Decay to equilibrium of the filament end density along the leading edge of the lamellipodium

TL;DR

This paper models actin filament dynamics at the lamellipodium edge, demonstrating exponential convergence to steady states through Lyapunov functional analysis, considering branching, capping, and lateral flow processes.

Contribution

It introduces a nonlinear model incorporating branching, capping, and flow, and proves exponential decay to equilibrium for high branching rates.

Findings

Existence of nontrivial steady states for high branching rates

Exponential convergence to steady states proven mathematically

Model captures key actin filament dynamics at the cell edge

Abstract

A model for the dynamics of actin filament ends along the leading edge of the lamellipodium is analyzed. It contains accounts of nucleation by branching, of deactivation by capping, and of lateral flow along the leading edge by polymerization. A nonlinearity arises from a Michaelis-Menten type modeling of the branching process. For branching rates large enough compared to capping rates, the existence and stability of nontrivial steady states is investigated. The main result is exponential convergence to nontrivial steady states, proven by investigating the decay of an appropriate Lyapunov functional.