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

Abstract

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, $p_0=P_0/\sqrt{A_0}$. For $p_0>p_{\ast }(6)=\sqrt{8\sqrt{3}}\approx 3.72$, with $p_{\ast }(6)$ 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 $p_0<p_{\ast }(6)$, 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.

References in corpus (6)

• Non-affine response: jammed packings versus spring networks
• Experimental Measures of Affine and Non-affine Deformation in Granular Shear
• Linear Viscoelastic Properties of the Vertex Model for Epithelial Tissues
• Shear-driven solidification and nonlinear elasticity in epithelial tissues
• Active T1 transitions in cellular networks
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Cited by in corpus (5)

• Morphological instability at topological defects in a three-dimensional vertex model for spherical epithelia
• Designing the pressure-dependent shear modulus using tessellated granular metamaterials
• Spectral approaches to stress relaxation in epithelial monolayers
• Geometry of T1 transitions in epithelia
• Cell divisions imprint long lasting elastic strain fields in epithelial tissues