Team:Imperial College London/Modelling/Output/Results and Conclusion

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Modelling Overview | Detection Model | Signaling Model | Fast Response Model | Interactions
A major part of the project consisted of modelling each module. This enabled us to decide which ideas we should implement. Look at the Fast Response page for a great example of how modelling has made a major impact on our design!
Objectives | Description | Results | Constants | MATLAB Code
Results
During the construction of the model and after the whole models were completed they were tested and their behaviour observed and analysed. The most common method that we used for analysing our models was parameter analysis. This involves taking one particular parameter, varying it across several orders of magnitude and observing the results. This was important during the development of the models so that we could determine which parameters our models are most sensitive to. Then efforts would be made to determine these paramteres in the labs or take extra care in looking for them in research papers. In order to see the parameter sensitivity analysis results click on the button below:
Preliminary Parameter Sensitivity Analysis

1. Parameter sensitivity performed on penultimate amplification level

The observations below are based on models that were implemented up to the production of dioxygenase. Catechol was not added to the system, so conclusions are based on the concentrations of dioxygenase. The results of the full models, including the colour output, are included in section 2 below.

Challenges

Despite being quite thoughtful in translating our models into MatLab, unexpected outputs were observed. For example, some species were generating negative concentrations. In particular, for initial dioxygenase concentrations smaller than 10-4 mol/dm3 the results were inconclusive since they were oscillating around zero. We realised that this could be due to the ode-solver that we were using initially (ode45 in Matlab).

Trying to correct this problem with the ode-solver, the following precuations were implemented:

  • NonNegative function in Matlab preventing solver from reaching negative values - still some marginally negative values show
  • Scaling - all the values were scaled up by a factor of 106 as working on small numbers could be problematic for Matlab. Once the result is generated by the solver the resulting matrix is scaled back down by 106.

When we entered the real production and degradation rates into our model, we again obtained negative values. This was due to our set of differential equations being stiff. Since ode45 cannot solve stiff differential equations, we had to switch to using ode15s - an ODE-solver designed to handle stiff equations.

Model pre-A

This is the result of the simulation of simple production of Dioxygenase. It can be seen that the concentration will tend towards a final value of approximately 8×10-6 mol/dm3. This final value is dependent on the production and degradation rate(which has been estimated for all of the models).

On the website E.coli Statistics [http://gchelpdesk.ualberta.ca/CCDB/cgi-bin/STAT_NEW.cgi] it is stated that number of ribosomal proteins per cell is 900,000. In a cellular volume of order of 1μm3 = 10-15dm3=10-15L, the above number of ribosomes converts to 1.5×10-3mol/L. This means that a concentration of 10-5mol/dm3 is not out of scale and there is even some room for manipulation.

IC Model preA.png
Figure 1. Results of the Matlab simulation of Model preA.

Sensitivity of Model preA

Parameter Sensitivity
Production rate 1 order of magnitude change in protein production results in 1 magnitude change in final protein concentration
Degradation rate 1 order of magnitude change in degradation rate results in 1 magnitude change in final protein concentration (degradation increases then protein concentration decreases), as well as it corresponds to 1 order of magnitude in time in seconds (increase in degradation decreases time taken to reach plateau).

Model A

Sensitivity of Model A

We want to determine how our system reacts if different parameters are changed. This is to find out which parameters our system is very sensitive to.

IC Comparison A, preA.png
Figure 2. Comparison between Model pre-A and Model A. Initial concentration of split Dioxygenase: 10-5 mol/dm3. Bear in mind that Dioxygenase is a penultimate compound to the colour compound.

Parameter Sensitivity
Initial concentration of split Dioxygenase Change of one order of magnitude in the initial concentration, c0, gives change of one order of magnitude in the output concentration (range: 1 > c0 > 10-5). Sensitivity is lost for extremely high or low values. The minimum concentration for any observable amplification to happen is 10-5 mol/dm3. Note that the obtained threshold value is very close to the maximum value that can be generated in the cell according to Model pre-A. Note that initially simple production has advantage over amplified system which takes some time to take off.
Km Change of one order of magnitude results in change of output concentration by one order of magnitude (0.01 < Km < 100). The bigger the value of Km, the smaller the "final concentration" will be. At values smaller than 0.01, the sensitivity is affected. For higher values than 100 the sensitivity is slightly higher than the change of order of magnitude.
(Also, it was found that the absolute value of k1 and k2 entered into Matlab does not change the outcome as long as the ratio between them (Km≈k2/k1) is kept constant. This is important when simulating (in case entering very high values for k1 and k2 slows down the simulation).
kcat kcat is proportional to dioxygenase production (1-to-1 sensitivity for all values) for an initial concentration of 0.01 mol/dm3. For very high initial concentrations, the system is very sensitive to changes in kcat.
Hence, it would be very beneficial for reliability of this model to determine kcat in the wet-lab experiment.
Production rate of TEV 1-1 sensitivity for most values. At some point the system’s response is limited by the initial concentration of sD, so for very high TEV production rates not much change is observed.
Production rate of split Dioxygenase Not much influence on 1-step amplification. However, the value seems to be crucial for simple production of Dioxygenase (1-1 order of magnitude sensitivity).
Degradation rates Sensitive within the relevant range. Not very sensitive for values smaller than 10-6. For high degradation rates (1 > degradation rate > 0.01): unexplainable behaviour.

