<h1>Construction of the model</h1>

      <a href="http://2011..org/wiki/images/d/d1/Rapport_Equations.pdf">

      <img src="http://2011.iGEM.org/wiki/skins/common/images/icons/fileicon-pdf.png" style="float: left; height: 20px; width: auto">

    </div>

    <h2>Establishment of the equation</h2>

      <li>Biological Model</li>

  A toggle switch consists of two genes, each coding for a protein

  that represses the expression of the other gene. This double

  repression system ensures the basic function of a toggle switch: a

  bistable system which can be switched from one state to the other

  by putting, in our example, some IPTG or aTc molecules in the

  medium.

  The biological system we are trying to implement is more complex,

  on both the biological and physical side. However, the toggle switch  

  model is basically the same. In the study of the toggle switch itself,

  the system can be reduced to a simple two-state subsystem that we will

  then use for the rest of our modelling. However, our toggle switch model

  toggle switch itself is not influenced by the rest of the system,

  if we do not consider the RsmA regulatory system that will be

  modeled at the very end of our work. Thus we will be able to model

  the toggle switch independently and then build the rest of the

  model upon this basis.

 

      <li>Mathematical Model</li>

In order to get the state of the bacteria, we need to compute the concentration of both

repressors. To obtain these variations, we develop a system of ordinary differential equations (ODES)

which governs the behavior of the Toggle Switch.

      <center>

      $      

      \frac{d[TetR]}{dt} = \frac{k_{pLac}.[pLac]_{tot}}{1 +  (\frac{[lacI]}{K_{pLac} + \frac{K_{pLac}.[IPTG]}{K_{lacI-IPTG}}.})^\beta} - \delta_{TetR}.[TetR]$

      <br/><br/>

      $\frac{d[lacI]}{dt} = \frac{k_{pTet}.[pTet]_{tot}}{1 +  (\frac{[tetR]}{K_{pTet} + \frac{K_{pTet}.[aTc]}{K_{TetR-aTc}}.})^\gamma} - \delta_{lacI}.[lacI]$

      </center>

With this ODE system, we follow the evolution of the concentration of the two toggle switch repressors,

TetR and LacI. This gives us the state of the bacteria. A demonstration of this ODE system is available

in the previous pdf file. We briefly explain the equations below.

  </p>

  <p>

Both equations are developed following the same approach. There is a term of production and a term of

degradation. The parameters $k_{pLac}.[pLac]_{tot}$ represent the protein synthesis rate from the pLac promoter.

K_{pLac} and K_{lacI−IPTG} are the dissociation constants of LacI and pLac, and LacI and IPTG, respectively.

The parameters β and γ denote the cooperativity of repression, that is, the number of repressors bound

to the promoter.  

  </p>

  <p>

According to these equations, the rate of variation of repressor TetR is inhibited by the repressor LacI.

The degree of inhibition is modulated by the IPTG concentration. Reciprocally in the second equation,  

the rate of variation of LacI depends on TetR, whose negative effect is modulated by aTc.

  </p>

  <p><strong>

This indicates that the two equations are coupled. And also that only one repressor could be

predominant, as shown later.

  </strong></p>

    </p>

  These two equations can be easily computed with a differential solver. We can precisely estimate the effects of each parameter.

  <h2>Establishment of the equation</h2>  

      <li>Mathematical Model</li>

In a second part, we define a model able to characterize the <a href="https://2011.igem.org/Team:Grenoble/Projet/Design/quorum">quorum sensing module</a> of our genetic network,

which involves the quorum sensing genes CinI and CinR, as well as the signaling molecule AHL.<br/>

In order to model the Quorum Sensing module, we detail below the different reactions taking place

inside this module. For reasons that will become clear later, we focus on a particular area of our

device, where neighboring bacteria will have a different behavior, although they carry the same

genetic circuit.

  </p>

  <img src="https://static.igem.org/mediawiki/2011/a/a5/QS_details_2.png" class="centerwide" style="bow-shadow: non"/>

  <div class="legend"><strong>Figure 1:</strong>Mechanism of Quorum Sensing diffusion at the boundary</div>

  <p>

<ol>

<li>the production of the Quorum Sensing molecule</li>

<li>the secretion of the molecule</li>

<li>the diffusion of the molecule in the medium</li>

<li>the penetration of the molecule</li>

<li>the complexation of the molecule with its receptor</li>

<li>the activation of the coloration</li>

</ol>

  </p>

  <p>

To model these mecanisms, we need to follow the evolution of the following quantities

<ul>

<li>$[QS_i]$ concentration of the intracellular Quorum Sensing molecule.</li>

<li>$[QS_e]$ concentration of the extracellular Quorum Sensing molecule.</li>

<li>$[cinR]$ concentration of the Quorum Sensing receptor.</li>

<li>$[cinI]$ concentration of the Quorum Sensing producer enzyme.</li>

<li>$[cinR_{comp}]$ concentration of the complexe cinR-QS.</li>

</ul>

  </p>

  <p>

  </p>

  <center>

  $\frac{d[QS_i]}{dt} = \eta([QS_e]-[QS_i])-\delta_{QSi} [QS_i] + k_{QS-production}'[cinI]$<br/><br/>

  $\frac{d[QS_e]}{dt} = \rho v_c\eta([QS_i]-[QS_e])-\delta_{QSe} [QS_e] + D_{diff}\frac{\partial^2 [QS_e]}{\partial x^2}$<br/><br/>

  $\frac{d[cinR_{free}]}{dt} = \frac{k_{pLac}.[pLac]_{tot}}{1 +  (\frac{[lacI_{total}]}{K_{pLac} + \frac{K_{pLac}.[IPTG]}{K_{lacI-IPTG}}.})^\beta} - \delta_{cinR}[cinR_{free}] - V_{complexation}$<br/><br/>

  $\frac{d[cinR_{comp}]}{dt} = K_{comp}([cinR_{free}].[QS_i])$<br/><br/>

  </center>

  <p>

$\delta_{QSi}[QSi]$, the degradation and $k_{QS−production}[cinI]$, the production of Quorum Sensing molecule

In the second equation, expressing the evolution of extracellular Quorum Sensing, there is no production term, but a spatial diffusion term

$D_{diff}\frac{\partial^2 [QS_e]}{\partial x^2}$.

  </p>

  <p>

\[V_{complexation} = K_{comp}[QS_i][CinR_{free}]\]  

with $K_{comp}$ the affinity constant of AHL for cinR.

  </p>

  <p>

on $[cinR]$ and $[QS_i] which is the limitant factor$.

  </p>

  <p><strong>

  </strong></p>

<li>Definition of the matrix</li>