Chapter 8.1 Reaction-Diffusion Models of Pattern Formation in Developmental Biology

Abstract

In this paper we present mathematical approaches to understand a symmetry break and formation of spatially heterogenous structures during development. We focus on the models given by reaction-diffusion equations and approach the question of possible mechanisms of development of spatially heterogeneous structures. We discuss two mechanisms of pattern formation: diffusion-driven instability (Turing instability) and a hysteresis-driven mechanism, and demonstrate their possibilities and constraints in explaining different aspects of structure formation in cell systems. Depending on the type of nonlinearities, we show the existence of Turing patterns, the maxima of which may be of the spike or plateau type, and the existence of transition layer stationary solutions. These concepts are discussed on example of morphogenesis of the fresh water polyp Hydra, which is a model organism in developmental biology.