Team:uOttawa/drylab

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                 <p>In a biological context, stable states refer to states of stable gene expression levels. This means that expression levels tend to progress until it reaches a certain stable expression level, depending on the initial conditions. These states also are resistant to modifications of expression by changing regulatory proteins, loss of genetic product, etc. In a modeling context, it refers to the trajectory of the system (or group) of equations to converge to a steady or stable state. Mathematically speaking, stable states occur at intersections of nullclines, but simply, they attract the trajectory of the system of equations. To analyze model stability, one can use phase plane and bifurcation analyses. Phase plane analyses are used to analyze the dynamics of the stability of the system of equations, at a given set of parameters, and bifurcation analyses are used to analyze the change in stability with regards to modifications of parameters.</p>
                 <p>In a biological context, stable states refer to states of stable gene expression levels. This means that expression levels tend to progress until it reaches a certain stable expression level, depending on the initial conditions. These states also are resistant to modifications of expression by changing regulatory proteins, loss of genetic product, etc. In a modeling context, it refers to the trajectory of the system (or group) of equations to converge to a steady or stable state. Mathematically speaking, stable states occur at intersections of nullclines, but simply, they attract the trajectory of the system of equations. To analyze model stability, one can use phase plane and bifurcation analyses. Phase plane analyses are used to analyze the dynamics of the stability of the system of equations, at a given set of parameters, and bifurcation analyses are used to analyze the change in stability with regards to modifications of parameters.</p>
                 <figure>
                 <figure>
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                     <img src="" alt="">
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                     <img src="https://static.igem.org/mediawiki/2014/d/d7/Uo2014-mod2.png" alt="">
                     <p>Phase plane analysis of GEV and rTTA.</p>
                     <p>Phase plane analysis of GEV and rTTA.</p>
                 </figure>
                 </figure>
                 <p>In a phase plane analysis, we investigate the stability of two components (molecules like GEV and rtTA) with respect to each other by plotting the amount of the first component (GEV) against the amount of the second (rtTa). For an example, take Figure 2. Each line represents a single simulation, with a unique set of initial conditions (starting amounts) where the blue progressing to red in the line represents the progress of the simulation from the beginning to the end. The phase plane, in essence, is a summary of multiple simulations of the system of equations with different initial conditions. This system of equations tend to progress to three stable states (the groupings of red ends). Stable states are also known as attractor sites or "sinks", because, like a sink, these sites pull in the trajectories of the components as seen by the phase plane.</p>
                 <p>In a phase plane analysis, we investigate the stability of two components (molecules like GEV and rtTA) with respect to each other by plotting the amount of the first component (GEV) against the amount of the second (rtTa). For an example, take Figure 2. Each line represents a single simulation, with a unique set of initial conditions (starting amounts) where the blue progressing to red in the line represents the progress of the simulation from the beginning to the end. The phase plane, in essence, is a summary of multiple simulations of the system of equations with different initial conditions. This system of equations tend to progress to three stable states (the groupings of red ends). Stable states are also known as attractor sites or "sinks", because, like a sink, these sites pull in the trajectories of the components as seen by the phase plane.</p>
                 <figure>
                 <figure>
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                     <img src="" alt="">
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                     <img src="https://static.igem.org/mediawiki/2014/3/30/Uo2014-mod3.png" alt="">
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                     <p>Bifurcation analysis of GEV and rTTA with respect to parameter <i>a1</i>.</p>
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                     <p>Bifurcation analysis of GEV and rTTA with respect to parameter <i>k9</i>.</p>
                 </figure>
                 </figure>
                 <p>In bifurcation analyses, we investigate the change in stable states with regards to the parameters in order to test robustness (the range of each parameter where the model can still predict tristability) and sensitivity (which parameter can cause the most change in the stable states). The bifurcation diagram is very similar to phase planes; in essence, a phase plane is conducted at each parameter value. At each value, multiple simulations of the system equations are conducted but only the last several values of the simulation (red regions in phase plane) are plotted. One can observe the progress of the stable points as one modifies the parameter. Figure 3 best demonstrates the concept of bifurcation analysis. In this example, we can say that the stable state with high expression of GEV is greatly affected by the parameter. </p>
                 <p>In bifurcation analyses, we investigate the change in stable states with regards to the parameters in order to test robustness (the range of each parameter where the model can still predict tristability) and sensitivity (which parameter can cause the most change in the stable states). The bifurcation diagram is very similar to phase planes; in essence, a phase plane is conducted at each parameter value. At each value, multiple simulations of the system equations are conducted but only the last several values of the simulation (red regions in phase plane) are plotted. One can observe the progress of the stable points as one modifies the parameter. Figure 3 best demonstrates the concept of bifurcation analysis. In this example, we can say that the stable state with high expression of GEV is greatly affected by the parameter. </p>
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                 </ul>
                 </ul>
                 <figure>
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                     <img src="" alt="">
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                     <img src="https://static.igem.org/mediawiki/2014/thumb/a/a9/Uo2014-eqng.png/800px-Uo2014-eqng.png" alt="">
                     <p>Graphical representation of how each equation was constructed.</p>
