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Journal Article

The role of non-affine deformations in the elastic behavior of the cellular vertex model


Staddon,  Michael F.
Max Planck Institute for the Physics of Complex Systems, Max Planck Society;

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Staddon, M. F., Hernandez, A., Bowick, M. J., Moshe, M., & Marchetti, M. C. (2023). The role of non-affine deformations in the elastic behavior of the cellular vertex model. Soft Matter, 19(17), 3080-3091. doi:10.1039/d2sm01580c.

Cite as: https://hdl.handle.net/21.11116/0000-000D-2FE7-B
The vertex model of 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, or target, area, A(0), and perimeter, P-0. The model exhibits a rigidity transition driven by geometric incompatibility as tuned by the target shape index, p(0) = P-0 /root A(0). For p0 > p* with p*(6) the perimeter of a regular hexagon of unit area, a cell can simultaneously attain both the preferred area and preferred perimeter. As a result, the tissue is in a mechanically soft compatible state, with 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 the linear elastic moduli of an ordered vertex model. We analyze a relaxation step that includes non-affine deformations, leading to a softer response than previously reported. The origin of the vanishing shear and Young's moduli in the compatible state is the presence of zero-energy deformations of cell shape. The bulk modulus exhibits a jump discontinuity at the transition and can be lower in the rigid state than in the fluid-like state. The Poisson's ratio can become negative 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.