MARK KAC SEMINAR

Morphogenesis, the formation of biological shape and pattern during embryonic development, is a topic of intensive experimental investigation, so the participating cell types and molecular signals continue to be characterized in great detail. Yet this data only partly tells biologists how molecules and cells interact dynamically to construct a biological tissue. Cell-based models simulate the behavior of individual cells and associated extracellular materials to predict the resulting collective cell behaviors that drive morphogenesis. I will present some recent developments on a widely-used lattice-based, stochastic cell-based model, called the Cellular Potts model. In particular, I will show results on the role of chemical and mechanical cell-cell communication during the formation of network-like structures, occurring, e.g., during blood vessel formation, and the formation of branched organs of epithelial origin, e.g., during the growth of mammary glands and kidneys. I will discuss this model in detail and conclude by suggesting some interesting continuum and stochastic mathematical problems that our simulations suggest.

I will consider a new dynamics inspired by the Kac ring model and designed to model diffusion in a generic random medium. When the disorder is quenched, the dynamics shares some common features with Hamiltonian dynamics. In particular, it is deterministic, periodic and reversible. We show how to derive macroscopic diffusion laws with large probability with respect to the disorder in two cases. In the first case one of the sides of the system is taken to infinity. In the second case, we consider the dynamics in dimension d larger then or equal to 5. In that context, Fick's law is obtained by using (self-) avoiding properties of random walks in high dimension.