                     <p>Graphical representation of how each equation was constructed.</p>
                 </figure>
                 </figure>
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                 <figure class="full">
                 <figure class="full">
                     <img src="https://static.igem.org/mediawiki/2014/b/b5/Uo2014-phase3.png" alt="">
                     <img src="https://static.igem.org/mediawiki/2014/b/b5/Uo2014-phase3.png" alt="">
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                     <p>Individual simulations of stochastic model for design one of the construct. The amount of GEV (green) and rtTa (blue) found in the nucleus was plotted against time of the simulation (minutes). The simulations were run with 50 β-estradiol and 5 aTc molecules (LEFT), 50 β-estradiol and 50 aTc molecules (MIDDLE) and 5 β-estradiol and 50 aTc molecules (RIGHT). All simulations were implemented with identical parameters found on the parameter page. </p>
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                     <p>Individual simulations of stochastic model for design one of the construct. The amount of GEV (green) and rtTa (blue) found in the nucleus was plotted against time of the simulation (minutes). The simulations were run with 50 beta-estradiol and 5 aTc molecules (LEFT), 50 beta-estradiol and 50 aTc molecules (MIDDLE) and 5 beta-estradiol and 50 aTc molecules (RIGHT). All simulations were implemented with identical parameters found on the parameter page. </p>
                 </figure>
                 </figure>
                 <p>The results for the stochastic model of design two are similar to those obtained from the models of design one. The three conditions (beta-estradiol &gt; aTc, beta-estradiol = aTc, beta-estradiol &lt; aTc) resulted in three distinct stable states with respective amounts of GEV and rtTa. These results are promising and suggest that it might be possible to obtain tristability with the design two construct in vitro. </p>
                 <p>The results for the stochastic model of design two are similar to those obtained from the models of design one. The three conditions (beta-estradiol &gt; aTc, beta-estradiol = aTc, beta-estradiol &lt; aTc) resulted in three distinct stable states with respective amounts of GEV and rtTa. These results are promising and suggest that it might be possible to obtain tristability with the design two construct in vitro. </p>
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                     <p>Individual simulations of stochastic model for design one of the construct. The amount of GEV (green) and rtTa (blue) found in the nucleus was plotted against time of the simulation (minutes). The simulations were run with 50 beta-estradiol and 5 aTc molecules (RIGHT), 50 beta-estradiol and 50 aTc molecules (MIDDLE) and 5 beta-estradiol and 50 aTc molecules (LEFT). All simulations were implemented with identical parameters found on the parameter page.</p>
                     <p>Individual simulations of stochastic model for design one of the construct. The amount of GEV (green) and rtTa (blue) found in the nucleus was plotted against time of the simulation (minutes). The simulations were run with 50 beta-estradiol and 5 aTc molecules (RIGHT), 50 beta-estradiol and 50 aTc molecules (MIDDLE) and 5 beta-estradiol and 50 aTc molecules (LEFT). All simulations were implemented with identical parameters found on the parameter page.</p>
                 </figure>
                 </figure>
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                <figure><img src="https://static.igem.org/mediawiki/2014/f/f1/Uo2014-mod4.png"></figure>
                 <p>Although the deterministic models demonstrate that obtaining a tristable system with the promoters currently being used may not be possible, the stochastic models do show a possibility of using the system to attain three stable states. The issue is that model data is not reliable unless the models have been validated with experimental data from in vitro, so the immediate next step is to validate our model results with experimental results from the constructs. With validation, the parameters of the system can be modified to better reflect the in vitro system. Following validation and model optimization, more rigorous stability analyses can be conducted on the deterministic and stochastic models to obtain a better insight of tristability. Once the system is shown to be tristable, then analyses can be conducted to study the dynamics of the inducible switch.</p>
                 <p>Although the deterministic models demonstrate that obtaining a tristable system with the promoters currently being used may not be possible, the stochastic models do show a possibility of using the system to attain three stable states. The issue is that model data is not reliable unless the models have been validated with experimental data from in vitro, so the immediate next step is to validate our model results with experimental results from the constructs. With validation, the parameters of the system can be modified to better reflect the in vitro system. Following validation and model optimization, more rigorous stability analyses can be conducted on the deterministic and stochastic models to obtain a better insight of tristability. Once the system is shown to be tristable, then analyses can be conducted to study the dynamics of the inducible switch.</p>
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<h2>Additional data</h2>
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<p>Download additional figures in the following documents:</p>
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<ul>
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<li><a href="https://static.igem.org/mediawiki/2014/9/92/Uo2014-design1results.pdf" target="_blank">Additional design 1 results</a></li>
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<li><a href="https://static.igem.org/mediawiki/2014/b/b4/Uo2014-stochastic1dbifurcations.pdf" target="_blank">Stochastic model 1D bifurcations</a></li>
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<li><a href="https://static.igem.org/mediawiki/2014/1/15/Uo2014-stochastic2dbifurcations.pdf" target="_blank">Stochastic model 2D bifurcations</a></li>
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<li><a href="https://static.igem.org/mediawiki/2014/5/59/Uo2014-stochasticphase.pdf" target="_blank">Stochastic model phase diagrams</a></li>
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<li><a href="https://static.igem.org/mediawiki/2014/b/b1/Uo2014-stochasticpopulation.pdf" target="_blank">Stochastic model population vs time charts</a></li>
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<li><a href="https://static.igem.org/mediawiki/2014/4/41/Uo2014-modaux.pdf" target="_blank">Auxiliary data</a></li>
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</ul>
             </div>
             </div>
             <div class="pane" id="pane-refs" hidden>
             <div class="pane" id="pane-refs" hidden>

