Universality in the Mechanical Behavior of Vertex Models for Biological Tissues

Simple vertex models, where the cell shape is defined as a network of edges and vertices, have made useful predictions about the collective behavior of confluent biological tissues, including rigidity transitions. Quite a few different versions of vertex models have appeared in the literature, and they propose substantial differences in how the mechanical energy depends on vertex positions, yet all seem to make correct predictions. To understand how this is possible, we search for universality in the emergent mechanical behavior—including the shear modulus defined in the limit of zero-strain rate and the viscoelastic response at finite strain rates—of six different vertex models. We identify a class of models with a well-defined shear modulus, and demonstrate that these models all exhibit a crossover from a soft or floppy regime to a stiff regime. We find that the observed cell shape index (the ratio of the cell perimeter to the square root of the cell area) is a good observable order parameter for the crossover, even though the input model parameter that controls the crossover is different in each model. We also find that the finite-strain-rate viscoelastic response of all models exhibits a universal scaling with frequency, following the Zener model in the rigid phase and Burgers model in the fluid phase. The observation that a geometric observable like cell shape governs the transition, rather than a topological observable such as coordination number, suggests that second-order rigidity may be a good framework for explaining this universality. It generates an empirical prediction that a broad class of vertex models with second-order rigidity transitions should also exhibit these universal mechanical features, and helps to explain why many different vertex models are able to robustly predict these features in experiments.