Team:uOttawa/drylab

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                 <h1>Results</h1>
                 <h1>Results</h1>
                 <p>The deterministic models were tested by conducting phase plane analyses on each set of equations for each design (an explanation for phase plane analysis can be found on the introduction page). With phase plane analyses, one can observe the stability of the system of equations (group of equations) and how many stable states exist. For our purposes, we expect to see three stable states (three distinct red areas) in the diagrams and the amounts of GEV and rtTa associated with the stable states. The current model data for design one demonstrate that the system is monostable (only one stable state). Each set of equations (multiplicative, additive and singular terms, which are described in the equations page) were tested and yielded very similar results. The multiplicative and singular term equations stabilized around no expression of GEV and rtTA (GEV=0, rtTa=0) and the additive equations stabilized around 8x1011 GEV molecules and somewhere in between 1x108¬ and 1x105 rtTa molecules (not 0 rtTA molecules given the log transformed phase plane analysis).</p>
                 <p>The deterministic models were tested by conducting phase plane analyses on each set of equations for each design (an explanation for phase plane analysis can be found on the introduction page). With phase plane analyses, one can observe the stability of the system of equations (group of equations) and how many stable states exist. For our purposes, we expect to see three stable states (three distinct red areas) in the diagrams and the amounts of GEV and rtTa associated with the stable states. The current model data for design one demonstrate that the system is monostable (only one stable state). Each set of equations (multiplicative, additive and singular terms, which are described in the equations page) were tested and yielded very similar results. The multiplicative and singular term equations stabilized around no expression of GEV and rtTA (GEV=0, rtTa=0) and the additive equations stabilized around 8x1011 GEV molecules and somewhere in between 1x108¬ and 1x105 rtTa molecules (not 0 rtTA molecules given the log transformed phase plane analysis).</p>
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                 <figure><img src="https://static.igem.org/mediawiki/2014/2/27/Uo2014-phase1.png" alt="">
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                 <figure class="full"><img src="https://static.igem.org/mediawiki/2014/2/27/Uo2014-phase1.png" alt="">
                     <p>Phase plane and log-phase plane diagrams of the three sets of equations (Multiplicative, Additive and Singular terms) for design one of tristable switch. Each line represents one simulation of the equations with a unique set of initial conditions and the progress of the simulation is denoted by the progress from blue (beginning) to red (end). All simulations were run for 4000 steps and all parameters of the equations may be found in parameters page.<br>LEFT: phase plane analyses of the multiplicative equations <br>MIDDLE: phase plane analyses of the additive equations. <br>RIGHT: phase plane analyses of the singular term equations. <br>TOP ROW: non-transformed phase plane analyses. <br>BOTTOM ROW: log transformed (log(rtTa) vs log(GEV)) phase plane analyses.</p></figure>
                     <p>Phase plane and log-phase plane diagrams of the three sets of equations (Multiplicative, Additive and Singular terms) for design one of tristable switch. Each line represents one simulation of the equations with a unique set of initial conditions and the progress of the simulation is denoted by the progress from blue (beginning) to red (end). All simulations were run for 4000 steps and all parameters of the equations may be found in parameters page.<br>LEFT: phase plane analyses of the multiplicative equations <br>MIDDLE: phase plane analyses of the additive equations. <br>RIGHT: phase plane analyses of the singular term equations. <br>TOP ROW: non-transformed phase plane analyses. <br>BOTTOM ROW: log transformed (log(rtTa) vs log(GEV)) phase plane analyses.</p></figure>
                 <p>The results for the analyses of the model for design two were very similar to those of design one showing only monostability of the system. The difference between the designs was the position of the stable states in the multiplicative and singular term equations. In design one, those equations stabilized at no expression of GEV and rtTa (GEV=0, rtTa=0), but for the second design, they stabilized at equal expression of both, around 2x108 molecules of GEV and rtTa. Both non-transformed and log transformed phase planes suggest that tristability may not be obtained with the current promoters and system designs being used in the wet lab, but the model needs to be validated with experimental results before accepting these results.</p>
                 <p>The results for the analyses of the model for design two were very similar to those of design one showing only monostability of the system. The difference between the designs was the position of the stable states in the multiplicative and singular term equations. In design one, those equations stabilized at no expression of GEV and rtTa (GEV=0, rtTa=0), but for the second design, they stabilized at equal expression of both, around 2x108 molecules of GEV and rtTa. Both non-transformed and log transformed phase planes suggest that tristability may not be obtained with the current promoters and system designs being used in the wet lab, but the model needs to be validated with experimental results before accepting these results.</p>
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                 <figure><img src="https://static.igem.org/mediawiki/2014/b/b7/Uo2014-phase2.png" alt="">
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                 <figure class="full"><img src="https://static.igem.org/mediawiki/2014/b/b7/Uo2014-phase2.png" alt="">
                     <p>Phase plane and log-phase plane diagrams of the three sets of equations (Multiplicative, Additive and Singular terms) for design two of tristable switch. See previous figure for a legend.</p></figure>
                     <p>Phase plane and log-phase plane diagrams of the three sets of equations (Multiplicative, Additive and Singular terms) for design two of tristable switch. See previous figure for a legend.</p></figure>
                 <p>The stochastic models were tested by conducting individual simulations with identical starting amounts of GEV and rtTa, but varied the initial amount of beta-estradiol and aTc. Three conditions of initial activator molecules amounts were tested: beta-estradiol &gt; aTc, beta-estradiol = aTc, beta-estradiol &lt; aTc. Each condition resulted in different and distinct stable states where the initial amount of activator molecule dictated which protein would be more expressed. In the case of higher beta-estradiol amount, the amount of GEV found in the nucleus was higher than the amount of rtTa and vice versa for the case of lower beta-estradiol amount. In the case of equal amounts, there were roughly equal amounts of GEV and rtTa transcriptional factors found in the nucleus. These results are promising and suggest that it might be possible to obtain tristability with the design one construct in vitro.</p>
                 <p>The stochastic models were tested by conducting individual simulations with identical starting amounts of GEV and rtTa, but varied the initial amount of beta-estradiol and aTc. Three conditions of initial activator molecules amounts were tested: beta-estradiol &gt; aTc, beta-estradiol = aTc, beta-estradiol &lt; aTc. Each condition resulted in different and distinct stable states where the initial amount of activator molecule dictated which protein would be more expressed. In the case of higher beta-estradiol amount, the amount of GEV found in the nucleus was higher than the amount of rtTa and vice versa for the case of lower beta-estradiol amount. In the case of equal amounts, there were roughly equal amounts of GEV and rtTa transcriptional factors found in the nucleus. These results are promising and suggest that it might be possible to obtain tristability with the design one construct in vitro.</p>
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                 <figure>
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                 <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="">
                     <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>
                     <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>
                 </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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                 <figure>
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                 <figure class="full">
                     <img src="https://static.igem.org/mediawiki/2014/e/ea/Uo2014-phase4.png" alt="">
                     <img src="https://static.igem.org/mediawiki/2014/e/ea/Uo2014-phase4.png" alt="">
                     <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>

