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

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== Our collaborations ==
 
== Our collaborations ==
  
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[[File:T--Aix-Marseille--Toulouse_logo.jpg|link=https://2016.igem.org/Team:Toulouse_France|300px|350px|center]]
 
[[File:T--Aix-Marseille--Toulouse_logo.jpg|link=https://2016.igem.org/Team:Toulouse_France|300px|350px|center]]
  
−
===='''Introduction'''====
+
====Introduction====
 
We conceived a model in order to handle questions concerning the following situation:
 
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.
 
A bacterial growth is carried out in a bioreactor, continually supplied in substrate.
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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!
 
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====
  
 
=====Development of the model=====
 
=====Development of the model=====
  
−
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.
+
In 1967 Fredrickson et al. <ref name="Fredrickson1967" group="toulouse">[https://pdfs.semanticscholar.org/1873/0fa936b17078cfe2b0ab1f74d44eae002758.pdf Fredrickson AG, Ramkrishna D, Tsuchiya H (1967) Statistics and dynamics of correct pro- caryotic cell populations. Mathematical Biosciences 1: 327–374.]</ref> 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 master equations of change
+
From this starting point they develop a pair of *master equations of
−
to describe the evolution of the population:
+
change* to describe the evolution of the population:  
  
−
<html>
+
<nowiki>
−
 
+
\begin{equation}
−
<div lang="latex">
+
\begin{gathered}
−
\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))]
+
\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'
−
  = 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) </div>(1)<div lang="latex">
+
- (D+\sigma (\mathbf{z,c})) W_{\mathbf{Z}}(\mathbf{z},t)\end{gathered}$$
−
\\
+
\end{equation}
−
\\
+
\begin{equation}
−
<br/>
+
\frac{d\mathbf{c}}{dt}  
−
</div><div lang="latex">
+
= D(\mathbf{c_f} - \mathbf{c} ) + \mathbf{\gamma}  
−
\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 </div>(2)
+
\cdotp \int  \bar{\mathbf{R}}(\mathbf{z,c}) W_{\mathbf{Z}}(\mathbf{z},t)\mathrm{d}v
−
 
+
\end{equation}
−
</html>
+
</nowiki>
  
 
In these equations the various symbols are as follows:
 
In these equations the various symbols are as follows:
  
 
{| class="wikitable"
 
{| class="wikitable"
−
|<div lang="latex">\mathbf{z}</div>
+
|$\mathbf{z}$
 
| Vector for internal state of a bacteria.
 
| Vector for internal state of a bacteria.
 
|-
 
|-
−
|<div lang="latex">\mathbf{c}</div>
+
|$\mathbf{c}$
 
| Time dependant vector for conditions.
 
| Time dependant vector for conditions.
 
|-
 
|-
−
|<div lang="latex"> W_\mathbf{z} (\mathbf{z},t)</div>
+
|$W_\mathbf{z} (\mathbf{z},t)$
 
| Distribution of bacteria in '''z''', t space.
 
| Distribution of bacteria in '''z''', t space.
 
|-
 
|-
−
|<div lang="latex">\overline{\mathbf{R}}(\mathbf{z},c)</div>
+
|$\overline{\mathbf{R}}(\mathbf{z},c)$
 
| The expected value or the reaction rate vector in '''z''', t space.
 
| The expected value or the reaction rate vector in '''z''', t space.
 
|-
 
|-
−
|<div lang="latex">\sigma (\mathbf{z,c})</div>
+
|$\sigma (\mathbf{z,c})$
 
| Rate of fision for bacteria in '''z''', '''c''' space.
 
| Rate of fision for bacteria in '''z''', '''c''' space.
 
|-
 
|-
−
|<div lang="latex">p(\mathbf{z,z',c}))</div>
+
|$p(\mathbf{z,z',c}))$
 
| Partitioning probability of generating a child in state z from a parent in state '''z''''.
 
| Partitioning probability of generating a child in state z from a parent in state '''z''''.
 
