I have implemented the coupled drift-diffusion equations using the simple numerical method tested recently.

For the first tests (program: invade3), the general parameter settings were as follows:

len=100.0;

dz=1.0;

timeSim=1000.0;

TMX=5;

As the initial conditions, the density profile of the guiding substance is set zero g(z,t=0) = 0. The cell density profile c(z,t=0) is set to a half-Gaussian of variance=1, with the maximum at the left boundary.

Run A: Nonmoving, insensitive cells. Effect of guide production, diffusion and decay

The cells remain in their initial distribution (because of D_c=0), sharply localized at the left boundary.

The guiding substance is constantly produced by the stationary cells (almost a point source). It also diffuses and decays. This leads quickly to a stationary, exponential concentration profile:

Run B: Additional slow cell diffusion, no sensitivity

Keeping all other parameters constant, the cell diffusion constant is next set to D_c=0.1. The cells, being not sensitive to the guide, now show the expected Gaussian diffusion profiles:

The initially exponential profile of the guiding substance is now gradually transforming into a Gaussian profile. This is due to the finite memory of the guide distribution and its constant re-production by the cells, which themselves assume a Gaussian distribution:

Correction: In the above figure, the green and blue curve corresponds to t=200 and t=400, respectively.

Run C: Adding sensitivity to the cells

Keeping all other parameters as in run B, we now set the sensitivity parameter to sens=3. Interestingly, the cells, starting from the initial Gaussian, now develop an approximately exponential profile:

The distribution of the guiding substance develops a non-Gaussian tail as well:

Correction: In the above figure, the lowest two curves correspond to t=200 and t=400, respectively.

The preliminary interpretation is as follows: Before the slow cell diffusion is setting in, the faster dynamics of the guiding substance is developing a concentration gradient. This gradient causes an effective back drift of the cells.

Conclusion: The chemotaxis model can produce approximately exponential invasion profiles, as observed in the experiments.

Idea for analytical case:

In the limit of D_g being much larger than D_c, the stationary g(z) should become close to a perfect exponential with a large spatial decay constant. If this decay constant exceeds the width of the stationary cell profile, it can (for those small z) be Taylor-approximated by a linear profile. The gradient of a linear profile is constant, corresponding to backward drift with constant velocity. This can be shown analytically to result in an exponential cell density profile.

It also seems mathematically reasonable that an exponential ansatz for both g(z) and c(z) can self-consistently solve the stationary coupled drift-diffusion equations (containing only derivatives and products of g(z) and c(z)) in the limit D_g>>D_c.

Note that this limit is also biologically reasonable, because small signalling molecules can diffuse much more easily through a dense matrix than the huge living cells.

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