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Nicolas Meunier

Publications and source records attributed to Nicolas Meunier.

At least 19 recordsLinked to original sources
arXiv2025

Mathematical model for collective migration on a viscoelastic collagen network

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.

math.AP↗
arXiv2025

Existence of traveling wave for a coupled incompressible Darcy's free boundary model with undercooling effect and surface tension

In this paper, we present a cell motility model that takes into account the cell membrane effect. The model introduced is an incompressible Darcy free boundary problem. This model involves a nonlinear term in the boundary condition to model the action of the membrane. This term can be seen as a undercooling effect of the membrane on the cell. It also implies a destabilizing nonlinear term in the boundary condition, depending on polarity markers and modeling the active character of the cytoskeleton. First, we study the linear stability of the steady state and prove that above a threshold, the disk is linearly unstable. This analysis highlights the stabilizing effect of undercooling. Then, using a bifurcation argument, we prove the existence of traveling waves that describe a persistent motion in cell migration and justify the relevance of the model.

math.AP↗
arXiv2024

Traveling Motility of Actin Lamellar Fragments Under spontaneous symmetry breaking

Cell motility is connected to the spontaneous symmetry breaking of a circular shape. In https://doi.org/10.1103/PhysRevLett.110.078102, Blanch-Mercader and Casademunt perfomed a nonlinear analysis of the minimal model proposed by Callan and Jones https://doi.org/10.1103/PhysRevLett.100.258106 and numerically conjectured the existence of traveling solutions once that symmetry is broken. In this work, we prove analytically that conjecture by means of nonlinear bifurcation techniques.

math.AP↗
arXiv2023

Concentration in an advection-diffusion model with diffusion coefficient depending on the past trajectory

We consider a drift-diffusion model, with an unknown function depending on the spatial variable and an additional structural variable, the amount of ingested lipid. The diffusion coefficient depends on this additional variable. The drift acts on this additional variable, with a power-law coefficient of the additional variable and a localization function in space. It models the dynamics of a population of macrophage cells. Lipids are located in a given region of space; when cells pass through this region, they internalize some lipids. This leads to a problem whose mathematical novelty is the dependence of the diffusion coefficient on the past trajectory. We discuss global existence and blow-up of the solution.

math.AP↗
arXiv2023

Well posedness for systems of self-propelled particles

This paper deals with the existence and uniqueness of solutions to kinetic equations describing alignment of self-propelled particles. The particularity of these models is that the velocity variable is not on the euclidean space but constrained on the unit sphere (the self-propulsion constraint). Two related equations are considered : the first one in which the alignment mechanism is nonlocal, using an observation kernel depending on the space variable, and a second form which is purely local, corresponding to the principal order of a scaling limit of the first one. We prove local existence and uniqueness of weak solutions in both cases for bounded initial conditions (in space and velocity) with finite total mass. The solution is proven to depend continuously on the initial data in $L^p$ spaces with finite p. In the case of a bounded kernel of observation, we obtain that the solution is global in time. Finally by exploiting the fact that the second equation corresponds to the principal order of a scaling limit of the first one we deduce a Cauchy theory for an approximate problem approaching the second one.

math.AP↗
arXiv2023

Convergence, concentration and critical mass phenomena in a model of cell motion with boundary signal production

We consider a model of cell motion with boundary signal production which describes some aspects of eukaryotic cell migration. Generic polarity markers located in the cell are transported by actin which they help to polymerize. This leads to a problem whose mathematical novelty is the nonlinear and nonlocal destabilizing term in the boundary condition. We provide a detailed study of the qualitative properties of this model, namely local and global existence, convergence and blow-up of solutions. We start with a complete analysis of local existence-uniqueness in Lebesgue spaces. This turns out to be particularly relevant, in view of the mass conservation property and of the existence of $L^p$ Liapunov functionals, also obtained in this paper. With the help of this local theory, we next study the global existence and convergence of solutions. In particular, in the case of quadratic nonlinearity, for any space dimension, we find an explicit, sharp mass threshold for global existence vs.~finite time blow-up of solutions. The proof is delicate, based on the possiblity to control the solution by means of the entropy function via an $\eps$-regularity type argument. This critical mass phenomenon is somehow reminiscent of the well-known situation for the $2d$ Keller-Segel system. For nonlinearitities with general power growth, under a suitable smallness condition on the initial data, we show that solutions exist globally and converge exponentially to a constant. As for the possibility of blow-up for large initial data, it turns out to occur only for nonlinearities with quadratic or superquadratic growth, whereas all solutions are shown to be global and bounded in the subquadratic case, thus revealing the existence of a sharp critical exponent for blow-up. Finally, we analyse some aspects of the blow-up asymptotics of solutions in time and space.

