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Traveling Motility of Actin Lamellar Fragments Under spontaneous symmetry breaking

Claudia García, Martina Magliocca, Nicolas Meunier

TL;DR

This work rigorously proves the existence of traveling-wave solutions in a minimal actin-motility model modeled as a free-boundary Darcy flow. By reformulating the moving boundary with a conformal map and Hilbert transform, the authors place the problem on a fixed disk and apply Crandall–Rabinowitz bifurcation theory. Spectral analysis identifies bifurcation points at $β_m=\frac{2}{m(m+1)}$ with a one-dimensional kernel, and the transversality condition is verified to obtain local branches of nontrivial traveling waves with $m$-fold symmetry. The results provide a mathematically rigorous foundation for traveling actin-driven motility and clarify how boundary curvature and surface tension influence symmetry breaking and pattern formation in the model.

Abstract

Cell motility is connected to the spontaneous symmetry breaking of a circular shape. In https://doi.org/10.1103/PhysRevLett.110.078102, Blanch-Mercader and Casademunt perfomed a nonlinear analysis of the minimal model proposed by Callan and Jones https://doi.org/10.1103/PhysRevLett.100.258106 and numerically conjectured the existence of traveling solutions once that symmetry is broken. In this work, we prove analytically that conjecture by means of nonlinear bifurcation techniques.

Traveling Motility of Actin Lamellar Fragments Under spontaneous symmetry breaking

TL;DR

This work rigorously proves the existence of traveling-wave solutions in a minimal actin-motility model modeled as a free-boundary Darcy flow. By reformulating the moving boundary with a conformal map and Hilbert transform, the authors place the problem on a fixed disk and apply Crandall–Rabinowitz bifurcation theory. Spectral analysis identifies bifurcation points at with a one-dimensional kernel, and the transversality condition is verified to obtain local branches of nontrivial traveling waves with -fold symmetry. The results provide a mathematically rigorous foundation for traveling actin-driven motility and clarify how boundary curvature and surface tension influence symmetry breaking and pattern formation in the model.

Abstract

Cell motility is connected to the spontaneous symmetry breaking of a circular shape. In https://doi.org/10.1103/PhysRevLett.110.078102, Blanch-Mercader and Casademunt perfomed a nonlinear analysis of the minimal model proposed by Callan and Jones https://doi.org/10.1103/PhysRevLett.100.258106 and numerically conjectured the existence of traveling solutions once that symmetry is broken. In this work, we prove analytically that conjecture by means of nonlinear bifurcation techniques.
Paper Structure (11 sections, 13 theorems, 130 equations)

This paper contains 11 sections, 13 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Rest state
  4. Linear stability of the rest state $P_0$
  5. Reformulation of the problem and reduction to the fixed boundary problem
  6. Function spaces and well-definition of $F$
  7. Crandall-Rabinowitz theorem
  8. Spectral study
  9. Kernel and range study
  10. Transversal condition
  11. Main result

Key Result

Theorem 1.1

For any $m\geq 2$, there exists $\xi\in I\mapsto (\gamma_{\xi},D_{\xi})$, with $D_{\xi}$ a $m$-fold symmetric domain, defining a traveling wave solution to eq:model-P--eq:model-bc-v with some constant speed.

Theorems & Definitions (26)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Proposition 2.6
  • ...and 16 more