Equas-Réponse pour Quentin
(rem: actuellement, en labo, pBAD remplace T7)
$\underline{ara}$ (inducteur) active la transcription d'ARN via pBAD :
\begin{equation}
\Rightarrow \left( v_{s_{j}} \dfrac{a}{a + K_{j}} \right) .
\end{equation}
$\underline{glu}$ (répresseur) réprime la transcription d'ARN via pBAD :
\begin{equation}
\Rightarrow \left( v_{s_{j}} \dfrac{K_{j}}{\mathrm{g} + K_{j}} \right)
\end{equation}
$\underline{\hspace{0.05cm}équas \hspace{0.2cm} chimq\hspace{0.05cm}}$, en processus élémentaires :
\begin{align*}
A \hspace{0.3cm}+\hspace{0.3cm} B \hspace{0.3cm}&\rightarrow^{v_{a}}\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*}
tu as les relations
$\hspace{0.3cm}\dfrac{v_{D_{3}}}{2} + v_{D_{4}} = 2 v_{d_{3}} \hspace{0.25cm}$(ajout de 2 A)$ \hspace{0.1cm}$
et $\hspace{0.3cm}\dfrac{v_{D_{4}}}{2} + v_{D_{3}} = 2 v_{d_{4}} \hspace{0.25cm}$(ajout de 2 T)$ \hspace{0.1cm}$,
donc tu peux récrire plus simplement les deux équas de récupération de A et T selon:
\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}
voilou.
Toxin-Antitoxin Systems
2.1. Components and 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 3a : 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's bound to DNA.
When DNA double strand is broken, there is activation of SOS emergency signals. 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. It's 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 3b : 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.
2.2. Mathematical 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 related to 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'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 $\hspace{0.04cm}\small\mathcal{K}\normalsize_{j}\hspace{0.02cm}$ 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'll 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.
-
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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- Université Libre de Bruxelles -