math.AP↗
arXiv2022

Traveling wave solution for a coupled incompressible Darcy's free boundary problem with surface tension

We study an incompressible Darcy's free boundary problem, recently introduced in [22]. Our goal is to prove the existence of non-trivial traveling wave solutions and thus validate the interest of this model to describe cell motility. The model equations include a convection diffusion equation for the polarity marker concentration and the incompressible Darcy's equation. The mathematical novelty of this problem is the nonlinear destabilizing term in the boundary condition that describes the active character of the cell cytoskeleton. We first study the linear stability of this problem and we show that, above a well precise threshold, the disk becomes linearly unstable. By using two different approaches we prove existence of traveling wave solutions, which describes persistent motion of a biological cell. One is explicit, by construction. The other is established implicitly, as the one bifurcating from stationary solution.

math.AP↗
arXiv2020

Lyapunov functions, Identities and the Cauchy problem for the Hele-Shaw equation

This article is devoted to the study of the Hele-Shaw equation. We introduce an approach inspired by the water-wave theory. Starting from a reduction to the boundary, introducing the Dirichlet to Neumann operator and exploiting various cancellations, we exhibit parabolic evolution equations for the horizontal and vertical traces of the velocity on the free surface. This allows to quasi-linearize the equations in a very simple way. By combining these exact identities with convexity inequalities, we prove the existence of hidden Lyapunov functions of different natures. We also deduce from these identities and previous works on the water wave problem a simple proof of the well-posedness of the Cauchy problem. The analysis contains two side results of independent interest. Firstly, we give a principle to derive estimates for the modulus of continuity of a PDE under general assumptions on the flow. Secondly we prove a convexity inequality for the Dirichlet to Neumann operator.

math.AP↗
arXiv2019

Analysis of a non-local and non-linear Fokker-Planck model for cell crawling migration

Cell movement has essential functions in development, immunity and cancer. Various cell migration patterns have been reported and a general rule has recently emerged, the so-called UCSP (Universal Coupling between cell Speed and cell Persistence), [30]. This rule says that cell persistence, which quantifies the straightness of trajectories, is robustly coupled to migration speed. In [30], the advection of polarity cues by a dynamic actin cytoskeleton undergoing flows at the cellular scale was proposed as a first explanation of this universal coupling. Here, following ideas proposed in [30], we present and study a simple model to describe motility initiation in crawling cells. It consists of a non-linear and non-local Fokker-Planck equation, with a coupling involving the trace value on the boundary. In the one-dimensional case we characterize the following behaviours: solutions are global if the mass is below the critical mass, and they can blow-up in finite time above the critical mass. In addition, we prove a quantitative convergence result using relative entropy techniques.

math.AP↗
arXiv2019

A stochastic model for cell adhesion to the vascular wall

This paper deals with the adhesive interaction arising between a cell circulating in the blood flow and the vascular wall. The purpose of this work is to investigate the effect of the blood flow velocity on the cell dynamics, and in particular on its possible adhesion to the vascular wall. We formulate a model that takes into account the stochastic variability of the formation of bonds, and the influence of the cell velocity on the binding dynamics: the faster the cell goes, the more likely existing bonds are to disassemble. The model is based on a nonlinear birth-and-death-like dynamics, in the spirit of Joffe and Metivier (1986); Ethier and Kurtz (2009). We prove that, under different scaling regimes, the cell velocity follows either an ordinary differential equation or a stochastic differential equation, that we both analyse. We obtain both the identification of a shear-velocity threshold associated with the transition from cell sliding and its firm adhesion, and the expression of the cell mean stopping time as a function of its adhesive dynamics.

math.AP↗
arXiv2018

A stochastic model for protrusion activity

In this work we approach cell migration under a large-scale assumption, so that the system reduces to a particle in motion. Unlike classical particle models, the cell displacement results from its internal activity: the cell velocity is a function of the (discrete) protrusive forces exerted by filopodia on the substrate. Cell polarisation ability is modeled in the feedback that the cell motion exerts on the protrusion rates: faster cells form preferentially protrusions in the direction of motion. By using the mathematical framework of structured population processes previously developed to study population dynamics [Fournier and M{é}l{é}ard, 2004], we introduce rigorously the mathematical model and we derive some of its fundamental properties. We perform numerical simulations on this model showing that different types of trajectories may be obtained: Brownian-like, persistent, or intermittent when the cell switches between both previous regimes. We find back the trajectories usually described in the literature for cell migration.

