Shape regulation generates elastic interaction between living cells

The organization of live cells to tissues is associated with the mechanical interaction between cells, which is mediated through their elastic environment. We model cells as spherical active force dipoles surrounded by an infinite elastic matrix, and analytically evaluate the interaction energy for different scenarios of their regulatory behavior. We obtain attraction for homeostatic (set point) forces and repulsion for homeostatic displacements. When the translational motion of the cells is regulated, the interaction energy decays with distance as $1/d^4$, while when it is not regulated the energy decays as $1/d^6$. This arises from the same reasons as the van der Waals interaction between induced electric dipoles.