http://2010.igem.org/wiki/index.php?title=Special:Contributions&feed=atom&limit=500&target=Brychan&year=&month=2010.igem.org - User contributions [en]2024-03-28T18:17:38ZFrom 2010.igem.orgMediaWiki 1.16.5http://2010.igem.org/Team:Aberdeen_ScotlandTeam:Aberdeen Scotland2010-10-27T19:31:56Z<p>Brychan: </p>
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<h1>Why ayeSwitch?</h1><br />
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Over the course of the summer, the University of Aberdeen iGEM team engineered a novel genetic toggle switch in yeast which is regulated at the translational level and allows mutually exclusive expression of either green or cyan fluorescent protein. Using cell cytometry (FACS) and fluorimetry, we successfully demonstrated gene expression and translational regulation of a fusion of mRNA binding proteins and fluorescent proteins. Deterministic and stochastic models including experimental results and published parameter values predicted that the probability of successful bistability for the switch is 0.96%, but that this can theoretically be improved to a maximum of 51.27% by limiting the variation range of the most sensitive parameters. The models also predicted that to generate switch-like behaviour, co-operative binding of the mRNA binding protein to its mRNA stem loop was essential. These results suggest that a translationally regulated genetic toggle switch is a viable and novel engineering concept applicable to medicinal, environmental and technological problems.<br />
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<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Project_Overview">Your ayeSwitch experience begins here&nbsp;<img src="https://static.igem.org/mediawiki/2010/3/36/Right_arrow.png"></a><br />
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<h1>Our Sponsors:</h1> <br />
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Aberdeen iGEM 2010 gratefully acknowledges the financial support of the following organisations:<br />
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{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/Stochastic_ModelTeam:Aberdeen Scotland/Stochastic Model2010-10-27T19:29:30Z<p>Brychan: </p>
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<h1>The Stochastic Model</h1><br />
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<p> Stochastic modelling can be used when modelling biochemical processes and intracellular dynamics <a href="#ref2"><sup style="font-size:10px">[2]</sup></a>. It is also widely used in many other areas such as physics, economics, geophysical systems and even music. </p> <br />
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<h3>Overview of the Deterministic Model</h3><br />
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<p>A first look at how our system behaved involved modelling it deterministically. The deterministic model is based on constructing a set of reaction-rate equations to describe the biochemical pathways being studied. These equations are non-linear ordinary differential equations (ODEs). The concentrations of chemical species are the variables and the parameters are the reaction rate constants. Solving the ODEs gives how the resulting concentrations of the chemical species change in time <a href="#ref2"><sup style="font-size:10px">[2]</sup></a><sup style="font-size:10px">,</sup><a href="#ref3"><sup style="font-size:10px">[3]</sup></a>. <br />
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In a deterministic system, the time evolution is considered to be continuous and predictable. Strictly speaking, the time evolution of a chemically reacting system is not a continuous process because molecular population levels can only change by discrete integer amounts. In order to predict the molecular population levels at a future time by means of a deterministic approach, we have to take into account the precise positions and velocities of all the molecules in the system <a href="#ref1"><sup style="font-size:10px">[1]</sup></a>, which becomes practically impossible.<br />
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In general, concentrations are only defined for large numbers of molecules. In this situation, when numbers change by one or two units in a reaction, these changes can be treated differentially. Also, when the number of molecules is large, any two reactions can take place at the same time. The system of ODEs thus represents a collection of reactions occurring simultaneously <a href="#ref3"><sup style="font-size:10px">[3]</sup></a>. <br />
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The deterministic model has some disadvantages however. It is unable to describe the fluctuations in the molecular population levels, which can become very important if the numbers of the molecules involved in the reactions are very low <a href="#ref1"><sup style="font-size:10px">[1]</sup></a>.</p><br />
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<h3>Introducing the Stochastic Model</h3><br />
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<p> The stochastic model is a way of following the dynamics of individual molecules <a href="#ref1"><sup style="font-size:10px">[4]</sup></a>. In addition to the deterministic simulation, we also perform the stochastic simulation to see if the potential fluctuations due to the low numbers of molecules involved in our system play a major role. We assume that reactions cannot happen simultaneously and that they do not happen continuously throughout time. There is also now a probability attached to when a reaction will happen and which reaction will occur. <br />
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We consider the time evolution of a chemical system to be a discrete process instead of a continuous, deterministic process. The stochastic model thinks of the time evolution as a random process with probabilities as variables <a href="#ref1"><sup style="font-size:10px">[1]</sup></a><sup style="font-size:10px">,</sup><a href="#ref2"><sup style="font-size:10px">[2]</sup></a>. The stochastic model involves the same decomposition of a pathway into elementary reactions. However, here we are looking at numbers of molecules instead of concentrations <a href="#ref2"><sup style="font-size:10px">[2]</sup></a>. The connection with molecular concentrations appears when, in the stochastic model, averages are taken over many cells. These averages satisfy the same equations as the concentrations. Thus the behaviour of concentrations can be interpreted as that of a population average, provided that fluctuations around the average are small <a href="#ref3"><sup style="font-size:10px">[3]</sup></a>.</p><br />
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<h3>How to Model Systems Stochastically</h3><br />
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<p>Modelling a system of differential equations stochastically can be done using the Gillespie Algorithm which was devised by Daniel T. Gillespie in 1977 <a href="#ref1"><sup style="font-size:10px">[1]</sup></a>. However, Gillespie also devised a variation on his method which is known as the tau-leaping method. This method results in faster simulations because it is less exact than the actual Gillespie Algorithm as it uses much larger time steps and therefore has less calculations to perform.<br />
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Due to time constraints we modelled our system using the tau-leaping model. <br />
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The tau-leaping method proceeds as follows: </p><br />
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<OL><br />
<LI>Specify a time step, tau.<br />
<LI>Multiply each term of each differential equation by tau.<br />
<LI>Apply Poisson distribution function to each result from step 2.<br />
<LI>Generate a random number using the Poisson distribution created in step 3.<br />
<LI>Calculate the new number of molecules by adding or subtracting each random number from step 4, from the initial number of molecules. Adding if the change has a positive influence on the system, subtracting if it has a negative influence on the system. <br />
<LI>Repeat steps 1-5 for a specified number of reactions, N.<br />
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<p>This algorithm allowed us to view what our model should look like stochastically. The only problem is that we do not know exactly how many molecules we are starting with. However, this is a problem which affects the deterministic model as well as the stochastic model. In both models, the initial concentrations or number of molecules needs to be specified. Until we know how many molecules we begin with, this program will only give us an idea of how our system could evolve. We decided to start our simulations with only one molecule of each of the mRNAs and proteins and watch how the system evolved from there.</p><br />
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<h3>Results</h3><br />
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<p>The deterministic model describes the average change in concentration over time. Since the stochastic model deals with individual particles we would expect the stochastic model to follow the trend of the deterministic model but there would be fluctuations around the average value. If we were to take the average of these particles over time, it would look like the deterministic model. The results showed exactly what we expected. The stochastic model follows the trend set by the deterministic model but it fluctuates, sometimes wildly, around the average. <br />
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One thing to note is that the deterministic model sometimes suggests that we will not see bistability. However, because of the fluctuations around the average in the stochastic model, one fluctuation could push us into the realm of bistability.<br />
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In the pictures below the stochastic model is on the left and the deterministic model is on the right. One can clearly see the similarity in the overall shape and it is clear that the deterministic model is just the average of the stochastic model.<br />
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<b>Scenario 1: GAL and METH present, winner GFP</b><br />
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When both galactose and methionine are present, the methionine inhibits the production of the CFP and the galactose encourages the production of the GFP. With nothing to inhibit the production of GFP, the GFP dominates. <br />
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<b>Scenario 2: no METH and GAL present, winner – either, depends on other parameters!</b><br />
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When no methionine is present, CFP can be produced. Also, when GAL is present GFP can be produced. When both proteins are produced, they will both inhibit each other. In this scenario, the dominant protein will depend on the parameters of the system. The modelling we have done suggests that the parameters which affect the system are the transcription/translation rates (λ values) and the binding coefficients (K values). As shown in the above graphs, the parameters of our system dictate that CFP will win. <br />
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<b>Scenario 3: no METH and no GAL present, winner CFP</b><br />
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No methionine present leads to CFP being produced. No galactose present means that no GFP will be produced. With nothing to inhibit the CFP production, CFP will dominate. <br />
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Also, a simulation was run with variable galactose and methionine over time. The aim of this was to model the switching behaviour of our system. The picture below shows the results of the stochastic simulation. What we see, is a clear switching behaviour, but only if we actively remove one protein before adding the other.<br />
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<img src="https://static.igem.org/mediawiki/2010/2/21/Stoch_switch.jpg" width="700" height="470"> <br />
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<p>As mentioned previously, when GAL and no METH are present in the system (at 80000 iterations), the winning protein will depend heavily on the parameters of the system. Our experience has shown us that the values for the transcription/translation rates (λ values) and the dissociation constants (K values) strongly affect how the system behaves and thus determines what protein will dominate.</p></p><br />
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<h3>Conclusions</h3><br />
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<p>The stochastic model is a more accurate method of modelling a biochemical system where low numbers of molecules are involved. The stochastic model follows the actions of individual molecules whereas the ODEs based model describes how concentrations change continuously in time. The deterministic model is basically just the average of the stochastic model and does not represent any fluctuations due to the individual molecules present. </p><br />
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<h3>References</h3><br />
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<a name="ref1"></a><br />
<p><sup style="font-size:10px">[1]</sup> Gillespie, D.T. (1977), Exact Stochastic Simulation of Coupled Chemical Reactions, <i>The Journal of Physical Chemistry</i>, Vol. 81, No. 25.</p><br />
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<a name="ref2"></a><br />
<p><sup style="font-size:10px">[2]</sup> Ullah, M., Schmidt, H., Cho, K.-H. and Wolkenhauer, O. (2006), Deterministic modelling and stochastic simulation of biochemical pathways using MATLAB, <i>IEE Proc.-Syst. Biol.</i>, Vol. 153, No. 2.</p><br />
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<a name="ref3"></a><br />
<p><sup style="font-size:10px">[3]</sup> Hayot, F. (2008), Single Cell experiments and Gillespie’s algorithm. Retrieved from <a href="http://tsb.mssm.edu/summerschool/images/4/4d/HayotSlides.pdf">http://tsb.mssm.edu/summerschool/images/4/4d/HayotSlides.pdf</a> </p><br />
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<a name="ref4"></a><br />
<p><sup style="font-size:10px">[4]</sup> Department of Computational & Applied Mathematics, Rice University, Modeling and Simulation of Reaction Networks. Retrieved from <a href="http://www.caam.rice.edu/~caam210/reac/lec.html">http://www.caam.rice.edu/~caam210/reac/lec.html</a> </p><br />
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{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/Fixed_PointsTeam:Aberdeen Scotland/Fixed Points2010-10-27T19:27:07Z<p>Brychan: </p>
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<h1>Fixed Points</h1><br />
<p>Fixed points are the points where an equation’s rate of change, or its slope, is zero. There are three main types of equilibrium points: stable, unstable, and saddle-node points. A stable equilibrium is a value which the function converges towards, whereas an unstable equilibrium is a value the function will diverge from. A saddle-node is where the function both converges to and diverges from, the equilibrium, depending on the direction from which we approach the point (S. Strogatz, Nonlinear dynamics and Chaos).</p><br />
<br><br />
<h3>Why are fixed points important?</h3><br />
<p>Finding fixed points is the first step in analyzing the stability of a system. In particular, we are interested in the nature of our system’s bistability and how it changes with a variation of the parameters, i.e., its bifurcation analysis. This analysis can be conveyed to the biologists to minimize experimental guessing.<br />
</p><br />
<br><br />
<h3>How are fixed points calculated?</h3><br />
<p>Fixed points are determined by setting all differential equations in a system equal to zero and solving for the variable being analyzed. The direct method of calculating equilibrium points is to find the roots of the system of equations using built-in root-finding functions such as fzero in MATLAB, and similar functions in Maple and C. The indirect method of calculating the equilibrium points is to plot and find the intersections of these nullclines, which represent equilibrium points.</p><br />
<br />
<br />
<h3>Results</h3><br />
<p>We were able to find the fixed points of the system analytically (for small Hill coefficients) and computationally. These points were used for bifurcation analysis, and for analyzing the probability that our system would exhibit bistable behavior.</p><br />
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<br><br />
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<h1>Nullclines</h1><br />
<p>In a system of differential equations, the nullclines are the solution curves for which all of the differential equations are equal to zero (Wikipedia).</p><br />
<br><br />
<h3>Why are nullclines important?</h3><br />
<p>The intersections of the nullclines give the equilibrium points of the system of differential equations. From graphs of the nullclines, it is possible to infer whether or not a system will be bistable. If the nullclines only intersect in one place the system is not bistable, since there is one single equilibrium point. If there are more than two intersections, the middle equilibrium point is often an unstable saddle point.</p><br />
<br><br />
<h3>How are the nullclines calculated?</h3><br />
<p>Just as in calculating the fixed points, we set the governing differential equations of the system equal to zero and plot the curves generated.</p><br />
<center><br />
<img src="https://static.igem.org/mediawiki/2010/e/e4/Equations3.png"><br />
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<h3>Simulations involved</h3><br />
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<p><img src="https://static.igem.org/mediawiki/2010/1/18/Nullclines_diagram1.png"/></p><br />
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<p><b>Figure 1.</b>This figure is a plot of the nullclines of the differential equations for CFP and GFP, where we solve them for CFP as a function of GFP. In this plot, both Hill coefficients are two, ie. n2 and n4 are both 2. The nullclines cross over at three fixed points, where the middle is a saddle-node fixed point. This is an ‘ideal’ bistability plot.<br />
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<p><img src="https://static.igem.org/mediawiki/2010/5/53/Nullclines_diagram2.png"></p><br />
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<p><b>Figure 2.</b>This figure is a plot, again, of the nullclines of the differential equations for CFP and GFP. In this plot, both Hill coefficients are one. Note that the nullclines only cross once, resulting in only one fixed point and hence no possibility of bistability. This result allowed us to tell the biologists that the toggle switch would definitely not work if both of the Hill coefficients were one.<br />
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<p><img src="https://static.igem.org/mediawiki/2010/d/da/Nullclines_diagram3.png"/></p><br />
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<p><b>Figure 3.</b> Figure 3: This figure is of CFP as a function of GFP for the two nullclines when the initial galactose and copper concentrations are changed. From this we can see how the fixed points change with different initial galactose and methionine concentrations. The blue line represents the GFP = CFP line. The further from this line equilibrium points are, the harder it will be to switch between stable states.</p><br />
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<h3>Results</h3><br />
<p>We focused our efforts on plotting the nullclines of the system for a range of Hill coefficient combinations in order to get a general idea of which combinations would most likely produce robust bistability. We found that all combinations gave bistability, except when both Hill coefficients were one. The optimal Hill coefficient combination occurred when both Hill coefficients were two. We passed this information onto the biology team, letting them know that if we wanted the system to successfully switch, we could not have both Hill coefficients with a value of one.<br />
</p><br />
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{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/Fixed_PointsTeam:Aberdeen Scotland/Fixed Points2010-10-27T19:26:36Z<p>Brychan: </p>
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<h1>Fixed Points</h1><br />
<p>Fixed points are the points where an equation’s rate of change, or its slope, is zero. There are three main types of equilibrium points: stable, unstable, and saddle-node points. A stable equilibrium is a value which the function converges towards, whereas an unstable equilibrium is a value the function will diverge from. A saddle-node is where the function both converges to and diverges from, the equilibrium, depending on the direction from which we approach the point (S. Strogatz, Nonlinear dynamics and Chaos).</p><br />
<br><br />
<h3>Why are fixed points important?</h3><br />
<p>Finding fixed points is the first step in analyzing the stability of a system. In particular, we are interested in the nature of our system’s bistability and how it changes with a variation of the parameters, i.e., its bifurcation analysis. This analysis can be conveyed to the biologists to minimize experimental guessing.<br />
</p><br />
<br><br />
<h3>How are fixed points calculated?</h3><br />
<p>Fixed points are determined by setting all differential equations in a system equal to zero and solving for the variable being analyzed. The direct method of calculating equilibrium points is to find the roots of the system of equations using built-in root-finding functions such as fzero in MATLAB, and similar functions in Maple and C. The indirect method of calculating the equilibrium points is to plot and find the intersections of these nullclines, which represent equilibrium points.</p><br />
<br />
<br />
<h3>Results</h3><br />
<p>We were able to find the fixed points of the system analytically (for small Hill coefficients) and computationally. These points were used for bifurcation analysis, and for analyzing the probability that our system would exhibit bistable behavior.</p><br />
<br><br />
<br><br />
<br />
<h1>Nullclines</h1><br />
<p>In a system of differential equations, the nullclines are the solution curves for which all of the differential equations are equal to zero (Wikipedia).</p><br />
<br><br />
<h3>Why are nullclines important?</h3><br />
<p>The intersections of the nullclines give the equilibrium points of the system of differential equations. From graphs of the nullclines, it is possible to infer whether or not a system will be bistable. If the nullclines only intersect in one place the system is not bistable, since there is one single equilibrium point. If there are more than two intersections, the middle equilibrium point is often an unstable saddle point.</p><br />
<br><br />
<h3>How are the nullclines calculated?</h3><br />
<p>Just as in calculating the fixed points, we set the governing differential equations of the system equal to zero and plot the curves generated.</p><br />
<center><br />
<img src="https://static.igem.org/mediawiki/2010/e/e4/Equations3.png"><br />
</center><br />
<br><br />
<h3>Simulations involved</h3><br />
<br />
<br />
<center><br />
<table><br />
<tr><br />
<td><br />
<p><img src="https://static.igem.org/mediawiki/2010/1/18/Nullclines_diagram1.png"/></p><br />
</td><br />
<td><br />
<p><b>Figure 1.</b>This figure is a plot of the nullclines of the differential equations for CFP and GFP, where we solve them for CFP as a function of GFP. In this plot, both Hill coefficients are two, ie. n2 and n4 are both 2. The nullclines cross over at three fixed points, where the middle is a saddle-node fixed point. This is an ‘ideal’ bistability plot.<br />
</p><br />
</td><br />
</tr><br />
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<td><br />
<p><img src="https://static.igem.org/mediawiki/2010/5/53/Nullclines_diagram2.png"></p><br />
</td><br />
<td><br />
<p><b>Figure 2.</b>This figure is a plot, again, of the nullclines of the differential equations for CFP and GFP. In this plot, both Hill coefficients are one. Note that the nullclines only cross once, resulting in only one fixed point and hence no possibility of bistability. This result allowed us to tell the biologists that the toggle switch would definitely not work if both of the Hill coefficients were one.<br />
</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<p><img src="https://static.igem.org/mediawiki/2010/d/da/Nullclines_diagram3.png"/></p><br />
</td><br />
<td><br />
<p><b>Figure 3.</b> Figure 3: This figure is of CFP as a function of GFP for the two nullclines when the initial galactose and copper concentrations are changed. From this we can see how the fixed points change with different initial galactose and methionine concentrations. The blue line represents the GFP = CFP line. The further from this line equilibrium points are, the harder it will be to switch between stable states.</p><br />
</td><br />
</tr><br />
</table><br />
<br><br />
<h3>Results</h3><br />
<p>We focused our efforts on plotting the nullclines of the system for a range of Hill coefficient combinations in order to get a general idea of which combinations would most likely produce robust bistability. We found that all combinations gave bistability, except when both Hill coefficients were one. The optimal Hill coefficient combination occurred when both Hill coefficients were two. We passed this information onto the biology team, letting them know that if we wanted the system to successfully switch, we could not have both Hill coefficients with a value of one.<br />
</p><br />
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{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/ModelingTeam:Aberdeen Scotland/Modeling2010-10-27T19:24:25Z<p>Brychan: </p>
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<h1>Introduction to the Modelling of the ayeSwitch</h1><br />
<p>This page is an introduction to the different equations and techniques that we used to design and predict the behaviour of the ayeSwitch.<br />
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<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Equations"><h3><font color="blue">Equations</font></h3></a><br />
<p>In this section we describe the process of developing a basic mathematical model for the ayeSwitch based on the promotion and inhibition behaviour necessary for mutual repression. We developed a set of four differential equations, one to model each of the two mRNAs and two proteins that are the active components of our system.</p><br />
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<h3><a href="https://2010.igem.org/Team:Aberdeen_Scotland/Fixed_Points"><font color="blue">Nullclines and Fixed Points</font></a></h3><br />
<p>In this section we describe how we used fixed point analysis to predict the equilibrium state(s) of the ayeSwitch system for different parameters. Three or more equilibrium points will give us bistability and the possibility of switching. </p><br />
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<h3><a href="https://2010.igem.org/Team:Aberdeen_Scotland/Bifurcation"><font color="blue">Bifurcation</font></a></h3><br />
<p>Bifurcation analysis allows us to track the number and relative position of the equilibrium points of our system for different parameters. This section describes in detail how bifurcation analysis can help us determine the optimal range for our parameters. </p><br />
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<h3><a href="https://2010.igem.org/Team:Aberdeen_Scotland/Stochastic_Model"><font color="blue">Stochastic Model</font></a></h3><br />
<p>There are two ways to model our system - deterministically and stochastically. Both methods have their advantages and disadvantages depending on the system in question. In our system, it was more beneficial to model the system stochastically due to the low numbers of molecules involved. This section describes in detail both methods, their advantages and disadvantages and how they are used to model our system.</p><br />
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<h3><a href="https://2010.igem.org/Team:Aberdeen_Scotland/Probability"><font color="blue">Parameter Space Analysis</font></a></h3><br />
<p>This section describes in detail how we analysed the parameter space for our system. The results of this will show when bistability is possible and when it is not. Using this information, we can determine the optimal parameter ranges for our system. </p><br />
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<h3><a href="https://2010.igem.org/Team:Aberdeen_Scotland/Curve_Fitting"><font color="blue">Determination of the Hill coefficient n<sub style="font-size:10px">1</sub></font></a></h3><br />
<p>The Hill coefficient relating to the CFP/MS2 stem loop association is assumed to be around 2 due to the number of stem loops present. This section details the method we used to calculate this value more accurately, and what the result means in terms of the parameter space analysis.</p><br />
<h3><a href="https://2010.igem.org/Team:Aberdeen_Scotland/Evolution"><font color="blue">Directed Evolution</font></a></h3><br />
<p>Here we describe one of the possible ways to improve our switch which we would have been able to attempt given more time. The results show an optimized version of our original system. </p><br />
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{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/TeamMembersTeam:Aberdeen Scotland/TeamMembers2010-10-27T16:40:33Z<p>Brychan: </p>
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<h4>Krystal Annand</h4><br />
<p><b>BSc Chemistry: University of Aberdeen</b><br />
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<p>Krystal has just completed her degree in Chemistry.</p><br />
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<h4> Brychan Cromwell</h4><br />
<p><b>MEng Chemical Engineering (4th Year): University of Aberdeen</b><br />
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I am currently in my fourth year of Chemical Engineering degree. As well as being on the iGem team, I am in charge of the presentation for Aberdeen's Formula Student team (Tau Racing), working on my thesis on fuel injection systems in car engines, president of Aberdeen University Engineering Society and the ExxonMobil Student Ambassador for Aberdeen Uni. <br />
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I am also very much looking forward to going to Boston and can't wait for the Jamboree!</p><br />
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<h4> Lisa Dryburgh</h4><br />
<p><b>BSc Hons Maths and Physics (4th Year): University of Aberdeen</b><br />
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I've just started my 4th and final year of my honours degree. Also just started an honours project on Game Theory which is challenging but also interesting! I loved being part of iGEM and spent most of my summer in front of a computer whilst listening to Margaret-Ann singing Disney! :D </p></td> <br />
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<h4>Joseph Hoare</h4><br />
<p><b>BSc Hons Immunology (4th Year): University of Aberdeen</b><br />
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Joseph is currently finishing his degree at Aberdeen.</p><br />
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<h4>Justyna Kucia</h4><br />
<p><b>MSci Biotechnology (Applied Molecular Biology) with Industrial Placement: University of Aberdeen</b><br />
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I am in my forth year of Biotechnology degree. This year I am doing my industrial placement at MedImmune in Cambridge and I work on monoclonal antibody aggregation project.<br />
I really enjoyed iGEM project. This was definitely one of the best experience undergraduate students can get :) I can’t wait to meet all the teams in Boston and see their great projects and ideas!<br />
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<h4>Stephen Lam</h4><br />
<p><b>University of Aberdeen</b><br />
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Stephen is in his fourth year of study of Immunology with a year’s Industrial placement at Aberdeen University. He is currently on placement at Novartis, Basel and investigating B cell activation signalling pathways. He joined the iGEM team to develop his knowledge of molecular biology and laboratory skills. His interests include ceilidh dancing, badminton and learning German. He used to play Triangle for his local band 'The Duffel Coats'</p><br />
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<h4>Christina McLeman</h4><br />
<p><b>BSc Hons Geology (2:1, 2006), BSc Physics (1st, 2010), PhD Physics, Maths and Hydrology (1st Year): University of Aberdeen</b><br />