|-
 
|-
−
|<div lang="latex">\nabla_\mathbf{z}\cdot\mathbf{V}</div>
+
|$\nabla_\mathbf{z}\cdot\mathbf{V}$
−
|<div lang="latex">\sum \frac{\partial}{\partial z_i}\mathbf{V}_i</div>
+
|$\sum \frac{\partial}{\partial z_i}\mathbf{V}_i$
 
|-
 
|-
−
|<div lang="latex">\mathrm{d}v'</div>
+
|$\mathrm{d}v'$
 
| Integral over state space v'
 
| Integral over state space v'
 
|-
 
|-
−
| D
+
| $D$
 
| Dilution rate of the culture (for femrenters).
 
| Dilution rate of the culture (for femrenters).
 
|-
 
|-
−
|<div lang="latex">\beta</div>
+
|$\beta$
 
| Stochiometric matrix for cellular substances.
 
| Stochiometric matrix for cellular substances.
 
|-
 
|-
−
|<div lang="latex">\gamma</div>
+
|$\gamma$
 
| Stochiometric matrix for extra-cellular substances.
 
| Stochiometric matrix for extra-cellular substances.
 
|-
 
|-
 
|}
 
|}
  
−
<html>
+
{| class="wikitable"
−
With these relations:
+
|$\bar{\dot{\mathbf{V}}}(\mathbf{z,c}) = \mathbf{\beta} \cdotp \bar{\mathbf{R}}(\mathbf{z,c}) $
−
<div lang="latex">\dot{\mathbf{V}}(\mathbf{z,c}) = \mathbf{\beta} \cdotp \mathbf{R}(\mathbf{z,c})</div> The expected internal state change rate vector.
+
|The expected internal state change rate vector.
−
<div lang="latex">\\ -\mathbf{\gamma} \cdotp \bar{\mathbf{R}}(\mathbf{z,c})</div> The expected consumation of substances in the environment by a cell in state <b>z</b>.<br/>
+
|-
 +
|$ -\mathbf{\gamma} \cdotp \bar{\mathbf{R}}(\mathbf{z,c}) $
 +
|The expected consumation of substances in the environment by a cell in state $\mathbf{z}$ .
 +
|}
  
−
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 <div lang="latex">\beta, \gamma, \mathbf{R}(\mathbf{z,c}), \sigma (\mathbf{z',c}) and p(\mathbf{z,z',c})</div> need to be defined for the problem considered. Finally the inital conditions <div lang="latex">W_{\mathbf{Z}}(\mathbf{z},t)</div> and <div lang="latex">c_0</div> and growth conditions D and <div lang="latex">c_f</div> need to be fixed.
 
  
−
<p>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:</p>
+
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.
  
−
<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><br/><br/>
+
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:
  
−
In this internal state vector <div lang="latex">v0</div> is a mesure of the growth of the bacteria,encompassing such things as size, number of chromosomes and mass, <div lang="latex">v1</div> and <div lang="latex">v2</div> represent 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.
+
$$\mathbf{z} = \begin{bmatrix} z_0 \\ z_1 \\ z_2 \end{bmatrix} =  
 +
\begin{bmatrix}\textrm{Cell maturity} \\ \textrm{count of plasmid 1} \\ \textrm{count of plasmid 2} \end{bmatrix}$$
  
−
<p>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:</p>
+
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.
  
−
<div lang="latex">\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}} \\</div>(3)<br/><br/>
+
For the rates of change of the internal state vector then we propose for
−
Here <div lang="latex">\mu _{max}</div> is the maximum growth rate <div lang="latex">hr^{-1}: $\mu (\mathbf{z,S})</div> the growth rate ; <div lang="latex">K_S</div> is the Monod constant in g/l for the substrate; <div lang="latex">K_{z_1}</div> is the inhibition constant for plamid number 1 in (plasmids per cell)<div lang="latex">^{m_1}</div>, and <div lang="latex">m_1</div> the Hill coefficient for the cooperativity of inhibition. <div lang="latex">K_{z_2}</div> and <div lang="latex">m_2</div> represent the same parameters for plasmid 2.
+
the bacterial maturity to extend the development presented in Shene et
 +
al. <ref name="Shene2003" group="toulouse">[https://dx.doi.org/10.1007/s00449-002-0313-x Shene, C., Andrews, B.A. & Asenjo, J.A. Bioprocess Biosyst Eng (2003) 25: 333. doi:10.1007/s00449-002-0313-x]</ref> to include 2 plasmids and incorporate the cell
 +
maturity as a state variable. This gives:
  
