"
Team:ETH Zurich/Modeling/Analytical Approximations
From 2013.igem.org
| Line 16: | Line 16: | ||
\begin{align} | \begin{align} | ||
\frac{\partial AHL(\textbf{r},t)}{\partial t} = C_{agar} \cdot D_{AHL} \nabla^{2} AHL(\textbf{r},t) + DF \cdot (\alpha_{AHL} \cdot [LuxI](\textbf{r},t)-d_{AHL,i} \cdot [AHL](\textbf{r},t))_{MineCell} + DF \cdot (-d_{AHL,i} \cdot [AHL](\textbf{r},t))_{ReceiverCell}- d_{AHL,e} \cdot [AHL](\textbf{r},t) | \frac{\partial AHL(\textbf{r},t)}{\partial t} = C_{agar} \cdot D_{AHL} \nabla^{2} AHL(\textbf{r},t) + DF \cdot (\alpha_{AHL} \cdot [LuxI](\textbf{r},t)-d_{AHL,i} \cdot [AHL](\textbf{r},t))_{MineCell} + DF \cdot (-d_{AHL,i} \cdot [AHL](\textbf{r},t))_{ReceiverCell}- d_{AHL,e} \cdot [AHL](\textbf{r},t) | ||
| + | \end{align} | ||
<br clear="all"/> | <br clear="all"/> | ||
{{:Team:ETH_Zurich/templates/footer}} | {{:Team:ETH_Zurich/templates/footer}} | ||
Revision as of 19:53, 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(\textbf{r},t)}{\partial t} = Diff(AHL(\textbf{r},t),\textbf{r}) + R(AHL(\textbf{r},t)) \end{align}
And a less general form, as derived for the reaction-diffusion model for AHL:
\begin{align}
\frac{\partial AHL(\textbf{r},t)}{\partial t} = C_{agar} \cdot D_{AHL} \nabla^{2} AHL(\textbf{r},t) + DF \cdot (\alpha_{AHL} \cdot [LuxI](\textbf{r},t)-d_{AHL,i} \cdot [AHL](\textbf{r},t))_{MineCell} + DF \cdot (-d_{AHL,i} \cdot [AHL](\textbf{r},t))_{ReceiverCell}- d_{AHL,e} \cdot [AHL](\textbf{r},t)
\end{align}
