Modeling

CONTENTS:

• Cell Movement in Microscopic and Macroscopic Aspects
• Front propagation for Cell Growth
• Density Dependent Motility
• Full Model of Density Dependent Motility
• Modeling Results

Cell Movement in Microscopic and Macroscopic Aspect

Basically, we considered the movement of a cell as a process of random walk, since an individual E.coli cell was always in the states between moving and tumbling (Fig.1).Similar to Brownian particles, the random walk of E.coli followed the Einstein-Smoluchowski theory [1].

Firstly, for the one dimension of random walk, let x (t) denote the position of an E.coli cell at time t, given that its position coincided with the point x=0 at time t=0. And we assumed an E.coli cell moves an average distance l—in either the positive or negative direction of the x-axis—each step (during a time (τ) between two tumbling states). The probability that the cell was found at the point x at time t was now equal to the probability that, in a series of n (=t/τ) successive moving steps, the cell made m more steps in the positive direction of the axis than in the negative direction. The desired probability was given by the binomial expression [1, 2]

To simplify equation (1.1), for m<<n, we obtain

Taking x to be a continuous variable, we can obtain

where

,

v= l/τ was the average speed of the cell movement, and f=1/τ was the tumbling frequency of E.coli.

In two dimensions, the square of the distance from the origin to the point (x, y) was r2=x2+y2; therefore

For a macroscopic view, D was defined as the diffusion coefficient. For a simple diffusion process, it is easy to write an diffusion equation to describe the density ρ distribution.

Normally, as the swimming speed of E.coli is about 20um/s and the tumbling frequency is about 1 Hz, the diffusion coefficient is about 200um2/s [3].

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Front propagation for cell growth

If we only considered the wild type E.coli that has the diffusion effect and growth effect, it came up with the model Fisher-Kolmogorov equation which was firstly developed by R. A. Fisher and A. N. Kolmogorov [4].

(2.1)

where ρ was the cell density, D was the diffusion coefficient, γ0 was growth rate, ρs was saturation density.

Here, due to the symmetry, we investigated equation (2.1) in case of cylindrical domains with u depending only on radius. Numerical study with an initially condition of Gaussian function as Fig.2 can give a solution shown in Fig.2, while analytical solution can refer to reference [5]. Based on the solution, it was found that the speed of the front propagation of the pattern was proportional to the square root of the product of diffusion coefficient D and growth rate γ. As this migration speeds of the cell pattern at different cheZ expression levels can be obtained from experiments, and the growth rate was also measured, the diffusion coefficient D at different cheZ expression levels were also known then. And the cell density can be transformed to the brightness that we observed in the experiments (see the brightness model part), the quantity ρ can be compared with the experiments. Therefore, each quantity in equation (4) can be compared with the experiments.

If we looked at the wild type E.coli which only had random walk and growth, the equation (2.1) was successful to describe its behavior. As shown in Fig.2, a droplet of wild type E.coli in the center of the plate will from a round expanding pattern. And most regions in this pattern except for some near in boundary seems to be uniform, which was the same as the result of the model (Fig.2).

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