Section: Application Domains

Modeling cell dynamics

Migration and proliferation of epithelial cell sheets are the two keystone aspects of the collective cell dynamics in most biological processes such as morphogenesis, embryogenesis, cancer and wound healing. It is then of utmost importance to understand their underlying mechanisms.

The cells in epithelial sheets (monolayers) maintain strong cell-cell contact during their collective migration. Although it is well known that under some experimental conditions apical and basal sites play distinctive important roles during the migration, as well as the substrate itself [130] , we consider here biological experiments where the apico-basal polarization does not take place. Thus, the cell monolayer can be considered as a 2 dimensional continuous structure. These epithelial monolayers, as the Madin-Darby Canine Kidney (MDCK) cells [53] , [90] , are universally used as multicellular models to study the migratory mechanisms.

Semilinear reaction-diffusion equations are widely used to give a phenomenological description of the temporal and spatial changes occurring within cell populations that undergo scattering (moving), spreading (expanding cell surface) and proliferation. We have followed the same methodology and contributed to assess the validity of such approaches in different settings (cell sheets  [101] , dorsal closure  [49] , actin organization  [48] ). However, epithelial cell-sheet movement is complex enough to undermine most of the mathematical approaches based on locality, that is mainly traveling wavefront-like partial differential equations. In [89] it is shown that MDCK cells extend cryptic lamellipodia to drive the migration, several rows behind the wound edge. In [125] MDCK monolayers are shown to exhibit similar non-local behavior (long range velocity fields, very active border-localized leader cells).

Our aim is to start from a mesoscopic description of interaction of the cells (at the cell-cell level, including the F-actin, but not e.g. the migration-related protein scale). Considering cells as independent anonymous agents, we plan to investigate the use of mathematical techniques adapted from the mean-field game theory. Otherwise, looking at them as interacting particles, we will use a multi-agent approach (at least for the actin dynamics). We intend also to consider approaches stemming from compartment-based simulation in the spirit of those developed in   [87] , [92] , [94] .