Hence, the system is sensitive to most of the constants (given a particular range of values). The most crucial one, however, seems to be the initial concentration of split Dioxygenase and kcat.

Model B

Sensitivity of Model B

This model has shown very similar sensitivity results as Model A, eg.:

  • Initial Concentration

The initial concentration of split Dioxygenase, c0, determines whether the system is amplifying.

Model A vs. B

Running both models with the same initial conditions (c0=10-5 mol/dm3),it has been noted that Model B does not generate a siginificant amplfication over Model A. Hence, it would be more sensible to integrate a one step amplification module into our system.

IC Comparison prea, a, b-1-.png
Figure 3. Comparison between Models pre-A, A and B. Bear in mind that this graph shows dioxygenase concentrations which is at penultimate stage to the colour compound.

2. Parameter sensitivity performed on the complete models

The important information about the coloured compound (i.e. the product of the last enzymatic reaction) is toxic to the cells ([http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6V78-4XXNV4Y-4-7&_cdi=5836&_user=217827&_pii=S0048969709011668&_orig=search&_coverDate=02%2F01%2F2010&_sk=995919994&view=c&wchp=dGLbVzz-zSkzk&_valck=1&md5=f8dca6227c29db659ddbeb588ad115e7&ie=/sdarticle.pdf],[http://www.msk.or.kr/jsp/downloadPDF1.jsp?fileName=393-15.pdf]). It is suggested that product of by-product of catechol-dioxygenase reaction destroys the cell membrane by inhibiting lipid peroxidation. It causes significant changes in the structure and functioning of membrane components (e.g. disruption of membrane potential, removal of lipids and proteins, loss of magnesium and calcium ions). These effects cause the loss of membrane functions, leading to cell death. In order to determine as least qualitatively the influence of cell death on performance of the amplifiers, 2 versions of models have been designed. Once version is neglcting the cell death while the other models instantaneous cell death with addition of catechol. Cell death in our models means that production of any proteins stops, but the reactions are still allowed to perform.

Challenges

Once we have implemented the colour change into the models, we noticed that there are some odd results. After adding catechol, some concentrations were reaching negative values. We checked our equations and constants but could not find a mistake. Hence, we concluded that there must be something wrong with the way that MatLab evaluates the equation or deals with the numbers. The problem seems to originate from the very rapid concentration change of catechol which disrupts the whole system.

Tried solutions:

  • Implementation in TinkerCell

We hoped that TinkerCell imposes non-negative conditions on its solutions. Hence, we implemented the whole amplification model (including coloured output) in TinkerCell. However, we realized that TinkerCell does not deal well with very high or low numbers (For example, values higher than 105 are not acceptable - this is important since our rate constants (k1) are usually bigger than 105. Also, the low degradation rates (10-9) result in a zero output line). However, TinkerCell can still be used for testing that our Matlab programs behave the way we anticipated (by using default parameter values of 1), as well as producing illustrative diagrams of our system.

  • Varying ODE solver options in MatLab

We had a close look at the ODE solver options in MatLab. However, we were already using the one that produced the most reasonable results (ode15s). We found that decreasing Relative and Absolute tolerances (to values as small as 10-15) significantly improved the simulation. However, this is not an ultimate solution as in the simulations negative numbers still appear (order of 10-15). We decided that such small negative concentrations were acceptable.

  • Using SimBiology

We hoped that SimBiology could be more suited for our modelling than using ODE-solvers, so we implemented our models into SimBiology. This package offers an interactive user interface similar to Tinker Cell, but uses MatLab to simulate. Initially, we confirmed that our simple production model (Model PreA) and 1-step amplification model (Model A) implemented in SimBiology generated exactly the same results as our ODE equation based models. The interface allowed us to have clearer control over parameters. It also allowed modelling special events, for example, adding catechol at certain point in time. Previously we had to split simulation into to two parts.

