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Multiscale analysis of a kinetic equation for mechanotaxis

Benoît Perthame, Francesco Salvarani, Shugo Yasuda

TL;DR

The paper addresses mechanotaxis by introducing a kinetic equation that includes an acceleration term representing substrate-driven forces, bridging mesoscopic cell dynamics and macroscopic population evolution. Through a formal multiscale analysis, it derives several diffusion-type macroscopic limits under different friction and tumbling time scales, revealing how substrate signaling and a velocity-dependent mobility shape density evolution. It establishes entropy-dissipation properties, a unique stationary state, and conditions for exponential convergence, and analyzes pattern formation via linear instability and one-dimensional steady states, including Dirac-concentration scenarios as substrate diffusion vanishes. Numerical simulations corroborate the theory, exhibiting periodic, phase-separated, and Dirac-like patterns and highlighting metastable transitions, thereby providing a comprehensive framework for multiscale mechanotaxis with substrate mechanics and guiding future nonlinear and applied work.

Abstract

We present a new kinetic equation for cell migration driven by mechanical interactions with the substrate, an effect not previously captured in kinetic models, and essential for explaining observed collective behaviors such as those in bacterial colonies. The model introduces an acceleration term that accounts for the dynamics of motile cells undergoing mechanotaxis, where extracellular signals modulate the forces arising from cell-substrate interactions. From this formulation, we derive a family of macroscopic limit equations and analyze their principal properties. In particular, we examine linear stability and pattern formation ability through theoretical analysis, supported by numerical simulations.

Multiscale analysis of a kinetic equation for mechanotaxis

TL;DR

The paper addresses mechanotaxis by introducing a kinetic equation that includes an acceleration term representing substrate-driven forces, bridging mesoscopic cell dynamics and macroscopic population evolution. Through a formal multiscale analysis, it derives several diffusion-type macroscopic limits under different friction and tumbling time scales, revealing how substrate signaling and a velocity-dependent mobility shape density evolution. It establishes entropy-dissipation properties, a unique stationary state, and conditions for exponential convergence, and analyzes pattern formation via linear instability and one-dimensional steady states, including Dirac-concentration scenarios as substrate diffusion vanishes. Numerical simulations corroborate the theory, exhibiting periodic, phase-separated, and Dirac-like patterns and highlighting metastable transitions, thereby providing a comprehensive framework for multiscale mechanotaxis with substrate mechanics and guiding future nonlinear and applied work.

Abstract

We present a new kinetic equation for cell migration driven by mechanical interactions with the substrate, an effect not previously captured in kinetic models, and essential for explaining observed collective behaviors such as those in bacterial colonies. The model introduces an acceleration term that accounts for the dynamics of motile cells undergoing mechanotaxis, where extracellular signals modulate the forces arising from cell-substrate interactions. From this formulation, we derive a family of macroscopic limit equations and analyze their principal properties. In particular, we examine linear stability and pattern formation ability through theoretical analysis, supported by numerical simulations.
Paper Structure (17 sections, 7 theorems, 155 equations, 5 figures)

This paper contains 17 sections, 7 theorems, 155 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The kinetic model for mechanotaxis
  3. Literature review
  4. Structure of the paper
  5. Macroscopic limits
  6. Friction dominating
  7. Strong friction, diffusion limit.
  8. Fast tumbling, moderate friction
  9. The general scaling as $m=O(\frac{1}{\lambda})$
  10. Fast tumbling, very small friction
  11. Bounds for the macroscopic equation
  12. Pattern formation
  13. Linear instability
  14. Steady periodic solutions in 1D
  15. Dirac concentration and periodic patterns
  16. ...and 2 more sections

Key Result

Proposition 1

Being given $T>0$, with the condition on $v:=v(S)$ the strong solutions of Basic, posed in $\mathbb{R}^d$, with initial data $\rho^0 >0$, satisfy the following entropy estimate for all $t\leq T$:

Figures (5)

  • Figure 1: Time evolutions of $\rho$ at different values of $\alpha$. The parameter $D_S=0.01$ is fixed.
  • Figure 2: Time evolutions of $\rho$ at different values of $D_S$. The parameter $\alpha=1$ is fixed. The critial wave lengths are $2\pi/k_\mathrm{c}=0.770$ for (a), $2\pi/k_\mathrm{c}=0.385$ for (b), and $2\pi/k_\mathrm{c}=0.192$ for (c)
  • Figure 3: Comparison of the steady solutions of $S$ obtained by the finite volume scheme (\ref{['1dfv']}) [blue solid lines] and those obtained by the semi-analytical method (\ref{['eq_DsSx']}) [red dashed lines]. The parameters are the same as in Fig. \ref{['fig:rho_Ds']}.
  • Figure 4: Time evolutions of $\rho$ [in (a)] and $E=\frac{1}{L}\int \rho\ln\rho \,\mathrm{d}x$ [in (b)] for $v(S)$ written as Eq. (\ref{['eq_vS_atan']}). The parameters are set as $\chi$=0.02, $\delta$=0.01, $D_S$=0.001, and $\alpha=0$, which gives $2\pi/k_{\mathrm{c}}=0.20$.
  • Figure 5: Time evolutions of $\rho$ [in (a)] and $E=\frac{1}{L}\int \rho\ln\rho \,\mathrm{d}x$ [in (b)] for $v(S)$ written as Eq. (\ref{['eq_vS_atan']}). The parameters are set as $\chi$=0.012, $\delta$=0.01, $D_S$=0.001, and $\alpha=0$, which gives $2\pi/k_{\mathrm{c}}=0.44$.

Theorems & Definitions (8)

  • Proposition 1: Entropy control
  • Proposition 2: Free energy decay
  • Proposition 3
  • Lemma 4
  • Lemma 5
  • Proposition 6
  • Corollary 7: Exponential convergence
  • Remark 8