Viscoelastic properties of tissues in the vertex model

Epithelial cell sheets have been studied extensively in various contexts ranging from embryonic development to cancer metastasis, where collective groups of cells can drastically reorganize and move over substantial distances like a fluid. The collective behavior of epithelial cells is commonly simulated with the vertex model. It was previously demonstrated that the vertex model can describe both the solid- and fluid-like behavior by tuning the target cell-shape parameter p₀=P₀/sqrt[A₀], where P₀ and A₀ are the preferred perimeter and area of cells, respectively. The shear modulus is finite for low values of p₀ and it vanishes beyond the critical value of the cell-shape parameter (p₀>p_c). However, the viscoelastic properties of tissues in the vertex model have not been yet studied systematically. Here, we performed rheological tests by applying an oscillatory shear strain and measuring the shear stress. We found that tissues can be described with the standard linear solid model in the solid phase (p₀<p_c) and with Burgers material model in the fluid phase (p₀>p_c). In both regimes, the values of elastic spring constants decrease towards 0 and the relaxation timescales diverge as p₀ approaches the critical value p_c.