  Experiments show that when a monolayer of cells cultured on an elastic
substrate is subject to a cyclic stretch, cells tend to re-orient either
perpendicularly or at an oblique angle with respect to the main direction of
the stretch. Due to stochastic effects, however, the distribution of angles
achieved by the cells is broader and, experimentally, histrograms over the
interval [0, 90] are reported. Here we will determine the evolution and the
stationary state of probability density functions describing the statistical
distribution of the orientations of the cells using Fokker-Planck equations
derived from microscopic rules for the evolution of the orientation of the
cell. As a first attempt, we shall use a stochastic differential equation
related to a very general elastic energy and we will show that the results of
the time integration and of the stationary state of the related forward
Fokker-Planck equation compare very well with experimental results obtained by
different researchers. Then, in order to model more accurately the microscopic
process of cell re-orientation, we consider discrete in time random processes
that allow to recover Fokker- Planck equations through the well known technique
of quasi-invariant limit. In particular, we shall introduce a non-local rule
related to the evaluation of the state of stress experienced by the cell
extending its protrusions, and a model of re-orientation as a result of an
optimal control internally activated by the cell. Also in the latter case the
results match very well with experiments.
