p>
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<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
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==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. <br />
<br />
We can clearly see that the model produces bistable behaviour. The steady-state level of GFP remains for a lower cell density that it did on turning on, with the cells turning off at a lower cell density of approximately 1x10<sup>5</sup> CFU. Clearly our model produces at least some form of the bistability which we were looking for it to produce. More important however is to find a reason why this bistability occurs, what its effects are, and how we can manipulate it effectively.<br />
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[[Image:GFPvcell desnity.jpg|800px]]<br />
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'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
Figure 2 gives us an insight into how some of the chemicals taking part in the quorum sensing reactions change with cell density. Concentration of GFP has been plotted again on this graph to allow the concentrations of the other chemicals to be compared at key points. We can see that at the same cell density that GFP concentration begins to rise there is a similar rise in the concentrations of LuxR and LuxI in the system. Similarly there is a slightly delayed, and markedly lower increase in the concentration of LuxI/LuxR and GFP mRNAs present. The mRNA concentrations also reach a steady state at a lower cell density than the other chemicals. We make the assumption that as the cell density increases, the associated increase in HSL cause and increase also in HSL-LuxR, which then results in a greater concentration of the mRNA (and hence the proteins themselves). Although not shown on these figures, the HSL-LuxR concentration mirrors that of the mRNA exactly, and so remains at a low concentration of approximately 100-200 molecules/plasmid/cell. This fits the known theory which says that the HSL-LuxR complex is highly unstable, and will therefore only ever exists in small amounts.<br />
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[[Image:CellGrowth(alldata).jpg|800px]]<br />
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'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
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<br />
Direct comparisons should be made between figures 2 and 3, which show the cell growth and cell death phases of the same simulation. Clearly the bistability extends to chemicals other than GFP, with the LuxI and LuxR concentrations following a similar curve as GFP. The concentrations of both of these start to drop at a lower cell density than in figure 2, but the exact reason for this is hard to define.<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
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<br />
Possible reasons include:<br />
* Since chemicals take time to degrade there is an excess which remains present in the system until it is used in a reaction or degrades<br />
*<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have. This was done by performing a series of "parameter tests". The methods used and the results obtained can be found below.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests. For our tests we chose a value of 20 molecules/plasmid/cell, as an indication that the system has begun switching on, as experience had shown that this was an appropriate value.<br />
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[[Image:Bistabilitymeasure.jpg|center]]<br />
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'''Figure 4: Our method of bistability measurement'''<br />
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==Testing our switch==<br />
We decided that there were some key parameters which we would like to test since experimental knowledge and our experience of modelling the system suggested that they would be of interest. These were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on, otherwise there would be no bistability whatsoever. This was done by running our model many times, each time changing the value of the parameter under investigation. We then looked for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing the results for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR complex is key to the entire quorum sensing circuit and as such we would expect it to play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa. This is what we would expect, since when the association rate is high HSL-LuxR will accumulate much faster and so promote transcription much faster also. Similarly a high dissociation rate will result in very few HSL-LuxR molecules present to promote transcription and so GFP production will be less pronounced. There is a definite central plateau on the graph in which the rates of association and dissociation are such that the system switches on and this is the parameter space in which we were to perform our tests.<br />
<br />
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[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
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'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure 6). This would make biological sense, since if we have HSL being produced in the cells at a high rate there will be a plentiful supply with which the LuxR can bind. Similarly a low production rate or high degradation rate will result in a shortage of HSL.<br />
<br />
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[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
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'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The next stage of testing was to run our simulation through these two ranges of values and observe the values produced for ΔCell density. The results of these tests are shown below. From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system. Thinking through these two results, logical conclusions can be drawn. With a high value for kConv, the saturation concentration of HSL is reached much quicker, and so when the cell death cycle is invoked, the difference between the two graphs is smaller than if a low value had been used. The key point to think about is that kConv does not change the final concentrations of any of the chemicals, rather it decreases the cell density required to reach those final concentrations.<br />
<br />
Looking now to the HSL degradation rate, a higher degradation rate will have the opposite effect of a high conversion rate, causing the system to take a longer time to reach the steady state. Thus when the cell death cycle is invoked a greater level of bistability will be displayed. <br />
<br />
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[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
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'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
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[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
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'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
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===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure 9.<br />
<br />
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[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
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'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading at such a high rate that there is not enough binding between it and the HSL to sustain the quorum sensing network, and thus the system never switches on during the cell growth cycle. Before this point there is an exponential relationship between the level of bistability and the LuxR degradation rate. The explanation for this we hypothesise is the same as was the case for the HSL degradation, in that if the degradation rate is high the system takes longer to reach the equilibrium and so the bistability is greater.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
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'''Figure 10: Point of up regulation as a function of RBS efficency and PoPs'''<br />
<br />
From this graphic we can see that, as predicted, the cell density at which the system turned on decreased as the PoPs and RBS efficiency were increased. The PoPs and the RBS efficency have almost the exact same effect on the point of up-regulation, hence the almost symmetrical plot. If viewed from above (as shown below) we can see that for very low PoPs & RBS efficiencies (<0.15) the system does not turn on at all.<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
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'''Figure 11: Point of up regulation as a function of RBS efficency and PoPs (from above)'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12: Delta cell density as a function of RBS efficeny and PoPs'''<br />
<br />
This plot confirms our initial hypothesis: changing the RBS expression and PoPs alters the "level of bistability" of the system. If analaysed (easier from above shown in the plot below) there is a very discernable blank spot at high PoPs and RBS efficencies. At this point the results are "Not A Number" which indicates that there is a malfuction in the model. This phonomena is explored further below.<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
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'''Figure 13: Delta cell density as a function of RBS efficeny and PoPs (from above)'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-28 01-52-10.png|800px]]<br />
<br />
'''Figure 14: HSL, HSL-LuxR, LuxR Concentration for "Haywire" cell (High PoPs & RBS efficency)'''<br />
<br />
There are three phases in this graphic. The intial section shows the concentrations gradually increasing as the cell density is increased. Having switched on, the system then pleatues as predicted in low PoPs & RBS efficency models. However, in contrast to these models, the plateau then switches away from the steady state concentrations and reaches exponential values. Eventually this becomes a timestep issue, however further study is needed to understand why this happens intiially. Regardless, this happens at an extremely high cell density which in reality is never reached, and therefore our model should be giving realistic results when not in this parameter space. <br />
<br />
=Conclusions=<br />
Our third modelling attempt could certainly be seen as the most successful and most useful for the purposes of our project. This model produced the switching behaviour we were looking for, and allowed us to perform detailed parameter tests on the system. What we have actually achieved is a tunable system, whereby the user can adjust the level of bistability to suit a particular need.</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-28T02:47:00Z<p>Ally m: /* LuxR degradation */</p>
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<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
</html><br />
<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. <br />
<br />
We can clearly see that the model produces bistable behaviour. The steady-state level of GFP remains for a lower cell density that it did on turning on, with the cells turning off at a lower cell density of approximately 1x10<sup>5</sup> CFU. Clearly our model produces at least some form of the bistability which we were looking for it to produce. More important however is to find a reason why this bistability occurs, what its effects are, and how we can manipulate it effectively.<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
Figure 2 gives us an insight into how some of the chemicals taking part in the quorum sensing reactions change with cell density. Concentration of GFP has been plotted again on this graph to allow the concentrations of the other chemicals to be compared at key points. We can see that at the same cell density that GFP concentration begins to rise there is a similar rise in the concentrations of LuxR and LuxI in the system. Similarly there is a slightly delayed, and markedly lower increase in the concentration of LuxI/LuxR and GFP mRNAs present. The mRNA concentrations also reach a steady state at a lower cell density than the other chemicals. We make the assumption that as the cell density increases, the associated increase in HSL cause and increase also in HSL-LuxR, which then results in a greater concentration of the mRNA (and hence the proteins themselves). Although not shown on these figures, the HSL-LuxR concentration mirrors that of the mRNA exactly, and so remains at a low concentration of approximately 100-200 molecules/plasmid/cell. This fits the known theory which says that the HSL-LuxR complex is highly unstable, and will therefore only ever exists in small amounts.<br />
<br />
[[Image:CellGrowth(alldata).jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
Direct comparisons should be made between figures 2 and 3, which show the cell growth and cell death phases of the same simulation. Clearly the bistability extends to chemicals other than GFP, with the LuxI and LuxR concentrations following a similar curve as GFP. The concentrations of both of these start to drop at a lower cell density than in figure 2, but the exact reason for this is hard to define.<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Possible reasons include:<br />
* Since chemicals take time to degrade there is an excess which remains present in the system until it is used in a reaction or degrades<br />
*<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have. This was done by performing a series of "parameter tests". The methods used and the results obtained can be found below.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests. For our tests we chose a value of 20 molecules/plasmid/cell, as an indication that the system has begun switching on, as experience had shown that this was an appropriate value.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test since experimental knowledge and our experience of modelling the system suggested that they would be of interest. These were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on, otherwise there would be no bistability whatsoever. This was done by running our model many times, each time changing the value of the parameter under investigation. We then looked for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing the results for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR complex is key to the entire quorum sensing circuit and as such we would expect it to play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa. This is what we would expect, since when the association rate is high HSL-LuxR will accumulate much faster and so promote transcription much faster also. Similarly a high dissociation rate will result in very few HSL-LuxR molecules present to promote transcription and so GFP production will be less pronounced. There is a definite central plateau on the graph in which the rates of association and dissociation are such that the system switches on and this is the parameter space in which we were to perform our tests.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure 6). This would make biological sense, since if we have HSL being produced in the cells at a high rate there will be a plentiful supply with which the LuxR can bind. Similarly a low production rate or high degradation rate will result in a shortage of HSL.<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The next stage of testing was to run our simulation through these two ranges of values and observe the values produced for ΔCell density. The results of these tests are shown below. From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system. Thinking through these two results, logical conclusions can be drawn. With a high value for kConv, the saturation concentration of HSL is reached much quicker, and so when the cell death cycle is invoked, the difference between the two graphs is smaller than if a low value had been used. The key point to think about is that kConv does not change the final concentrations of any of the chemicals, rather it decreases the cell density required to reach those final concentrations.<br />
<br />
Looking now to the HSL degradation rate, a higher degradation rate will have the opposite effect of a high conversion rate, causing the system to take a longer time to reach the steady state. Thus when the cell death cycle is invoked a greater level of bistability will be displayed. <br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure 9.<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading at such a high rate that there is not enough binding between it and the HSL to sustain the quorum sensing network, and thus the system never switches on during the cell growth cycle. Before this point there is an exponential relationship between the level of bistability and the LuxR degradation rate. The explanation for this we hypothesise is the same as was the case for the HSL degradation, in that if the degradation rate is high the system takes longer to reach the equilibrium and so the bistability is greater.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10: Point of up regulation as a function of RBS efficency and PoPs'''<br />
<br />
From this graphic we can see that, as predicted, the cell density at which the system turned on decreased as the PoPs and RBS efficiency were increased. The PoPs and the RBS efficency have almost the exact same effect on the point of up-regulation, hence the almost symmetrical plot. If viewed from above (as shown below) we can see that for very low PoPs & RBS efficiencies (<0.15) the system does not turn on at all.<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11: Point of up regulation as a function of RBS efficency and PoPs (from above)'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12: Delta cell density as a function of RBS efficeny and PoPs'''<br />
<br />
This plot confirms our initial hypothesis: changing the RBS expression and PoPs alters the "level of bistability" of the system. If analaysed (easier from above shown in the plot below) there is a very discernable blank spot at high PoPs and RBS efficencies. At this point the results are "Not A Number" which indicates that there is a malfuction in the model. This phonomena is explored further below.<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13: Delta cell density as a function of RBS efficeny and PoPs (from above)'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-28 01-52-10.png|800px]]<br />
<br />
'''Figure 14: HSL, HSL-LuxR, LuxR Concentration for "Haywire" cell (High PoPs & RBS efficency)'''<br />
<br />
There are three phases in this graphic. The intial section shows the concentrations gradually increasing as the cell density is increased. Having switched on, the system then pleatues as predicted in low PoPs & RBS efficency models. However, in contrast to these models, the plateau then switches away from the steady state concentrations and reaches exponential values. Eventually this becomes a timestep issue, however further study is needed to understand why this happens intiially. Regardless, this happens at an extremely high cell density which in reality is never reached, and therefore our model should be giving realistic results when not in this parameter space. <br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_1Team:St Andrews/project/modelling/model 12010-10-28T02:40:12Z<p>Ally m: /* Conclusions */</p>
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<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 1: Single Cell without Feedback </h1><br />
</html><br />
<br />
= Theory & assumptions =<br />
<br />
The first method we attempted was to put the user in control of the change in HSL within the cell, and take any HSL produced and remove it from the system entirely. In this fashion a gradual increase in cell density was simulated by a linear “ramp up” of the HSL in the cell; a gradual decrease (presumably due to cell death) was simulated by a linear “ramp down” in HSL. The activation of the cell was measured by plotting the GFP concentration. The practicalities of our code can be viewed [[Team:St_Andrews/project/modelling/downloads|here]].<br />
<br />
<br />
<br />
[[Image:FullDisconnected.jpg]] <br />
<br />
'''Figure 1: Schematic diagram of the LuxR quorum sensing circuit'''<br />
<br />
==Parameters ==<br />
In finding the parameters for our model we searched through a great number of scientific papers in order to collate the various pieces of data which we required. Ultimately however, the most useful resource in our search proved to be the previous work done by other iGEM teams, notably Aberdeen’s Pico Plumber project of 2009. A comprehensive list of all rate constants used is given in the table below:<br />
<br />
<br />
[[Image:Parametertablev.2.jpg]]<br />
<br />
=Results=<br />
In order to simulate the effect of the cell colony growing in number we first set the change in HSL concentration to a fixed value and ran the model until a steady state was reached. It can be clearly seen from figure that the level of GFP in the cell reaches a point where the concentration starts to sharply increase before a plateau is reached and there is no further activity. This is the switch-on, where the cell has moved from a down-regulated to an up-regulated state and starts to produce greater amounts of HSL, LuxR, LuxI and GFP. <br />
<br />
<br />
<br />
[[Image:LuxRSingleCellNoLoopHSLRampUpDowncurve.jpg|800px]]<br />
<br />
'''Figure 2: Graph of the HSL input over time''' <br />
<br />
<br />
<br />
As figure 2 shows, when ramping the HSL down we chose a rate which was significantly slower than that used when performing a ramp up in HSL. <br />
<br />
<br />
<br />
[[Image:LuxRSingleCellNoLoopRampUp2Down6.jpg|800px]]<br />
<br />
'''Figure 3: Graph of GFP v HSL'''<br />
<br />
<br />
<br />
The final values of all the molecules are RNAs are then substituted into the model as the initial values for the next run and the HSL is reduced at a fixed rate, producing the data shown in figure. It can be seen that the up-regulation is maintained for a much lower HSL threshold than when the first run. In fact, the cell never fully returns to a down-regulated state again. After trying to achieve this through decreasing the rate at which the HSL is taken away and increasing the number of increments for which the model is run, it was decided that our model simply did not produce the behaviour we were looking for. Figure shows how the concentrations of the various molecules and the mRNAs changes over the transition from down- to up-regulated states. The model predicts low concentrations of the HSL-LuxR complex which is theoretically predicted, but in general the results were unsatisfactory as to drawing any real conclusions about the causes or otherwise of the bistability.<br />
<br />
<br />
<br />
[[Image:LuxRSingleCellNoLoopRampUp2Large.jpg|800px]]<br />
<br />
'''Figure 4: Graph showing the variation in the concentrations of all the components of the system with time during the HSL up ramp'''<br />
<br />
<br />
<br />
It should be noted that while figure 4 actually contains plots of the mRNA concentrations and of the HSL-LuxR complex, they are so small when compared to the other components that they cannot be appropriately scaled.<br />
<br />
<br />
<br />
[[Image:LuxRSingleCellNoLoopRampUp2Small.jpg|800px]]<br />
<br />
'''Figure 5: Graph showing the change in concentrations of HSL-LuxR and mRNA with time over the HSL up ramp'''<br />
<br />
=Conclusions=<br />
Our first model was met with some success in that switching behaviour did seem to be present. However, after further analysis we found that the 'bistability' was not true to what we would expect and so was disregarded. In essence this model was too simplistic for any quantitative purpose and so further work was continued.<br />
<br />
=References=<br />
[1] Goryachev, A.B., D.J. Toh and T. Lee, “System analysis of a quorum sensing network: Design constraints imposed by the functional requirements, network topology and kinetic constant.” BioSystems 2006 '''83''', 178-187<br />
<br />
[2] Aberdeen 2009, "Parameters" iGEM wiki, <http://2009.igem.org/Team:Aberdeen_Scotland/parameters><br />
<br />
[3] Alon, Uri. “An Introduction to Systems Biology Design Principles of Biological Circiuts.” London: Chapman & Hall/CRC, 2007<br />
<br />
[4] Subhayu Basu, “A synthetic multicellular system for programmed pattern formation.” Nature April 2005, '''434''', 1130-1134</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_2Team:St Andrews/project/modelling/model 22010-10-28T02:29:39Z<p>Ally m: /* Conclusions */</p>
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<br />
<html><br />
<h1> Model 2: Single Cell With Loop </h1><br />
</html><br />
<br />
=Theory & assumptions=<br />
Our second model introduced the concept of an external environment into which the HSL could diffuse into. This was assumed to be of infinite volume and thus the concentration of HSL in this region would always be zero, regardless of the quantity of HSL leaving the cell. <br />
In order to implement this modification we produced a new differential to desrcibe the change in HSL.<br />
<br />
<br />
[[Image:HSL2.jpg|900px]]<br />
<br />
'''Figure 1: Differential equation describing HSL change'''<br />
<br />
<br />
The creation of HSL is dictated by kConv, which is associated with the ability of LuxI to catalyse the conversion of resources into HSL. The concentration is reduced by the association with LuxR and by natural degredation. It is also affected by the degredation of molecules in the HSL-LuxR complex in the same way that LuxR is. We also introduce a diffusion term which accounts for HSL diffusing out of the cell due to a concentration gradient between the cell and the external environment. <br />
<br />
The code on which this model is based can be found [[Team:St_Andrews/project/modelling/downloads|here]].<br />
<br />
=Results=<br />
The switching on behaviour produced by this model is shown in figure . This switching on can be interpreted as the cell turning itself on due to the HSL it produces not diffusing out of the cell at a high enough rate. Therefore there is a considerable concentration of HSL remaining in the cell to maintain the feedback loop and produce more HSL. Thus regardless of the initial concentrations our cell always switches on. This does not match with the true biological scenario at all. Quorum sensing is designed such that cells will only become up-regulated when present at a certain critical cell density. Therefore the suggestion made by our results that a single cell can self-activate undermines this principle entirely.<br />
<br />
<br />
[[Image:LuxRSingleCellLoopRampUp(correctedmRNA).jpg|800px]]<br />
<br />
'''Figure 2: Graph of GFP concentration against time'''<br />
<br />
<br />
There is an apparent switching on of the system, but this is in fact not what we would expect seeing as our model is for a single cell in a surrounding environment of infinite volume. Therefore any HSL produced by the cell should become diluted to the point where the concentration externally can be said to be negligible.<br />
<br />
<br />
[[Image:LuxRSingleCellLoopRampUp8th(Large).jpg|800px]]<br />
<br />
'''Figure 3: Graph of GFP, LuxI, LuxR, HSL-LuxR and mRNA concentrations against time'''<br />
<br />
<br />
We can see from figure 3 that the model predicts huge concentrations of the HSL-LuxR complex to be present in the cell, which strongly contradicts previous work. The complex is very unstable and should therefore exist in only small concentrations within the cell. This gives us some idea as to the fallibility of this model. It is also of note that in general the order of magnitude if the concentrations output from this model are a few below that from [[Team:St_Andrews/project/modelling/model_1|model 1]].<br />
<br />
[[Image:LuxRSingleCellLoopRampUpmRNA.jpg|800px]]<br />
<br />
'''Figure 4: Graph of mRNA concentration against time'''<br />
<br />
<br />
The concentrations of mRNA are equal for GFP, LuxR and LuxI, and whatsmore are comparable to those seen in our previous model.<br />
<br />
=Conclusions=<br />
Our model demonstrated switching behaviour and also contained some concept of diffusion. However, it also came with its fair share of problems. Among these was the fact that a single cell seems to switch itself on regardless of the initial conditions or of the rate constants. In addition there was no concept of other cells. As such we felt this particular model was unsuitable for use and began work on an improved version.</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/downloadsTeam:St Andrews/project/modelling/downloads2010-10-28T02:16:45Z<p>Ally m: </p>
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<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
==Model code==<br />
<br />
The C++ code below forms the framework for all of our projects. Each project used a slightly tweaked version of the files given here, however to keep things straightforward we have uploaded only the generic code.<br />
<br />
[[Media:StAndrewsMainFile.txt|Main body of our code]]<br />
<br />
[[Media:StAndrewsRungeKuttaSolver.txt|Runge-Kutta solver code]]<br />
<br />
[[Media:StAndrewsHeaderFile.txt|Header file]]<br />
<br />
[[Media:StAndrewsDifferentialEquations.txt|Differential equation code]]<br />
<br />
[[Media:StAndrewsCellGrowth.txt|Cell Growth function code]]</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/downloadsTeam:St Andrews/project/modelling/downloads2010-10-28T02:06:36Z<p>Ally m: </p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
==Models==<br />
<br />
The C++ code below forms the framework for all of our projects. Each project used a slightly tweaked version of the files given here, however to keep things straightforward we have uploaded only the generic code.<br />
<br />
[[Media:StAndrewsMainFile.txt|Main body of our code]]<br />
<br />
[[Media:StAndrewsRungeKuttaSolver.txt|Runge-Kutta solver code]]<br />
<br />
[[Media:StAndrewsHeaderFile.txt|Header file]]<br />
<br />
[[Media:StAndrewsDifferentialEquations.txt|Differential equation code]]<br />
<br />
[[Media:StAndrewsCellGrowth.txt|Cell Growth function code]]</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/models/RK4Team:St Andrews/project/modelling/models/RK42010-10-28T02:02:07Z<p>Ally m: </p>
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<br />
<html><br />
<h1> Differential Solver (RK4) </h1><br />
</html><br />
<br />
=Introduction=<br />
Our method of solving differential equations is based on Fourth Order Runge-Kutta Method. This technique is the most widely used way of numerically solving differential equations and various methods of implementation were looked at. Most of our coding has been based on the work of Aberdeen 2009 iGEM team who used the same method in their modelling. We would like to thank the team for their work and making it available to others like us for future use.<br />
<br />
=Mathematical basis=<br />
<br />
While the strict mathematical derivation of the Runge-Kutta method is available (see [http://www.ss.ncu.edu.tw/~lyu/lecture_files_en/lyu_NSSP_Notes/Lyu_NSSP_AppendixC.pdf here for details] ), we omit it here and instead give a brief explanation of the principles behind the technique.<br />
<br />
There are several key formulae in the Fourth Order Runge-Kutta algorithm, which are:<br />
<br />
<br />
[[Image:Runge kutta eqns.jpg]]<br />
<br />
'''Figure 1: Runge-Kutta equations'''<br />
<br />
<br />
The iteration of the x-values is done very simply adding a fixed step-size (h) at each iteration, thus resulting in a constant increase in the x-value according to the value of h chosen for a particular purpose.<br />
<br />
The y-value iteration contains a much more elegant set of relationships. It should be clear to you that the y-iteration formula is in fact a weighted average of the four value k values, k<sub>1</sub>, k<sub>2</sub>, k<sub>3</sub> and k<sub>4</sub>. Even at a first glance it should also be apparent that a 'weight' of 2/6 is given to k<sub>2</sub> and k<sub>3</sub> while a smaller weighting of 1/6 is attributed to k<sub>1</sub> and k<sub>2</sub>. What then do these k-value correspond to geometrically?<br />
<br />
<br />
Look first at k<sub>1</sub>. It is equal to the function f(x<sub>n</sub>,y<sub>n</sub>) multiplied by the step-size h, which is just the Euler prediction for Δy. It is the vertical distance from the previous point to the next Euler-predicted point.<br />
