Symmetry-Breaking Bifurcations for Compartmental Reaction Kinetics Coupled by Two Bulk Diffusing Species with Comparable Diffusivities in 2-D

Abstract

For a 2-D coupled PDE-ODE bulk-cell model, we investigate symmetry-breaking bifurcations that can emerge when two bulk diffusing species are coupled to two-component nonlinear intracellular reactions that are restricted to occur only within a disjoint collection of small circular compartments, or "cells", of a common small radius that are confined in a bounded 2-D domain. Outside of the union of these cells, the two bulk species with comparable diffusivities and bulk degradation rates diffuse and globally couple the spatially segregated intracellular reactions through Robin boundary conditions across the cell boundaries, which depend on certain membrane reaction rates. In the singular limit of a small common cell radius, we construct steady-state solutions for the bulk-cell model and formulate a nonlinear matrix eigenvalue problem that determines the linear stability properties of the steady-states. For a certain spatial arrangement of cells for which the steady-state and linear stability analysis become highly tractable, we construct a symmetric steady-state solution where the steady-states of the intracellular species are the same for each cell. As regulated by the ratio of the membrane reaction rates on the cell boundaries, we show for various specific prototypical intracellular reactions, and for a specific twocell arrangement, that our 2-D coupled PDE-ODE model admits symmetry-breaking bifurcations from this symmetric steady-state, leading to linearly stable asymmetric patterns, even when the bulk diffusing species have comparable or possibly equal diffusivities. Overall, our analysis shows that symmetry-breaking bifurcations can occur without the large diffusivity ratio requirement for the bulk diffusing species as is well-known from a Turing stability analysis applied to a spatially uniform steady-state for typical two-component activator-inhibitor systems. Instead, for our theoretical compartmental-reaction diffusion bulk-cell model, our analysis shows that the emergence of stable asymmetric steady-states can be controlled by the ratio of the membrane reaction rates for the two species. Bifurcation theoretic results for symmetric and asymmetric steady-state patterns obtained from our asymptotic theory are confirmed with full numerical PDE simulations.

Turing pattern formation in reaction-cross-diffusion systems with a bilayer geometry

“…We also generalize the system to consider chemotaxis and other nonlinear reactiondiffusion effects present in both layers. Another class of related models recently studied in the literature are the so-called compartmental models [51,50] where two or more reaction-diffusion compartments are spatially coupled through a passive diffusive bulk.…”