Collaborations
Our collaborations
Introduction
We conceived a model in order to handle questions concerning the following situation:
A bacterial growth is carried out in a bioreactor, continually supplied in substrate.
Bacteria can possess 2 type of plasmids:
- plasmids 1 carry the toxin A gene and the anti-toxin B gene
- plasmids 2 carry the toxin B gene and the anti-toxin A gene
So during the growth each bacterium can have no plasmids, either one type of plasmids, or both types.
The aim of the model is to assess the evolution of plasmids throughout the culture, to determine which parameters can matter in the loss of those plasmids, and to precise what are the probabilities for a bacteria to loose its plasmids during cell division.
As a plasmids can be a disadvantage for growth (energy spent into replicating processes) or a advantage (protection against a toxin) this question is hard to answer. But in this situation, where one type of plasmid can influence on the presence of the other type of plasmid in (and reciprocally) in the same bacteria, the question become too tough to answer and only a mathematical model can resolve such a interrogation!
Equations
EQUATIONS DE 1 A 17
To creat our model, we had to determine some equations.
\frac{\partial}{\partial t} W_\mathbf{z} (\mathbf{z},t) + \nabla_\mathbf{z}\cdot[(\beta\cdot\overline{\mathbf{R}}(\mathbf{z},c)W_\mathbf{z}(\mathbf{z},t))] = 2 \int \sigma (\mathbf{z',c}) p(\mathbf{z,z',c}) W_{\mathbf{Z}}(\mathbf{z'},t)\mathrm{d}v' - (D+\sigma (\mathbf{z,c})) W_{\mathbf{Z}}(\mathbf{z},t) (1)
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\frac{d\mathbf{c}}{dt} = D(\mathbf{c_f} - \mathbf{c} ) + \mathbf{\gamma}\cdotp \int \bar{\mathbf{R}}(\mathbf{z,c}) W_{\mathbf{Z}}(\mathbf{z},t)\mathrm{d}v (2)
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div lang="latex">
\begin{tabular}{p{0.10\linewidth}p{0.75\linewidth}}
\mathbf{z} & Vector for internal state of a bacteria.\\
\mathbf{c} & Time dependant vector for conditions.\\
W_{\mathbf{Z}}(\mathbf{z},t) & Distribution of bacteria in $\mathbf{z}$ space at time $t$.\\
\bar{\mathbf{R}}(\mathbf{z,c}) & The expected value or the reaction rate vector function of $\mathbf{z}$ and \mathbf{c}$.\\
\sigma (\mathbf{z',c}) & Rate of fision for bacteria as a scalar function of $ \mathbf{z,c} $.\\
p(\mathbf{z,z',c}) & Partitioning probability of generating a child in state $\mathbf{z}$ from a parent
in state $\mathbf{z'}$ given the conditions $\mathbf{c}$.\\
\nabla_{\mathbf{Z}}\cdot \mathbf{V} & $\sum \frac{\partial}{\partial z_i}\mathbf{V}_i $\\
\textrm{d}v' & Integral over state space $v'$ .\\
D & Dilution rate of the culture (for fermenters). \\
\beta & Stochiometric matrix for cellular substances.\\
\gamma & Stochiometric matrix for extra-cellular substances.\\
\end{tabular}\\
