We saw in the last section in the discussion of the liquid experiments and their model system that patterns consisting of a random arrangement of spots will probably appear on a short timescale in the liquid medium experiments but eventually the aggregates fade and homogeneity again obtains. We suggested that this fading is probably due to saturation of the chemotactic response. Basically since cellular production of chemoat-tractant is not countered by any form of chemoattractant degradation (or inhibition), the amount of chemoattractant in the dish increases continuously. As a result, the chemo-tactic response eventually saturates, and diffusion takes over.
We also noted that the usual linear analysis about a uniform steady state is not possible so we have to develop a different analysis to study the pattern formation dynamics. The method (Tyson et al., 1999) we develop is very much intuitive rather than exact, but as we shall see it is nevertheless informative and qualitatively predictive and explains how transient patterns of randomly or circularly arranged spots can appear in a chemo-taxis model and in experiment. We also give some numerical solutions to compare with the analytical predictions. For all of the analysis and simulations we assume zero flux boundary conditions which reflect the experimental situation.
We start with the simplest model for the liquid experiments which is just the semisolid phase model with zero proliferation of cells, zero degradation of chemoattractant and uniform distribution of stimulant, s, which is neither consumed nor degraded and is thus just another parameter here. In these circumstances (5.14)-(5.16) become d n dt
where n and c are respectively the density of cells and concentration of chemoattractant.
For simplicity the analysis we carry out is for the one-dimensional case where V2 = d2/dx2. Although we carry out the analysis for a one-dimensional domain the results can be extended with only minor changes to two dimensions (like what we did in Chapter 2 when investigating reaction diffusion pattern formation). We nondimension-alise the equations by setting n c s * k5 so *
no k2 so k2
which gives, on dropping the asterisks for algebraic simplicity, the nondimensional equations d u d2U d — = a—T — a —
dt dx2 dx dv
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