10:00 AM – 10:45 AM, Tuesday, October 16, 2018; Room: 114 McAllister bldg.

**Models of Cell Migration**

Nicolas Meunier, Université d’Evry Val d’Essonne, Evry Cedex, France

We study two types of models describing the motility of eukaryotic cells. The first model type describes protrusion activity by using stochastic processes. The cell velocity is a function of the (discrete) protrusive forces exerted by filopodia on the substrate. The key property of this system is 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 Méléard, 2004], we introduce the mathematical model and we derive some of its fundamental properties. We find back the trajectories usually described in the literature for cell migration.

The second model type is a non-linear free boundary problem for a Hele-Shaw type system of PDEs. We perform linear stability analysis and some numerical simulations. In this approach, the orientation of the actin filament network appears through boundary terms in a fluid framework. The marker concentration obeys to a non-linear and non-local convection-diffusion equation, where the convection field corresponds to the fluid advection field. We find a bifurcation explaining, how varying a single (physical) parameter allows the cell to switch from rest to motion. From the mathematical viewpoint, we consider the situation where the cell is 1D and rigid and we obtain global existence or apparition of a singularity in finite time, non-trivial steady states, long-time convergence.

The work was done jointly with C. Etchegaray, I. Lavi and R. Voituriez.

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