7. Initiation of Cellular Slime Mold Aggregation Viewed as an Instability

The initiation of cellular slime mold aggregation is identified as the onset of a self-organized linear instability of a simplified reaction-diffusion model system for the slime mold amoeba density and the concentration of the extracellular chemical acrasin produced by them to which they are chemotactically attracted. To derive these governing equations a general balance law must be deduced employing the divergence, Stokes, and Green’s theorems which are the subject of a pastoral interlude. The initial conditions are satisfied by means of Fourier integrals, introduced by another pastoral interlude that deduces the relevant formula and in so doing also includes the concept of Laplace transforms. The factors that favor the initiation of such aggregation are predicted by examining the linear instability criterion. The problem considers the equivalent normal-mode linear stability analysis of a slightly more general four-component model system explicitly including the two other dependent variables: Namely, the enzyme acrasinase, a second chemical produced by the amoeba that degrades the acrasin to a product to which they are not chemotactically attracted, and an intermediate complex formed by the interaction of these two chemicals in a reversible equilibrium reaction.