Initial Layer Analysis for a Linkage Density in Cell Adhesion Mechanisms

Milisic, Vuk

doi:10.1051/proc/201862108

Cited by 4 publications

(11 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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).…”

Section: Existence Of Minimizers and A Priori Estimates : The Discretsupporting

confidence: 81%

“…This latter unknown is singular : ∂ t ρ ε is a measure that converges, when ε goes to zero, to ∂ t ρ 0 , the time derivative of the population of linkages and to a Dirac mass located near the origin in time for all ages. Since the problem is here space dependent, these results extend and complete the initial layer analysis performed in 15 .…”

Section: Introductionsupporting

confidence: 65%

“…x ∈ Ω, a > 0,t = 0 (5.1) and we defineρ 0,ε (x, a, t) :=ρ 0 (x, a, t/ε). As in 15 , we obtain at the microscopic level global existence and a priori bounds : Proof. The proof of the existence and uniqueness part is easy and follows the same ideas as in 12,15 where one shall only manage the x dependence in addition.…”

Section: Study Of the Initial Layer And Convergence Of Continuous Andmentioning

confidence: 96%

See 2 more Smart Citations

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

Milisic

2020

Journal of Functional Analysis

View full text Add to dashboard Cite

In the context of cell motility modelling and more particularly related to the Filament Based Lamelipodium Model 18,10,11 , this work deals with a rigorous mathematical proof of convergence between solutions of two problems : we start from a microscopic description of adhesions using a delayed and constrained vector valued equation with spacial diffusion and show the convergence towards the corresponding friction limit. The convergence is performed with respect to the bond characteristic lifetime ε whose inverse is also proportional to the stifness of the bonds. The originality of this work is the extension of gradient flow techniques to our setting. Namely, the discrete finite difference term in the gradient flow energy is here replaced by a delay term which complicates greatly the mathematical analysis. Contrarily to the standard approach 2,19 , compactness in time is not provided by the energy minimization process : a series of past times are taken into account in our discrete energy. A supplementary equation on the time derivative is obtained requiring uniform estimate with respect to ε of the Lagrange multiplier and provides compactness. Due to the non-linearity induced by the constraint, a specific stability estimate useful in our previous works, is not at hand here. Numerical simulations even showed that this estimate does not hold. Nevertheless, transposing our delay operator, we succeed in proving convergence under slightly weaker hypotheses. The result relies on a careful initial layer analysis, extending 15 to the space dependent setting.

show abstract

Section: Existence Of Minimizers and A Priori Estimates : The Discretsupporting

confidence: 81%

Section: Introductionsupporting

confidence: 65%

Section: Study Of the Initial Layer And Convergence Of Continuous Andmentioning

confidence: 96%

See 1 more Smart Citation

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

Milisic

2020

Journal of Functional Analysis

View full text Add to dashboard Cite

show abstract

“…This parameter is assumed to be small and the fact that the elasticity is O(ε −1 ), is a scaling assumption required for a non-vanishing effect of adhesions in the limit ε → 0. Our works construct various tools in order to handle rigorously this asymptotic [14,15,16,17]. In addition, concerning adhesion forces, a similar Ansatz was performed formally in a somehow different mechanical setting in [23,8].…”

Section: Biological and Mathematical Settingsmentioning

confidence: 99%

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

Milisic

Oelz

2018

Journal of Differential Equations

View full text Add to dashboard Cite

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.

show abstract

“…Here, we consider a rescaled non-linear version of the previous model in the spirit of [15,11,12,13,14,10,9], namely we study the problem : find z ε ∈ Lip([0, T ]) solution of…”

Section: Introductionmentioning

confidence: 99%

Asymptotic limits for a non-linear integro-differential equation modelling leukocytes' rolling on arterial walls

Milisic¹,

Schmeiser²

2020

Preprint

Self Cite

View full text Add to dashboard Cite

We consider a non-linear integro-differential model describing z, the position of the cell center on the real line presented in [5]. We introduce a new ε-scaling and we prove rigorously the asymptotics when ε goes to zero. We show that this scaling characterizes the long-time behavior of the solutions of our problem in the cinematic regime (i.e. the velocity ż tends to a limit). The convergence results are first given when ψ, the elastic energy associated to linkages, is convex and regular (the second order derivative of ψ is bounded). In the absence of blood flow, when ψ, is quadratic, we compute the final position z ∞ to which we prove that z tends. We then build a rigorous mathematical framework for ψ being convex but only Lipschitz. We extend convergence results with respect to ε to this case when ψ admits a finite number of jumps. In the last part, we show that in the constant force case (see Model 3 in [5], i.e. ψ is the absolute value), we solve explicitly the problem and recover the above asymptotic results.

show abstract

Initial Layer Analysis for a Linkage Density in Cell Adhesion Mechanisms

Cited by 4 publications

References 5 publications

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

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

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

Asymptotic limits for a non-linear integro-differential equation modelling leukocytes' rolling on arterial walls

Contact Info

Product

Resources

About