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I finished my undergraduate degree in physics this year and moved straight into doing a PhD at the University of Aberdeen. My research is interdisciplinary between physics, maths and hydrology. I am looking at applying mathematical models to river networks in order to study their fractal properties. I hope to find a link between the geometric and physical properties in order to predict and control river behaviour. I also take extra classes in maths and evening classes in German. I really enjoyed working on the iGEM project over the summer months and I am looking forward to presenting our results at the iGEM jamboree in Boston this November!<br />
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<h4>Ben Porter</h4><br />
<p><b>BSc Hons Physics (4th Year): University of Aberdeen</b><br />
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My fourth year of physics has just begun, which has brought a medley of new challenges for me to get stuck into. The best of these is my honours project where I'll be putting solid oxides through the path of fire to test their mettle (or performing ac impedance spectroscopy if you listen to my supervisor). iGEM was an experience without comparison and I'm looking forward to joining all of the other teams in Boston. </p><br />
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<h4>Maragaret-Ann Seger</h4><br />
<p><b>BS Electrical Engineering (2nd Year): Olin College</b><br />
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Margaret-Ann attends Olin College.</p><br />
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<h4>Liz Threlkeld</h4><br />
<p><b>BS Maths and Biology (2nd Year): Olin College</b><br />
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I am currently a second year student at Olin College of Engineering in Needham, Massachusetts. I haven't declared a major yet, but it will be a sweet self-designed combination of engineering, biology, and applied mathematics when I get around to it. But really, there are only three things you need to know about me: I have eight cats, I am awkward at all else, and I have a freakish obsession with math."</p><br />
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{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/Fixed_PointsTeam:Aberdeen Scotland/Fixed Points2010-10-27T16:39:00Z<p>Brychan: </p>
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<h1>Fixed Points</h1><br />
<p>Fixed points are the points where an equation’s rate of change, or its slope, is zero. There are three main types of equilibrium points: stable, unstable, and saddle-node points. A stable equilibrium is a value which the function converges towards, whereas an unstable equilibrium is a value the function will diverge from. A saddle-node is where the function both converges to and diverges from, the equilibrium, depending on the direction from which we approach the point (S. Strogatz, Nonlinear dynamics and Chaos).</p><br />
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<h3>Why are fixed points important?</h3><br />
<p>Finding fixed points is the first step in analyzing the stability of a system. In particular, we are interested in the nature of our system’s bistability and how it changes with a variation of the parameters, i.e., its bifurcation analysis. This analysis can be conveyed to the biologists to minimize experimental guessing.<br />
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<h3>How are fixed points calculated?</h3><br />
<p>Fixed points are determined by setting all differential equations in a system equal to zero and solving for the variable being analyzed. The direct method of calculating equilibrium points is to find the roots of the system of equations using built-in root-finding functions such as fzero in MATLAB, and similar functions in Maple and C. The indirect method of calculating the equilibrium points is to plot and find the intersections of these nullclines, which represent equilibrium points.</p><br />
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<h3>Results</h3><br />
<p>We were able to find the fixed points of the system analytically (for small Hill coefficients) and computationally. These points were used for bifurcation analysis, and for analyzing the probability that our system would exhibit bistable behavior.</p><br />
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<h1>Nullclines</h1><br />
<p>In a system of differential equations, the nullclines are the solution curves for which all of the differential equations are equal to zero (Wikipedia).</p><br />
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<h3>Why are nullclines important?</h3><br />
<p>The intersections of the nullclines give the equilibrium points of the system of differential equations. From graphs of the nullclines, it is possible to infer whether or not a system will be bistable. If the nullclines only intersect in one place the system is not bistable, since there is one single equilibrium point. If there are more than two intersections, the middle equilibrium point is often an unstable saddle point.</p><br />
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<h3>How are the nullclines calculated?</h3><br />
<p>Just as in calculating the fixed points, we set the governing differential equations of the system equal to zero and plot the curves generated.</p><br />
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<h3>Simulations involved</h3><br />
<p>Just as in calculating the equilibrium points, we set the governing differential equations of the system equal to zero and plot the curves generated.</p><br />
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<p><b>Figure 1.</b>This figure is a plot of the nullclines of the differential equations for CFP and GFP, where we solve them for CFP as a function of GFP. In this plot, both Hill coefficients are two, ie. n2 and n4 are both 2. The nullclines cross over at three fixed points, where the middle is a saddle-node fixed point. This is an ‘ideal’ bistability plot.<br />
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<p><img src="https://static.igem.org/mediawiki/2010/5/53/Nullclines_diagram2.png"></p><br />
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<p><b>Figure 2.</b>This figure is a plot, again, of the nullclines of the differential equations for CFP and GFP. In this plot, both Hill coefficients are one. Note that the nullclines only cross once, resulting in only one fixed point and hence no possibility of bistability. This result allowed us to tell the biologists that the toggle switch would definitely not work if both of the Hill coefficients were one.<br />
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<p><b>Figure 3.</b> Figure 3: This figure is of CFP as a function of GFP for the two nullclines when the initial galactose and copper concentrations are changed. From this we can see how the fixed points change with different initial galactose and methionine concentrations. The blue line represents the GFP = CFP line. The further from this line equilibrium points are, the harder it will be to switch between stable states.</p><br />
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<h3>Results</h3><br />
<p>We focused our efforts on plotting the nullclines of the system for a range of Hill coefficient combinations in order to get a general idea of which combinations would most likely produce robust bistability. We found that all combinations gave bistability, except when both Hill coefficients were one. The optimal Hill coefficient combination occurred when both Hill coefficients were two. We passed this information onto the biology team, letting them know that if we wanted the system to successfully switch, we could not have both Hill coefficients with a value of one.<br />
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{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/cssTeam:Aberdeen Scotland/css2010-10-27T16:31:51Z<p>Brychan: </p>
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</html></div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/EquationsTeam:Aberdeen Scotland/Equations2010-10-27T16:22:15Z<p>Brychan: Undo revision 187480 by Brychan (Talk)</p>
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<br />
<h1>Equations</h1><br />
<br />
<p>Here we define the equations and parameters that describe the novel genetic toggle switch that works at the translational level. The switch allows mutually exclusive expression of either green fluorescent protein (GFP) or cyan fluorescent protein (CFP). The synthetic biological circuit is represented in Fig 1.</p> <br />
<br><br />
<center><br />
<img src="https://static.igem.org/mediawiki/2010/f/ff/Toggle_switch.jpg"><br />
<br><br />
<br><br />
<p style="font-size:10px">Figure 1: Translation of DNA to mRNA.</p><br />
</center><br />
<br><br />
<br />
<p>We can regulate the system when we add galactose or methionine. Galactose will bind to the GAL promoter and activate the transcription of M1, allowing the system to express GFP. If we remove methionine from the system instead of adding galactose, it will bind to the MET1 promoter, the transcription of M2 will be activated, leading to the expression of CFP.</p><br />
<br><br />
<p>From Fig 1 it can be seen that there is mutual inhibition of the translation of the two mRNAs. That is because the translated proteins can bind to the corresponding stem loop structures on the opposing construct.</p><br />
<br><br />
<p>For our initial conditions, we began with more GFP than CFP and thus the production of CFP was inhibited. When methionine was added removed from the system, the rate of CFP production will increase and decrease for GFP. Eventually, we will see more CFP than GFP so the system will have switched. Once we have more CFP than GFP, galactose can then be added to switch back to an expression of GFP. </p><br />
<br><br />
<p>The N-Peptide and GFP strand has two MS2-Stem loops as we discovered that one single loop would not inhibit the production of CFP enough to achieve our switch.</p><br />
<br />
<br><br />
<h3>Equation 1</h3><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="center"><img src="https://static.igem.org/mediawiki/2010/9/90/Dm1.jpg"></div><br />
</td><br />
<td><br />
<div align="right"><p><b>(1)</b></p></div><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<p>This is the equation for the rate of change of the mRNA that is transcribed from the galactose promoter. The three terms represent production, degradation, and dilution respectively.</p><br />
<br><br />
<p>[GAL] represents the concentration of galactose that is added to the system. When galactose is added it binds to the promoter and activates the transcription of M1.<br />
<br><br />
<br><br />
[M1] is the concentration of mRNA that translates the N-peptide and GFP.</p><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="right"><b><p>Parameter<p></b></div><br />
</td><br />
<td><br />
<p><b>Description</b></p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>λ<sub style="font-size:10px">1</sub>:<p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of transcription of the DNA that encodes for the production of N peptide and GFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>μ<sub style="font-size:10px">1</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of degradation of mRNA</p><br />
</td><br />
</tr><br />
<tr><br />
<td> <br />
<div align="right"><b><p>n<sub style="font-size:10px">1</sub>:<p></b></div><br />
</td><br />
<td><br />
<p>Hill coefficient for the association between the galactose and the GAL promoter<p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>K<sub style="font-size:10px">1</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Dissociation constant for the GAL promoter</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>T:</p></b></div><br />
</td><br />
<td><br />
<p>Time constant representing rate of cellular division</p><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<br />
<h3>Equation 2</h3><br />
<br><br />
<br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="center"><img src="https://static.igem.org/mediawiki/2010/7/7e/DGFP.jpg"></div><br />
</td><br />
<td><br />
<div align="right"><p><b>(2)</b></p></div><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<p>This is the equation for the rate of change of protein that is translated from the mRNA for GFP. The three terms represent production, degradation, and dilution respectively.</p><br />
<br><br />
<p>[M1] is the concentration of mRNA that translates the N-peptide GFP.<br />
<br><br />
<br><br />
[GFP] represents the concentration of N-peptide and GFP.<br />
<br><br />
<br><br />
[CFP] represents the concentration of the MS2-protein and CFP.</p><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="right"><b><p>Parameter:<p></b></div><br />
</td><br />
<td><br />
<p><b>Description</b></p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>λ<sub style="font-size:10px">2</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of translation of the mRNA that encodes for the production of N-peptide and GFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>μ<sub style="font-size:10px">2</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of degradation of the GFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td> <br />
<div align="right"><b><p>n<sub style="font-size:10px">2</sub>:<p></b></div><br />
</td><br />
<td><br />
<p>Hill coefficient of the CFP/MS2 stem loop association<p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>K<sub style="font-size:10px">2</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Dissociation constant for the MS2-CFP protein to MS2 loop</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>T:</p></b></div><br />
</td><br />
<td><br />
<p>Time constant representing rate of cellular division</p><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br />
<br><br />
<h3>Equation 3</h3><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="center"><img src="https://static.igem.org/mediawiki/2010/2/23/Dm2.jpg"></div><br />
</td><br />
<td><br />
<div align="right"><p><b>(3)</b></p></div><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<br />
<p>This is the equation for the rate of change of the mRNA that is transcribed from the copper promoter. The three terms represent production, degradation, and dilution respectively.</p><br />
<br><br />
<p>[Cu<sup style="font-size:10px">2+</sup>] is the concentration of the copper added to the system that binds to the CUP1 promoter and activates the transcription of M2.<br />
<br><br />
<br><br />
[M2] represents the concentration of mRNA that translates the MS2-protein and CFP. </p><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="right"><b><p>Parameter</p></b></div><br />
</td><br />
<td><br />
<p><b>Description</b></p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>λ<sub style="font-size:10px">3</sup>:<p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of transcription of the DNA that encodes for the production of the MS2-protein and CFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>μ<sub style="font-size:10px">3</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of degradation of mRNA</p><br />
</td><br />
</tr><br />
<tr><br />
<td> <br />
<div align="right"><b><p>n<sub style="font-size:10px">3</sub>:<p></b></div><br />
</td><br />
<td><br />
<p>Hill coefficient of the association between copper and the CUP1 promoter<p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>K<sub style="font-size:10px">3</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Dissociation constant for Copper promoter</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>T:</p></b></div><br />
</td><br />
<td><br />
<p>Time constant representing rate of cellular division</p><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br />
<br><br />
<br />
<h3>Equation 4</h3><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="center"><img src="https://static.igem.org/mediawiki/2010/d/d4/DCFP.jpg"></div><br />
</td><br />
<td><br />
<div align="right"><p><b>(4)</b></p></div><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<br />
<p>This is the equation for the rate of change of protein that is translated from the mRNA for CFP. The three terms represent production, degradation, and dilution respectively.</p><br />
<br><br />
<p>[M2] is the concentration of mRNA that translates to MS2-protein and CFP.<br />
<br><br />
<br><br />
[GFP] represents the concentration of the N-peptide and GFP.<br />
<br><br />
<br><br />
[CFP] represents the concentration of the MS2-protein and CFP.</p><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="right"><b><p>Parameters<p></b></div><br />
</td><br />
<td><br />
<p><b>Description</b></p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>λ<sub style="font-size:10px">4</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of translation of the mRNA that encodes for the production of MS2-protein and CFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>μ<sub style="font-size:10px">4</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of degradation of the CFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td> <br />
<div align="right"><b><p>n<sub style="font-size:10px">4</sub>:<p></b></div><br />
</td><br />
<td><br />
<p>Hill coefficient of the GFP/Bbox stem loop association<p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>K<sub style="font-size:10px">4</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Dissociation constant for the N-Pep-GFP protein to the Bbox-stem loop</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>T:</p></b></div><br />
</td><br />
<td><br />
<p>time constant representing rate of cellular division</p><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<h1>Parameter Study</h1><br />
<p>The parameter values were first estimated based on the literature <sup style="font-size:10px">[1]</sup> and after the first estimation, a possible range of variation for each parameter was assigned, also based on literature. Then, we studied the bistability of the model depending on the parameter values that were varied in the above mentioned ranges. For more information, see <a href="https://2010.igem.org/Team:Aberdeen_Scotland/Probability">Parameter Space Analysis</a> and <a href="https://2010.igem.org/Team:Aberdeen_Scotland/Evolution">Directed Evolution</a>.</p><br />
<br><br />
<h1>Modification of the construct</h1><br />
<br />
<p>Some experimental difficulties were encountered with the copper construct which led to the use of a methionine promoter to substitute it. Methionine acts as an inhibitor of the promoter, so that equation 3 had to be substituted by the following equation:<p><br />
<br><br />
<center><br />
<img src="https://static.igem.org/mediawiki/2010/f/f2/Meth.png"><br />
<br><br><br />
<img src="https://static.igem.org/mediawiki/2010/7/75/MET_toggle_switch.png"><br />
</center><br />
<br><br />
<p>The behaviour of the switch can then be summarise in the following table:</p><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="right"><b><p>What is present in the system</p></b></div><br />
</td><br />
<td><br />
<p><b>Protein(s) produced</b></p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><p>Galactose and Methionine</p></div><br />
</td><br />
<td><br />
<p>GFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><p>Galactose only</p></div><br />
</td><br />
<td><br />
<p>GFP, CFP (doses dependent)</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><p>Methionine only</p></div><br />
</td><br />
<td><br />
<p>No GFP or CFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><p>No Galactose and no Methionine</p></div><br />
</td><br />
<td><br />
<p>CFP</p><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<br />
<h1>References</h1><br />
<br />
<p><sup style="font-size:10px">[1]</sup> Beyer A, Hollunder J, Nasheuer HP, Wilhelm T. (2004), Post-transcriptional expression regulation in the yeast Saccharomyces cerevisiae on a genomic scale, <i>Mol Cell Proteomics.</i>, Vol. 3, No.11, pp. 1083-1092.</p><br />
<br><br />
<p><sup style="font-size:10px">[2]</sup> Alon, U. (2006), An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall. </p><br />
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<html><br />
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<h1>Equations</h1><br />
<br />
<p>Here we define the equations and parameters that describe the novel genetic toggle switch that works at the translational level. The switch allows mutually exclusive expression of either green fluorescent protein (GFP) or cyan fluorescent protein (CFP). The synthetic biological circuit is represented in Fig 1.</p> <br />
<br><br />
<center><br />
<img src="https://static.igem.org/mediawiki/2010/f/ff/Toggle_switch.jpg"><br />
<br><br />
<br><br />
<p style="font-size:10px">Figure 1: Translation of DNA to mRNA.</p><br />
</center><br />
<br><br />
<br />
<p>We can regulate the system when we add galactose or methionine. Galactose will bind to the GAL promoter and activate the transcription of M1, allowing the system to express GFP. If we remove methionine from the system instead of adding galactose, it will bind to the MET1 promoter, the transcription of M2 will be activated, leading to the expression of CFP.</p><br />
<br><br />
<p>From Fig 1 it can be seen that there is mutual inhibition of the translation of the two mRNAs. That is because the translated proteins can bind to the corresponding stem loop structures on the opposing construct.</p><br />
<br><br />
<p>For our initial conditions, we began with more GFP than CFP and thus the production of CFP was inhibited. When methionine was added removed from the system, the rate of CFP production will increase and decrease for GFP. Eventually, we will see more CFP than GFP so the system will have switched. Once we have more CFP than GFP, galactose can then be added to switch back to an expression of GFP. </p><br />
<br><br />
<p>The N-Peptide and GFP strand has two MS2-Stem loops as we discovered that one single loop would not inhibit the production of CFP enough to achieve our switch.</p><br />
<br />
<br><br />
<h3>Equation 1</h3><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="center"><img src="https://static.igem.org/mediawiki/2010/9/90/Dm1.jpg"></div><br />
</td><br />
<td><br />
<div align="right"><p><b>(1)</b></p></div><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<p>This is the equation for the rate of change of the mRNA that is transcribed from the galactose promoter. The three terms represent production, degradation, and dilution respectively.</p><br />
<br><br />
<p>[GAL] represents the concentration of galactose that is added to the system. When galactose is added it binds to the promoter and activates the transcription of M1.<br />
<br><br />
<br><br />
[M1] is the concentration of mRNA that translates the N-peptide and GFP.</p><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="right"><b><p>Parameter<p></b></div><br />
</td><br />
<td><br />
<p><b>Description</b></p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>λ<sub style="font-size:10px">1</sub>:<p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of transcription of the DNA that encodes for the production of N peptide and GFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>μ<sub style="font-size:10px">1</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of degradation of mRNA</p><br />
</td><br />
</tr><br />
<tr><br />
<td> <br />
<div align="right"><b><p>n<sub style="font-size:10px">1</sub>:<p></b></div><br />
</td><br />
<td><br />
<p>Hill coefficient for the association between the galactose and the GAL promoter<p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>K<sub style="font-size:10px">1</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Dissociation constant for the GAL promoter</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>T:</p></b></div><br />
</td><br />
<td><br />
<p>Time constant representing rate of cellular division</p><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<br />
<h3>Equation 2</h3><br />
<br><br />
<br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="center"><img src="https://static.igem.org/mediawiki/2010/7/7e/DGFP.jpg"></div><br />
</td><br />
<td><br />
<div align="right"><p><b>(2)</b></p></div><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<p>This is the equation for the rate of change of protein that is translated from the mRNA for GFP. The three terms represent production, degradation, and dilution respectively.</p><br />
<br><br />
<p>[M1] is the concentration of mRNA that translates the N-peptide GFP.<br />
<br><br />
<br><br />
[GFP] represents the concentration of N-peptide and GFP.<br />
<br><br />
<br><br />
[CFP] represents the concentration of the MS2-protein and CFP.</p><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="right"><b><p>Parameter:<p></b></div><br />
</td><br />
<td><br />
<p><b>Description</b></p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>λ<sub style="font-size:10px">2</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of translation of the mRNA that encodes for the production of N-peptide and GFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>μ<sub style="font-size:10px">2</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of degradation of the GFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td> <br />
<div align="right"><b><p>n<sub style="font-size:10px">2</sub>:<p></b></div><br />
</td><br />
<td><br />
<p>Hill coefficient of the CFP/MS2 stem loop association<p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>K<sub style="font-size:10px">2</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Dissociation constant for the MS2-CFP protein to MS2 loop</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>T:</p></b></div><br />
</td><br />
<td><br />
<p>Time constant representing rate of cellular division</p><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br />
<br><br />
<br><br />
<center><br />
<img src="https://static.igem.org/mediawiki/2010/f/f2/Meth.png"><br />
<br><br><br />
<img src="https://static.igem.org/mediawiki/2010/7/75/MET_toggle_switch.png"><br />
</center><br />
<br><br />
<p>The behaviour of the switch can then be summarise in the following table:</p><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="right"><b><p>What is present in the system</p></b></div><br />
</td><br />
<td><br />
<p><b>Protein(s) produced</b></p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><p>Galactose and Methionine</p></div><br />
</td><br />
<td><br />
<p>GFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><p>Galactose only</p></div><br />
</td><br />
<td><br />
<p>GFP, CFP (doses dependent)</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><p>Methionine only</p></div><br />
</td><br />
<td><br />
<p>No GFP or CFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><p>No Galactose and no Methionine</p></div><br />
</td><br />
<td><br />
<p>CFP</p><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<br />
<br><br />
<br />
<h3>Equation 4</h3><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="center"><img src="https://static.igem.org/mediawiki/2010/d/d4/DCFP.jpg"></div><br />
</td><br />
<td><br />
<div align="right"><p><b>(4)</b></p></div><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<br />
<p>This is the equation for the rate of change of protein that is translated from the mRNA for CFP. The three terms represent production, degradation, and dilution respectively.</p><br />
<br><br />
<p>[M2] is the concentration of mRNA that translates to MS2-protein and CFP.<br />
<br><br />
<br><br />
[GFP] represents the concentration of the N-peptide and GFP.<br />
<br><br />
<br><br />
[CFP] represents the concentration of the MS2-protein and CFP.</p><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="right"><b><p>Parameters<p></b></div><br />
</td><br />
<td><br />
<p><b>Description</b></p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>λ<sub style="font-size:10px">4</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of translation of the mRNA that encodes for the production of MS2-protein and CFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>μ<sub style="font-size:10px">4</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of degradation of the CFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td> <br />
<div align="right"><b><p>n<sub style="font-size:10px">4</sub>:<p></b></div><br />
</td><br />
<td><br />
<p>Hill coefficient of the GFP/Bbox stem loop association<p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>K<sub style="font-size:10px">4</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Dissociation constant for the N-Pep-GFP protein to the Bbox-stem loop</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>T:</p></b></div><br />
</td><br />
<td><br />
<p>time constant representing rate of cellular division</p><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<h1>Parameter Study</h1><br />
<p>The parameter values were first estimated based on the literature <sup style="font-size:10px">[1]</sup> and after the first estimation, a possible range of variation for each parameter was assigned, also based on literature. Then, we studied the bistability of the model depending on the parameter values that were varied in the above mentioned ranges. For more information, see <a href="https://2010.igem.org/Team:Aberdeen_Scotland/Probability">Parameter Space Analysis</a> and <a href="https://2010.igem.org/Team:Aberdeen_Scotland/Evolution">Directed Evolution</a>.</p><br />
<br><br />
<h1>Modification of the construct</h1><br />
<br />
<p>Some experimental difficulties were encountered with the copper construct which led to the use of a methionine promoter to substitute it. Methionine acts as an inhibitor of the promoter, so that equation 3 had to be substituted by the following equation:<p><br />
<br><br />
<center><br />
<img src="https://static.igem.org/mediawiki/2010/f/f2/Meth.png"><br />
<br><br><br />
<img src="https://static.igem.org/mediawiki/2010/7/75/MET_toggle_switch.png"><br />
</center><br />
<br><br />
<p>The behaviour of the switch can then be summarise in the following table:</p><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="right"><b><p>What is present in the system</p></b></div><br />
</td><br />
<td><br />
<p><b>Protein(s) produced</b></p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><p>Galactose and Methionine</p></div><br />
</td><br />
<td><br />
<p>GFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><p>Galactose only</p></div><br />
</td><br />
<td><br />
<p>GFP, CFP (doses dependent)</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><p>Methionine only</p></div><br />
</td><br />
<td><br />
<p>No GFP or CFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><p>No Galactose and no Methionine</p></div><br />
</td><br />
<td><br />
<p>CFP</p><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<br />
<h1>References</h1><br />
<br />
<p><sup style="font-size:10px">[1]</sup> Beyer A, Hollunder J, Nasheuer HP, Wilhelm T. (2004), Post-transcriptional expression regulation in the yeast Saccharomyces cerevisiae on a genomic scale, <i>Mol Cell Proteomics.</i>, Vol. 3, No.11, pp. 1083-1092.</p><br />
<br><br />
<p><sup style="font-size:10px">[2]</sup> Alon, U. (2006), An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall. </p><br />
<br />
<br />
<br />
<br><br><br />
<hr><br />
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<tr><br />
<td><br />
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</td><br />
<td align="right"><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Fixed_Points">Continue to Nullclines and Fixed Points&nbsp;&nbsp;<img src="https://static.igem.org/mediawiki/2010/3/36/Right_arrow.png"></a><br />
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{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/EquationsTeam:Aberdeen Scotland/Equations2010-10-27T16:15:30Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<br />
<html><br />
<br />
<h1>Equations</h1><br />
<br />
<p>Here we define the equations and parameters that describe the novel genetic toggle switch that works at the translational level. The switch allows mutually exclusive expression of either green fluorescent protein (GFP) or cyan fluorescent protein (CFP). The synthetic biological circuit is represented in Fig 1.</p> <br />
<br><br />
<center><br />
<img src="https://static.igem.org/mediawiki/2010/f/ff/Toggle_switch.jpg"><br />
<br><br />
<br><br />
<p style="font-size:10px">Figure 1: Translation of DNA to mRNA.</p><br />
</center><br />
<br><br />
<br />
<p>We can regulate the system when we add galactose or methionine. Galactose will bind to the GAL promoter and activate the transcription of M1, allowing the system to express GFP. If we remove methionine from the system instead of adding galactose, it will bind to the MET1 promoter, the transcription of M2 will be activated, leading to the expression of CFP.</p><br />
<br><br />