<br />
<br />
Then consider k<sub>2</sub>. It should be noted that the x-value at which the function is being evaluated is halfway across the prediction interval and the y-value consists of the current y-value plus half of the Euler predicted Δy. So then the function is evaluated at a point lying halfway between the current point and the Euler-predicted next point. When we evaluate the function at this point what is produced is an estimate of the slope of the solution curve at this halfway point on the curve. Multiplying this result by h gives a prediction of the y-jump made by the solution across the entire interval. However, this prediction is made not on the slope of the solution at the left end of the interval but on the estimated slope halfway to the next Euler-predicted point.<br />
<br />
<br />
A comparison can be made between k<sub>3</sub> and k<sub>2</sub>, with the only difference in their formulation being the replacement of k<sub>1</sub> by k<sub>2</sub>. Here the function evaluated at this point gives an estimate of the solution slope at the midpoint of the prediction interval but one which is based in the y-jump predicted by k<sub>2</sub> rather than the Euler-predicted one. When multiplied by h we get a further estimate of the y-jump made by the solution across the whole interval.<br />
<br />
<br />
Finally, k<sub>4</sub> evaluates the function at the right-hand side of the prediction interval. The y-value at which the function is evaluated, y<sub>n</sub>+k<sub>3</sub> is an estimate of the y-value at the right-hand side based on the estimate of the y-jump made by k<sub>3</sub>. In its entirety, k<sub>4</sub> gives a final estimate of the y-jump made by the solution across the entire width of the prediction interval.<br />
<br />
<br />
So to provide a summary: each k<sub>i</sub> yields an estimate of the y-jump made by the solution across the entire width of the prediction interval h, with each using the previous k<sub>i</sub> to make its estimate. The Runge-Kutta formula can be viewed as the y-value of the current point plus a weighted average of four different predictions for the slope.<br />
<br />
<br />
[[Image:Picture 4.png|600px]]<br />
<br />
'''Figure 2: Runge-Kutta technique'''<br />
=Implementation=<br />
The code which performs the operations in the Runge-Kutta algorithm is shown below. <br />
<br />
The operation of the algorithm is thus; k<sub>1</sub> is calculated from the value stored from the solved differential equation which is stored in the array "ode", multiplied by our step size h which can be chosen by the user. This process is looped round for each variable to produce an array with an element for variable in the system. Values for y<sub>2</sub> are then calculated using the value for k<sub>1</sub> and the y value from the previous iteration, which is initially zero. Again this process is looped round for each variable. Similar processes are followed for the other values, and finally the y value if found by averaging the calculated k-values. The Runge-Kutta function is called for every increment the model is run.<br />
<br />
void RungeKutta(double y[], double dy[]){<br />
double k1[var], k2[var], k3[var], k4[var], y2[var], y3[var], y4[var], *pode;<br />
double h = 0.025;<br />
pode = ode(y,dy);<br />
<br />
//Calculates k1 (for all variables) -> The slope at the start of the interval (h)<br />
for (int i = 0; i < var; ++i) {<br />
k1[i] = *(pode + i) * h;<br />
}<br />
<br />
// Calculates k2 (for all variables) -> The slope at the midpoint of the interval (h),<br />
for (int i = 0; i < var; ++i) {<br />
y2[i] =y[i] + k1[i] * 0.5 ;<br />
}<br />
pode = ode(y2,dy);<br />
for (int i = 0; i < var; ++i) {<br />
k2[i] = *(pode + i) * h;<br />
}<br />
<br />
// Calculates k3 (for all variables) -> The slope at the midpoint of... using the y value (y2) determined from k2.<br />
for (int i = 0; i < var; i++) {<br />
y3[i]=y[i] + k2[i] * 0.5;<br />
}<br />
pode = ode(y3,dy);<br />
for (int i = 0; i < var; i++) {<br />
k3[i] = *(pode + i) * h;<br />
}<br />
<br />
// Calculates k4 (for all variables) -> The slope at the end of the interval (h).<br />
for (int i = 0; i < var; i++) {<br />
y4[i]= y[i] + k3[i];<br />
}<br />
pode=ode(y4,dy);<br />
for (int i = 0; i < var; i++) {<br />
k4[i] = *(pode + i) * h;<br />
}<br />
<br />
// Calculates the new y values (for all variables).<br />
for (int i = 0; i < var; i++){<br />
y[i] +=((k1[i] + 2*k2[i] + 2*k3[i] + k4[i])/6);<br />
}<br />
}<br />
<br />
=References=<br />
[1] Barker.C.A, "Numerical Methods for Solving Differential Equations,The Runge-Kutta Method, Theoretical Introduction", San Joaquin Delta College, 2009 http://calculuslab.deltacollege.edu/ODE/7-C-3/7-C-3-h.html, [Accessed 13 September 2010]<br />
<br />
[2] "Numerical Recipes in Fortran 77: The Art of Scientific Computing", Cambridge University Press, 1992, pp 704-707</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modellingTeam:St Andrews/project/modelling2010-10-28T01:52:44Z<p>Ally m: </p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Modelling Background </h1><br />
</html><br />
<br />
Hello and welcome to the University of St Andrews 2010 iGEM modelling pages. Here we present our work on the mathematical and computational modelling of the V.Cholerae bacterium and its quorum sensing system. What follows is an overall description of our work and our methodologies. For a more in depth description of our individual models please refer to our [[Team:St_Andrews/project/modelling/models| models page]] and to our [[Team:St_Andrews/project/modelling/downloads|downloads page]] where you can freely download and review our work<br />
<br />
<html><br />
<center><br />
<object width="480" height="385"><param name="movie" value="http://www.youtube.com/v/SZoUAoAFqmg?fs=1&amp;hl=en_US&amp;rel=0"></param><param name="allowFullScreen" value="true"></param><param name="allowscriptaccess" value="always"></param><embed src="http://www.youtube.com/v/SZoUAoAFqmg?fs=1&amp;hl=en_US&amp;rel=0" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="480" height="385"></embed></object><br />
</center><br />
</html><br />
<br />
'''A short animation created to explain the basic biological principles behind our modelling'''<br />
<br />
= From the Biology... =<br />
== Quorum sensing in E.Coli: The LuxR system ==<br />
The LuxR quorum sensing system is common among the vibrio family, and the lux system of the luminescent bacterium V.fischeri has been studied in great detail. This gave us a plentiful supply of scientific papers and review articles on which to base our understanding of the system.<br />
The LuxR system is underpinned by three quorum sensing molecules, those being the signalling molecule HSL; the HSL synthase LuxI and the transcriptional regulator LuxR. HSL may be generated in the cell by two means, either diffusion through the cell membrane from the external environment or being produced within the cell. Throughout the development of our models we considered a number of ways of trying to replicate this, and the details of these differenent techniques can be found in further sections. Whilst in the cell it may bind to LuxR, and two HSL molecules form a tetramer with two complementary LuxR molecules to form a complex which can then activate the lux operon. In the wild-type, this operon contains the genes coding for LuxI and those ultimately responsible for GFP production, with the luxR gene being located upstream and being constitutively produced. However, in our re-engineered operon the luxR gene lies downstream of the lux operon, thus its expression is also promoted. Hence once the promoter is activated, there is an increased production of LuxI, LuxR and also of GFP, which can therefore be used as an indicator of the promoter activity within the cell. Once transcription has been completed, the LuxI catalyses the production of more HSL from resources within the cell (which we have assumed exist in a sufficiently high concentration that they can be considered infinite). This may leave the cell due to a concentration gradient if the cell density in the medium is low, or otherwise will be used again in the loop, causing a positive feedback effect which will cause a high increase in the concentration of HSL. In this way the single bacterium can communicate with those surrounding it in order to evaluate whether it exists in sufficient numbers to overcome the immune system.<br />
<br />
==Quorum sensing in V.Cholerae: The LuxPQ system ==<br />
The V.cholerae quorum sensing circuit is considerably more complex than that of the LuxR system, consisting as it does of two circuits working in tandem. These are the CqsS and the LuxPQ circuits, which communicate with other bacterium and relay this information to a shared component, LuxU which ultimately controls the expression of virulence factors. <br />
<br />
[[Image:CholeraQS diagram.jpg|800px]]<br />
<br />
'''Figure 1: Schematic diagram of the LuxPQ and CqsS quorum sensing systems of V. cholerae'''<br />
<br />
<br />
<br />
In the CqsS circuit, the CAI-1 (cholera autoinducer) molecule attaches to a dimerised complex of two CqsS molecules which exist in the inner membrane of the bacterium. When the CqsS dimer has not CAI-1 attached, it acts as a kinase and transfers phosphate groups down to LuxU which in turn phosphorylates LuxO. Similarly, another autoinducer molecule, AI-2 can attach to LuxPQ which is also in the inner membrane of the bacterium. When unattached the LuxPQ also phosphorylates LuxU in the same manner as CqsS and do the two systems are connected here. When phosphorylated, LuxO acts as a transcriptional regulator in the transcription of 4 sRNAs (small RNAs) which, in conjunction with the protein Hfq inhibit the production of what we have named the 'master regulator' HapR. HapR inhibits the expression of virulence factors, so at low cell density i.e. when CqsS and LuxPQ are kinases, HapR is repressed, and virulence factors are expressed.<br />
<br />
<br />
Converse to this, if CAI-1 attaches to CqsS or AI-2 to LuxPQ, the receptors change to phosphatases and remove phosphate groups from the LuxU, which removes them from LuxO. Thus the 4 sRNAs are not transcribed and there is no inhibition of HapR, such that virulence factors are no longer expressed. Instead, HapR promotes the production of a protein which cuts the V.Cholerae from the gut wall and allows it to pass out of the body in order to infect a different host. In this way, at low cell density the cholera causes infection in the host until reaching a critical density at which the behaviour changes and the bacteria exit the system. The internal dynamics of the system are more complex than this however. HapR attaches to its own promoter and thus is self-repressing, as is LuxO. HapR also promotes the transcription of the sRNAs and the sRNAs repress the transcription of LuxO. The system also has many unknown factors at play, however, and in combination with the lack of available rate constants makes it an ominous prospect to accurately model.<br />
<br />
== Components ==<br />
Our project can be divided into two main components: the engineering of a bistable switch into the LuxR quorum sensing system, and the integration of CqsA into the LuxR circuit. The aim of the modelling side of the project was to treat these two tasks independently and on their completion construct a combined model. However, this initial aim was proven to be almost impossible due to the lack of rate constants for the cholera system, which has only been understood in its full complexity relatively recently [1]. In order to reach a compromise, we have built a number of qualitatively accurate models for the bistable LuxR system, and outlined a framework of differential equations for the cholera system which are correct at the time of writing, and which requires more rate constants to be of further use. The work done on the bistable switch included an investigation into why exactly such a configuration of genes exhibits hysteretic behavior, and what parameters are of importance in determining the “level” of bistability of the system. <br />
<br />
<br />
The purpose of our modelling is to accurately replicate the behaviour of the bistable switch in order to allow our E.coli to be tuned so that they switch off only when we are sure that all the V.cholerae have left the system and the host is free of infection.<br />
<br />
<br />
= To the Computer=<br />
== Under the hood ==<br />
All our models are based upon a series of ordinary differential equations (ODEs) each of which have been derived from either prior research papers or our own research. These equations are solved computationally via the [[Team:St_Andrews/project/modelling/models/RK4|Fourth Order Runge-Kutta Method]] (RK4) - the classical iterative method of approximating numerical ODEs. Seeking full control of the implementation of our model we decided against the use of mathematical packages and instead decided upon using a fully-fledged programming language to develop our models. Initially we decided upon GNU R, a programming language and environment for statistical computing. While R allowed for the rapid development of our initial models it was found that R lacked the fine grain control and the raw machine efficiency that we so required. A key problem that we encountered was that the majority of ODE solver libraries available for R were not in fact written in R, instead they were written in FORTRAN and C and the R component of the library meerly passed data to and from the external program. This was problematic as it did not allow us to access and modify the ODE solving algorithm which, in order to acquire the full set of data pertinent to any of or models and thus made future use of R unfeasible. Instead we turned to C++, a far lower level language than R hoping for greater power of design. Being a compiled language (with an efficient compiler) C++ programs proved far faster than their R counterparts and offered a far finer degree of control. Initially we developed our own RK4 based ODE solver and put to work developing all our models as C++ programs. Output of the each of the models was processed and graphed using the gnuplot graphing utility. The first step in developing a usable model of the behaviour of the bistable switch was to manufacture a working model of the LuxR quorum sensing circuit (See our animation for an overview of how the circuit works). Through discussions with the biologists, we were able to generate a circuit diagram which allowed us to better visualise the workings of the system and split it up into individual reactions which we could then assign differential equations to. Ongoing testing of the model, in conjunction with further insights into the biology led to tweaking and adjustments, leading to further models with added degrees of complexity and greater realism. The various models and their associated results can be found in the [[Team:St_Andrews/project/modelling/models|models]] section. Our ultimate goal is to create a model accurately predicting the results of adding samples of our engineered E.coli into the human gut in the event of a cholera infection.<br />
<br />
==What we used ==<br />
In line with the nature of iGEM all of the software used in development of our models was Free software, operating on both Windows and Mac OS systems. We wish to thank each of the following communities for producing such high quality software:<br />
<br />
* [http://gcc.gnu.org/ GCC], used to compile our C++ programs<br />
* [http://www.vim.org vim], a text editor used to write programs prior to compiling them<br />
* [http://www.gnuplot.info gnuplot], command-line plotting software used to produce our plots<br />
* [http://www.codeblocks.org/ Code::Blocks], a development environment used for creating our programs<br />
<br />
= Download =<br />
In a continuing commitment to Free Software, all of our models are released under the [http://www.gnu.org/licenses/gpl.html GNU General Public License Version 3]. If you would like to view the code for our software it is available in the [[Team:St_Andrews/project/modelling/downloads|downloads]] page.<br />
<br />
= Models =<br />
Full detail of our models can be found at [[Team:St_Andrews/project/modelling/models|"Models"]].</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modellingTeam:St Andrews/project/modelling2010-10-28T01:50:01Z<p>Ally m: </p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Modelling Background </h1><br />
</html><br />
<br />
Hello and welcome to the University of St Andrews 2010 iGEM modelling pages. Here we present our work on the mathematical and computational modelling of the V.Cholerae bacterium and its quorum sensing system. What follows is an overall description of our work and our methodologies. For a more in depth description of our individual models please refer to our [[Team:St_Andrews/project/modelling/models| models page]] and to our [[Team:St_Andrews/project/modelling/downloads|downloads page]] where you can freely download and review our work<br />
<br />
<html><br />
<center><br />
<object width="480" height="385"><param name="movie" value="http://www.youtube.com/v/SZoUAoAFqmg?fs=1&amp;hl=en_US&amp;rel=0"></param><param name="allowFullScreen" value="true"></param><param name="allowscriptaccess" value="always"></param><embed src="http://www.youtube.com/v/SZoUAoAFqmg?fs=1&amp;hl=en_US&amp;rel=0" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="480" height="385"></embed></object><br />
</center><br />
</html><br />
<br />
'''A short animation created to explain the basic biological principles behind our modelling'''<br />
<br />
= From the Biology... =<br />
== Quorum sensing in E.Coli: The LuxR system ==<br />
The LuxR quorum sensing system is common among the vibrio family, and the lux system of the luminescent bacterium V.fischeri has been studied in great detail. This gave us a plentiful supply of scientific papers and review articles on which to base our understanding of the system.<br />
The LuxR system is underpinned by three quorum sensing molecules, those being the signalling molecule HSL; the HSL synthase LuxI and the transcriptional regulator LuxR. HSL may be generated in the cell by two means, either diffusion through the cell membrane from the external environment or being produced within the cell. Throughout the development of our models we considered a number of ways of trying to replicate this, and the details of these differenent techniques can be found in further sections. Whilst in the cell it may bind to LuxR, and two HSL molecules form a tetramer with two complementary LuxR molecules to form a complex which can then activate the lux operon. In the wild-type, this operon contains the genes coding for LuxI and those ultimately responsible for GFP production, with the luxR gene being located upstream and being constitutively produced. However, in our re-engineered operon the luxR gene lies downstream of the lux operon, thus its expression is also promoted. Hence once the promoter is activated, there is an increased production of LuxI, LuxR and also of GFP, which can therefore be used as an indicator of the promoter activity within the cell. Once transcription has been completed, the LuxI catalyses the production of more HSL from resources within the cell (which we have assumed exist in a sufficiently high concentration that they can be considered infinite). This may leave the cell due to a concentration gradient if the cell density in the medium is low, or otherwise will be used again in the loop, causing a positive feedback effect which will cause a high increase in the concentration of HSL. In this way the single bacterium can communicate with those surrounding it in order to evaluate whether it exists in sufficient numbers to overcome the immune system.<br />
<br />
==Quorum sensing in V.Cholerae: The LuxPQ system ==<br />
The V.cholerae quorum sensing circuit is considerably more complex than that of the LuxR system, consisting as it does of two circuits working in tandem. These are the CqsS and the LuxPQ circuits, which communicate with other bacterium and relay this information to a shared component, LuxU which ultimately controls the expression of virulence factors. <br />
<br />
[[Image:CholeraQS diagram.jpg|800px]]<br />
<br />
'''Figure 1: Schematic diagram of the LuxPQ and CqsS quorum sensing systems of V. cholerae'''<br />
<br />
<br />
<br />
In the CqsS circuit, the CAI-1 (cholera autoinducer) molecule attaches to a dimerised complex of two CqsS molecules which exist in the inner membrane of the bacterium. When the CqsS dimer has not CAI-1 attached, it acts as a kinase and transfers phosphate groups down to LuxU which in turn phosphorylates LuxO. Similarly, another autoinducer molecule, AI-2 can attach to LuxPQ which is also in the inner membrane of the bacterium. When unattached the LuxPQ also phosphorylates LuxU in the same manner as CqsS and do the two systems are connected here. When phosphorylated, LuxO acts as a transcriptional regulator in the transcription of 4 sRNAs (small RNAs) which, in conjunction with the protein Hfq inhibit the production of what we have named the 'master regulator' HapR. HapR inhibits the expression of virulence factors, so at low cell density i.e. when CqsS and LuxPQ are kinases, HapR is repressed, and virulence factors are expressed.<br />
<br />
<br />
Converse to this, if CAI-1 attaches to CqsS or AI-2 to LuxPQ, the receptors change to phosphatases and remove phosphate groups from the LuxU, which removes them from LuxO. Thus the 4 sRNAs are not transcribed and there is no inhibition of HapR, such that virulence factors are no longer expressed. Instead, HapR promotes the production of a protein which cuts the V.Cholerae from the gut wall and allows it to pass out of the body in order to infect a different host. In this way, at low cell density the cholera causes infection in the host until reaching a critical density at which the behaviour changes and the bacteria exit the system. The internal dynamics of the system are more complex than this however. HapR attaches to its own promoter and thus is self-repressing, as is LuxO. HapR also promotes the transcription of the sRNAs and the sRNAs repress the transcription of LuxO. The system also has many unknown factors at play, however, and in combination with the lack of available rate constants makes it an ominous prospect to accurately model.<br />
<br />
== Components ==<br />
Our project can be divided into two main components: the engineering of a bistable switch into the LuxR quorum sensing system, and the integration of CqsA into the LuxR circuit. The aim of the modelling side of the project was to treat these two tasks independently and on their completion construct a combined model. However, this initial aim was proven to be almost impossible due to the lack of rate constants for the cholera system, which has only been understood in its full complexity relatively recently [1]. In order to reach a compromise, we have built a number of qualitatively accurate models for the bistable LuxR system, and outlined a framework of differential equations for the cholera system which are correct at the time of writing, and which requires more rate constants to be of further use. The work done on the bistable switch included an investigation into why exactly such a configuration of genes exhibits hysteretic behavior, and what parameters are of importance in determining the “level” of bistability of the system. <br />
<br />
<br />
The purpose of our modelling is to accurately replicate the behaviour of the bistable switch in order to allow our E.coli to be tuned so that they switch off only when we are sure that all the V.cholerae have left the system and the host is free of infection.<br />
<br />
<br />
= To the Computer=<br />
== Under the hood ==<br />
All our models are based upon a series of ordinary differential equations (ODEs) each of which have been derived from either prior research papers or our own research. These equations are solved computationally via the [[Team:St_Andrews/project/modelling/models/RK4|Fourth Order Runge-Kutta Method]] (RK4) - the classical iterative method of approximating numerical ODEs. Seeking full control of the implementation of our model we decided against the use of mathematical packages and instead decided upon using a fully-fledged programming language to develop our models. Initially we decided upon GNU R, a programming language and environment for statistical computing. While R allowed for the rapid development of our initial models it was found that R lacked the fine grain control and the raw machine efficiency that we so required. A key problem that we encountered was that the majority of ODE solver libraries available for R were not in fact written in R, instead they were written in FORTRAN and C and the R component of the library meerly passed data to and from the external program. This was problematic as it did not allow us to access and modify the ODE solving algorithm which, in order to acquire the full set of data pertinent to any of or models and thus made future use of R unfeasible. Instead we turned to C++, a far lower level language than R hoping for greater power of design. Being a compiled language (with an efficient compiler) C++ programs proved far faster than their R counterparts and offered a far finer degree of control. Initially we developed our own RK4 based ODE solver and put to work developing all our models as C++ programs. Output of the each of the models was processed and graphed using the gnuplot graphing utility. The first step in developing a usable model of the behaviour of the bistable switch was to manufacture a working model of the LuxR quorum sensing circuit (See our animation for an overview of how the circuit works). Through discussions with the biologists, we were able to generate a circuit diagram which allowed us to better visualise the workings of the system and split it up into individual reactions which we could then assign differential equations to. Ongoing testing of the model, in conjunction with further insights into the biology led to tweaking and adjustments, leading to further models with added degrees of complexity and greater realism. The various models and their associated results can be found in the [[Team:St_Andrews/project/modelling/models|models]] section. Our ultimate goal is to create a model accurately predicting the results of adding samples of our engineered E.coli into the human gut in the event of a cholera infection.<br />
<br />
==What we used ==<br />
In line with the nature of iGEM all of the software used in development of our models was Free software, operating on both Windows and Mac OS systems. We wish to thank each of the following communities for producing such high quality software:<br />
<br />
* [http://gcc.gnu.org/ GCC], used to compile our C++ programs<br />
* [http://www.vim.org vim], a text editor used to write programs prior to compiling them<br />
* [http://www.gnuplot.info gnuplot], command-line plotting software used to produce our plots<br />
* [http://www.codeblocks.org/ Code::Blocks], a development environment used for creating our programs<br />
<br />
= Download =<br />
In a continuing commitment to Free Software, all of our models are released under the [http://www.gnu.org/licenses/gpl.html GNU General Public License Version 3]. If you would like to view the code for our software it is available in the [[Team:St_Andrews/project/modelling/downloads|downloads]] page.<br />
<br />
= Models =<br />
Full detail of our models can be found at [[Team:St_Andrews/project/modelling/models|"Models"]].</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modellingTeam:St Andrews/project/modelling2010-10-28T01:35:10Z<p>Ally m: </p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Modelling Background </h1><br />
</html><br />
<br />
Hello and welcome to the University of St Andrews 2010 iGEM modelling pages. Here we present our work on the mathematical and computational modelling of the V.Cholerae bacterium and its quorum sensing system. What follows is an overall description of our work and our methodologies. For a more in depth description of our individual models please refer to our [[Team:St_Andrews/project/modelling/models| models page]] and to our [[Team:St_Andrews/project/modelling/downloads|downloads page]] where you can freely download and review our work<br />
<br />
<html><br />
<center><br />
<object width="480" height="385"><param name="movie" value="http://www.youtube.com/v/SZoUAoAFqmg?fs=1&amp;hl=en_US&amp;rel=0"></param><param name="allowFullScreen" value="true"></param><param name="allowscriptaccess" value="always"></param><embed src="http://www.youtube.com/v/SZoUAoAFqmg?fs=1&amp;hl=en_US&amp;rel=0" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="480" height="385"></embed></object><br />
</center><br />
</html><br />
<br />
'''A short animation created to explain the basic biological principles behind our modelling'''<br />
<br />
= From the Biology... =<br />
== Quorum sensing in E.Coli: The LuxR system ==<br />
The LuxR quorum sensing system is common among the vibrio family, and the lux system of the luminescent bacterium V.fischeri has been studied in great detail. This gave us a plentiful supply of scientific papers and review articles on which to base our understanding of the system.<br />
The LuxR system is underpinned by three quorum sensing molecules, those being the signalling molecule HSL; the HSL synthase LuxI and the transcriptional regulator LuxR. HSL may be generated in the cell by two means, either diffusion through the cell membrane from the external environment or being produced within the cell. Throughout the development of our models we considered a number of ways of trying to replicate this, and the details of these differenent techniques can be found in further sections. Whilst in the cell it may bind to LuxR, and two HSL molecules form a tetramer with two complementary LuxR molecules to form a complex which can then activate the lux operon. In the wild-type, this operon contains the genes coding for LuxI and those ultimately responsible for GFP production, with the luxR gene being located upstream and being constitutively produced. However, in our re-engineered operon the luxR gene lies downstream of the lux operon, thus its expression is also promoted. Hence once the promoter is activated, there is an increased production of LuxI, LuxR and also of GFP, which can therefore be used as an indicator of the promoter activity within the cell. Once transcription has been completed, the LuxI catalyses the production of more HSL from resources within the cell (which we have assumed exist in a sufficiently high concentration that they can be considered infinite). This may leave the cell due to a concentration gradient if the cell density in the medium is low, or otherwise will be used again in the loop, causing a positive feedback effect which will cause a high increase in the concentration of HSL. In this way the single bacterium can communicate with those surrounding it in order to evaluate whether it exists in sufficient numbers to overcome the immune system.<br />
<br />
==Quorum sensing in V.Cholerae: The LuxPQ system ==<br />
The V.cholerae quorum sensing circuit is considerably more complex than that of the LuxR system, consisting as it does of two circuits working in tandem. These are the CqsS and the LuxPQ circuits, which communicate with other bacterium and relay this information to a shared component, LuxU which ultimately controls the expression of virulence factors. <br />
<br />
[[Image:CholeraQS diagram.jpg|800px]]<br />
<br />
'''Figure 1: Schematic diagram of the LuxPQ and CqsS quorum sensing systems of V. cholerae'''<br />
<br />
<br />
<br />
In the CqsS circuit, the CAI-1 (cholera autoinducer) molecule attaches to a dimerised complex of two CqsS molecules which exist in the inner membrane of the bacterium. When the CqsS dimer has not CAI-1 attached, it acts as a kinase and transfers phosphate groups down to LuxU which in turn phosphorylates LuxO. Similarly, another autoinducer molecule, AI-2 can attach to LuxPQ which is also in the inner membrane of the bacterium. When unattached the LuxPQ also phosphorylates LuxU in the same manner as CqsS and do the two systems are connected here. When phosphorylated, LuxO acts as a transcriptional regulator in the transcription of 4 sRNAs (small RNAs) which, in conjunction with the protein Hfq inhibit the production of what we have named the 'master regulator' HapR. HapR inhibits the expression of virulence factors, so at low cell density i.e. when CqsS and LuxPQ are kinases, HapR is repressed, and virulence factors are expressed.<br />
