Team:ULB-Brussels/Modelling/TA-System

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    Université Libre de Bruxelles  **/                 -->
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<section style="text-align: justify; margin: 50px">
<section style="text-align: justify; margin: 50px">
 +
<h1>Toxin-Antitoxin Systems</h1>
<h1>Toxin-Antitoxin Systems</h1>
 +
 +
<h2>$\small 2.1.$ $\small Components$ $\small and$ $\small Diagrams$</h2>
 +
 +
<section style="text-align: justify; margin: 15px"></section>
<p>Two type II TA systems are investigating in our project.  
<p>Two type II TA systems are investigating in our project.  
-
The first consists of ccdB (the toxin, T) and ccdA (the antitoxin, A) and for the second, these are Kid (T) and Kis (A):</p>
+
The $\small\mathtt{1}\normalsize^{st}$ consists of ccdB (the toxin, $\mathbb{T}\hspace{0.04cm}$) and ccdA (the antitoxin, $\mathbb{A}\hspace{0.02cm}$) and  
 +
these are Kid ($\hspace{0.02cm}\mathbb{T}\hspace{0.04cm}$) and Kis ($\hspace{0.02cm}\mathbb{A}\hspace{0.012cm}$) for the $\small\mathtt{2}\normalsize^{nd}$:</p>
 +
<section style="text-align: justify; margin: 25px"></section>
 +
<h3 style="color: #4169E1">2.1.1) CcdBA</h3>
<section style="text-align: justify; margin: 50px">
<section style="text-align: justify; margin: 50px">
-
<h3>2.1) CcdBA</h3>
+
<section style="text-align: justify; margin: -25px"></section>
<p>One of the most studied and characterized TA systems, CcdBA involves two principal components : ccdB and its antidote ccdA.
<p>One of the most studied and characterized TA systems, CcdBA involves two principal components : ccdB and its antidote ccdA.
-
As we explained in the introduction page of our project, ccdB is an inhibitor of the DNA gyrase, so it binds the subunit A of the DNA gyrase complex when it's bound to DNA.
+
</section>
-
When DNA double strand is broken, there is activation of emergency signals (SOS system blocks cellular division in bacteria). If the DNA gyrase cannot protect itself by a mutation (some events are possible, but very rare) or if the antidote is degraded (very frequent because ccdA is unstable in comparison with ccdA), the death of a bacterium in unavoidable. </p>
+
-
\begin{equation}
+
<section style="text-align: justify; margin: -45px"></section>
-
\hspace{-0.12cm}\wp\hspace{0.02cm}\equiv[p2A]\\
+
<section style="text-align: justify; margin: 50px">
-
\mathbb{A}\hspace{0.02cm}\equiv[ccdA]\\
+
<center>
-
\mathbb{B}\hspace{0.02cm}\equiv[ccdB]\\
+
<img src="https://static.igem.org/mediawiki/2014/6/66/MightyColi-Sch-v2-Coli-0.png">
-
\mathbb{C}\hspace{0.02cm}\equiv[(ccdA)_{2}-(ccdB)_{2}]\\
+
</center></p>
-
a\equiv[ara] \hspace{0.12cm}|\hspace{0.12cm} g\equiv[glu]\\
+
<font size="1"><b>Figure m4a </b>: This diagram illustrates the production of F2A peptid and GFP fluorescent protein controlled by PSK, in the case of TA ccdBA system in E.Coli. </font>
-
\mathring{x}= \dfrac{dx}{dt}
+
</section>
-
\end{equation}
+
 
 +
As we explained in the introduction page of our project, ccdB is an inhibitor of the DNA gyrase, so it binds the subunit A of the DNA gyrase complex when it is bound to DNA.
 +
When DNA double strand is broken, there is activation of SOS emergency signals. Here, the point is: if the DNA gyrase cannot protect itself against ccdB by a mutation (some events are possible, but very rare) or if the antidote is degraded (very frequent because ccdA is unstable in comparison with ccdB), the death of a bacterium in unavoidable. It is why we usually say that bacteria are addicted to the antitoxin to survive.</p>
 +
 