−
For plasmid replication rate we propose, again following Shene et al. [?], the empirical relationship :<br/><br/>
+
<nowiki>
−
<div lang="latex"> \dot{z}_1 (\mathbf{z},S) = k_1 \frac{\mu (\mathbf{z},S)}{K_1 + \mu (\mathbf{z},S)} ( z_{1_{max}} - z_1 ) if z_1 \geq 1.0 or  \dot{z}_1 (\mathbf{z},S) = 0 if 0.0 \leq z_1 < 1.0</div> (4)<br/><br/>
+
\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}
 +
</nowiki>
  
−
This relation, and equivalent one for plasmid number 2 <div lang="latex">\dot{z}_2 (\mathbf{z,S})</div> 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 <div lang="latex">z_{1_{max}}</div>. This introduces the parameters <div lang="latex">k_1</div> and <div lang="latex">K_1</div> which are respectively the plasmid replication rate (in <div lang="latex">hr^{-1}</div>) and the inhibition constant (also in <div lang="latex">hr^{-1}</div>).<br/>
+
Here $ \mu _{max}$ is the maximum growth rate $hr^{-1}$:
−
The inhibition constant reduces plasmid replication rate at slower growth rates.
+
$\mu (\mathbf{z,S})$ 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.
  
−
Notice that here we have directly defined <div lang="latex">\bar{\dot{\mathbf{V}}}(\mathbf{z,c})</div> rather than <div lang="latex">\beta </div> and <div lang="latex">\bar{\mathbf{R}}(\mathbf{z,c})</div>.<br/>
+
For plasmid replication rate we propose, again following Shene et
−
For the growth yield we propose :<br/><br/>
+
al. <ref name="Shene2003" group="toulouse" />, the empirical relationship :
  
−
<div lang="latex">\gamma \cdotp \bar{\mathbf{R}}(\mathbf{z,c}) = \alpha \mu(\mathbf{z},S)</div>(5)<br/><br/>
+
<nowiki>
 +
\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}
 +
</nowiki>
  
−
Where alpha is the growth yield in <div lang="latex">g/l/cell</div>. The remaining functions and parameters in equations 1 and 2 are the division rate <div lang="latex">\sigma (\mathbf{z,c})</div> and the partitioning 2 function <div lang="latex">p(\mathbf{z,z',c})</div>. There is less consensus in the litterature for an at least empirically appropriate form for these equations. To remain simple we propose:<br/><br/>
+
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.
  
−
<div lang="latex">\sigma (\mathbf{z,c}) = \sigma \times H[2.0] = 0  if z_0 < 2.0 \\ \sigma (\mathbf{z,c}) = \sigma \times H[2.0] = \sigma if z_0 \geq 2.0</div>(6)<br/><br/>
+
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 :
  
−
Here we assume that there is a fixed rate of division <div lang="latex">\sigma</div> once cells are big enough to divide (<div lang="latex">H[]</div> is the Heaviside function).<br/><br/>
+
<nowiki>
 +
\begin{equation}
 +
\gamma \cdotp \bar{\mathbf{R}}(\mathbf{z,c}) = \alpha \mu(\mathbf{z},S)
 +
\end{equation}
 +
</nowiki>
  
−
<div lang="latex">p(\mathbf{z,z',c}) = p(z_0,z'_0) \times p(z_1,z'_1) \times p(z_2,z'_2)</div> (7)<br/><br/>
+
The remaining functions
−
<div lang="latex">p(z_0,z'_0) = \delta_{z_0,\frac{z'_0}{2}} = 1 if  z_0 = z'_0/2.0 \\
+
and parameters in equations 1 and 2 are the division rate
−
p(z_0,z'_0) = \delta_{z_0,\frac{z'_0}{2}} = 0 if z_0 \ne z'_0/2.0.</div> (8)<br/><br/>
+
$\sigma (\mathbf{z,c})$ and the partitioning function
−
<div lang="latex">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 !}</div>(9)<br/><br/>
+
$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:
  
