On the stationary state analysis of reaction-diffusion mechanisms for biological pattern formation

We present a biologically plausible two-variable reaction-diffusion model for the developing vertebrate limb, for which we postulate the existence of a stationary solution. A consequence of this assumption is that the stationary state depends on only a single concentration-variable. Under these circumstances, features of potential biological significance, such as the dependence of the steady-state concentration profile of this variable on parameters such as tissue size and shape, can be studied without detailed information about the rate functions. As the existence and stability of stationary solutions, which must be assumed for any biochemical system governing morphogenesis, cannot be investigated without such information, an analysis is made of the minimal requirements for stable, stationary non-uniform solutions in a general class of reaction-diffusion systems. We discuss the strategy of studying stationary-state properties of systems that are incompletely specified. Where abrupt transitions between successive compartment-sizes occur, as in the developing limb, we argue that it is reasonable to model pattern reorganization as a sequence of independent stationary states.