<br />
<br />
Converse to this, if CAI-1 attaches to CqsS or AI-2 to LuxPQ, the receptors change to phosphatases and remove phosphate groups from the LuxU, which removes them from LuxO. Thus the 4 sRNAs are not transcribed and there is no inhibition of HapR, such that virulence factors are no longer expressed. Instead, HapR promotes the production of a protein which cuts the V.Cholerae from the gut wall and allows it to pass out of the body in order to infect a different host. In this way, at low cell density the cholera causes infection in the host until reaching a critical density at which the behaviour changes and the bacteria exit the system. The internal dynamics of the system are more complex than this however. HapR attaches to its own promoter and thus is self-repressing, as is LuxO. HapR also promotes the transcription of the sRNAs and the sRNAs repress the transcription of LuxO. The system also has many unknown factors at play, however, and in combination with the lack of available rate constants makes it an ominous prospect to accurately model.<br />
<br />
== Components ==<br />
Our project can be divided into two main components: the engineering of a bistable switch into the LuxR quorum sensing system, and the integration of CqsA into the LuxR circuit. The aim of the modelling side of the project was to treat these two tasks independently and on their completion construct a combined model. However, this initial aim was proven to be almost impossible due to the lack of rate constants for the cholera system, which has only been understood in its full complexity relatively recently [1]. In order to reach a compromise, we have built a number of qualitatively accurate models for the bistable LuxR system, and outlined a framework of differential equations for the cholera system which are correct at the time of writing, and which requires more rate constants to be of further use. The work done on the bistable switch included an investigation into why exactly such a configuration of genes exhibits hysteretic behavior, and what parameters are of importance in determining the “level” of bistability of the system. <br />
<br />
<br />
The purpose of our modelling is to accurately replicate the behaviour of the bistable switch in order to allow our E.coli to be tuned so that they switch off only when we are sure that all the V.cholerae have left the system and the host is free of infection.<br />
<br />
<br />
= To the Computer=<br />
== Under the hood ==<br />
All our models are based upon a series of ordinary differential equations (ODEs) each of which have been derived from either prior research papers or our own research. These equations are solved computationally via the [[Team:St_Andrews/project/modelling/models/RK4|Fourth Order Runge-Kutta Method]] (RK4) - the classical iterative method of approximating numerical ODEs. Seeking full control of the implementation of our model we decided against the use of mathematical packages and instead decided upon using a fully-fledged programming language to develop our models. Initially we decided upon GNU R, a programming language and environment for statistical computing. While R allowed for the rapid development of our initial models it was found that R lacked the fine grain control and the raw machine efficiency that we so required. A key problem that we encountered was that the majority of ODE solver libraries available for R were not in fact written in R, instead they were written in FORTRAN and C and the R component of the library meerly passed data to and from the external program. This was problematic as it did not allow us to access and modify the ODE solving algorithm which, in order to acquire the full set of data pertinent to any of or models and thus made future use of R unfeasible. Instead we turned to C++, a far lower level language than R hoping for greater power of design. Being a compiled language (with an efficient compiler) C++ programs proved far faster than their R counterparts and offered a far finer degree of control. Initially we developed our own RK4 based ODE solver and put to work developing all our models as C++ programs. Output of the each of the models was processed and graphed using the gnuplot graphing utility. The first step in developing a usable model of the behaviour of the bistable switch was to manufacture a working model of the LuxR quorum sensing circuit (See our animation for an overview of how the circuit works). Through discussions with the biologists, we were able to generate a circuit diagram which allowed us to better visualise the workings of the system and split it up into individual reactions which we could then assign differential equations to. Ongoing testing of the model, in conjunction with further insights into the biology led to tweaking and adjustments, leading to further models with added degrees of complexity and greater realism. The various models and their associated results can be found in the [[Team:St_Andrews/project/modelling/models|models]] section. Our ultimate goal is to create a model accurately predicting the results of adding samples of our engineered E.coli into the human gut in the event of a cholera infection.<br />
<br />
==What we used ==<br />
In line with the nature of iGEM all of the software used in development of our models was Free software, operating on both Windows and Mac OS systems. We wish to thank each of the following communities for producing such high quality software:<br />
<br />
* [http://gcc.gnu.org/ GCC], used to compile our C++ programs<br />
* [http://www.vim.org vim], a text editor used to write programs prior to compiling them<br />
* [http://www.gnuplot.info gnuplot], command-line plotting software used to produce our plots<br />
* [http://www.codeblocks.org/ Code::Blocks], a development environment used for creating our programs<br />
<br />
= Download =<br />
In a continuing commitment to Free Software, all of our models are released under the [http://www.gnu.org/licenses/gpl.html GNU General Public License Version 3]. If you would like to view the code for our software it is available in the [[Team:St_Andrews/project/modelling/downloads|downloads]] page.<br />
<br />
= Models =<br />
Full detail of our models can be found at [[Team:St_Andrews/project/modelling/models|"Models"]].</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modellingTeam:St Andrews/project/modelling2010-10-28T01:34:04Z<p>Ally m: </p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Modelling Background </h1><br />
</html><br />
<br />
Hello and welcome to the University of St Andrews 2010 iGEM modelling pages. Here we present our work on the mathematical and computational modelling of the V.Cholerae bacterium and its quorum sensing system. What follows is an overall description of our work and our methodologies. For a more in depth description of our individual models please refer to our [[Team:St_Andrews/project/modelling/models| models page]] and to our [[Team:St_Andrews/project/modelling/downloads|downloads page]] where you can freely download and review our work<br />
<br />
<html><br />
<center><br />
<object width="480" height="385"><param name="movie" value="http://www.youtube.com/v/SZoUAoAFqmg?fs=1&amp;hl=en_US&amp;rel=0"></param><param name="allowFullScreen" value="true"></param><param name="allowscriptaccess" value="always"></param><embed src="http://www.youtube.com/v/SZoUAoAFqmg?fs=1&amp;hl=en_US&amp;rel=0" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="480" height="385"></embed></object><br />
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<br />
'''A short animation created to explain the basic biological principles behind our modelling'''<br />
<br />
= From the Biology... =<br />
== Quorum sensing in E.Coli: The LuxR system ==<br />
The LuxR quorum sensing system is common among the vibrio family, and the lux system of the luminescent bacterium V.fischeri has been studied in great detail. This gave us a plentiful supply of scientific papers and review articles on which to base our understanding of the system.<br />
The LuxR system is underpinned by three quorum sensing molecules, those being the signalling molecule HSL; the HSL synthase LuxI and the transcriptional regulator LuxR. HSL may be generated in the cell by two means, either diffusion through the cell membrane from the external environment or being produced within the cell. Throughout the development of our models we considered a number of ways of trying to replicate this, and the details of these differenent techniques can be found in further sections. Whilst in the cell it may bind to LuxR, and two HSL molecules form a tetramer with two complementary LuxR molecules to form a complex which can then activate the lux operon. In the wild-type, this operon contains the genes coding for LuxI and those ultimately responsible for GFP production, with the luxR gene being located upstream and being constitutively produced. However, in our re-engineered operon the luxR gene lies downstream of the lux operon, thus its expression is also promoted. Hence once the promoter is activated, there is an increased production of LuxI, LuxR and also of GFP, which can therefore be used as an indicator of the promoter activity within the cell. Once transcription has been completed, the LuxI catalyses the production of more HSL from resources within the cell (which we have assumed exist in a sufficiently high concentration that they can be considered infinite). This may leave the cell due to a concentration gradient if the cell density in the medium is low, or otherwise will be used again in the loop, causing a positive feedback effect which will cause a high increase in the concentration of HSL. In this way the single bacterium can communicate with those surrounding it in order to evaluate whether it exists in sufficient numbers to overcome the immune system.<br />
<br />
==Quorum sensing in V.Cholerae: The LuxPQ system ==<br />
The V.cholerae quorum sensing circuit is considerably more complex than that of the LuxR system, consisting as it does of two circuits working in tandem. These are the CqsS and the LuxPQ circuits, which communicate with other bacterium and relay this information to a shared component, LuxU which ultimately controls the expression of virulence factors. <br />
<br />
[[Image:CholeraQS diagram.jpg|800px]]<br />
<br />
'''Figure 1: Schematic diagram of the LuxPQ and CqsS quorum sensing systems of V. cholerae'''<br />
<br />
<br />
<br />
In the CqsS circuit, the CAI-1 (cholera autoinducer) molecule attaches to a dimerised complex of two CqsS molecules which exist in the inner membrane of the bacterium. When the CqsS dimer has not CAI-1 attached, it acts as a kinase and transfers phosphate groups down to LuxU which in turn phosphorylates LuxO. Similarly, another autoinducer molecule, AI-2 can attach to LuxPQ which is also in the inner membrane of the bacterium. When unattached the LuxPQ also phosphorylates LuxU in the same manner as CqsS and do the two systems are connected here. When phosphorylated, LuxO acts as a transcriptional regulator in the transcription of 4 sRNAs (small RNAs) which, in conjunction with the protein Hfq inhibit the production of what we have named the 'master regulator' HapR. HapR inhibits the expression of virulence factors, so at low cell density i.e. when CqsS and LuxPQ are kinases, HapR is repressed, and virulence factors are expressed.<br />
<br />
<br />
Converse to this, if CAI-1 attaches to CqsS or AI-2 to LuxPQ, the receptors change to phosphatases and remove phosphate groups from the LuxU, which removes them from LuxO. Thus the 4 sRNAs are not transcribed and there is no inhibition of HapR, such that virulence factors are no longer expressed. Instead, HapR promotes the production of a protein which cuts the V.Cholerae from the gut wall and allows it to pass out of the body in order to infect a different host. In this way, at low cell density the cholera causes infection in the host until reaching a critical density at which the behaviour changes and the bacteria exit the system. The internal dynamics of the system are more complex than this however. HapR attaches to its own promoter and thus is self-repressing, as is LuxO. HapR also promotes the transcription of the sRNAs and the sRNAs repress the transcription of LuxO. The system also has many unknown factors at play, however, and in combination with the lack of available rate constants makes it an ominous prospect to accurately model.<br />
<br />
== Components ==<br />
Our project can be divided into two main components: the engineering of a bistable switch into the LuxR quorum sensing system, and the integration of CqsA into the LuxR circuit. The aim of the modelling side of the project was to treat these two tasks independently and on their completion construct a combined model. However, this initial aim was proven to be almost impossible due to the lack of rate constants for the cholera system, which has only been understood in its full complexity relatively recently [1]. In order to reach a compromise, we have built a number of qualitatively accurate models for the bistable LuxR system, and outlined a framework of differential equations for the cholera system which are correct at the time of writing, and which requires more rate constants to be of further use. The work done on the bistable switch included an investigation into why exactly such a configuration of genes exhibits hysteretic behavior, and what parameters are of importance in determining the “level” of bistability of the system. <br />
<br />
<br />
The purpose of our modelling is to accurately replicate the behaviour of the bistable switch in order to allow our E.coli to be tuned so that they switch off only when we are sure that all the V.cholerae have left the system and the host is free of infection.<br />
<br />
<br />
= To the Computer=<br />
== Under the hood ==<br />
All our models are based upon a series of ordinary differential equations (ODEs) each of which have been derived from either prior research papers or our own research. These equations are solved computationally via the [[Team:St_Andrews/project/modelling/models/RK4|Fourth Order Runge-Kutta Method]] (RK4) - the classical iterative method of approximating numerical ODEs. Seeking full control of the implementation of our model we decided against the use of mathematical packages and instead decided upon using a fully-fledged programming language to develop our models. Initially we decided upon GNU R, a programming language and environment for statistical computing. While R allowed for the rapid development of our initial models it was found that R lacked the fine grain control and the raw machine efficiency that we so required. A key problem that we encountered was that the majority of ODE solver libraries available for R were not in fact written in R, instead they were written in FORTRAN and C and the R component of the library meerly passed data to and from the external program. This was problematic as it did not allow us to access and modify the ODE solving algorithm which, in order to acquire the full set of data pertinent to any of or models and thus made future use of R unfeasible. Instead we turned to C++, a far lower level language than R hoping for greater power of design. Being a compiled language (with an efficient compiler) C++ programs proved far faster than their R counterparts and offered a far finer degree of control. Initially we developed our own RK4 based ODE solver and put to work developing all our models as C++ programs. Output of the each of the models was processed and graphed using the gnuplot graphing utility. The first step in developing a usable model of the behaviour of the bistable switch was to manufacture a working model of the LuxR quorum sensing circuit (See our animation for an overview of how the circuit works). Through discussions with the biologists, we were able to generate a circuit diagram which allowed us to better visualise the workings of the system and split it up into individual reactions which we could then assign differential equations to. Ongoing testing of the model, in conjunction with further insights into the biology led to tweaking and adjustments, leading to further models with added degrees of complexity and greater realism. The various models and their associated results can be found in the [[Team:St_Andrews/project/modelling/models|models]] section. Our ultimate goal is to create a model accurately predicting the results of adding samples of our engineered E.coli into the human gut in the event of a cholera infection.<br />
<br />
==What we used ==<br />
In line with the nature of iGEM all of the software used in development of our models was Free software, operating on both Windows and Mac OS systems. We wish to thank each of the following communities for producing such high quality software:<br />
<br />
* [http://gcc.gnu.org/ GCC], used to compile our C++ programs<br />
* [http://www.vim.org vim], a text editor used to write programs prior to compiling them<br />
* [http://www.gnuplot.info gnuplot], command-line plotting software used to produce our plots<br />
* [http://www.codeblocks.org/ Code::Blocks], a development environment used for creating our programs<br />
<br />
= Download =<br />
In a continuing commitment to Free Software, all of our models are released under the [http://www.gnu.org/licenses/gpl.html GNU General Public License Version 3]. If you would like to view the code for our software it is available in the [[Team:St_Andrews/project/modelling/downloads|downloads]] page.<br />
<br />
= Models =<br />
Full detail of our models can be found at [[Team:St_Andrews/project/modelling/models|"Models"]].</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modellingTeam:St Andrews/project/modelling2010-10-28T01:32:00Z<p>Ally m: </p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
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<html><br />
<h1> Modelling Background </h1><br />
</html><br />
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<center><br />
<object width="480" height="385"><param name="movie" value="http://www.youtube.com/v/SZoUAoAFqmg?fs=1&amp;hl=en_US&amp;rel=0"></param><param name="allowFullScreen" value="true"></param><param name="allowscriptaccess" value="always"></param><embed src="http://www.youtube.com/v/SZoUAoAFqmg?fs=1&amp;hl=en_US&amp;rel=0" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="480" height="385"></embed></object><br />
</center><br />
</html><br />
<br />
Hello and welcome to the University of St Andrews 2010 iGEM modelling pages. Here we present our work on the mathematical and computational modelling of the V.Cholerae bacterium and its quorum sensing system. What follows is an overall description of our work and our methodologies. For a more in depth description of our individual models please refer to our [[Team:St_Andrews/project/modelling/models| models page]] and to our [[Team:St_Andrews/project/modelling/downloads|downloads page]] where you can freely download and review our work <br />
<br />
= From the Biology... =<br />
== Quorum sensing in E.Coli: The LuxR system ==<br />
The LuxR quorum sensing system is common among the vibrio family, and the lux system of the luminescent bacterium V.fischeri has been studied in great detail. This gave us a plentiful supply of scientific papers and review articles on which to base our understanding of the system.<br />
The LuxR system is underpinned by three quorum sensing molecules, those being the signalling molecule HSL; the HSL synthase LuxI and the transcriptional regulator LuxR. HSL may be generated in the cell by two means, either diffusion through the cell membrane from the external environment or being produced within the cell. Throughout the development of our models we considered a number of ways of trying to replicate this, and the details of these differenent techniques can be found in further sections. Whilst in the cell it may bind to LuxR, and two HSL molecules form a tetramer with two complementary LuxR molecules to form a complex which can then activate the lux operon. In the wild-type, this operon contains the genes coding for LuxI and those ultimately responsible for GFP production, with the luxR gene being located upstream and being constitutively produced. However, in our re-engineered operon the luxR gene lies downstream of the lux operon, thus its expression is also promoted. Hence once the promoter is activated, there is an increased production of LuxI, LuxR and also of GFP, which can therefore be used as an indicator of the promoter activity within the cell. Once transcription has been completed, the LuxI catalyses the production of more HSL from resources within the cell (which we have assumed exist in a sufficiently high concentration that they can be considered infinite). This may leave the cell due to a concentration gradient if the cell density in the medium is low, or otherwise will be used again in the loop, causing a positive feedback effect which will cause a high increase in the concentration of HSL. In this way the single bacterium can communicate with those surrounding it in order to evaluate whether it exists in sufficient numbers to overcome the immune system.<br />
<br />
==Quorum sensing in V.Cholerae: The LuxPQ system ==<br />
The V.cholerae quorum sensing circuit is considerably more complex than that of the LuxR system, consisting as it does of two circuits working in tandem. These are the CqsS and the LuxPQ circuits, which communicate with other bacterium and relay this information to a shared component, LuxU which ultimately controls the expression of virulence factors. <br />
<br />
[[Image:CholeraQS diagram.jpg|800px]]<br />
<br />
'''Figure 1: Schematic diagram of the LuxPQ and CqsS quorum sensing systems of V. cholerae'''<br />
<br />
<br />
<br />
In the CqsS circuit, the CAI-1 (cholera autoinducer) molecule attaches to a dimerised complex of two CqsS molecules which exist in the inner membrane of the bacterium. When the CqsS dimer has not CAI-1 attached, it acts as a kinase and transfers phosphate groups down to LuxU which in turn phosphorylates LuxO. Similarly, another autoinducer molecule, AI-2 can attach to LuxPQ which is also in the inner membrane of the bacterium. When unattached the LuxPQ also phosphorylates LuxU in the same manner as CqsS and do the two systems are connected here. When phosphorylated, LuxO acts as a transcriptional regulator in the transcription of 4 sRNAs (small RNAs) which, in conjunction with the protein Hfq inhibit the production of what we have named the 'master regulator' HapR. HapR inhibits the expression of virulence factors, so at low cell density i.e. when CqsS and LuxPQ are kinases, HapR is repressed, and virulence factors are expressed.<br />
<br />
<br />
Converse to this, if CAI-1 attaches to CqsS or AI-2 to LuxPQ, the receptors change to phosphatases and remove phosphate groups from the LuxU, which removes them from LuxO. Thus the 4 sRNAs are not transcribed and there is no inhibition of HapR, such that virulence factors are no longer expressed. Instead, HapR promotes the production of a protein which cuts the V.Cholerae from the gut wall and allows it to pass out of the body in order to infect a different host. In this way, at low cell density the cholera causes infection in the host until reaching a critical density at which the behaviour changes and the bacteria exit the system. The internal dynamics of the system are more complex than this however. HapR attaches to its own promoter and thus is self-repressing, as is LuxO. HapR also promotes the transcription of the sRNAs and the sRNAs repress the transcription of LuxO. The system also has many unknown factors at play, however, and in combination with the lack of available rate constants makes it an ominous prospect to accurately model.<br />
<br />
== Components ==<br />
Our project can be divided into two main components: the engineering of a bistable switch into the LuxR quorum sensing system, and the integration of CqsA into the LuxR circuit. The aim of the modelling side of the project was to treat these two tasks independently and on their completion construct a combined model. However, this initial aim was proven to be almost impossible due to the lack of rate constants for the cholera system, which has only been understood in its full complexity relatively recently [1]. In order to reach a compromise, we have built a number of qualitatively accurate models for the bistable LuxR system, and outlined a framework of differential equations for the cholera system which are correct at the time of writing, and which requires more rate constants to be of further use. The work done on the bistable switch included an investigation into why exactly such a configuration of genes exhibits hysteretic behavior, and what parameters are of importance in determining the “level” of bistability of the system. <br />
<br />
<br />
The purpose of our modelling is to accurately replicate the behaviour of the bistable switch in order to allow our E.coli to be tuned so that they switch off only when we are sure that all the V.cholerae have left the system and the host is free of infection.<br />
<br />
<br />
= To the Computer=<br />
== Under the hood ==<br />
All our models are based upon a series of ordinary differential equations (ODEs) each of which have been derived from either prior research papers or our own research. These equations are solved computationally via the [[Team:St_Andrews/project/modelling/models/RK4|Fourth Order Runge-Kutta Method]] (RK4) - the classical iterative method of approximating numerical ODEs. Seeking full control of the implementation of our model we decided against the use of mathematical packages and instead decided upon using a fully-fledged programming language to develop our models. Initially we decided upon GNU R, a programming language and environment for statistical computing. While R allowed for the rapid development of our initial models it was found that R lacked the fine grain control and the raw machine efficiency that we so required. A key problem that we encountered was that the majority of ODE solver libraries available for R were not in fact written in R, instead they were written in FORTRAN and C and the R component of the library meerly passed data to and from the external program. This was problematic as it did not allow us to access and modify the ODE solving algorithm which, in order to acquire the full set of data pertinent to any of or models and thus made future use of R unfeasible. Instead we turned to C++, a far lower level language than R hoping for greater power of design. Being a compiled language (with an efficient compiler) C++ programs proved far faster than their R counterparts and offered a far finer degree of control. Initially we developed our own RK4 based ODE solver and put to work developing all our models as C++ programs. Output of the each of the models was processed and graphed using the gnuplot graphing utility. The first step in developing a usable model of the behaviour of the bistable switch was to manufacture a working model of the LuxR quorum sensing circuit (See our animation for an overview of how the circuit works). Through discussions with the biologists, we were able to generate a circuit diagram which allowed us to better visualise the workings of the system and split it up into individual reactions which we could then assign differential equations to. Ongoing testing of the model, in conjunction with further insights into the biology led to tweaking and adjustments, leading to further models with added degrees of complexity and greater realism. The various models and their associated results can be found in the [[Team:St_Andrews/project/modelling/models|models]] section. Our ultimate goal is to create a model accurately predicting the results of adding samples of our engineered E.coli into the human gut in the event of a cholera infection.<br />
<br />
==What we used ==<br />
In line with the nature of iGEM all of the software used in development of our models was Free software, operating on both Windows and Mac OS systems. We wish to thank each of the following communities for producing such high quality software:<br />
<br />
* [http://gcc.gnu.org/ GCC], used to compile our C++ programs<br />
* [http://www.vim.org vim], a text editor used to write programs prior to compiling them<br />
* [http://www.gnuplot.info gnuplot], command-line plotting software used to produce our plots<br />
* [http://www.codeblocks.org/ Code::Blocks], a development environment used for creating our programs<br />
<br />
= Download =<br />
In a continuing commitment to Free Software, all of our models are released under the [http://www.gnu.org/licenses/gpl.html GNU General Public License Version 3]. If you would like to view the code for our software it is available in the [[Team:St_Andrews/project/modelling/downloads|downloads]] page.<br />
<br />
= Models =<br />
Full detail of our models can be found at [[Team:St_Andrews/project/modelling/models|"Models"]].</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modellingTeam:St Andrews/project/modelling2010-10-28T01:28:22Z<p>Ally m: </p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Modelling Background </h1><br />
</html><br />
<br />
<html><br />
<center><br />
<object width="425" height="344"><param name="movie" value="http://www.youtube.com/v/SZoUAoAFqmg?hl=en&fs=1"></param><param name="allowFullScreen" value="true"></param><param name="allowscriptaccess" value="always"></param><embed src="http://www.youtube.com/v/SZoUAoAFqmg?hl=en&fs=1" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="344"></embed></object><br />
</center><br />
</html><br />
<br />
Hello and welcome to the University of St Andrews 2010 iGEM modelling pages. Here we present our work on the mathematical and computational modelling of the V.Cholerae bacterium and its quorum sensing system. What follows is an overall description of our work and our methodologies. For a more in depth description of our individual models please refer to our [[Team:St_Andrews/project/modelling/models| models page]] and to our [[Team:St_Andrews/project/modelling/downloads|downloads page]] where you can freely download and review our work <br />
<br />
= From the Biology... =<br />
== Quorum sensing in E.Coli: The LuxR system ==<br />
The LuxR quorum sensing system is common among the vibrio family, and the lux system of the luminescent bacterium V.fischeri has been studied in great detail. This gave us a plentiful supply of scientific papers and review articles on which to base our understanding of the system.<br />
The LuxR system is underpinned by three quorum sensing molecules, those being the signalling molecule HSL; the HSL synthase LuxI and the transcriptional regulator LuxR. HSL may be generated in the cell by two means, either diffusion through the cell membrane from the external environment or being produced within the cell. Throughout the development of our models we considered a number of ways of trying to replicate this, and the details of these differenent techniques can be found in further sections. Whilst in the cell it may bind to LuxR, and two HSL molecules form a tetramer with two complementary LuxR molecules to form a complex which can then activate the lux operon. In the wild-type, this operon contains the genes coding for LuxI and those ultimately responsible for GFP production, with the luxR gene being located upstream and being constitutively produced. However, in our re-engineered operon the luxR gene lies downstream of the lux operon, thus its expression is also promoted. Hence once the promoter is activated, there is an increased production of LuxI, LuxR and also of GFP, which can therefore be used as an indicator of the promoter activity within the cell. Once transcription has been completed, the LuxI catalyses the production of more HSL from resources within the cell (which we have assumed exist in a sufficiently high concentration that they can be considered infinite). This may leave the cell due to a concentration gradient if the cell density in the medium is low, or otherwise will be used again in the loop, causing a positive feedback effect which will cause a high increase in the concentration of HSL. In this way the single bacterium can communicate with those surrounding it in order to evaluate whether it exists in sufficient numbers to overcome the immune system.<br />