 +
<section style="text-align: justify; margin: 25px"></section>
 +
<h3 style="color: #4169E1">2.1.2) Kid/Kis</h3>
 +
<section style="text-align: justify; margin: 50px">
 +
<section style="text-align: justify; margin: -25px"></section>
 +
 
 +
The same is relevant about the second TA system studied. In this case, the two principal components are Kid and its antidote Kis and the parameters are chosen a little bit different than in the first system.</p>
 +
</section>
 +
 
 +
<section style="text-align: justify; margin: -45px"></section>
 +
<section style="text-align: justify; margin: 50px">
 +
<center>
 +
<img src="https://static.igem.org/mediawiki/2014/5/5a/MightyColi-Sch-v2-Cerev-0.png">
 +
</center></p>
 +
<font size="1"><b>Figure m4b </b>: This diagram illustrates the production of F2A peptid and GFP fluorescent protein controlled by PSK (post-segregational killing), in the case of TA Kid/Kis system in S.Cerevisae. </font>
 +
</section>
 +
 
 +
 
 +
<h2>$\small 2.2.$ $\small Mathematical$ $\small Modelling$</h2>
 +
 
 +
<!--MathTex Raccourcis-->
 +
$\newcommand{\AA}{\mathbb{A}}
 +
\newcommand{\CC}{\mathbb{C}}
 +
\newcommand{\TT}{\mathbb{T}}
 +
\newcommand{\GG}{\mathbb{G}}
 +
\newcommand{\KK}{\small\mathcal{K}\normalsize}$
 +
 
