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 J (that is, the amount of substance that flows through a given surface per unit of time mol m^-2s^-1) is given by:

where φ is the concentration (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 (∇φ=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 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 S. At a given time instant, the cell has an amount of substance c (and then a concentration c\V).

If the cells and time step Δt 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:

and then, the amount of substance that diffusses from i to j:

The following figure illustrates the basic elements of the simulation. The flux J between cells is computed by the difference of concentrations. Then, this flux is used to compute the amount of substance that will flow to the neighbour cell. The amount is proportional to the flux and the common surface between cells.

Bacteria motion

The main actuator of E. coli is a flagellar motor that can rotate clockwise or counterclockwise. Through a set of transmembrane receptors proteins, E. coli is able to detect chemoattractants. Moreover, this detection influences the motion of the flagellar motor [Topp and Gallivan, 2007].

E. Coli has two main motion modes, which we will name:

1. Random Walk
2. Gradient climbing

Random walk mode

When no gradient of chemoattractant is present, E. coli is in random walk mode. In this case, the bacteria performs smooth runs followed by tumbles.

Mathematically, we will model this as a Brownian motion:

where x is the position of the bacteria and v is the velocity. This (vector) velocity is randomly sampled from a normal distribution of zero mean and a certain covariance matrix that models the potential directions and speeds of the velocity vector when tumbling. Usually, this covariance matrix is selected so that all directions are equally likely.

The bacteria will move for a time Δ t_rm with constant velocity v(t). At the next time instant the bacteria tumbles, selecting a new velocity vector (randomly).

Gradient climbing behaviour

When a positive difference of concentration (a gradient) on the chemoattractant is detected, the bacteria enters into a new mode that we will call gradient climbing.

In this mode, the flagellar motor tends to move counterclockwise; as a result, the smooth runs last for more time, and the tumbling frequency decreases.

In order to model that, we use the same model than above, but a larger tumbling interval (smaller tumbling frequency) Δ t_gc;

[Topp and Gallivan, 2007] Topp, S. and Gallivan, J. (2007). Guiding Bacteria with Small Molecules and RNA. J. Am. Chem. Soc., 129:6807–6811.

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