Difference between revisions of "Team:Aix-Marseille/Collaborations"

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<div lang="latex">$$\mathbf{z} =  \begin{bmatrix} z_0 \\ z_1 \\ z_2 \end{bmatrix} = \begin{bmatrix}\textrm{Cell maturity} \\ \textrm{count of plasmid 1} \\ \textrm{count of plasmid 2} \end{bmatrix}$$ \\</div>
 
<div lang="latex">$$\mathbf{z} =  \begin{bmatrix} z_0 \\ z_1 \\ z_2 \end{bmatrix} = \begin{bmatrix}\textrm{Cell maturity} \\ \textrm{count of plasmid 1} \\ \textrm{count of plasmid 2} \end{bmatrix}$$ \\</div>
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<div lang="latex">
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In 1967 Fredrickson \textit{et al.} \cite{Fredrickson1967} studied mathematically development of a
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bacterial population, under the assumptions of a large population of independant bacteria in a well
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mixed solution of constant volume.
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The large population ensures that for the population the expectation value is a good estimate of the
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average.
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The bacteria being independant ensures that the behaviour of each individual depends only on its internal
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state $\mathbf{z}$ and the conditions $\mathbf{c}$ which are the same for all individuals.
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The volume is well mixed so the conditions $\mathbf{c}$ which are the same everywhere.
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The volume is constant so that the population caracteristics can be evaulated by integration over the volume.
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In their development $\mathbf{z}$ and $\mathbf{c}$ are considered to be arbitrary vector quantities.
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From this starting point they develop a pair of \textit{master equations of change} to describe the evolution
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of the population:
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\begin{multline}
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\frac{\partial}{\partial t} W_{\mathbf{Z}}(\mathbf{z},t)
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+ \nabla_{\mathbf{Z}} \cdotp [(\mathbf{\beta} \cdotp \bar{\mathbf{R}}(\mathbf{z,c}) W_{\mathbf{Z}}(\mathbf{z},t)] \\
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= 2 \int \sigma (\mathbf{z',c}) p(\mathbf{z,z',c}) W_{\mathbf{Z}}(\mathbf{z'},t)\mathrm{d}v'
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- (D+\sigma (\mathbf{z,c})) W_{\mathbf{Z}}(\mathbf{z},t)
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\end{multline}
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\begin{equation}
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\frac{d\mathbf{c}}{dt}
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= D(\mathbf{c_f} - \mathbf{c} ) + \mathbf{\gamma}
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\cdotp \int  \bar{\mathbf{R}}(\mathbf{z,c}) W_{\mathbf{Z}}(\mathbf{z},t)\mathrm{d}v
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\end{equation}
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In these equations the various symbols are as follows:\\
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\begin{tabular}{p{0.10\linewidth}p{0.75\linewidth}}
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$\mathbf{z}                          $& Vector for internal state of a bacteria.\\
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$\mathbf{c}                          $& Time dependant vector for conditions.\\
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$W_{\mathbf{Z}}(\mathbf{z},t)        $& Distribution of bacteria in $\mathbf{z}$ space at time $t$.\\
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$\bar{\mathbf{R}}(\mathbf{z,c})      $& The expected value or the reaction rate vector function of $\mathbf{z}$ and $\mathbf{c}$.\\
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$\sigma (\mathbf{z',c})              $& Rate of fision for bacteria as a scalar function of $ \mathbf{z,c} $.\\
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$p(\mathbf{z,z',c})                  $& Partitioning probability of generating a child in state $\mathbf{z}$ from a parent
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                                        in state $\mathbf{z'}$ given the conditions $\mathbf{c}$.\\
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$\nabla_{\mathbf{Z}}\cdot \mathbf{V} $& $\sum \frac{\partial}{\partial z_i}\mathbf{V}_i $\\
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$\textrm{d}v'                        $& Integral over state space $v'$ .\\
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$D                                  $& Dilution rate of the culture (for fermenters). \\
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$\beta                              $& Stochiometric matrix for cellular substances.\\
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$\gamma                              $& Stochiometric matrix for extra-cellular substances.\\
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\end{tabular}\\
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\\With these relations:\\
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\begin{tabular}{p{0.09\linewidth}p{0.16\linewidth}p{0.63\linewidth}}
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$\bar{\dot{\mathbf{V}}}(\mathbf{z,c})$ &$= \mathbf{\beta} \cdotp \bar{\mathbf{R}}(\mathbf{z,c}) $&The expected internal state change rate vector.\\
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&$ -\mathbf{\gamma} \cdotp \bar{\mathbf{R}}(\mathbf{z,c}) $&The expected consumation of substances in the environment by a cell in state $\mathbf{z}$ .\\
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\end{tabular}
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\\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.
