Uniqueness and traveling waves in a cell motility model

TL;DR

This paper analyzes a non-local evolution equation modeling cell motility, proving solution uniqueness in certain regimes, establishing traveling wave existence under asymmetry conditions, and numerically exploring their stability and dynamics.

Contribution

It provides the first proof of solution uniqueness in the subcritical regime and demonstrates traveling wave existence in the supercritical regime with asymmetry, complemented by numerical simulations.

Findings

Uniqueness of solutions in subcritical regime

Existence of traveling waves in supercritical regime with asymmetry

Numerical evidence of traveling wave instability and complex cell motions

Abstract

We study a non-linear and non-local evolution equation for curves obtained as the sharp interface limit of a phase-field model for crawling motion of eukaryotic cells on a substrate. We establish uniqueness of solutions to the sharp interface limit equation in the so-called subcritical parameter regime. The proof relies on a Gr\"{o}nwall estimate for a specially chosen weighted $L^2$ norm. Next, as persistent motion of crawling cells is of central interest to biologists we study the existence of traveling wave solutions. We prove that traveling wave solutions exist in the supercritical parameter regime provided the non-linear term of the sharp interface limit equation possesses certain asymmetry (related, e.g., to myosin contractility). Finally, we numerically investigate traveling wave solutions and simulate their dynamics. Due to non-uniqueness of solutions of the sharp interface limit equation we simulate a related, singularly perturbed PDE system which is uniquely solvable. Our simulations predict instability of traveling wave solutions and capture both bipedal wandering cell motion as well as rotating cell motion; these behaviors qualitatively agree with recent experimental and theoretical fidings.