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 for a single mine colony.

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}

We were interested in the AHL gradient that can be established by a single colony ( located at $x = 0$) on an agar plate, ending up with a less general form of 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} - d_{AHL,e} \cdot [AHL] \end{align}

As boundary condition, we considered that the concentration at the mine colony stays constant, i.e. we are assuming cells are at the steady state:

\begin{align} [AHL](x=0,t) = [AHL]_{MineCell,ss} = \frac{\alpha_{AHL} \cdot [LuxI]_{ss}} {d_{AHL,i}+\eta \cdot \left(1-\frac{\eta_{ext}}{d_{AHL,e}+\eta_{ext}}\right)} \end{align}

Solution at the steady state

To solve the KPP equation for the steady state, we set to zero the time derivative:

\begin{align} \frac{\partial AHL}{\partial t} = C_{agar} \cdot D_{AHL} \frac{\partial^2 AHL}{\partial x^2} - d_{AHL,e} \cdot [AHL] = 0 \end{align} \begin{align} \frac{\partial^2 AHL}{\partial x^2} = \frac{d_{AHL,e}} {C_{agar} \cdot D_{AHL}} \cdot [AHL] \end{align}

This ordinary differential equation can be solved, integrating wrt. to x twice:

\begin{align} [AHL] = [AHL]_{MineCell,ss} \cdot e^{-\sqrt{\frac{d_{AHL,e}} {C_{agar} \cdot D_{AHL}}} \cdot x} \end{align}