<br />
==Quorum sensing in V.Cholerae: The LuxPQ system ==<br />
The V.cholerae quorum sensing circuit is considerably more complex than that of the LuxR system, consisting as it does of two circuits working in tandem. These are the CqsS and the LuxPQ circuits, which communicate with other bacterium and relay this information to a shared component, LuxU which ultimately controls the expression of virulence factors. <br />
<br />
[[Image:CholeraQS diagram.jpg|800px]]<br />
<br />
'''Figure 1: Schematic diagram of the LuxPQ and CqsS quorum sensing systems of V. cholerae'''<br />
<br />
<br />
<br />
In the CqsS circuit, the CAI-1 (cholera autoinducer) molecule attaches to a dimerised complex of two CqsS molecules which exist in the inner membrane of the bacterium. When the CqsS dimer has not CAI-1 attached, it acts as a kinase and transfers phosphate groups down to LuxU which in turn phosphorylates LuxO. Similarly, another autoinducer molecule, AI-2 can attach to LuxPQ which is also in the inner membrane of the bacterium. When unattached the LuxPQ also phosphorylates LuxU in the same manner as CqsS and do the two systems are connected here. When phosphorylated, LuxO acts as a transcriptional regulator in the transcription of 4 sRNAs (small RNAs) which, in conjunction with the protein Hfq inhibit the production of what we have named the 'master regulator' HapR. HapR inhibits the expression of virulence factors, so at low cell density i.e. when CqsS and LuxPQ are kinases, HapR is repressed, and virulence factors are expressed.<br />
<br />
<br />
Converse to this, if CAI-1 attaches to CqsS or AI-2 to LuxPQ, the receptors change to phosphatases and remove phosphate groups from the LuxU, which removes them from LuxO. Thus the 4 sRNAs are not transcribed and there is no inhibition of HapR, such that virulence factors are no longer expressed. Instead, HapR promotes the production of a protein which cuts the V.Cholerae from the gut wall and allows it to pass out of the body in order to infect a different host. In this way, at low cell density the cholera causes infection in the host until reaching a critical density at which the behaviour changes and the bacteria exit the system. The internal dynamics of the system are more complex than this however. HapR attaches to its own promoter and thus is self-repressing, as is LuxO. HapR also promotes the transcription of the sRNAs and the sRNAs repress the transcription of LuxO. The system also has many unknown factors at play, however, and in combination with the lack of available rate constants makes it an ominous prospect to accurately model.<br />
<br />
== Components ==<br />
Our project can be divided into two main components: the engineering of a bistable switch into the LuxR quorum sensing system, and the integration of CqsA into the LuxR circuit. The aim of the modelling side of the project was to treat these two tasks independently and on their completion construct a combined model. However, this initial aim was proven to be almost impossible due to the lack of rate constants for the cholera system, which has only been understood in its full complexity relatively recently [1]. In order to reach a compromise, we have built a number of qualitatively accurate models for the bistable LuxR system, and outlined a framework of differential equations for the cholera system which are correct at the time of writing, and which requires more rate constants to be of further use. The work done on the bistable switch included an investigation into why exactly such a configuration of genes exhibits hysteretic behavior, and what parameters are of importance in determining the “level” of bistability of the system. <br />
<br />
<br />
The purpose of our modelling is to accurately replicate the behaviour of the bistable switch in order to allow our E.coli to be tuned so that they switch off only when we are sure that all the V.cholerae have left the system and the host is free of infection.<br />
<br />
<br />
= To the Computer=<br />
== Under the hood ==<br />
All our models are based upon a series of ordinary differential equations (ODEs) each of which have been derived from either prior research papers or our own research. These equations are solved computationally via the [[Team:St_Andrews/project/modelling/models/RK4|Fourth Order Runge-Kutta Method]] (RK4) - the classical iterative method of approximating numerical ODEs. Seeking full control of the implementation of our model we decided against the use of mathematical packages and instead decided upon using a fully-fledged programming language to develop our models. Initially we decided upon GNU R, a programming language and environment for statistical computing. While R allowed for the rapid development of our initial models it was found that R lacked the fine grain control and the raw machine efficiency that we so required. A key problem that we encountered was that the majority of ODE solver libraries available for R were not in fact written in R, instead they were written in FORTRAN and C and the R component of the library meerly passed data to and from the external program. This was problematic as it did not allow us to access and modify the ODE solving algorithm which, in order to acquire the full set of data pertinent to any of or models and thus made future use of R unfeasible. Instead we turned to C++, a far lower level language than R hoping for greater power of design. Being a compiled language (with an efficient compiler) C++ programs proved far faster than their R counterparts and offered a far finer degree of control. Initially we developed our own RK4 based ODE solver and put to work developing all our models as C++ programs. Output of the each of the models was processed and graphed using the gnuplot graphing utility. The first step in developing a usable model of the behaviour of the bistable switch was to manufacture a working model of the LuxR quorum sensing circuit (See our animation for an overview of how the circuit works). Through discussions with the biologists, we were able to generate a circuit diagram which allowed us to better visualise the workings of the system and split it up into individual reactions which we could then assign differential equations to. Ongoing testing of the model, in conjunction with further insights into the biology led to tweaking and adjustments, leading to further models with added degrees of complexity and greater realism. The various models and their associated results can be found in the [[Team:St_Andrews/project/modelling/models|models]] section. Our ultimate goal is to create a model accurately predicting the results of adding samples of our engineered E.coli into the human gut in the event of a cholera infection.<br />
<br />
==What we used ==<br />
In line with the nature of iGEM all of the software used in development of our models was Free software, operating on both Windows and Mac OS systems. We wish to thank each of the following communities for producing such high quality software:<br />
<br />
* [http://gcc.gnu.org/ GCC], used to compile our C++ programs<br />
* [http://www.vim.org vim], a text editor used to write programs prior to compiling them<br />
* [http://www.gnuplot.info gnuplot], command-line plotting software used to produce our plots<br />
* [http://www.codeblocks.org/ Code::Blocks], a development environment used for creating our programs<br />
<br />
= Download =<br />
In a continuing commitment to Free Software, all of our models are released under the [http://www.gnu.org/licenses/gpl.html GNU General Public License Version 3]. If you would like to view the code for our software it is available in the [[Team:St_Andrews/project/modelling/downloads|downloads]] page.<br />
<br />
= Models =<br />
Full detail of our models can be found at [[Team:St_Andrews/project/modelling/models|"Models"]].</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modellingTeam:St Andrews/project/modelling2010-10-28T01:25:22Z<p>Ally m: Undo revision 204027 by Ally m (Talk)</p>
<hr />
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<h1> Modelling Background </h1><br />
</html><br />
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<br />
Hello and welcome to the University of St Andrews 2010 iGEM modelling pages. Here we present our work on the mathematical and computational modelling of the V.Cholerae bacterium and its quorum sensing system. What follows is an overall description of our work and our methodologies. For a more in depth description of our individual models please refer to our [[Team:St_Andrews/project/modelling/models| models page]] and to our [[Team:St_Andrews/project/modelling/downloads|downloads page]] where you can freely download and review our work <br />
<br />
= From the Biology... =<br />
== Quorum sensing in E.Coli: The LuxR system ==<br />
The LuxR quorum sensing system is common among the vibrio family, and the lux system of the luminescent bacterium V.fischeri has been studied in great detail. This gave us a plentiful supply of scientific papers and review articles on which to base our understanding of the system.<br />
The LuxR system is underpinned by three quorum sensing molecules, those being the signalling molecule HSL; the HSL synthase LuxI and the transcriptional regulator LuxR. HSL may be generated in the cell by two means, either diffusion through the cell membrane from the external environment or being produced within the cell. Throughout the development of our models we considered a number of ways of trying to replicate this, and the details of these differenent techniques can be found in further sections. Whilst in the cell it may bind to LuxR, and two HSL molecules form a tetramer with two complementary LuxR molecules to form a complex which can then activate the lux operon. In the wild-type, this operon contains the genes coding for LuxI and those ultimately responsible for GFP production, with the luxR gene being located upstream and being constitutively produced. However, in our re-engineered operon the luxR gene lies downstream of the lux operon, thus its expression is also promoted. Hence once the promoter is activated, there is an increased production of LuxI, LuxR and also of GFP, which can therefore be used as an indicator of the promoter activity within the cell. Once transcription has been completed, the LuxI catalyses the production of more HSL from resources within the cell (which we have assumed exist in a sufficiently high concentration that they can be considered infinite). This may leave the cell due to a concentration gradient if the cell density in the medium is low, or otherwise will be used again in the loop, causing a positive feedback effect which will cause a high increase in the concentration of HSL. In this way the single bacterium can communicate with those surrounding it in order to evaluate whether it exists in sufficient numbers to overcome the immune system.<br />
<br />
==Quorum sensing in V.Cholerae: The LuxPQ system ==<br />
The V.cholerae quorum sensing circuit is considerably more complex than that of the LuxR system, consisting as it does of two circuits working in tandem. These are the CqsS and the LuxPQ circuits, which communicate with other bacterium and relay this information to a shared component, LuxU which ultimately controls the expression of virulence factors. <br />
<br />
[[Image:CholeraQS diagram.jpg|800px]]<br />
<br />
'''Figure 1: Schematic diagram of the LuxPQ and CqsS quorum sensing systems of V. cholerae'''<br />
<br />
<br />
<br />
In the CqsS circuit, the CAI-1 (cholera autoinducer) molecule attaches to a dimerised complex of two CqsS molecules which exist in the inner membrane of the bacterium. When the CqsS dimer has not CAI-1 attached, it acts as a kinase and transfers phosphate groups down to LuxU which in turn phosphorylates LuxO. Similarly, another autoinducer molecule, AI-2 can attach to LuxPQ which is also in the inner membrane of the bacterium. When unattached the LuxPQ also phosphorylates LuxU in the same manner as CqsS and do the two systems are connected here. When phosphorylated, LuxO acts as a transcriptional regulator in the transcription of 4 sRNAs (small RNAs) which, in conjunction with the protein Hfq inhibit the production of what we have named the 'master regulator' HapR. HapR inhibits the expression of virulence factors, so at low cell density i.e. when CqsS and LuxPQ are kinases, HapR is repressed, and virulence factors are expressed.<br />
<br />
<br />
Converse to this, if CAI-1 attaches to CqsS or AI-2 to LuxPQ, the receptors change to phosphatases and remove phosphate groups from the LuxU, which removes them from LuxO. Thus the 4 sRNAs are not transcribed and there is no inhibition of HapR, such that virulence factors are no longer expressed. Instead, HapR promotes the production of a protein which cuts the V.Cholerae from the gut wall and allows it to pass out of the body in order to infect a different host. In this way, at low cell density the cholera causes infection in the host until reaching a critical density at which the behaviour changes and the bacteria exit the system. The internal dynamics of the system are more complex than this however. HapR attaches to its own promoter and thus is self-repressing, as is LuxO. HapR also promotes the transcription of the sRNAs and the sRNAs repress the transcription of LuxO. The system also has many unknown factors at play, however, and in combination with the lack of available rate constants makes it an ominous prospect to accurately model.<br />
<br />
== Components ==<br />
Our project can be divided into two main components: the engineering of a bistable switch into the LuxR quorum sensing system, and the integration of CqsA into the LuxR circuit. The aim of the modelling side of the project was to treat these two tasks independently and on their completion construct a combined model. However, this initial aim was proven to be almost impossible due to the lack of rate constants for the cholera system, which has only been understood in its full complexity relatively recently [1]. In order to reach a compromise, we have built a number of qualitatively accurate models for the bistable LuxR system, and outlined a framework of differential equations for the cholera system which are correct at the time of writing, and which requires more rate constants to be of further use. The work done on the bistable switch included an investigation into why exactly such a configuration of genes exhibits hysteretic behavior, and what parameters are of importance in determining the “level” of bistability of the system. <br />
<br />
<br />
The purpose of our modelling is to accurately replicate the behaviour of the bistable switch in order to allow our E.coli to be tuned so that they switch off only when we are sure that all the V.cholerae have left the system and the host is free of infection.<br />
<br />
<br />
= To the Computer=<br />
== Under the hood ==<br />
All our models are based upon a series of ordinary differential equations (ODEs) each of which have been derived from either prior research papers or our own research. These equations are solved computationally via the [[Team:St_Andrews/project/modelling/models/RK4|Fourth Order Runge-Kutta Method]] (RK4) - the classical iterative method of approximating numerical ODEs. Seeking full control of the implementation of our model we decided against the use of mathematical packages and instead decided upon using a fully-fledged programming language to develop our models. Initially we decided upon GNU R, a programming language and environment for statistical computing. While R allowed for the rapid development of our initial models it was found that R lacked the fine grain control and the raw machine efficiency that we so required. A key problem that we encountered was that the majority of ODE solver libraries available for R were not in fact written in R, instead they were written in FORTRAN and C and the R component of the library meerly passed data to and from the external program. This was problematic as it did not allow us to access and modify the ODE solving algorithm which, in order to acquire the full set of data pertinent to any of or models and thus made future use of R unfeasible. Instead we turned to C++, a far lower level language than R hoping for greater power of design. Being a compiled language (with an efficient compiler) C++ programs proved far faster than their R counterparts and offered a far finer degree of control. Initially we developed our own RK4 based ODE solver and put to work developing all our models as C++ programs. Output of the each of the models was processed and graphed using the gnuplot graphing utility. The first step in developing a usable model of the behaviour of the bistable switch was to manufacture a working model of the LuxR quorum sensing circuit (See our animation for an overview of how the circuit works). Through discussions with the biologists, we were able to generate a circuit diagram which allowed us to better visualise the workings of the system and split it up into individual reactions which we could then assign differential equations to. Ongoing testing of the model, in conjunction with further insights into the biology led to tweaking and adjustments, leading to further models with added degrees of complexity and greater realism. The various models and their associated results can be found in the [[Team:St_Andrews/project/modelling/models|models]] section. Our ultimate goal is to create a model accurately predicting the results of adding samples of our engineered E.coli into the human gut in the event of a cholera infection.<br />
<br />
==What we used ==<br />
In line with the nature of iGEM all of the software used in development of our models was Free software, operating on both Windows and Mac OS systems. We wish to thank each of the following communities for producing such high quality software:<br />
<br />
* [http://gcc.gnu.org/ GCC], used to compile our C++ programs<br />
* [http://www.vim.org vim], a text editor used to write programs prior to compiling them<br />
* [http://www.gnuplot.info gnuplot], command-line plotting software used to produce our plots<br />
* [http://www.codeblocks.org/ Code::Blocks], a development environment used for creating our programs<br />
<br />
= Download =<br />
In a continuing commitment to Free Software, all of our models are released under the [http://www.gnu.org/licenses/gpl.html GNU General Public License Version 3]. If you would like to view the code for our software it is available in the [[Team:St_Andrews/project/modelling/downloads|downloads]] page.<br />
<br />
= Models =<br />
Full detail of our models can be found at [[Team:St_Andrews/project/modelling/models|"Models"]].</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modellingTeam:St Andrews/project/modelling2010-10-28T01:24:37Z<p>Ally m: Undo revision 204038 by Ally m (Talk)</p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Modelling Background </h1><br />
</html><br />
<br />
Hello and welcome to the University of St Andrews 2010 iGEM modelling pages. Here we present our work on the mathematical and computational modelling of the V.Cholerae bacterium and its quorum sensing system. What follows is an overall description of our work and our methodologies. For a more in depth description of our individual models please refer to our [[Team:St_Andrews/project/modelling/models| models page]] and to our [[Team:St_Andrews/project/modelling/downloads|downloads page]] where you can freely download and review our work <br />
<br />
<html><br />
<center><br />
<object width="425" height="344"><param name="movie" value="http://www.youtube.com/v/SZoUAoAFqmg?hl=en&fs=1"></param><param name="allowFullScreen" value="true"></param><param name="allowscriptaccess" value="always"></param><embed src="http://www.youtube.com/v/SZoUAoAFqmg?hl=en&fs=1" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="344" center></embed></object><br />
</center><br />
</html><br />
'''A short animation created to explain the basic operation of the quorum sensing circuit we focussed our research on'''<br />
<br />
= From the Biology... =<br />
== Quorum sensing in E.Coli: The LuxR system ==<br />
The LuxR quorum sensing system is common among the vibrio family, and the lux system of the luminescent bacterium V.fischeri has been studied in great detail. This gave us a plentiful supply of scientific papers and review articles on which to base our understanding of the system.<br />
The LuxR system is underpinned by three quorum sensing molecules, those being the signalling molecule HSL; the HSL synthase LuxI and the transcriptional regulator LuxR. HSL may be generated in the cell by two means, either diffusion through the cell membrane from the external environment or being produced within the cell. Throughout the development of our models we considered a number of ways of trying to replicate this, and the details of these differenent techniques can be found in further sections. Whilst in the cell it may bind to LuxR, and two HSL molecules form a tetramer with two complementary LuxR molecules to form a complex which can then activate the lux operon. In the wild-type, this operon contains the genes coding for LuxI and those ultimately responsible for GFP production, with the luxR gene being located upstream and being constitutively produced. However, in our re-engineered operon the luxR gene lies downstream of the lux operon, thus its expression is also promoted. Hence once the promoter is activated, there is an increased production of LuxI, LuxR and also of GFP, which can therefore be used as an indicator of the promoter activity within the cell. Once transcription has been completed, the LuxI catalyses the production of more HSL from resources within the cell (which we have assumed exist in a sufficiently high concentration that they can be considered infinite). This may leave the cell due to a concentration gradient if the cell density in the medium is low, or otherwise will be used again in the loop, causing a positive feedback effect which will cause a high increase in the concentration of HSL. In this way the single bacterium can communicate with those surrounding it in order to evaluate whether it exists in sufficient numbers to overcome the immune system.<br />
<br />
==Quorum sensing in V.Cholerae: The LuxPQ system ==<br />
The V.cholerae quorum sensing circuit is considerably more complex than that of the LuxR system, consisting as it does of two circuits working in tandem. These are the CqsS and the LuxPQ circuits, which communicate with other bacterium and relay this information to a shared component, LuxU which ultimately controls the expression of virulence factors. <br />
<br />
[[Image:CholeraQS diagram.jpg|800px]]<br />
<br />
'''Figure 1: Schematic diagram of the LuxPQ and CqsS quorum sensing systems of V. cholerae'''<br />
<br />
<br />
<br />
In the CqsS circuit, the CAI-1 (cholera autoinducer) molecule attaches to a dimerised complex of two CqsS molecules which exist in the inner membrane of the bacterium. When the CqsS dimer has not CAI-1 attached, it acts as a kinase and transfers phosphate groups down to LuxU which in turn phosphorylates LuxO. Similarly, another autoinducer molecule, AI-2 can attach to LuxPQ which is also in the inner membrane of the bacterium. When unattached the LuxPQ also phosphorylates LuxU in the same manner as CqsS and do the two systems are connected here. When phosphorylated, LuxO acts as a transcriptional regulator in the transcription of 4 sRNAs (small RNAs) which, in conjunction with the protein Hfq inhibit the production of what we have named the 'master regulator' HapR. HapR inhibits the expression of virulence factors, so at low cell density i.e. when CqsS and LuxPQ are kinases, HapR is repressed, and virulence factors are expressed.<br />
<br />
<br />
Converse to this, if CAI-1 attaches to CqsS or AI-2 to LuxPQ, the receptors change to phosphatases and remove phosphate groups from the LuxU, which removes them from LuxO. Thus the 4 sRNAs are not transcribed and there is no inhibition of HapR, such that virulence factors are no longer expressed. Instead, HapR promotes the production of a protein which cuts the V.Cholerae from the gut wall and allows it to pass out of the body in order to infect a different host. In this way, at low cell density the cholera causes infection in the host until reaching a critical density at which the behaviour changes and the bacteria exit the system. The internal dynamics of the system are more complex than this however. HapR attaches to its own promoter and thus is self-repressing, as is LuxO. HapR also promotes the transcription of the sRNAs and the sRNAs repress the transcription of LuxO. The system also has many unknown factors at play, however, and in combination with the lack of available rate constants makes it an ominous prospect to accurately model.<br />
<br />
== Components ==<br />
Our project can be divided into two main components: the engineering of a bistable switch into the LuxR quorum sensing system, and the integration of CqsA into the LuxR circuit. The aim of the modelling side of the project was to treat these two tasks independently and on their completion construct a combined model. However, this initial aim was proven to be almost impossible due to the lack of rate constants for the cholera system, which has only been understood in its full complexity relatively recently [1]. In order to reach a compromise, we have built a number of qualitatively accurate models for the bistable LuxR system, and outlined a framework of differential equations for the cholera system which are correct at the time of writing, and which requires more rate constants to be of further use. The work done on the bistable switch included an investigation into why exactly such a configuration of genes exhibits hysteretic behavior, and what parameters are of importance in determining the “level” of bistability of the system. <br />
<br />
<br />
The purpose of our modelling is to accurately replicate the behaviour of the bistable switch in order to allow our E.coli to be tuned so that they switch off only when we are sure that all the V.cholerae have left the system and the host is free of infection.<br />
<br />
<br />
= To the Computer=<br />
== Under the hood ==<br />
All our models are based upon a series of ordinary differential equations (ODEs) each of which have been derived from either prior research papers or our own research. These equations are solved computationally via the [[Team:St_Andrews/project/modelling/models/RK4|Fourth Order Runge-Kutta Method]] (RK4) - the classical iterative method of approximating numerical ODEs. Seeking full control of the implementation of our model we decided against the use of mathematical packages and instead decided upon using a fully-fledged programming language to develop our models. Initially we decided upon GNU R, a programming language and environment for statistical computing. While R allowed for the rapid development of our initial models it was found that R lacked the fine grain control and the raw machine efficiency that we so required. A key problem that we encountered was that the majority of ODE solver libraries available for R were not in fact written in R, instead they were written in FORTRAN and C and the R component of the library meerly passed data to and from the external program. This was problematic as it did not allow us to access and modify the ODE solving algorithm which, in order to acquire the full set of data pertinent to any of or models and thus made future use of R unfeasible. Instead we turned to C++, a far lower level language than R hoping for greater power of design. Being a compiled language (with an efficient compiler) C++ programs proved far faster than their R counterparts and offered a far finer degree of control. Initially we developed our own RK4 based ODE solver and put to work developing all our models as C++ programs. Output of the each of the models was processed and graphed using the gnuplot graphing utility. The first step in developing a usable model of the behaviour of the bistable switch was to manufacture a working model of the LuxR quorum sensing circuit (See our animation for an overview of how the circuit works). Through discussions with the biologists, we were able to generate a circuit diagram which allowed us to better visualise the workings of the system and split it up into individual reactions which we could then assign differential equations to. Ongoing testing of the model, in conjunction with further insights into the biology led to tweaking and adjustments, leading to further models with added degrees of complexity and greater realism. The various models and their associated results can be found in the [[Team:St_Andrews/project/modelling/models|models]] section. Our ultimate goal is to create a model accurately predicting the results of adding samples of our engineered E.coli into the human gut in the event of a cholera infection.<br />
<br />
==What we used ==<br />
In line with the nature of iGEM all of the software used in development of our models was Free software, operating on both Windows and Mac OS systems. We wish to thank each of the following communities for producing such high quality software:<br />
<br />
* [http://gcc.gnu.org/ GCC], used to compile our C++ programs<br />
* [http://www.vim.org vim], a text editor used to write programs prior to compiling them<br />
* [http://www.gnuplot.info gnuplot], command-line plotting software used to produce our plots<br />
* [http://www.codeblocks.org/ Code::Blocks], a development environment used for creating our programs<br />
<br />
= Download =<br />
In a continuing commitment to Free Software, all of our models are released under the [http://www.gnu.org/licenses/gpl.html GNU General Public License Version 3]. If you would like to view the code for our software it is available in the [[Team:St_Andrews/project/modelling/downloads|downloads]] page.<br />
<br />
= Models =<br />
Full detail of our models can be found at [[Team:St_Andrews/project/modelling/models|"Models"]].</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modellingTeam:St Andrews/project/modelling2010-10-28T01:23:25Z<p>Ally m: </p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Modelling Background </h1><br />
</html><br />
<br />
Hello and welcome to the University of St Andrews 2010 iGEM modelling pages. Here we present our work on the mathematical and computational modelling of the V.Cholerae bacterium and its quorum sensing system. What follows is an overall description of our work and our methodologies. For a more in depth description of our individual models please refer to our [[Team:St_Andrews/project/modelling/models| models page]] and to our [[Team:St_Andrews/project/modelling/downloads|downloads page]] where you can freely download and review our work <br />
<br />
<html><br />
<center><br />
<object width="425" height="344"><param name="movie" value="http://www.youtube.com/v/SZoUAoAFqmg?hl=en&fs=1"></param><param name="allowFullScreen" value="true"></param><param name="allowscriptaccess" value="always"></param><embed src="http://www.youtube.com/v/SZoUAoAFqmg?hl=en&fs=1" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="344" center></embed></object><br />
</center><br />
</html><br />
<br />
'''A short animation created to explain the basic operation of the quorum sensing circuit we focussed our research on'''<br />
<br />
= From the Biology... =<br />
== Quorum sensing in E.Coli: The LuxR system ==<br />
The LuxR quorum sensing system is common among the vibrio family, and the lux system of the luminescent bacterium V.fischeri has been studied in great detail. This gave us a plentiful supply of scientific papers and review articles on which to base our understanding of the system.<br />
