Analysis of non-overlapping models with a weighted infinite delay

TL;DR

This paper models cell motility using a delayed elastic force framework, proving convergence to a friction model as microscopic linkages turnover rate approaches zero, with implications for realistic external load assumptions.

Contribution

It introduces a novel delayed and constrained model for cell motility and proves its convergence to a friction model under realistic biological assumptions.

Findings

Convergence of the delayed elastic model to a friction model as linkage turnover tends to zero.

Development of energy estimates accounting for delay effects.

Establishment of well-posedness and compactness of the discretized problem.

Abstract

The framework of this article is cell motility modeling. Approximating cells as rigid spheres we take into account for both non-penetration and adhesions forces. Adhesions are modeled as a memory-like microscopic elastic forces. This leads to a delayed and constrained vector valued system of equations. We prove that the solution of these equations converges when {\epsilon}, the linkages turnover parameter, tends to zero to the a constrained model with friction. We discretize the problem and penalize the constraints to get an uncon?strained minimization problem. The well-posedness of the constrained problem is obtained by letting the penalty parameter to tend to zero. Energy estimates `a la De Giorgi are derived accounting for delay. Thanks to these estimates and the convexity of the constraints, we obtain compactness uniformly with respect to the discretisation step and {\epsilon}, this is the mathematically involved part of the article. Considering that the characteristic bonds lifetime goes to zero, we recover a friction model comparable to [Venel et al, ESAIM, 2011] but under more realistic assumptions on the external load, this part being also one of the challenging aspects of the work