Revision as of 14:38, 17 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. Mathemtically 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 a1.

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.

References

Orth, Peter, Dirk Schnappinger, Phaik-Eng Sum, George A. Ellestad, Wolfgang Hillen, Wolfram Saenger, and Winfried Hinrichs. "Crystal Structure of the Tet Repressor in Complex with a Novel Tetracycline, 9-(N, N-dimethylglycylamido)-6-demethyl-6-deoxy-tetracycline1." Journal of Molecular Biology 285.2 (1999): 455-61. Print.

Takahashi, M. Kinetic and Equilibrium Characterization of the Tet Repressor-Tetracycline complex by Fluorescence Measurements. J.Mol.Bio (1986) 187: 341-348

McIsaac, R., et al. Synthetic gene expression perturbation systems with rapid tunable single-gene specificity in yeast. Nuc.Acids.Research (2013) 41(4) doi:10.1093/nar/gks1313

McIsaac, R., et al. Fast-acting and nearly gratuitous induction of gene expression and protein depletion in Saccahromyces cerevisae. Molecular Biology of the Cell (2011) 22(4) : 4447-4459 doi:10.1093/nar/gks1313

Sneppen K., Krishna, S., Semesy, S. Simplified models of biological networks. Annu.Rev.Biophys (2010), 39:43-59