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Team:ETH Zurich/Modeling/Analytical Approximations
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<h1><b>Steady State AHL Gradient Approximation</b></h1> | <h1><b>Steady State AHL Gradient Approximation</b></h1> | ||
| - | In the reaction-diffusion model we have formalized the gradient establishment and we solved the resulting partial differential equation numerically. Now, the aim is to derive a suitable analytical approximation. | + | <p align = "justify">In the reaction-diffusion model we have formalized the gradient establishment and we solved the resulting partial differential equation numerically. Now, the aim is to derive a suitable analytical approximation. </p> |
<h1> Kolmogorov-Petrovsky-Piskounov Equation </h1> | <h1> Kolmogorov-Petrovsky-Piskounov Equation </h1> | ||
| - | A reaction–diffusion equation concerning the concentration of a single molecule in one spatial dimension is known as the Kolmogorov-Petrovsky-Piskounov Equation. In our case, the equation for AHL has the following general structure: | + | <p align = "justify">A reaction–diffusion equation concerning the concentration of a single molecule in one spatial dimension is known as the Kolmogorov-Petrovsky-Piskounov Equation. In our case, the equation for AHL has the following general structure:</p> |
\begin{align} | \begin{align} | ||
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\end{align} | \end{align} | ||
| - | And a less general form, as derived for the reaction-diffusion model for AHL: | + | <p align = "justify">And a less general form, as derived for the reaction-diffusion model for AHL:</p> |
\begin{align} | \begin{align} | ||
\frac{\partial AHL}{\partial t} = C_{agar} \cdot D_{AHL} \frac{\partial^2 AHL}{\partial x^2} + DF \cdot (\alpha_{AHL} \cdot [LuxI]-d_{AHL,i} \cdot [AHL]) - d_{AHL,e} \cdot [AHL] | \frac{\partial AHL}{\partial t} = C_{agar} \cdot D_{AHL} \frac{\partial^2 AHL}{\partial x^2} + DF \cdot (\alpha_{AHL} \cdot [LuxI]-d_{AHL,i} \cdot [AHL]) - d_{AHL,e} \cdot [AHL] | ||
\end{align} | \end{align} | ||
| + | |||
| + | <p align = "justify"> For the analytical approximation of the AHL gradient in one dimension, we consider the boundary condition that the concentration at the reservoir (located at z = 0) stays constant: | ||
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<br clear="all"/> | <br clear="all"/> | ||
{{:Team:ETH_Zurich/templates/footer}} | {{:Team:ETH_Zurich/templates/footer}} | ||
Revision as of 20:48, 24 October 2013
Steady State AHL Gradient Approximation
In the reaction-diffusion model we have formalized the gradient establishment and we solved the resulting partial differential equation numerically. Now, the aim is to derive a suitable analytical approximation.
Kolmogorov-Petrovsky-Piskounov Equation
A reaction–diffusion equation concerning the concentration of a single molecule in one spatial dimension is known as the Kolmogorov-Petrovsky-Piskounov Equation. In our case, the equation for AHL has the following general structure:
\begin{align} \frac{\partial AHL(x,t)}{\partial t} = Diff(AHL(x,t),x) + R(AHL(x,t)) \end{align}
And a less general form, as derived for the reaction-diffusion model for AHL:
\begin{align} \frac{\partial AHL}{\partial t} = C_{agar} \cdot D_{AHL} \frac{\partial^2 AHL}{\partial x^2} + DF \cdot (\alpha_{AHL} \cdot [LuxI]-d_{AHL,i} \cdot [AHL]) - d_{AHL,e} \cdot [AHL] \end{align}
For the analytical approximation of the AHL gradient in one dimension, we consider the boundary condition that the concentration at the reservoir (located at z = 0) stays constant:
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