Team:Slovenia/ModelingQuantitativeModel

From 2012.igem.org

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<h1>Quantitative model and stability analysis</h1>
<h1>Quantitative model and stability analysis</h1>
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<h2>Model description and parameter determination</h2>
 
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<p>In order to be able to predict the quantitative behavior of <i>in vivo</i> application of the genetic switches, we set up a quantitative model of a Transcription-activator like effector (TAL) based bistable switches. We apply experimental measurements for individual parts and predict their behavior when applied into a single system.  The essential characteristics of a switch are <i>responsiveness</i> (reaching a stable state with the introduction of the corresponding inducer), <i>stability</i> (not leaving the state it after inducer removal) and <i>robustness</i> (expressing equivalent qualitative behavior under a wide range of conditions). In the following subsections, modeling approaches for individual parts are described.</p>
<p>In order to be able to predict the quantitative behavior of <i>in vivo</i> application of the genetic switches, we set up a quantitative model of a Transcription-activator like effector (TAL) based bistable switches. We apply experimental measurements for individual parts and predict their behavior when applied into a single system.  The essential characteristics of a switch are <i>responsiveness</i> (reaching a stable state with the introduction of the corresponding inducer), <i>stability</i> (not leaving the state it after inducer removal) and <i>robustness</i> (expressing equivalent qualitative behavior under a wide range of conditions). In the following subsections, modeling approaches for individual parts are described.</p>
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<p>To get a better idea of the functioning of the model, an interactive web application was developed and is available <a href="https://2012.igem.org/Team:Slovenia/InteractiveSimulations">here</a>.</p>
<p>To get a better idea of the functioning of the model, an interactive web application was developed and is available <a href="https://2012.igem.org/Team:Slovenia/InteractiveSimulations">here</a>.</p>
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<h2>Model description and parameter determination</h2>
<h3>TAL:KRAB repressor constructs</h3>
<h3>TAL:KRAB repressor constructs</h3>
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<h3>Inducible system</h3>
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<p>The inducible system in both switches is built upon two different inducer molecules. In the mutual repressor switch, we employ a <i>pristinamycin-inducible protein (PIP) with KRAB repressor domain</i>. After the introduction of <i>pristinamycin</i>, PIP:KRAB is inactivated and thus unbound from its binding site at the pCMV promoter. The concentration of active PIP:KRAB is hence inversely proportional to the pristinamycin concentration:</p>
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<p>Consequently, the transcription rate of the corresponding promoter is proportional to the concentration of active PIP:KRAB :</p>
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<p>where k is the maximum transcription rate, f scaling factor and alpha the fraction of inactivated transcription rate. The increasing concentration of active PIP:KRAB causes the resulting rate r to approach the basal rate, with half the maximum rate concentration determined by the threshold K. In an analogue manner, the erythromycin and E:KRAB inducer system was modeled.</p>
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<h2>Stability analysis</h2>
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<p>For the purpose of stability analysis, the simulation was run multiple times with varying pairs of parameters and producing a two dimensional <i>parameter map</i>. For example, with varying plasmid input dosages (other things being equal), one or the other state might become not reachable. For an illustrative example, see Figure 2.</p>
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<p>For every instance of the simulation (a colored point in Figure 2), the outcome results were examined. During the course of a simulation, a steady state is defined as a point in phase space (all possible pairs of the two reporter concentrations) where concentrations of <i>both reporter protein concentrations do not change</i>, i.e.:</p>
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<p>The list of steady state points is achieved and analyzed for each instance of a simulation. A <i>valid state A</i> is defined when BFP concentration crosses a predefined threshold <i>th</i> while at the same time the mCitrine concentration is below <i>th</i>. <i>Valid state B</i> is defined analogously for mCitrine. If both stable states are reached during a simulation instance run, the system is labeled bistable.</p>
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<p>If at any stable state points, both concentrations cross the threshold simultaneously or remain below threshold simultaneously, the state point and the system are defined ambiguous. In case no stable state points are found during the simulation, the system is labeled as not valid.</p>
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<p>Summing up, five possible outcomes for each simulation instance were defined and placed on the parameter map (using separate colors to differentiate between them):
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<ul>
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<li><div style="font-color:green;">Bistable; The system reached both stable states on a given time interval.</dic></li>
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<li><div style="font-color:green;">Monostable A; The system reached expressed stable state A on a given time interval.</div></li>
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<li><div style="font-color:green;">Monostable B; The system only reached stable state B on a given time interval.</div></li>
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<li><div style="font-color:green;">Ambigous; The system expressed reached states simultaneously during a given time interval.</div></li>
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<li><div style="font-color:green;">Not valid; The system did not reach any stable states during a given time interval.</div></li>
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</ul>
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</p>
<h2 style="color:grey;">References</h2>
<h2 style="color:grey;">References</h2>

Revision as of 01:03, 25 September 2012


Quantitative model and stability analysis

In order to be able to predict the quantitative behavior of in vivo application of the genetic switches, we set up a quantitative model of a Transcription-activator like effector (TAL) based bistable switches. We apply experimental measurements for individual parts and predict their behavior when applied into a single system. The essential characteristics of a switch are responsiveness (reaching a stable state with the introduction of the corresponding inducer), stability (not leaving the state it after inducer removal) and robustness (expressing equivalent qualitative behavior under a wide range of conditions). In the following subsections, modeling approaches for individual parts are described.

A variety of well-established models in the literature were examined, which resulted in choosing ordinary differential Hill equations model, presented in (Tigges et al., 2009). The former work presents a tunable synthetic mammalian oscillator, whose model was readjusted to present a switch structure. On top of that, the host choice of mammalian cells and pristinamycin-based inducible system corresponds to our system.

