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

Gervy Marie Angeles, Jared Barber, Christian Schmeiser

TL;DR

A strongly simplified version of the Filament Based Lamellipodium Model is considered, showing a destabilizing effect of the repulsion, leading to a pitchfork bifurcation from a trivial steady state to traveling wave solutions.

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, leading to a pitchfork bifurcation from a trivial steady state to 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.

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

TL;DR

A strongly simplified version of the Filament Based Lamellipodium Model is considered, showing a destabilizing effect of the repulsion, leading to a pitchfork bifurcation from a trivial steady state to traveling wave solutions.

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, leading to a pitchfork bifurcation from a trivial steady state to 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.
Paper Structure (10 sections, 3 theorems, 107 equations, 7 figures, 1 table)

This paper contains 10 sections, 3 theorems, 107 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model derivation
  3. Linearization around a trivial steady state
  4. Remark.
  5. Normal form reduction close to the bifurcation point
  6. Case 1. $\frac{1}{2}<\gamma<1$ (high tension).
  7. Case 2. $0<\gamma<\frac{1}{2}$ (low tension).
  8. Remark.
  9. Numerical Simulations
  10. Discussion

Key Result

Lemma 1

Let $\beta>0$, $0<\gamma<1$, and $a_I,b_I,\psi_I\in L^2(0,1)$. Then eq:bif1a, eq:bif1--eq:bif1bdy, eq:bif1-IC has a unique solution $(a,b,\psi)\in C([0,\infty), L^2(0,1))^3$ satisfying for $t\ge 0$, with $c(\gamma), \lambda_1(\gamma)>0$.

Figures (7)

  • Figure 1: Evolution of filament strip in Example 1.1 with $\gamma = 3/4$ and $\beta=1.01\beta_0$ (high tension, thick band). The trivial steady state is stable.
  • Figure 2: Evolution of filament strip in Example 1.2 with $\gamma = 3/4$ and $\beta=0.99\beta_0$ (high tension, thin band). Stability is transferred to the bifurcating steady state \ref{['eq:ad8a']}-\ref{['eq:ad8b']}.
  • Figure 3: Evolution of filament strip in Example 1.3 with $\gamma = 1/4$ and $\beta=0.99\beta_0$ (low tension, thin band).
  • Figure 4: Subcritical bifurcation showing bistability for $\gamma=1/4$ and $\beta=1.01\beta_0$. Left: Relaxation to the trivial steady state from \ref{['eq:myic2a']}. Right: Evolution toward a bifurcating steady state from \ref{['eq:myic2b']}.
  • Figure 5: Plot of the bifurcation behavior showing the speed of the filament ensemble vs. $\beta \in (0.045,0.14)$ at $t=10^6$ with $\gamma=3/4$ (high tension) and $\beta_0=0.0919$. Gray points correspond to unstable steady state solutions while blue correspond to stable steady state solutions.
  • ...and 2 more figures

Theorems & Definitions (11)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof : Proof of Theorem \ref{['thm:wp']}
  • Example 1: Small perturbation from the trivial steady state
  • Example 1.1: $\beta> \beta_0$ -- thick band
  • Example 1.2: $\gamma = 3/4\,,$ $\beta =0.99\beta_0$ -- high tension, thin band
  • Example 1.3: $\gamma = 1/4\,,$ $\beta =0.99\beta_0$ -- low tension, thin band
  • ...and 1 more