Epithelial cell sheets have been studied extensively in various contexts, such as embryonic development, wound healing and cancer metastasis, where collective groups of cells dramatically reorganize and move over substantial distances like a fluid. The collective behavior of epithelial sheets is commonly simulated with the vertex model, which exhibits both solid- and fluid-like behaviors. In the vertex model tissues are described as a confluent layer of 2D polygons, where the deformation energy costs are associated with deviation of the area Ai and the perimeter Pi of cell i from their target values A0 and P0, respectively. It was previously demonstrated, that the solid- and fluid-like behavior in the vertex model can be obtained by  tuning the target cell-shape parameter p0=P0/sqrt[A0], where for low values of p0 the effective shear modulus is finite, which vanishes beyond a critical value of the target cell-shape parameter pc (=3.72 for tissues with hexagonal cells).  However, the viscoelastic properties of cell tissues within the vertex model have not been yet studied systematically. Here, we performed rheological tests by deforming the simulation box with an oscillatory shear and measuring the resultant shear stress. For low values of the target cell-shape parameter (p0<pc), we found that tissues can be described with the standard linear solid model. On the other hand, we found that for high values of the target cell-shape parameter (p0>pc) tissues can be described with the Maxwell material model. In both regimes the values of elastic spring constants decrease towards 0 and the relaxation time-scales diverge as the target cell-shape parameter p0 approaches the critical value pc.

Viscoelastic properties of tissues in the vertex model

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