Universal features of rigidity transitions in vertex models for biological tissues

Abstract

AbstractFor confluent or nearly confluent tissues, simulations and analyses of simple vertex models – where the cell shape is defined as a network of edges and vertices – have proven useful for making predictions about mechanics and collective behavior, including rigidity transitions. Scientists have developed many versions of vertex models with different assumptions. Here we investigate a set of vertex models which differ in the functional form that describes how energy depends on the vertex positions. We analyze whether such models have a well-defined shear modulus in the limit of zero applied strain rate, and if so how that modulus depends on model parameters. We identify a class of models with a well-defined shear modulus, and demonstrate that these models do exhibit a cross-over from a soft or floppy regime to a stiff regime as a function of a single control parameter, similar to the standard vertex model. Moreover, and perhaps more surprisingly, the rigidity crossover is associated with a crossover in the observed cell shape index (the ratio of the cell perimeter to the square root of the cell area), just as in the standard vertex model, even though the control parameters are different. This suggests that there may be a broad class of vertex models with these universal features, and helps to explain why vertex models are able to robustly predict these features in experiments.