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:
In 1967 Fredrickson \textit{et al.} \cite{Fredrickson1967} studied mathematically development of a
bacterial population, under the assumptions of a large population of independant bacteria in a well
mixed solution of constant volume.
The large population ensures that for the population the expectation value is a good estimate of the
average.
The bacteria being independant ensures that the behaviour of each individual depends only on its internal
state $\mathbf{z}$ and the conditions $\mathbf{c}$ which are the same for all individuals.
The volume is well mixed so the conditions $\mathbf{c}$ which are the same everywhere.
The volume is constant so that the population caracteristics can be evaulated by integration over the volume.
In their development $\mathbf{z}$ and $\mathbf{c}$ are considered to be arbitrary vector quantities.
From this starting point they develop a pair of \textit{master equations of change} to describe the evolution
of the population:
\begin{multline}
\frac{\partial}{\partial t} W_{\mathbf{Z}}(\mathbf{z},t)
+ \nabla_{\mathbf{Z}} \cdotp [(\mathbf{\beta} \cdotp \bar{\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)
\end{multline}
\begin{equation}
\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
\end{equation}
In these equations the various symbols are as follows:\\
\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}\\
\\With these relations:\\
\begin{tabular}{p{0.09\linewidth}p{0.16\linewidth}p{0.63\linewidth}}
$\bar{\dot{\mathbf{V}}}(\mathbf{z,c})$ &$= \mathbf{\beta} \cdotp \bar{\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 $\mathbf{z}$ .\\
\end{tabular}
\\Thus for a particular problem in hand it is necessary to chose $\mathbf{z}$ and $\mathbf{c}$ that represent the state of cells and the media.
Then the matrices and functions $\beta$, $\gamma$, $\bar{\mathbf{R}}(\mathbf{z,c})$, $\sigma (\mathbf{z',c})$ 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 $\mathbf{c}_0 $ and growth conditions $D$ and $\mathbf{c}_f $ need to be fixed.
For the problem in hand, plasmid maintenance during growth with 2 different 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:
$z_0$ is a mesure of the growth of the bacteria, encompassing such things as size, number of chromosomes and mass;
$z_1$ and $z_2$ represent the number of copies of each plasmid.
For the external conditions we propose simply the substrate concentration $S$.
The maturity parameter has a minimum value 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
\textit{et al.} \cite{Shene2003} to include 2 plasmids and incorporate the cell maturity as a state variable. This gives:
\begin{equation}
\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}}
\end{equation}
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.
For plasmid replication rate we propose, again following Shene \textit{et al.} \cite{Shene2003}, the empirical relationship :
\begin{equation}
\dot{z}_1 (\mathbf{z},S) =
\begin{cases}
k_1 \frac{\mu (\mathbf{z},S)}{K_1 + \mu (\mathbf{z},S)} ( z_{1_{max}} - z_1 ) & \text{if } z_1 \geq 1.0 \\
0 & \text{if } 0.0 \leq z_1 < 1.0 \\
\end{cases}
\end{equation}
This relation, and equivalent one for plasmid number 2 $\dot{z}_2 (\mathbf{z,S})$ is designed to satisfy the boundary conditions
of no reproduction if there is less than 1 plasmid in the cell, and a maximum copy number of $z_{1_{max}}$. This introduces the
parameters $k_1$ and $K_1$ which are respectively the plasmid replication rate (in $hr^{-1}$) and the inhibition constant (also in $hr^{-1}$).
The inhibition constant reduces plasmid replication rate at slower growth rates.
Notice that here we have directly defined $\bar{\dot{\mathbf{V}}}(\mathbf{z,c})$ rather than $\beta $ and $\bar{\mathbf{R}}(\mathbf{z,c})$.
For the growth yield we propose :
\begin{equation}
\gamma \cdotp \bar{\mathbf{R}}(\mathbf{z,c}) = \alpha \mu(\mathbf{z},S)
\end{equation}
Where alpha is the growth yield in $g/l/cell$. The remaining functions and parameters in equations 1 and 2 are the
division rate $\sigma (\mathbf{z,c})$ and the partitioning function $p(\mathbf{z,z',c}) $. There is less consensus in the
litterature for an at least empirically appropriate form for these equations. To remain simple we propose:
\begin{equation}
\sigma (\mathbf{z,c}) = \sigma \times H[2.0] =
\begin{cases}
0 & \text{if } z_0 < 2.0 \\
\sigma & \text{if } z_0 \geq 2.0
\end{cases}
\end{equation}
Here we assume that there is a fixed rate of division $\sigma $ once cells are big enough to divide ($H[]$ is the Heaviside function).
\begin{equation}
p(\mathbf{z,z',c}) = p(z_0,z'_0) \times p(z_1,z'_1) \times p(z_2,z'_2)
\end{equation}
\begin{equation}
p(z_0,z'_0) = \delta_{z_0,\frac{z'_0}{2}} = \begin{cases}
1 & \text{if } z_0 = z'_0/2.0 \\
0 & \text{if } z_0 \ne z'_0/2.0.
\end{cases}
\end{equation}
\begin{equation}
p(z_1,z'_1) = 0.5^{z'_1} \begin{pmatrix} z_1 \\ z'_1 \\ \end{pmatrix} = 0.5^{z'_1} \frac{z'_1 !}{(z'_1-z_1)! z_1 !}
\end{equation}
In these equations we assume that the partitioning of the three internal state variables are independant.
That cells divide exactly in half, that is the maturity parameter is exactly halved when the cells divide
($\delta$ is a Kronecker delta function).
That the two plasmids segregate independantly and as individual plasmids according to a binomial distribution.
These assumptions are probably the most suspect in the model.
This initial version of the model has no contention, that is $z_1$ and $z_2$ do not influence the growth rate $\mu $.
In order to develop the model for the system envisaged this needs to be introduced.
Progamming code
Code de françois à ajouter
Results
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.
★ ALERT!
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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.
Which other teams can we work with?
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.
Here are some suggestions for projects you could work on with other teams:
- Improve the function of another team's BioBrick Part or Device
- Characterize another team's part
- Debug a construct
- Model or simulating another team's system
- Test another team's software
- Help build and test another team's hardware project
- Mentor a high-school team
