Three-dimensional Turing patterns: Stability, implications in biological modeling, and equilibrium

Turing patterns are structures that can form spontaneously in systems of reacting and diffusing chemicals. Since the 1950s, when Alain Turing first put forth the theoretical considerations, there has been a vast amount of literature on the subject. Interestingly however, most simulations have been done in two spatial dimensions. We consider some aspects of three dimensional Turing patterns. We first study the stability of Turing patterns in an n-dimensional cube for n>1. Generalizing a classical result of Ermentrout about spots and stripes in two dimensions, it is shown that under appropriate assumptions only sheet-like or nodule-like structures can be stable in an n-dimensional cube. The stability results are applied to a model of skeletal pattern formation in the vertebrate limb. We then investigate the isoconcentration surfaces corresponding to the equilibrium concentration of the reaction kinetics in Turing patterns. We call these surfaces equilibrium concentration surfaces (EC surfaces). They are the interfaces between the regions of ``high'' and ``low'' concentrations. EC surfaces have been investigated numerically before. Remarkably, they are often very well approximated by certain minimal surfaces. We give alternate characterizations of EC surfaces by means of two variational principles, one of them being that they are optimal for diffusive transport. Several examples of EC surfaces are considered. Parts of this talk are based on joint work with M. Alber (Notre Dame), H.G.E. Hentschel (Emory) and B. Kazmierczak (IFTR, Warsaw).