Conclusions

  • Changing time when catechol is added

If Catechol is added before t= 500s, then the coloured output will reach its threshold value faster by 1-step amplification. If Catechol is added when t>500s, then the coloured output will increase (marginally) faster through the amplification step in Model A (2-step amplifier). However, then there does not seem to be a appreciable difference between the two models (Model preA and Model A). These observations are true for intial concentration of dioxygenase equal to 10-5mol/dm3. However, we noticed that if the initial concentration is raised to 10-4mol/dm3, then Model A can be more beneficial than Model preA after only 100 seconds. Note that time of adding catechol does not exactly correspond to “cross-section” in Figure 3 as the systems still work after adding catechol, so it can be added a bit before the “cross section”. However, there is no point in delaying addition of catechol into system just prove that 2-step amplification could work.

IC Cathecol model before cross.png
Figure 4. Concentration of coloured compound for catechol being added before the "cross-section" is reached (check Figure 3 for definition of "cross-section").
IC Cathecol model at cross.png
Figure 5. Concentration of coloured compound for catechol being added when the cross-section is reached.
IC Cathecol model after cross.png
Figure 6. Concentration of coloured compound for catechol being added after the cross-section has been reached.
  • Changing concentration of catechol added

There seem to be 3 regions of catechol concentration that influence the system in different ways. Whatever, the initial conditions and other parameters are, those 3 regions exist, but their boundaries move around. For one particular sequence of initial conditions, these regions were: c>1M, 1M>c>0.01M, 0.01M>c. Varying the initial concentration of catechol within the highest region does not result in any change of colour output response (It is possible that all enzymes are occupied and the solution is over saturated with catechol). In the middle region the catechol concentration influences the amplfication. Amplification decreased when the concentration tends towards the 0.01M border. When the lowest region is entered, there is no difference in output production by Model A and preA.

  • Cell death

The coloured product of catechol kills cells by destroying the cell membrane. However, we do not know how quickly the cells will die. Therefore, we examined two different cases: immediate cell death and negligible cell death (i.e. cells death is negligible because it takes too long)

Running the simluation in Matlab (not Simbiology!), our conclusions are:

  1. Immediate cell death slows down production of coloured output. Depending on the threshold concentration this can delay the detectable response by a few minutes.
  2. If Catechol is added before t=1000s, then cell death slows down the response considerably.
  3. In case of cells being modelled as alive, the difference between the 2-step amplified and the 1-step amplified model is smaller than it is in case of cell death.

Since it appears that the time of cell death is important, we decided to discuss this issue with our team members. Referring to this paper [1] we decided that cell death induced by catechool is a very slow process (we estimate that it will take a few hours) in comparison to the time scale that we are interested in (several seconds to minutes). Hence, our final conclusions will be on the model neglecting cell death.

IC alive cells.png
Figure 7. Colour response model for cells being kept alive. The four graphs correspond to 4 species of the last enzymatic reaction that leads to colour showing up.
IC dead cells.png
Figure 8. Colour response model with the same parameters as Figure 7 for cells being killed instantaneously by catechol. The four graphs correspond to 4 species of the last enzymatic reaction that leads to colour showing up.

References

  1. Chen, H. et al (2009) Toxicity of three phenolic compounds and their mixtures on the gram-positive bacteria Bacillus subtilis in the aquatic environment. Science of the Total Environment. [Online] 408(2010), 1043-1049. Available from: http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6V78-4XXNV4Y-4-7&_cdi=5836&_user=217827&_pii=S0048969709011668&_orig=search&_coverDate=02%2F01%2F2010&_sk=995919994&view=c&wchp=dGLbVzz-zSkzk&_valck=1&md5=f8dca6227c29db659ddbeb588ad115e7&ie=/sdarticle.pdf [Accessed 6th September 2010]
  2. Park, S., Ko, Y. & Kim, C. (2001) Toxic Effects of Catechol and 4-Chlorobenzoate Stresses on Bacterial Cells. The Journal of Microbiology. [Online] 39(3), 206-212. Available from: http://www.msk.or.kr/jsp/downloadPDF1.jsp?fileName=393-15.pdf [Accessed 6th September 2010]
  3. Habibi Nazhad, B. (2008) E. coli Statistics. [Online] Available from: http://gchelpdesk.ualberta.ca/CCDB/cgi-bin/STAT_NEW.cgi [Accessed 6th September 2010]



Finally, 2 parameters were determined as crucial to the designs and their collaborative influence on the models is discussed in the Conclusions section.

