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Existence of traveling wave for a coupled incompressible Darcy's free boundary model with undercooling effect and surface tension

Claire Alamichel, Nicolas Meunier

TL;DR

This work develops an incompressible Darcy free-boundary model for cell motility that incorporates membrane undercooling and surface tension, coupling polarization markers to edge dynamics. By performing linear stability analysis and Fourier decomposition, it identifies a threshold $\chi_c^* $ separating stable resting states from instability, and proves, via Crandall–Rabinowitz bifurcation, the existence of a one-parameter family of traveling waves with fixed area, describing persistent motion. The results show the membrane undercooling stabilizes the resting state and shapes the traveling wave domain, providing a rigorous mathematical justification for spontaneous persistent cell migration within a biophysically motivated framework. These insights extend prior free-boundary motility models by incorporating polarity-marker coupling and a nonlinear membrane boundary condition, with implications for understanding polarization-driven motility in biological systems.

Abstract

In this paper, we present a cell motility model that takes into account the cell membrane effect. The model introduced is an incompressible Darcy free boundary problem. This model involves a nonlinear term in the boundary condition to model the action of the membrane. This term can be seen as a undercooling effect of the membrane on the cell. It also implies a destabilizing nonlinear term in the boundary condition, depending on polarity markers and modeling the active character of the cytoskeleton. First, we study the linear stability of the steady state and prove that above a threshold, the disk is linearly unstable. This analysis highlights the stabilizing effect of undercooling. Then, using a bifurcation argument, we prove the existence of traveling waves that describe a persistent motion in cell migration and justify the relevance of the model.

Existence of traveling wave for a coupled incompressible Darcy's free boundary model with undercooling effect and surface tension

TL;DR

This work develops an incompressible Darcy free-boundary model for cell motility that incorporates membrane undercooling and surface tension, coupling polarization markers to edge dynamics. By performing linear stability analysis and Fourier decomposition, it identifies a threshold separating stable resting states from instability, and proves, via Crandall–Rabinowitz bifurcation, the existence of a one-parameter family of traveling waves with fixed area, describing persistent motion. The results show the membrane undercooling stabilizes the resting state and shapes the traveling wave domain, providing a rigorous mathematical justification for spontaneous persistent cell migration within a biophysically motivated framework. These insights extend prior free-boundary motility models by incorporating polarity-marker coupling and a nonlinear membrane boundary condition, with implications for understanding polarization-driven motility in biological systems.

Abstract

In this paper, we present a cell motility model that takes into account the cell membrane effect. The model introduced is an incompressible Darcy free boundary problem. This model involves a nonlinear term in the boundary condition to model the action of the membrane. This term can be seen as a undercooling effect of the membrane on the cell. It also implies a destabilizing nonlinear term in the boundary condition, depending on polarity markers and modeling the active character of the cytoskeleton. First, we study the linear stability of the steady state and prove that above a threshold, the disk is linearly unstable. This analysis highlights the stabilizing effect of undercooling. Then, using a bifurcation argument, we prove the existence of traveling waves that describe a persistent motion in cell migration and justify the relevance of the model.
Paper Structure (8 sections, 13 theorems, 121 equations)

This paper contains 8 sections, 13 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Biological context and modeling
  3. On traveling waves
  4. Main results
  5. Characterization of traveling waves
  6. Study of the resting state
  7. Existence of traveling waves
  8. Conclusion and perspectives

Key Result

Proposition 1.2

Let $V\geq0$ be given. If $c$ and $P$ are solutions of eq:tw_def associated with $V$ then $c$ and $P$ are of the form: where $\left(x,y\right) \in \tilde{\Omega}$ and $M \geq 0$ is the total quantity of markers. Furthermore, $\partial \tilde{\Omega}$ is characterized by the curvature equation, given for all $\left(x,y\right) \in \partial \tilde{\Omega}$:

Theorems & Definitions (26)

  • Definition 1.1
  • Proposition 1.2
  • Remark 1.3
  • Proposition 1.4
  • Theorem 1.5: Stability of the resting state
  • Theorem 1.6: Existence of traveling waves
  • proof : Proof of \ref{['prop:characterization_tw']}
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 16 more