Competition and boundary formation in heterogeneous media: Application to neuronal differentiation

TL;DR

This paper models gene expression dynamics during nerve cell differentiation using reaction-diffusion equations, revealing how sharp boundaries between neuronal types form in inhomogeneous media through competition and diffusion effects.

Contribution

It introduces a novel analysis of boundary formation in heterogeneous media, linking reaction-diffusion models to neural differentiation and providing a precise characterization of boundary location.

Findings

Unique monotonic stationary solution exists in the small diffusion limit.

The neural tissue is split into two distinct regions at a boundary point.

Boundary location is determined by a zero-speed traveling wave condition.

Abstract

We analyze an inhomogeneous system of coupled reaction-diffusion equations representing the dynamics of gene expression during differentiation of nerve cells. The outcome of this developmental phase is the formation of distinct functional areas separated by sharp and smooth boundaries. It proceeds through the competition between the expression of two genes whose expression is driven by monotonic gradients of chemicals, and the products of gene expression undergo local diffusion and drive gene expression in neighboring cells. The problem therefore falls in a more general setting of species in competition within a non-homogeneous medium. We show that in the limit of arbitrarily small diffusion, there exists a unique monotonic stationary solution, which splits the neural tissue into two winner-take-all parts at a precise boundary point: on both sides of the boundary, different neuronal types are present. In order to further characterize the location of this boundary, we use a blow-up of the system and define a traveling wave problem parametrized by the position within the monotonic gradient: the precise boundary location is given by the unique point in space at which the speed of the wave vanishes.