The LuxR system is underpinned by three quorum sensing molecules, those being the signalling molecule HSL; the HSL synthase LuxI and the transcriptional regulator LuxR. HSL may be generated in the cell by two means, either diffusion through the cell membrane from the external environment or being produced within the cell. Throughout the development of our models we considered a number of ways of trying to replicate this, and the details of these differenent techniques can be found in further sections. Whilst in the cell it may bind to LuxR, and two HSL molecules form a tetramer with two complementary LuxR molecules to form a complex which can then activate the lux operon. In the wild-type, this operon contains the genes coding for LuxI and those ultimately responsible for GFP production, with the luxR gene being located upstream and being constitutively produced. However, in our re-engineered operon the luxR gene lies downstream of the lux operon, thus its expression is also promoted. Hence once the promoter is activated, there is an increased production of LuxI, LuxR and also of GFP, which can therefore be used as an indicator of the promoter activity within the cell. Once transcription has been completed, the LuxI catalyses the production of more HSL from resources within the cell (which we have assumed exist in a sufficiently high concentration that they can be considered infinite). This may leave the cell due to a concentration gradient if the cell density in the medium is low, or otherwise will be used again in the loop, causing a positive feedback effect which will cause a high increase in the concentration of HSL. In this way the single bacterium can communicate with those surrounding it in order to evaluate whether it exists in sufficient numbers to overcome the immune system.<br />
<br />
==Quorum sensing in V.Cholerae: The LuxPQ system ==<br />
The V.cholerae quorum sensing circuit is considerably more complex than that of the LuxR system, consisting as it does of two circuits working in tandem. These are the CqsS and the LuxPQ circuits, which communicate with other bacterium and relay this information to a shared component, LuxU which ultimately controls the expression of virulence factors. <br />
<br />
[[Image:CholeraQS diagram.jpg|800px]]<br />
<br />
'''Figure 1: Schematic diagram of the LuxPQ and CqsS quorum sensing systems of V. cholerae'''<br />
<br />
<br />
<br />
In the CqsS circuit, the CAI-1 (cholera autoinducer) molecule attaches to a dimerised complex of two CqsS molecules which exist in the inner membrane of the bacterium. When the CqsS dimer has not CAI-1 attached, it acts as a kinase and transfers phosphate groups down to LuxU which in turn phosphorylates LuxO. Similarly, another autoinducer molecule, AI-2 can attach to LuxPQ which is also in the inner membrane of the bacterium. When unattached the LuxPQ also phosphorylates LuxU in the same manner as CqsS and do the two systems are connected here. When phosphorylated, LuxO acts as a transcriptional regulator in the transcription of 4 sRNAs (small RNAs) which, in conjunction with the protein Hfq inhibit the production of what we have named the 'master regulator' HapR. HapR inhibits the expression of virulence factors, so at low cell density i.e. when CqsS and LuxPQ are kinases, HapR is repressed, and virulence factors are expressed.<br />
<br />
<br />
Converse to this, if CAI-1 attaches to CqsS or AI-2 to LuxPQ, the receptors change to phosphatases and remove phosphate groups from the LuxU, which removes them from LuxO. Thus the 4 sRNAs are not transcribed and there is no inhibition of HapR, such that virulence factors are no longer expressed. Instead, HapR promotes the production of a protein which cuts the V.Cholerae from the gut wall and allows it to pass out of the body in order to infect a different host. In this way, at low cell density the cholera causes infection in the host until reaching a critical density at which the behaviour changes and the bacteria exit the system. The internal dynamics of the system are more complex than this however. HapR attaches to its own promoter and thus is self-repressing, as is LuxO. HapR also promotes the transcription of the sRNAs and the sRNAs repress the transcription of LuxO. The system also has many unknown factors at play, however, and in combination with the lack of available rate constants makes it an ominous prospect to accurately model.<br />
<br />
== Components ==<br />
Our project can be divided into two main components: the engineering of a bistable switch into the LuxR quorum sensing system, and the integration of CqsA into the LuxR circuit. The aim of the modelling side of the project was to treat these two tasks independently and on their completion construct a combined model. However, this initial aim was proven to be almost impossible due to the lack of rate constants for the cholera system, which has only been understood in its full complexity relatively recently [1]. In order to reach a compromise, we have built a number of qualitatively accurate models for the bistable LuxR system, and outlined a framework of differential equations for the cholera system which are correct at the time of writing, and which requires more rate constants to be of further use. The work done on the bistable switch included an investigation into why exactly such a configuration of genes exhibits hysteretic behavior, and what parameters are of importance in determining the “level” of bistability of the system. <br />
<br />
<br />
The purpose of our modelling is to accurately replicate the behaviour of the bistable switch in order to allow our E.coli to be tuned so that they switch off only when we are sure that all the V.cholerae have left the system and the host is free of infection.<br />
<br />
<br />
= To the Computer=<br />
== Under the hood ==<br />
All our models are based upon a series of ordinary differential equations (ODEs) each of which have been derived from either prior research papers or our own research. These equations are solved computationally via the [[Team:St_Andrews/project/modelling/models/RK4|Fourth Order Runge-Kutta Method]] (RK4) - the classical iterative method of approximating numerical ODEs. Seeking full control of the implementation of our model we decided against the use of mathematical packages and instead decided upon using a fully-fledged programming language to develop our models. Initially we decided upon GNU R, a programming language and environment for statistical computing. While R allowed for the rapid development of our initial models it was found that R lacked the fine grain control and the raw machine efficiency that we so required. A key problem that we encountered was that the majority of ODE solver libraries available for R were not in fact written in R, instead they were written in FORTRAN and C and the R component of the library meerly passed data to and from the external program. This was problematic as it did not allow us to access and modify the ODE solving algorithm which, in order to acquire the full set of data pertinent to any of or models and thus made future use of R unfeasible. Instead we turned to C++, a far lower level language than R hoping for greater power of design. Being a compiled language (with an efficient compiler) C++ programs proved far faster than their R counterparts and offered a far finer degree of control. Initially we developed our own RK4 based ODE solver and put to work developing all our models as C++ programs. Output of the each of the models was processed and graphed using the gnuplot graphing utility. The first step in developing a usable model of the behaviour of the bistable switch was to manufacture a working model of the LuxR quorum sensing circuit (See our animation for an overview of how the circuit works). Through discussions with the biologists, we were able to generate a circuit diagram which allowed us to better visualise the workings of the system and split it up into individual reactions which we could then assign differential equations to. Ongoing testing of the model, in conjunction with further insights into the biology led to tweaking and adjustments, leading to further models with added degrees of complexity and greater realism. The various models and their associated results can be found in the [[Team:St_Andrews/project/modelling/models|models]] section. Our ultimate goal is to create a model accurately predicting the results of adding samples of our engineered E.coli into the human gut in the event of a cholera infection.<br />
<br />
==What we used ==<br />
In line with the nature of iGEM all of the software used in development of our models was Free software, operating on both Windows and Mac OS systems. We wish to thank each of the following communities for producing such high quality software:<br />
<br />
* [http://gcc.gnu.org/ GCC], used to compile our C++ programs<br />
* [http://www.vim.org vim], a text editor used to write programs prior to compiling them<br />
* [http://www.gnuplot.info gnuplot], command-line plotting software used to produce our plots<br />
* [http://www.codeblocks.org/ Code::Blocks], a development environment used for creating our programs<br />
<br />
= Download =<br />
In a continuing commitment to Free Software, all of our models are released under the [http://www.gnu.org/licenses/gpl.html GNU General Public License Version 3]. If you would like to view the code for our software it is available in the [[Team:St_Andrews/project/modelling/downloads|downloads]] page.<br />
<br />
= Models =<br />
Full detail of our models can be found at [[Team:St_Andrews/project/modelling/models|"Models"]].</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modellingTeam:St Andrews/project/modelling2010-10-28T01:23:03Z<p>Ally m: </p>
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<h1> Modelling Background </h1><br />
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Hello and welcome to the University of St Andrews 2010 iGEM modelling pages. Here we present our work on the mathematical and computational modelling of the V.Cholerae bacterium and its quorum sensing system. What follows is an overall description of our work and our methodologies. For a more in depth description of our individual models please refer to our [[Team:St_Andrews/project/modelling/models| models page]] and to our [[Team:St_Andrews/project/modelling/downloads|downloads page]] where you can freely download and review our work <br />
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'''A short animation created to explain the basic operation of the quorum sensing circuit we focussed our research on'''<br />
<br />
= From the Biology... =<br />
== Quorum sensing in E.Coli: The LuxR system ==<br />
The LuxR quorum sensing system is common among the vibrio family, and the lux system of the luminescent bacterium V.fischeri has been studied in great detail. This gave us a plentiful supply of scientific papers and review articles on which to base our understanding of the system.<br />
The LuxR system is underpinned by three quorum sensing molecules, those being the signalling molecule HSL; the HSL synthase LuxI and the transcriptional regulator LuxR. HSL may be generated in the cell by two means, either diffusion through the cell membrane from the external environment or being produced within the cell. Throughout the development of our models we considered a number of ways of trying to replicate this, and the details of these differenent techniques can be found in further sections. Whilst in the cell it may bind to LuxR, and two HSL molecules form a tetramer with two complementary LuxR molecules to form a complex which can then activate the lux operon. In the wild-type, this operon contains the genes coding for LuxI and those ultimately responsible for GFP production, with the luxR gene being located upstream and being constitutively produced. However, in our re-engineered operon the luxR gene lies downstream of the lux operon, thus its expression is also promoted. Hence once the promoter is activated, there is an increased production of LuxI, LuxR and also of GFP, which can therefore be used as an indicator of the promoter activity within the cell. Once transcription has been completed, the LuxI catalyses the production of more HSL from resources within the cell (which we have assumed exist in a sufficiently high concentration that they can be considered infinite). This may leave the cell due to a concentration gradient if the cell density in the medium is low, or otherwise will be used again in the loop, causing a positive feedback effect which will cause a high increase in the concentration of HSL. In this way the single bacterium can communicate with those surrounding it in order to evaluate whether it exists in sufficient numbers to overcome the immune system.<br />
<br />
==Quorum sensing in V.Cholerae: The LuxPQ system ==<br />
The V.cholerae quorum sensing circuit is considerably more complex than that of the LuxR system, consisting as it does of two circuits working in tandem. These are the CqsS and the LuxPQ circuits, which communicate with other bacterium and relay this information to a shared component, LuxU which ultimately controls the expression of virulence factors. <br />
<br />
[[Image:CholeraQS diagram.jpg|800px]]<br />
<br />
'''Figure 1: Schematic diagram of the LuxPQ and CqsS quorum sensing systems of V. cholerae'''<br />
<br />
<br />
<br />
In the CqsS circuit, the CAI-1 (cholera autoinducer) molecule attaches to a dimerised complex of two CqsS molecules which exist in the inner membrane of the bacterium. When the CqsS dimer has not CAI-1 attached, it acts as a kinase and transfers phosphate groups down to LuxU which in turn phosphorylates LuxO. Similarly, another autoinducer molecule, AI-2 can attach to LuxPQ which is also in the inner membrane of the bacterium. When unattached the LuxPQ also phosphorylates LuxU in the same manner as CqsS and do the two systems are connected here. When phosphorylated, LuxO acts as a transcriptional regulator in the transcription of 4 sRNAs (small RNAs) which, in conjunction with the protein Hfq inhibit the production of what we have named the 'master regulator' HapR. HapR inhibits the expression of virulence factors, so at low cell density i.e. when CqsS and LuxPQ are kinases, HapR is repressed, and virulence factors are expressed.<br />
<br />
<br />
Converse to this, if CAI-1 attaches to CqsS or AI-2 to LuxPQ, the receptors change to phosphatases and remove phosphate groups from the LuxU, which removes them from LuxO. Thus the 4 sRNAs are not transcribed and there is no inhibition of HapR, such that virulence factors are no longer expressed. Instead, HapR promotes the production of a protein which cuts the V.Cholerae from the gut wall and allows it to pass out of the body in order to infect a different host. In this way, at low cell density the cholera causes infection in the host until reaching a critical density at which the behaviour changes and the bacteria exit the system. The internal dynamics of the system are more complex than this however. HapR attaches to its own promoter and thus is self-repressing, as is LuxO. HapR also promotes the transcription of the sRNAs and the sRNAs repress the transcription of LuxO. The system also has many unknown factors at play, however, and in combination with the lack of available rate constants makes it an ominous prospect to accurately model.<br />
<br />
== Components ==<br />
Our project can be divided into two main components: the engineering of a bistable switch into the LuxR quorum sensing system, and the integration of CqsA into the LuxR circuit. The aim of the modelling side of the project was to treat these two tasks independently and on their completion construct a combined model. However, this initial aim was proven to be almost impossible due to the lack of rate constants for the cholera system, which has only been understood in its full complexity relatively recently [1]. In order to reach a compromise, we have built a number of qualitatively accurate models for the bistable LuxR system, and outlined a framework of differential equations for the cholera system which are correct at the time of writing, and which requires more rate constants to be of further use. The work done on the bistable switch included an investigation into why exactly such a configuration of genes exhibits hysteretic behavior, and what parameters are of importance in determining the “level” of bistability of the system. <br />
<br />
<br />
The purpose of our modelling is to accurately replicate the behaviour of the bistable switch in order to allow our E.coli to be tuned so that they switch off only when we are sure that all the V.cholerae have left the system and the host is free of infection.<br />
<br />
<br />
= To the Computer=<br />
== Under the hood ==<br />
All our models are based upon a series of ordinary differential equations (ODEs) each of which have been derived from either prior research papers or our own research. These equations are solved computationally via the [[Team:St_Andrews/project/modelling/models/RK4|Fourth Order Runge-Kutta Method]] (RK4) - the classical iterative method of approximating numerical ODEs. Seeking full control of the implementation of our model we decided against the use of mathematical packages and instead decided upon using a fully-fledged programming language to develop our models. Initially we decided upon GNU R, a programming language and environment for statistical computing. While R allowed for the rapid development of our initial models it was found that R lacked the fine grain control and the raw machine efficiency that we so required. A key problem that we encountered was that the majority of ODE solver libraries available for R were not in fact written in R, instead they were written in FORTRAN and C and the R component of the library meerly passed data to and from the external program. This was problematic as it did not allow us to access and modify the ODE solving algorithm which, in order to acquire the full set of data pertinent to any of or models and thus made future use of R unfeasible. Instead we turned to C++, a far lower level language than R hoping for greater power of design. Being a compiled language (with an efficient compiler) C++ programs proved far faster than their R counterparts and offered a far finer degree of control. Initially we developed our own RK4 based ODE solver and put to work developing all our models as C++ programs. Output of the each of the models was processed and graphed using the gnuplot graphing utility. The first step in developing a usable model of the behaviour of the bistable switch was to manufacture a working model of the LuxR quorum sensing circuit (See our animation for an overview of how the circuit works). Through discussions with the biologists, we were able to generate a circuit diagram which allowed us to better visualise the workings of the system and split it up into individual reactions which we could then assign differential equations to. Ongoing testing of the model, in conjunction with further insights into the biology led to tweaking and adjustments, leading to further models with added degrees of complexity and greater realism. The various models and their associated results can be found in the [[Team:St_Andrews/project/modelling/models|models]] section. Our ultimate goal is to create a model accurately predicting the results of adding samples of our engineered E.coli into the human gut in the event of a cholera infection.<br />
<br />
==What we used ==<br />
In line with the nature of iGEM all of the software used in development of our models was Free software, operating on both Windows and Mac OS systems. We wish to thank each of the following communities for producing such high quality software:<br />
<br />
* [http://gcc.gnu.org/ GCC], used to compile our C++ programs<br />
* [http://www.vim.org vim], a text editor used to write programs prior to compiling them<br />
* [http://www.gnuplot.info gnuplot], command-line plotting software used to produce our plots<br />
* [http://www.codeblocks.org/ Code::Blocks], a development environment used for creating our programs<br />
<br />
= Download =<br />
In a continuing commitment to Free Software, all of our models are released under the [http://www.gnu.org/licenses/gpl.html GNU General Public License Version 3]. If you would like to view the code for our software it is available in the [[Team:St_Andrews/project/modelling/downloads|downloads]] page.<br />
<br />
= Models =<br />
Full detail of our models can be found at [[Team:St_Andrews/project/modelling/models|"Models"]].</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_5Team:St Andrews/project/modelling/model 52010-10-28T01:19:06Z<p>Ally m: </p>
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<h1> V. Cholerae Quorum Sensing Model </h1><br />
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<br />
This component of our modelling work was conceived while in dialogue with the biologists, who wanted to understand how our E.Coli bacteria would interact with cholera while in the human gut. Having searched through the scientific literature, we found that no real body of work had been undertaken to fully model the V.Cholerae quorum sensing system, even in isolation. It seemed that the specific biological pathways linking the signalling molecule to increased promotion of the quorum sensing operon had only been recently understood [http://www.molbio.princeton.edu/index.php?option=content&task=view&id=27] and as such had not yet been computationally examined. With the prospect of a hitherto unexplored area of research to investigate, we embarked on the task of understanding these complex pathways and developing equations to describe their dynamics.<br />
<br />
Below you will find the code (written in C++) for our V.Cholerae model. Having developed our set of equations, we soon discovered that it was not possible to make any worthwhile progress with this project, due to the distinct lack of rate constants available. However, we still think our code will be useful for future teams investigating this line of research and as such decided to upload it under a GNU general purpose license. <br />
<br />
[[Media:CholeraDifferentialEquations.txt| V.Cholerae Quorum Sensing network differential equations ]]<br />
<br />
[[Media:CholeraMain.txt| V.Cholerae Quorum Sensing network code]]</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_5Team:St Andrews/project/modelling/model 52010-10-28T01:17:16Z<p>Ally m: </p>
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<h1> V. Cholerae Quorum Sensing Model </h1><br />
</html><br />
<br />
This component of our modelling work was conceived while in dialogue with the biologists, who wanted to understand how our E.Coli bacteria would interact with cholera while in the human gut. Having searched through the scientific literature, we found that no real body of work had been undertaken to fully model the V.Cholerae quorum sensing system, even in isolation. It seemed that the specific biological pathways linking the signalling molecule to increased promotion of the quorum sensing operon had only been recently understood [[http://www.molbio.princeton.edu/index.php?option=content&task=view&id=27]] and as such had not yet been computationally examined. With the prospect of a hitherto unexplored area of research to investigate, we embarked on the task of understanding these complex pathways and developing equations to describe their dynamics.<br />
<br />
Below you will find the code (written in C++) for our V.Cholerae model. Having developed our set of equations, we soon discovered that it was not possible to make any worthwhile progress with this project, due to the distinct lack of rate constants available. However, we still think our code will be useful for future teams investigating this line of research and as such decided to upload it under a GNU general purpose license. <br />
<br />
[[Media:CholeraDifferentialEquations.txt| V.Cholerae Quorum Sensing network differential equations ]]<br />
<br />
[[Media:CholeraMain.txt| V.Cholerae Quorum Sensing network code]]</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_5Team:St Andrews/project/modelling/model 52010-10-28T01:15:21Z<p>Ally m: </p>
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<h1> V. Cholerae Quorum Sensing Model </h1><br />
</html><br />
<br />
This component of our modelling work was conceived while in dialogue with the biologists, who wanted to understand how our e-coli bacteria would interact with cholera while in the gut. Having searched through the scientific literature, we found that no real body of work had been undertaken to fully model the cholera quorum sensing system even in isolation. It seemed that the specific biological pathways linking the signalling molecule to increased promotion of the quorum sensing operon had only been recently understood [[http://www.molbio.princeton.edu/index.php?option=content&task=view&id=27]] and as such had not yet been computationally examined. With the prospect of a hitherto unexplored area of research to investigate, we embarked on the task of understanding these complex pathways, and developing equations to describe their dynamics.<br />
<br />
Below you will find the code (written in C++) for our cholera model. Having developed our set of equations, we soon discovered that it was not possible to make any worthwhile progress with this project, due to the distinct lack of rate constants available. However, we still think our code will be useful for future teams investigating this line of research, and as such decided to upload it under a GNU general purpose license. <br />
<br />
[[Media:CholeraDifferentialEquations.txt| V.Cholerae Quorum Sensing network differential equations ]]<br />
<br />
[[Media:CholeraMain.txt| V.Cholerae Quorum Sensing network code]]</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-28T00:58:31Z<p>Ally m: /* Promoter efficiency and affinity */</p>
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<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
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<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. <br />
<br />
We can clearly see that the model produces bistable behaviour. The steady-state level of GFP remains for a lower cell density that it did on turning on, with the cells turning off at a lower cell density of approximately 1x10<sup>5</sup> CFU. Clearly our model produces at least some form of the bistability which we were looking for it to produce. More important however is to find a reason why this bistability occurs, what its effects are, and how we can manipulate it effectively.<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
Figure 2 gives us an insight into how some of the chemicals taking part in the quorum sensing reactions change with cell density. Concentration of GFP has been plotted again on this graph to allow the concentrations of the other chemicals to be compared at key points. We can see that at the same cell density that GFP concentration begins to rise there is a similar rise in the concentrations of LuxR and LuxI in the system. Similarly there is a slightly delayed, and markedly lower increase in the concentration of LuxI/LuxR and GFP mRNAs present. The mRNA concentrations also reach a steady state at a lower cell density than the other chemicals. We make the assumption that as the cell density increases, the associated increase in HSL cause and increase also in HSL-LuxR, which then results in a greater concentration of the mRNA (and hence the proteins themselves). Although not shown on these figures, the HSL-LuxR concentration mirrors that of the mRNA exactly, and so remains at a low concentration of approximately 100-200 molecules/plasmid/cell. This fits the known theory which says that the HSL-LuxR complex is highly unstable, and will therefore only ever exists in small amounts.<br />
<br />
[[Image:CellGrowth(alldata).jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
Direct comparisons should be made between figures 2 and 3, which show the cell growth and cell death phases of the same simulation. Clearly the bistability extends to chemicals other than GFP, with the LuxI and LuxR concentrations following a similar curve as GFP. The concentrations of both of these start to drop at a lower cell density than in figure 2, but the exact reason for this is hard to define.<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Possible reasons include:<br />
* Since chemicals take time to degrade there is an excess which remains present in the system until it is used in a reaction or degrades<br />
*<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have. This was done by performing a series of "parameter tests". The methods used and the results obtained can be found below.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests. For our tests we chose a value of 20 molecules/plasmid/cell, as an indication that the system has begun switching on, as experience had shown that this was an appropriate value.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test since experimental knowledge and our experience of modelling the system suggested that they would be of interest. These were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on, otherwise there would be no bistability whatsoever. This was done by running our model many times, each time changing the value of the parameter under investigation. We then looked for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing the results for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR complex is key to the entire quorum sensing circuit and as such we would expect it to play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa. This is what we would expect, since when the association rate is high HSL-LuxR will accumulate much faster and so promote transcription much faster also. Similarly a high dissociation rate will result in very few HSL-LuxR molecules present to promote transcription and so GFP production will be less pronounced. There is a definite central plateau on the graph in which the rates of association and dissociation are such that the system switches on and this is the parameter space in which we were to perform our tests.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure 6). This would make biological sense, since if we have HSL being produced in the cells at a high rate there will be a plentiful supply with which the LuxR can bind. Similarly a low production rate or high degradation rate will result in a shortage of HSL.<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The next stage of testing was to run our simulation through these two ranges of values and observe the values produced for ΔCell density. The results of these tests are shown below. From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system. Thinking through these two results, logical conclusions can be drawn. With a high value for kConv, the saturation concentration of HSL is reached much quicker, and so when the cell death cycle is invoked, the difference between the two graphs is smaller than if a low value had been used. The key point to think about is that kConv does not change the final concentrations of any of the chemicals, rather it decreases the cell density required to reach those final concentrations.<br />
<br />
Looking now to the HSL degradation rate, a higher degradation rate will have the opposite effect of a high conversion rate, causing the system to take a longer time to reach the steady state. Thus when the cell death cycle is invoked a greater level of bistability will be displayed. <br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure .<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading at such a high rate that there is not enough binding between it and the HSL to sustain the quorum sensing network, and thus the system never switches on during the cell growth cycle. Before this point there is an exponential relationship between the level of bistability and the LuxR degradation rate. The explanation for this we hypothesise is the same as was the case for the HSL degradation, in that if the degradation rate is high the system takes longer to reach the equilibrium and so the bistability is greater.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-28 01-52-10.png|800px]]<br />
<br />
'''Figure 14'''<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/File:Snapshot_2010-10-28_01-52-10.pngFile:Snapshot 2010-10-28 01-52-10.png2010-10-28T00:54:18Z<p>Ally m: St Andrews</p>
<hr />
<div>St Andrews</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-27T23:58:42Z<p>Ally m: /* LuxR degradation */</p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
</html><br />
<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. <br />
<br />
We can clearly see that the model produces bistable behaviour. The steady-state level of GFP remains for a lower cell density that it did on turning on, with the cells turning off at a lower cell density of approximately 1x10<sup>5</sup> CFU. Clearly our model produces at least some form of the bistability which we were looking for it to produce. More important however is to find a reason why this bistability occurs, what its effects are, and how we can manipulate it effectively.<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
