Anomalous elasticity of cellular tissue vertex model

Vertex Models, as used to describe cellular tissue, have an energy controlled by deviations of each cell area and perimeter from target values. The constrained nonlinear relation between area and perimeter leads to new mechanical response. Here we provide a mean-field treatment of a highly simplified model: a uniform network of regular polygons with no topological rearrangements. Since all polygons deform in the same way, we only need to analyze the ground states and the response to deformations of a single polygon (cell). The model exhibits the known transition between a fluid/compatible state, where the cell can accommodate both target area and perimeter, and a rigid/incompatible state. %The rigid solid-like state has a single gapped ground state. We calculate and measure the mechanical resistance to various deformation protocols and discover that at the onset of rigidity, where a single zero-energy ground-state exists, %We show that in the incompatible state, where a single frustrated ground-state exists, linear elasticity fails to describe the mechanical response to even infinitesimal deformations. In particular we identify a breakdown of reciprocity expressed via different moduli for compressive and tensile loads, implying non-analyticity of the energy functional. We give a pictorial representation in configuration space that reveals that the complex elastic response of the Vertex Model arises from the presence of two distinct sets of reference states (associated with target area and target perimeter).