Latest revision as of 03:24, 18 October 2014

Modelling

What do we do in computational biological modeling? Simply put, we translate biology into mathematics and back again. It may be to predict, to confirm or to study. Whatever the reason may be, as modelers, our goal is to turn the biology into mathematical terms, which are more easily manipulated, tested and analyzed, and deduce the biological meaning from the results. Above all, our main goal is to work with and support the members conducting the biological research in the laboratory.

The system we designed and built this year is known as a tristable switch (a more detailed description of the biology can be found in the project section). In essence, the system is a two-gene construct with mutual repression and self-activation that should result in three stable states. A diagram of the system is seen in Figure 1. The two genes code for transcriptional factors, GEV and rtTA, which act as self-activators and repressors in the presence of activator molecules (beta-estradiol and aTc, respectively). These small molecules, in effect, act as inducible switches which allow them to control which transcriptional factor is active. Our objectives for the model are to design mathematical models representative of the system in order to 1) predict the stable points of the systems and at what concentrations of beta-estradiol and aTc they occur and 2) to study the dynamics of inducing the switch between stable states. But what does it mean to have stable states?

Simplified visual diagram of genetic construct. Mutual inhibition and self activation are mediated by beta-estradiol and aTc when GEV and rtTa are involved, respectively.

In a biological context, stable states refer to states of stable gene expression levels. This means that expression levels tend to progress until it reaches a certain stable expression level, depending on the initial conditions. These states also are resistant to modifications of expression by changing regulatory proteins, loss of genetic product, etc. In a modeling context, it refers to the trajectory of the system (or group) of equations to converge to a steady or stable state. Mathematically speaking, stable states occur at intersections of nullclines, but simply, they attract the trajectory of the system of equations. To analyze model stability, one can use phase plane and bifurcation analyses. Phase plane analyses are used to analyze the dynamics of the stability of the system of equations, at a given set of parameters, and bifurcation analyses are used to analyze the change in stability with regards to modifications of parameters.

Phase plane analysis of GEV and rTTA.

In a phase plane analysis, we investigate the stability of two components (molecules like GEV and rtTA) with respect to each other by plotting the amount of the first component (GEV) against the amount of the second (rtTa). For an example, take Figure 2. Each line represents a single simulation, with a unique set of initial conditions (starting amounts) where the blue progressing to red in the line represents the progress of the simulation from the beginning to the end. The phase plane, in essence, is a summary of multiple simulations of the system of equations with different initial conditions. This system of equations tend to progress to three stable states (the groupings of red ends). Stable states are also known as attractor sites or "sinks", because, like a sink, these sites pull in the trajectories of the components as seen by the phase plane.

Bifurcation analysis of GEV and rTTA with respect to parameter k9.

In bifurcation analyses, we investigate the change in stable states with regards to the parameters in order to test robustness (the range of each parameter where the model can still predict tristability) and sensitivity (which parameter can cause the most change in the stable states). The bifurcation diagram is very similar to phase planes; in essence, a phase plane is conducted at each parameter value. At each value, multiple simulations of the system equations are conducted but only the last several values of the simulation (red regions in phase plane) are plotted. One can observe the progress of the stable points as one modifies the parameter. Figure 3 best demonstrates the concept of bifurcation analysis. In this example, we can say that the stable state with high expression of GEV is greatly affected by the parameter.

We just went through why we are modeling and how we're going to use it. The question now is what we are going to use to model? For modeling biological systems, one of the first choices is whether to use a deterministic or stochastic model. In essence, a deterministic model assumes that all variables of the model can account for the majority of the biological behavior of the system, ignoring the inherent variability of the system. Stochastic models take this variability into account by introducing a random factor, but this often makes the system difficult to analyze. We decided to pursue the design of an ordinary differential equation (ODE) deterministic model and a Gillespie-based stochastic model.