Existence of traveling wave for a coupled incompressible Darcy's free boundary model with undercooling effect and surface tension

TL;DR

This paper proves the existence of traveling wave solutions in a coupled incompressible Darcy free boundary model for cell motility, incorporating membrane effects, undercooling, and surface tension, highlighting stability and instability conditions.

Contribution

It introduces a novel cell motility model with nonlinear boundary conditions accounting for membrane effects and proves the existence of traveling wave solutions.

Findings

Disk becomes unstable above a certain threshold.

Undercooling stabilizes the steady state.

Traveling waves model persistent cell motion.

Abstract

In this paper, we present a cell motility model that takes into account the cell membrane effect. The model introduced is an incompressible Darcy free boundary problem. This model involves a nonlinear term in the boundary condition to model the action of the membrane. This term can be seen as a undercooling effect of the membrane on the cell. It also implies a destabilizing nonlinear term in the boundary condition, depending on polarity markers and modeling the active character of the cytoskeleton. First, we study the linear stability of the steady state and prove that above a threshold, the disk is linearly unstable. This analysis highlights the stabilizing effect of undercooling. Then, using a bifurcation argument, we prove the existence of traveling waves that describe a persistent motion in cell migration and justify the relevance of the model.