Section: Scientific Foundations

Multiscale modeling and computations

Spatial complexity: collective motion of cells

The collective motion of cells (bacteries on a gel or endothelial cells during angiogenesis) is a fascinating subject, that involves a combinaison of random walk and chemotaxis. The modeling of these problems is still active, since the pioneering works of Keller and Segel, and the mathematical study of the arising equations is a very active area of research.

Vincent Calvez focuses its effort on the following questions:

• Mathematical analysis of the Keller-Segel model

  [In collaboration with J.A. Carrillo and J. Rosado (UAB, Barcelona)]

  Following McCann 1997 and Otto 2001, we interpret the classical Keller-Segel system for chemotaxis as a gradient flow in the Wasserstein space. The free-energy functional turns out to be homogeneous. This viewpoint helps to understand better blow-up mechanisms, and to derive rates of convergence towards self-similar profiles. We investigate more precisely linear diffusion, porous medium diffusion and fast diffusion in competition with various interaction kernels.

  [In collaboration with N. Meunier (Paris 5) and R. Voituriez (Paris 6)]

  Another project consists in analyzing some variant of the Keller-Segel system when the chemoattractant is secreted at the boundary of the domain. This is motivated by modeling issues in cell polarization.

• Kinetic models for bacterial collective motion

  [In collaboration with N. Bournaveas (Univ. Edinburgh)]

  We shed a new light on the theoretical analysis of kinetic equations for chemotaxis: we exhibited a peculiar example subject to a critical mass phenomenon as for the two-dimensional Keller-Segel system. This shows a posteriori that previous attempts to show global existence were in fact borderline with respect to this critical example.

  [In collaboration with N. Bournaveas (Univ. Edinburgh), B. Perthame (Paris 6), A. Buguin, Jonathan Saragosti and P. Silberzan (Institut Curie, Paris)]

  We have developed a macroscopic model for bacterial traveling pulses based on a mesoscopic description of the run-and-tumble process. We are able to capture some key features observed in the experiments (asymmetric profile, speed of the pulse).

Figure 1. Collective motion of cells

Complex rheology

To investigate the growth of a tumor it is crucial to have a correct description of its mechanical aspects. Tumoral and normal cells may be seen as a complex fluid, with complex rheology.

Numerical investigations of complex flows is studied by P. Vigneaux who develops new numerical schemes fo Bingham type flows.