<p>From Fig 1 it can be seen that there is mutual inhibition of the translation of the two mRNAs. That is because the translated proteins can bind to the corresponding stem loop structures on the opposing construct.</p><br />
<br><br />
<p>For our initial conditions, we began with more GFP than CFP and thus the production of CFP was inhibited. When methionine was added removed from the system, the rate of CFP production will increase and decrease for GFP. Eventually, we will see more CFP than GFP so the system will have switched. Once we have more CFP than GFP, galactose can then be added to switch back to an expression of GFP. </p><br />
<br><br />
<p>The N-Peptide and GFP strand has two MS2-Stem loops as we discovered that one single loop would not inhibit the production of CFP enough to achieve our switch.</p><br />
<br />
<br><br />
<h3>Equation 1</h3><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="center"><img src="https://static.igem.org/mediawiki/2010/9/90/Dm1.jpg"></div><br />
</td><br />
<td><br />
<div align="right"><p><b>(1)</b></p></div><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<p>This is the equation for the rate of change of the mRNA that is transcribed from the galactose promoter. The three terms represent production, degradation, and dilution respectively.</p><br />
<br><br />
<p>[GAL] represents the concentration of galactose that is added to the system. When galactose is added it binds to the promoter and activates the transcription of M1.<br />
<br><br />
<br><br />
[M1] is the concentration of mRNA that translates the N-peptide and GFP.</p><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="right"><b><p>Parameter<p></b></div><br />
</td><br />
<td><br />
<p><b>Description</b></p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>λ<sub style="font-size:10px">1</sub>:<p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of transcription of the DNA that encodes for the production of N peptide and GFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>μ<sub style="font-size:10px">1</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of degradation of mRNA</p><br />
</td><br />
</tr><br />
<tr><br />
<td> <br />
<div align="right"><b><p>n<sub style="font-size:10px">1</sub>:<p></b></div><br />
</td><br />
<td><br />
<p>Hill coefficient for the association between the galactose and the GAL promoter<p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>K<sub style="font-size:10px">1</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Dissociation constant for the GAL promoter</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>T:</p></b></div><br />
</td><br />
<td><br />
<p>Time constant representing rate of cellular division</p><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<br />
<h3>Equation 2</h3><br />
<br><br />
<br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="center"><img src="https://static.igem.org/mediawiki/2010/7/7e/DGFP.jpg"></div><br />
</td><br />
<td><br />
<div align="right"><p><b>(2)</b></p></div><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<p>This is the equation for the rate of change of protein that is translated from the mRNA for GFP. The three terms represent production, degradation, and dilution respectively.</p><br />
<br><br />
<p>[M1] is the concentration of mRNA that translates the N-peptide GFP.<br />
<br><br />
<br><br />
[GFP] represents the concentration of N-peptide and GFP.<br />
<br><br />
<br><br />
[CFP] represents the concentration of the MS2-protein and CFP.</p><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="right"><b><p>Parameter:<p></b></div><br />
</td><br />
<td><br />
<p><b>Description</b></p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>λ<sub style="font-size:10px">2</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of translation of the mRNA that encodes for the production of N-peptide and GFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>μ<sub style="font-size:10px">2</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of degradation of the GFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td> <br />
<div align="right"><b><p>n<sub style="font-size:10px">2</sub>:<p></b></div><br />
</td><br />
<td><br />
<p>Hill coefficient of the CFP/MS2 stem loop association<p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>K<sub style="font-size:10px">2</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Dissociation constant for the MS2-CFP protein to MS2 loop</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>T:</p></b></div><br />
</td><br />
<td><br />
<p>Time constant representing rate of cellular division</p><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br />
<br><br />
<h3>Equation 3</h3><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="center"><img src="https://static.igem.org/mediawiki/2010/2/23/Dm2.jpg"></div><br />
</td><br />
<td><br />
<div align="right"><p><b>(3)</b></p></div><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<br />
<p>This is the equation for the rate of change of the mRNA that is transcribed from the copper promoter. The three terms represent production, degradation, and dilution respectively.</p><br />
<br><br />
<p>[Cu<sup style="font-size:10px">2+</sup>] is the concentration of the copper added to the system that binds to the CUP1 promoter and activates the transcription of M2.<br />
<br><br />
<br><br />
[M2] represents the concentration of mRNA that translates the MS2-protein and CFP. </p><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="right"><b><p>Parameter</p></b></div><br />
</td><br />
<td><br />
<p><b>Description</b></p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>λ<sub style="font-size:10px">3</sup>:<p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of transcription of the DNA that encodes for the production of the MS2-protein and CFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>μ<sub style="font-size:10px">3</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of degradation of mRNA</p><br />
</td><br />
</tr><br />
<tr><br />
<td> <br />
<div align="right"><b><p>n<sub style="font-size:10px">3</sub>:<p></b></div><br />
</td><br />
<td><br />
<p>Hill coefficient of the association between copper and the CUP1 promoter<p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>K<sub style="font-size:10px">3</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Dissociation constant for Copper promoter</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>T:</p></b></div><br />
</td><br />
<td><br />
<p>Time constant representing rate of cellular division</p><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br />
<br><br />
<br />
<h3>Equation 4</h3><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="center"><img src="https://static.igem.org/mediawiki/2010/d/d4/DCFP.jpg"></div><br />
</td><br />
<td><br />
<div align="right"><p><b>(4)</b></p></div><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<br />
<p>This is the equation for the rate of change of protein that is translated from the mRNA for CFP. The three terms represent production, degradation, and dilution respectively.</p><br />
<br><br />
<p>[M2] is the concentration of mRNA that translates to MS2-protein and CFP.<br />
<br><br />
<br><br />
[GFP] represents the concentration of the N-peptide and GFP.<br />
<br><br />
<br><br />
[CFP] represents the concentration of the MS2-protein and CFP.</p><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="right"><b><p>Parameters<p></b></div><br />
</td><br />
<td><br />
<p><b>Description</b></p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>λ<sub style="font-size:10px">4</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of translation of the mRNA that encodes for the production of MS2-protein and CFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>μ<sub style="font-size:10px">4</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of degradation of the CFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td> <br />
<div align="right"><b><p>n<sub style="font-size:10px">4</sub>:<p></b></div><br />
</td><br />
<td><br />
<p>Hill coefficient of the GFP/Bbox stem loop association<p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>K<sub style="font-size:10px">4</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Dissociation constant for the N-Pep-GFP protein to the Bbox-stem loop</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>T:</p></b></div><br />
</td><br />
<td><br />
<p>time constant representing rate of cellular division</p><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<h1>Parameter Study</h1><br />
<p>The parameter values were first estimated based on the literature <sup style="font-size:10px">[1]</sup> and after the first estimation, a possible range of variation for each parameter was assigned, also based on literature. Then, we studied the bistability of the model depending on the parameter values that were varied in the above mentioned ranges. For more information, see <a href="https://2010.igem.org/Team:Aberdeen_Scotland/Probability">Parameter Space Analysis</a> and <a href="https://2010.igem.org/Team:Aberdeen_Scotland/Evolution">Directed Evolution</a>.</p><br />
<br><br />
<h1>Modification of the construct</h1><br />
<br />
<p>Some experimental difficulties were encountered with the copper construct which led to the use of a methionine promoter to substitute it. Methionine acts as an inhibitor of the promoter, so that equation 3 had to be substituted by the following equation:<p><br />
<br><br />
<center><br />
<img src="https://static.igem.org/mediawiki/2010/f/f2/Meth.png"><br />
<br><br><br />
<img src="https://static.igem.org/mediawiki/2010/7/75/MET_toggle_switch.png"><br />
</center><br />
<br><br />
<p>The behaviour of the switch can then be summarise in the following table:</p><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="right"><b><p>What is present in the system</p></b></div><br />
</td><br />
<td><br />
<p><b>Protein(s) produced</b></p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><p>Galactose and Methionine</p></div><br />
</td><br />
<td><br />
<p>GFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><p>Galactose only</p></div><br />
</td><br />
<td><br />
<p>GFP, CFP (doses dependent)</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><p>Methionine only</p></div><br />
</td><br />
<td><br />
<p>No GFP or CFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><p>No Galactose and no Methionine</p></div><br />
</td><br />
<td><br />
<p>CFP</p><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<br />
<h1>References</h1><br />
<br />
<p><sup style="font-size:10px">[1]</sup> Beyer A, Hollunder J, Nasheuer HP, Wilhelm T. (2004), Post-transcriptional expression regulation in the yeast Saccharomyces cerevisiae on a genomic scale, <i>Mol Cell Proteomics.</i>, Vol. 3, No.11, pp. 1083-1092.</p><br />
<br><br />
<p><sup style="font-size:10px">[2]</sup> Alon, U. (2006), An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall. </p><br />
<br />
<br />
<br />
<br><br><br />
<hr><br />
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<tr><br />
<td><br />
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</td><br />
<td align="right"><br />
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{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/EquationsTeam:Aberdeen Scotland/Equations2010-10-27T16:14:44Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<br />
<html><br />
<br />
<h1>Equations</h1><br />
<br />
<p>Here we de�ne the equations and parameters that describe the novel genetic toggle switch that works at the translational level. The switch allows mutually exclusive expression of either green fluorescent protein (GFP) or cyan fluorescent protein (CFP). The synthetic biological circuit is represented in Fig 1.</p> <br />
<br><br />
<center><br />
<img src="https://static.igem.org/mediawiki/2010/f/ff/Toggle_switch.jpg"><br />
<br><br />
<br><br />
<p style="font-size:10px">Figure 1: Translation of DNA to mRNA.</p><br />
</center><br />
<br><br />
<br />
<p>We can regulate the system when we add galactose or methionine. Galactose will bind to the GAL promoter and activate the transcription of M1, allowing the system to express GFP. If we remove methionine from the system instead of adding galactose, it will bind to the MET1 promoter, the transcription of M2 will be activated, leading to the expression of CFP.</p><br />
<br><br />
<p>From Fig 1 it can be seen that there is mutual inhibition of the translation of the two mRNAs. That is because the translated proteins can bind to the corresponding stem loop structures on the opposing construct.</p><br />
<br><br />
<p>For our initial conditions, we began with more GFP than CFP and thus the production of CFP was inhibited. When methionine was added removed from the system, the rate of CFP production will increase and decrease for GFP. Eventually, we will see more CFP than GFP so the system will have switched. Once we have more CFP than GFP, galactose can then be added to switch back to an expression of GFP. </p><br />
<br><br />
<p>The N-Peptide and GFP strand has two MS2-Stem loops as we discovered that one single loop would not inhibit the production of CFP enough to achieve our switch.</p><br />
<br />
<br><br />
<h3>Equation 1</h3><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="center"><img src="https://static.igem.org/mediawiki/2010/9/90/Dm1.jpg"></div><br />
</td><br />
<td><br />
<div align="right"><p><b>(1)</b></p></div><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<p>This is the equation for the rate of change of the mRNA that is transcribed from the galactose promoter. The three terms represent production, degradation, and dilution respectively.</p><br />
<br><br />
<p>[GAL] represents the concentration of galactose that is added to the system. When galactose is added it binds to the promoter and activates the transcription of M1.<br />
<br><br />
<br><br />
[M1] is the concentration of mRNA that translates the N-peptide and GFP.</p><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="right"><b><p>Parameter<p></b></div><br />
</td><br />
<td><br />
<p><b>Description</b></p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>λ<sub style="font-size:10px">1</sub>:<p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of transcription of the DNA that encodes for the production of N peptide and GFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>μ<sub style="font-size:10px">1</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of degradation of mRNA</p><br />
</td><br />
</tr><br />
<tr><br />
<td> <br />
<div align="right"><b><p>n<sub style="font-size:10px">1</sub>:<p></b></div><br />
</td><br />
<td><br />
<p>Hill coefficient for the association between the galactose and the GAL promoter<p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>K<sub style="font-size:10px">1</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Dissociation constant for the GAL promoter</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>T:</p></b></div><br />
</td><br />
<td><br />
<p>Time constant representing rate of cellular division</p><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<br />
<h3>Equation 2</h3><br />
<br><br />
<br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="center"><img src="https://static.igem.org/mediawiki/2010/7/7e/DGFP.jpg"></div><br />
</td><br />
<td><br />
<div align="right"><p><b>(2)</b></p></div><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<p>This is the equation for the rate of change of protein that is translated from the mRNA for GFP. The three terms represent production, degradation, and dilution respectively.</p><br />
<br><br />
<p>[M1] is the concentration of mRNA that translates the N-peptide GFP.<br />
<br><br />
<br><br />
[GFP] represents the concentration of N-peptide and GFP.<br />
<br><br />
<br><br />
[CFP] represents the concentration of the MS2-protein and CFP.</p><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="right"><b><p>Parameter:<p></b></div><br />
</td><br />
<td><br />
<p><b>Description</b></p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>λ<sub style="font-size:10px">2</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of translation of the mRNA that encodes for the production of N-peptide and GFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>μ<sub style="font-size:10px">2</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of degradation of the GFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td> <br />
<div align="right"><b><p>n<sub style="font-size:10px">2</sub>:<p></b></div><br />
</td><br />
<td><br />
<p>Hill coefficient of the CFP/MS2 stem loop association<p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>K<sub style="font-size:10px">2</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Dissociation constant for the MS2-CFP protein to MS2 loop</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>T:</p></b></div><br />
</td><br />
<td><br />
<p>Time constant representing rate of cellular division</p><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br />
<br><br />
<h3>Equation 3</h3><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="center"><img src="https://static.igem.org/mediawiki/2010/2/23/Dm2.jpg"></div><br />
</td><br />
<td><br />
<div align="right"><p><b>(3)</b></p></div><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<br />
<p>This is the equation for the rate of change of the mRNA that is transcribed from the copper promoter. The three terms represent production, degradation, and dilution respectively.</p><br />
<br><br />
<p>[Cu<sup style="font-size:10px">2+</sup>] is the concentration of the copper added to the system that binds to the CUP1 promoter and activates the transcription of M2.<br />
<br><br />
<br><br />
[M2] represents the concentration of mRNA that translates the MS2-protein and CFP. </p><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="right"><b><p>Parameter</p></b></div><br />
</td><br />
<td><br />
<p><b>Description</b></p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>λ<sub style="font-size:10px">3</sup>:<p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of transcription of the DNA that encodes for the production of the MS2-protein and CFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>μ<sub style="font-size:10px">3</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of degradation of mRNA</p><br />
</td><br />
</tr><br />
<tr><br />
<td> <br />
<div align="right"><b><p>n<sub style="font-size:10px">3</sub>:<p></b></div><br />
</td><br />
<td><br />
<p>Hill coefficient of the association between copper and the CUP1 promoter<p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>K<sub style="font-size:10px">3</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Dissociation constant for Copper promoter</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>T:</p></b></div><br />
</td><br />
<td><br />
<p>Time constant representing rate of cellular division</p><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br />
<br><br />
<br />
<h3>Equation 4</h3><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="center"><img src="https://static.igem.org/mediawiki/2010/d/d4/DCFP.jpg"></div><br />
</td><br />
<td><br />
<div align="right"><p><b>(4)</b></p></div><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<br />
<p>This is the equation for the rate of change of protein that is translated from the mRNA for CFP. The three terms represent production, degradation, and dilution respectively.</p><br />
<br><br />
<p>[M2] is the concentration of mRNA that translates to MS2-protein and CFP.<br />
<br><br />
<br><br />
[GFP] represents the concentration of the N-peptide and GFP.<br />
<br><br />
<br><br />
[CFP] represents the concentration of the MS2-protein and CFP.</p><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="right"><b><p>Parameters<p></b></div><br />
</td><br />
<td><br />
<p><b>Description</b></p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>λ<sub style="font-size:10px">4</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of translation of the mRNA that encodes for the production of MS2-protein and CFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>μ<sub style="font-size:10px">4</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Constant representing rate of degradation of the CFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td> <br />
<div align="right"><b><p>n<sub style="font-size:10px">4</sub>:<p></b></div><br />
</td><br />
<td><br />
<p>Hill coefficient of the GFP/Bbox stem loop association<p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>K<sub style="font-size:10px">4</sub>:</p></b></div><br />
</td><br />
<td><br />
<p>Dissociation constant for the N-Pep-GFP protein to the Bbox-stem loop</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><b><p>T:</p></b></div><br />
</td><br />
<td><br />
<p>time constant representing rate of cellular division</p><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<h1>Parameter Study</h1><br />
<p>The parameter values were first estimated based on the literature <sup style="font-size:10px">[1]</sup> and after the first estimation, a possible range of variation for each parameter was assigned, also based on literature. Then, we studied the bistability of the model depending on the parameter values that were varied in the above mentioned ranges. For more information, see <a href="https://2010.igem.org/Team:Aberdeen_Scotland/Probability">Parameter Space Analysis</a> and <a href="https://2010.igem.org/Team:Aberdeen_Scotland/Evolution">Directed Evolution</a>.</p><br />
<br><br />
<h1>Modification of the construct</h1><br />
<br />
<p>Some experimental difficulties were encountered with the copper construct which led to the use of a methionine promoter to substitute it. Methionine acts as an inhibitor of the promoter, so that equation 3 had to be substituted by the following equation:<p><br />
<br><br />
<center><br />
<img src="https://static.igem.org/mediawiki/2010/f/f2/Meth.png"><br />
<br><br><br />
<img src="https://static.igem.org/mediawiki/2010/7/75/MET_toggle_switch.png"><br />
</center><br />
<br><br />
<p>The behaviour of the switch can then be summarise in the following table:</p><br />
<br><br />
<div align="center"><br />
<table><br />
<tr><br />
<td><br />
<div align="right"><b><p>What is present in the system</p></b></div><br />
</td><br />
<td><br />
<p><b>Protein(s) produced</b></p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><p>Galactose and Methionine</p></div><br />
</td><br />
<td><br />
<p>GFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><p>Galactose only</p></div><br />
</td><br />
<td><br />
<p>GFP, CFP (doses dependent)</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><p>Methionine only</p></div><br />
</td><br />
<td><br />
<p>No GFP or CFP</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<div align="right"><p>No Galactose and no Methionine</p></div><br />
</td><br />
<td><br />
<p>CFP</p><br />
</td><br />
</tr><br />
</table><br />
</div><br />
<br><br />
<br />
<h1>References</h1><br />
<br />
<p><sup style="font-size:10px">[1]</sup> Beyer A, Hollunder J, Nasheuer HP, Wilhelm T. (2004), Post-transcriptional expression regulation in the yeast Saccharomyces cerevisiae on a genomic scale, <i>Mol Cell Proteomics.</i>, Vol. 3, No.11, pp. 1083-1092.</p><br />
<br><br />
<p><sup style="font-size:10px">[2]</sup> Alon, U. (2006), An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall. </p><br />
<br />
<br />
<br />
<br><br><br />
<hr><br />
<table class="nav"><br />
<tr><br />
<td><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Modeling"><img src="https://static.igem.org/mediawiki/2010/8/8e/Left_arrow.png">&nbsp;&nbsp;Return to the Modelling Summary</a><br />
</td><br />
<td align="right"><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Fixed_Points">Continue to Nullclines and Fixed Points&nbsp;&nbsp;<img src="https://static.igem.org/mediawiki/2010/3/36/Right_arrow.png"></a><br />
</td><br />
</tr><br />
</table><br />
<br />
<br />
<br />
<br />
</html><br />
{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/Fixed_PointsTeam:Aberdeen Scotland/Fixed Points2010-10-27T16:07:29Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<html><br />
<h1>Fixed Points</h1><br />
<p>Fixed points are the points where an equation’s rate of change, or its slope, is zero. There are three main types of equilibrium points: stable, unstable, and saddle-node points. A stable equilibrium is a value which the function converges towards, whereas an unstable equilibrium is a value the function will diverge from. A saddle-node is where the function both converges to and diverges from, the equilibrium, depending on the direction from which we approach the point (S. Strogatz, Nonlinear dynamics and Chaos).</p><br />
<br><br />
<h3>Why are fixed points important?</h3><br />
<p>Finding fixed points is the first step in analyzing the stability of a system. In particular, we are interested in the nature of our system’s bistability and how it changes with a variation of the parameters, i.e., its bifurcation analysis. This analysis can be conveyed to the biologists to minimize experimental guessing.<br />
</p><br />
<br><br />
<h3>How are fixed points calculated?</h3><br />
<p>Fixed points are determined by setting all differential equations in a system equal to zero and solving for the variable being analyzed. The direct method of calculating equilibrium points is to find the roots of the system of equations using built-in root-finding functions such as fzero in MATLAB, and similar functions in Maple and C. The indirect method of calculating the equilibrium points is to plot and find the intersections of these nullclines, which represent equilibrium points.</p><br />
<br />
<br />
<h3>Results</h3><br />
<p>We were able to find the fixed points of the system analytically (for small Hill coefficients) and computationally. These points were used for bifurcation analysis, and for analyzing the probability that our system would exhibit bistable behavior.</p><br />
<br><br />
<br><br />
<br />
<h1>Nullclines</h1><br />
<p>In a system of differential equations, the nullclines are the solution curves for which all of the differential equations are equal to zero (Wikipedia).</p><br />
<br><br />
<h3>Why are nullclines important?</h3><br />
<p>The intersections of the nullclines give the equilibrium points of the system of differential equations. From graphs of the nullclines, it is possible to infer whether or not a system will be bistable. If the nullclines only intersect in one place the system is not bistable, since there is one single equilibrium point. If there are more than two intersections, the middle equilibrium point is often an unstable saddle point.</p><br />
<br><br />
<h3>How are the nullclines calculated?</h3><br />
<p>Just as in calculating the fixed points, we set the governing differential equations of the system equal to zero and plot the curves generated.</p><br />
<br><br />
<h3>Simulations involved</h3><br />
<p>Just as in calculating the equilibrium points, we set the governing differential equations of the system equal to zero and plot the curves generated.</p><br />
<center> equations </center><br />
<center><br />
<table><br />
<tr><br />
<td><br />
<p><img src="https://static.igem.org/mediawiki/2010/1/18/Nullclines_diagram1.png"/></p><br />
</td><br />
<td><br />
<p><b>Figure 1.</b>This figure is a plot of the nullclines of the differential equations for CFP and GFP, where we solve them for CFP as a function of GFP. In this plot, both Hill coefficients are two, ie. n2 and n4 are both 2. The nullclines cross over at three fixed points, where the middle is a saddle-node fixed point. This is an ‘ideal’ bistability plot.<br />
</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<p><img src="https://static.igem.org/mediawiki/2010/5/53/Nullclines_diagram2.png"></p><br />
</td><br />
<td><br />
<p><b>Figure 2.</b>This figure is a plot, again, of the nullclines of the differential equations for CFP and GFP. In this plot, both Hill coefficients are one. Note that the nullclines only cross once, resulting in only one fixed point and hence no possibility of bistability. This result allowed us to tell the biologists that the toggle switch would definitely not work if both of the Hill coefficients were one.<br />
</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<p><img src="https://static.igem.org/mediawiki/2010/d/da/Nullclines_diagram3.png"/></p><br />
</td><br />
<td><br />
<p><b>Figure 3.</b> Figure 3: This figure is of CFP as a function of GFP for the two nullclines when the initial galactose and copper concentrations are changed. From this we can see how the fixed points change with different initial galactose and methionine concentrations. The blue line represents the GFP = CFP line. The further from this line equilibrium points are, the harder it will be to switch between stable states.</p><br />
</td><br />
</tr><br />
</table><br />
<br><br />
<h3>Results</h3><br />
<p>We focused our efforts on plotting the nullclines of the system for a range of Hill coefficient combinations in order to get a general idea of which combinations would most likely produce robust bistability. We found that all combinations gave bistability, except when both Hill coefficients were one. The optimal Hill coefficient combination occurred when both Hill coefficients were two. We passed this information onto the biology team, letting them know that if we wanted the system to successfully switch, we could not have both Hill coefficients with a value of one.<br />
</p><br />
<hr><br />
<table class="nav"><br />
<tr><br />
<td><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Equations"><img src="https://static.igem.org/mediawiki/2010/8/8e/Left_arrow.png">&nbsp;&nbsp;Return to Equations</a><br />
</td><br />
<td align="right"><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Bifurcation">Continue to Bifurcation&nbsp;&nbsp;<img src="https://static.igem.org/mediawiki/2010/3/36/Right_arrow.png"></a><br />
</td><br />
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{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/Fixed_PointsTeam:Aberdeen Scotland/Fixed Points2010-10-27T16:07:03Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<html><br />
<h1>Fixed Points</h1><br />
<p>Fixed points are the points where an equation’s rate of change, or its slope, is zero. There are three main types of equilibrium points: stable, unstable, and saddle-node points. A stable equilibrium is a value which the function converges towards, whereas an unstable equilibrium is a value the function will diverge from. A saddle-node is where the function both converges to and diverges from, the equilibrium, depending on the direction from which we approach the point (S. Strogatz, Nonlinear dynamics and Chaos).</p><br />
<br><br />
<h3>Why are fixed points important?</h3><br />
<p>Finding fixed points is the first step in analyzing the stability of a system. In particular, we are interested in the nature of our system’s bistability and how it changes with a variation of the parameters, i.e., its bifurcation analysis. This analysis can be conveyed to the biologists to minimize experimental guessing.<br />
</p><br />
<br><br />
<h3>How are fixed points calculated?</h3><br />
<p>Fixed points are determined by setting all differential equations in a system equal to zero and solving for the variable being analyzed. The direct method of calculating equilibrium points is to find the roots of the system of equations using built-in root-finding functions such as fzero in MATLAB, and similar functions in Maple and C. The indirect method of calculating the equilibrium points is to plot and find the intersections of these nullclines, which represent equilibrium points.</p><br />
<br />
<br />
<h3>Results</h3><br />
<p>We were able to find the fixed points of the system analytically (for small Hill coefficients) and computationally. These points were used for bifurcation analysis, and for analyzing the probability that our system would exhibit bistable behavior.</p><br />
<br><br />
<br><br />
<br />
<h1>Nullclines</h1><br />
<p>In a system of differential equations, the nullclines are the solution curves for which all of the differential equations are equal to zero (Wikipedia).</p><br />
<br><br />
<h3>Why are nullclines important?</h3><br />
<p>The intersections of the nullclines give the equilibrium points of the system of differential equations. From graphs of the nullclines, it is possible to infer whether or not a system will be bistable. If the nullclines only intersect in one place the system is not bistable, since there is one single equilibrium point. If there are more than two intersections, the middle equilibrium point is often an unstable saddle point.</p><br />
<br><br />
<h3>How are the nullclines calculated?</h3><br />
<p>Just as in calculating the fixed points, we set the governing differential equations of the system equal to zero and plot the curves generated.</p><br />
<br><br />
<h3>Simulations involved</h3><br />
