The Emergence of Spatial Patterns for Compartmental Reaction Kinetics Coupled by Two Bulk Diffusing Species with Comparable Diffusivities

TL;DR

This paper demonstrates that spatial patterns can form in a 1-D coupled PDE-ODE model with two diffusing species of similar diffusivities, challenging the traditional activator-inhibitor paradigm, and explores stability and oscillations of these patterns.

Contribution

It introduces a novel 1-D coupled PDE-ODE model showing stable pattern formation with comparable diffusivities, extending Turing pattern theory to more realistic biological scenarios.

Findings

Stable asymmetric steady states emerge with equal diffusivities.

Pattern formation occurs via symmetry-breaking bifurcations.

Oscillatory instabilities can be induced depending on kinetics.

Abstract

Originating from the pioneering study of Alan Turing, the bifurcation analysis predicting spatial pattern formation from a spatially uniform state for diffusing morphogens or chemical species that interact through nonlinear reactions is a central problem in many chemical and biological systems. From a mathematical viewpoint, one key challenge with this theory for two component systems is that stable spatial patterns can typically only occur from a spatially uniform state when a slowly diffusing "activator" species reacts with a much faster diffusing "inhibitor" species. However, from a modeling perspective, this large diffusivity ratio requirement for pattern formation is often unrealistic in biological settings since different molecules tend to diffuse with similar rates in extracellular spaces. As a result, one key long-standing question is how to robustly obtain pattern formation in the biologically realistic case where the time scales for diffusion of the interacting species are comparable. For a coupled 1-D bulk-compartment theoretical model, we investigate the emergence of spatial patterns for the scenario where two bulk diffusing species with comparable diffusivities are coupled to nonlinear reactions that occur only in localized "compartments", such as on the boundaries of a 1-D domain. The exchange between the bulk medium and the spatially localized compartments is modeled by a Robin boundary condition with certain binding rates. As regulated by these binding rates, we show for various specific nonlinearities that our 1-D coupled PDE-ODE model admits symmetry-breaking bifurcations, leading to linearly stable asymmetric steady-state patterns, even when the bulk diffusing species have equal diffusivities. Depending on the form of the nonlinear kinetics, oscillatory instabilities can also be triggered. Moreover, the analysis is extended to treat a periodic chain of compartments.