Travelling waves in a free boundary mechanobiological model of an epithelial tissue

TL;DR

This paper investigates a free boundary model of epithelial tissue migration, revealing unique traveling wave solutions influenced by mechanical interactions, with implications for understanding tissue invasion and retreat.

Contribution

It introduces a novel free boundary mechanobiological model with analytical and numerical analysis of traveling waves, highlighting the role of mechanical states in invasion dynamics.

Findings

Traveling waves have well-defined fronts unlike classical reaction-diffusion waves.

Wave speed and boundary density are derived analytically.

Invasion or retreat depends solely on cell compression or extension.

Abstract

We consider a free boundary model of epithelial cell migration with logistic growth and nonlinear diffusion induced by mechanical interactions. Using numerical simulations, phase plane and perturbation analysis, we find and analyse travelling wave solutions with negative, zero, and positive wavespeeds. Unlike classical travelling wave solutions of reaction-diffusion equations, the travelling wave solutions that we explore have a well-defined front and are not associated with a heteroclinic orbit in the phase plane. We find leading order expressions for both the wavespeed and the density at the free boundary. Interestingly, whether the travelling wave solution invades or retreats depends only on whether the carrying capacity density corresponds to cells being in compression or extension.