q-bio.CB↗
arXiv2018

Crawling migration under chemical signalling: a stochastic particle model

Cell migration is a fundamental process involved in physiological phenomena such as the immune response and morphogenesis, but also in pathological processes, such as the development of tumor metastasis. These functions are effectively ensured because cells are active systems that adapt to their environment. In this work, we consider a migrating cell as an active particle, where its intracellular activity is responsible for motion. Such system was already modeled in a previous model where the protrusion activity of the cell was described by a stochastic Markovian jump process. The model was proven able to capture the diversity in observed trajectories. Here, we add a description of the effect of an external chemical attractive signal on the protrusion dynamics, that may vary in time. We show that the resulting stochastic model is a well-posed non-homogeneous Markovian process, and provide cell trajectories in different settings, illustrating the effects of the signal on long-term trajectories.

q-bio.CB↗
arXiv2018

Numerical solutions of a 2D fluid problem coupled to a nonlinear non-local reaction-advection-diffusion problem for cell crawling migration in a discoidal domain

In this work, we present a numerical scheme for the approximate solutions of a 2D crawling cell migration problem. The model, defined on a non-deformable discoidal domain, consists in a Darcy fluid problem coupled with a Poisson problem and a reaction-advection-diffusion problem. Moreover, the advection velocity depends on boundary values, making the problem nonlinear and non local. \parFor a discoidal domain, numerical solutions can be obtained using the finite volume method on the polar formulation of the model. Simulations show that different migration behaviours can be captured.

math.NA↗
arXiv2017

Modelling the inflammatory process in atherosclerosis: a nonlinear renewal equation

We present here a population structured model to describe the dynamics of macrophage cells. The model involves the interactions between modified LDL, monocytes/macrophages, cytokines and foam cells. The key assumption is that the individual macrophage dynamics depends on the amount of lipoproteins it has internalized. The obtained renewal equation is coupled with an ODE describing the lipoprotein dynamics. We first prove global existence and uniqueness for the nonlinear and nonlocal system. We then study long time asymp-totics in a particular case describing silent plaques which undergo periodic rupture and repair. Finally we study long time asymptotics for the nonlinear renewal equation obtained when considering the steady state of the ODE. and we prove that ....

math.AP↗
arXiv2016

Long time behavior in locally activated random walks

We consider a 1-dimensional Brownian motion whose diffusion coefficient varies when it crosses the origin. We study the long time behavior and we establish different regimes, depending on the variations of the diffusion coefficient: emergence of a non-Gaussian mul-tipeaked probability distribution and a dynamical transition to an absorbing static state. We compute the generator and we study the partial differential equation which involves its adjoint. We discuss global existence and blow-up of the solution to this latter equation.

math.PR↗
arXiv2013

Cell polarisation model : the 1D case

We study the dynamics of a one-dimensional non-linear and non-local drift-di usion equation set in the half-line, with the coupling involving the trace value on the boundary. The initial mass M of the density determines the behaviour of the equation: attraction to self similar pro le, to a steady state of nite time blow up for supercritical mass. Using the logarithmic Sobolev and the HWI inequalities we obtain a rate of convergence for the cases subcritical and critical mass. Moreover, we prove a comparison principle on the equation obtained after space integration. This concentration-comparison principle allows proving blow-up of solutions for large initial data without any monotonicity assumption on the initial data.

math.AP↗
arXiv2013

Numerical simulation of the dynamics of molecular markers involved in cell polarisation

A cell is polarised when it has developed a main axis of organisation through the reorganisation of its cytosqueleton and its intracellular organelles. Polarisation can occur spontaneously or be triggered by external signals, like gradients of signaling molecules ... In this work, we study mathematical models for cell polarisation. These models are based on nonlinear convection-diffusion equations. The nonlinearity in the transport term expresses the positive loop between the level of protein concentration localised in a small area of the cell membrane and the number of new proteins that will be convected to the same area. We perform numerical simulations and we illustrate that these models are rich enough to describe the apparition of a polarisome.

math.AP↗