 +
<h3 style="color: #4169E1">2.2.1) Equations from the Diagrams</h3>
 +
 
 +
<section style="text-align: justify; margin: -30px"></section>
 +
<section style="text-align: justify; margin: 50px">
 +
\begin{align*}
 +
\mathbb{A}\hspace{0.02cm}&\equiv[antitoxin] &
 +
\mathbb{C}\hspace{0.02cm}&\equiv[\small TA- \normalsize complex]\\[0.1cm]
 +
\mathbb{T}\hspace{0.02cm}&\equiv[toxin] & 
 +
\mathbb{G}\hspace{0.012cm}&\equiv[\small GFP \normalsize]\\[-0.3cm]
 +
a\hspace{0.03cm}&\equiv[\mathrm{ara}] &
 +
\mathrm{g}\hspace{0.05cm}&\equiv[glu] \hspace{1.45cm}
 +
\mathring{x}=\dfrac{dx}{dt}
 +
\end{align*}
-
In presence of arabinose, AraC activates the transcription of RNAm (catalysed by RNApoly):
+
In presence of arabinose, AraC activates the transcription of RNA$\hspace{0.01cm}_{\textbf{m}}$ (catalysed by RNA$\hspace{0.02cm}_{\textbf{poly}}$) :
\begin{array}.
\begin{array}.
-
  \hspace{0.06cm}\mathring{\wp} &=& v_{s_{0}} \dfrac{a}{a + K_{0}} - \hspace{0.05cm}v_{d_{0}} \hspace{0.04cm}\wp \\[0.05cm]
+
  \hspace{0.02cm}\mathring{\AA} &=& v_{s_{1}} \dfrac{a}{a + \KK_{\mathtt{1}}} - \hspace{0.05cm}v_{d_{1}} \hspace{0.01cm}\AA\hspace{0.02cm} - \hspace{0.05cm}v_{a} \hspace{0.02cm}(\AA\TT)\hspace{0.02cm} + \hspace{0.05cm}v_{d_{3}}\hspace{0.02cm} \CC\hspace{0.03cm} \\[0.1cm]
-
\hspace{0.02cm}\mathring{\mathbb{A}} &=&  v_{s_{1}} \dfrac{a}{a + K_{1}} . \dfrac{\wp}{\wp + K_{3}} - \hspace{0.05cm}v_{a} \hspace{0.02cm}(\mathbb{A}\mathbb{B})\hspace{0.02cm} + \hspace{0.05cm}v_{d_{3}}\hspace{0.02cm} \mathbb{C}\hspace{0.03cm} - \hspace{0.05cm}v_{d_{1}} \hspace{0.01cm}\mathbb{A} \\[0.1cm]
+
%
-
  \hspace{0.02cm}\mathring{\mathbb{B}} &=&  v_{s_{2}} \dfrac{a}{a + K_{2}} . \dfrac{\wp}{\wp + K_{4}} - \hspace{0.05cm}v_{a} \hspace{0.02cm}(\mathbb{A}\mathbb{B})\hspace{0.02cm} + \hspace{0.05cm}v_{d_{4}} \hspace{0.02cm}\mathbb{C}\hspace{0.03cm} - v_{d_{2}} \hspace{0.03cm}\mathbb{B} \\[0.15cm]
+
  \hspace{0.02cm}\mathring{\TT} &=&  v_{s_{2}} \dfrac{a}{a + \KK_{2}} - \hspace{0.05cm}v_{d_{2}} \hspace{0.03cm}\TT\hspace{0.02cm} - \hspace{0.05cm}v_{a} \hspace{0.02cm}(\AA\TT)\hspace{0.02cm} + \hspace{0.05cm}v_{d_{4}} \hspace{0.02cm}\CC\hspace{0.03cm} \\[0.1cm]
-
  \mathring{\mathbb{C}} &=& v_{a} \hspace{0.02cm}(\mathbb{A}\mathbb{B}) - (v_{d_{3}}+v_{d_{4}}) \hspace{0.05cm}\mathbb{C}
+
%
 +
  \hspace{0.02cm}\mathring{\CC} &=& v_{a} \hspace{0.02cm}(\AA\TT)\hspace{0.02cm} - \hspace{0.02cm}(v_{d_{3}}+v_{d_{4}}) \hspace{0.05cm}\CC\\[0.25cm]
 +
%
 +
\hspace{0.02cm}\mathring{\GG} &=& v_{s_{3}} \dfrac{a}{a + \KK_{3}} - \hspace{0.05cm}v_{d_{5}} \hspace{0.04cm}\GG
\end{array}
\end{array}
In presence of glucose, AraC becomes a repressor of the promotor pBAD, so the three first differential equations (for  
In presence of glucose, AraC becomes a repressor of the promotor pBAD, so the three first differential equations (for  
-
$\hspace{0.1cm}\wp\hspace{0.04cm}$, $\hspace{0.04cm}\mathbb{A}\hspace{0.04cm}$ $\small\&\normalsize$ $\hspace{0.04cm}\mathbb{B}\hspace{0.1cm}$)  
+
$\hspace{0.1cm}\AA\hspace{0.04cm}$, $\hspace{0.04cm}\TT\hspace{0.04cm}$ $\small\&\normalsize$ $\hspace{0.04cm}\GG\hspace{0.1cm}$)  
are modificated by the change :
are modificated by the change :
\begin{equation}
\begin{equation}
-
\left(  v_{s_{j}} \dfrac{a}{a + K_{j}} \right) \rightarrow \left( v_{s_{j}} \dfrac{K_{j}}{g + K_{j}} \right).
+
\left(  v_{s_{j}} \dfrac{a}{a + \KK_{j}} \right) \rightarrow \left( v_{s_{j}} \dfrac{\KK_{j}}{\mathrm{g} + \KK_{j}} \right).
\end{equation}
\end{equation}
-
NB: We have chosen a Michaelis-Menten kinetics, maybe a higher Hill coefficient would be desirable.</p>
+
NB:$\hspace{0.06cm}$  We have chosen a Michaelis-Menten kinetics, maybe a higher Hill coefficient would be desirable.</p>
-
Because we'll preserve some fragment of the population, it's necessary to controle its level. In practical, different parameters are introduced in the mathematical model to describe all the configurations of the biological system (in the equations above, the parameters are the constants $\hspace{0.04cm}K_{j}\hspace{0.02cm}$ and the velocities $\hspace{0.04cm}v_{j}\hspace{0.06cm}$).  
+
Because we will preserve some fragment of the population, it is necessary to controle its level. In practical, different parameters are introduced in the mathematical model to describe all the configurations of the biological system (in the equations above, the parameters are the constants $\hspace{0.04cm}\small\mathcal{K}\normalsize_{j}\hspace{0.02cm}$ and the velocities $\hspace{0.04cm}v_{j}\hspace{0.06cm}$).
-
These parameters influence the global dynamics of the TA system, with or without an additional proline with p2A.</p>
+
These parameters influence the global dynamics of the TA system, with or without an additional proline via p2A.</p>
By modelling and by comparison with experiments, we hope to obtain finally a model close to the reality.</p>
By modelling and by comparison with experiments, we hope to obtain finally a model close to the reality.</p>
 +
</section>
-
<h3>2.2) Kis/Kid</h3>
+
<h3 style="color: #4169E1">2.2.2) Stationary State and Stability</h3>
-
The same is relevant about the second TA system studied. In this case, the two principal components are Kid and its antidote Kis and the parameters are chosen a little bit different than in the first system.</p>
+
<section style="text-align: justify; margin: -30px"></section>
 +
<section style="text-align: justify; margin: 50px">
 +
At the stationary state, the quantities $\hspace{0.1cm}\mathbb{A}\hspace{0.04cm}$, $\hspace{0.04cm}\mathbb{C}\hspace{0.04cm}$, $\hspace{0.04cm}\mathbb{T}\hspace{0.04cm}$, $\hspace{0.04cm}\mathbb{G}\hspace{0.1cm}$
 +
don't fluctuate in time. We will distinguish it with the symbol hat ^.</p>
 +
 
