Rheology of the vertex model of tissues: Simple shear and oscillatory geometries

Most tissue properties and biological processes in tissues are dependent on or are regulated by the local mechanical stresses: Examples are cell division and differentiation, or, at larger scales, tissue morphogensis. In this paper, we theoretically study the rheology of two-dimensional epithelial tissues described by a discrete vertexlike model of actively dividing cells. We use a general analytical coarse-grained continuum formulation, which allows for explicit calculation of the mechanical stresses and the cell size and shape in various geometries. We discuss two different scenarios, a tissue under an applied simple shear and a supported tissue on an oscillating substrate. We show that this model predicts most often a shear-thinning behavior under a constant shear rate and, in certain circumstances, a crossover from shear thickening at low shear rates to shear thinning at high shear rates. We give an analytical expression for the tissue response in an oscillating strain experiment in the linear regime and calculate it numerically in the nonlinear regime. When the tissue is supported by an oscillating substrate, it reorients depending on frequency and the Poisson ratio of the substrate. Reorientation can be gradual or sharp, depending on the tissue's reference perimeter and the substrate Poisson ratio and on frequency. We also show the existence of a tricritical point in the orientation phase space.