Implementation of cellular bulk stresses in vertex models of biological tissues

Vertex models describe biological tissues as tilings of polygons. In standard vertex models, the tissue dynamics result from a balance between isotropic stresses, which are associated with the bulk of the cells, and tensions associated with cell-cell interfaces. However, in this framework it is less obvious how to describe anisotropic stresses arising from the bulk of cells. In epithelia, such bulk anisotropic stresses could arise for instance through medial myosin fluctuations. Two recent publications-Tlili et al. (Proc Natl Acad Sci USA 116(51):25430-25439, 2019) and Comelles et al. (eLife 10:e57730, 2021)-have proposed different schemes to implement bulk anisotropic stresses in vertex models. Here we show that while both schemes transform in the same way under affine deformations, they lead to significantly different tissue dynamics. Our results are consistent with the interpretation that the Tilli et al. scheme describes bulk stresses that are uniform within each cell, while the Comelles et al. scheme corresponds to non-uniform bulk stresses. Finally, we wondered whether a standard vertex model can be fully expressed in terms of bulk cellular stresses alone. We find that, in general, neither scheme can mimic the vertex forces created by cell-cell interface tensions.