<p>Just as in calculating the equilibrium points, we set the governing differential equations of the system equal to zero and plot the curves generated.</p><br />
<center> equations </center><br />
<center><br />
<table><br />
<tr><br />
<td><br />
<p><img src="https://static.igem.org/mediawiki/2010/1/18/Nullclines_diagram1.png"/></p><br />
</td><br />
<td><br />
<p><b>Figure 1.</b>This figure is a plot of the nullclines of the differential equations for CFP and GFP, where we solve them for CFP as a function of GFP. In this plot, both Hill coefficients are two, ie. n2 and n4 are both 2. The nullclines cross over at three fixed points, where the middle is a saddle-node fixed point. This is an ‘ideal’ bistability plot.<br />
</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<p><img src="https://static.igem.org/mediawiki/2010/5/53/Nullclines_diagram2.png"></p><br />
</td><br />
<td><br />
<p><b>Figure 2.</b>This figure is a plot, again, of the nullclines of the differential equations for CFP and GFP. In this plot, both Hill coefficients are one. Note that the nullclines only cross once, resulting in only one fixed point and hence no possibility of bistability. This result allowed us to tell the biologists that the toggle switch would definitely not work if both of the Hill coefficients were one.<br />
</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<p><img src="https://static.igem.org/mediawiki/2010/d/da/Nullclines_diagram3.png"/></p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<p><b>Figure 3.</b> Figure 3: This figure is of CFP as a function of GFP for the two nullclines when the initial galactose and copper concentrations are changed. From this we can see how the fixed points change with different initial galactose and methionine concentrations. The blue line represents the GFP = CFP line. The further from this line equilibrium points are, the harder it will be to switch between stable states.</p><br />
</td><br />
</tr><br />
</table><br />
<br><br />
<h3>Results</h3><br />
<p>We focused our efforts on plotting the nullclines of the system for a range of Hill coefficient combinations in order to get a general idea of which combinations would most likely produce robust bistability. We found that all combinations gave bistability, except when both Hill coefficients were one. The optimal Hill coefficient combination occurred when both Hill coefficients were two. We passed this information onto the biology team, letting them know that if we wanted the system to successfully switch, we could not have both Hill coefficients with a value of one.<br />
</p><br />
<hr><br />
<table class="nav"><br />
<tr><br />
<td><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Equations"><img src="https://static.igem.org/mediawiki/2010/8/8e/Left_arrow.png">&nbsp;&nbsp;Return to Equations</a><br />
</td><br />
<td align="right"><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Bifurcation">Continue to Bifurcation&nbsp;&nbsp;<img src="https://static.igem.org/mediawiki/2010/3/36/Right_arrow.png"></a><br />
</td><br />
</tr><br />
</table><br />
</html><br />
{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/Fixed_PointsTeam:Aberdeen Scotland/Fixed Points2010-10-27T16:03:38Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<html><br />
<h1>Fixed Points</h1><br />
<p>Fixed points are the points where an equation’s rate of change, or its slope, is zero. There are three main types of equilibrium points: stable, unstable, and saddle-node points. A stable equilibrium is a value which the function converges towards, whereas an unstable equilibrium is a value the function will diverge from. A saddle-node is where the function both converges to and diverges from, the equilibrium, depending on the direction from which we approach the point (S. Strogatz, Nonlinear dynamics and Chaos).</p><br />
<br><br />
<h3>Why are fixed points important?</h3><br />
<p>Finding fixed points is the first step in analyzing the stability of a system. In particular, we are interested in the nature of our system’s bistability and how it changes with a variation of the parameters, i.e., its bifurcation analysis. This analysis can be conveyed to the biologists to minimize experimental guessing.<br />
</p><br />
<br><br />
<h3>How are fixed points calculated?</h3><br />
<p>Fixed points are determined by setting all differential equations in a system equal to zero and solving for the variable being analyzed. The direct method of calculating equilibrium points is to find the roots of the system of equations using built-in root-finding functions such as fzero in MATLAB, and similar functions in Maple and C. The indirect method of calculating the equilibrium points is to plot and find the intersections of these nullclines, which represent equilibrium points.</p><br />
<br />
<br />
<h3>Results</h3><br />
<p>We were able to find the fixed points of the system analytically (for small Hill coefficients) and computationally. These points were used for bifurcation analysis, and for analyzing the probability that our system would exhibit bistable behavior.</p><br />
<br><br />
<br><br />
<br />
<h1>Nullclines</h1><br />
<p>In a system of differential equations, the nullclines are the solution curves for which all of the differential equations are equal to zero (Wikipedia).</p><br />
<br><br />
<h3>Why are nullclines important?</h3><br />
<p>The intersections of the nullclines give the equilibrium points of the system of differential equations. From graphs of the nullclines, it is possible to infer whether or not a system will be bistable. If the nullclines only intersect in one place the system is not bistable, since there is one single equilibrium point. If there are more than two intersections, the middle equilibrium point is often an unstable saddle point.</p><br />
<br><br />
<h3>How are the nullclines calculated?</h3><br />
<p>Just as in calculating the fixed points, we set the governing differential equations of the system equal to zero and plot the curves generated.</p><br />
<br><br />
<h3>Simulations involved</h3><br />
<p>Just as in calculating the equilibrium points, we set the governing differential equations of the system equal to zero and plot the curves generated.</p><br />
<center> equations </center><br />
<center><br />
<table><br />
<tr><br />
<td><br />
<p><img src="https://static.igem.org/mediawiki/2010/1/18/Nullclines_diagram1.png"/></p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<p><b>Figure 1.</b> Figure 1: This figure is a plot of the nullclines of the differential equations for CFP and GFP, where we solve them for CFP as a function of GFP. In this plot, both Hill coefficients are two, ie. n2 and n4 are both 2. The nullclines cross over at three fixed points, where the middle is a saddle-node fixed point. This is an ‘ideal’ bistability plot.<br />
</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<p><img src="https://static.igem.org/mediawiki/2010/5/53/Nullclines_diagram2.png"></p><br />
</td><br />
</tr><br />
>tr><br />
<td><br />
<p><b>Figure 2.</b> Figure 2: This figure is a plot, again, of the nullclines of the differential equations for CFP and GFP. In this plot, both Hill coefficients are one. Note that the nullclines only cross once, resulting in only one fixed point and hence no possibility of bistability. This result allowed us to tell the biologists that the toggle switch would definitely not work if both of the Hill coefficients were one.<br />
</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<p><b>Figure 3.</b> Figure 3: This figure is of CFP as a function of GFP for the two nullclines when the initial galactose and copper concentrations are changed. From this we can see how the fixed points change with different initial galactose and methionine concentrations. The blue line represents the GFP = CFP line. The further from this line equilibrium points are, the harder it will be to switch between stable states.<br />
</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<p><img src="https://static.igem.org/mediawiki/2010/d/da/Nullclines_diagram3.png"/></p><br />
</td><br />
</tr><br />
</table><br />
<br><br />
<h3>Results</h3><br />
<p>We focused our efforts on plotting the nullclines of the system for a range of Hill coefficient combinations in order to get a general idea of which combinations would most likely produce robust bistability. We found that all combinations gave bistability, except when both Hill coefficients were one. The optimal Hill coefficient combination occurred when both Hill coefficients were two. We passed this information onto the biology team, letting them know that if we wanted the system to successfully switch, we could not have both Hill coefficients with a value of one.<br />
</p><br />
<hr><br />
<table class="nav"><br />
<tr><br />
<td><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Equations"><img src="https://static.igem.org/mediawiki/2010/8/8e/Left_arrow.png">&nbsp;&nbsp;Return to Equations</a><br />
</td><br />
<td align="right"><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Bifurcation">Continue to Bifurcation&nbsp;&nbsp;<img src="https://static.igem.org/mediawiki/2010/3/36/Right_arrow.png"></a><br />
</td><br />
</tr><br />
</table><br />
</html><br />
{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/Fixed_PointsTeam:Aberdeen Scotland/Fixed Points2010-10-27T16:00:40Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<html><br />
<h1>Fixed Points</h1><br />
<p>Fixed points are the points where an equation’s rate of change, or its slope, is zero. There are three main types of equilibrium points: stable, unstable, and saddle-node points. A stable equilibrium is a value which the function converges towards, whereas an unstable equilibrium is a value the function will diverge from. A saddle-node is where the function both converges to and diverges from, the equilibrium, depending on the direction from which we approach the point (S. Strogatz, Nonlinear dynamics and Chaos).</p><br />
<br><br />
<h3>Why are fixed points important?</h3><br />
<p>Finding fixed points is the first step in analyzing the stability of a system. In particular, we are interested in the nature of our system’s bistability and how it changes with a variation of the parameters, i.e., its bifurcation analysis. This analysis can be conveyed to the biologists to minimize experimental guessing.<br />
</p><br />
<br><br />
<h3>How are fixed points calculated?</h3><br />
<p>Fixed points are determined by setting all differential equations in a system equal to zero and solving for the variable being analyzed. The direct method of calculating equilibrium points is to find the roots of the system of equations using built-in root-finding functions such as fzero in MATLAB, and similar functions in Maple and C. The indirect method of calculating the equilibrium points is to plot and find the intersections of these nullclines, which represent equilibrium points.</p><br />
<br />
<br />
<h3>Results</h3><br />
<p>We were able to find the fixed points of the system analytically (for small Hill coefficients) and computationally. These points were used for bifurcation analysis, and for analyzing the probability that our system would exhibit bistable behavior.</p><br />
<br><br />
<br><br />
<br />
<h1>Nullclines</h1><br />
<p>In a system of differential equations, the nullclines are the solution curves for which all of the differential equations are equal to zero (Wikipedia).</p><br />
<br><br />
<h3>Why are nullclines important?</h3><br />
<p>The intersections of the nullclines give the equilibrium points of the system of differential equations. From graphs of the nullclines, it is possible to infer whether or not a system will be bistable. If the nullclines only intersect in one place the system is not bistable, since there is one single equilibrium point. If there are more than two intersections, the middle equilibrium point is often an unstable saddle point.</p><br />
<br><br />
<h3>How are the nullclines calculated?</h3><br />
<p>Just as in calculating the fixed points, we set the governing differential equations of the system equal to zero and plot the curves generated.</p><br />
<br><br />
<h3>Simulations involved</h3><br />
<p>Just as in calculating the equilibrium points, we set the governing differential equations of the system equal to zero and plot the curves generated.</p><br />
<center> equations </center><br />
<center><br />
<table><br />
<tr><br />
<td><br />
<p><img src="https://static.igem.org/mediawiki/2010/1/18/Nullclines_diagram1.png"/></p><br />
</td><br />
<td><br />
<p><b>Figure 1.</b> Figure 1: This figure is a plot of the nullclines of the differential equations for CFP and GFP, where we solve them for CFP as a function of GFP. In this plot, both Hill coefficients are two, ie. n2 and n4 are both 2. The nullclines cross over at three fixed points, where the middle is a saddle-node fixed point. This is an ‘ideal’ bistability plot.<br />
</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<p><img src="https://static.igem.org/mediawiki/2010/5/53/Nullclines_diagram2.png"></p><br />
</td><br />
<td><br />
<p><b>Figure 2.</b> Figure 2: This figure is a plot, again, of the nullclines of the differential equations for CFP and GFP. In this plot, both Hill coefficients are one. Note that the nullclines only cross once, resulting in only one fixed point and hence no possibility of bistability. This result allowed us to tell the biologists that the toggle switch would definitely not work if both of the Hill coefficients were one.<br />
</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<p><b>Figure 3.</b> Figure 3: This figure is of CFP as a function of GFP for the two nullclines when the initial galactose and copper concentrations are changed. From this we can see how the fixed points change with different initial galactose and methionine concentrations. The blue line represents the GFP = CFP line. The further from this line equilibrium points are, the harder it will be to switch between stable states.<br />
</p><br />
</td><br />
<td><br />
<p><img src="https://static.igem.org/mediawiki/2010/d/da/Nullclines_diagram3.png"/></p><br />
</td><br />
</tr><br />
</table><br />
<br><br />
<h3>Results</h3><br />
<p>We focused our efforts on plotting the nullclines of the system for a range of Hill coefficient combinations in order to get a general idea of which combinations would most likely produce robust bistability. We found that all combinations gave bistability, except when both Hill coefficients were one. The optimal Hill coefficient combination occurred when both Hill coefficients were two. We passed this information onto the biology team, letting them know that if we wanted the system to successfully switch, we could not have both Hill coefficients with a value of one.<br />
</p><br />
<hr><br />
<table class="nav"><br />
<tr><br />
<td><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Equations"><img src="https://static.igem.org/mediawiki/2010/8/8e/Left_arrow.png">&nbsp;&nbsp;Return to Equations</a><br />
</td><br />
<td align="right"><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Bifurcation">Continue to Bifurcation&nbsp;&nbsp;<img src="https://static.igem.org/mediawiki/2010/3/36/Right_arrow.png"></a><br />
</td><br />
</tr><br />
</table><br />
</html><br />
{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/File:Nullclines_diagram3.pngFile:Nullclines diagram3.png2010-10-27T15:59:55Z<p>Brychan: </p>
<hr />
<div></div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/Fixed_PointsTeam:Aberdeen Scotland/Fixed Points2010-10-27T15:59:29Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<html><br />
<h1>Fixed Points</h1><br />
<p>Fixed points are the points where an equation’s rate of change, or its slope, is zero. There are three main types of equilibrium points: stable, unstable, and saddle-node points. A stable equilibrium is a value which the function converges towards, whereas an unstable equilibrium is a value the function will diverge from. A saddle-node is where the function both converges to and diverges from, the equilibrium, depending on the direction from which we approach the point (S. Strogatz, Nonlinear dynamics and Chaos).</p><br />
<br><br />
<h3>Why are fixed points important?</h3><br />
<p>Finding fixed points is the first step in analyzing the stability of a system. In particular, we are interested in the nature of our system’s bistability and how it changes with a variation of the parameters, i.e., its bifurcation analysis. This analysis can be conveyed to the biologists to minimize experimental guessing.<br />
</p><br />
<br><br />
<h3>How are fixed points calculated?</h3><br />
<p>Fixed points are determined by setting all differential equations in a system equal to zero and solving for the variable being analyzed. The direct method of calculating equilibrium points is to find the roots of the system of equations using built-in root-finding functions such as fzero in MATLAB, and similar functions in Maple and C. The indirect method of calculating the equilibrium points is to plot and find the intersections of these nullclines, which represent equilibrium points.</p><br />
<br />
<br />
<h3>Results</h3><br />
<p>We were able to find the fixed points of the system analytically (for small Hill coefficients) and computationally. These points were used for bifurcation analysis, and for analyzing the probability that our system would exhibit bistable behavior.</p><br />
<br><br />
<br><br />
<br />
<h1>Nullclines</h1><br />
<p>In a system of differential equations, the nullclines are the solution curves for which all of the differential equations are equal to zero (Wikipedia).</p><br />
<br><br />
<h3>Why are nullclines important?</h3><br />
<p>The intersections of the nullclines give the equilibrium points of the system of differential equations. From graphs of the nullclines, it is possible to infer whether or not a system will be bistable. If the nullclines only intersect in one place the system is not bistable, since there is one single equilibrium point. If there are more than two intersections, the middle equilibrium point is often an unstable saddle point.</p><br />
<br><br />
<h3>How are the nullclines calculated?</h3><br />
<p>Just as in calculating the fixed points, we set the governing differential equations of the system equal to zero and plot the curves generated.</p><br />
<br><br />
<h3>Simulations involved</h3><br />
<p>Just as in calculating the equilibrium points, we set the governing differential equations of the system equal to zero and plot the curves generated.</p><br />
<center> equations </center><br />
<center><br />
<table><br />
<tr><br />
<td><br />
<p><img src="https://static.igem.org/mediawiki/2010/1/18/Nullclines_diagram1.png"/></p><br />
</td><br />
<td><br />
<p><b>Figure 1.</b> Figure 1: This figure is a plot of the nullclines of the differential equations for CFP and GFP, where we solve them for CFP as a function of GFP. In this plot, both Hill coefficients are two, ie. n2 and n4 are both 2. The nullclines cross over at three fixed points, where the middle is a saddle-node fixed point. This is an ‘ideal’ bistability plot.<br />
</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<p><img src="https://static.igem.org/mediawiki/2010/5/53/Nullclines_diagram2.png"></p><br />
</td><br />
<td><br />
<p><b>Figure 2.</b> Figure 2: This figure is a plot, again, of the nullclines of the differential equations for CFP and GFP. In this plot, both Hill coefficients are one. Note that the nullclines only cross once, resulting in only one fixed point and hence no possibility of bistability. This result allowed us to tell the biologists that the toggle switch would definitely not work if both of the Hill coefficients were one.<br />
</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<p><b>Figure 3.</b> Figure 3: This figure is of CFP as a function of GFP for the two nullclines when the initial galactose and copper concentrations are changed. From this we can see how the fixed points change with different initial galactose and methionine concentrations. The blue line represents the GFP = CFP line. The further from this line equilibrium points are, the harder it will be to switch between stable states.<br />
</p><br />
</td><br />
<td><br />
nullclines3.png<br />
</td><br />
</tr><br />
</table><br />
<br><br />
<h3>Results</h3><br />
<p>We focused our efforts on plotting the nullclines of the system for a range of Hill coefficient combinations in order to get a general idea of which combinations would most likely produce robust bistability. We found that all combinations gave bistability, except when both Hill coefficients were one. The optimal Hill coefficient combination occurred when both Hill coefficients were two. We passed this information onto the biology team, letting them know that if we wanted the system to successfully switch, we could not have both Hill coefficients with a value of one.<br />
</p><br />
<hr><br />
<table class="nav"><br />
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<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Equations"><img src="https://static.igem.org/mediawiki/2010/8/8e/Left_arrow.png">&nbsp;&nbsp;Return to Equations</a><br />
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{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/File:Nullclines_diagram2.pngFile:Nullclines diagram2.png2010-10-27T15:58:19Z<p>Brychan: </p>
<hr />
<div></div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/Fixed_PointsTeam:Aberdeen Scotland/Fixed Points2010-10-27T15:57:46Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<html><br />
<h1>Fixed Points</h1><br />
<p>Fixed points are the points where an equation’s rate of change, or its slope, is zero. There are three main types of equilibrium points: stable, unstable, and saddle-node points. A stable equilibrium is a value which the function converges towards, whereas an unstable equilibrium is a value the function will diverge from. A saddle-node is where the function both converges to and diverges from, the equilibrium, depending on the direction from which we approach the point (S. Strogatz, Nonlinear dynamics and Chaos).</p><br />
<br><br />
<h3>Why are fixed points important?</h3><br />
<p>Finding fixed points is the first step in analyzing the stability of a system. In particular, we are interested in the nature of our system’s bistability and how it changes with a variation of the parameters, i.e., its bifurcation analysis. This analysis can be conveyed to the biologists to minimize experimental guessing.<br />
</p><br />
<br><br />
<h3>How are fixed points calculated?</h3><br />
<p>Fixed points are determined by setting all differential equations in a system equal to zero and solving for the variable being analyzed. The direct method of calculating equilibrium points is to find the roots of the system of equations using built-in root-finding functions such as fzero in MATLAB, and similar functions in Maple and C. The indirect method of calculating the equilibrium points is to plot and find the intersections of these nullclines, which represent equilibrium points.</p><br />
<br />
<br />
<h3>Results</h3><br />
<p>We were able to find the fixed points of the system analytically (for small Hill coefficients) and computationally. These points were used for bifurcation analysis, and for analyzing the probability that our system would exhibit bistable behavior.</p><br />
<br><br />
<br><br />
<br />
<h1>Nullclines</h1><br />
<p>In a system of differential equations, the nullclines are the solution curves for which all of the differential equations are equal to zero (Wikipedia).</p><br />
<br><br />
<h3>Why are nullclines important?</h3><br />
<p>The intersections of the nullclines give the equilibrium points of the system of differential equations. From graphs of the nullclines, it is possible to infer whether or not a system will be bistable. If the nullclines only intersect in one place the system is not bistable, since there is one single equilibrium point. If there are more than two intersections, the middle equilibrium point is often an unstable saddle point.</p><br />
<br><br />
<h3>How are the nullclines calculated?</h3><br />
<p>Just as in calculating the fixed points, we set the governing differential equations of the system equal to zero and plot the curves generated.</p><br />
<br><br />
<h3>Simulations involved</h3><br />
<p>Just as in calculating the equilibrium points, we set the governing differential equations of the system equal to zero and plot the curves generated.</p><br />
<center> equations </center><br />
<br />
<table><br />
<tr><br />
<td><br />
<p><img src="https://static.igem.org/mediawiki/2010/1/18/Nullclines_diagram1.png"/></p><br />
</td><br />
<td><br />
<p><b>Figure 1.</b> Figure 1: This figure is a plot of the nullclines of the differential equations for CFP and GFP, where we solve them for CFP as a function of GFP. In this plot, both Hill coefficients are two, ie. n2 and n4 are both 2. The nullclines cross over at three fixed points, where the middle is a saddle-node fixed point. This is an ‘ideal’ bistability plot.<br />
</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
nullclines2.png<br />
</td><br />
<td><br />
<p><b>Figure 2.</b> Figure 2: This figure is a plot, again, of the nullclines of the differential equations for CFP and GFP. In this plot, both Hill coefficients are one. Note that the nullclines only cross once, resulting in only one fixed point and hence no possibility of bistability. This result allowed us to tell the biologists that the toggle switch would definitely not work if both of the Hill coefficients were one.<br />
</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<p><b>Figure 3.</b> Figure 3: This figure is of CFP as a function of GFP for the two nullclines when the initial galactose and copper concentrations are changed. From this we can see how the fixed points change with different initial galactose and methionine concentrations. The blue line represents the GFP = CFP line. The further from this line equilibrium points are, the harder it will be to switch between stable states.<br />
</p><br />
</td><br />
<td><br />
nullclines3.png<br />
</td><br />
</tr><br />
</table><br />
<br><br />
<h3>Results</h3><br />
<p>We focused our efforts on plotting the nullclines of the system for a range of Hill coefficient combinations in order to get a general idea of which combinations would most likely produce robust bistability. We found that all combinations gave bistability, except when both Hill coefficients were one. The optimal Hill coefficient combination occurred when both Hill coefficients were two. We passed this information onto the biology team, letting them know that if we wanted the system to successfully switch, we could not have both Hill coefficients with a value of one.<br />
</p><br />
<hr><br />
<table class="nav"><br />
<tr><br />
<td><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Equations"><img src="https://static.igem.org/mediawiki/2010/8/8e/Left_arrow.png">&nbsp;&nbsp;Return to Equations</a><br />
</td><br />
<td align="right"><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Bifurcation">Continue to Bifurcation&nbsp;&nbsp;<img src="https://static.igem.org/mediawiki/2010/3/36/Right_arrow.png"></a><br />
</td><br />
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</table><br />
</html><br />
{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/File:Nullclines_diagram1.pngFile:Nullclines diagram1.png2010-10-27T15:57:01Z<p>Brychan: </p>
<hr />
<div></div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/Fixed_PointsTeam:Aberdeen Scotland/Fixed Points2010-10-27T15:56:25Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<html><br />
<h1>Fixed Points</h1><br />
<p>Fixed points are the points where an equation’s rate of change, or its slope, is zero. There are three main types of equilibrium points: stable, unstable, and saddle-node points. A stable equilibrium is a value which the function converges towards, whereas an unstable equilibrium is a value the function will diverge from. A saddle-node is where the function both converges to and diverges from, the equilibrium, depending on the direction from which we approach the point (S. Strogatz, Nonlinear dynamics and Chaos).</p><br />
<br><br />
<h3>Why are fixed points important?</h3><br />
<p>Finding fixed points is the first step in analyzing the stability of a system. In particular, we are interested in the nature of our system’s bistability and how it changes with a variation of the parameters, i.e., its bifurcation analysis. This analysis can be conveyed to the biologists to minimize experimental guessing.<br />
</p><br />
<br><br />
<h3>How are fixed points calculated?</h3><br />
<p>Fixed points are determined by setting all differential equations in a system equal to zero and solving for the variable being analyzed. The direct method of calculating equilibrium points is to find the roots of the system of equations using built-in root-finding functions such as fzero in MATLAB, and similar functions in Maple and C. The indirect method of calculating the equilibrium points is to plot and find the intersections of these nullclines, which represent equilibrium points.</p><br />
<br />
<br />
<h3>Results</h3><br />
<p>We were able to find the fixed points of the system analytically (for small Hill coefficients) and computationally. These points were used for bifurcation analysis, and for analyzing the probability that our system would exhibit bistable behavior.</p><br />
<br><br />
<br><br />
<br />
<h1>Nullclines</h1><br />
<p>In a system of differential equations, the nullclines are the solution curves for which all of the differential equations are equal to zero (Wikipedia).</p><br />
<br><br />
<h3>Why are nullclines important?</h3><br />
<p>The intersections of the nullclines give the equilibrium points of the system of differential equations. From graphs of the nullclines, it is possible to infer whether or not a system will be bistable. If the nullclines only intersect in one place the system is not bistable, since there is one single equilibrium point. If there are more than two intersections, the middle equilibrium point is often an unstable saddle point.</p><br />
<br><br />
<h3>How are the nullclines calculated?</h3><br />
<p>Just as in calculating the fixed points, we set the governing differential equations of the system equal to zero and plot the curves generated.</p><br />
<br><br />
<h3>Simulations involved</h3><br />
<p>Just as in calculating the equilibrium points, we set the governing differential equations of the system equal to zero and plot the curves generated.</p><br />
<center> equations </center><br />
<br />
<table><br />
<tr><br />
<td><br />
nullclines1.png<br />
</td><br />
<td><br />
<p><b>Figure 1.</b> Figure 1: This figure is a plot of the nullclines of the differential equations for CFP and GFP, where we solve them for CFP as a function of GFP. In this plot, both Hill coefficients are two, ie. n2 and n4 are both 2. The nullclines cross over at three fixed points, where the middle is a saddle-node fixed point. This is an ‘ideal’ bistability plot.<br />
</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
nullclines2.png<br />
</td><br />
<td><br />
<p><b>Figure 2.</b> Figure 2: This figure is a plot, again, of the nullclines of the differential equations for CFP and GFP. In this plot, both Hill coefficients are one. Note that the nullclines only cross once, resulting in only one fixed point and hence no possibility of bistability. This result allowed us to tell the biologists that the toggle switch would definitely not work if both of the Hill coefficients were one.<br />
</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<p><b>Figure 3.</b> Figure 3: This figure is of CFP as a function of GFP for the two nullclines when the initial galactose and copper concentrations are changed. From this we can see how the fixed points change with different initial galactose and methionine concentrations. The blue line represents the GFP = CFP line. The further from this line equilibrium points are, the harder it will be to switch between stable states.<br />
</p><br />
</td><br />
<td><br />
nullclines3.png<br />
</td><br />
</tr><br />
</table><br />
<br><br />
<h3>Results</h3><br />
<p>We focused our efforts on plotting the nullclines of the system for a range of Hill coefficient combinations in order to get a general idea of which combinations would most likely produce robust bistability. We found that all combinations gave bistability, except when both Hill coefficients were one. The optimal Hill coefficient combination occurred when both Hill coefficients were two. We passed this information onto the biology team, letting them know that if we wanted the system to successfully switch, we could not have both Hill coefficients with a value of one.<br />
</p><br />
<hr><br />
<table class="nav"><br />
<tr><br />
<td><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Equations"><img src="https://static.igem.org/mediawiki/2010/8/8e/Left_arrow.png">&nbsp;&nbsp;Return to Equations</a><br />
