A nonlinear elastic description of cell preferential orientations over a stretched substrate

The active response of cells to mechanical cues due to their interaction with the environment has been of increasing interest, since it is involved in many physiological phenomena, pathologies, and in tissue engineering. In particular, several experiments have shown that, if a substrate with overlying cells is cyclically stretched, they will reorient to reach a well-defined angle between their major axis and the main stretching direction. Recent experimental findings, also supported by a linear elastic model, indicated that the minimization of an elastic energy might drive this reorientation process. Motivated by the fact that a similar behaviour is observed even for high strains, in this paper we address the problem in the framework of finite elasticity, in order to study the presence of nonlinear effects. We find that, for a very large class of constitutive orthotropic models and with very general assumptions, there is a single linear relationship between a parameter describing the biaxial deformation and cos^(2) theta(eq), where theta(eq) is the orientation angle of the cell, with the slope of the line depending on a specific combination of four parameters that characterize the nonlinear constitutive equation. We also study the effect of introducing a further dependence of the energy on the anisotropic invariants related to the square of the Cauchy-Green strain tensor. This leads to departures from the linear relationship mentioned above, that are again critically compared with experimental data.