Reaction-Diffusion Patterns on Growing Domains

Abstract

The reaction-diffusion (Turing) mechanism is one of the simplest and most elegant theories for biological pattern formation. The recent experimental realisation of Turing patterns in chemical systems has fostered renewed interest in reaction-diffusion theory, however, its relevance to many biological problems has been questioned because of the perceived failure of the mechanism to generate patterns reliably. A recent paper suggesting the involvement of reaction-diffusion in fish skin patterns has implicated domain growth as an important mechanism controlling pattern selection. In this thesis we present a systematic study of the effects of domain growth on reaction-diffusion patterns, and discuss the implications for reliable pattern generation.

Starting from the postulate that tissue growth rates will be locally determined, we derive general evolution equations for reaction-diffusion on growing domains as a problem in kinematics. We argue that the biologically plausible scenario is to consider domain growth on a longer timescale than pattern formation. Then it is found that the solution goes through a sequence of recognisable (quasi-steady) patterns. Using symmetry arguments relating different pattern modes we show that for uniform domain growth the solution evolves by frequency-doubling, the regular splitting or insertion of peaks in the pattern. For pattern formation in two spatial dimensions domain growth is found to select rectangular lattices, rather than the hexagonal planform that is preferred on the fixed domain. For nonuniform growth the local tissue expansion rate varies across the domain and splitting or insertion may be restricted to regions of the domain where the growth is sufficiently fast.

The behaviour of solutions can be studied asymptotically and peak splitting and insertion are shown to occur according to the form of the reaction nullclines. We highlight a novel behaviour, frequency-tripling, where both mechanisms operate simultaneously, which is realised when quadratic terms are absent from the reaction kinetics. Any particular pattern in a sequence remains established until the domain is sufficiently large that a transition to a higher pattern mode occurs. This presents a degree of scale invariance. The pattern which persists finally is not strongly dependent on the final domain size, and hence domain growth can provide a mechanism for reliable pattern selection.