−
In these equations we assume that the partitioning of the three internal state variables are independant.
+
<nowiki>
−
That cells divide exactly in half, that is the maturity parameter is exactly halved when the cells divide
+
\begin{equation}
−
(<div lang="latex">\delta</div> is a Kronecker delta function).
+
\sigma (\mathbf{z,c}) = \sigma \times H[2.0] =
−
That the two plasmids segregate independantly and as individual plasmids according to a binomial distribution.  
+
\begin{cases}
−
These assumptions are probably the most suspect in the model.
+
  0      & \text{if } z_0 < 2.0 \\
 +
  \sigma & \text{if } z_0 \geq 2.0
 +
\end{cases}
 +
\end{equation}
 +
</nowiki>
  
−
This initial version of the model has no contention, that is <div lang="latex">z_1</div> and <div lang="latex">z_2</div> do not influence the growth rate <div lang="latex">\mu </div>.
+
Here we assume that there is a fixed rate of division
−
In order to develop the model for the system envisaged this needs to be introduced.<br/>
+
$\sigma $ once cells are big enough to divide ($H[]$ is the Heaviside
 +
function).
  
−
Substituting into the equations 1 and 2 we obtain:<br/><br/>
+
<nowiki>
 +
\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}
 +
</nowiki>
  
−
<div lang="latex">\frac{\partial}{\partial t} W_{\mathbf{Z}}(\mathbf{z},t)  
+
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.
 +
 
 +
Substituting into the equations 1 and 2 we obtain:
 +
 
 +
<nowiki>
 +
\begin{multline}
 +
  \frac{\partial}{\partial t} W_{\mathbf{Z}}(\mathbf{z},t)  
 
  + [\nabla_{\mathbf{Z}} \cdotp \bar{\dot{\mathbf{Z}}}(\mathbf{z},S)] W_{\mathbf{Z}}(\mathbf{z},t)
 
  + [\nabla_{\mathbf{Z}} \cdotp \bar{\dot{\mathbf{Z}}}(\mathbf{z},S)] W_{\mathbf{Z}}(\mathbf{z},t)
−
  + \sum_i \bar{\dot{z_i}} \times \frac{\partial}{\partial z_i} W_{\mathbf{Z}}(\mathbf{z},t)\\ </div>
+
  + \sum_i \bar{\dot{z_i}} \times \frac{\partial}{\partial z_i} W_{\mathbf{Z}}(\mathbf{z},t) \\
−
<div lang="latex">= 2 \sigma \int_{z'_0>2.0} \delta_{z_0,\frac{z'_0}{2}} \times 0.5^{z'_1+z'_2} \times \begin{pmatrix} z_1 \\ z'_1 \\ \end{pmatrix} \times \begin{pmatrix} z_2 \\ z'_2 \\ \end{pmatrix} \times
+
= 2 \sigma \int_{z'_0>2.0} \delta_{z_0,\frac{z'_0}{2}} \times 0.5^{z'_1+z'_2} \times \begin{pmatrix} z_1 \\ z'_1 \\ \end{pmatrix}  
−
  W_{\mathbf{Z}}(\mathbf{z'},t)\mathrm{d}v' \\</div>
+
\times \begin{pmatrix} z_2 \\ z'_2 \\ \end{pmatrix} \times
−
<div lang="latex">- (D+\sigma H[2.0] W_{\mathbf{Z}}(\mathbf{z},t))</div>(10)<br/><br/>
+
  W_{\mathbf{Z}}(\mathbf{z'},t)\mathrm{d}v' \\
−
<div lang="latex"> \frac{d\mathbf{c}}{dt}  
+
- (D+\sigma H[2.0] W_{\mathbf{Z}}(\mathbf{z},t))
−
  = D(c_f - c ) - \alpha \int  \mu (\mathbf{z},S) W_{\mathbf{Z}}(\mathbf{z},t)\mathrm{d}vv</div>(11)<br/><br/>
+
\end{multline}
−
</html>
+
\begin{equation}
 +
\frac{d\mathbf{c}}{dt}  
 +
  = D(c_f - c ) - \alpha \int  \mu (\mathbf{z},S) W_{\mathbf{Z}}(\mathbf{z},t)\mathrm{d}v
 +
\end{equation}
 +
</nowiki>
  