 +
First, let's define some new quantities:
 +
\begin{align*}
 +
k_{j}\hspace{0.02cm}&\equiv \dfrac{a}{a + \KK_{\mathtt{j}}} \hspace{0.3cm}&
 +
\Gamma\hspace{0.02cm}&\equiv v_s{_{1}} k_{1} - v_s{_{2}} k_{2} \\[0.3cm]
 +
\Delta\hspace{0.02cm}&\equiv \dfrac{v_d{_{4}} - v_d{_{3}}}{v_d{_{4}} + v_d{_{3}}} \hspace{0.3cm}&
 +
\Omega\hspace{0.02cm}&\equiv \dfrac{\CC}{v_{a}} \\[0.1cm]
 +
\end{align*}
 +
 
 +
Assuming there's only arabinose and combining the two first equations,
 +
we obtain the stationary concentration of the toxin like a function of the concentration of the antitoxin, and the reverse too:
 +
\begin{equation}
 +
\hat{\AA} = \dfrac{\Gamma + v_d{_{2}} \hspace{0.06cm} \hat{\TT}}{v_{c} \hspace{0.06cm} \Delta \hspace{0.04cm} \hat{\TT} + v_d{_{1}}} \hspace{0.6cm}
 +
%
 +
\hat{\TT} = \dfrac{\Gamma - v_d{_{1}} \hspace{0.06cm} \hat{\AA}}{v_{c} \hspace{0.06cm} \Delta \hspace{0.04cm} \hat{\AA} - v_d{_{2}}} \\[0.1cm]
 +
\end{equation}
 +
 
 +
 
 +
If the four constants $\hspace{0.05cm}\KK_{j}\hspace{0.05cm}$ are the same,
 +
$\hspace{0.12cm} k_{1}=k_{2}=k_{3}=k_{4}\equiv \hspace{0.04cm}k\hspace{0.12cm}$,</p>
 +
 
 +
$\hspace{0.12cm} \hat{\CC} = \dfrac{v_{c} \hspace{0.06cm} \hat{\AA}\hspace{0.04cm}\hat{\TT}}{v_d{_{4}} + v_d{_{3}}} \hspace{0.12cm}$ by the thirth equation $\hspace{0.06cm}$
 +
 
 +
and $\hspace{0.12cm} \hat{\GG} = \dfrac{v_s{_{3}}}{v_d{_{5}}} k \hspace{0.12cm}$ by the fourth equation.</p>
 +
 