To get a better idea of the functioning of the model, an interactive web application was developed and is available here.

Model description and parameter determination

TAL:KRAB repressor constructs

For repressor constructs, the percentage of remaining transcription rate comparing to the unrepressed reporter plasmid was measured. Intuitively, the remaining rate was directly proportional to the repressor/reporter ratio, but not in a linear fashion. To approximate this relation for TAL:KRAB constructs, the following analytical function was fitted to the measurements:

where x is repressor/reporter plasmid ratio, and a, b, c are parameters subject to least square error fitting. This produced decreasing function for remaining transcription rate q, depicted in Figure 1a. The achieved percentage of remaining transcription rate q was directly employed in repressed promoter rate equation. The dynamics were modeled using Hill equations for repression and activation (Alon, 2007). For example, the effect of a repressor on transcription rate is modeled as a factor in a rate equation:

Intuitively, the unrepressed transcription rate r is inversely proportional to the concentration of TAL-A:KRAB protein with non-linearity coefficient n. The remaining rate percentage q was used to determine the s constant, which determines the concentration at which half of the unrepressed rate is reached, after the concentration threshold t is crossed.

TAL:VP16 activator constructs

Using a similar approach to that described above, we fitted the data achieved for activator constructs TAL-A:VP16 and TAL-B:VP16. As expected, the transcription rate increased proportionally with increasing activator/reporter ratio of plasmids transfected, again in a non-linear fashion. To approximate the behavior, the following increasing function was fitted:

where x is activator/reporter plasmid radio and c, b are parameters subject to least square error fitting. The plot of the function is shown in Figure 1b. The factor of activation a is again directly employed as a rate equation factor for activated promoter:

where x is activator/reporter plasmid radio and c, b are parameters subject to least square error fitting. The plot of the function is shown in Figure 1b. The factor of activation a is again directly employed as a rate equation factor for activated promoter: where the increasing concentration of TAL-B:VP16 protein reaches half the maximum activation rate a when crossing the concentration threshold t, with non-linearity exponent n. Figure 1: Functions fitted to experimental data for repressor and activator constructs. (a) Data measured for TAL-A:KRAB (blue) and TAL-B:KRAB (red) constructs and averaged afterwards (black). (b) Data measured for TAL-A:VP16 (blue) and TAL-B:VP16 (red) constructs and averaged afterwards (black).

Inducible system

The inducible system in both switches is built upon two different inducer molecules. In the mutual repressor switch, we employ a pristinamycin-inducible protein (PIP) with KRAB repressor domain. After the introduction of pristinamycin, PIP:KRAB is inactivated and thus unbound from its binding site at the pCMV promoter. The concentration of active PIP:KRAB is hence inversely proportional to the pristinamycin concentration:

Consequently, the transcription rate of the corresponding promoter is proportional to the concentration of active PIP:KRAB :

where k is the maximum transcription rate, f scaling factor and alpha the fraction of inactivated transcription rate. The increasing concentration of active PIP:KRAB causes the resulting rate r to approach the basal rate, with half the maximum rate concentration determined by the threshold K. In an analogue manner, the erythromycin and E:KRAB inducer system was modeled.

Stability analysis

For the purpose of stability analysis, the simulation was run multiple times with varying pairs of parameters and producing a two dimensional parameter map. For example, with varying plasmid input dosages (other things being equal), one or the other state might become not reachable. For an illustrative example, see Figure 2.

For every instance of the simulation (a colored point in Figure 2), the outcome results were examined. During the course of a simulation, a steady state is defined as a point in phase space (all possible pairs of the two reporter concentrations) where concentrations of both reporter protein concentrations do not change, i.e.:

The list of steady state points is achieved and analyzed for each instance of a simulation. A valid state A is defined when BFP concentration crosses a predefined threshold th while at the same time the mCitrine concentration is below th. Valid state B is defined analogously for mCitrine. If both stable states are reached during a simulation instance run, the system is labeled bistable.

If at any stable state points, both concentrations cross the threshold simultaneously or remain below threshold simultaneously, the state point and the system are defined ambiguous. In case no stable state points are found during the simulation, the system is labeled as not valid.

Summing up, five possible outcomes for each simulation instance were defined and placed on the parameter map (using separate colors to differentiate between them):

  • Bistable; The system reached both stable states on a given time interval.
  • Monostable A; The system reached expressed stable state A on a given time interval.
  • Monostable B; The system only reached stable state B on a given time interval.
  • Ambigous; The system expressed reached states simultaneously during a given time interval.
  • Not valid; The system did not reach any stable states during a given time interval.

References

Alon U. (2007), An introduction to systems biology: design principles of biological circuits. Chapman & Hall/CRC.

Chatterjee A., Kaznessis Y., and Hu W. (2006) Tweaking biological switches through a better understanding of bistability behavior. Current opinion in biotechnology, 19(5):475-81.

Gardner T., Cantor C., and Collins J. (2000) Construction of a genetic toggle switch in Escherichia coli. Nature, 403(6767):339-42.

Batard P., and Wurm F. (2001) Transfer of high copy number plasmid into mammalian cells by calcium phosphate transfection. Gene: An international Journal on Genes and Genomes, 270:61-68.

Malphettes L., and Fussenegger M. (2006) Impact of RNA interference on gene networks. Metabolic engineering, 8(6):672-83.

Tigges M., Marquez-Lago T., Stelling J., and Fussenegger M. (2009) A tunable synthetic mammalian oscillator. Nature, 457(7227):309-12.

Zakharova A., Kurths J., Vadivasova T., and Koseska A. (2011) Analysing dynamical behavior of cellular networks via stochastic bifurcations. PloS one, 6(5):e19696.


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