Many living cells exhibit the ability to move during several biological processes: embryogenesis, vasculogenesis and metastatasis are relevant examples.

The inner mechanism of locomotion has been investigated in a number of experimental studies, providing the impulse for theoretical models that have been developed in recent years. In particular, the motion of fish keratocytes on a flat substrate has attracted many efforts, because these cells exhibit a distinct bistable behavior: a cell can be rounded at rest, or it can travel at a characteristic speed with constant shape, the transition from one state to the other being driven to a sufficiently large mechanical or chemotactical perturbation.

Mechanics plays a major role in this behavior. Inverse methods based on a variational approach and finite elements allow a determination of the produced stress field on the basis of a partial knowledge of the observed displacement of the substrate.

On the basis of what is known about the internal machinery of locomotion (the "actomyosin treadmilling"), I will discuss some mathematical models that can explain the observed dynamics: the known pattern of active stress and the spatial concentration fields of actin and myosin.

Special emphasis is devoted to the balance of mass species in the cell as modulated by the active stress.

Davide Ambrosi is full professor of Mathematical Physics at MOX-Department of Mathematics, Politecnico di Milano. Prof. Ambrosi has a PhD in Aerodynamics and he previously covered the position of researcher at CRS4 in Sardinia and of associate professor at Politecnico di Torino.

His main current area of research is mathematical modeling of biological phenomena from a continuum mechanics perspective, such as electrophysiology and mechanics of the heart, tumor growth, morphogenesis, mixture theory and multi-phase flows, homogenization, and mechanics of the cell.