Tissue mechanics: Insight from continuum mechanics for revised vertex models

Two-dimensional vertex models have been used extensively to study and simulate the mechanical behavior of monolayer tissues, a simplified approach that allows for detailed study of structure and disorder in large systems. The most common 2D vertex models use generic elasticity terms whose validity has not been closely scrutinized. Starting from three-dimensional continuum modeling of cell deformation demonstrated to agree with experimental data on MDCK epithelial cells, we systematically develop consistent new approaches to effective 2D elasticity. Translating continuum material properties of cell bulk and cortex into relationships between elastic energy and 2D geometric parameters such as perimeter length, we show that the commonly assumed harmonic perimeter elasticity needs modification in most situations relevant to confluent tissues. Depending on stiffness and compressibility of the bulk and cortex continuum materials, and on the cortical thickness, the derived leading-order description of perimeter elasticity becomes anharmonic and results in quantitatively and qualitatively different behavior as cells deform under active stresses and adhesive forces.