In this page, the modeling part will be detailed with previews about our biological system and the results with numerical simulations of bacterial populations as well as structural components of the chosen plasmids.

Afterwards, these simulations will be compared with experimental results. In parallel, an estimation of the production in sub-populations cells in bioreactors by coupling the recombinant protein with an essential protein and a numerical estimation will be purposed. Now, we're beginning to perform it.

Population Dynamics Model

The growth of bacteria involves ... a Population Dynamics Model can be fitted in our system. Theorically, two approaches have been planned:

1.1) By Probabilities

When some new plasmids are genetically introduced into the cytoplasm of E.Coli bacteria, this doesn't garantee that the future copies will contain it. Indeed, these plasmids can be lost after cell division or replication, so it's interesting to study a model based on the different possibilities of plasmid combinations in bacteria, like in the studies of mutations in animals. A typical example of a similar way is found if we study the mutations of the eyes color in a family, by vertical genes transfer. In this case, there's a horizontal genes transfer too, originated by the plasmids.

A Probabilistic Model is util because easily undertsood, but necessits some assumptions.

1.2) By Logistic Equation

The Logistic Equation was initially introduced during the beginning of the XIXth Century, by the belgian mathematician P.F. Verhulst. Now, this equation is mainly used in Population Dynamics Models, especially in Biological Sciences.

Toxin-Antitoxin Systems

Two type II TA systems are investigating in our project.

The $\hspace{0.12cm}\small\mathtt{1}\normalsize^{st}$ consists of ccdB (the toxin, T) and ccdA (the antitoxin, A) and for the $\hspace{0.12cm}\small\mathtt{2}\normalsize^{nd}$ these are Kid (T) and Kis (A):

2.1) CcdBA

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. 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.

\begin{equation} \hspace{-0.12cm}\wp\hspace{0.02cm}\equiv[p\hspace{0.03cm} 2A]\\ \mathbb{A}\hspace{0.02cm}\equiv[ccdA]\\ \mathbb{B}\hspace{0.02cm}\equiv[ccdB]\\ \mathbb{C}\hspace{0.02cm}\equiv[(ccdA)_{2}-(ccdB)_{2}]\\ a\equiv[\mathrm{ara}] \hspace{0.12cm}|\hspace{0.12cm} \mathrm{g}\equiv[glu]\\[0.12cm] \mathring{x}= \dfrac{dx}{dt} \end{equation} In presence of arabinose, AraC activates the transcription of RNAm (catalysed by RNApoly): \begin{array}. \hspace{0.06cm}\mathring{\wp} &=& v_{s_{0}} \dfrac{a}{a + \small\mathcal{K}\normalsize_{0}} - \hspace{0.05cm}v_{d_{0}} \hspace{0.04cm}\wp \\[0.05cm] \hspace{0.02cm}\mathring{\mathbb{A}} &=& v_{s_{1}} \dfrac{a}{a + \small\mathcal{K}\normalsize_{\mathtt{1}}} . \dfrac{\wp}{\wp + \small\mathcal{K}\normalsize_{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 + \small\mathcal{K}\normalsize_{2}} . \dfrac{\wp}{\wp + \small\mathcal{K}\normalsize_{\mathtt{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] \mathring{\mathbb{C}} &=& v_{a} \hspace{0.02cm}(\mathbb{A}\mathbb{B}) - (v_{d_{3}}+v_{d_{4}}) \hspace{0.05cm}\mathbb{C} \end{array} In presence of glucose, AraC becomes a repressor of the promotor pBAD, so the differential equations for $\hspace{0.1cm}\wp\hspace{0.04cm}$, $\hspace{0.04cm}\mathbb{A}\hspace{0.04cm}$ $\small\&\normalsize$ $\hspace{0.04cm}\mathbb{B}\hspace{0.1cm}$ are modificated by the change: \begin{equation} \left( v_{s_{j}} \dfrac{a}{a + \small\mathcal{K}\normalsize_{j}} \right) \rightarrow \left( v_{s_{j}} \dfrac{\small\mathcal{K}\normalsize_{j}}{\mathrm{g} + \small\mathcal{K}\normalsize_{j}} \right). \end{equation} NB: 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 with p2A.

By modelling and by comparison with experiments, we hope to obtain finally a model that correctly describes our TA system.

2.2) Kis/Kid

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.