title: Reaction-diffusion systems and external morphogen gradients: the two-dimensional case, with an application to skeletal pattern formation.
abstract: We investigate a reaction-diffusion system consisting of an activator and an inhibitor in a two-dimensional domain. There is a morphogen gradient in the domain. The production of the activator depends on the concentration of the morphogen. Mathematically, this leads to reaction-diffusion equations with explicitly space-dependent terms. It is well known that in the absence of an external morphogen, the system can produce either spots or stripes via the Turing bifurcation. We derive first-order expansions for the possible patterns in the presence of an external morphogen and show how both stripes and spots are affected. This work generalizes previous one-dimensional results to two dimensions. Specifically, we consider the quasi-one-dimensional case of a thin rectangular domain and the case of a square domain. We apply the results to a model of skeletal pattern formation in vertebrate limbs. In the framework of reaction-diffusion models, our results suggest a simple explanation for some recent experimental findings in the mouse limb which are much harder to explain in positional-information-type models.
authors: Department of Mathematics, Western Washington University, Bellingham, WA 98225, USA. glimmt@wwu.edu
authors: Tilmann
authors: T
authors: Glimm
authors: Jianying
authors: J
authors: Zhang
authors: Yun-Qiu
authors: YQ
authors: Shen
authors: Stuart A
authors: SA
authors: Newman
mesh_terms: D000818:Animals; D004058:Diffusion; D005121:Extremities / Q000196:embryology; D053823:Hedgehog Proteins / Q000502:physiology; D017880:Limb Deformities, Congenital / Q000209:etiology; D051379:Mice; D008954:Models, Biological; D009024:Morphogenesis / Q000502:physiology
chemical_list: D053823:Hedgehog Proteins; C465570:Shh protein, mouse