Matrix rigidity optimizes the polarization of stem cells

We present a theoretical model and experiments to explain the non-monotonic dependence of stress-fiber polarization in stem cells on matrix rigidity. The theory generalizes the treatment of elastic inclusions to ``living'' inclusions (cells) whose active polarizability, unlike non-living matter, depends on the feedback of cellular forces that develop in response to matrix stresses. We demonstrate experimentally that the stress fibers in adult mesenchymal stem cells, generally orient parallel to the long axis of the cells, with an anisotropy that depends non-monotonically on substrate stiffness. Consistent with these experiments, our theory predicts that the magnitude of the cellular force increases monotonically with the matrix rigidity while the polarization anisotropy shows a maximum that depends on the cell shape and the elastic modulus of the medium. These findings offer a mechanical correlate for the observation that stem cell differentiation optimizes in a range of matrix rigidities that depends on the tissue type.