Figure 2 gives us an insight into how some of the chemicals taking part in the quorum sensing reactions change with cell density. Concentration of GFP has been plotted again on this graph to allow the concentrations of the other chemicals to be compared at key points. We can see that at the same cell density that GFP concentration begins to rise there is a similar rise in the concentrations of LuxR and LuxI in the system. Similarly there is a slightly delayed, and markedly lower increase in the concentration of LuxI/LuxR and GFP mRNAs present. The mRNA concentrations also reach a steady state at a lower cell density than the other chemicals. We make the assumption that as the cell density increases, the associated increase in HSL cause and increase also in HSL-LuxR, which then results in a greater concentration of the mRNA (and hence the proteins themselves). Although not shown on these figures, the HSL-LuxR concentration mirrors that of the mRNA exactly, and so remains at a low concentration of approximately 100-200 molecules/plasmid/cell. This fits the known theory which says that the HSL-LuxR complex is highly unstable, and will therefore only ever exists in small amounts.<br />
<br />
[[Image:CellGrowth(alldata).jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
Direct comparisons should be made between figures 2 and 3, which show the cell growth and cell death phases of the same simulation. Clearly the bistability extends to chemicals other than GFP, with the LuxI and LuxR concentrations following a similar curve as GFP. The concentrations of both of these start to drop at a lower cell density than in figure 2, but the exact reason for this is hard to define.<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Possible reasons include:<br />
* Since chemicals take time to degrade there is an excess which remains present in the system until it is used in a reaction or degrades<br />
*<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have. This was done by performing a series of "parameter tests". The methods used and the results obtained can be found below.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests. For our tests we chose a value of 20 molecules/plasmid/cell, as an indication that the system has begun switching on, as experience had shown that this was an appropriate value.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test since experimental knowledge and our experience of modelling the system suggested that they would be of interest. These were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on, otherwise there would be no bistability whatsoever. This was done by running our model many times, each time changing the value of the parameter under investigation. We then looked for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing the results for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR complex is key to the entire quorum sensing circuit and as such we would expect it to play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa. This is what we would expect, since when the association rate is high HSL-LuxR will accumulate much faster and so promote transcription much faster also. Similarly a high dissociation rate will result in very few HSL-LuxR molecules present to promote transcription and so GFP production will be less pronounced. There is a definite central plateau on the graph in which the rates of association and dissociation are such that the system switches on and this is the parameter space in which we were to perform our tests.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure 6). This would make biological sense, since if we have HSL being produced in the cells at a high rate there will be a plentiful supply with which the LuxR can bind. Similarly a low production rate or high degradation rate will result in a shortage of HSL.<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The next stage of testing was to run our simulation through these two ranges of values and observe the values produced for ΔCell density. The results of these tests are shown below. From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system. Thinking through these two results, logical conclusions can be drawn. With a high value for kConv, the saturation concentration of HSL is reached much quicker, and so when the cell death cycle is invoked, the difference between the two graphs is smaller than if a low value had been used. The key point to think about is that kConv does not change the final concentrations of any of the chemicals, rather it decreases the cell density required to reach those final concentrations.<br />
<br />
Looking now to the HSL degradation rate, a higher degradation rate will have the opposite effect of a high conversion rate, causing the system to take a longer time to reach the steady state. Thus when the cell death cycle is invoked a greater level of bistability will be displayed. <br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure .<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading at such a high rate that there is not enough binding between it and the HSL to sustain the quorum sensing network, and thus the system never switches on during the cell growth cycle. Before this point there is an exponential relationship between the level of bistability and the LuxR degradation rate. The explanation for this we hypothesise is the same as was the case for the HSL degradation, in that if the degradation rate is high the system takes longer to reach the equilibrium and so the bistability is greater.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13'''<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-27T23:28:50Z<p>Ally m: /* Implementation */</p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
</html><br />
<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. <br />
<br />
We can clearly see that the model produces bistable behaviour. The steady-state level of GFP remains for a lower cell density that it did on turning on, with the cells turning off at a lower cell density of approximately 1x10<sup>5</sup> CFU. Clearly our model produces at least some form of the bistability which we were looking for it to produce. More important however is to find a reason why this bistability occurs, what its effects are, and how we can manipulate it effectively.<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
Figure 2 gives us an insight into how some of the chemicals taking part in the quorum sensing reactions change with cell density. Concentration of GFP has been plotted again on this graph to allow the concentrations of the other chemicals to be compared at key points. We can see that at the same cell density that GFP concentration begins to rise there is a similar rise in the concentrations of LuxR and LuxI in the system. Similarly there is a slightly delayed, and markedly lower increase in the concentration of LuxI/LuxR and GFP mRNAs present. The mRNA concentrations also reach a steady state at a lower cell density than the other chemicals. We make the assumption that as the cell density increases, the associated increase in HSL cause and increase also in HSL-LuxR, which then results in a greater concentration of the mRNA (and hence the proteins themselves). Although not shown on these figures, the HSL-LuxR concentration mirrors that of the mRNA exactly, and so remains at a low concentration of approximately 100-200 molecules/plasmid/cell. This fits the known theory which says that the HSL-LuxR complex is highly unstable, and will therefore only ever exists in small amounts.<br />
<br />
[[Image:CellGrowth(alldata).jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
Direct comparisons should be made between figures 2 and 3, which show the cell growth and cell death phases of the same simulation. Clearly the bistability extends to chemicals other than GFP, with the LuxI and LuxR concentrations following a similar curve as GFP. The concentrations of both of these start to drop at a lower cell density than in figure 2, but the exact reason for this is hard to define.<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Possible reasons include:<br />
* Since chemicals take time to degrade there is an excess which remains present in the system until it is used in a reaction or degrades<br />
*<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have. This was done by performing a series of "parameter tests". The methods used and the results obtained can be found below.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests. For our tests we chose a value of 20 molecules/plasmid/cell, as an indication that the system has begun switching on, as experience had shown that this was an appropriate value.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test since experimental knowledge and our experience of modelling the system suggested that they would be of interest. These were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on, otherwise there would be no bistability whatsoever. This was done by running our model many times, each time changing the value of the parameter under investigation. We then looked for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing the results for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR complex is key to the entire quorum sensing circuit and as such we would expect it to play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa. This is what we would expect, since when the association rate is high HSL-LuxR will accumulate much faster and so promote transcription much faster also. Similarly a high dissociation rate will result in very few HSL-LuxR molecules present to promote transcription and so GFP production will be less pronounced. There is a definite central plateau on the graph in which the rates of association and dissociation are such that the system switches on and this is the parameter space in which we were to perform our tests.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure 6). This would make biological sense, since if we have HSL being produced in the cells at a high rate there will be a plentiful supply with which the LuxR can bind. Similarly a low production rate or high degradation rate will result in a shortage of HSL.<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The next stage of testing was to run our simulation through these two ranges of values and observe the values produced for ΔCell density. The results of these tests are shown below. From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system. Thinking through these two results, logical conclusions can be drawn. With a high value for kConv, the saturation concentration of HSL is reached much quicker, and so when the cell death cycle is invoked, the difference between the two graphs is smaller than if a low value had been used. The key point to think about is that kConv does not change the final concentrations of any of the chemicals, rather it decreases the cell density required to reach those final concentrations.<br />
<br />
Looking now to the HSL degradation rate, a higher degradation rate will have the opposite effect of a high conversion rate, causing the system to take a longer time to reach the steady state. Thus when the cell death cycle is invoked a greater level of bistability will be displayed. <br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure .<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading before an can bind to the HSL so the system does not switch on.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13'''<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-27T22:44:16Z<p>Ally m: /* HSL production & degradation */</p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
</html><br />
<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. <br />
<br />
We can clearly see that the model produces bistable behaviour. The steady-state level of GFP remains for a lower cell density that it did on turning on, with the cells turning off at a lower cell density of approximately 1x10<sup>5</sup> CFU. Clearly our model produces at least some form of the bistability which we were looking for it to produce. More important however is to find a reason why this bistability occurs, what its effects are, and how we can manipulate it effectively.<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
Figure 2 gives us an insight into how some of the chemicals taking part in the quorum sensing reactions change with cell density. Concentration of GFP has been plotted again on this graph to allow the concentrations of the other chemicals to be compared at key points. We can see that at the same cell density that GFP concentration begins to rise there is a similar rise in the concentrations of LuxR and LuxI in the system. Similarly there is a slightly delayed, and markedly lower increase in the concentration of LuxI/LuxR and GFP mRNAs present. The mRNA concentrations also reach a steady state at a lower cell density than the other chemicals. We make the assumption that as the cell density increases, the associated increase in HSL cause and increase also in HSL-LuxR, which then results in a greater concentration of the mRNA (and hence the proteins themselves). Although not shown on these figures, the HSL-LuxR concentration mirrors that of the mRNA exactly, and so remains at a low concentration of approximately 100-200 molecules/plasmid/cell. This fits the known theory which says that the HSL-LuxR complex is highly unstable, and will therefore only ever exists in small amounts.<br />
<br />
[[Image:CellGrowth(alldata).jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
Direct comparisons should be made between figures 2 and 3, which show the cell growth and cell death phases of the same simulation. Clearly the bistability extends to chemicals other than GFP, with the LuxI and LuxR concentrations following a similar curve as GFP. The concentrations of both of these start to drop at a lower cell density than in figure 2, but the exact reason for this is hard to define.<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Possible reasons include:<br />
* Since chemicals take time to degrade there is an excess which remains present in the system until it is used in a reaction or degrades<br />
*<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have. This was done by performing a series of "parameter tests". The methods used and the results obtained can be found below.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests. For our tests we chose a value of 20 molecules/plasmid/cell, as an indication that the system has begun switching on, as experience had shown that this was an appropriate value.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test since experimental knowledge and our experience of modelling the system suggested that they would be of interest. These were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on, otherwise there would be no bistability whatsoever. This was done by running our model many times, each time changing the value of the parameter under investigation. We then looked for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing the results for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR complex is key to the entire quorum sensing circuit and as such we would expect it to play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa. This is what we would expect, since when the association rate is high HSL-LuxR will accumulate much faster and so promote transcription much faster also. Similarly a high dissociation rate will result in very few HSL-LuxR molecules present to promote transcription and so GFP production will be less pronounced. There is a definite central plateau on the graph in which the rates of association and dissociation are such that the system switches on and this is the parameter space in which we were to perform our tests.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure 6). This would make biological sense, since if we have HSL being produced in the cells at a high rate there will be a plentiful supply with which the LuxR can bind. Similarly a low production rate or high degradation rate will result in a shortage of HSL.<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The next stage of testing was to run our simulation through these two ranges of values and observe the values produced for ΔCell density. The results of these tests are shown below. From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system. Thinking through these two results, logical conclusions can be drawn. If kConv is high, the system will generate large amounts of HSL during the transition from OFF to ON, and so when it comes to the transition from ON to OFF, there is still a high concentration of HSL present in the environment. Hence the system maintains the same steady state until this HSL supply diminishes. <br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure .<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading before an can bind to the HSL so the system does not switch on.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13'''<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-27T22:26:12Z<p>Ally m: /* Parameter Testing */</p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
</html><br />
<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. <br />
<br />
We can clearly see that the model produces bistable behaviour. The steady-state level of GFP remains for a lower cell density that it did on turning on, with the cells turning off at a lower cell density of approximately 1x10<sup>5</sup> CFU. Clearly our model produces at least some form of the bistability which we were looking for it to produce. More important however is to find a reason why this bistability occurs, what its effects are, and how we can manipulate it effectively.<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
Figure 2 gives us an insight into how some of the chemicals taking part in the quorum sensing reactions change with cell density. Concentration of GFP has been plotted again on this graph to allow the concentrations of the other chemicals to be compared at key points. We can see that at the same cell density that GFP concentration begins to rise there is a similar rise in the concentrations of LuxR and LuxI in the system. Similarly there is a slightly delayed, and markedly lower increase in the concentration of LuxI/LuxR and GFP mRNAs present. The mRNA concentrations also reach a steady state at a lower cell density than the other chemicals. We make the assumption that as the cell density increases, the associated increase in HSL cause and increase also in HSL-LuxR, which then results in a greater concentration of the mRNA (and hence the proteins themselves). Although not shown on these figures, the HSL-LuxR concentration mirrors that of the mRNA exactly, and so remains at a low concentration of approximately 100-200 molecules/plasmid/cell. This fits the known theory which says that the HSL-LuxR complex is highly unstable, and will therefore only ever exists in small amounts.<br />
<br />
[[Image:CellGrowth(alldata).jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
Direct comparisons should be made between figures 2 and 3, which show the cell growth and cell death phases of the same simulation. Clearly the bistability extends to chemicals other than GFP, with the LuxI and LuxR concentrations following a similar curve as GFP. The concentrations of both of these start to drop at a lower cell density than in figure 2, but the exact reason for this is hard to define.<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Possible reasons include:<br />
* Since chemicals take time to degrade there is an excess which remains present in the system until it is used in a reaction or degrades<br />
*<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have. This was done by performing a series of "parameter tests". The methods used and the results obtained can be found below.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests. For our tests we chose a value of 20 molecules/plasmid/cell, as an indication that the system has begun switching on, as experience had shown that this was an appropriate value.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test since experimental knowledge and our experience of modelling the system suggested that they would be of interest. These were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on, otherwise there would be no bistability whatsoever. This was done by running our model many times, each time changing the value of the parameter under investigation. We then looked for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing the results for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR complex is key to the entire quorum sensing circuit and as such we would expect it to play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa. This is what we would expect, since when the association rate is high HSL-LuxR will accumulate much faster and so promote transcription much faster also. Similarly a high dissociation rate will result in very few HSL-LuxR molecules present to promote transcription and so GFP production will be less pronounced. There is a definite central plateau on the graph in which the rates of association and dissociation are such that the system switches on and this is the parameter space in which we were to perform our tests.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure 6). This would make biological sense, since if we have HSL being produced in the cells at a high rate there will be a plentiful supply with which the LuxR can bind. Similarly a low production rate or high degradation rate will result in a shortage of HSL.<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The results of these tests are shown below.<br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
<br />
From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system.<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure .<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading before an can bind to the HSL so the system does not switch on.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13'''<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-27T22:13:50Z<p>Ally m: /* Parameter Testing */</p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
</html><br />
<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. <br />
<br />
We can clearly see that the model produces bistable behaviour. The steady-state level of GFP remains for a lower cell density that it did on turning on, with the cells turning off at a lower cell density of approximately 1x10<sup>5</sup> CFU. Clearly our model produces at least some form of the bistability which we were looking for it to produce. More important however is to find a reason why this bistability occurs, what its effects are, and how we can manipulate it effectively.<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
Figure 2 gives us an insight into how some of the chemicals taking part in the quorum sensing reactions change with cell density. Concentration of GFP has been plotted again on this graph to allow the concentrations of the other chemicals to be compared at key points. We can see that at the same cell density that GFP concentration begins to rise there is a similar rise in the concentrations of LuxR and LuxI in the system. Similarly there is a slightly delayed, and markedly lower increase in the concentration of LuxI/LuxR and GFP mRNAs present. The mRNA concentrations also reach a steady state at a lower cell density than the other chemicals. We make the assumption that as the cell density increases, the associated increase in HSL cause and increase also in HSL-LuxR, which then results in a greater concentration of the mRNA (and hence the proteins themselves). Although not shown on these figures, the HSL-LuxR concentration mirrors that of the mRNA exactly, and so remains at a low concentration of approximately 100-200 molecules/plasmid/cell. This fits the known theory which says that the HSL-LuxR complex is highly unstable, and will therefore only ever exists in small amounts.<br />
<br />
[[Image:CellGrowth(alldata).jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
Direct comparisons should be made between figures 2 and 3, which show the cell growth and cell death phases of the same simulation. Clearly the bistability extends to chemicals other than GFP, with the LuxI and LuxR concentrations following a similar curve as GFP. The concentrations of both of these start to drop at a lower cell density than in figure 2, but the exact reason for this is hard to define.<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Possible reasons include:<br />
* Since chemicals take time to degrade there is an excess which remains present in the system until it is used in a reaction or degrades<br />
*<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have. This was done by performing a series of "parameter tests". The methods used and the results obtained can be found below.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests. For our tests we chose a value of 20 molecules/plasmid/cell, as an indication that the system has begun switching on, as experience had shown that this was an appropriate value.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test since experimental knowledge and our experience of modelling the system suggested that they would be of interest. These were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on, otherwise there would be no bistability whatsoever. This was done by running our model many times, each time changing the value of the parameter under investigation. We then looked for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing the results for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR complex is key to the entire quorum sensing circuit and as such we would expect it to play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa. This is what we would expect, since when the association rate is high HSL-LuxR will accumulate much faster and so promote transcription much faster also. Similarly a high dissociation rate will result in very few HSL-LuxR molecules present to promote transcription and so GFP production will be less pronounced. There is a definite central plateau on the graph in which the rates of association and dissociation are such that the system switches on and this is the parameter space in which we were to perform our tests.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure 6).<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The results of these tests are shown below.<br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
<br />
From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system.<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure .<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading before an can bind to the HSL so the system does not switch on.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13'''<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-27T21:54:26Z<p>Ally m: /* Parameter Testing */</p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
</html><br />
<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. <br />
<br />
We can clearly see that the model produces bistable behaviour. The steady-state level of GFP remains for a lower cell density that it did on turning on, with the cells turning off at a lower cell density of approximately 1x10<sup>5</sup> CFU. Clearly our model produces at least some form of the bistability which we were looking for it to produce. More important however is to find a reason why this bistability occurs, what its effects are, and how we can manipulate it effectively.<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
Figure 2 gives us an insight into how some of the chemicals taking part in the quorum sensing reactions change with cell density. Concentration of GFP has been plotted again on this graph to allow the concentrations of the other chemicals to be compared at key points. We can see that at the same cell density that GFP concentration begins to rise there is a similar rise in the concentrations of LuxR and LuxI in the system. Similarly there is a slightly delayed, and markedly lower increase in the concentration of LuxI/LuxR and GFP mRNAs present. The mRNA concentrations also reach a steady state at a lower cell density than the other chemicals. We make the assumption that as the cell density increases, the associated increase in HSL cause and increase also in HSL-LuxR, which then results in a greater concentration of the mRNA (and hence the proteins themselves). Although not shown on these figures, the HSL-LuxR concentration mirrors that of the mRNA exactly, and so remains at a low concentration of approximately 100-200 molecules/plasmid/cell. This fits the known theory which says that the HSL-LuxR complex is highly unstable, and will therefore only ever exists in small amounts.<br />
<br />
[[Image:CellGrowth(alldata).jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
Direct comparisons should be made between figures 2 and 3, which show the cell growth and cell death phases of the same simulation. Clearly the bistability extends to chemicals other than GFP, with the LuxI and LuxR concentrations following a similar curve as GFP. The concentrations of both of these start to drop at a lower cell density than in figure 2, but the exact reason for this is hard to define.<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Possible reasons include:<br />
* Since chemicals take time to degrade there is an excess which remains present in the system until it is used in a reaction or degrades<br />
*<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have. This was done by performing a series of "parameter tests". The methods used and the results obtained can be found below.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests. For our tests we chose a value of 20 molecules/plasmid/cell, as an indication that the system has begun switching on, as experience had shown that this was an appropriate value.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test since experimental knowledge and our experience of modelling the system suggested that they would be of interest. These were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on, otherwise there would be no bistability whatsoever. This was done by running our model many times, each time changing the value of the parameter under investigation. We then looked for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing the results for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR complex is key to the entire quorum sensing circuit and as such we would expect it to play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa. This is what we would expect, since when the association rate is high HSL-LuxR will accumulate much faster and so promote transcription much faster also. Similarly a high dissociation rate will result in very few HSL-LuxR molecules present to promote transcription and so GFP production will be less pronounced.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure 6).<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The results of these tests are shown below.<br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
<br />
From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system.<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure .<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading before an can bind to the HSL so the system does not switch on.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13'''<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-27T21:37:13Z<p>Ally m: /* Parameter Testing */</p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
</html><br />
<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. <br />
<br />
We can clearly see that the model produces bistable behaviour. The steady-state level of GFP remains for a lower cell density that it did on turning on, with the cells turning off at a lower cell density of approximately 1x10<sup>5</sup> CFU. Clearly our model produces at least some form of the bistability which we were looking for it to produce. More important however is to find a reason why this bistability occurs, what its effects are, and how we can manipulate it effectively.<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
