Multiscale PDE problems for active biosystems

This is an easy introductory talk to the mathematics of active biosystems. Before jumping into mathematics we present experimental videos of striking biological phenomena that motivate our study. The talk consists of three parts. First we give a brief introduction to homogenization theory which is a powerful mathematical tool for studying multiscale problems. Next we describe a PDE model of collective behavior of swimming bacteria and present mathematical results on effective viscosity of bacterial suspensions. Finally we present a phase field PDE model of a crawling cell on a substrate and describe its surprising features such as self-sustained motion (traveling wave solutions) and spontaneous breaking of symmetry. In conclusion, we briefly describe the concept of interdisciplinary mathematics.