Mean-field interactions between living cells in linear and nonlinear elastic matrices

Living cells respond to mechanical changes in the matrix surrounding them by applying contractile forces that are in turn transmitted to distant cells. We calculate the mechanical work that each cell performs in order to deform the matrix, and study how that energy changes when a contracting cell is surrounded by other cells with similar properties and behavior. We consider two simple effective geometries for the spatial arrangement of cells, with spherical and with cylindrical symmetries, and model the presence of neighboring cells by imposing zero-displacement at some distance from the cell, which represents the surface of symmetry between neighboring cells. We analytically calculate the resulting interaction energy in linear elastic matrices, and study its dependence on the geometry, on the cell’s stiffness, and on the cell’s regulatory behavior. In nonlinear, strain stiffening matrices, we obtain numerical solutions and complement them by asymptotic analytical approximations.