A Selection Criterion for Patterns in Reaction-Diffusion Systems

TL;DR

This paper introduces an analytic criterion for predicting whether reaction-diffusion systems will form spots or stripes, aiding the design of biological experiments and understanding pattern formation mechanisms.

Contribution

It provides a new, rigorously proven criterion for pattern selection in reaction-diffusion systems, extending previous scalar equation results to more complex models.

Findings

Analytic criterion for pattern prediction in reaction-diffusion systems

Rigorous proof in certain cases, numerical analysis in complex settings

Application potential in biological experiment design

Abstract

Alan Turing's work in Morphogenesis has received wide attention during the past 60 years. The central idea behind his theory is that two chemically interacting diffusible substances are able to generate stable spatial patterns, provided certain conditions are met. Turing's proposal has already been confirmed as a pattern formation mechanism in several chemical and biological systems and, due to their wide applicability, there is a great deal of interest in deciphering how to generate specific patterns under controlled conditions. However, techniques allowing one to predict what kind of spatial structure will emerge from Turing systems, as well as generalized reaction-diffusion systems, remain unknown. Here, we consider a generalized reaction diffusion system on a planar domain and provide an analytic criterion to determine whether spots or stripes will be formed. It is motivated by the existence of an associated energy function that allows bringing in the intuition provided by phase transitions phenomena. This criterion is proved rigorously in some situations, generalizing well known results for the scalar equation where the pattern selection process can be understood in terms of a potential. In more complex settings it is investigated numerically. Our criterion can be applied to efficiently design Biotechnology and Developmental Biology experiments, or simplify the analysis of hypothesized morphogenetic models.