</td><br />
<td align="right"><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Bifurcation">Continue to Bifurcation&nbsp;&nbsp;<img src="https://static.igem.org/mediawiki/2010/3/36/Right_arrow.png"></a><br />
</td><br />
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</table><br />
</html><br />
{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/cssTeam:Aberdeen Scotland/css2010-10-27T15:43:21Z<p>Brychan: </p>
<hr />
<div><html><br />
<style type="text/css"><br />
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</html></div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/BifurcationTeam:Aberdeen Scotland/Bifurcation2010-10-27T15:38:56Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<br />
<html><br />
<h1>Bifurcation Analysis</h1><br />
<br />
<table><br />
<tr><br />
<td><p><br />
The literal meaning of bifurcation is, “division into two parts, branches, or aspects” (Answers). In mathematics, this ‘division’ is in reference to when</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<p>a non-linear equation suddenly changes its behaviour qualitatively when a parameter of the system is varied. The point at which bifurcation occurs </p><br />
</td><br />
<tr><br />
</tr><br />
<td><br />
<p>is referred to as the ‘bifurcation point’.</p><br />
</td><br />
</tr><br />
</table><br />
<br />
</p><br />
<br><br />
<h3>what is bistability?</h3><br />
<p>If a system is bistable it can be stable in two distinct states. There are also stable oscillations, where the state of the system is stable but not at rest.</p><br />
<br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/d/da/Bifurcation1diagram.png"/><br />
<br />
<p style="font-size:10px"><b>Figure 1.</b>This is a graphical representation of bistability. Ball 2 can be equally<br>'happy' in either the 'one' or 'three' positions. Once it falls into one of these states,<br>ball 2 will not be inclined to leave this state, hence making it a "stable state".</p></center><br />
<br />
<br><br />
<br />
<h3>Why are they important?</h3><br />
<br />
<p>Analyzing the stability of a system is key to understand its behaviour. In biological terms, bistability represents the potential for our system to switch between the expression of green- and cyan-fluorescent proteins. Our goal is to analyze the parameter space in order to determine which parameters are needed to allow for bistability in the system.</p><br />
<br />
<br><br />
<p>A bifurcation diagram has branches which define the possible states a system can be in. By analyzing the bifurcation plots, we can determine how far apart the two “branches”, or stable states, are. The further apart the branches, the harder it will be to attain a switch between the two stable states, however the switch also becomes more stable with increased distance between the branches. Therefore, we want to manipulate the parameters in the system to guarantee robust bistability.</p><br />
<br><br />
<h3>How do you analyse the stability of a system?</h3><br />
<br />
<p>In order to analyze the stability of the system, we utilized XPP Auto, C programming, and MATLAB. We will attempt to detail the math behind what the program does computationally.<br />
<br><br />
<br><br />
The process involved with general stability analysis is as follows. We have taken a simple example system that is easy to analyze in order to demonstrate the general process:<br />
<br><br />
<br><br />
Take the following example system of two differential equations:</p><br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/5/59/Bistability_method1.png"/></center><br />
<br><br />
<br />
<h4><p>1. Set the differential equations to zero:</p></h4><br />
<br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/e/e0/Bistability_method2.png"/></center><br />
<br><br />
<br />
<h4><p>Find the fixed points of the system of differential equations</p></h4><br />
<p>There is a fixed point at x = y = 0. By observation, we can see that there are fixed points at x = 0, y = 2, and x = 3, y=0. Finally, by solving this system of equations, we see that there is also a fixed point at x=1 and y=1.<br />
</p><br />
<br><br />
<br />
<h4><p>3. Evaluate the Jacobian matrix of the system at these fixed points:</p></h4><br />
<br />
<p>If we label the top function f1 and the bottom f2, the Jacobian of the system will be of the form:</p><br />
<br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/6/6e/Bistability_method3.png"/></center><br />
<br><br />
<br />
<p>When we evaluate the Jacobian matrix for our example system of equations, we get:</p><br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/9/95/Bistability_method4.png"/></center><br />
<br />
<br><br />
<br />
<h4><p>4. Using the Jacobian matrix, find the corresponding eigenvalues:</p></h4><br />
<br />
<p><u>Definition</u>: An eigenvalue is a value λ such that Ax = λx, where x is said to be an eigenvector of A. In order to solve for λ, solve the following equation<br />
</p><br />
<br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/a/ac/Bistability_method5.png"/></center><br />
<br><br />
<br />
<p>where det is the determinant of the matrix and I is the identity matrix. This is known as the characteristic equation of a matrix.</p><br />
<br><br />
<br />
<p>In the case of our example system, the eigenvalues are:</p><br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/4/4f/Bistability_method6.png"/></center><br />
<br><br />
<br />
<h4><p>5. Take the signs of these eigenvalues in order to analyse stability:</p></h4><br />
<br />
<p>The eigenvalues of a system give important information about the stability of the system. The stability of each of the fixed points can be analyzed based on the following criteria:</p><br />
<br><br />
<br />
<center><table><br />
<tr><br />
<td><br />
Positive real and complex parts<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real and complex parts <br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Positive real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable spiral (spiraling out)<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable spiral (spiraling inwards)<br />
</td><br />
</tr><br />
</table></center><br />
<br />
<br><br />
<p>So, for our example system, it can be deduced that the stability at the fixed points is as follows:</p><br />
<br />
<br><br />
<center><br />
One positive and one negative eigenvalue at (0,0) means that <b>(0,0)</b> is a <b>saddle node</b><br />
<br><br />
Two negative eigenvalues at (0,2) means that <b>(0,2)</b> is a <b>stable node </b><br />
<br><br />
Two negative eigenvalues at (3,0) means that <b>(3,0)</b> is a <b>stable node </b><br />
<br><br />
One positive and one negative eigevalue at (1,1) means that <b>(1,1)</b> is a <b>saddle node</b></center><br />
<br><br />
<br />
<h3>The simulations involved</h3><br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/5/50/Bifurcation_diagram2.png"/><br />
<p style="font-size:10px"><b>Figure 2.</b> Figure 2: This is a bifurcation diagram that we generated using Maple. It ranges over the parameter space of <br />
<br><br />
P1 and C (protein 1, or GFP, and a dimensionless group of parameters) and plots the fixed points of the system at those values. <br />
</p></center><br />
<br><br />
<br />
<p> Where there are three fixed points the system will exhibit bistability. We also know that the further apart the two outer ‘prongs’ the more stable the two definitive fixed states will be, however this makes it more difficult to switch between states.</p><br />
<br />
<br><br><br />
<hr><br />
<table class="nav"><br />
<tr><br />
<td><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Fixed_Points"><img src="https://static.igem.org/mediawiki/2010/8/8e/Left_arrow.png">&nbsp;&nbsp;Return to Nullclines and Fixed Points</a><br />
</td><br />
<td align="right"><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Stochastic_Model">Continue to the Stochastic Model&nbsp;&nbsp;<img src="https://static.igem.org/mediawiki/2010/3/36/Right_arrow.png"></a><br />
</td><br />
</tr><br />
</table><br />
<br />
</html><br />
{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/BifurcationTeam:Aberdeen Scotland/Bifurcation2010-10-27T15:38:31Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<br />
<html><br />
<br />
<table><tr><td><h1>Bifurcation Analysis<br />
<br></h1></td></tr></table><br />
<br />
<table><br />
<tr><br />
<td><p><br />
The literal meaning of bifurcation is, “division into two parts, branches, or aspects” (Answers). In mathematics, this ‘division’ is in reference to when</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<p>a non-linear equation suddenly changes its behaviour qualitatively when a parameter of the system is varied. The point at which bifurcation occurs </p><br />
</td><br />
<tr><br />
</tr><br />
<td><br />
<p>is referred to as the ‘bifurcation point’.</p><br />
</td><br />
</tr><br />
</table><br />
<br />
</p><br />
<br><br />
<h3>what is bistability?</h3><br />
<p>If a system is bistable it can be stable in two distinct states. There are also stable oscillations, where the state of the system is stable but not at rest.</p><br />
<br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/d/da/Bifurcation1diagram.png"/><br />
<br />
<p style="font-size:10px"><b>Figure 1.</b>This is a graphical representation of bistability. Ball 2 can be equally<br>'happy' in either the 'one' or 'three' positions. Once it falls into one of these states,<br>ball 2 will not be inclined to leave this state, hence making it a "stable state".</p></center><br />
<br />
<br><br />
<br />
<h3>Why are they important?</h3><br />
<br />
<p>Analyzing the stability of a system is key to understand its behaviour. In biological terms, bistability represents the potential for our system to switch between the expression of green- and cyan-fluorescent proteins. Our goal is to analyze the parameter space in order to determine which parameters are needed to allow for bistability in the system.</p><br />
<br />
<br><br />
<p>A bifurcation diagram has branches which define the possible states a system can be in. By analyzing the bifurcation plots, we can determine how far apart the two “branches”, or stable states, are. The further apart the branches, the harder it will be to attain a switch between the two stable states, however the switch also becomes more stable with increased distance between the branches. Therefore, we want to manipulate the parameters in the system to guarantee robust bistability.</p><br />
<br><br />
<h3>How do you analyse the stability of a system?</h3><br />
<br />
<p>In order to analyze the stability of the system, we utilized XPP Auto, C programming, and MATLAB. We will attempt to detail the math behind what the program does computationally.<br />
<br><br />
<br><br />
The process involved with general stability analysis is as follows. We have taken a simple example system that is easy to analyze in order to demonstrate the general process:<br />
<br><br />
<br><br />
Take the following example system of two differential equations:</p><br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/5/59/Bistability_method1.png"/></center><br />
<br><br />
<br />
<h4><p>1. Set the differential equations to zero:</p></h4><br />
<br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/e/e0/Bistability_method2.png"/></center><br />
<br><br />
<br />
<h4><p>Find the fixed points of the system of differential equations</p></h4><br />
<p>There is a fixed point at x = y = 0. By observation, we can see that there are fixed points at x = 0, y = 2, and x = 3, y=0. Finally, by solving this system of equations, we see that there is also a fixed point at x=1 and y=1.<br />
</p><br />
<br><br />
<br />
<h4><p>3. Evaluate the Jacobian matrix of the system at these fixed points:</p></h4><br />
<br />
<p>If we label the top function f1 and the bottom f2, the Jacobian of the system will be of the form:</p><br />
<br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/6/6e/Bistability_method3.png"/></center><br />
<br><br />
<br />
<p>When we evaluate the Jacobian matrix for our example system of equations, we get:</p><br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/9/95/Bistability_method4.png"/></center><br />
<br />
<br><br />
<br />
<h4><p>4. Using the Jacobian matrix, find the corresponding eigenvalues:</p></h4><br />
<br />
<p><u>Definition</u>: An eigenvalue is a value λ such that Ax = λx, where x is said to be an eigenvector of A. In order to solve for λ, solve the following equation<br />
</p><br />
<br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/a/ac/Bistability_method5.png"/></center><br />
<br><br />
<br />
<p>where det is the determinant of the matrix and I is the identity matrix. This is known as the characteristic equation of a matrix.</p><br />
<br><br />
<br />
<p>In the case of our example system, the eigenvalues are:</p><br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/4/4f/Bistability_method6.png"/></center><br />
<br><br />
<br />
<h4><p>5. Take the signs of these eigenvalues in order to analyse stability:</p></h4><br />
<br />
<p>The eigenvalues of a system give important information about the stability of the system. The stability of each of the fixed points can be analyzed based on the following criteria:</p><br />
<br><br />
<br />
<center><table><br />
<tr><br />
<td><br />
Positive real and complex parts<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real and complex parts <br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Positive real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable spiral (spiraling out)<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable spiral (spiraling inwards)<br />
</td><br />
</tr><br />
</table></center><br />
<br />
<br><br />
<p>So, for our example system, it can be deduced that the stability at the fixed points is as follows:</p><br />
<br />
<br><br />
<center><br />
One positive and one negative eigenvalue at (0,0) means that <b>(0,0)</b> is a <b>saddle node</b><br />
<br><br />
Two negative eigenvalues at (0,2) means that <b>(0,2)</b> is a <b>stable node </b><br />
<br><br />
Two negative eigenvalues at (3,0) means that <b>(3,0)</b> is a <b>stable node </b><br />
<br><br />
One positive and one negative eigevalue at (1,1) means that <b>(1,1)</b> is a <b>saddle node</b></center><br />
<br><br />
<br />
<h3>The simulations involved</h3><br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/5/50/Bifurcation_diagram2.png"/><br />
<p style="font-size:10px"><b>Figure 2.</b> Figure 2: This is a bifurcation diagram that we generated using Maple. It ranges over the parameter space of <br />
<br><br />
P1 and C (protein 1, or GFP, and a dimensionless group of parameters) and plots the fixed points of the system at those values. <br />
</p></center><br />
<br><br />
<br />
<p> Where there are three fixed points the system will exhibit bistability. We also know that the further apart the two outer ‘prongs’ the more stable the two definitive fixed states will be, however this makes it more difficult to switch between states.</p><br />
<br />
<br><br><br />
<hr><br />
<table class="nav"><br />
<tr><br />
<td><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Fixed_Points"><img src="https://static.igem.org/mediawiki/2010/8/8e/Left_arrow.png">&nbsp;&nbsp;Return to Nullclines and Fixed Points</a><br />
</td><br />
<td align="right"><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Stochastic_Model">Continue to the Stochastic Model&nbsp;&nbsp;<img src="https://static.igem.org/mediawiki/2010/3/36/Right_arrow.png"></a><br />
</td><br />
</tr><br />
</table><br />
<br />
</html><br />
{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/BifurcationTeam:Aberdeen Scotland/Bifurcation2010-10-27T15:37:55Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<br />
<html><br />
<br />
<table><tr><td><h1>Bifurcation Analysis</h1></td></tr></table><br />
<br />
<table><br />
<tr><br />
<td><p><br />
The literal meaning of bifurcation is, “division into two parts, branches, or aspects” (Answers). In mathematics, this ‘division’ is in reference to when</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<p>a non-linear equation suddenly changes its behaviour qualitatively when a parameter of the system is varied. The point at which bifurcation occurs </p><br />
</td><br />
<tr><br />
</tr><br />
<td><br />
<p>is referred to as the ‘bifurcation point’.</p><br />
</td><br />
</tr><br />
</table><br />
<br />
</p><br />
<br><br />
<h3>what is bistability?</h3><br />
<p>If a system is bistable it can be stable in two distinct states. There are also stable oscillations, where the state of the system is stable but not at rest.</p><br />
<br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/d/da/Bifurcation1diagram.png"/><br />
<br />
<p style="font-size:10px"><b>Figure 1.</b>This is a graphical representation of bistability. Ball 2 can be equally<br>'happy' in either the 'one' or 'three' positions. Once it falls into one of these states,<br>ball 2 will not be inclined to leave this state, hence making it a "stable state".</p></center><br />
<br />
<br><br />
<br />
<h3>Why are they important?</h3><br />
<br />
<p>Analyzing the stability of a system is key to understand its behaviour. In biological terms, bistability represents the potential for our system to switch between the expression of green- and cyan-fluorescent proteins. Our goal is to analyze the parameter space in order to determine which parameters are needed to allow for bistability in the system.</p><br />
<br />
<br><br />
<p>A bifurcation diagram has branches which define the possible states a system can be in. By analyzing the bifurcation plots, we can determine how far apart the two “branches”, or stable states, are. The further apart the branches, the harder it will be to attain a switch between the two stable states, however the switch also becomes more stable with increased distance between the branches. Therefore, we want to manipulate the parameters in the system to guarantee robust bistability.</p><br />
<br><br />
<h3>How do you analyse the stability of a system?</h3><br />
<br />
<p>In order to analyze the stability of the system, we utilized XPP Auto, C programming, and MATLAB. We will attempt to detail the math behind what the program does computationally.<br />
<br><br />
<br><br />
The process involved with general stability analysis is as follows. We have taken a simple example system that is easy to analyze in order to demonstrate the general process:<br />
<br><br />
<br><br />
Take the following example system of two differential equations:</p><br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/5/59/Bistability_method1.png"/></center><br />
<br><br />
<br />
<h4><p>1. Set the differential equations to zero:</p></h4><br />
<br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/e/e0/Bistability_method2.png"/></center><br />
<br><br />
<br />
<h4><p>Find the fixed points of the system of differential equations</p></h4><br />
<p>There is a fixed point at x = y = 0. By observation, we can see that there are fixed points at x = 0, y = 2, and x = 3, y=0. Finally, by solving this system of equations, we see that there is also a fixed point at x=1 and y=1.<br />
</p><br />
<br><br />
<br />
<h4><p>3. Evaluate the Jacobian matrix of the system at these fixed points:</p></h4><br />
<br />
<p>If we label the top function f1 and the bottom f2, the Jacobian of the system will be of the form:</p><br />
<br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/6/6e/Bistability_method3.png"/></center><br />
<br><br />
<br />
<p>When we evaluate the Jacobian matrix for our example system of equations, we get:</p><br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/9/95/Bistability_method4.png"/></center><br />
<br />
<br><br />
<br />
<h4><p>4. Using the Jacobian matrix, find the corresponding eigenvalues:</p></h4><br />
<br />
<p><u>Definition</u>: An eigenvalue is a value λ such that Ax = λx, where x is said to be an eigenvector of A. In order to solve for λ, solve the following equation<br />
</p><br />
<br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/a/ac/Bistability_method5.png"/></center><br />
<br><br />
<br />
<p>where det is the determinant of the matrix and I is the identity matrix. This is known as the characteristic equation of a matrix.</p><br />
<br><br />
<br />
<p>In the case of our example system, the eigenvalues are:</p><br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/4/4f/Bistability_method6.png"/></center><br />
<br><br />
<br />
<h4><p>5. Take the signs of these eigenvalues in order to analyse stability:</p></h4><br />
<br />
<p>The eigenvalues of a system give important information about the stability of the system. The stability of each of the fixed points can be analyzed based on the following criteria:</p><br />
<br><br />
<br />
<center><table><br />
<tr><br />
<td><br />
Positive real and complex parts<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real and complex parts <br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Positive real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable spiral (spiraling out)<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable spiral (spiraling inwards)<br />
</td><br />
</tr><br />
</table></center><br />
<br />
<br><br />
<p>So, for our example system, it can be deduced that the stability at the fixed points is as follows:</p><br />
<br />
<br><br />
<center><br />
One positive and one negative eigenvalue at (0,0) means that <b>(0,0)</b> is a <b>saddle node</b><br />
<br><br />
Two negative eigenvalues at (0,2) means that <b>(0,2)</b> is a <b>stable node </b><br />
<br><br />
Two negative eigenvalues at (3,0) means that <b>(3,0)</b> is a <b>stable node </b><br />
<br><br />
One positive and one negative eigevalue at (1,1) means that <b>(1,1)</b> is a <b>saddle node</b></center><br />
<br><br />
<br />
<h3>The simulations involved</h3><br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/5/50/Bifurcation_diagram2.png"/><br />
<p style="font-size:10px"><b>Figure 2.</b> Figure 2: This is a bifurcation diagram that we generated using Maple. It ranges over the parameter space of <br />
<br><br />
P1 and C (protein 1, or GFP, and a dimensionless group of parameters) and plots the fixed points of the system at those values. <br />
</p></center><br />
<br><br />
<br />
<p> Where there are three fixed points the system will exhibit bistability. We also know that the further apart the two outer ‘prongs’ the more stable the two definitive fixed states will be, however this makes it more difficult to switch between states.</p><br />
<br />
<br><br><br />
<hr><br />
<table class="nav"><br />
<tr><br />
<td><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Fixed_Points"><img src="https://static.igem.org/mediawiki/2010/8/8e/Left_arrow.png">&nbsp;&nbsp;Return to Nullclines and Fixed Points</a><br />
</td><br />
<td align="right"><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Stochastic_Model">Continue to the Stochastic Model&nbsp;&nbsp;<img src="https://static.igem.org/mediawiki/2010/3/36/Right_arrow.png"></a><br />
</td><br />
</tr><br />
</table><br />
<br />
</html><br />
{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/BifurcationTeam:Aberdeen Scotland/Bifurcation2010-10-27T15:36:23Z<p>Brychan: </p>
<hr />
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{{:Team:Aberdeen_Scotland/Title}}<br />
<br />
<html><br />
<br />
<h1>Bifurcation Analysis</h1><br />
<br />
<table><br />
<tr><br />
<td><p><br />
The literal meaning of bifurcation is, “division into two parts, branches, or aspects” (Answers). In mathematics, this ‘division’ is in reference to when</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<p>a non-linear equation suddenly changes its behaviour qualitatively when a parameter of the system is varied. The point at which bifurcation occurs </p><br />
</td><br />
<tr><br />
</tr><br />
<td><br />
<p>is referred to as the ‘bifurcation point’.</p><br />
</td><br />
</tr><br />
</table><br />
<br />
</p><br />
<br><br />
<h3>what is bistability?</h3><br />
<p>If a system is bistable it can be stable in two distinct states. There are also stable oscillations, where the state of the system is stable but not at rest.</p><br />
<br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/d/da/Bifurcation1diagram.png"/><br />
<br />
<p style="font-size:10px"><b>Figure 1.</b>This is a graphical representation of bistability. Ball 2 can be equally<br>'happy' in either the 'one' or 'three' positions. Once it falls into one of these states,<br>ball 2 will not be inclined to leave this state, hence making it a "stable state".</p></center><br />
<br />
<br><br />
<br />
<h3>Why are they important?</h3><br />
<br />
<p>Analyzing the stability of a system is key to understand its behaviour. In biological terms, bistability represents the potential for our system to switch between the expression of green- and cyan-fluorescent proteins. Our goal is to analyze the parameter space in order to determine which parameters are needed to allow for bistability in the system.</p><br />
<br />
<br><br />
<p>A bifurcation diagram has branches which define the possible states a system can be in. By analyzing the bifurcation plots, we can determine how far apart the two “branches”, or stable states, are. The further apart the branches, the harder it will be to attain a switch between the two stable states, however the switch also becomes more stable with increased distance between the branches. Therefore, we want to manipulate the parameters in the system to guarantee robust bistability.</p><br />
<br><br />
<h3>How do you analyse the stability of a system?</h3><br />
<br />
<p>In order to analyze the stability of the system, we utilized XPP Auto, C programming, and MATLAB. We will attempt to detail the math behind what the program does computationally.<br />
<br><br />
<br><br />
The process involved with general stability analysis is as follows. We have taken a simple example system that is easy to analyze in order to demonstrate the general process:<br />
<br><br />
<br><br />
Take the following example system of two differential equations:</p><br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/5/59/Bistability_method1.png"/></center><br />
<br><br />
<br />
<h4><p>1. Set the differential equations to zero:</p></h4><br />
<br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/e/e0/Bistability_method2.png"/></center><br />
<br><br />
<br />
<h4><p>Find the fixed points of the system of differential equations</p></h4><br />
<p>There is a fixed point at x = y = 0. By observation, we can see that there are fixed points at x = 0, y = 2, and x = 3, y=0. Finally, by solving this system of equations, we see that there is also a fixed point at x=1 and y=1.<br />
</p><br />
<br><br />
<br />
<h4><p>3. Evaluate the Jacobian matrix of the system at these fixed points:</p></h4><br />
<br />
<p>If we label the top function f1 and the bottom f2, the Jacobian of the system will be of the form:</p><br />
<br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/6/6e/Bistability_method3.png"/></center><br />
<br><br />
<br />
<p>When we evaluate the Jacobian matrix for our example system of equations, we get:</p><br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/9/95/Bistability_method4.png"/></center><br />
<br />
<br><br />
<br />
<h4><p>4. Using the Jacobian matrix, find the corresponding eigenvalues:</p></h4><br />
<br />
<p><u>Definition</u>: An eigenvalue is a value λ such that Ax = λx, where x is said to be an eigenvector of A. In order to solve for λ, solve the following equation<br />
</p><br />
<br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/a/ac/Bistability_method5.png"/></center><br />
<br><br />
<br />
<p>where det is the determinant of the matrix and I is the identity matrix. This is known as the characteristic equation of a matrix.</p><br />
<br><br />
<br />
<p>In the case of our example system, the eigenvalues are:</p><br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/4/4f/Bistability_method6.png"/></center><br />
<br><br />
<br />
<h4><p>5. Take the signs of these eigenvalues in order to analyse stability:</p></h4><br />
<br />
<p>The eigenvalues of a system give important information about the stability of the system. The stability of each of the fixed points can be analyzed based on the following criteria:</p><br />
<br><br />
<br />
<center><table><br />
<tr><br />
<td><br />
Positive real and complex parts<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real and complex parts <br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Positive real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable spiral (spiraling out)<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable spiral (spiraling inwards)<br />
</td><br />
</tr><br />
</table></center><br />
<br />
<br><br />
<p>So, for our example system, it can be deduced that the stability at the fixed points is as follows:</p><br />
<br />
<br><br />
<center><br />
One positive and one negative eigenvalue at (0,0) means that <b>(0,0)</b> is a <b>saddle node</b><br />
<br><br />
Two negative eigenvalues at (0,2) means that <b>(0,2)</b> is a <b>stable node </b><br />
<br><br />
Two negative eigenvalues at (3,0) means that <b>(3,0)</b> is a <b>stable node </b><br />
<br><br />
One positive and one negative eigevalue at (1,1) means that <b>(1,1)</b> is a <b>saddle node</b></center><br />
<br><br />
<br />
<h3>The simulations involved</h3><br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/5/50/Bifurcation_diagram2.png"/><br />
<p style="font-size:10px"><b>Figure 2.</b> Figure 2: This is a bifurcation diagram that we generated using Maple. It ranges over the parameter space of <br />
<br><br />
P1 and C (protein 1, or GFP, and a dimensionless group of parameters) and plots the fixed points of the system at those values. <br />
</p></center><br />
<br><br />
<br />
<p> Where there are three fixed points the system will exhibit bistability. We also know that the further apart the two outer ‘prongs’ the more stable the two definitive fixed states will be, however this makes it more difficult to switch between states.</p><br />
<br />
<br><br><br />
<hr><br />
<table class="nav"><br />
<tr><br />
<td><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Fixed_Points"><img src="https://static.igem.org/mediawiki/2010/8/8e/Left_arrow.png">&nbsp;&nbsp;Return to Nullclines and Fixed Points</a><br />
</td><br />
<td align="right"><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Stochastic_Model">Continue to the Stochastic Model&nbsp;&nbsp;<img src="https://static.igem.org/mediawiki/2010/3/36/Right_arrow.png"></a><br />
</td><br />
</tr><br />
</table><br />
<br />
</html><br />
{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/BifurcationTeam:Aberdeen Scotland/Bifurcation2010-10-27T15:29:08Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<br />
<html><br />
<br />
<h1>Bifurcation Analysis</h1><br />
<br />
<p>The literal meaning of bifurcation is, “division into two parts, branches, or aspects” (Answers). In mathematics, this ‘division’ is in reference to when a non-linear equation suddenly changes its behaviour qualitatively when a parameter of the system is varied. The point at which bifurcation occurs is referred to as the ‘bifurcation point’.<br />
</p><br />
<br><br />
<h3>what is bistability?</h3><br />
<p>If a system is bistable it can be stable in two distinct states. There are also stable oscillations, where the state of the system is stable but not at rest.</p><br />
<br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/d/da/Bifurcation1diagram.png"/><br />
<br />
<p style="font-size:10px"><b>Figure 1.</b>This is a graphical representation of bistability. Ball 2 can be equally<br>'happy' in either the 'one' or 'three' positions. Once it falls into one of these states,<br>ball 2 will not be inclined to leave this state, hence making it a "stable state".</p></center><br />
<br />
<br><br />
<br />
<h3>Why are they important?</h3><br />
<br />
<p>Analyzing the stability of a system is key to understand its behaviour. In biological terms, bistability represents the potential for our system to switch between the expression of green- and cyan-fluorescent proteins. Our goal is to analyze the parameter space in order to determine which parameters are needed to allow for bistability in the system.</p><br />
<br />
<br><br />
<p>A bifurcation diagram has branches which define the possible states a system can be in. By analyzing the bifurcation plots, we can determine how far apart the two “branches”, or stable states, are. The further apart the branches, the harder it will be to attain a switch between the two stable states, however the switch also becomes more stable with increased distance between the branches. Therefore, we want to manipulate the parameters in the system to guarantee robust bistability.</p><br />
<br><br />
<h3>How do you analyse the stability of a system?</h3><br />
<br />
