Interaction of Turing patterns with an external linear morphogen gradient

We investigate a simple generic model of a reaction–diffusion system consisting of an activator and an inhibitor molecule in the presence of a linear morphogen gradient. We assume that this morphogen gradient is established independently of the reaction–diffusion system and acts by increasing the production of the activator proportional to the morphogen concentration. The model is motivated by several existing models in developmental biology in which a Turing patterning mechanism is proposed and various chemical gradients are known to be important for development. Mathematically, this leads to reaction– diffusion equations with explicit spatial dependence. We investigate how the Turing pattern is affected, if it exists. We also show that in the parameter range where a Turing pattern is not possible, the system may nevertheless produce ‘Turing-like’ patterns. Mathematics Subject Classification: 92C15, 35K57, 35B32, 37B55 (Some figures in this article are in colour only in the electronic version) 1. General introduction One of the most fundamental problems in developmental biology is the question of morphogenesis, or pattern formation; that is, by what mechanisms cells self-organize into highly complex spatial distributions giving rise to tissues and organs during embryonic development. While tremendous advances have been made in the last 50 years, there is still debate over basic mechanisms and their relationships. 2 Author to whom any correspondence should be addressed. 3 Joint first authors of this article. 0951-7715/09/102541+20$30.00 © 2009 IOP Publishing Ltd and London Mathematical Society Printed in the UK 2541