Active self-polarization of contractile cells in asymmetrically shaped domains

Mechanical forces generated by contractile cells allow the cells to sense their environment and to interact with other cells. By locally pulling on their environment, cells can sense and respond to mechanical features such as the local stress (or strain), the shape of a cellular domain, and the surrounding rigidity; at the same time, they also modify the mechanical state of the system. This creates a mechanical feedback loop that can result in self-polarization of cells. In this paper, we present a quantitative mechanical model that predicts the self-polarization of cells in spheroidally shaped domains, comprising contractile cells and an elastic matrix, that are embedded in a three-dimensional, cell-free gel. The theory is based on a generalization of the known results for passive inclusions in solids to include the effects of cell activity. We use the active cellular susceptibility tensor presented by Zemel [Phys. Rev. Lett. 97, 128103 (2006)] to calculate the polarization response and hence the elastic stress field developed by the cells in the cellular domain. The cell polarization is analyzed as a function of the shape and the elastic moduli of the cellular domain compared with the cell-free surrounding material. Consistent with experiment, our theory predicts the development of a stronger contractile force for cells in a gel that is surrounded by a large, cell-free material whose elastic modulus is stiffer than that of the gel that contains the cells. This provides a quantitative explanation of the differences in the development of cellular forces as observed in free and fixed gels. In the case of an asymmetrically shaped (spheroidal) domain of cells, we show that the anisotropic elastic field within the domain leads to a spontaneous self-polarization of the cells along the long axis of the domain.