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.
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