Steady-State Patterns in a Reaction Diffusion System with Mixed Boundary Conditions

Abstract

A number of models for pattern formation and regulation are based on the hypothesis that a diffusible morphogen supplies positional information that can be interpreted by cells. Such models fall into two main classes: those in which pattern arises from distributed sources and/or sinks of the morphogens, and those which can spontaneously produce pattern via the interaction of reaction and transport. In source-sink models, specialized cells maintain the concentration of the morphogen at fixed levels, and given a suitable distribution of sources and sinks, a tissue can be proportioned into any number of cell types with a threshold interpretation mechanism. However, the spatial pattern established is strongly dependent on the distances between the sources and sinks, and additional hypotheses must be invoked to ensure that the pattern is invariant under changes in the scale of the system. This is most easily seen in a one-dimensional system with a source at one end and a sink at the other. If the ends are held at c ₀ and c ₁ respectively, then the morphogen distribution is given by c(x)=(c ₁ −c ₀)(x/L) + c ₀, and so the flux through the system must vary as 1/L. Thus the homeostatic mechanism that maintains the boundary concentrations at fixed levels must be able to vary the production or consumption of morphogen over a wide range.