Mathematical model for collective migration on a viscoelastic collagen network
Nicolas Meunier, Andrei Tarfulea
TL;DR
The paper analyzes a regularized PDE-ODE model for self-generated directional migration on a viscoelastic collagen network, coupling deformation $S$ to cell position $x_c(t)$ through $\left(\frac{1}{\beta^2}+\frac{\alpha}{\beta^2}\partial_t-\partial_{xx}^2\right)S=\frac{\alpha\gamma}{\beta^2} g_\varepsilon(x-x_c(t))$ with $\dot{x}_c(t)= -\eta \partial_x S(t,x_c(t))$, where $g_\varepsilon$ is a Gaussian-like regularization. By nondimensionalizing, the authors reduce to the essential parameter $\eta$ and study stationary states and traveling pulses: stationary states exist for all parameters, while traveling pulses exist only for large $\eta$ with velocity $v_c$ determined implicitly by an integral condition. They prove global well-posedness in $W^{k,\infty}$ spaces, rigorously construct the traveling pulse and the stationary state, and establish exponential stability of traveling pulses for large stiffness and exponential convergence to stationary states for small stiffness. The results provide a rigorous foundation for understanding how self-generated gradients on a viscoelastic substrate can drive persistent collective migration and clarify the stiffness threshold governing wave-like behavior. The methods combine explicit Fourier-based constructions, rescaling to nondimensional form, and barrier arguments to obtain sharp stability results relevant to biomechanical gradient generation.
Abstract
In this paper, we study a model of self-generated directional cell migration on viscoelastic substrates in the absence of apparent intrinsic polarity. This model, first proposed in \cite{Clark}, was observed numerically to manifest traveling pulse solutions for sufficiently large collagen stiffness, leading to a persistent collective migration. Here we provide a rigorous mathematical framework for the model, finding the exact stationary states and conditional traveling pulse. We also prove global well-posed in $W^{k,\infty}$ spaces, local stability of the traveling pulse for high stiffness, and exponential convergence to the stationary state for low stiffness.