 +
-</p>
 +
If $\hspace{0.06cm}v_d{_{4}} >> v_d{_{3}}$, $\hspace{0.14cm} \Delta \hspace{0.02cm}\simeq \mathtt{1} \hspace{0.04cm}> \mathtt{0}.\hspace{0.06cm}$
 +
And if we approximate $\hspace{0.06cm} v_d{_{1}} \simeq v_d{_{4}} \hspace{0.02cm}\equiv V \hspace{0.06cm},
 +
\hspace{0.06cm} v_d{_{2}} \simeq v_d{_{3}} \hspace{0.02cm}\equiv v \hspace{0.06cm}$,
 +
the notation is nicely simplified with $\hspace{0.06cm} V > v .\hspace{0.06cm}$
 +
In a biological sense, this condition can be viewed as the antitoxin is unstable compared with the toxin.</p>
 +
 
 +
\begin{equation}
 +
\Delta = \dfrac{V-v}{V+v} \hspace{0.02cm}\simeq\hspace{0.04cm} \mathtt{1}
 +
\end{equation}
 +
 
 +
\begin{equation}
 +
\hat{\AA} = \dfrac{\Gamma + v \hspace{0.06cm} \hat{\TT}}{v_{c} \hspace{0.06cm}  \hat{\TT} + V} \hspace{0.6cm}
 +
%
 +
\hat{\TT} = \dfrac{\Gamma - V \hspace{0.06cm} \hat{\AA}}{v_{c} \hspace{0.06cm}  \hat{\AA} - v} \\[0.1cm]
 +
\end{equation}
 +
<br>
 +
</section>
 +
 
 +
 
 +
<!-- start temporary comment
 +
<section style="text-align: justify; margin: -30px"></section>
 +
<h3 style="color: #4169E1">Chemical Response</h3>
 +
 
 +
<section style="text-align: justify; margin: -30px"></section>
 +
<section style="text-align: left; margin: 50px">
 +
* (rem: now, in labo, pBAD remplaces T7)</p>
 +
</section>
 +
<section style="text-align: justify; margin: -60px"></section>
 +
<section style="text-align: justify; margin: 80px">
 +
$\underline{ara}$ (inductor) activates the ARN transcription via pBAD :
 +
\begin{equation}
 +
\Rightarrow \left(  v_{s_{j}} \dfrac{a}{a + K_{j}} \right) .
 +
\end{equation}
 +
$\underline{glu}$ (repressor) represses the ARN transcription via pBAD:
 +
\begin{equation}
 +
\Rightarrow \left( v_{s_{j}} \dfrac{K_{j}}{\mathrm{g} + K_{j}} \right)
 +
\end{equation}
 +
$\hspace{0.05cm}chemical \hspace{0.2cm} equas\hspace{0.05cm}$ in elementary processes :
 +
\begin{align*}
 +
n\hspace{0.15cm} (A \hspace{0.25cm}+\hspace{0.3cm} T) \hspace{0.15cm}&\rightarrow^{v_{a/n}}\hspace{0.3cm} C &
 +
\hspace{0.5cm} A \hspace{0.3cm}&\rightarrow^{v_{d_{1}}} \hspace{0.3cm} X \\
 +
C \hspace{0.3cm}-\hspace{0.3cm} A \hspace{0.3cm}&\rightarrow^{v_{D_{4}}} \hspace{0.3cm} 2T \hspace{0.3cm}+\hspace{0.3cm} A &
 +
\hspace{0.5cm} T \hspace{0.3cm}&\rightarrow^{v_{d_{2}}} \hspace{0.3cm} Y \\
 +
C \hspace{0.3cm}-\hspace{0.3cm} T \hspace{0.3cm}&\rightarrow^{v_{D_{3}}} \hspace{0.3cm} 2A \hspace{0.3cm}+\hspace{0.3cm} T &
 +
\hspace{0.5cm}  G \hspace{0.3cm}&\rightarrow^{v_{d_{5}}} \hspace{0.3cm} Z \\
 +
\end{align*}
 +
this signify that
 +
$\hspace{0.3cm}\dfrac{v_{D_{3}}}{2} + v_{D_{4}} = 2 v_{d_{3}} \hspace{0.25cm}$<i>(increasing of 2 A)</i>$ \hspace{0.2cm}$
 +
$\small \& \normalsize$ $\hspace{0.2cm}\dfrac{v_{D_{4}}}{2} + v_{D_{3}} = 2 v_{d_{4}} \hspace{0.25cm}$<i>(increasing of 2 T)</i>$ \hspace{0.1cm}$; </p>
 +
so this is similar to the two recuperation equations for A and T like:
 +
\begin{array} .
 +
\Longrightarrow &C \hspace{0.3cm}\rightarrow^{v_{d_{3}}} \hspace{0.3cm} A \\
 +
\Longrightarrow &C \hspace{0.3cm}\rightarrow^{v_{d_{4}}} \hspace{0.3cm} T
 +
\end{array}
 +
-->
 +
</section>
 +
<!-- <section style="text-align: right">voilou.</section> -->
 +
<section style="text-align: justify; margin: -50px"></section>
 +
<!-- temporary comment ended-->
 +
 
