Phase-Field Model of Cell Motility: Traveling Waves and Sharp Interface Limit

TL;DR

This paper analyzes a PDE model of cell motility, deriving its sharp interface limit, and investigates traveling wave solutions and complex interface behaviors through rigorous and numerical methods.

Contribution

It formally derives the sharp interface limit of a cell motility PDE model and explores traveling wave solutions and interface dynamics.

Findings

Sharp interface limit describes volume-preserving curvature-driven motion.

Traveling waves emerge when physical parameters exceed critical thresholds.

Numerical simulations reveal interface velocity discontinuities and hysteresis.

Abstract

This letter is concerned with asymptotic analysis of a PDE model for motility of a eukaryotic cell on a substrate. This model was introduced in [1], where it was shown numerically that it successfully reproduces experimentally observed phenomena of cell-motility such as a discontinuous onset of motion and shape oscillations. The model consists of a parabolic PDE for a scalar phase-field function coupled with a vectorial parabolic PDE for the actin filament network (cytoskeleton). We formally derive the sharp interface limit (SIL), which describes the motion of the cell membrane and show that it is a volume preserving curvature driven motion with an additional nonlinear term due to adhesion to the substrate and protrusion by the cytoskeleton. In a 1D model problem we rigorously justify the SIL, and, using numerical simulations, observe some surprising features such as discontinuity of interface velocities and hysteresis. We show that nontrivial traveling wave solutions appear when the key physical parameter exceeds a certain critical value and the potential in the equation for phase field function possesses certain asymmetry.