Investigating the Turing conditions for diffusion-driven instability in the presence of a binding immobile substrate.

Turing's diffusion-driven instability for the standard two species reaction-diffusion system is only achievable under well-known and rather restrictive conditions on both the diffusion rates and the kinetic parameters, which necessitates the pairing of a self-activator with a self-inhibitor. In this study we generalize the standard two-species model by considering the case where the reactants can bind to an immobile substrate, for instance extra-cellular matrix, and investigate the influence of this dynamics on Turing's diffusion-driven instability. Such systems have been previously studied on the grounds that binding of the self-activator to a substrate may effectively reduce its diffusion rate and thus induce a Turing instability for species with equal diffusion coefficients, as originally demonstrated by Lengyel and Epstein (1992) under the assumption that the bound state dynamics occurs on a fast timescale. We, however, analyse the full system without any separation of timescales and demonstrate that the full system also allows a relaxation of the standard constraints on the reaction kinetics for the Turing instability, increasing the type of interactions that could give rise to spatial patterning. In particular, we show that two self-activators can undertake a diffusively driven instability in the presence of a binding immobile substrate, highlighting that the interactions required of a putative biological Turing instability need not be associated with a self-activator-self-inhibitor morphogen pair.

A didactical note on the advantage of using two parameters in Hopf bifurcation studies.

In order to maximize the information that a linearized stability analysis provides, one should work with two free parameters rather than one. Moreover, it is recommended to first consider coefficients in the characteristic equation as parameters and in a second step (try to) invert the map that defines the coefficients in terms of the parameters as they occur in the original equation. Our aim is to substantiate these claims by way of a delay equation example taken from the literature.