Elasticity of a three-dimensional cell vertex model of epithelia

Material properties of epithelial tissues play essential roles in morphogenesis and physiological function, yet how tissue-scale viscoelasticity emerges from cellular-scale mechanics remains unclear. Here, we investigate the elastic mechanics of a three-dimensional cell vertex model for epithelial monolayers. By analyzing infinitesimal affine deformations around a regular hexagonal-prism equilibrium state, we derive analytical expressions for the in-plane elastic moduli. We show that the model produces a near-zero in-plane Poisson's ratio over a broad region of parameter space, thereby accounting for a characteristic mechanical property reported in cultured epithelial monolayers. Numerical simulations further confirm that the theoretical Poisson's ratio remains accurate under finite, biologically relevant strains. In addition, we show that the morphological bistability between squamous-like and columnar-like states is associated with distinct elastic responses. Our results indicate that three-dimensional vertex models can account not only for epithelial morphogenesis but also for their material properties.