Models

1. The difussion of the chemoattractant through the medium.
2. The motion of the bacterias through the medium due to the gradient on the chemoattractant concentration.
3. The chemoattractant generation within the bacteria.

Chemoattractant Diffusion

The basic equations for the diffusion of the chemoattractant in the medium are the Fick laws of diffusion, which govern the variation of the concentration of a substance within a medium.

The flux (that is, the amount of substance that flows through a given surface per unit of time $\frac{mol}{m^2 s}$) is given by:

where $\phi$ is the concentration ($\frac{mol}{m^{3}}$) in a given point. $D$ is a constant called the diffusion coefficient, and that depends on the medium .

Basically, the equation states that the is directed towards places with lower concentration (thus the minus sign). If the concentration is constant in the space $ \Delta \phi = \mathbf{0}$ there is no flux.

If the flux is known, it is possible to determine the amount of substance that goes through a small surface $d\mathbf{S}$ and a small amount of time $dt$

In order to simulate the diffusion, we define the environment and discretize it in very small cells. Each cell determines a given volume $V$, and has a surface $\mathbf{S}$. At a given time instant, the cell has an amount of substance $c$ (and then a concentration $\frac{c}{V}$).

If the cells and time step $\Delta t$ are small, we can consider that the gradient of concentration can be approximated though the differences in concentration between a cell $i$ and 4 (or 8) neighbors $j$. Thus:

Bacteria motion

Chemical reactions

Modeling Tools