Nicolas Meunier’s research while affiliated with National Council for Scientific Research, Lebanon and other places

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.

We consider a system of two nonlocal and nonlinear partial differential equations that describe some aspects of yeast cell-cell communication. We study local and global existence and uniqueness of solutions. We consider mild solutions and we perform bilinear and trilinear fixed point arguments in suitable functional spaces.

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.

In this work, we develop a model to describe some aspects of communication between yeast cells. It consists in a coupled system of two one-dimensional non-linear advection-diffusion equations in which the advective field is given by the Hilbert transform. We give some sufficient condition for the blow-up in finite time of the coupled system (formation of a singularity). We provide a biological interpretation of these mathematical results.

In this work, we propose and compare three numerical methods to handle the one-phase Hele-Shaw problem with surface tension in dimension two by using three variational approaches in the spirit of the seminal works \cite{Otto, Gia_Otto}.

We propose to model certain aspects of the dynamics of a macrophage that moves randomly in a one dimensional space in arterial wall tissue and grows by accumulating localized lipid particles, thus reducing its motility. This phenomenon has been observed in the context of atherosclerotic plaque formation. For this purpose, we use a system of stochastic differential equations satisfied by the position and diffusion coefficient of a Brownian particle whose diffusion coefficient is modified at each visit to the origin and with a dumping coefficient. The novelty of the model, with respect to Bénichou et al. (Phys Rev E 85(2):021137, 2012), Meunier et al. (Acta Appl Math 161:107–126, 2019), is to include offloading of lipids through the dumping term. We find explicit necessary and sufficient conditions for macrophage trapping in the locally enriched region.

In this work, we propose and compare three numerical methods to handle the one-phase Hele-Shaw problem with surface tension in dimension two by using three variational approaches in the spirit of the seminal works \cite{Otto, Gia_Otto}.

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.

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.