 +
 
 +
<!-- <h2>$\small 2.3.$ $\small Resolving$ $\small the$ $\small System$ </h2> (...)
 +
 
 +
<h2>2.4. Graphs </h2> (...)
 +
 
 +
\begin{array}.
 +
 
 +
\end{array} -->
 +
 
 +
</table>
 +
 
 +
 
 +
</tr>
 +
<font color="white">
 +
</font>
 +
<!-- previous and next pages -->
 +
<tr style="background-color:rgb(204,214,234);"><td>
 +
<section style="text-align: left">
 +
<a href="https://2014.igem.org/Team:ULB-Brussels/Modelling/Population-Dynamics"><b> < Population Dynamics </b></a>
</section>
</section>
 +
<section style="text-align: right">
 +
<a href="https://2014.igem.org/Team:ULB-Brussels/Modelling/2A-Peptid"><b> 2A Pep > </b></a>
 +
</section></tr>
<tr><td><br/><br/></td></tr>
<tr><td><br/><br/></td></tr>
-
</table></th></tr>
+
</th></tr>
</div>
</div>

Latest revision as of 19:23, 15 October 2014

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \newcommand{\MyColi}{{\small Mighty\hspace{0.12cm}Coli}} \newcommand{\Stabi}{\small Stabi}$ $\newcommand{\EColi}{\small E.coli} \newcommand{\SCere}{\small S.cerevisae}\\[0cm] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \newcommand{\PI}{\small PI}$ $\newcommand{\Igo}{\Large\mathcal{I}} \newcommand{\Tgo}{\Large\mathcal{T}} \newcommand{\Ogo}{\Large\mathcal{O}} ~$ Example of a hierarchical menu in CSS

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Retrieved from "http://2014.igem.org/Team:ULB-Brussels/Modelling/TA-System"




- Université Libre de Bruxelles -



Toxin-Antitoxin Systems

$\small 2.1.$ $\small Components$ $\small and$ $\small Diagrams$

Two type II TA systems are investigating in our project. The $\small\mathtt{1}\normalsize^{st}$ consists of ccdB (the toxin, $\mathbb{T}\hspace{0.04cm}$) and ccdA (the antitoxin, $\mathbb{A}\hspace{0.02cm}$) and these are Kid ($\hspace{0.02cm}\mathbb{T}\hspace{0.04cm}$) and Kis ($\hspace{0.02cm}\mathbb{A}\hspace{0.012cm}$) for the $\small\mathtt{2}\normalsize^{nd}$:

2.1.1) CcdBA

One of the most studied and characterized TA systems, CcdBA involves two principal components : ccdB and its antidote ccdA.

Figure m4a : This diagram illustrates the production of F2A peptid and GFP fluorescent protein controlled by PSK, in the case of TA ccdBA system in E.Coli.
As we explained in the introduction page of our project, ccdB is an inhibitor of the DNA gyrase, so it binds the subunit A of the DNA gyrase complex when it is bound to DNA. When DNA double strand is broken, there is activation of SOS emergency signals. Here, the point is: if the DNA gyrase cannot protect itself against ccdB by a mutation (some events are possible, but very rare) or if the antidote is degraded (very frequent because ccdA is unstable in comparison with ccdB), the death of a bacterium in unavoidable. It is why we usually say that bacteria are addicted to the antitoxin to survive.

2.1.2) Kid/Kis

The same is relevant about the second TA system studied. In this case, the two principal components are Kid and its antidote Kis and the parameters are chosen a little bit different than in the first system.

