Emergence of traveling waves and their stability in a free boundary model of cell motility

TL;DR

This paper presents a mathematical model of cell motility using a Hele-Shaw free boundary approach, revealing how stationary solutions become unstable and bifurcate into traveling waves, with stability analyzed through linear methods.

Contribution

It introduces a novel two-dimensional free boundary model coupling Darcy law and Keller-Segel system for cell motility, analyzing bifurcations and stability of traveling wave solutions.

Findings

Radially symmetric solutions become unstable at a critical myosin mass.

Traveling wave solutions bifurcate from stationary solutions.

Stability transitions occur via generalized eigenvectors.

Abstract

We introduce a two-dimensional Hele-Shaw type free boundary model for motility of eukaryotic cells on substrates. The key ingredients of this model are the Darcy law for overdamped motion of the cytoskeleton gel (active gel) coupled with advection-diffusion equation for myosin density leading to elliptic-parabolic Keller-Segel system. This system is supplemented with Hele-Shaw type boundary conditions: Young-Laplace equation for pressure and continuity of velocities. We first show that radially symmetric stationary solutions become unstable and bifurcate to traveling wave solutions at a critical value of the total myosin mass. Next we perform linear stability analysis of these traveling wave solutions and identify the type of bifurcation (sub- or supercritical). Our study sheds light on the mathematics underlying instability/stability transitions in this model. Specifically, we show that these transitions occur via generalized eigenvectors of the linearized operator.