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

Yoichiro Mori, Alexandra Jilkine, Leah Edelstein-Keshet

Research output: Contribution to journalArticlepeer-review

60 Scopus citations

Abstract

We describe and analyze a bistable reaction-diffusion 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). Wavepinning 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 bifurcation and a pitchfork bifurcation.

Original languageEnglish (US)
Pages (from-to)1401-1427
Number of pages27
JournalSIAM Journal on Applied Mathematics
Volume71
Issue number4
DOIs
StatePublished - 2011

Keywords

  • Bistable reaction-diffusion system
  • Cell polarization
  • Mass conservation
  • Rho GTPases
  • Stationary front
  • Wave-pinning

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