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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:

First Fick Law

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

First Fick Law

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:

First Fick Law

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

First Fick Law

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.

Basic parameters of the diffusion simulation.

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:

First Fick Law

where x is the position of the bacteria and v is the velocity. This (vector) velocity is is defined as:

First Fick Law

where u(t) is an unitary vector and s is the speed. The vector u(t) is randomly sampled from the unit circle. The speed is randomly sampled from a normal distribution:

First Fick Law

where μ is the mean displacement velocity of E.Coli, and σ models the variations from this mean velocity.

The bacteria will move for a time Δ trm 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 tgc;

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

Modeling Tools

Donwload Bacterial Crowding Simulation

The Dry Lab has created a program using Java to simulate the previous equations and dynamics. The program has a visual interface developed using the graphics library PulpCore.

You can download the bacterial crowding simulation here. You will find a readme file explaining how to execute it. Moreover, the source code is also included.

The simulation allows to control several parameters, like the diffusion coefficient, the velocity of the bacterias, and so on. Moreover, the simulation shows visually the evolution of the concentration of chemoattractant and the position of the bacterias. Below you can see the evolution of a simulation in 5 different time steps. The bacterias are represented in green, while the concrentration is depicted using colors:

Bacterial Crowding simulation iteration 1 Bacterial Crowding simulation iteration 2 Bacterial Crowding simulation iteration 3 Bacterial Crowding simulation iteration 4 Bacterial Crowding simulation iteration 5