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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<p>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.</p>
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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.
  
 
<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>
 
<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>
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<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}} (3)\\</div><br/>
 
<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}} (3)\\</div><br/>
 
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.
 
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.
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For plasmid replication rate we propose, again following Shene et al. [?], the empirical relationship :
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<div lang="latex">  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 \\</div>
 
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Revision as of 15:23, 17 October 2016