Figure m4b : This diagram illustrates the production of F2A peptid and GFP fluorescent protein controlled by PSK (post-segregational killing), in the case of TA Kid/Kis system in S.Cerevisae.

$\small 2.2.$ $\small Mathematical$ $\small Modelling$

$\newcommand{\AA}{\mathbb{A}} \newcommand{\CC}{\mathbb{C}} \newcommand{\TT}{\mathbb{T}} \newcommand{\GG}{\mathbb{G}} \newcommand{\KK}{\small\mathcal{K}\normalsize}$

2.2.1) Equations from the Diagrams

\begin{align*} \mathbb{A}\hspace{0.02cm}&\equiv[antitoxin] & \mathbb{C}\hspace{0.02cm}&\equiv[\small TA- \normalsize complex]\\[0.1cm] \mathbb{T}\hspace{0.02cm}&\equiv[toxin] & \mathbb{G}\hspace{0.012cm}&\equiv[\small GFP \normalsize]\\[-0.3cm] a\hspace{0.03cm}&\equiv[\mathrm{ara}] & \mathrm{g}\hspace{0.05cm}&\equiv[glu] \hspace{1.45cm} \mathring{x}=\dfrac{dx}{dt} \end{align*} In presence of arabinose, AraC activates the transcription of RNA$\hspace{0.01cm}_{\textbf{m}}$ (catalysed by RNA$\hspace{0.02cm}_{\textbf{poly}}$) : \begin{array}. \hspace{0.02cm}\mathring{\AA} &=& v_{s_{1}} \dfrac{a}{a + \KK_{\mathtt{1}}} - \hspace{0.05cm}v_{d_{1}} \hspace{0.01cm}\AA\hspace{0.02cm} - \hspace{0.05cm}v_{a} \hspace{0.02cm}(\AA\TT)\hspace{0.02cm} + \hspace{0.05cm}v_{d_{3}}\hspace{0.02cm} \CC\hspace{0.03cm} \\[0.1cm] % \hspace{0.02cm}\mathring{\TT} &=& v_{s_{2}} \dfrac{a}{a + \KK_{2}} - \hspace{0.05cm}v_{d_{2}} \hspace{0.03cm}\TT\hspace{0.02cm} - \hspace{0.05cm}v_{a} \hspace{0.02cm}(\AA\TT)\hspace{0.02cm} + \hspace{0.05cm}v_{d_{4}} \hspace{0.02cm}\CC\hspace{0.03cm} \\[0.1cm] % \hspace{0.02cm}\mathring{\CC} &=& v_{a} \hspace{0.02cm}(\AA\TT)\hspace{0.02cm} - \hspace{0.02cm}(v_{d_{3}}+v_{d_{4}}) \hspace{0.05cm}\CC\\[0.25cm] % \hspace{0.02cm}\mathring{\GG} &=& v_{s_{3}} \dfrac{a}{a + \KK_{3}} - \hspace{0.05cm}v_{d_{5}} \hspace{0.04cm}\GG \end{array} In presence of glucose, AraC becomes a repressor of the promotor pBAD, so the three first differential equations (for $\hspace{0.1cm}\AA\hspace{0.04cm}$, $\hspace{0.04cm}\TT\hspace{0.04cm}$ $\small\&\normalsize$ $\hspace{0.04cm}\GG\hspace{0.1cm}$) are modificated by the change : \begin{equation} \left( v_{s_{j}} \dfrac{a}{a + \KK_{j}} \right) \rightarrow \left( v_{s_{j}} \dfrac{\KK_{j}}{\mathrm{g} + \KK_{j}} \right). \end{equation} NB:$\hspace{0.06cm}$ We have chosen a Michaelis-Menten kinetics, maybe a higher Hill coefficient would be desirable.

Because we will preserve some fragment of the population, it is necessary to controle its level. In practical, different parameters are introduced in the mathematical model to describe all the configurations of the biological system (in the equations above, the parameters are the constants $\hspace{0.04cm}\small\mathcal{K}\normalsize_{j}\hspace{0.02cm}$ and the velocities $\hspace{0.04cm}v_{j}\hspace{0.06cm}$). These parameters influence the global dynamics of the TA system, with or without an additional proline via p2A.

