Boundary-Bound Reactions: Pattern Formation with and without Hydrodynamics

Abstract

Pattern formation through chemical morphogenesis has received a significant amount of attention since Alan Turing's original analysis in a Chemical Basis for Morphogenesis. In that article, Turing showed the surprising result that pattern formation in bulk reaction-diffusion systems could be created through a diffusion-driven instability. More recently, researchers have studied active-membrane models; these models split the domain into two regions: a membrane where the reaction occurs and the bulk region where the reagents can diffuse and decay. Generally, the complexity of active-membrane models is far greater than boundary-bound reactions, where the primary source of reagents comes from boundary fluxes. These capture the behaviour of reaction systems where an enzyme or catalyst is necessary for the reaction to proceed, and we coat it along the boundary of the domain. We study these boundary-bound reactions, with specific applications to the control of pattern formation and fluid flow pattern formation.

Our investigation of boundary-bound reactions was originally motivated by the work of Balazs et al. (2020) in coupling chemical oscillations of boundary-bound reactions with the Boussinesq approximation of fluid flow. We begin by presenting a general overview of pattern formation without flow and obtain necessary conditions for pattern formation in terms of the reaction parameters and linear instability. To motivate our results, we consider the Schnakenberg system in two settings: the first demonstrates how boundary-bound reactions can be used to control pattern formation and chemical oscillations, and the second to study fluid flow pattern formation. In our analysis of fluid flow pattern formation, we begin by comparing and contrasting the behaviour of the boundary-bound model against the classical bulk reaction-diffusion problem without flow. Before, coupling it to the Boussinesq approximation for fluid flow. There we demonstrate how the diffusive instabilities change the behaviour of buoyancy-driven instabilities. Overall, these results provide possible mechanisms for controllable pattern formation, chemical signalling, and flows.