Section: Application Domains

Cell motion

Several processes are employed by cells to communicate, regulate and control their movements, and generate collective motion. Among them, chemotaxis is the phenomenon by which cells direct their active motion in response to an external chemical (or physical) agent. In chemotaxis, cells not only respond but can also produce the chemical agent, leading to a feedback loop. Understanding this phenomenon is a major challenge for describing the collective behaviour of cells. Many mathematical models have been proposed at different scales, yielding a good description of cell aggregation. In collaboration with biophysicists at Institut Curie in Paris, we develop and study (N. Bournaveas, V. Calvez, S. Gutiérrez and B. Perthame, Global existence for a kinetic model of chemotaxis via dispersion and Strichartz estimates, Comm. PDE, 2008) mathematical models based on kinetic equations for bacterial travelling waves in a microchannel. These models have shown a remarkable quantitative agreement with experimental observations.

Cell motion arises also in the growth of solid tumours, which can be described through cell population models or multiphase flows (J. Ranft et al, Fluidization of tissues by cell division and apoptosis, PNAS, 2010 and L. Preziosi and A. Tosin, Multiphase modelling of tumour growth and extracellular matrix interaction: mathematical tools and applications, J. Math. Biol., 2009.). This is a very active subject because several bio-chemico-physical mechanisms are at work; for instance motion can arise from pressure forces resulting from cell divisions and from active cell motility. At the smaller scale stochastic agent-based models of tumour cells invading the tumour environment or blood vessels are considered (I. Ramis-Conde et al., J. Phys. Biol., 2009), and allow to represent detailed behaviours and interactions. At a larger scale, free boundary problems are widely used, e.g., for image-based prediction because of the reduced number of parameters (Works by O. Saut, T. Colin, A. Iollo, N. Ayache, J. Lowengrub). Asymptotic analysis makes a link between these different mechanistic models [91] .

One other setting where we will study cell motion is epithelial gap closure, a form of collective cell migration that is a very widespread phenomenon both during development and adult life - it is essential for both the formation and for the maintenance of epithelial layers. Due to their importance, in vivo wound healing and morphogenetic movements involving closure of holes in epithelia have been the object of many studies (including some involving members of this project like [57] ). Several theoretical models have also been proposed recently for the advancement of tissue covering unoccupied areas (see, for instance, [58] ). It is particularly interesting to study epithelial gap closure in vivo. However, the complexity of the process and the difficulty to measure relevant quantities directly and to control the parameters in vivo, lead biologists to seek alternative systems where epithelial gap closure can be studied under better-defined and better-controlled conditions. We extended our work from in vivo studies to in vitro situations taking advantage of a collaboration with the group of Benoît Ladoux who performed experiments on cell monolayers of human keratinocytes and of MDCK cells. We could single out some similar geometry dependence of the wound closure strategies between these two settings, indicating the existence of conserved mechanisms that should be widespread across living beings. In our model we consider viscous behaviour in the tissue and some simple friction with the substrate, plus boundary terms associated to cable and lamellipodial forces. The numerical simulations obtained using this model are in good agreement with the experimental results [30] , [27] .