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:
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!
In 1967 Fredrickson et al. studied mathematically development of a bacterialpopulation, under the assumptions of a large population of independant bacteriain a well mixed solution of constant volume. The large population ensures thatfor the population the expectation value is a good estimate of the average.The bacteria being independant ensures that the behaviour of each individualdepends only on its internal state z and the conditions c which are the samefor all individuals. The volume is well mixed so the conditions c which are thesame everywhere. The volume is constant so that the population caracteristicscan be evaulated by integration over the volume.
From this starting point they develop a pair of master equations of change to describe the evolution of the population:
In these equations the various symbols are as follows:
\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 z, t space. |
\overline{\mathbf{R}}(\mathbf{z},c)
|
The expected value or the reaction rate vector in z, t space. |
\sigma (\mathbf{z,c})
|
Rate of fision for bacteria in z, c space. |
p(\mathbf{z,z',c}))
|
Partitioning probability of generating a child in state z from a parent in state z'. |
\nabla_\mathbf{z}\cdot\mathbf{V}
|
\sum \frac{\partial}{\partial z_i}\mathbf{V}_i
|
\mathrm{d}v'
|
Integral over state space v' |
| D | Dilution rate of the culture (for femrenters). |
\beta
|
Stochiometric matrix for cellular substances. |
\gamma
|
Stochiometric matrix for extra-cellular substances. |
With these relations:
Thus for a particular problem in hand it is necessary to chose z and c that represent the state of cells and the media. Then the matrices and functions
Code de françois à ajouter
METTRE LES COURBE
In order to help the iGEM team Bordeaux, we had to read the paper « Dynamics of plasmid transfer on surfaces » and collect and organize some data about their experiments on plasmids transfer. Thanks to this, the iGEM team Bordeaux can compare those results to their own computaionnal results.
The iGEM team Pretoria 2016 hepled us focusing on the socio-economic and political issues facing the current platinum sector, including the Marikana strikes.
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