Asymptotic and bifurcation analysis of wave-pinning in a reaction-diffusion model for cell polarization

TL;DR

This paper analyzes a reaction-diffusion model explaining wave-pinning in cell polarization, using asymptotic and bifurcation analysis to understand how stationary fronts form and depend on system parameters.

Contribution

It provides a mathematical explanation of wave-pinning phenomena in cell polarization models through asymptotic and bifurcation analysis, highlighting the conditions for wave pinning and its loss.

Findings

Wave-pinning explained by mass conservation, nonlinear kinetics, and diffusion differences.

Predicted wave speed and pinned position using matched asymptotic analysis.

Identified bifurcation types leading to loss of wave-pinning.

Abstract

We describe and analyze a bistable reaction-diffusion (RD) model for two interconverting chemical species that exhibits a phenomenon of wave-pinning: a wave of activation of one of the species is initiated at one end of the domain, moves into the domain, decelerates, and eventually stops inside the domain, forming a stationary front. The second ("inactive") species is depleted in this process. This behavior arises in a model for chemical polarization of a cell by Rho GTPases in response to stimulation. The initially spatially homogeneous concentration profile (representative of a resting cell) develops into an asymmetric stationary front profile (typical of a polarized cell). Wave-pinning here is based on three properties: (1) mass conservation in a finite domain, (2) nonlinear reaction kinetics allowing for multiple stable steady states, and (3) a sufficiently large difference in diffusion of the two species. Using matched asymptotic analysis, we explain the mathematical basis of wave-pinning, and predict the speed and pinned position of the wave. An analysis of the bifurcation of the pinned front solution reveals how the wave-pinning regime depends on parameters such as rates of diffusion and total mass of the species. We describe two ways in which the pinned solution can be lost depending on the details of the reaction kinetics: a saddle-node or a pitchfork bifurcation.