<p>In order to analyze the stability of the system, we utilized XPP Auto, C programming, and MATLAB. We will attempt to detail the math behind what the program does computationally.<br />
<br><br />
<br><br />
The process involved with general stability analysis is as follows. We have taken a simple example system that is easy to analyze in order to demonstrate the general process:<br />
<br><br />
<br><br />
Take the following example system of two differential equations:</p><br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/5/59/Bistability_method1.png"/></center><br />
<br><br />
<br />
<h4><p>1. Set the differential equations to zero:</p></h4><br />
<br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/e/e0/Bistability_method2.png"/></center><br />
<br><br />
<br />
<h4><p>Find the fixed points of the system of differential equations</p></h4><br />
<p>There is a fixed point at x = y = 0. By observation, we can see that there are fixed points at x = 0, y = 2, and x = 3, y=0. Finally, by solving this system of equations, we see that there is also a fixed point at x=1 and y=1.<br />
</p><br />
<br><br />
<br />
<h4><p>3. Evaluate the Jacobian matrix of the system at these fixed points:</p></h4><br />
<br />
<p>If we label the top function f1 and the bottom f2, the Jacobian of the system will be of the form:</p><br />
<br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/6/6e/Bistability_method3.png"/></center><br />
<br><br />
<br />
<p>When we evaluate the Jacobian matrix for our example system of equations, we get:</p><br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/9/95/Bistability_method4.png"/></center><br />
<br />
<br><br />
<br />
<h4><p>4. Using the Jacobian matrix, find the corresponding eigenvalues:</p></h4><br />
<br />
<p><u>Definition</u>: An eigenvalue is a value λ such that Ax = λx, where x is said to be an eigenvector of A. In order to solve for λ, solve the following equation<br />
</p><br />
<br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/a/ac/Bistability_method5.png"/></center><br />
<br><br />
<br />
<p>where det is the determinant of the matrix and I is the identity matrix. This is known as the characteristic equation of a matrix.</p><br />
<br><br />
<br />
<p>In the case of our example system, the eigenvalues are:</p><br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/4/4f/Bistability_method6.png"/></center><br />
<br><br />
<br />
<h4><p>5. Take the signs of these eigenvalues in order to analyse stability:</p></h4><br />
<br />
<p>The eigenvalues of a system give important information about the stability of the system. The stability of each of the fixed points can be analyzed based on the following criteria:</p><br />
<br><br />
<br />
<center><table><br />
<tr><br />
<td><br />
Positive real and complex parts<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real and complex parts <br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Positive real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable spiral (spiraling out)<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable spiral (spiraling inwards)<br />
</td><br />
</tr><br />
</table></center><br />
<br />
<br><br />
<p>So, for our example system, it can be deduced that the stability at the fixed points is as follows:</p><br />
<br />
<br><br />
<center><br />
One positive and one negative eigenvalue at (0,0) means that <b>(0,0)</b> is a <b>saddle node</b><br />
<br><br />
Two negative eigenvalues at (0,2) means that <b>(0,2)</b> is a <b>stable node </b><br />
<br><br />
Two negative eigenvalues at (3,0) means that <b>(3,0)</b> is a <b>stable node </b><br />
<br><br />
One positive and one negative eigevalue at (1,1) means that <b>(1,1)</b> is a <b>saddle node</b></center><br />
<br><br />
<br />
<h3>The simulations involved</h3><br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/5/50/Bifurcation_diagram2.png"/><br />
<p style="font-size:10px"><b>Figure 2.</b> Figure 2: This is a bifurcation diagram that we generated using Maple. It ranges over the parameter space of <br />
<br><br />
P1 and C (protein 1, or GFP, and a dimensionless group of parameters) and plots the fixed points of the system at those values. <br />
</p></center><br />
<br><br />
<br />
<p> Where there are three fixed points the system will exhibit bistability. We also know that the further apart the two outer ‘prongs’ the more stable the two definitive fixed states will be, however this makes it more difficult to switch between states.</p><br />
<br />
<br><br><br />
<hr><br />
<table class="nav"><br />
<tr><br />
<td><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Fixed_Points"><img src="https://static.igem.org/mediawiki/2010/8/8e/Left_arrow.png">&nbsp;&nbsp;Return to Nullclines and Fixed Points</a><br />
</td><br />
<td align="right"><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Stochastic_Model">Continue to the Stochastic Model&nbsp;&nbsp;<img src="https://static.igem.org/mediawiki/2010/3/36/Right_arrow.png"></a><br />
</td><br />
</tr><br />
</table><br />
<br />
</html><br />
{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/BifurcationTeam:Aberdeen Scotland/Bifurcation2010-10-27T15:27:39Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<br />
<html><br />
<br />
<h1>Bifurcation Analysis</h1><br />
<br />
<p>The literal meaning of bifurcation is, “division into two parts, branches, or aspects” (Answers). In mathematics, this ‘division’ is in reference to when a non-linear equation suddenly changes its behaviour qualitatively when a parameter of the system is varied. The point at which bifurcation occurs is referred to as the ‘bifurcation point’.<br />
</p><br />
<br><br />
<h3>what is bistability?</h3><br />
<p>If a system is bistable it can be stable in two distinct states. There are also stable oscillations, where the state of the system is stable but not at rest.</p><br />
<br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/d/da/Bifurcation1diagram.png"/><br />
<br />
<p style="font-size:10px"><b>Figure 1.</b>This is a graphical representation of bistability. Ball 2 can be equally<br>'happy' in either the 'one' or 'three' positions. Once it falls into one of these states,<br>ball 2 will not be inclined to leave this state, hence making it a "stable state".</p></center><br />
<br />
<br><br />
<br />
<h3>Why are they important?</h3><br />
<br />
<p>Analyzing the stability of a system is key to understand its behaviour. In biological terms, bistability represents the potential for our system to switch between the expression of green- and cyan-fluorescent proteins. Our goal is to analyze the parameter space in order to determine which parameters are needed to allow for bistability in the system.</p><br />
<br />
<br><br />
<p>A bifurcation diagram has branches which define the possible states a system can be in. By analyzing the bifurcation plots, we can determine how far apart the two “branches”, or stable states, are. The further apart the branches, the harder it will be to attain a switch between the two stable states, however the switch also becomes more stable with increased distance between the branches. Therefore, we want to manipulate the parameters in the system to guarantee robust bistability.</p><br />
<br><br />
<h3>How do you analyse the stability of a system?</h3><br />
<br />
<p>In order to analyze the stability of the system, we utilized XPP Auto, C programming, and MATLAB. We will attempt to detail the math behind what the program does computationally.<br />
<br><br />
<br><br />
The process involved with general stability analysis is as follows. We have taken a simple example system that is easy to analyze in order to demonstrate the general process:<br />
<br><br />
<br><br />
Take the following example system of two differential equations:</p><br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/5/59/Bistability_method1.png"/></center><br />
<br><br />
<br />
<h4><p>1. Set the differential equations to zero:</p></h4><br />
<br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/e/e0/Bistability_method2.png"/></center><br />
<br><br />
<br />
<h4><p>Find the fixed points of the system of differential equations</p></h4><br />
<p>There is a fixed point at x = y = 0. By observation, we can see that there are fixed points at x = 0, y = 2, and x = 3, y=0. Finally, by solving this system of equations, we see that there is also a fixed point at x=1 and y=1.<br />
</p><br />
<br><br />
<br />
<h4><p>3. Evaluate the Jacobian matrix of the system at these fixed points:</p></h4><br />
<br />
<p>If we label the top function f1 and the bottom f2, the Jacobian of the system will be of the form:</p><br />
<br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/6/6e/Bistability_method3.png"/></center><br />
<br><br />
<br />
<p>When we evaluate the Jacobian matrix for our example system of equations, we get:</p><br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/9/95/Bistability_method4.png"/></center><br />
<br />
<br><br />
<br />
<h4><p>4. Using the Jacobian matrix, find the corresponding eigenvalues:</p></h4><br />
<br />
<p><u>Definition</u>: An eigenvalue is a value λ such that Ax = λx, where x is said to be an eigenvector of A. In order to solve for λ, solve the following equation<br />
</p><br />
<br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/a/ac/Bistability_method5.png"/></center><br />
<br><br />
<br />
<p>where det is the determinant of the matrix and I is the identity matrix. This is known as the characteristic equation of a matrix.</p><br />
<br><br />
<br />
<p>In the case of our example system, the eigenvalues are:</p><br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/4/4f/Bistability_method6.png"/></center><br />
<br><br />
<br />
<h4><p>5. Take the signs of these eigenvalues in order to analyse stability:</p></h4><br />
<br />
<p>The eigenvalues of a system give important information about the stability of the system. The stability of each of the fixed points can be analyzed based on the following criteria:</p><br />
<br><br />
<br />
<center><table><br />
<tr><br />
<td><br />
Positive real and complex parts<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real and complex parts <br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Positive real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable spiral (spiraling out)<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable spiral (spiraling inwards)<br />
</td><br />
</tr><br />
</table></center><br />
<br />
<br><br />
<p>So, for our example system, it can be deduced that the stability at the fixed points is as follows:</p><br />
<br />
<br><br />
<center><br />
One positive and one negative eigenvalue at (0,0) means that <b>(0,0)</b> is a <b>saddle node</b><br />
<br><br />
Two negative eigenvalues at (0,2) means that <b>(0,2)</b> is a <b>stable node </b><br />
<br><br />
Two negative eigenvalues at (3,0) means that <b>(3,0)</b> is a <b>stable node </b><br />
<br><br />
One positive and one negative eigevalue at (1,1) means that <b>(1,1)</b> is a <b>saddle node</b></center><br />
<br><br />
<br />
<h3>The simulations involved</h3><br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/5/50/Bifurcation_diagram2.png"/><br />
<p style="font-size:10px"><b>Figure 2.</b> Figure 2: This is a bifurcation diagram that we generated using Maple. It ranges over the parameter space of P1 and C (protein 1, or GFP, and copper respectively) and plots the fixed points of the system at those values. <br />
</p></center><br />
<br><br />
<br />
<p> Where there are three fixed points the system will exhibit bistability. We also know that the further apart the two outer ‘prongs’ the more stable the two definitive fixed states will be, however this makes it more difficult to switch between states.</p><br />
<br />
<br><br><br />
<hr><br />
<table class="nav"><br />
<tr><br />
<td><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Fixed_Points"><img src="https://static.igem.org/mediawiki/2010/8/8e/Left_arrow.png">&nbsp;&nbsp;Return to Nullclines and Fixed Points</a><br />
</td><br />
<td align="right"><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Stochastic_Model">Continue to the Stochastic Model&nbsp;&nbsp;<img src="https://static.igem.org/mediawiki/2010/3/36/Right_arrow.png"></a><br />
</td><br />
</tr><br />
</table><br />
<br />
</html><br />
{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/BifurcationTeam:Aberdeen Scotland/Bifurcation2010-10-27T15:24:47Z<p>Brychan: </p>
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{{:Team:Aberdeen_Scotland/Title}}<br />
<br />
<html><br />
<br />
<h1>Bifurcation Analysis</h1><br />
<br />
<p>The literal meaning of bifurcation is, “division into two parts, branches, or aspects” (Answers). In mathematics, this ‘division’ is in reference to when a non-linear equation suddenly changes its behaviour qualitatively when a parameter of the system is varied. The point at which bifurcation occurs is referred to as the ‘bifurcation point’.<br />
</p><br />
<br><br />
<h3>what is bistability?</h3><br />
<p>If a system is bistable it can be stable in two distinct states. There are also stable oscillations, where the state of the system is stable but not at rest.</p><br />
<br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/d/da/Bifurcation1diagram.png"/><br />
<br />
<p style="font-size:10px"><b>Figure 1.</b>This is a graphical representation of bistability. Ball 2 can be equally<br>'happy' in either the 'one' or 'three' positions. Once it falls into one of these states,<br>ball 2 will not be inclined to leave this state, hence making it a "stable state".</p></center><br />
<br />
<br><br />
<br />
<h3>Why are they important?</h3><br />
<br />
<p>Analyzing the stability of a system is key to understand its behaviour. In biological terms, bistability represents the potential for our system to switch between the expression of green- and cyan-fluorescent proteins. Our goal is to analyze the parameter space in order to determine which parameters are needed to allow for bistability in the system.</p><br />
<br />
<br><br />
<p>A bifurcation diagram has branches which define the possible states a system can be in. By analyzing the bifurcation plots, we can determine how far apart the two “branches”, or stable states, are. The further apart the branches, the harder it will be to attain a switch between the two stable states, however the switch also becomes more stable with increased distance between the branches. Therefore, we want to manipulate the parameters in the system to guarantee robust bistability.</p><br />
<br><br />
<h3>How do you analyse the stability of a system?</h3><br />
<br />
<p>In order to analyze the stability of the system, we utilized XPP Auto, C programming, and MATLAB. We will attempt to detail the math behind what the program does computationally.<br />
<br><br />
<br><br />
The process involved with general stability analysis is as follows. We have taken a simple example system that is easy to analyze in order to demonstrate the general process:<br />
<br><br />
<br><br />
Take the following example system of two differential equations:</p><br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/5/59/Bistability_method1.png"/></center><br />
<br><br />
<br />
<h4><p>1. Set the differential equations to zero:</p></h4><br />
<br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/e/e0/Bistability_method2.png"/></center><br />
<br><br />
<br />
<h4><p>Find the fixed points of the system of differential equations</p></h4><br />
<p>There is a fixed point at x = y = 0. By observation, we can see that there are fixed points at x = 0, y = 2, and x = 3, y=0. Finally, by solving this system of equations, we see that there is also a fixed point at x=1 and y=1.<br />
</p><br />
<br><br />
<br />
<h4><p>3. Evaluate the Jacobian matrix of the system at these fixed points:</p></h4><br />
<br />
<p>If we label the top function f1 and the bottom f2, the Jacobian of the system will be of the form:</p><br />
<br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/6/6e/Bistability_method3.png"/></center><br />
<br><br />
<br />
<p>When we evaluate the Jacobian matrix for our example system of equations, we get:</p><br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/9/95/Bistability_method4.png"/></center><br />
<br />
<br><br />
<br />
<h4><p>4. Using the Jacobian matrix, find the corresponding eigenvalues:</p></h4><br />
<br />
<p><u>Definition</u>: An eigenvalue is a value λ such that Aλ=λx, where x is said to be an eigenvector of A. In order to solve for λ, solve the following equation:</p><br />
<br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/a/ac/Bistability_method5.png"/></center><br />
<br><br />
<br />
<p>where det is the determinant of the matrix and I is the identity matrix. This is known as the the characteristic equation of a matrix.</p><br />
<br><br />
<br />
<p>In the case of our example system, the eigenvalues are:</p><br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/4/4f/Bistability_method6.png"/></center><br />
<br><br />
<br />
<h4><p>5. Take the signs of these eigenvalues in order to analyse stability:</p></h4><br />
<br />
<p>The eigenvalues of a system give important information about the stability of the system. The stability of each of the fixed points can be analysed based on the following criteria:</p><br />
<br><br />
<br />
<center><table><br />
<tr><br />
<td><br />
Positive real and complex parts<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real and complex parts <br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Positive real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable spiral (spiraling out)<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable spiral (spiraling inwards)<br />
</td><br />
</tr><br />
</table></center><br />
<br />
<br><br />
<p>So, for our example system, it can be deduced that the stability at the fixed points is as follows:</p><br />
<br />
<br><br />
<center><br />
One positive and one negative eigenvalue at (0,0) means that <b>(0,0)</b> is a <b>saddle node</b><br />
<br><br />
Two negative eigenvalues at (0,2) means that <b>(0,2)</b> is a <b>stable node </b><br />
<br><br />
Two negative eigenvalues at (3,0) means that <b>(3,0)</b> is a <b>stable node </b><br />
<br><br />
One positive and one negative eigevalue at (1,1) means that <b>(1,1)</b> is a <b>saddle node</b></center><br />
<br><br />
<br />
<h3>The simulations involved</h3><br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/5/50/Bifurcation_diagram2.png"/><br />
<p style="font-size:10px"><b>Figure 2.</b> This is a bifurcation diagram that we generated using Maple. It ranges<br>over the parameter space of P1 and C (protein 1, or GFP, and a dimensionless<br>group of parameters) and plots the fixed points of the system at those values.</p></center><br />
<br><br />
<br />
<p>Where there are three fixed points for a given value of C, the system will exhibit bistability. We also know that the further apart the two outer 'prongs' are, the more stable the two definitive states will be. However, this makes it more difficult to switch between states.</p><br />
<br />
<br><br><br />
<hr><br />
<table class="nav"><br />
<tr><br />
<td><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Fixed_Points"><img src="https://static.igem.org/mediawiki/2010/8/8e/Left_arrow.png">&nbsp;&nbsp;Return to Nullclines and Fixed Points</a><br />
</td><br />
<td align="right"><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Stochastic_Model">Continue to the Stochastic Model&nbsp;&nbsp;<img src="https://static.igem.org/mediawiki/2010/3/36/Right_arrow.png"></a><br />
</td><br />
</tr><br />
</table><br />
<br />
</html><br />
{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/BifurcationTeam:Aberdeen Scotland/Bifurcation2010-10-27T15:20:48Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<br />
<html><br />
<br />
<h1>Bifurcation Analysis</h1><br />
<br />
<p>The literal meaning of bifurcation is, “division into two parts, branches, or aspects” (Answers). In mathematics, this ‘division’ is in reference to when a non-linear equation suddenly changes its behaviour qualitatively when a parameter of the system is varied. The point at which bifurcation occurs is referred to as the ‘bifurcation point’.<br />
</p><br />
<br><br />
<h3>what is bistability?</h3><br />
<p>If a system is bistable it can be stable in two distinct states. There are also stable oscillations, where the state of the system is stable but not at rest.</p><br />
<br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/d/da/Bifurcation1diagram.png"/><br />
<br />
<p style="font-size:10px"><b>Figure 1.</b>This is a graphical representation of bistability. Ball 2 can be equally<br>'happy' in either the 'one' or 'three' positions. Once it falls into one of these states,<br>ball 2 will not be inclined to leave this state, hence making it a "stable state".</p></center><br />
<br />
<br><br />
<br />
<h3>Why are they important?</h3><br />
<br />
<p>Analyzing the stability of a system is key to understand its behaviour. In biological terms, bistability represents the potential for our system to switch between the expression of green- and cyan-fluorescent proteins. Our goal is to analyze the parameter space in order to determine which parameters are needed to allow for bistability in the system.</p><br />
<br />
<br><br />
<p>A bifurcation diagram has branches which define the possible states a system can be in. By analyzing the bifurcation plots, we can determine how far apart the two “branches”, or stable states, are. The further apart the branches, the harder it will be to attain a switch between the two stable states, however the switch also becomes more stable with increased distance between the branches. Therefore, we want to manipulate the parameters in the system to guarantee robust bistability.</p><br />
<br><br />
<h3>How do you analyse the stability of a system?</h3><br />
<br />
<p>In order to analyze the stability of the system, we utilized XPP Auto, C programming, and MATLAB. We will attempt to detail the math behind what the program does computationally.<br />
<br><br />
<br><br />
The process involved with general stability analysis is as follows. We have taken a simple example system that is easy to analyze in order to demonstrate the general process:<br />
<br><br />
<br><br />
Take the following example system of two differential equations:</p><br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/5/59/Bistability_method1.png"/></center><br />
<br><br />
<br />
<h4><p>1. Set the differential equations to zero:</p></h4><br />
<br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/e/e0/Bistability_method2.png"/></center><br />
<br><br />
<br />
<h4><p>2. There is a fixed point at x = y = 0. By observation, we can see that there are fixed points at x = 0, y = 2, and x = 3, y=0. Finally, by solving this system of equations, we see that there is also a fixed point at x=1 and y=1.<br />
</p></h4><br />
<br />
<p>There is a fixed point at x=y=0. By observation, we can see that there are fixed points at x=0, y=2, and x=3, y=0. Finally, by solving this system of equations, we see that there is also a fixed point at x=1 and y=1.</p><br />
<br><br />
<br />
<h4><p>3. Evaluate the Jacobian matrix of the system at these fixed points:</p></h4><br />
<br />
<p>If we label the top function <i>f1</i> and the bottom <i>f2</i>, the Jacobian matrix of the system will be of the form:</p><br />
<br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/6/6e/Bistability_method3.png"/></center><br />
<br><br />
<br />
<p>When we evaluate the Jacobian matrix for our example system of equations, we get:</p><br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/9/95/Bistability_method4.png"/></center><br />
<br />
<br><br />
<br />
<h4><p>4. Using the Jacobian matrix, find the corresponding eigenvalues:</p></h4><br />
<br />
<p><u>Definition</u>: An eigenvalue is a value λ such that Aλ=λx, where x is said to be an eigenvector of A. In order to solve for λ, solve the following equation:</p><br />
<br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/a/ac/Bistability_method5.png"/></center><br />
<br><br />
<br />
<p>where det is the determinant of the matrix and I is the identity matrix. This is known as the the characteristic equation of a matrix.</p><br />
<br><br />
<br />
<p>In the case of our example system, the eigenvalues are:</p><br />
<br><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/4/4f/Bistability_method6.png"/></center><br />
<br><br />
<br />
<h4><p>5. Take the signs of these eigenvalues in order to analyse stability:</p></h4><br />
<br />
<p>The eigenvalues of a system give important information about the stability of the system. The stability of each of the fixed points can be analysed based on the following criteria:</p><br />
<br><br />
<br />
<center><table><br />
<tr><br />
<td><br />
Positive real and complex parts<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real and complex parts <br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Positive real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable spiral (spiraling out)<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable spiral (spiraling inwards)<br />
</td><br />
</tr><br />
</table></center><br />
<br />
<br><br />
<p>So, for our example system, it can be deduced that the stability at the fixed points is as follows:</p><br />
<br />
<br><br />
<center><br />
One positive and one negative eigenvalue at (0,0) means that <b>(0,0)</b> is a <b>saddle node</b><br />
<br><br />
Two negative eigenvalues at (0,2) means that <b>(0,2)</b> is a <b>stable node </b><br />
<br><br />
Two negative eigenvalues at (3,0) means that <b>(3,0)</b> is a <b>stable node </b><br />
<br><br />
One positive and one negative eigevalue at (1,1) means that <b>(1,1)</b> is a <b>saddle node</b></center><br />
<br><br />
<br />
<h3>The simulations involved</h3><br />
<br><br />
<center><img src="https://static.igem.org/mediawiki/2010/5/50/Bifurcation_diagram2.png"/><br />
<p style="font-size:10px"><b>Figure 2.</b> This is a bifurcation diagram that we generated using Maple. It ranges<br>over the parameter space of P1 and C (protein 1, or GFP, and a dimensionless<br>group of parameters) and plots the fixed points of the system at those values.</p></center><br />
<br><br />
<br />
<p>Where there are three fixed points for a given value of C, the system will exhibit bistability. We also know that the further apart the two outer 'prongs' are, the more stable the two definitive states will be. However, this makes it more difficult to switch between states.</p><br />
<br />
<br><br><br />
<hr><br />
<table class="nav"><br />
<tr><br />
<td><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Fixed_Points"><img src="https://static.igem.org/mediawiki/2010/8/8e/Left_arrow.png">&nbsp;&nbsp;Return to Nullclines and Fixed Points</a><br />
</td><br />
<td align="right"><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Stochastic_Model">Continue to the Stochastic Model&nbsp;&nbsp;<img src="https://static.igem.org/mediawiki/2010/3/36/Right_arrow.png"></a><br />
</td><br />
</tr><br />
</table><br />
<br />
</html><br />
{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/HeaderTeam:Aberdeen Scotland/Header2010-10-27T15:16:28Z<p>Brychan: </p>
<hr />
<div><html><br />
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<h3 class="maintitle">University of Aberdeen - ayeSwitch</h3><br />
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</html></div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/StabilityTeam:Aberdeen Scotland/Stability2010-10-26T13:23:26Z<p>Brychan: </p>
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{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/StabilityTeam:Aberdeen Scotland/Stability2010-10-26T11:28:27Z<p>Brychan: </p>
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<h1>Bistability Analysis</h1><br />
<p>The literal meaning of bifurcation is,m "division into two parts, branches, or aspects" (Answers). In mathematics, this 'division' is in reference to when a non-linear equation suddenly switches from having one solution to having multiple solutions when a parameter of the system is varied. The point at which bifurcation is referred to as the 'bifurcation point.' If a system is bistable it can be stable, or at rest in two distinct states.</p><br />
<br><br />
<center><br />
<p>bistability1.png</p><br />
<p><b>figure 1.</b> This is a graphical representation of bistability. Ball 2 can be equally 'happy' in either the 'one' or the 'three' positions. Once it falls into one of these states, ball 2 will not bin inclined to leave this state, hence making it a 'stable state.'</p><br />
</center><br />
<br><br />
<h3>Why are they important?</h3><br />
<p>Analysing the stability of a system is ket to the success of a project. In biological terms, bistability represetns the potential for our system to switch between the expression fo gree and cyan fluarescent proteins. Our goal is to analyse the parameter space in order to determione which paramenters andre needed to allow for bistability in the system.<br />
<br><br />
a bifurcation diagram has branches which define the possbile states a system can rest in. By analysing the bifurcation plots, we can determing how far apart the two 'branches,' or stable states, are. the further apart the branches, the harder it will be to attacin a switch between the two stable states, however the switch also becomes more stable with increased distance between the branchs. Therefore, we want to manipulate the parameters in the system to guarantee stable bistability.<br />
<br><br />
<h3>How do you analyse the stability of a system?</h3><br />
<p>In order to analyse the stability of the system, we utilised XPPAuto, C programming and MATLAB. We will attempt to detail the maths behind what the programme does computationally.<br />
<br><br />
The process involved with general stability is as follws. We have taken a simple example system that is easy to analyse in order to demonstrate the general process:<br />
<br><br />
Take the following example system of two differential equations:<br />
<br><br />
bistability2.png<br />
<b>1. Set the differential equations equal to zero:</b><br />
<br><br />
bistability3.png<br />
<br><br />
<b>2. Find the fixed points of the system of differential equations:</b><br />
There is a fixed point at x=y=0. By observation, we can see that there are fixed points at x=0, y=2, and x=3, y=0. Finally, by solving this system of equations, we see that there is also a fixed point at x=1 and y=1.<br />
<br><br />
<b>3. Evaluate the Jacobian matrix of the system at these fixed points:</b><br />
<br><br />
If we label the top fuction <i>f1</i> and the bottom <i>f2</i>, the Jacobian matrix of the system will be of the firm:<br />
<br><br />
bistability4.png<br />
<br><br />
When we evaluate the Jacobian matrix for our example system of equations, we get:<br />
<br><br />
bistability5.png<br />
<br><br />
<b>4. Using the Jacobian matrix, find the corresponding eigenvalues:</b><br />
<u>Definition</u>: an eigenvalue is a value λ such that Ax = λx, where x is said to be an eigenvector of A. In order to solve for λ, solve the following equation:<br />
<br><br />
bistability6.png<br />
<br><br />
Where det is the deterinant of the matrix and I is the identity matrix. THis is known as the characteristic equation of a matrix.<br />
<br><br />
In the case of our example system, the eigenvalues are:<br />
<br><br />
<center>bistability7.png</center><br />
<br><br />
<b>5. Take the signs of these eigenvalues in order to analyse the stability:</b><br />
<br><br />
The eigenvalues of a system give important information about the stability of the system. The stability of each of the fixed points can be analysed based on the following criteria:<br />
<center><br />
<table><br />
<tr><br />
<td><br />
Positive real and complex parts<br />
</td><br />
<td><br />
= unstable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real and complex parts<br />
</td><br />
<td><br />
= stable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Positive real part<br />
</td><br />
<td><br />
= unstable spiral (spiraling out)<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real part<br />
</td><br />
<td><br />
stable spiral (spiralling in)<br />
</td><br />
</tr><br />
</table><br />
</center><br />
<br><br />
So, for our example system, it can be deduced that the stability at the fixed points is as follows:<br />
<br><br />
<center><br />
<table><br />
<tr><br />
<td><br />
<left>One positive and one negative eigenvalue at (0,0) means that</left><br />
</td><br />
<td><br />
<b>(0,0)</b> is a <b>saddle node</b><br />
</td><br />
<td><br />
</tr><br />
<tr><br />
<td><br />
<left>Two negative eigenvalues at (0,2) means that</left><br />
</td><br />
<td><br />
<b>(0,2)</b> is a <b>stable node</b><br />
</td><br />
</tr><br />
<tr><br />
<td> <br />
<left>Two negative eigenvalues at (3,0) means that</left><br />
</td><br />
<td><br />
<b>(3,0)</b> is a <b>stable node</b><br />
</tr><br />
<tr><br />
<td><br />
