The Analysis of Turing Instability in Reaction-diffusion Systems Using a Single Diffuser

Abstract

The dynamics of reaction-diffusion systems with a single diffusible molecule is of great interest in the study of recent synthetic biology. In this paper, we analyze the spatio-temporal dynamics of such systems. Specifically, we classify the spatio-temporal patterns from a control theoretic viewpoint and analytically derive the Turing instability conditions that correspond to the classified patterns. To this end, we first show that the reaction-diffusion systems with a single diffuser can be represented by a SISO system with a constant feedback gain. This then boils the Turing instability analysis down to the root locus analysis problem and allows us to derive the instability conditions using control theoretic tools. As a result, we can point out that at least three reacting molecules are necessary to produce physically plausible spatial patterns unlike the classical activator-inhibitor models with two diffusible molecules. These results are demonstrated on an extended Gray-Scott model with a single diffuser.