Pitchfork bifurcation and traveling waves for a planar ensemble of rigid filaments with repulsive interaction

TL;DR

This paper analyzes a simplified model of cell filaments with repulsive interactions, revealing a pitchfork bifurcation that leads to traveling wave solutions and complex dynamics, supported by bifurcation analysis and numerical simulations.

Contribution

It introduces a simplified model capturing the destabilizing effect of filament repulsion and characterizes the bifurcation structure and resulting traveling wave solutions.

Findings

Repulsive interactions destabilize the system.

A pitchfork bifurcation leads to traveling wave solutions.

Numerical simulations show periodic and chaotic dynamics.

Abstract

The so-called Filament Based Lamellipodium Model is a complex modeling framework for a very heterogeneous chemo-mechanical system of cell biology. It contains a model for Coulomb repulsion between filaments, whose effect on the stability of the system has been unclear. In this work, a strongly simplified version of the model is considered, showing a destabilizing effect of the repulsion. This instability results in a pitchfork bifurcation with an additional rotational symmetry, leading to a two-dimensional bifurcating manifold of traveling wave solutions. The simplified model is derived, its linearization around the trivial steady state is analyzed, and a formal bifurcation analysis is carried out. It is shown that the pitchfork bifurcation maybe super- or sub-critical. Time dependent numerical simulations illustrate these results and provide additional, more global information on the emergence of periodic and chaotic dynamics by secondary bifurcations.