Epithelial mechanics in a three-dimensional cell vertex model with local junctional constraints

Epithelial monolayers must accommodate deformation while maintaining continuous tissue coverage, but how their material properties emerge from three-dimensional cell mechanics remains incompletely understood. Among several three-dimensional cell vertex formulations, we build on a minimal model that captures experimentally observed features of epithelial monolayers, including shape bistability. The geometric constraint introduced in the model, however, is limited to regular cells undergoing uniform deformation. Here, we extend this model by replacing the original geometric constraint with a local energy based on the geometry of cell-cell junctions, allowing general vertex motions while retaining analytical tractability for regular cell packings and enabling simulations of disordered tissues. Using this framework, we derive analytical expressions for the in-plane elastic moduli and Poisson's ratio and characterize the shear modulus across the tissue thickness. We show that the model reproduces a near-zero Poisson's ratio observed in cultured epithelia. Numerical simulations show that this response persists beyond uniform deformation and remains robust to structural disorder over the timescales examined. Structural disorder also broadens the spectrum of internal relaxation modes and thus introduces slower modes to the frequency-dependent mechanical response. Together, these results establish a tractable three-dimensional framework for understanding how epithelial tissue mechanics emerges from cell-level geometry and interactions under realistic deformations and structural disorder.