Impact of cell shapes on their dynamics and fate

Nicolas Meunier
Vuk Milišić


Cells (as immune cells or cancer cells) are deformable and their shapes adapt and play an important role in their evolution and their pathological/physiological properties. We consider here mesoscale and macroscale aspects of cells’ shapes and the corresponing immune responses. Control questions related to the shape are to be considered as well and their consequences on the immune response.

Background and proposed research

In tissues, cells can take many different shapes, by attaching to structures or other cells or by moving. The mechanisms by which this adaptation occurs and the long-term consequences that these repeated shape changes have on physiology and pathology remain largely unknown. It has been observed that changes in the shape of cells and organelles induce reversible and irreversible modifications of their behavior and function.

In this project, we investigate if cells’ shape, imposed by their physical confinement in tissues, have an influence on their evolution in the spirit of [1245678910]. Our multi-scale approach, takes into account the randomness of the environment, the intrinsic activity and the shape of the cell.

Position of the project as it relates to the state of the art

This project focus on the modeling and mathematical study at different scales of cell’s shape and the impact of these changes on its dynamics and fate. In particular, we aim to develop multi-scale models of the immune response and of the cell fate for cancer cells, integrating both the biophysics of the cellular scale as well as the complex geometry of our organs [15]. A particularity of this project is to take into account a conceptual aspect little studied in biology : the memory effect in the cell.The main difficulties comes from the consideration of different spatial and temporal scales.

Mesoscopic scale (cell scale)

In the spirit of [1112] we will seek to determine the biophysical mechanisms that impact cell shapes that in turn induce different cell trajectories and assess their optimality.

Recently a model was introduced in [961] to describe cell motility. The novelty of this model lies in the coupling in the boundary term of the Hele-Shaw model with a PDE, stated on the free evolving domain, which describes its organelles and cytoskeleton. Despite a large number of studies of cell motility the determination of cellular trajectories remains a largely open problem.

The objectives here are to: (i) construct a coupled free-boundary model integrating the biophysical phenomena involved in dendritic cell motility (antigen-myosin competition); (ii) characterize its behavior: stationary solutions, linear stability, bifurcation, traveling waves (a hallmark of cell motility); (iii) perform numerical simulations in nontrivial geometries, to do this it will be necessary to add stochastic perturbations to the model; (iv) add the memory effect.

Macroscopic scale (that of adaptive immune response)

This part of the project focuses on the macroscopic scale (that of adaptive immune response). We will first look at the optimal strategy that a population of dendritic cells must adopt to find the highest number of antigens of unknown locations. This refers to the “first pass time” (TPP) in random walk theory. Despite a large number of studies, the determination of TPPs in confined geometries remains a largely open problem.

The objectives here are to: (i) study one-dimensional SDE models with self repulsion or self attraction, which act as toy models for the manner in which the dendritic cells explore the body, in order to obtain explicit bounds on propagation speed (see [13] in the deterministic case); (ii) use an additional variable, the “local time,” to construct an integro-differential PDE to describe the dynamics of a population of cells with memory; (iii) study the scale invariances and long time behaviour for this new PDE.

Methodology and work plan:

In this project, we will study mathematically and numerically nonlinear and nonlocal convection diffusion equations on the whole space, a fixed domain, or a free-boundary domain with a particular interest in asymptotic behavior (e.g., traveling waves) and the effects of randomness. We will be interested in these same questions for models structured by the local time spent at some positions [314], which will be a way to quantity the antigens internalized or the physical confinement in tissues.


[1]   T. Alazard, M. Magliocca and N. Meunier. Traveling wave solution for a coupled incompressible Darcy’s free boundary problem with surface tension.

[2]   T. Alazard, N. Meunier, and D. Smets. Lyapounov functions, identities and the Cauchy problem for the Hele-Shaw equation. Commun. Math. Phys., 2020.

[3]   O. Benichou, N. Meunier, S. Redner et R. Voituriez. Non-Gaussianity and dynamical trapping in locally activated random walks, Phys. Rev. E, 2012.

[4]   V. Calvez, T. Lepoutre, N. Meunier and N. Muller. Non-linear analysis of a model for yeast cell communication, ESAIM: Mathematical modeling and Numerical Analysis, 2020.

[5]   A. Cucchi, C. Etchegaray, N. Meunier, L. Navoret and L. Sabbagh. Cell migration in complex environments : chemotaxis and topographical obstacles, ESAIM: Proc, 2020.

[6]   A. Cucchi, A. Mellet and N. Meunier. A Cahn-Hilliard model for cell motility, SIAM J. Math. Anal., 2020.

[7]   A. Cucchi, A. Mellet and N. Meunier. Self polarization and traveling wave in a model for cell crawling migration, DCDS, 2022.

[8]   C. Etchegaray, N. Meunier and R. Voituriez. Analysis of a non-local and non-linear Fokker-Planck model for cell migration, SIAM J. Appl. Math., 2017.

[9]   I. Lavi, N. Meunier, R. Voituriez and J. Casademunt. Motility and morphodynamics of confined cells, Phys. Rev. E., 2020.

[10]   T. Lepoutre and N. Meunier. Analysis of a model of cell crawling migration. Com. Math. Sci., 2022.

[11]   P. Maiuri, J.F. Rupprecht, S. Wieser, V. Ruprecht, O. Bénichou, N. Carpi, M. Coppey, S. De Beco, N. Gov, C.P. Heisenberg, C. Lage Crespo, F. Lautenschlaeger, M. Le Berre, A.M. Lennon-Dumenil, M. Raab, H.R. Thiam, M. Piel, M. Sixt, R. Voituriez, Actin flows mediate a universal coupling between cell speed and cell persistence. Cell 161(2), 2015.

[12]   P. Maiuri, E. Terriac, P. Paul-Gilloteaux, T. Vignaud, K. McNally, J. Onuffer, K. Thorn, P.A. Nguyen, N. Georgoulia, D. Soong, D., et al., The First World Cell Race. Current Biology 22(17), 2012.

[13]   V. Milisic, C. Schmeiser. Asymptotic limits for a non-linear integro-differential equation modelling leukocytes’ rolling on arterial walls, Nonlinearity (35):843-869, 2022.

[14]   C. Mouhot, N. Meunier and R. Roux. Long time behavior in locally activated random walks, Com. Math. Sci., 2019.

[15]   P. Vargas, et. al. Innate control of actin nucleation determines two distinct migration behaviours in dendritic cells, Nat Cell Biol., 18(1):43-53, Jan 2016.

Nicolas Meunier
Université d’Évry Val d’Essonne
Laboratoire de Mathématiques et Modélisation d’Évry
23 bvd de France
91 037 Évry Cedex
Office : IBGBI 114
E-mail Address:

Vuk Milisic
Chargé de recherche CNRS
Unversité Bretagne Occidentale 
Laboratoire de Mathématiques de Bretagne Atlantique
6 avenue Le Gorgeu, CS 93837
29238 BREST cedex 3
Office : H108
E-mail Address: