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
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:
\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)
\\
\\
\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)
\\
\\
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:
\dot{\mathbf{V}}(\mathbf{z,c}) = \mathbf{\beta} \cdotp \mathbf{R}(\mathbf{z,c})
The expected internal state change rate vector.
\\ -\mathbf{\gamma} \cdotp \bar{\mathbf{R}}(\mathbf{z,c})
The expected consumation of substances in the environment by a cell in state
z.
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
\beta, \gamma, \mathbf{R}(\mathbf{z,c}), \sigma (\mathbf{z',c}) and p(\mathbf{z,z',c})
need to be defined for the problem considered. Finally the inital conditions
W_{\mathbf{Z}}(\mathbf{z},t)
and
c_0
and growth conditions D and
c_f
need to be fixed.
For the problem in hand, plasmid maintenance during growth with 2 dif-
ferent plasmids, and attempting to find a simple solution to the problem we propose a 3 variable internal state vector:
$$\mathbf{z} = \begin{bmatrix} z_0 \\ z_1 \\ z_2 \end{bmatrix} = \begin{bmatrix}\textrm{Cell maturity} \\ \textrm{count of plasmid 1} \\ \textrm{count of plasmid 2} \end{bmatrix}$$ \\
In this internal state vector v0 is a mesure of the growth of the bacteria,encompassing such things as size, number of chromosomes and mass, v1 and v2represent the number of copies of each plasmid. For the external conditions wepropose simply the substrate concentration S. The maturity has a minimumvalue of 1 and must increase to 2 before division can occur.
For the rates of change of the internal state vector then we propose for the bacterial maturity to extend the development presented in Shene et al. [?] to include 2 plasmids and incorporate cell maturity as a state variable. This gives:
\dot{z}_0 (\mathbf{z},S) = \mu = \mu _{max} \frac{S}{K_S+S} \frac{K_{z_1}}{K_{z_1}+z_1^{m_1}} \frac{K_{z_2}}{K_{z_2}+z_2^{m_2}} (3)\\
Here
\mu _{max}
is the maximum growth rate $hr^{-1}$: $\mu (\mathbf{z,S})$ the growth rate ; $K_S$ is the Monod constant in $g/l$ for the substrate; $K_{z_1}$ is the inhibition constant for plamid number 1 in (plasmids per cell)$^{m_1}$, and $m_1$ the Hill coefficient for the cooperativity of inhibition. $K_{z_2}$ and $m_2$ represent the same parameters for plasmid 2.
The iGEM team Pretoria 2016 hepled us focusing on the socio-economic and political issues facing the current platinum sector, including the Marikana strikes.
Sharing and collaboration are core values of iGEM. We encourage you to reach out and work with other teams on difficult problems that you can more easily solve together.
You can work with any other team in the competition, including software, hardware, high school and other tracks. You can also work with non-iGEM research groups, but they do not count towards the iGEM team collaboration silver medal criterion.
In order to meet the silver medal criteria on helping another team, you must complete this page and detail the nature of your collaboration with another iGEM team.
