Elastic Moduli of the Vertex Model

The vertex model of confluent epithelia describes the apical surface of a tissue as a tiling of polygonal cells, with a mechanical energy governed by deviations in cell shape from preferred $A_0$, and perimeter, $P_0$. Even in the absence of topological rearrangements, the model exhibits a solid-solid rigidity transition driven by geometric incompatibility and tuned by the target shape index of the cell, $p_0 = P_0 / \sqrt{A_0}$. For $p_0 > p_*(6) = \sqrt{8 \sqrt{3}} \approx 3.72$, with $p_*(6)$ the perimeter of a regular hexagon of unit area, a cell can simultaneously attain both the preferred area and perimeter. As a result, the tissue is in a mechanically soft compatible state, with a manifold of degenerate zero-energy states and zero shear and Young's moduli. For $p_0 < p_*(6)$, it is geometrically impossible for any cell to realize the preferred area and perimeter simultaneously, and the tissue is in an incompatible rigid solid state. Using a mean-field approach, we present a complete analytical calculation of both the ground states and the linear elastic moduli of an ordered vertex model. In addition to the standard affine transformations, we analyze a relaxation step that allows for non-affine deformations, leading to a softening of the system. The origin of the vanishing shear and Young's moduli in the compatible state is the presence of zero-energy deformations of cell shape within the manifold of degenerate ground states. The bulk modulus exhibits a jump discontinuity at the transition and, for some parameters, can be lower in the rigid state than in the fluid-like state. The Poisson ratio becomes negative at small $p_0$ and intermediate contractility, which lowers the bulk and Young's moduli. Our work provides a unified treatment of linear elasticity for the vertex model and demonstrates that this linear response is protocol-dependent.