Mathematical Modeling of Cell Collective Motion Triggered By Self-Generated Gradients

TL;DR

This paper reviews mathematical models of cell collective movement driven by self-generated chemical gradients, focusing on traveling wave solutions, stability, and biological implications, supported by numerical simulations.

Contribution

It introduces and analyzes models of self-generated gradient-driven cell migration, including extensions with explicit wave speeds and discusses stability and open problems.

Findings

Existence of traveling wave solutions in models

Explicit wave speeds in extended models

Discussion of stability and speed determinacy mechanisms

Abstract

Self-generated gradients have atttracted a lot of attention in the recent biological literature. It is considered as a robust strategy for a group of cells to find its way during a long journey. This note is intended to discuss various scenarios for modeling traveling waves of cells that constantly deplete a chemical cue, and so create their own signaling gradient all along the way. We begin with one famous model by Keller and Segel for bacterial chemotaxis. We present the model and the construction of the traveling wave solutions. We also discuss the limitation of this approach, and review some subsequent work addressing stability issues. Next, we review two relevant extensions, which are supported by biological experiments. They both admit traveling wave solutions with an explicit value for the wave speed. We conclude by discussing some open problems and perspectives, and particularly a striking mechanism of speed determinacy occurring at the back of the wave. All the results presented in this note are illustrated by numerical simulations.