 
=====Objectives=====
 
=====Objectives=====
  
−
The aim of studying the behaviour of this model is to investigate how growth conditions <div lang="latex">D, S_f</div> and time modulate the development of the population in state space <div lang="latex">W_{\mathbf{Z}}(\mathbf{z},t)</div>. In particular we are interested in finding how the number of bacteria without plasmids  <div lang="latex">W_{\mathbf{Z}}([z_0,0,0],t)</div> in the culture progresses, and how this depends on the various parameters in the model.
+
The aim of studying the behaviour of this model is to investigate how  
 +
growth conditions $D, S_f$ and time modulate the development of the population  
 +
in state space $W_{\mathbf{Z}}(\mathbf{z},t)$. In particular we are interested in
 +
finding how the number of bacteria without plasmids  $W_{\mathbf{Z}}([z_0,0,0],t)$ in  
 +
the culture progresses, and how this depends on the various parameters in the model.
  
 
=====Proposition: Adding contention=====
 
=====Proposition: Adding contention=====
  
−
<html>
+
We need to introduce a modification to equation 3 (the state dependant
 +
growth rate) in which the equilibrium between $z_1$ and $z_2$ features.
 +
This modification is to take into account that each toxin should be
 +
inhibited by anti-toxin for maximal growth. A possibility is that we
 +
consider that for each toxin anti-toxin pair:
  
−
We need to introduce a modification to equation 3 (the state dependant growth rate) in which the equilibrium between <div lang="latex">z_1</div> and <div lang="latex">z_2</div> features.
+
* the genes produce toxin at a constant rate dependant on the numberof copies $k_1\times z_1$,
−
This modification is to take into account that each toxin should be inhibited by anti-toxin for maximal growth.
+
* the anti-toxin genes produce anti-toxin at a rate dependant on the number of copies $k_2\times z_2$,
−
A possibility is that we consider that for each toxin anti-toxin pair:
+
* anti-toxin instantly and irreversible kills the toxin in a stochiometric manner.
 +
* undestroyed toxin disappears at a constant rate, by dilution and other pathways $k_3$,
 +
* the concentration of toxin is at a dynamic steady state, i.e. the rates of production and disappearance are equal.
 +
* growth in inhibited in an exponential manner by free toxin with a characteristic $IC_{50}$.
  
−
<ul>
+
This model of toxin anti-toxin interaction takes into account the
−
  <li>item the genes produce toxin at a constant rate dependant on the number of copies <div lang="latex">k_1\times z_1</div>,</li>
+
bacteriostatic nature of most such toxins, and the presence of
−
  <li>item the anti-toxin genes produce anti-toxin at a rate dependant on the number of copies <div lang="latex">k_2\times z_2</div>,</li>
+
measurable $IC_{50}$ values. Clearly a more realsitic model would need
−
  <li>item anti-toxin instantly and irreversible kills the toxin in a stochiometric manner.</li>
+
to take into account cell volume, protein synthesis rates etc.
−
  <li>item undestroyed toxin disappears at a constant rate, by dilution and other pathways <div lang="latex">k_3</div>,</li>
+
Nevertheless, this simple model gives:
−
  <li>item the concentration of toxin is at a dynamic steady state, i.e. the rates of production and disappearance are equal.</li>
+
<nowiki>
−
  <li>item growth in inhibited in an exponential manner by free toxin with a characteristic <div lang="latex">IC_{50}</div>.</li>
+
$$
−
</ul>  
+
\begin{align}
 +
\mu &= \mu_{0} \times e^{-\frac{[Toxin]}{IC_{50}}} \\
 +
[Toxin] &= max(0,\frac{k_1z_1-k_2z_2}{k_3}) \\
 +
\mu &= \mu_{0} \times min(1.0,e^{-k_a(z_1-k_bz_2)}) \\
 +
k_a &= \frac{k_1}{k_3 \times IC_{50}} \\
 +
k_b &= \frac{k_2}{k_1}
 +
\end{align}
 +
$$
 +
</nowiki>
  