Conclusions
Having explored the sensitivity of our models at 2 amplification levels and after many values were constrained by lab results or biological ranges, 2 parameters have been recognised as crucial. Their values are decisive regarding the design of the 1 and 2 output amplification systems. Hence, one should pay most attention to them, when trying to assemble similar systems:
  • Rate of production of species

This is one of the parameters that could have been very poorly estimated in our models. Initially, all our models have been tested for values which we thought could correspond to high copy plasmids in E.Coli (rate= 2.4998×10-9 M/s). 2 step-amplification performs worse than 1-step amplification given biologically relevant concentrations of catechol (and considering other constraints). However, our system is implemented in Bacillus subtilis which expresses genes very weakly. If its expression rate of dioxygenase is 1 or 2 magnitudes lower than in the case of E.Coli, then 2-step amplification seems to be 4 minutes faster than 1-step amplification (see figure below).

Amplification.png
Figure 9. Four graphs represent species present in the last enzymatic reaction, which leads to colour output. Note that time=0 corresponds to when the first transcription happens upon binding of the transcription factor to DNA. These results are generated for parameter values that we thought are reflecting our lab situation. The bottom right-hand graph shows how our 2-step amplification system outperforms 1-step amplification.

If you forgot the elements of the system, have a look at diagrams hidden in the button.

Reminder of Amplification Step diagrams
Simple Production.png
Simple production upon activation of arbitrary colour output by transcription and translation indicated by the blue arrow.
1-step amplification.png
Dioxygenase (C230) is simply produced. Upon activation at time t=0, it acts on catechol (cat.) to produce yellow output - muconic acid. Catechol is not shown to be produced by cell as it is added by person at arbitrary time.
2-step amplification.png
The species that are shown in front of vertical line which indicates beginning of experiment mean that they have been accumulated beforehand in the cell. TEV protease activates inactive dioxygenase which acts on catechol to produce colour.
3-step amplification.png
This diagram introduces inactive split TEV protease attached to a coiled-coil as the third amplification step. Both inactive compounds have active site for TEV to activate tehm which results in multiple possibilities of action.




This means that one needs to be careful when considering in which species the system would be used. In a chassis that efficiently expresses proteins (e.g. E.Coli) we would recommend that 1-step amplifier gives the best performance (if the same output is used). On the other hand, in a chassis that is not very good in expressing proteins like Bacillus subtilis we would recommend implementing the 2-step amplification system (if the same output is used). One could argue that degradation constants could have a similar role to production rates. However, degradation rates influence the magnitude of the plateau that is being reached (in simple transcription and translation). Degradaation rates also determine how fast the plateau will be reached. The influence on timing discards degradation constants from having a big impact on the system.

  • Catalytic constant

It should be noticed that kcat of Dioxygenase (kcat=52) is 4 magnitudes higher than kcat of TEV (kcat=0.16) . This is a tremendous difference and it emphasises the efficiency of the 1-step amplifier. Basically, in our combination of enzymes the second one – TEV – does not have much to improve on. It is even harder for 2-step amplifier to be efficient taking into account the slow action of the TEV protease. To analyze the importance of the kcat values please observe the graphs below.

Kcat1.png
Figure 10. The colour output concentration changing with time. Note that time=0 corresponds to when the first transcription happens upon binding of the transcription factor to DNA. This result is generated for the following catalytic constant values: kTEV=1.6 s-1 kC2,3O=0.52 s-1. Compare with Figure 11.
Kcat2.png
Figure 11. The colour output concentration changing with time. Note that time=0 corresponds to when the first transcription happens upon binding of the transcription factor to DNA. This result is generated for the following catalytic constant values: kTEV=1.6 s-1 kC2,3O=52 s-1.

From the above graphs it can be seen that if the situation regarding kcat values was reversed 2-step amplifier would perform much better than the 1 step amplification system even for high copy plasmids like E.Coli! Clearly, 3 step amplification does not perform well under the given conditions, due to too much delay being introduced by the third amplification step. Hence, if one was to design the amplification module for different colour outputs and would not be satisfied with the speed of response generated by 1-step amplification, it would be recommended to choose the first enzyme 1 (refer to picture below) to have a kcat value that is at least 2 magnitudes higher than the kcat value of enzyme 2 (for high expressing chassis).

  • Combined effects of kcat and rate of production

The two factors can decide whether the 2-amplifier system could perform better than the 1-step amplifier. The effect of these two factors can be combined to generate even more efficient systems. If one was using a chassis of medium expression rate, then choosing enzymes with 1 magnitude difference in kcat would result in the 2-step amplifier performing better.

Click here for the constants of this model...
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