Mathematical model for collective migration on a viscoelastic collagen network

TL;DR

This paper rigorously analyzes a mathematical model of collective cell migration on viscoelastic collagen, revealing conditions for stable stationary states and traveling pulses, and establishing well-posedness and stability results.

Contribution

It provides a rigorous mathematical framework for a previously proposed model, including exact stationary states, traveling pulses, and stability analysis based on collagen stiffness.

Findings

Existence of stationary states and traveling pulses.

Global well-posedness in $W^{k, abla ext{infty}}$ spaces.

Stability results depending on collagen stiffness.

Abstract

In this paper, we study a model of self-generated directional cell migration on viscoelastic substrates in the absence of apparent intrinsic polarity. This model, first proposed in \cite{Clark}, was observed numerically to manifest traveling pulse solutions for sufficiently large collagen stiffness, leading to a persistent collective migration. Here we provide a rigorous mathematical framework for the model, finding the exact stationary states and conditional traveling pulse. We also prove global well-posed in $W^{k,\infty}$ spaces, local stability of the traveling pulse for high stiffness, and exponential convergence to the stationary state for low stiffness.