−
This model of toxin anti-toxin interaction takes into account the bacteriostatic nature of most such toxins,  
+
Incorporating 2 toxin anti toxin pairs, if we assume
−
and the presence of measurable <div lang="latex">IC_{50}</div> values.
+
that the production rate ratio is the same for both, $k_b$, is
−
Clearly a more realsitic model would need to take into account cell volume, protein synthesis rates etc.
+
independant of the system we have:
−
Nevertheless, this simple model gives:<br/><br/>
+
  
−
<div lang="latex">\mu = \mu_{0} \times e^{-\frac{[Toxin]}{IC_{50}}} \\</div>(12)<br/><br/>
+
<nowiki>
−
<div lang="latex">[Toxin] = max(0,\frac{k_1z_1-k_2z_2}{k_3}) \\</div>(13)<br/><br/>
+
\begin{equation}
−
<div lang="latex">\mu = \mu_{0} \times min(1.0,e^{-k_a(z_1-k_bz_2)}) \\</div>(14)<br/><br/>
+
  \mu = \mu_{0} \times min(1,e^{-k_a(z_1-k_bz_2)})) \times min(1,e^{-k_a(z_2-k_bz_1)}))
−
<div lang="latex">k_a = \frac{k_1}{k_3 \times IC_{50}} \\</div>(15)<br/><br/>
+
\end{equation}
−
<div lang="latex">k_b = \frac{k_2}{k_1}</div>(16)<br/><br/>
+
</nowiki>
  
−
Incorporating 2 toxin anti toxin pairs,
+
where $\mu_0$ is given by equation 3. For each toxin the efficiency
−
if we assume that the production rate ratio is the same for both, <div lang="latex">k_b</div>, is independant of the system we have:<br/><br/>
+
parameter is a measure of ratio of toxin accumulation in cells with one
 +
gene copy and without anti-toxin to the $IC_{50}$.
  
−
<div lang="latex">\mu = \mu_{0} \times min(1,e^{-k_a(z_1-k_bz_2)})) \times min(1,e^{-k_a(z_2-k_bz_1)}))</div>(17)<br/><br/>
+
====Program====
−
 
+
−
where <div lang="latex">\mu_0</div> is given by equation 3.
+
−
For each toxin the efficiency parameter is a measure of ratio of toxin accumulation in cells with one gene copy and without anti-toxin to the <div lang="latex">IC_{50}</div>.
+
−
 
+
−
</html>
+
−
 
+
−
===='''Progamming code'''====
+
  
 
Code de françois à ajouter
 
Code de françois à ajouter
  
−
===='''Results'''====
+
====Results====
 
METTRE LES COURBE
 
METTRE LES COURBE
 +
 +
====Bibliography====
 +
<references group="toulouse"/>
  
 
=== Data recovery [https://2016.igem.org/Team:Bordeaux Bordeaux 2016] ===
 
=== Data recovery [https://2016.igem.org/Team:Bordeaux Bordeaux 2016] ===
Line 219: Line 326:
  
 
The iGEM team Pretoria 2016 hepled us focusing on the socio-economic and political issues facing the current platinum sector, including the Marikana strikes.
 
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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<h3>★  ALERT! </h3>
 
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<p>This page is used by the judges to evaluate your team for the <a href="https://2016.igem.org/Judging/Medals">team collaboration silver medal criterion</a>. </p>
 
−
 
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<p> Delete this box in order to be evaluated for this medal. See more information at <a href="https://2016.igem.org/Judging/Evaluated_Pages/Instructions"> Instructions for Evaluated Pages </a>.</p>
 
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<p>
 
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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.
 
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</p>
 
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<h4> Which other teams can we work with? </h4>
 
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<p>
 
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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.
 
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</p>
 
−
 
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<p>
 
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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.
 
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</p>
 
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<p>
 
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Here are some suggestions for projects you could work on with other teams:
 
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</p>
 
−
 
−
<ul>
 
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<li> Improve the function of another team's BioBrick Part or Device</li>
 
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<li> Characterize another team's part </li>
 
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<li> Debug a construct </li>
 
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<li> Model or simulating another team's system </li>
 
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<li> Test another team's software</li>
 
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<li> Help build and test another team's hardware project</li>
 
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<li> Mentor a high-school team</li>
 
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</ul>
 

Revision as of 00:27, 18 October 2016