Non-Hookean Elastic Moduli of 2D Tissue Model

We study the peculiar mechanical response of a discrete vertex model (VM) of 2D epithelia. In the absence of T1 rearrangements, the VM exhibits a transition between a soft and a stiff solid tuned by the target shape index of the cells and associated with the onset of geometric incompatibility. By examining the response to a variety of deformations, we show that the stiff solid phase exhibits nonlinear elastic response that cannot be recast in the framework of conventional Hookean elasticity even in the limit of infinitesimal strain. The soft solid is characterized by floppy modes that can absorb strain. As a result, the linear mechanical response is not fully characterized by two independent Lamé coefficients, as in linear elasticity of a 2D isotropic solid, but depends on the specific imposed deformation. We show that this unconventional mechanical behavior is in agreement with a continuum version of the model developed recently by some of us.