Difference between revisions of "Team:Aix-Marseille/Collaborations"
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With these relations: | With these relations: | ||
| − | <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. | + | <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.<br/> |
| − | <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>. | + | <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/> |
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. | 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. | ||
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<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> | ||
| − | <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 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/> |
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. | ||
For plasmid replication rate we propose, again following Shene et al. [?], the empirical relationship :<br/> | For plasmid replication rate we propose, again following Shene et al. [?], the empirical relationship :<br/> | ||
| − | <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) | + | <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/> |
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/> | 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/> | ||
