Space dependent adhesion forces mediated by transient elastic linkages: New convergence and global existence results

Abstract

In the first part of this work we show the convergence with respect to an asymptotic parameter ε of a delayed heat equation. It represents a mathematical extension of works considered previously by the authors [14,15,16]. Namely, this is the first result involving delay operators approximating protein linkages coupled with a spatial elliptic second order operator. For the sake of simplicity we choose the Laplace operator, although more general results could be derived. The main arguments are (i) new energy estimates and (ii) a stability result extended from the previous work to this more involved context. They allow to prove convergence of the delay operator to a friction term together with the Laplace operator in the same asymptotic regime considered without the space dependence in [14]. In a second part we extend fixed-point results for the fully non-linear model introduced in [16] and prove global existence in time. This shows that the blow-up scenario observed previously does not occur. Since the latter result was interpreted as a rupture of adhesion forces, we discuss the possibility of bond breaking both from the analytic and numerical point of view.

From delayed and constrained minimizing movements to the harmonic map heat equation

“…where δZ 3.1. Positivity and convergence of the discrete solution ρ ε,∆ From Lemma 3.1 to Theorem 3.1, we extend results from previous works 12,16,15 to the discrete case. When needed, we characterize also some properties of ρ ε , the continuous solution of (1.3).…”

“…In 14 we proved that a non-linear fully coupled model could either have global solutions or, if the external load exceeds the microscopic adhesion capacity, blow-up could occur. More recently 16 , we extended these results adding space dependent adhesion and diffusion.…”

“…Proof. For the existence and uniqueness part, one proceeds as in Theorem 3.1 in 16 : as x is a mute parameter, for a.e. x ∈ Ω, there exists a solution ρ ε (x, ·, ·) ∈ C t ([0, T ]; L 1 a (R + )).…”

Initial Layer Analysis for a Linkage Density in Cell Adhesion Mechanisms

“…Then various mathematical problems were handled extending [5] : in [6], the authors weakened some of the hypotheses and rephrased the model in new variables, while in [7] a non-linear coupling was exposed and blow-up in finite time vs global existence was proved. More recently, a model closer to the original description in [9] is analyzed [4]. This led to new questions, this work aims to answer one of those.…”

“…The initial layer is a microscopic lifting of the initial condition of the difference ρ ε −ρ 0 , it removes the exponential decay occurring in S ε , which is visible on the first term of the right hand side in (4). As the boundary term in (1) is non local, the analysis is not straightforward, and relies strongly on the specific energy functional introduced in [5].…”