<left>One positive and one negative eigenvalue at (1,1) means that</left><br />
</td><br />
<td><br />
<b>(1,1)</b> is a <b>saddle node</b><br />
</td><br />
</tr><br />
</table><br />
</center><br />
<br><br />
<h3>Simulations involved</h3><br />
<br><br />
<center><br />
bistability8.png<br />
<p><b>Figure 2.</b> This is a bifurcation diagram that we generated using Maple. It ranges over the parameter space of P1 (GFP) and C (a dimensionless group of parameters) and plots the fixed points o the system at those values.<br />
</center<br />
Where there are three fixed points for a given value of C1, the system will exhibit bistability. we also know that the further apart the two outer 'prongs' the more stable the two definitive fixed states will be, however thsi makes it more difficult to switch between states.<br />
</p><br />
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</td><br />
<td align="right"><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Probability">Continue to Parameter Space Analysis&nbsp;&nbsp;<img src="https://static.igem.org/mediawiki/2010/3/36/Right_arrow.png"></a><br />
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{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/StabilityTeam:Aberdeen Scotland/Stability2010-10-26T11:26:21Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<html><br />
<h1>Bistability Analysis</h1><br />
<p>The literal meaning of bifurcation is,m "division into two parts, branches, or aspects" (Answers). In mathematics, this 'division' is in reference to when a non-linear equation suddenly switches from having one solution to having multiple solutions when a parameter of the system is varied. The point at which bifurcation is referred to as the 'bifurcation point.' If a system is bistable it can be stable, or at rest in two distinct states.</p><br />
<br><br />
<center><br />
<p>bistability1.png</p><br />
<p><b>figure 1.</b> This is a graphical representation of bistability. Ball 2 can be equally 'happy' in either the 'one' or the 'three' positions. Once it falls into one of these states, ball 2 will not bin inclined to leave this state, hence making it a 'stable state.'</p><br />
</center><br />
<br><br />
<h3>Why are they important?</h3><br />
<p>Analysing the stability of a system is ket to the success of a project. In biological terms, bistability represetns the potential for our system to switch between the expression fo gree and cyan fluarescent proteins. Our goal is to analyse the parameter space in order to determione which paramenters andre needed to allow for bistability in the system.<br />
<br><br />
a bifurcation diagram has branches which define the possbile states a system can rest in. By analysing the bifurcation plots, we can determing how far apart the two 'branches,' or stable states, are. the further apart the branches, the harder it will be to attacin a switch between the two stable states, however the switch also becomes more stable with increased distance between the branchs. Therefore, we want to manipulate the parameters in the system to guarantee stable bistability.<br />
<br><br />
<h3>How do you analyse the stability of a system?</h3><br />
<p>In order to analyse the stability of the system, we utilised XPPAuto, C programming and MATLAB. We will attempt to detail the maths behind what the programme does computationally.<br />
<br><br />
The process involved with general stability is as follws. We have taken a simple example system that is easy to analyse in order to demonstrate the general process:<br />
<br><br />
Take the following example system of two differential equations:<br />
<br><br />
bistability2.png<br />
<b>1. Set the differential equations equal to zero:</b><br />
<br><br />
bistability3.png<br />
<br><br />
<b>2. Find the fixed points of the system of differential equations:</b><br />
There is a fixed point at x=y=0. By observation, we can see that there are fixed points at x=0, y=2, and x=3, y=0. Finally, by solving this system of equations, we see that there is also a fixed point at x=1 and y=1.<br />
<br><br />
<b>3. Evaluate the Jacobian matrix of the system at these fixed points:</b><br />
<br><br />
If we label the top fuction <i>f1</i> and the bottom <i>f2</i>, the Jacobian matrix of the system will be of the firm:<br />
<br><br />
bistability4.png<br />
<br><br />
When we evaluate the Jacobian matrix for our example system of equations, we get:<br />
<br><br />
bistability5.png<br />
<br><br />
<b>4. Using the Jacobian matrix, find the corresponding eigenvalues:</b><br />
<u>Definition</u>: an eigenvalue is a value λ such that Ax = λx, where x is said to be an eigenvector of A. In order to solve for λ, solve the following equation:<br />
<br><br />
bistability6.png<br />
<br><br />
Where det is the deterinant of the matrix and I is the identity matrix. THis is known as the characteristic equation of a matrix.<br />
<br><br />
In the case of our example system, the eigenvalues are:<br />
<br><br />
<center>bistability7.png</center><br />
<br><br />
<b>5. Take the signs of these eigenvalues in order to analyse the stability:</b><br />
<br><br />
The eigenvalues of a system give important information about the stability of the system. The stability of each of the fixed points can be analysed based on the following criteria:<br />
<center><br />
<table><br />
<tr><br />
<td><br />
Positive real and complex parts<br />
</td><br />
<td><br />
= unstable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real and complex parts<br />
</td><br />
<td><br />
= stable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Positive real part<br />
</td><br />
<td><br />
= unstable spiral (spiraling out)<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real part<br />
</td><br />
<td><br />
stable spiral (spiralling in)<br />
</td><br />
</tr><br />
</table><br />
</center><br />
<br><br />
So, for our ixample system, it can be deduced that the stability at the fixed poitns is as follows:<br />
<br><br />
<center><br />
<table><br />
<tr><br />
<td><br />
One positive and one negative eigenvalue at (0,0) means that<br />
</td><br />
<td><br />
<b>(0,0)</b> is a <b>saddle node</b><br />
</td><br />
<td><br />
</tr><br />
<tr><br />
<td><br />
Two negative eigenvalues at (0,2) means that<br />
</td><br />
<td><br />
<b>(0,2)</b> is a <b>stable node</b><br />
</td><br />
</tr><br />
<tr><br />
<td> <br />
Two negative eigenvalues at (3,0) means that<br />
</td><br />
<td><br />
<b>(3,0)</b> is a <b>stable node</b><br />
</tr><br />
<tr><br />
<td><br />
One positive and one negative eigenvalue at (1,1) means that<br />
</td><br />
<td><br />
<b>(1,1)</b> is a <b>saddle node</b><br />
</td><br />
</tr><br />
</table><br />
</center><br />
<br><br />
<h3>Simulations involved</h3><br />
<br><br />
<center><br />
bistability8.png<br />
<p><b>Figure 2.</b> This is a bifurcation diagram that we generated using Maple. It ranges over the parameter space of P1 (GFP) and C (a dimensionless group of parameters) and plots the fixed points o the system at those values.<br />
</center<br />
Where there are three fixed points for a given value of C1, the system will exhibit bistability. we also know that the further apart the two outer 'prongs' the more stable the two definitive fixed states will be, however thsi makes it more difficult to switch between states.<br />
</p><br />
<br />
<br><br><br />
<br />
<hr><br />
<table class="nav"><br />
<tr><br />
<td><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Stochastic_Model"><img src="https://static.igem.org/mediawiki/2010/8/8e/Left_arrow.png">&nbsp;&nbsp;Return to the Stochastic Model</a><br />
</td><br />
<td align="right"><br />
<a href="https://2010.igem.org/Team:Aberdeen_Scotland/Probability">Continue to Parameter Space Analysis&nbsp;&nbsp;<img src="https://static.igem.org/mediawiki/2010/3/36/Right_arrow.png"></a><br />
</td><br />
</tr><br />
</table><br />
<br />
</html><br />
{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/StabilityTeam:Aberdeen Scotland/Stability2010-10-26T11:25:30Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<html><br />
<h1>Bistability Analysis</h1><br />
<p>The literal meaning of bifurcation is,m "division into two parts, branches, or aspects" (Answers). In mathematics, this 'division' is in reference to when a non-linear equation suddenly switches from having one solution to having multiple solutions when a parameter of the system is varied. The point at which bifurcation is referred to as the 'bifurcation point.' If a system is bistable it can be stable, or at rest in two distinct states.</p><br />
<br><br />
<center><br />
<p>bistability1.png</p><br />
<p><b>figure 1.</b> This is a graphical representation of bistability. Ball 2 can be equally 'happy' in either the 'one' or the 'three' positions. Once it falls into one of these states, ball 2 will not bin inclined to leave this state, hence making it a 'stable state.'</p><br />
</center><br />
<br><br />
<h3>Why are they important?</h3><br />
<p>Analysing the stability of a system is ket to the success of a project. In biological terms, bistability represetns the potential for our system to switch between the expression fo gree and cyan fluarescent proteins. Our goal is to analyse the parameter space in order to determione which paramenters andre needed to allow for bistability in the system.<br />
<br><br />
a bifurcation diagram has branches which define the possbile states a system can rest in. By analysing the bifurcation plots, we can determing how far apart the two 'branches,' or stable states, are. the further apart the branches, the harder it will be to attacin a switch between the two stable states, however the switch also becomes more stable with increased distance between the branchs. Therefore, we want to manipulate the parameters in the system to guarantee stable bistability.<br />
<br><br />
<h3>How do you analyse the stability of a system?</h3><br />
<p>In order to analyse the stability of the system, we utilised XPPAuto, C programming and MATLAB. We will attempt to detail the maths behind what the programme does computationally.<br />
<br><br />
The process involved with general stability is as follws. We have taken a simple example system that is easy to analyse in order to demonstrate the general process:<br />
<br><br />
Take the following example system of two differential equations:<br />
<br><br />
bistability2.png<br />
<b>1. Set the differential equations equal to zero:</b><br />
<br><br />
bistability3.png<br />
<br><br />
<b>2. Find the fixed points of the system of differential equations:</b><br />
There is a fixed point at x=y=0. By observation, we can see that there are fixed points at x=0, y=2, and x=3, y=0. Finally, by solving this system of equations, we see that there is also a fixed point at x=1 and y=1.<br />
<br><br />
<b>3. Evaluate the Jacobian matrix of the system at these fixed points:</b><br />
<br><br />
If we label the top fuction <i>f1</i> and the bottom <i>f2</i>, the Jacobian matrix of the system will be of the firm:<br />
<br><br />
bistability4.png<br />
<br><br />
When we evaluate the Jacobian matrix for our example system of equations, we get:<br />
<br><br />
bistability5.png<br />
<br><br />
<b>4. Using the Jacobian matrix, find the corresponding eigenvalues:</b><br />
<u>Definition</u>: an eigenvalue is a value λ such that Ax = λx, where x is said to be an eigenvector of A. In order to solve for λ, solve the following equation:<br />
<br><br />
bistability6.png<br />
<br><br />
Where det is the deterinant of the matrix and I is the identity matrix. THis is known as the characteristic equation of a matrix.<br />
<br><br />
In the case of our example system, the eigenvalues are:<br />
<br><br />
<center>bistability7.png</center><br />
<br><br />
<b>5. Take the signs of these eigenvalues in order to analyse the stability:</b><br />
<br><br />
The eigenvalues of a system give important information about the stability of the system. The stability of each of the fixed points can be analysed based on the following criteria:<br />
<center><br />
<table><br />
<tr><br />
<td><br />
Positive real and complex parts<br />
</td><br />
<td><br />
= unstable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real and complex parts<br />
</td><br />
<td><br />
= stable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Positive real part<br />
</td><br />
<td><br />
= unstable spiral (spiraling out)<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real part<br />
</td><br />
<td><br />
stable spiral (spiralling in)<br />
</td><br />
</tr><br />
</table><br />
</center><br />
<br><br />
So, for our ixample system, it can be deduced that the stability at the fixed poitns is as follows:<br />
<br><br />
<center><br />
<table><br />
<tr><br />
<td><br />
One positive and one negative eigenvalue at (0,0) means that<br />
</td><br />
<td><br />
<b>(0,0)</b> is a <b>saddle node</b><br />
</td><br />
<td><br />
</tr><br />
<tr><br />
<td><br />
Two negative eigenvalues at (0,2) means that<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<b>(0,2)</b> is a <b>stable node</b><br />
</td><br />
</tr><br />
<tr><br />
<td> <br />
Two negative eigenvalues at (3,0) means that<br />
</td><br />
<td><br />
<b>(3,0)</b> is a <b>stable node</b><br />
</tr><br />
<tr><br />
<td><br />
One positive and one negative eigenvalue at (1,1) means that<br />
</td><br />
<td><br />
<b>(1,1)</b> is a <b>saddle node</b><br />
</td><br />
</tr><br />
</table><br />
</center><br />
<br><br />
<h3>Simulations involved</h3><br />
<br><br />
<center><br />
bistability8.png<br />
<p><b>Figure 2.</b> This is a bifurcation diagram that we generated using Maple. It ranges over the parameter space of P1 (GFP) and C (a dimensionless group of parameters) and plots the fixed points o the system at those values.<br />
</center<br />
Where there are three fixed points for a given value of C1, the system will exhibit bistability. we also know that the further apart the two outer 'prongs' the more stable the two definitive fixed states will be, however thsi makes it more difficult to switch between states.<br />
</p><br />
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{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/StabilityTeam:Aberdeen Scotland/Stability2010-10-26T11:24:51Z<p>Brychan: </p>
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{{:Team:Aberdeen_Scotland/Title}}<br />
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<h1>Bistability Analysis</h1><br />
<p>The literal meaning of bifurcation is,m "division into two parts, branches, or aspects" (Answers). In mathematics, this 'division' is in reference to when a non-linear equation suddenly switches from having one solution to having multiple solutions when a parameter of the system is varied. The point at which bifurcation is referred to as the 'bifurcation point.' If a system is bistable it can be stable, or at rest in two distinct states.</p><br />
<br><br />
<center><br />
<p>bistability1.png</p><br />
<p><b>figure 1.</b> This is a graphical representation of bistability. Ball 2 can be equally 'happy' in either the 'one' or the 'three' positions. ~Once it calls into one of these states, ball 2 will not bin inclined to leave this state, hence making it a 'stable state.'</p><br />
<br><br />
<h3>Why are they important?</h3><br />
<p>Analysing the stability of a system is ket to the success of a project. In biological terms, bistability represetns the potential for our system to switch between the expression fo gree and cyan fluarescent proteins. Our goal is to analyse the parameter space in order to determione which paramenters andre needed to allow for bistability in the system.<br />
<br><br />
a bifurcation diagram has branches which define the possbile states a system can rest in. By analysing the bifurcation plots, we can determing how far apart the two 'branches,' or stable states, are. the further apart the branches, the harder it will be to attacin a switch between the two stable states, however the switch also becomes more stable with increased distance between the branchs. Therefore, we want to manipulate the parameters in the system to guarantee stable bistability.<br />
<br><br />
<h3>How do you analyse the stability of a system?</h3><br />
<p>In order to analyse the stability of the system, we utilised XPPAuto, C programming and MATLAB. We will attempt to detail the maths behind what the programme does computationally.<br />
<br><br />
The process involved with general stability is as follws. We have taken a simple example system that is easy to analyse in order to demonstrate the general process:<br />
<br><br />
Take the following example system of two differential equations:<br />
<br><br />
bistability2.png<br />
<b>1. Set the differential equations equal to zero:</b><br />
<br><br />
bistability3.png<br />
<br><br />
<b>2. Find the fixed points of the system of differential equations:</b><br />
There is a fixed point at x=y=0. By observation, we can see that there are fixed points at x=0, y=2, and x=3, y=0. Finally, by solving this system of equations, we see that there is also a fixed point at x=1 and y=1.<br />
<br><br />
<b>3. Evaluate the Jacobian matrix of the system at these fixed points:</b><br />
<br><br />
If we label the top fuction <i>f1</i> and the bottom <i>f2</i>, the Jacobian matrix of the system will be of the firm:<br />
<br><br />
bistability4.png<br />
<br><br />
When we evaluate the Jacobian matrix for our example system of equations, we get:<br />
<br><br />
bistability5.png<br />
<br><br />
<b>4. Using the Jacobian matrix, find the corresponding eigenvalues:</b><br />
<u>Definition</u>: an eigenvalue is a value λ such that Ax = λx, where x is said to be an eigenvector of A. In order to solve for λ, solve the following equation:<br />
<br><br />
bistability6.png<br />
<br><br />
Where det is the deterinant of the matrix and I is the identity matrix. THis is known as the characteristic equation of a matrix.<br />
<br><br />
In the case of our example system, the eigenvalues are:<br />
<br><br />
<center>bistability7.png</center><br />
<br><br />
<b>5. Take the signs of these eigenvalues in order to analyse the stability:</b><br />
<br><br />
The eigenvalues of a system give important information about the stability of the system. The stability of each of the fixed points can be analysed based on the following criteria:<br />
<center><br />
<table><br />
<tr><br />
<td><br />
Positive real and complex parts<br />
</td><br />
<td><br />
= unstable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real and complex parts<br />
</td><br />
<td><br />
= stable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Positive real part<br />
</td><br />
<td><br />
= unstable spiral (spiraling out)<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real part<br />
</td><br />
<td><br />
stable spiral (spiralling in)<br />
</td><br />
</tr><br />
</table><br />
</center><br />
<br><br />
So, for our ixample system, it can be deduced that the stability at the fixed poitns is as follows:<br />
<br><br />
<center><br />
<table><br />
<tr><br />
<td><br />
One positive and one negative eigenvalue at (0,0) means that<br />
</td><br />
<td><br />
<b>(0,0)</b> is a <b>saddle node</b><br />
</td><br />
<td><br />
</tr><br />
<tr><br />
<td><br />
Two negative eigenvalues at (0,2) means that<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<b>(0,2)</b> is a <b>stable node</b><br />
</td><br />
</tr><br />
<tr><br />
<td> <br />
Two negative eigenvalues at (3,0) means that<br />
</td><br />
<td><br />
<b>(3,0)</b> is a <b>stable node</b><br />
</tr><br />
<tr><br />
<td><br />
One positive and one negative eigenvalue at (1,1) means that<br />
</td><br />
<td><br />
<b>(1,1)</b> is a <b>saddle node</b><br />
</td><br />
</tr><br />
</table><br />
</center><br />
<br><br />
<h3>Simulations involved</h3><br />
<br><br />
<center><br />
bistability8.png<br />
<p><b>Figure 2.</b> This is a bifurcation diagram that we generated using Maple. It ranges over the parameter space of P1 (GFP) and C (a dimensionless group of parameters) and plots the fixed points o the system at those values.<br />
</center<br />
Where there are three fixed points for a given value of C1, the system will exhibit bistability. we also know that the further apart the two outer 'prongs' the more stable the two definitive fixed states will be, however thsi makes it more difficult to switch between states.<br />
</p><br />
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{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/Fixed_PointsTeam:Aberdeen Scotland/Fixed Points2010-10-26T10:28:59Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<html><br />
<h1>Fixed Points</h1><br />
<p>Fixed points are the points where and equation's rate of change, or slope, is zero. there are three main types of equilibrium points: stable, unstable, and saddle-node points. A stable equilibrium is a value towards which the function converges, whereas an unstable equilibrium is a value away from which the function will diverge. A saddle-node is here the function both converges towards and diverges away from (Wikipedia).</p><br />
<br><br />
<h3>Why are fixed points important?</h3><br />
<p>fixed points are determined by setting all differential equations in a system equal to zero and solving for the variable being analysed. The numerical method for directly calculating fixed points is to find the roots of the system of equations using built-in root finding function such as <i>fzero</i> in MATLAB or similar functions in Maple and C. The indirect method is to plot and find the intersections of these nullclines, which represent fixed points.</p><br />
<br><br />
<br />
<h3>Results</h3><br />
<p>We were able to find the equilibrium points of the system analytically (for Hill coefficients) and computationally. These points were used for bifurcation analysis, and for analysing the probability that our system would exhibit istable behaviour.</p><br />
<br><br />
<br><br />
<br />
<h1>Nullclines</h1><br />
<p>In a system of differential equations, the nullclines are the solution curves for which all of the differential equations are equal to zero (Wikipedia).</p><br />
<br><br />
<h3>Why are nullclines important?</h3><br />
<p>the intersections of the cullclines gives the equilibrium points of the system of differential equations. From graphs of the sullclines, it is obvious whether or not a system will be bistable. If the nullclines only intersect in one place the sestem is not viable. If there are more than two interesctions, the middle equilibrium point is often an unstable saddle point.</p><br />
<br><br />
<h3>How are the nullclines calculated?</h3><br />
<p>Just as in calculating the equilibrium points, we set the governing differential equations of the system equal to zero and plot the curves generated.</p><br />
<br><br />
<h3>Simulations involved</h3><br />
<table><br />
<tr><br />
<td><br />
nullclines1.png<br />
</td><br />
<td><br />
<p><b>Figure 1.</b> This figure is a plot of how much galactose (in green) and methionine (in blue) must be added to produce GFP and CFP. In this plot, both Hill coefficients are equal to two. The nullclines cross over at three equilibrium points, where the middle is a saddle-node fixed point. This is an 'ideal' bistability plot.</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
nullclines2.png<br />
</td><br />
<td><br />
<p><b>Figure 2.</b> This figure again shows the required amounts of galactose and methionine to produce GFP and CFP respectively. In this plot, both Hill cefficients are equal to one. Note that the nullclines only cross once, resulting in only one fixed point and hence no possibility of bistability. This result allowed us to tell the biologists that the toggle switch would definitely <i>not</i> work if both of the Hill coefficients were equal to one.</p><br />
</td><br />
</tr><br />
<tr><br />
<td><br />
<p><b>Figure 3.</b> This figure again shows the concentrations of galactose and methionine to produce GFP and CFP. In this graph, we experimented with setting the initial values of galactose and methionine and seeing how the intersection points are offset. The blue line represents the GFP = CFP line. The further from this line the equilibrium points are, the harder it will be to switch between stable states.</p><br />
</td><br />
<td><br />
nullclines3.png<br />
</td><br />
</tr><br />
</table><br />
<br><br />
<h3>Results</h3><br />
<p>We focused our efforts on plotting the mullclines of the system for a range of Hill coefficient combinations in order to get a general idea of which combination would most likely produce stable bistability. We cound that all combinations gave bistability, except when both Hill coefficents were equal to one. the optimal Hill coefficient combination occurred when both Hill coefficients were equal to two. We informed the biology team of this, and that is we wanted our system to successfully switch, we could not have both Hill coefficents equal to one.</p><br />
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{{:Team:Aberdeen_Scotland/Footer}}</div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/HeaderTeam:Aberdeen Scotland/Header2010-10-23T15:06:49Z<p>Brychan: </p>
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</html></div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/Fixed_PointsTeam:Aberdeen Scotland/Fixed Points2010-10-23T15:01:22Z<p>Brychan: </p>
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{{:Team:Aberdeen_Scotland/Title}}<br />
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<h1>Fixed Points</h1><br />
<p>Fixed points are the points where and equation's rate of change, or slope, is zero. there are three main types of equilibrium points: stable, unstable, and saddle-node points. A stable equilibrium is a value towards which the function converges, whereas an unstable equilibrium is a value away from which the function will diverge. A saddle-node is here the function both converges towards and diverges away from (Wikipedia).</p><br />
<br><br />
<h3>Why are fixed points important?</h3><br />
<p>fixed points are determined by setting all differential equations in a system equal to zero and solving for the variable being analysed. The numerical method for directly calculating fixed points is to find the roots of the system of equations using built-in root finding function such as <i>fzero</i> in MATLAB or similar functions in Maple and C. The indirect method is to plot and find the intersections of these nullclines, which represent fixed points.</p><br />
<br><br />
<br />
<h3>Results</h3><br />
<p>We were able to find the equilibrium points of the system analytically (for Hill coefficients) and computationally. These points were used for bifurcation analysis, and for analysing the probability that our system would exhibit istable behaviour.</p></div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/Fixed_PointsTeam:Aberdeen Scotland/Fixed Points2010-10-23T15:01:02Z<p>Brychan: </p>
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<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
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<h1>Fixed Points</h1><br />
<p>Fixed points are the points where and equation's rate of change, or slope, is zero. there are three main types of equilibrium points: stable, unstable, and saddle-node points. A stable equilibrium is a value towards which the function converges, whereas an unstable equilibrium is a value away from which the function will diverge. A saddle-node is here the function both converges towards and diverges away from (Wikipedia).</p><br />
<br><br />
<h3>Why are fixed points important?</h3><br />
<p>fixed points are determined by setting all differential equations in a system equal to zero and solving for the variable being analysed. The numerical method for directly calculating fixed points is to find the roots of the system of equations using built-in root finding function such as <i>fzero</i> in MATLAB or similar functions in Maple and C. The indirect method is to plot and find the intersections of these nullclines, which represent fixed points.</p><br />
<br><br />
<br />
<h3>Results</h3><br />
<p>We were able to find the equilibrium points of the system analytically (for Hill coefficients) and computationsally. These points were used for bifurcation analysis, and for analysing the probability that our system would exhibit istable behaviour.</p></div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/Fixed_PointsTeam:Aberdeen Scotland/Fixed Points2010-10-23T15:00:19Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<html><br />
<h1>Fixed Points</h1><br />
<p>Fixed points are the points hwere and equations rate of change, or slope, is zero. there are three main types of equilibrium points: stable, unstable, and saddle-node points. A stable equilibrium is a value towards which the function converges, whereas an unstable equilibrium is a value away from which the function will diverge. A saddle-node is here the function both converges towards and diverges away from (Wikipedia).</p><br />
<br><br />
<h3>Why are fixed points important?</h3><br />
<p>fixed points are determined by setting all differential equations in a system equal to zero and solving for the variable being analysed. The numerical method for directly calculating fixed points is to find the roots of the system of equations using built-in root finding function such as <i>fzero</i> in MATLAB or similar functions in Maple and C. The indirect method is to plot and find the intersections of these nullclines, which represent fixed points.</p><br />
<br><br />
<br />
<h3>Results</h3><br />
<p>We were able to find the equilibrium points of the system analytically (for Hill coefficients) and computationsally. These points were used for bifurcation analysis, and for analysing the probability that our system would exhibit istable behaviour.</p></div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/Fixed_PointsTeam:Aberdeen Scotland/Fixed Points2010-10-23T14:59:59Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<html><br />
<h1>Fixed Points</h1><br />
<p>Fixed points are the points hwere and equations rate of change, or slope, is zero. there are three main types of equilibrium points: stable, unstable, and saddle-node points. A stable equilibrium is a value towards which the function converges, whereas an unstable equilibrium is a value away from which the function will diverge. A saddle-node is here the function both converges towards and diverges away from (Wikipedia).</p><br />
<br><br />
<h3>Why are fixed points important?</h3><br />
<p>fixed points are determined by setting all differential equations in a system equal to zero and solving for the variable being analysed. The numerical method for directly calculating fixed points is to find the roots of the system of equations using built-in root finding function such as <i>fzero</i> in MATLAB or similar functions in Maple and C. The indirect method is to plot and find the intersections of these nullclines, which represent fixed points.</p><br />
<br><br />
<br />
<h3>The results we found</h3><br />
<p>We were able to find the equilibrium points of the system analytically (for Hill coefficients) and computationsally. These points were used for bifurcation analysis, and for analysing the probability that our system would exhibit istable behaviour.</p></div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/Fixed_PointsTeam:Aberdeen Scotland/Fixed Points2010-10-23T14:46:19Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<html><br />
<h1>Fixed Points</h1><br />
<p>Fixed points are the points hwere and equations rate of change, or slope, is zer. there are three main types of equilibrium points: stable, unstable, and saddle-node points. A stable equilibrium is a value towards which the function converges, whereas an unstable equilibrium is a value away from which the function will diverge. A saddle-node is here the function both converges towards and diverges away from (Wikipedia).</p><br />
<br><br />
<h3>Why are fixed points important?</h3><br />
<p>fixed points are determined by setting all differential equations in a system equal to zero and solving for the variable being analysed. The numerical method for directly calculating fixed points is to find the roots of the system of equations using built-in root finding function such as <i>fzero</i> in MATLAB or similar functions in Maple and C. The indirect method is to plot and find the intersections of these nullclines, which represent fixed points.</p><br />
<br><br />
<br />
<h3>The results we found</h3><br />
<p>We were able to find the equilibrium points of the system analytically (for Hill coefficients) and computationsally. These points were used for bifurcation analysis, and for analysing the probability that our system would exhibit istable behaviour.</p></div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/Fixed_PointsTeam:Aberdeen Scotland/Fixed Points2010-10-23T14:45:58Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<html><br />
<h1>Fixed Points</h1><br />