Figure 2 gives us an insight into how some of the chemicals taking part in the quorum sensing reactions change with cell density. Concentration of GFP has been plotted again on this graph to allow the concentrations of the other chemicals to be compared at key points. We can see that at the same cell density that GFP concentration begins to rise there is a similar rise in the concentrations of LuxR and LuxI in the system. Similarly there is a slightly delayed, and markedly lower increase in the concentration of LuxI/LuxR and GFP mRNAs present. The mRNA concentrations also reach a steady state at a lower cell density than the other chemicals. We make the assumption that as the cell density increases, the associated increase in HSL cause and increase also in HSL-LuxR, which then results in a greater concentration of the mRNA (and hence the proteins themselves). Although not shown on these figures, the HSL-LuxR concentration mirrors that of the mRNA exactly, and so remains at a low concentration of approximately 100-200 molecules/plasmid/cell. This fits the known theory which says that the HSL-LuxR complex is highly unstable, and will therefore only ever exists in small amounts.<br />
<br />
[[Image:CellGrowth(alldata).jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
Direct comparisons should be made between figures 2 and 3, which show the cell growth and cell death phases of the same simulation. Clearly the bistability extends to chemicals other than GFP, with the LuxI and LuxR concentrations following a similar curve as GFP. The concentrations of both of these start to drop at a lower cell density than in figure 2, but the exact reason for this is hard to define.<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Possible reasons include:<br />
* Since chemicals take time to degrade there is an excess which remains present in the system until it is used in a reaction or degrades<br />
*<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have. This was done by performing a series of "parameter tests". The methods used and the results obtained can be found below.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests. For our tests we chose a value of 20 molecules/plasmid/cell, as an indication that the system has begun switching on, as experience had shown that this was an appropriate value.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test since experimental knowledge and our experience of modelling the system suggested that they would be of interest. These were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on, otherwise there would be no bistability whatsoever. This was done by running our model many times, each time changing the value of the parameter under investigation. We then looked for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing the results for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR complex is key to the entire quorum sensing circuit and as such we would expect it to play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure ).<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The results of these tests are shown below.<br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
<br />
From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system.<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure .<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading before an can bind to the HSL so the system does not switch on.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13'''<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-27T21:24:29Z<p>Ally m: /* Parameter Testing */</p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
</html><br />
<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. <br />
<br />
We can clearly see that the model produces bistable behaviour. The steady-state level of GFP remains for a lower cell density that it did on turning on, with the cells turning off at a lower cell density of approximately 1x10<sup>5</sup> CFU. Clearly our model produces at least some form of the bistability which we were looking for it to produce. More important however is to find a reason why this bistability occurs, what its effects are, and how we can manipulate it effectively.<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
Figure 2 gives us an insight into how some of the chemicals taking part in the quorum sensing reactions change with cell density. Concentration of GFP has been plotted again on this graph to allow the concentrations of the other chemicals to be compared at key points. We can see that at the same cell density that GFP concentration begins to rise there is a similar rise in the concentrations of LuxR and LuxI in the system. Similarly there is a slightly delayed, and markedly lower increase in the concentration of LuxI/LuxR and GFP mRNAs present. The mRNA concentrations also reach a steady state at a lower cell density than the other chemicals. We make the assumption that as the cell density increases, the associated increase in HSL cause and increase also in HSL-LuxR, which then results in a greater concentration of the mRNA (and hence the proteins themselves). Although not shown on these figures, the HSL-LuxR concentration mirrors that of the mRNA exactly, and so remains at a low concentration of approximately 100-200 molecules/plasmid/cell. This fits the known theory which says that the HSL-LuxR complex is highly unstable, and will therefore only ever exists in small amounts.<br />
<br />
[[Image:CellGrowth(alldata).jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
Direct comparisons should be made between figures 2 and 3, which show the cell growth and cell death phases of the same simulation. Clearly the bistability extends to chemicals other than GFP, with the LuxI and LuxR concentrations following a similar curve as GFP. The concentrations of both of these start to drop at a lower cell density than in figure 2, but the exact reason for this is hard to define.<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Possible reasons include:<br />
* Since chemicals take time to degrade there is an excess which remains present in the system until it is used in a reaction or degrades<br />
*<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have. This was done by performing a series of "parameter tests". The methods used and the results obtained can be found below.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests. For our tests we chose a value of 20 molecules/plasmid/cell, as an indication that the system has begun switching on, as experience had shown that this was an appropriate value.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test since experimental knowledge and our experience of modelling the system suggested that they would be of interest. These were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on, otherwise there would be no bistability whatsoever. This was done by running our model many times, each time changing the value of the parameter under investigation. We then looked for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing the results for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR is the key to the entire quorum sensing circuit and as such should play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure ).<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The results of these tests are shown below.<br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
<br />
From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system.<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure .<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading before an can bind to the HSL so the system does not switch on.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13'''<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-27T21:18:39Z<p>Ally m: /* Parameter Testing */</p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
</html><br />
<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. <br />
<br />
We can clearly see that the model produces bistable behaviour. The steady-state level of GFP remains for a lower cell density that it did on turning on, with the cells turning off at a lower cell density of approximately 1x10<sup>5</sup> CFU. Clearly our model produces at least some form of the bistability which we were looking for it to produce. More important however is to find a reason why this bistability occurs, what its effects are, and how we can manipulate it effectively.<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
Figure 2 gives us an insight into how some of the chemicals taking part in the quorum sensing reactions change with cell density. Concentration of GFP has been plotted again on this graph to allow the concentrations of the other chemicals to be compared at key points. We can see that at the same cell density that GFP concentration begins to rise there is a similar rise in the concentrations of LuxR and LuxI in the system. Similarly there is a slightly delayed, and markedly lower increase in the concentration of LuxI/LuxR and GFP mRNAs present. The mRNA concentrations also reach a steady state at a lower cell density than the other chemicals. We make the assumption that as the cell density increases, the associated increase in HSL cause and increase also in HSL-LuxR, which then results in a greater concentration of the mRNA (and hence the proteins themselves). Although not shown on these figures, the HSL-LuxR concentration mirrors that of the mRNA exactly, and so remains at a low concentration of approximately 100-200 molecules/plasmid/cell. This fits the known theory which says that the HSL-LuxR complex is highly unstable, and will therefore only ever exists in small amounts.<br />
<br />
[[Image:CellGrowth(alldata).jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
Direct comparisons should be made between figures 2 and 3, which show the cell growth and cell death phases of the same simulation. Clearly the bistability extends to chemicals other than GFP, with the LuxI and LuxR concentrations following a similar curve as GFP. The concentrations of both of these start to drop at a lower cell density than in figure 2, but the exact reason for this is hard to define.<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Possible reasons include:<br />
* Since chemicals take time to degrade there is an excess which remains present in the system until it is used in a reaction or degrades<br />
*<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have. This was done by performing a series of "parameter tests". The methods used and the results obtained can be found below.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests. For our tests we chose a value of 20 molecules/plasmid/cell, as an indication that the system has begun switching on, as experience had shown that this was an appropriate value.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test since experimental knowledge and our experience of modelling the system suggested that they would be of interest. These were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on. This was done by running our model many times with different values of the parameter we wished to investigate, and looking at for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR is the key to the entire quorum sensing circuit and as such should play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure ).<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The results of these tests are shown below.<br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
<br />
From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system.<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure .<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading before an can bind to the HSL so the system does not switch on.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13'''<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-27T21:16:42Z<p>Ally m: /* Parameter Testing */</p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
</html><br />
<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. <br />
<br />
We can clearly see that the model produces bistable behaviour. The steady-state level of GFP remains for a lower cell density that it did on turning on, with the cells turning off at a lower cell density of approximately 1x10<sup>5</sup> CFU. Clearly our model produces at least some form of the bistability which we were looking for it to produce. More important however is to find a reason why this bistability occurs, what its effects are, and how we can manipulate it effectively.<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
Figure 2 gives us an insight into how some of the chemicals taking part in the quorum sensing reactions change with cell density. Concentration of GFP has been plotted again on this graph to allow the concentrations of the other chemicals to be compared at key points. We can see that at the same cell density that GFP concentration begins to rise there is a similar rise in the concentrations of LuxR and LuxI in the system. Similarly there is a slightly delayed, and markedly lower increase in the concentration of LuxI/LuxR and GFP mRNAs present. The mRNA concentrations also reach a steady state at a lower cell density than the other chemicals. We make the assumption that as the cell density increases, the associated increase in HSL cause and increase also in HSL-LuxR, which then results in a greater concentration of the mRNA (and hence the proteins themselves). Although not shown on these figures, the HSL-LuxR concentration mirrors that of the mRNA exactly, and so remains at a low concentration of approximately 100-200 molecules/plasmid/cell. This fits the known theory which says that the HSL-LuxR complex is highly unstable, and will therefore only ever exists in small amounts.<br />
<br />
[[Image:CellGrowth(alldata).jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
Direct comparisons should be made between figures 2 and 3, which show the cell growth and cell death phases of the same simulation. Clearly the bistability extends to chemicals other than GFP, with the LuxI and LuxR concentrations following a similar curve as GFP. The concentrations of both of these start to drop at a lower cell density than in figure 2, but the exact reason for this is hard to define.<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Possible reasons include:<br />
* Since chemicals take time to degrade there is an excess which remains present in the system until it is used in a reaction or degrades<br />
*<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have. This was done by performing a series of "parameter tests". The methods used and the results obtained can be found below.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests. For our tests we chose a value of 20 molecules/plasmid/cell, as an indication that the system has begun switching on, as experience had shown that this was an appropriate value.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center|800px]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test since experimental knowledge and our experience of modelling the system suggested that they would be of interest. These were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on. This was done by running our model many times with different values of the parameter we wished to investigate, and looking at for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR is the key to the entire quorum sensing circuit and as such should play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure ).<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The results of these tests are shown below.<br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
<br />
From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system.<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure .<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading before an can bind to the HSL so the system does not switch on.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13'''<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-27T21:10:50Z<p>Ally m: /* Parameter Testing */</p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
</html><br />
<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. <br />
<br />
We can clearly see that the model produces bistable behaviour. The steady-state level of GFP remains for a lower cell density that it did on turning on, with the cells turning off at a lower cell density of approximately 1x10<sup>5</sup> CFU. Clearly our model produces at least some form of the bistability which we were looking for it to produce. More important however is to find a reason why this bistability occurs, what its effects are, and how we can manipulate it effectively.<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
Figure 2 gives us an insight into how some of the chemicals taking part in the quorum sensing reactions change with cell density. Concentration of GFP has been plotted again on this graph to allow the concentrations of the other chemicals to be compared at key points. We can see that at the same cell density that GFP concentration begins to rise there is a similar rise in the concentrations of LuxR and LuxI in the system. Similarly there is a slightly delayed, and markedly lower increase in the concentration of LuxI/LuxR and GFP mRNAs present. The mRNA concentrations also reach a steady state at a lower cell density than the other chemicals. We make the assumption that as the cell density increases, the associated increase in HSL cause and increase also in HSL-LuxR, which then results in a greater concentration of the mRNA (and hence the proteins themselves). Although not shown on these figures, the HSL-LuxR concentration mirrors that of the mRNA exactly, and so remains at a low concentration of approximately 100-200 molecules/plasmid/cell. This fits the known theory which says that the HSL-LuxR complex is highly unstable, and will therefore only ever exists in small amounts.<br />
<br />
[[Image:CellGrowth(alldata).jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
Direct comparisons should be made between figures 2 and 3, which show the cell growth and cell death phases of the same simulation. Clearly the bistability extends to chemicals other than GFP, with the LuxI and LuxR concentrations following a similar curve as GFP. The concentrations of both of these start to drop at a lower cell density than in figure 2, but the exact reason for this is hard to define.<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Possible reasons include:<br />
* Since chemicals take time to degrade there is an excess which remains present in the system until it is used in a reaction or degrades<br />
*<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have. This was done by performing a series of "parameter tests". The methods used and the results obtained can be found below.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests. For our tests we chose a value of 20 molecules/plasmid/cell, as an indication that the system has begun switching on, as experience had shown that this was an appropriate value.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test, which were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on. This was done by running our model many times with different values of the parameter we wished to investigate, and looking at for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR is the key to the entire quorum sensing circuit and as such should play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure ).<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The results of these tests are shown below.<br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
<br />
From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system.<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure .<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading before an can bind to the HSL so the system does not switch on.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13'''<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-27T21:08:18Z<p>Ally m: /* Parameter Testing */</p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
</html><br />
<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. <br />
<br />
We can clearly see that the model produces bistable behaviour. The steady-state level of GFP remains for a lower cell density that it did on turning on, with the cells turning off at a lower cell density of approximately 1x10<sup>5</sup> CFU. Clearly our model produces at least some form of the bistability which we were looking for it to produce. More important however is to find a reason why this bistability occurs, what its effects are, and how we can manipulate it effectively.<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
Figure 2 gives us an insight into how some of the chemicals taking part in the quorum sensing reactions change with cell density. Concentration of GFP has been plotted again on this graph to allow the concentrations of the other chemicals to be compared at key points. We can see that at the same cell density that GFP concentration begins to rise there is a similar rise in the concentrations of LuxR and LuxI in the system. Similarly there is a slightly delayed, and markedly lower increase in the concentration of LuxI/LuxR and GFP mRNAs present. The mRNA concentrations also reach a steady state at a lower cell density than the other chemicals. We make the assumption that as the cell density increases, the associated increase in HSL cause and increase also in HSL-LuxR, which then results in a greater concentration of the mRNA (and hence the proteins themselves). Although not shown on these figures, the HSL-LuxR concentration mirrors that of the mRNA exactly, and so remains at a low concentration of approximately 100-200 molecules/plasmid/cell. This fits the known theory which says that the HSL-LuxR complex is highly unstable, and will therefore only ever exists in small amounts.<br />
<br />
[[Image:CellGrowth(alldata).jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
Direct comparisons should be made between figures 2 and 3, which show the cell growth and cell death phases of the same simulation. Clearly the bistability extends to chemicals other than GFP, with the LuxI and LuxR concentrations following a similar curve as GFP. The concentrations of both of these start to drop at a lower cell density than in figure 2, but the exact reason for this is hard to define.<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Possible reasons include:<br />
* Since chemicals take time to degrade there is an excess which remains present in the system until it is used in a reaction or degrades<br />
*<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have. This was done by performing a series of "parameter tests". The methods used and the results obtained can be found below.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test, which were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on. This was done by running our model many times with different values of the parameter we wished to investigate, and looking at for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR is the key to the entire quorum sensing circuit and as such should play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure ).<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The results of these tests are shown below.<br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
<br />
From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system.<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure .<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading before an can bind to the HSL so the system does not switch on.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13'''<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-27T21:03:35Z<p>Ally m: /* Results */</p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
</html><br />
<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. <br />
<br />
We can clearly see that the model produces bistable behaviour. The steady-state level of GFP remains for a lower cell density that it did on turning on, with the cells turning off at a lower cell density of approximately 1x10<sup>5</sup> CFU. Clearly our model produces at least some form of the bistability which we were looking for it to produce. More important however is to find a reason why this bistability occurs, what its effects are, and how we can manipulate it effectively.<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
Figure 2 gives us an insight into how some of the chemicals taking part in the quorum sensing reactions change with cell density. Concentration of GFP has been plotted again on this graph to allow the concentrations of the other chemicals to be compared at key points. We can see that at the same cell density that GFP concentration begins to rise there is a similar rise in the concentrations of LuxR and LuxI in the system. Similarly there is a slightly delayed, and markedly lower increase in the concentration of LuxI/LuxR and GFP mRNAs present. The mRNA concentrations also reach a steady state at a lower cell density than the other chemicals. We make the assumption that as the cell density increases, the associated increase in HSL cause and increase also in HSL-LuxR, which then results in a greater concentration of the mRNA (and hence the proteins themselves). Although not shown on these figures, the HSL-LuxR concentration mirrors that of the mRNA exactly, and so remains at a low concentration of approximately 100-200 molecules/plasmid/cell. This fits the known theory which says that the HSL-LuxR complex is highly unstable, and will therefore only ever exists in small amounts.<br />
<br />
[[Image:CellGrowth(alldata).jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
Direct comparisons should be made between figures 2 and 3, which show the cell growth and cell death phases of the same simulation. Clearly the bistability extends to chemicals other than GFP, with the LuxI and LuxR concentrations following a similar curve as GFP. The concentrations of both of these start to drop at a lower cell density than in figure 2, but the exact reason for this is hard to define.<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Possible reasons include:<br />
* Since chemicals take time to degrade there is an excess which remains present in the system until it is used in a reaction or degrades<br />
*<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test, which were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on. This was done by running our model many times with different values of the parameter we wished to investigate, and looking at for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR is the key to the entire quorum sensing circuit and as such should play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure ).<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The results of these tests are shown below.<br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
<br />
From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system.<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure .<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading before an can bind to the HSL so the system does not switch on.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13'''<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-27T20:54:36Z<p>Ally m: /* Results */</p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
</html><br />
<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. <br />
<br />
We can clearly see that the model produces bistable behaviour. The steady-state level of GFP remains for a lower cell density that it did on turning on, with the cells turning off at a lower cell density of approximately 1x10<sup>5</sup> CFU. Clearly our model produces at least some form of the bistability which we were looking for it to produce. More important however is to find a reason why this bistability occurs, what its effects are, and how we can manipulate it effectively.<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
Figure 2 gives us an insight into how some of the chemicals taking part in the quorum sensing reactions change with cell density. Concentration of GFP has been plotted again on this graph to allow the concentrations of the other chemicals to be compared at key points. We can see that at the same cell density that GFP concentration begins to rise there is a similar rise in the concentrations of LuxR and LuxI in the system. Similarly there is a slightly delayed, and markedly lower increase in the concentration of LuxI/LuxR and GFP mRNAs present. The mRNA concentrations also reach a steady state at a lower cell density than the other chemicals. We make the assumption that as the cell density increases, the associated increase in HSL cause and increase also in HSL-LuxR, which then results in a greater concentration of the mRNA (and hence the proteins themselves). Although not shown on these figures, the HSL-LuxR concentration mirrors that of the mRNA exactly, and so remains at a low concentration of approximately 100-200 molecules/plasmid/cell. This fits the known theory which says that the HSL-LuxR complex is highly unstable, and will therefore only ever exists in small amounts.<br />
<br />
[[Image:CellGrowth(alldata).jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
Direct comparisons should be made between figures 2 and 3, which show the cell growth and cell death phases of the same simulation. Clearly the bistability extends to chemicals other than GFP, with the LuxI and LuxR concentrations following a similar curve as GFP. The concentrations of both of these start to drop at a lower cell density than in figure 2, but the exact reason for this is hard to define.<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Info on HSL-LuxR concentration behaviour<br />
<br />
Graph with up and down data?<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test, which were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on. This was done by running our model many times with different values of the parameter we wished to investigate, and looking at for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR is the key to the entire quorum sensing circuit and as such should play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure ).<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The results of these tests are shown below.<br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
<br />
From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system.<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure .<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading before an can bind to the HSL so the system does not switch on.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13'''<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-27T20:44:43Z<p>Ally m: /* Results */</p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
</html><br />
<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. <br />
<br />
We can clearly see that the model produces bistable behaviour. The steady-state level of GFP remains for a lower cell density that it did on turning on, with the cells turning off at a lower cell density of approximately 1x10<sup>5</sup> CFU. Clearly our model produces at least some form of the bistability which we were looking for it to produce. More important however is to find a reason why this bistability occurs, what its effects are, and how we can manipulate it effectively.<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
Figure 2 gives us an insight into how some of the chemicals taking part in the quorum sensing reactions change with cell density. Concentration of GFP has been plotted again on this graph to allow the concentrations of the other chemicals to be compared at key points. We can see that at the same cell density that GFP concentration begins to rise there is a similar rise in the concentrations of LuxR and LuxI in the system. Similarly there is a slightly delayed, and markedly lower increase in the concentration of LuxI/LuxR and GFP mRNAs present. The mRNA concentrations also reach a steady state at a lower cell density than the other chemicals. We make the assumption that as the cell density increases, the associated increase in HSL cause and increase also in HSL-LuxR, which then results in a greater concentration of the mRNA (and hence the proteins themselves). Although not shown on these figures, the HSL-LuxR concentration mirrors that of the mRNA exactly, and so remains at a low concentration of approximately 100-200 molecules/plasmid/cell. This fits the known theory which says that the HSL-LuxR complex is highly unstable, and will therefore only ever exists in small amounts.<br />
<br />
[[Image:CellGrowth(alldata).jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Info on HSL-LuxR concentration behaviour<br />