By modelling and by comparison with experiments, we hope to obtain finally a model close to the reality.

2.2.2) Stationary State and Stability

At the stationary state, the quantities $\hspace{0.1cm}\mathbb{A}\hspace{0.04cm}$, $\hspace{0.04cm}\mathbb{C}\hspace{0.04cm}$, $\hspace{0.04cm}\mathbb{T}\hspace{0.04cm}$, $\hspace{0.04cm}\mathbb{G}\hspace{0.1cm}$ don't fluctuate in time. We will distinguish it with the symbol hat ^.

First, let's define some new quantities: \begin{align*} k_{j}\hspace{0.02cm}&\equiv \dfrac{a}{a + \KK_{\mathtt{j}}} \hspace{0.3cm}& \Gamma\hspace{0.02cm}&\equiv v_s{_{1}} k_{1} - v_s{_{2}} k_{2} \\[0.3cm] \Delta\hspace{0.02cm}&\equiv \dfrac{v_d{_{4}} - v_d{_{3}}}{v_d{_{4}} + v_d{_{3}}} \hspace{0.3cm}& \Omega\hspace{0.02cm}&\equiv \dfrac{\CC}{v_{a}} \\[0.1cm] \end{align*} Assuming there's only arabinose and combining the two first equations, we obtain the stationary concentration of the toxin like a function of the concentration of the antitoxin, and the reverse too: \begin{equation} \hat{\AA} = \dfrac{\Gamma + v_d{_{2}} \hspace{0.06cm} \hat{\TT}}{v_{c} \hspace{0.06cm} \Delta \hspace{0.04cm} \hat{\TT} + v_d{_{1}}} \hspace{0.6cm} % \hat{\TT} = \dfrac{\Gamma - v_d{_{1}} \hspace{0.06cm} \hat{\AA}}{v_{c} \hspace{0.06cm} \Delta \hspace{0.04cm} \hat{\AA} - v_d{_{2}}} \\[0.1cm] \end{equation} If the four constants $\hspace{0.05cm}\KK_{j}\hspace{0.05cm}$ are the same, $\hspace{0.12cm} k_{1}=k_{2}=k_{3}=k_{4}\equiv \hspace{0.04cm}k\hspace{0.12cm}$,

$\hspace{0.12cm} \hat{\CC} = \dfrac{v_{c} \hspace{0.06cm} \hat{\AA}\hspace{0.04cm}\hat{\TT}}{v_d{_{4}} + v_d{_{3}}} \hspace{0.12cm}$ by the thirth equation $\hspace{0.06cm}$ and $\hspace{0.12cm} \hat{\GG} = \dfrac{v_s{_{3}}}{v_d{_{5}}} k \hspace{0.12cm}$ by the fourth equation.

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If $\hspace{0.06cm}v_d{_{4}} >> v_d{_{3}}$, $\hspace{0.14cm} \Delta \hspace{0.02cm}\simeq \mathtt{1} \hspace{0.04cm}> \mathtt{0}.\hspace{0.06cm}$ And if we approximate $\hspace{0.06cm} v_d{_{1}} \simeq v_d{_{4}} \hspace{0.02cm}\equiv V \hspace{0.06cm}, \hspace{0.06cm} v_d{_{2}} \simeq v_d{_{3}} \hspace{0.02cm}\equiv v \hspace{0.06cm}$, the notation is nicely simplified with $\hspace{0.06cm} V > v .\hspace{0.06cm}$ In a biological sense, this condition can be viewed as the antitoxin is unstable compared with the toxin.

\begin{equation} \Delta = \dfrac{V-v}{V+v} \hspace{0.02cm}\simeq\hspace{0.04cm} \mathtt{1} \end{equation} \begin{equation} \hat{\AA} = \dfrac{\Gamma + v \hspace{0.06cm} \hat{\TT}}{v_{c} \hspace{0.06cm} \hat{\TT} + V} \hspace{0.6cm} % \hat{\TT} = \dfrac{\Gamma - V \hspace{0.06cm} \hat{\AA}}{v_{c} \hspace{0.06cm} \hat{\AA} - v} \\[0.1cm] \end{equation}
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