Universality in the Mechanical Behavior of Vertex Models for Biological Tissues

Abstract

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 of them 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 cross-over from a soft or floppy regime to a stiff regime. While the parameter that controls the crossover is different in each model, 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. 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. This suggests there is a broad class of vertex models with universal mechanical features, and helps to explain why many different vertex models are able to robustly predict these features in experiments.