<br />
Graph with up and down data?<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test, which were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on. This was done by running our model many times with different values of the parameter we wished to investigate, and looking at for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR is the key to the entire quorum sensing circuit and as such should play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure ).<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The results of these tests are shown below.<br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
<br />
From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system.<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure .<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading before an can bind to the HSL so the system does not switch on.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13'''<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-27T20:28:54Z<p>Ally m: /* Results */</p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
</html><br />
<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. <br />
<br />
We can clearly see that the model produces bistable behaviour. The steady-state level of GFP remains for a lower cell density that it did on turning on, with the cells turning off at a lower cell density of approximately 1x10<sup>5</sup> CFU. Clearly our model produces at least some form of the bistability which we were looking for it to produce. More important however is to find a reason why this bistability occurs, what its effects are, and how we can manipulate it effectively.<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
Figure 2 gives us an insight into how some of the chemicals taking part in the quorum sensing reactions change with cell density. Concentration of GFP has been plotted again on this graph to allow the concentrations of the other chemicals to be compared at key points. We can see that at the same cell density that GFP concentration begins to rise there is a similar rise in the concentrations of LuxR and LuxI in the system. Similarly there is a slightly delayed, and markedly lower increase in the concentration of LuxI/LuxR and GFP mRNAs present. The mRNA concentrations also reach a steady state at a lower cell density than the other chemicals.<br />
<br />
[[Image:CellGrowth(alldata).jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Info on HSL-LuxR concentration behaviour<br />
<br />
Graph with up and down data?<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test, which were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on. This was done by running our model many times with different values of the parameter we wished to investigate, and looking at for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR is the key to the entire quorum sensing circuit and as such should play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure ).<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The results of these tests are shown below.<br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
<br />
From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system.<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure .<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading before an can bind to the HSL so the system does not switch on.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13'''<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-27T20:22:07Z<p>Ally m: </p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
</html><br />
<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. <br />
<br />
We can clearly see that the model produces bistable behaviour. The steady-state level of GFP remains for a lower cell density that it did on turning on, with the cells turning off at a lower cell density of approximately 1x10<sup>5</sup> CFU. Clearly our model produces at least some form of the bistability which we were looking for it to produce. More important however is to find a reason why this bistability occurs, what its effects are, and how we can manipulate it effectively.<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
Figure 2 gives us an insight into how some of the chemicals taking part in the quorum sensing reactions change with cell density. Concentration of GFP has been plotted again on this graph to allow the concentrations of the other chemicals to be compared at key points. <br />
<br />
[[Image:CellGrowth(alldata).jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Info on HSL-LuxR concentration behaviour<br />
<br />
Graph with up and down data?<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test, which were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on. This was done by running our model many times with different values of the parameter we wished to investigate, and looking at for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR is the key to the entire quorum sensing circuit and as such should play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure ).<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The results of these tests are shown below.<br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
<br />
From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system.<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure .<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading before an can bind to the HSL so the system does not switch on.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13'''<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/File:CellDeath(alldata).jpgFile:CellDeath(alldata).jpg2010-10-27T20:04:08Z<p>Ally m: uploaded a new version of "Image:CellDeath(alldata).jpg"</p>
<hr />
<div></div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-27T19:59:12Z<p>Ally m: /* Results */</p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
</html><br />
<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. <br />
<br />
We can clearly see that the model produces bistable behaviour. The steady-state level of GFP remains for a lower cell density that it did on turning on, with the cells turning off at a lower cell density of approximately 1x10<sup>5</sup> CFU. Clearly our model produces at least some form of the bistability which we were looking for it to produce. More important however is to find a reason why this bistability occurs, what its effects are, and how we can manipulate it effectively.<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
Figure 2 gives us an insight into how some of the chemicals taking part in the quorum sensing reactions change <br />
<br />
[[Image:CellGrowth(alldata).jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Info on HSL-LuxR concentration behaviour<br />
<br />
Graph with up and down data?<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test, which were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on. This was done by running our model many times with different values of the parameter we wished to investigate, and looking at for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR is the key to the entire quorum sensing circuit and as such should play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure ).<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The results of these tests are shown below.<br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
<br />
From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system.<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure .<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading before an can bind to the HSL so the system does not switch on.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13'''<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/File:CellGrowth(alldata).jpgFile:CellGrowth(alldata).jpg2010-10-27T19:58:41Z<p>Ally m: St Andrews</p>
<hr />
<div>St Andrews</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-27T19:35:00Z<p>Ally m: /* Results */</p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
</html><br />
<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. <br />
<br />
We can clearly see that the model produces bistable behaviour. The steady-state level of GFP remains for a lower cell density that it did on turning on, with the cells turning off at a lower cell density of approximately 1x10<sup>5</sup> CFU. Clearly our model produces at least some form of the bistability which we were looking for it to produce. More important however is to find a reason why this bistability occurs, what its effects are, and how we can manipulate it effectively.<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
[[Image:CellGrowth(alldata) larger range.jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Info on HSL-LuxR concentration behaviour<br />
<br />
Graph with up and down data?<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test, which were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on. This was done by running our model many times with different values of the parameter we wished to investigate, and looking at for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR is the key to the entire quorum sensing circuit and as such should play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure ).<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The results of these tests are shown below.<br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
<br />
From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system.<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure .<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading before an can bind to the HSL so the system does not switch on.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13'''<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-27T19:32:47Z<p>Ally m: /* Results */</p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
</html><br />
<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. <br />
<br />
We can clearly see that the model produces bistable behaviour. The steady-state level of GFP remains for a lower cell density that it did on turning on, with the cells turning off at a lower cell density of approximately 1x10<sup>5</sup> CFU.<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
[[Image:CellGrowth(alldata) larger range.jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Info on HSL-LuxR concentration behaviour<br />
<br />
Graph with up and down data?<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test, which were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on. This was done by running our model many times with different values of the parameter we wished to investigate, and looking at for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR is the key to the entire quorum sensing circuit and as such should play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure ).<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The results of these tests are shown below.<br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
<br />
From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system.<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure .<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading before an can bind to the HSL so the system does not switch on.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13'''<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-27T19:27:03Z<p>Ally m: </p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
</html><br />
<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure 1. Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. At a cell density of between 1x10<sup>6</sup> and 1x10<sup>7</sup> colony forming units (CFU), there is an exponential increase in the GFP concentration, indicating an increase in the activity of the cells and a transition from the 'off' state to the 'on'. At a value of approximately 1x10<sup>8</sup> CFU the GFP concentration then begins to level out and the system reaches a steady state. At this point all the cells in the system would be seen to have turned out and any associated virulence factors would now be expressed. There is also a clear difference between the switch-on and switch-off thresholds which we hypothesise is caused by…<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
[[Image:CellGrowth(alldata) larger range.jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Info on HSL-LuxR concentration behaviour<br />
<br />
Graph with up and down data?<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test, which were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on. This was done by running our model many times with different values of the parameter we wished to investigate, and looking at for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR is the key to the entire quorum sensing circuit and as such should play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure ).<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The results of these tests are shown below.<br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
<br />
From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system.<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure .<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading before an can bind to the HSL so the system does not switch on.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13'''<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_2Team:St Andrews/project/modelling/model 22010-10-27T19:08:03Z<p>Ally m: /* Theory & assumptions */</p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 2: Single Cell With Loop </h1><br />
</html><br />
<br />
=Theory & assumptions=<br />
Our second model introduced the concept of an external environment into which the HSL could diffuse into. This was assumed to be of infinite volume and thus the concentration of HSL in this region would always be zero, regardless of the quantity of HSL leaving the cell. <br />
In order to implement this modification we produced a new differential to desrcibe the change in HSL.<br />
<br />
<br />
[[Image:HSL2.jpg|900px]]<br />
<br />
'''Figure 1: Differential equation describing HSL change'''<br />
<br />
<br />
The creation of HSL is dictated by kConv, which is associated with the ability of LuxI to catalyse the conversion of resources into HSL. The concentration is reduced by the association with LuxR and by natural degredation. It is also affected by the degredation of molecules in the HSL-LuxR complex in the same way that LuxR is. We also introduce a diffusion term which accounts for HSL diffusing out of the cell due to a concentration gradient between the cell and the external environment. <br />
<br />
The code on which this model is based can be found [[Team:St_Andrews/project/modelling/downloads|here]].<br />
<br />
=Results=<br />
The switching on behaviour produced by this model is shown in figure . This switching on can be interpreted as the cell turning itself on due to the HSL it produces not diffusing out of the cell at a high enough rate. Therefore there is a considerable concentration of HSL remaining in the cell to maintain the feedback loop and produce more HSL. Thus regardless of the initial concentrations our cell always switches on. This does not match with the true biological scenario at all. Quorum sensing is designed such that cells will only become up-regulated when present at a certain critical cell density. Therefore the suggestion made by our results that a single cell can self-activate undermines this principle entirely.<br />
<br />
<br />
[[Image:LuxRSingleCellLoopRampUp(correctedmRNA).jpg|800px]]<br />
<br />
'''Figure 2: Graph of GFP concentration against time'''<br />
<br />
<br />
There is an apparent switching on of the system, but this is in fact not what we would expect seeing as our model is for a single cell in a surrounding environment of infinite volume. Therefore any HSL produced by the cell should become diluted to the point where the concentration externally can be said to be negligible.<br />
<br />
<br />
[[Image:LuxRSingleCellLoopRampUp8th(Large).jpg|800px]]<br />
<br />
'''Figure 3: Graph of GFP, LuxI, LuxR, HSL-LuxR and mRNA concentrations against time'''<br />
<br />
<br />
We can see from figure 3 that the model predicts huge concentrations of the HSL-LuxR complex to be present in the cell, which strongly contradicts previous work. The complex is very unstable and should therefore exist in only small concentrations within the cell. This gives us some idea as to the fallibility of this model. It is also of note that in general the order of magnitude if the concentrations output from this model are a few below that from [[Team:St_Andrews/project/modelling/model_1|model 1]].<br />
<br />
[[Image:LuxRSingleCellLoopRampUpmRNA.jpg|800px]]<br />
<br />
'''Figure 4: Graph of mRNA concentration against time'''<br />
<br />
<br />
The concentrations of mRNA are equal for GFP, LuxR and LuxI, and whatsmore are comparable to those seen in our previous model.<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_2Team:St Andrews/project/modelling/model 22010-10-27T19:07:46Z<p>Ally m: </p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 2: Single Cell With Loop </h1><br />
</html><br />
<br />
=Theory & assumptions=<br />
Our second model introduced the concept of an external environment into which the HSL could diffuse into. This was assumed to be of infinite volume and thus the concentration of HSL in this region would always be zero, regardless of the quantity of HSL leaving the cell. <br />
In order to implement this modification we produced a new differential to desrcibe the change in HSL.<br />
<br />
<br />
[[Image:HSL2.jpg|900px]]<br />
<br />
'''Figure 1: Differential equation describing HSL change'''<br />
<br />
The creation of HSL is dictated by kConv, which is associated with the ability of LuxI to catalyse the conversion of resources into HSL. The concentration is reduced by the association with LuxR and by natural degredation. It is also affected by the degredation of molecules in the HSL-LuxR complex in the same way that LuxR is. We also introduce a diffusion term which accounts for HSL diffusing out of the cell due to a concentration gradient between the cell and the external environment. <br />
<br />
The code on which this model is based can be found [[Team:St_Andrews/project/modelling/downloads|here]].<br />
<br />
=Results=<br />
The switching on behaviour produced by this model is shown in figure . This switching on can be interpreted as the cell turning itself on due to the HSL it produces not diffusing out of the cell at a high enough rate. Therefore there is a considerable concentration of HSL remaining in the cell to maintain the feedback loop and produce more HSL. Thus regardless of the initial concentrations our cell always switches on. This does not match with the true biological scenario at all. Quorum sensing is designed such that cells will only become up-regulated when present at a certain critical cell density. Therefore the suggestion made by our results that a single cell can self-activate undermines this principle entirely.<br />
<br />
<br />
[[Image:LuxRSingleCellLoopRampUp(correctedmRNA).jpg|800px]]<br />
<br />
'''Figure 2: Graph of GFP concentration against time'''<br />
<br />
<br />
There is an apparent switching on of the system, but this is in fact not what we would expect seeing as our model is for a single cell in a surrounding environment of infinite volume. Therefore any HSL produced by the cell should become diluted to the point where the concentration externally can be said to be negligible.<br />
<br />
<br />
[[Image:LuxRSingleCellLoopRampUp8th(Large).jpg|800px]]<br />
<br />
'''Figure 3: Graph of GFP, LuxI, LuxR, HSL-LuxR and mRNA concentrations against time'''<br />
<br />
<br />
We can see from figure 3 that the model predicts huge concentrations of the HSL-LuxR complex to be present in the cell, which strongly contradicts previous work. The complex is very unstable and should therefore exist in only small concentrations within the cell. This gives us some idea as to the fallibility of this model. It is also of note that in general the order of magnitude if the concentrations output from this model are a few below that from [[Team:St_Andrews/project/modelling/model_1|model 1]].<br />
<br />
[[Image:LuxRSingleCellLoopRampUpmRNA.jpg|800px]]<br />
<br />
'''Figure 4: Graph of mRNA concentration against time'''<br />
<br />
<br />
The concentrations of mRNA are equal for GFP, LuxR and LuxI, and whatsmore are comparable to those seen in our previous model.<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_2Team:St Andrews/project/modelling/model 22010-10-27T19:07:18Z<p>Ally m: </p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 2: Single Cell With Loop </h1><br />
</html><br />
<br />
=Theory & assumptions=<br />
Our second model introduced the concept of an external environment into which the HSL could diffuse into. This was assumed to be of infinite volume and thus the concentration of HSL in this region would always be zero, regardless of the quantity of HSL leaving the cell. <br />
In order to implement this modification we produced a new differential to desrcibe the change in HSL.<br />
<br />
<br />
[[Image:HSL2.jpg|900px]]<br />
<br />
''Figure 1: Differential equation describing HSL change''<br />
<br />
The creation of HSL is dictated by kConv, which is associated with the ability of LuxI to catalyse the conversion of resources into HSL. The concentration is reduced by the association with LuxR and by natural degredation. It is also affected by the degredation of molecules in the HSL-LuxR complex in the same way that LuxR is. We also introduce a diffusion term which accounts for HSL diffusing out of the cell due to a concentration gradient between the cell and the external environment. <br />
<br />
The code on which this model is based can be found [[Team:St_Andrews/project/modelling/downloads|here]].<br />
<br />
=Results=<br />
The switching on behaviour produced by this model is shown in figure . This switching on can be interpreted as the cell turning itself on due to the HSL it produces not diffusing out of the cell at a high enough rate. Therefore there is a considerable concentration of HSL remaining in the cell to maintain the feedback loop and produce more HSL. Thus regardless of the initial concentrations our cell always switches on. This does not match with the true biological scenario at all. Quorum sensing is designed such that cells will only become up-regulated when present at a certain critical cell density. Therefore the suggestion made by our results that a single cell can self-activate undermines this principle entirely.<br />
<br />
<br />
[[Image:LuxRSingleCellLoopRampUp(correctedmRNA).jpg|800px]]<br />
<br />
'''Figure 2: Graph of GFP concentration against time'''<br />
<br />
<br />
There is an apparent switching on of the system, but this is in fact not what we would expect seeing as our model is for a single cell in a surrounding environment of infinite volume. Therefore any HSL produced by the cell should become diluted to the point where the concentration externally can be said to be negligible.<br />
<br />
<br />
[[Image:LuxRSingleCellLoopRampUp8th(Large).jpg|800px]]<br />
<br />
'''Figure 3: Graph of GFP, LuxI, LuxR, HSL-LuxR and mRNA concentrations against time'''<br />
<br />
<br />
We can see from figure 3 that the model predicts huge concentrations of the HSL-LuxR complex to be present in the cell, which strongly contradicts previous work. The complex is very unstable and should therefore exist in only small concentrations within the cell. This gives us some idea as to the fallibility of this model. It is also of note that in general the order of magnitude if the concentrations output from this model are a few below that from [[Team:St_Andrews/project/modelling/model_1|model 1]].<br />
<br />
[[Image:LuxRSingleCellLoopRampUpmRNA.jpg|800px]]<br />
<br />
'''Figure 4: Graph of mRNA concentration against time'''<br />
<br />
<br />
The concentrations of mRNA are equal for GFP, LuxR and LuxI, and whatsmore are comparable to those seen in our previous model.<br />
<br />
=Conclusions=</div>Ally mhttp://2010.igem.org/Team:St_Andrews/project/modelling/model_3Team:St Andrews/project/modelling/model 32010-10-27T19:05:45Z<p>Ally m: </p>
<hr />
<div>{{:Team:St_Andrews/defaulttemplate}}<br />
<br />
<html><br />
<h1> Model 3: Pseudo Multi Cell One Dimension </h1><br />
</html><br />
<br />
==Theory & assumptions==<br />
The results produced from both of our previous models were unsatisfactory, since neither gave us quantifiable data from which experimental comparisons could be made, although the switching on was clearly visible in both. Therefore it was decided that in order to properly capture the mechanism of bistability it was essential to have some element of cell number, and as such we must simulate the growth of a cell colony rather than artificially increase the amount of HSL in our external environment.<br />
Several crucial elements were added to the model to incorporate this new outlook: <br />
<br />
<br />
i. A new variable [[Team:St_Andrews/project/modelling/model_3/spatial|'Number of cells']] was added<br />
<br />
ii. Two new functions which simulate [[Team:St_Andrews/project/modelling/model_3/growthdeathfunctions|cell growth and cell death]]<br />
<br />
iii. A new paramter [[Team:St_Andrews/project/modelling/model_3/spatial|'Volume']]<br />
<br />
<br />
The number of cells is set at the beginning of the simulation by the user. Based on a doubling time of 20 minutes, the variable is increased every increment by the cell growth function. In order to simulate the increase in HSL due to these new cells, each new cell is accompanied by an increase in HSL which is equal to that being contributed by the cells already present in the system. (VOLUME THING IN DIFFERENTIALS). Similarly we included a cell death function which decreases the number of cells at a halving rate of 20 minutes in tandem to the growth rate.<br />
<br />
==Results==<br />
On running the model through the cell growth, stationary and cell death phases, the first example of ‘true bistability’ is produced, see figure . Since this model contains spatial dimensions we are able to determine a specific cell density at which the switch happens, and this is comparable to the value predicted by other models and by experimental evidence. There is also a clear difference between the switch-on and switch-off thresholds which we hypothesise is caused by…<br />
<br />
<br />
[[Image:GFPvcell desnity.jpg|800px]]<br />
<br />
'''Figure 1: Graph showing change in GFP concentration against cell density'''<br />
<br />
<br />
[[Image:CellGrowth(alldata) larger range.jpg|800px]]<br />
<br />
'''Figure 2: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from off to on'''<br />
<br />
<br />
[[Image:CellDeath(alldata).jpg|800px]]<br />
<br />
'''Figure 3: Graph showing change in GFP, LuxR, LuxI and mRNA concentrations for the transition from on to off'''<br />
<br />
<br />
Info on HSL-LuxR concentration behaviour<br />
<br />
Graph with up and down data?<br />
<br />
=Parameter Testing=<br />
The next stage in the development of our model was to probe which factors had an impact on the behaviour of the system and exactly what that impact was and what implications they have.<br />
<br />
==Measuring bistability==<br />
The key point which we wished to investigate was the bistability of our system, so in order to do this we had to develop a standard way in which we would measure the 'level' of bistability. The method decided upon was to take a reading of the cell density at a fixed value of GFP concentration during cell growth and similarly record the cell density at the same GFP concentration during cell death. Thus we would obtain a value Δ Cell density which gives us a useful quantity which we can use in our tests.<br />
<br />
<br />
[[Image:Bistabilitymeasure.jpg|center]]<br />
<br />
'''Figure 4: Our method of bistability measurement'''<br />
<br />
==Testing our switch==<br />
We decided that there were some key parameters which we would like to test, which were the rates of:<br />
* Association and dissociation<br />
* HSL production and degradation<br />
* LuxR degradation<br />
* Promoter binding site strengths<br />
<br />
==Implementation==<br />
In order to test the level of bistability, we first needed to find the parameter range for which the system switched on. This was done by running our model many times with different values of the parameter we wished to investigate, and looking at for which values the system switched on by the final concentration of GFP. We could then operate only within the operating range of the switch and run our model many times within this range, looking at the value of ΔCell density and comparing for different parameter values.<br />
<br />
===Association and dissociation of HSL-LuxR===<br />
The HSL-LuxR is the key to the entire quorum sensing circuit and as such should play an important role in the level of bistability of the system. Our initial tests showed unsurprising results, that at high dissociation rates the system requires a higher rate of association to switch on and vice-versa.<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 13-19-50.jpg|center]]<br />
<br />
'''Figure 5: Parameter sensitivity tests on rate of associations and dissociation on HSL and LuxR'''<br />
<br />
===HSL production & degradation===<br />
We reasoned that the rate at which HSL was both produced, and degraded within the system should have a significant effect on the bistability since this molecule is fundamental to the process of quorum sensing in E.coli. Indeed our initial tests found that if the conversion rate was too low, or the degradation rate too high, the system did not actually switch on (see figure ).<br />
<br />
<br />
[[Image:LuxRHSLInitialParameter.JPG|center|800px]]<br />
<br />
'''Figure 6: Results from initial parameter tests on rate of HSL production and degradation'''<br />
<br />
<br />
Our test indicated that our parameter test should be performed for values, 0.05 < kConvHSL < 0.25 and 0.0004 < kDegHSL < 0.0005. The results of these tests are shown below.<br />
<br />
<br />
[[Image:KConvHSLDeltaCellDensity.JPG]]<br />
<br />
'''Figure 7: Parameter sensitivity test on rate of HSL conversion'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 12-43-47.jpg]]<br />
<br />
'''Figure 8: Parameter sensitivity test on rate of HSL degradation'''<br />
<br />
<br />
From these plots we can clearly see that by increasing the HSL conversion rate we also decrease the bistability of the system and by increasing the HSL degradation rate we increase the bistability of the system.<br />
<br />
===LuxR degradation===<br />
Similar tests were performed on the effects of changing the rate of LuxR degradation. We also expected interesting results from these tests since the LuxR is the HSL binding molecule. Our initial test to find a suitable parameter range gave us the data shown in figure .<br />
<br />
<br />
[[Image:LuxR Degradation Initial Test.jpg|center|800px]]<br />
<br />
'''Figure 9: Parameter sensitivity test results on LuxR degradation rate'''<br />
<br />
<br />
There is a clear drop to zero in the bistability at a certain threshold level for LuxR degradation. This is due to the fact that the LuxR is degrading before an can bind to the HSL so the system does not switch on.<br />
<br />
===Promoter efficiency and affinity===<br />
There are two aspects of the quorum sensing network over which we as experimental scientists can control: the speed at which ribosomes translate our proteins and the binding of the promoter. In our model, these two phenomena correspond to the rate of translation kTranslation and kMaxProductionRNA. Thus we felt it would be very useful if we could characterise the beahvaiour of our switch on changing these parameters, since if the switch is to be put to any practical use it will be these parameters which would experimentally be changed. Thus we performed parameter tests changing these rates simultaneously to investigate the effects. <br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-06.jpg|center|800px]]<br />
<br />
'''Figure 10'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-52-30.jpg|center|800px]]<br />
<br />
'''Figure 11'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-47-00.jpg|center|800px]]<br />
<br />
'''Figure 12'''<br />
<br />
<br />
[[Image:Snapshot 2010-10-26 16-48-54.jpg|center|800px]]<br />
<br />
'''Figure 13'''<br />
<br />
=Conclusions=</div>Ally m