<p>fixed points are the points hwere and equations rate of change, or slope, is zer. there are three main types of equilibrium points: stable, unstable, and saddle-node points. A stable equilibrium is a value towards which the function converges, whereas an unstable equilibrium is a value away from which the function will diverge. A saddle-node is here the function both converges towards and diverges away from (Wikipedia).</p><br />
<br><br />
<h3>Why are fixed points important?</h3><br />
<p>fixed points are determined by setting all differential equations in a system equal to zero and solving for the variable being analysed. The numerical method for directly calculating fixed points is to find the roots of the system of equations using built-in root finding function such as <i>fzero</i> in MATLAB or similar functions in Maple and C. The indirect method is to plot and find the intersections of these nullclines, which represent fixed points.</p><br />
<br><br />
<br />
<h3>The results we found</h3><br />
<p>We were able to find the equilibrium points of the system analytically (for Hill coefficients) and computationsally. These points were used for bifurcation analysis, and for analysing the probability that our system would exhibit istable behaviour.</p></div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/BifurcationTeam:Aberdeen Scotland/Bifurcation2010-10-23T14:28:32Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<html><br />
<h1>Bifurcation Analysis</h1><br />
<p>The literal meaning of bifurcation is, "division into two parts, branches or aspects" (Answers). In mathematics, this 'division' is reference to when non-linear equation suddenly switches from having one solution to having multiple solution when a parameter of the system is varied. The point at which bifurcation occurs is referred to as the 'bifurcation point.' If a system is bistable it can be stable, or at rest in two distinct states.</p><br />
<center><img src="https://static.igem.org/mediawiki/2010/d/da/Bifurcation1diagram.png"/><br />
<p><b>Figure 1.</b>This is a graphical representation of bistability. Ball 2 ban be equally 'happy' in either the 'one' or 'three' positions. Once it falls into one of these states, ball 2 will not be inclined to leave this state, hence making it a "stable state."</p></center><br />
<h3>Why are they important?</h3><br />
<p>Analysing the stability of a system is a key to the success of a project. In biological terms, bistability represents the potential for out system to switch between the expression of green and cyan fluorescent proteins. Our goal is to analyse the parameter space in order to determine which parameters are needed to allow for bistability in the system.<br />
<br><br />
<br><br />
a bifurcation diagram has branches which define the possible states a system can rest in. By analysing the bifurcation plots, we can determine how far apart the two "branches," or stable states, are. The further apart the branches, the harder it will be to attain a switch between the two stable states, however the switch also becomes more stable with increased distance between the branches. Therefore, we want to manipulate the parameters in the system to guarantee stable bistability.</p><br />
<h3>How do you analyse the stability of a system?</h3><br />
<p>In order to analyse the stability of the system, we utilised XPP Auto, C programming, and MATLAB. we will attempt to detail the math behind what the programme does computationally. The process involved with general stability analysis is as follows. We have takes a simple example system that is easy to analyse in order to demonstrate the general process:<br />
<br><br />
<br><br />
Take the following example system of two differential equations:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/5/59/Bistability_method1.png"/></center><br />
<br />
<h4><p>1. Set the differential equations to zero:</p></h4><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/e/e0/Bistability_method2.png"/></center><br />
<br />
<h4><p>2. Find the fixed points of the system of differential equations:</p></h4><br />
<br />
<p>There is a fixed point at x=y=0. By observation, we can see that there are fixed points at x=0, y=2, and x=3, y=0. Finally, by solving this system of equation, we see that there is also a fixed point at x=1 and y=1.</p><br />
<br />
<h4><p>3. Evaluate the Jacobian matrix of the system at these fixed points:</p></h4><br />
<br />
<p>If we label the top function <i>f1</i> and the bottom <i>f2</i>, the Jacobian matrix of the system will be of the form:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/6/6e/Bistability_method3.png"/></center><br />
<br />
<p>When we evaluate the Jacobian matrix for our example system of equations, we get:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/9/95/Bistability_method4.png"/></center><br />
<br />
<h4><p>4. Using the Jacobian matrix, find the corresponding eigenvalues:</p></h4><br />
<br />
<p><u>Definition</u>: An eigenvalue is a value λ such that Aλ=λx, where x is said to be an eigenvector of A. In order to solve for λ, solve the following equation:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/a/ac/Bistability_method5.png"/></center><br />
<br />
<p>where det is the determinant of the matrix and I is the identity matrix. This is known as the the characteristic equation of a matrix.</p><br />
<br><br />
<p>In the case of our example system, the eigenvalues are:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/4/4f/Bistability_method6.png"/></center><br />
<br />
<h4><p>5. Take the signs of these eigenvalues in order to analyse stability:</p></h4><br />
<br />
<p>The eigenvalues of a system give important information about the stability of the system. The stability of each of the fixed points can be analysed based on the following criteria:</p><br />
<br />
<center><table><br />
<tr><br />
<td><br />
Positive real and complex parts<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real and complex parts <br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Positive real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable spiral (spiraling out)<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable spiral (spiraling inwards)<br />
</td><br />
</tr><br />
</table></center><br />
<br><br />
<p>So, for our example system, it can be deduced that the stability at the fixed points is as follows:</p><br />
<br />
<br><br />
<center><br />
One positive and one negative eigenvalue at (0,0) means that <b>(0,0)</b> is a <b>saddle node</b><br />
<br><br />
Two negative eigenvalues at (0,2) means that <b>(0,2)</b> is a <b>stable node</b><br />
<br><br />
Two negative eigenvalues at (3,0) means that <b>(3,0)</b> is a <b>stable node</b><br />
<br><br />
One positive and one negative eigevalue at (1,1) means that <b>(1,1)</b> is a <b>saddle node</b></center><br />
<br><br />
<br />
<h3>The simulations involved</h3><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/5/50/Bifurcation_diagram2.png"/><br />
<p><b>Figure 2.</b> This is a bifurcation diagram that we generated using Maple. It ranges over the parameter space of P1 and C (protein 1, or GFP, and a dimensionsless group of parameters) and plots the fixed points of the sysem at those values.</center><br />
<br><br />
<br />
<p>Where there are three fixed points for a given value of C, the system will exhibit bistability. We also know that the further apart the two outer 'prongs' that more stable the two definitive states will be, howeve thsi makes it more difficult to switch between states.</p><br />
<br />
</html></div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/BifurcationTeam:Aberdeen Scotland/Bifurcation2010-10-23T14:25:59Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<html><br />
<h1>Bifurcation Analysis</h1><br />
<p>The literal meaning of bifurcation is, "division into two parts, branches or aspects" (Answers). In mathematics, this 'division' is reference to when non-linear equation suddenly switches from having one solution to having multiple solution when a parameter of the system is varied. The point at which bifurcation occurs is referred to as the 'bifurcation point.' If a system is bistable it can be stable, or at rest in two distinct states.</p><br />
<center><img src="https://static.igem.org/mediawiki/2010/d/da/Bifurcation1diagram.png"/><br />
<p><b>Figure 1.</b>This is a graphical representation of bistability. Ball 2 ban be equally 'happy' in either the 'one' or 'three' positions. Once it falls into one of these states, ball 2 will not be inclined to leave this state, hence making it a "stable state."</p></center><br />
<h3>Why are they important?</h3><br />
<p>Analysing the stability of a system is a key to the success of a project. In biological terms, bistability represents the potential for out system to switch between the expression of green and cyan fluorescent proteins. Our goal is to analyse the parameter space in order to determine which parameters are needed to allow for bistability in the system.<br />
<br><br />
<br><br />
a bifurcation diagram has branches which define the possible states a system can rest in. By analysing the bifurcation plots, we can determine how far apart the two "branches," or stable states, are. The further apart the branches, the harder it will be to attain a switch between the two stable states, however the switch also becomes more stable with increased distance between the branches. Therefore, we want to manipulate the parameters in the system to guarantee stable bistability.</p><br />
<h3>How do you analyse the stability of a system?</h3><br />
<p>In order to analyse the stability of the system, we utilised XPP Auto, C programming, and MATLAB. we will attempt to detail the math behind what the programme does computationally. The process involved with general stability analysis is as follows. We have takes a simple example system that is easy to analyse in order to demonstrate the general process:<br />
<br><br />
<br><br />
Take the following example system of two differential equations:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/5/59/Bistability_method1.png"/></center><br />
<br />
<h4><p>1. Set the differential equations to zero:</p></h4><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/e/e0/Bistability_method2.png"/></center><br />
<br />
<h4><p>2. Find the fixed points of the system of differential equations:</p></h4><br />
<br />
<p>There is a fixed point at x=y=0. By observation, we can see that there are fixed points at x=0, y=2, and x=3, y=0. Finally, by solving this system of equation, we see that there is also a fixed point at x=1 and y=1.</p><br />
<br />
<h4><p>3. Evaluate the Jacobian matrix of the system at these fixed points:</p></h4><br />
<br />
<p>If we label the top function <i>f1</i> and the bottom <i>f2</i>, the Jacobian matrix of the system will be of the form:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/6/6e/Bistability_method3.png"/></center><br />
<br />
<p>When we evaluate the Jacobian matrix for our example system of equations, we get:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/9/95/Bistability_method4.png"/></center><br />
<br />
<h4><p>4. Using the Jacobian matrix, find the corresponding eigenvalues:</p></h4><br />
<br />
<p><u>Definition</u>: An eigenvalue is a value λ such that Aλ=λx, where x is said to be an eigenvector of A. In order to solve for λ, solve the following equation:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/a/ac/Bistability_method5.png"/></center><br />
<br />
<p>where det is the determinant of the matrix and I is the identity matrix. This is known as the the characteristic equation of a matrix.</p><br />
<br><br />
<p>In the case of our example system, the eigenvalues are:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/4/4f/Bistability_method6.png"/></center><br />
<br />
<h4><p>5. Take the signs of these eigenvalues in order to analyse stability:</p></h4><br />
<br />
<p>The eigenvalues of a system give important information about the stability of the system. The stability of each of the fixed points can be analysed based on the following criteria:</p><br />
<br />
<center><table><br />
<tr><br />
<td><br />
Positive real and complex parts<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real and complex parts <br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Positive real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable spiral (spiraling out)<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable spiral (spiraling inwards)<br />
</td><br />
</tr><br />
</table></center><br />
<br><br />
<p>So, for our example system, it can be deduced that the stability at the fixed points is as follows:</p><br />
<br />
<br><br />
<center><br />
One positive and one negative eigenvalue at (0,0) means that <b>(0,0)</b> is a <b>saddle node</b><br />
<br><br />
Two negative eigenvalues at (0,2) means that <b>(0,2)</b> is a <b>stable node</b><br />
<br><br />
Two negative eigenvalues at (3,0) means that <b>(3,0)</b> is a <b>stable node</b><br />
<br><br />
One positive and one negative eigevalue at (1,1) means that <b>(1,1)</b> is a <b>saddle node</b></center><br />
<br><br />
<br />
<h3>The simulations involved</h3><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/5/50/Bifurcation_diagram2.png"/><br />
<p><b>Figure 2.</b> This is a bifurcation diagram that we generated using Maple. It ranges over the parameter space of P1 and C (protein 1, or GFP, and a dimensionsless group of parameters) and plots the fixed points of hte sysem at those values.</center><br />
<br><br />
<br />
<p>Where there are three fixed points for a given value of C, the system will exhibit bistability. We also know that the further apart the two outer 'prongs' that mre stable the two definitive states will be, howeve thsi makes it more difficult to switch between states.</p><br />
<br />
</html></div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/BifurcationTeam:Aberdeen Scotland/Bifurcation2010-10-23T14:25:30Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<html><br />
<h1>Bifurcation Analysis</h1><br />
<p>The literal meaning of bifurcation is, "division into two parts, branches or aspects" (Answers). In mathematics, this 'division' is reference to when non-linear equation suddenly switches from having one solution to having multiple solution when a parameter of the system is varied. The point at which bifurcation occurs is referred to as the 'bifurcation point.' If a system is bistable it can be stable, or at rest in two distinct states.</p><br />
<center><img src="https://static.igem.org/mediawiki/2010/d/da/Bifurcation1diagram.png"/><br />
<p><b>Figure 1.</b>This is a graphical representation of bistability. Ball 2 ban be equally 'happy' in either the 'one' or 'three' positions. Once it falls into one of these states, ball 2 will not be inclined to leave this state, hence making it a "stable state."</p></center><br />
<h3>Why are they important?</h3><br />
<p>Analysing the stability of a system is a key to the success of a project. In biological terms, bistability represents the potential for out system to switch between the expression of green and cyan fluorescent proteins. Our goal is to analyse the parameter space in order to determine which parameters are needed to allow for bistability in the system.<br />
<br><br />
<br><br />
a bifurcation diagram has branches which define the possible states a system can rest in. By analysing the bifurcation plots, we can determine how far apart the two "branches," or stable states, are. The further apart the branches, the harder it will be to attain a switch between the two stable states, however the switch also becomes more stable with increased distance between the branches. Therefore, we want to manipulate the parameters in the system to guarantee stable bistability.</p><br />
<h3>How do you analyse the stability of a system?</h3><br />
<p>In order to analyse the stability of the system, we utilised XPP Auto, C programming, and MATLAB. we will attempt to detail the math behind what the programme does computationally. The process involved with general stability analysis is as follows. We have takes a simple example system that is easy to analyse in order to demonstrate the general process:<br />
<br><br />
<br><br />
Take the following example system of two differential equations:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/5/59/Bistability_method1.png"/></center><br />
<br />
<h4><p>1. Set the differential equations to zero:</p></h4><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/e/e0/Bistability_method2.png"/></center><br />
<br />
<h4><p>2. Find the fixed points of the system of differential equations:</p></h4><br />
<br />
<p>There is a fixed point at x=y=0. By observation, we can see that there are fixed points at x=0, y=2, and x=3, y=0. Finally, by solving this system of equation, we see that there is also a fixed point at x=1 and y=1.</p><br />
<br />
<h4><p>3. Evaluate the Jacobian matrix of the system at these fixed points:</p></h4><br />
<br />
<p>If we label the top function <i>f1</i> and the bottom <i>f2</i>, the Jacobian matrix of the system will be of the form:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/6/6e/Bistability_method3.png"/></center><br />
<br />
<p>When we evaluate the Jacobian matrix for our example system of equations, we get:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/9/95/Bistability_method4.png"/></center><br />
<br />
<h4><p>4. Using the Jacobian matrix, find the corresponding eigenvalues:</p></h4><br />
<br />
<p><u>Definition</u>: An eigenvalue is a value λ such that Aλ=λx, where x is said to be an eigenvector of A. In order to solve for λ, solve the following equation:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/a/ac/Bistability_method5.png"/></center><br />
<br />
<p>where det is the determinant of the matrix and I is the identity matrix. This is known as the the characteristic equation of a matrix.</p><br />
<br><br />
<p>In the case of our example system, the eigenvalues are:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/4/4f/Bistability_method6.png"/></center><br />
<br />
<h4><p>5. Take the signs of these eigenvalues in order to analyse stability:</p></h4><br />
<br />
<p>The eigenvalues of a system give important information about the stability of the system. The stability of each of the fixed points can be analysed based on the following criteria:</p><br />
<br />
<center><table><br />
<tr><br />
<td><br />
Positive real and complex parts<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real and complex parts <br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Positive real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable spiral (spiraling out)<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable spiral (spiraling inwards)<br />
</td><br />
</tr><br />
</table></center><br />
<br><br />
<p>So, for our example system, it can be deduced that the stability at the fixed points is as follows:</p><br />
<br />
<br><br />
<center><br />
One positive and one negative eigenvalue at (0,0) means that <b>(0,0)</b> is a <b>saddle node</b><br />
<br><br />
Two negative eigenvalues at (0,2) means that <b>(0,2)</b> is a <b>stable node</b><br />
<br><br />
Two negative eigenvalues at (3,0) means that <b>(3,0)</b> is a <b>stable node</b><br />
<br><br />
One positive and one negative eigevalue at (1,1) means that <b>(1,1)</b> is a <b>saddle node</b></center><br />
<br><br />
<br />
<h3>The simulations involved</h3><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/5/50/Bifurcation_diagram2.png"/><br />
<p><b>Figure 2.</b> This is a bifurcation diagram that we generated using Maple. It ranges over the parameter space of P1 and C (protein 1, or GFP, and a dimensionsless group of parameters) and plots the fixed points of hte sysem at those values.</center><br />
<br><br />
<br />
Where there are three fixed points for a given value of C, the system will exhibit bistability. We also know that the further apart the two outer 'prongs' that mre stable the two definitive states will be, howeve thsi makes it more difficult to switch between states.</p><br />
<br />
</html></div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/BifurcationTeam:Aberdeen Scotland/Bifurcation2010-10-23T14:22:56Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<html><br />
<h1>Bifurcation Analysis</h1><br />
<p>The literal meaning of bifurcation is, "division into two parts, branches or aspects" (Answers). In mathematics, this 'division' is reference to when non-linear equation suddenly switches from having one solution to having multiple solution when a parameter of the system is varied. The point at which bifurcation occurs is referred to as the 'bifurcation point.' If a system is bistable it can be stable, or at rest in two distinct states.</p><br />
<center><img src="https://static.igem.org/mediawiki/2010/d/da/Bifurcation1diagram.png"/><br />
<p><b>Figure 1.</b>This is a graphical representation of bistability. Ball 2 ban be equally 'happy' in either the 'one' or 'three' positions. Once it falls into one of these states, ball 2 will not be inclined to leave this state, hence making it a "stable state."</p></center><br />
<h3>Why are they important?</h3><br />
<p>Analysing the stability of a system is a key to the success of a project. In biological terms, bistability represents the potential for out system to switch between the expression of green and cyan fluorescent proteins. Our goal is to analyse the parameter space in order to determine which parameters are needed to allow for bistability in the system.<br />
<br><br />
<br><br />
a bifurcation diagram has branches which define the possible states a system can rest in. By analysing the bifurcation plots, we can determine how far apart the two "branches," or stable states, are. The further apart the branches, the harder it will be to attain a switch between the two stable states, however the switch also becomes more stable with increased distance between the branches. Therefore, we want to manipulate the parameters in the system to guarantee stable bistability.</p><br />
<h3>How do you analyse the stability of a system?</h3><br />
<p>In order to analyse the stability of the system, we utilised XPP Auto, C programming, and MATLAB. we will attempt to detail the math behind what the programme does computationally. The process involved with general stability analysis is as follows. We have takes a simple example system that is easy to analyse in order to demonstrate the general process:<br />
<br><br />
<br><br />
Take the following example system of two differential equations:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/5/59/Bistability_method1.png"/></center><br />
<br />
<h4><p>1. Set the differential equations to zero:</p></h4><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/e/e0/Bistability_method2.png"/></center><br />
<br />
<h4><p>2. Find the fixed points of the system of differential equations:</p></h4><br />
<br />
<p>There is a fixed point at x=y=0. By observation, we can see that there are fixed points at x=0, y=2, and x=3, y=0. Finally, by solving this system of equation, we see that there is also a fixed point at x=1 and y=1.</p><br />
<br />
<h4><p>3. Evaluate the Jacobian matrix of the system at these fixed points:</p></h4><br />
<br />
<p>If we label the top function <i>f1</i> and the bottom <i>f2</i>, the Jacobian matrix of the system will be of the form:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/6/6e/Bistability_method3.png"/></center><br />
<br />
<p>When we evaluate the Jacobian matrix for our example system of equations, we get:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/9/95/Bistability_method4.png"/></center><br />
<br />
<h4><p>4. Using the Jacobian matrix, find the corresponding eigenvalues:</p></h4><br />
<br />
<p><u>Definition</u>: An eigenvalue is a value λ such that Aλ=λx, where x is said to be an eigenvector of A. In order to solve for λ, solve the following equation:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/a/ac/Bistability_method5.png"/></center><br />
<br />
<p>where det is the determinant of the matrix and I is the identity matrix. This is known as the the characteristic equation of a matrix.</p><br />
<br><br />
<p>In the case of our example system, the eigenvalues are:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/4/4f/Bistability_method6.png"/></center><br />
<br />
<h4><p>5. Take the signs of these eigenvalues in order to analyse stability:</p></h4><br />
<br />
<p>The eigenvalues of a system give important information about the stability of the system. The stability of each of the fixed points can be analysed based on the following criteria:</p><br />
<br />
<center><table><br />
<tr><br />
<td><br />
Positive real and complex parts<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real and complex parts <br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Positive real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable spiral (spiraling out)<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable spiral (spiraling inwards)<br />
</td><br />
</tr><br />
</table></center><br />
<br><br />
<p>So, for our example system, it can be deduced that the stability at the fixed points is as follows:</p><br />
<br />
<br><br />
<center><br />
One positive and one negative eigenvalue at (0,0) means that <b>(0,0)</b> is a <b>saddle node</b><br />
<br><br />
Two negative eigenvalues at (0,2) means that <b>(0,2)</b> is a <b>stable node</b><br />
<br><br />
Two negative eigenvalues at (3,0) means that <b>(3,0)</b> is a <b>stable node</b><br />
<br><br />
One positive and one negative eigevalue at (1,1) means that <b>(1,1)</b> is a <b>saddle node</b></center><br />
<br><br />
<h3>The simulations involved<h3><br />
<center><img src="https://static.igem.org/mediawiki/2010/5/50/Bifurcation_diagram2.png"/><br />
<p><b>Figure 2.</b> This is a bifurcation diagram that we generated using Maple. It ranges over the parameter space of P1 and C (protein 1, or GFP, and a dimensionsless group of parameters) and plots the fixed points of hte sysem at those values.</center><br />
<br><br />
where thate are three fixed points for a given value of C, the system will exhibit bistability. We also know that the further apart the two outer 'prongs' that mre stable the two definitive states will be, howeve thsi makes it more difficult to switch between states.</p><br />
<br />
</html></div>Brychanhttp://2010.igem.org/Team:Aberdeen_Scotland/BifurcationTeam:Aberdeen Scotland/Bifurcation2010-10-23T14:22:24Z<p>Brychan: </p>
<hr />
<div>{{:Team:Aberdeen_Scotland/css}}<br />
{{:Team:Aberdeen_Scotland/Title}}<br />
<html><br />
<h1>Bifurcation Analysis</h1><br />
<p>The literal meaning of bifurcation is, "division into two parts, branches or aspects" (Answers). In mathematics, this 'division' is reference to when non-linear equation suddenly switches from having one solution to having multiple solution when a parameter of the system is varied. The point at which bifurcation occurs is referred to as the 'bifurcation point.' If a system is bistable it can be stable, or at rest in two distinct states.</p><br />
<center><img src="https://static.igem.org/mediawiki/2010/d/da/Bifurcation1diagram.png"/><br />
<p><b>Figure 1.</b>This is a graphical representation of bistability. Ball 2 ban be equally 'happy' in either the 'one' or 'three' positions. Once it falls into one of these states, ball 2 will not be inclined to leave this state, hence making it a "stable state."</p></center><br />
<h3>Why are they important?</h3><br />
<p>Analysing the stability of a system is a key to the success of a project. In biological terms, bistability represents the potential for out system to switch between the expression of green and cyan fluorescent proteins. Our goal is to analyse the parameter space in order to determine which parameters are needed to allow for bistability in the system.<br />
<br><br />
<br><br />
a bifurcation diagram has branches which define the possible states a system can rest in. By analysing the bifurcation plots, we can determine how far apart the two "branches," or stable states, are. The further apart the branches, the harder it will be to attain a switch between the two stable states, however the switch also becomes more stable with increased distance between the branches. Therefore, we want to manipulate the parameters in the system to guarantee stable bistability.</p><br />
<h3>How do you analyse the stability of a system?</h3><br />
<p>In order to analyse the stability of the system, we utilised XPP Auto, C programming, and MATLAB. we will attempt to detail the math behind what the programme does computationally. The process involved with general stability analysis is as follows. We have takes a simple example system that is easy to analyse in order to demonstrate the general process:<br />
<br><br />
<br><br />
Take the following example system of two differential equations:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/5/59/Bistability_method1.png"/></center><br />
<br />
<h4><p>1. Set the differential equations to zero:</p></h4><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/e/e0/Bistability_method2.png"/></center><br />
<br />
<h4><p>2. find the fixed points of the system of differential equations:</p></h4><br />
<br />
<p>There is a fixed point at x=y=0. By observation, we can see that there are fixed points at x=0, y=2, and x=3, y=0. Finally, by solving this system of equation, we see that there is also a fixed point at x=1 and y=1.</p><br />
<br />
<h4><p>3. Evaluate the Jacobian matrix of the system at these fixed points:</p></h4><br />
<br />
<p>If we label the top function <i>f1</i> and the bottom <i>f2</i>, the Jacobian matrix of the system will be of the form:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/6/6e/Bistability_method3.png"/></center><br />
<br />
<p>When we evaluate the Jacobian matrix for our example system of equations, we get:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/9/95/Bistability_method4.png"/></center><br />
<br />
<h4><p>4. Using the Jacobian matrix, find the corresponding eigenvalues:</p></h4><br />
<br />
<p><u>Definition</u>: An eigenvalue is a value λ such that Aλ=λx, where x is said to be an eigenvector of A. In order to solve for λ, solve the following equation:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/a/ac/Bistability_method5.png"/></center><br />
<br />
<p>where det is the determinant of the matrix and I is the identity matrix. This is known as the the characteristic equation of a matrix.</p><br />
<br><br />
<p>In the case of our example system, the eigenvalues are:</p><br />
<br />
<center><img src="https://static.igem.org/mediawiki/2010/4/4f/Bistability_method6.png"/></center><br />
<br />
<h4><p>5. Take the signs of these eigenvalues in order to analyse stability:</p></h4><br />
<br />
<p>The eigenvalues of a system give important information about the stability of the system. The stability of each of the fixed points can be analysed based on the following criteria:</p><br />
<br />
<center><table><br />
<tr><br />
<td><br />
Positive real and complex parts<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real and complex parts <br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Positive real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Unstable spiral (spiraling out)<br />
</td><br />
</tr><br />
<tr><br />
<td><br />
Negative real part<br />
</td><br />
<td><br />
:<br />
</td><br />
<td><br />
Stable spiral (spiraling inwards)<br />
</td><br />
</tr><br />
</table></center><br />
<br><br />
<p>So, for our example system, it can be deduced that the stability at the fixed points is as follows:</p><br />
<br />
<br><br />
<center><br />
One positive and one negative eigenvalue at (0,0) means that <b>(0,0)</b> is a <b>saddle node</b><br />
<br><br />
Two negative eigenvalues at (0,2) means that <b>(0,2)</b> is a <b>stable node</b><br />
<br><br />
Two negative eigenvalues at (3,0) means that <b>(3,0)</b> is a <b>stable node</b><br />
<br><br />
One positive and one negative eigevalue at (1,1) means that <b>(1,1)</b> is a <b>saddle node</b></center><br />
<br><br />
<h3>The simulations involved<h3><br />
<center><img src="https://static.igem.org/mediawiki/2010/5/50/Bifurcation_diagram2.png"/><br />
<p><b>Figure 2.</b> This is a bifurcation diagram that we generated using Maple. It ranges over the parameter space of P1 and C (protein 1, or GFP, and a dimensionsless group of parameters) and plots the fixed points of hte sysem at those values.</center><br />
<br><br />
where thate are three fixed points for a given value of C, the system will exhibit bistability. We also know that the further apart the two outer 'prongs' that mre stable the two definitive states will be, howeve thsi makes it more difficult to switch between states.</p><br />
<br />
</html></div>Brychanhttp://2010.igem.org/File:Bifurcation_diagram2.pngFile:Bifurcation diagram2.png2010-10-23T14:15:14Z<p>Brychan: </p>
<hr />
<div></div>Brychan