Polarization of concave domains by traveling wave pinning.

Slawomir Bialecki,Bogdan Kazmierczak,Tomasz Lipniacki,Scott W Mccue

doi:10.1371/journal.pone.0190372

Abstract

Pattern formation is one of the most fundamental yet puzzling phenomena in physics and biology. We propose that traveling front pinning into concave portions of the boundary of 3-dimensional domains can serve as a generic gradient-maintaining mechanism. Such a mechanism of domain polarization arises even for scalar bistable reaction-diffusion equations, and, depending on geometry, a number of stationary fronts may be formed leading to complex spatial patterns. The main advantage of the pinning mechanism, with respect to the Turing bifurcation, is that it allows for maintaining gradients in the specific regions of the domain. By linking the instant domain shape with the spatial pattern, the mechanism can be responsible for cellular polarization and differentiation.

Highlights

Pattern formation is a ubiquitous phenomenon in biology and is observed at different scales [1, 2]
The group of Edelstein-Keshet proposed an interesting mechanism in which traveling front in a bistable system decelerates and becomes stationary due to the global depletion of a fast diffusing variable as a consequence of front propagation [23]. This mechanism requires two equations with different diffusion coefficients. Another similar mechanism known as local excitation/global inhibition (LEGI), that arises in spatial systems exhibiting relaxation oscillations, provides spatial patterns fluctuating in time [17]
It is common that stable stationary solutions to equations of mathematical physics correspond to the minima of some potential

Summary

Introduction

Pattern formation is a ubiquitous phenomenon in biology and is observed at different scales [1, 2]. The most recognized is the Turing bifurcation theory [20], in which spatially nonuniform solutions arise in systems with two or more reaction–diffusion equations with different diffusion coefficients [2, 21]. The group of Edelstein-Keshet proposed an interesting mechanism in which traveling front in a bistable system decelerates and becomes stationary due to the global depletion of a fast diffusing variable as a consequence of front propagation [23]. This mechanism requires (at least) two equations with different diffusion coefficients. Domain polarization can arise in systems in which two opposing

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Conclusion