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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.
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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.
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For the problem in hand, plasmid maintenance during growth with 2 different plasmids, and attempting to find a simple solution
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to the problem we propose a 3 variable internal state vector:\\
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$$
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\mathbf{z} =  \begin{bmatrix} z_0 \\ z_1 \\ z_2 \end{bmatrix} =
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\begin{bmatrix}\textrm{Cell maturity} \\ \textrm{count of plasmid 1} \\ \textrm{count of plasmid 2} \end{bmatrix}
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$$ \\
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In this internal state vector:
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$z_0$ is a mesure of the growth of the bacteria, encompassing such things as size, number of chromosomes and mass;
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$z_1$ and $z_2$ represent the number of copies of each plasmid.
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For the external conditions we propose simply the substrate concentration $S$.
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The maturity parameter has a minimum value of 1 and must increase to 2 before division can occur.
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For the rates of change of the internal state vector then we propose for the bacterial maturity to extend the development presented in Shene
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\textit{et al.} \cite{Shene2003} to include 2 plasmids and incorporate the cell maturity as a state variable. This gives:
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\begin{equation}
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\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}}
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\end{equation}
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Here $ \mu _{max}$ is the maximum growth rate $hr^{-1}$:
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$\mu (\mathbf{z,S})$ the growth rate ;
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$K_S$ is the Monod constant in $g/l$ for the substrate;
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$K_{z_1}$ is the inhibition constant for plamid number 1 in (plasmids per cell)$^{m_1}$,
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and $m_1$ the Hill coefficient for the cooperativity of inhibition.
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$K_{z_2}$ and $m_2$ represent the same parameters for plasmid 2.
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For plasmid replication rate we propose, again following Shene \textit{et al.} \cite{Shene2003}, the empirical relationship :
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\begin{equation}
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\dot{z}_1 (\mathbf{z},S) =
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  \begin{cases}
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  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 \\
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  0 & \text{if } 0.0 \leq z_1 < 1.0 \\
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  \end{cases}         
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\end{equation}
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This relation, and equivalent one for plasmid number 2 $\dot{z}_2 (\mathbf{z,S})$ is designed to satisfy the boundary conditions
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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
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parameters $k_1$ and $K_1$ which are respectively the plasmid replication rate (in $hr^{-1}$) and the inhibition constant (also in $hr^{-1}$).
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The inhibition constant reduces plasmid replication rate at slower growth rates.
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Notice that here we have directly defined $\bar{\dot{\mathbf{V}}}(\mathbf{z,c})$ rather than $\beta $ and $\bar{\mathbf{R}}(\mathbf{z,c})$.
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For the growth yield we propose :
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\begin{equation}
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\gamma \cdotp \bar{\mathbf{R}}(\mathbf{z,c}) = \alpha \mu(\mathbf{z},S)
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\end{equation}
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Where alpha is the growth yield in $g/l/cell$. The remaining functions and parameters in equations 1 and 2 are the
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division rate $\sigma (\mathbf{z,c})$ and the partitioning function $p(\mathbf{z,z',c}) $. There is less consensus in the
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litterature for an at least empirically appropriate form for these equations. To remain simple we propose:
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\begin{equation}
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\sigma (\mathbf{z,c}) = \sigma \times H[2.0] =
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\begin{cases}
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  0      & \text{if } z_0 < 2.0 \\
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  \sigma & \text{if } z_0 \geq 2.0
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\end{cases}
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\end{equation}
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Here we assume that there is a fixed rate of division $\sigma $ once cells are big enough to divide ($H[]$ is the Heaviside function).
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\begin{equation}
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p(\mathbf{z,z',c}) = p(z_0,z'_0) \times p(z_1,z'_1) \times p(z_2,z'_2)
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\end{equation}
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\begin{equation}
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p(z_0,z'_0) = \delta_{z_0,\frac{z'_0}{2}} = \begin{cases}
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                1 & \text{if } z_0 = z'_0/2.0 \\
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                0 & \text{if } z_0 \ne z'_0/2.0.
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              \end{cases}
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\end{equation}
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\begin{equation}
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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 !}
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\end{equation}
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In these equations we assume that the partitioning of the three internal state variables are independant.
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That cells divide exactly in half, that is the maturity parameter is exactly halved when the cells divide
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($\delta$ is a Kronecker delta function).
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That the two plasmids segregate independantly and as individual plasmids according to a binomial distribution.
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These assumptions are probably the most suspect in the model.
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This initial version of the model has no contention, that is $z_1$ and $z_2$ do not influence the growth rate $\mu $.
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In order to develop the model for the system envisaged this needs to be introduced.
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Revision as of 15:12, 17 October 2016