  Cross-sections of cell shapes in a tissue monolayer typically resemble a
tiling of convex polygons. Yet, examples exist where the polygons are not
convex with curved cell-cell interfaces, as seen in the adaxial epidermis. To
date, two-dimensional vertex models predicting the structure and mechanics of
cell monolayers have been mostly limited to convex polygons. To overcome this
limitation, we introduce a framework to study curvy cell-cell interfaces at the
subcellular scale within vertex models by using a parameterized curve between
vertices that is expanded in a Fourier series and whose coefficients represent
additional degrees of freedom. This extension to non-convex polygons allows for
cells with same shape index, or dimensionless perimeter, to be, for example,
either elongated or globular with lobes. In the presence of applied,
anisotropic stresses, we find that local, subcellular curvature, or buckling,
can be energetically more favorable than larger scale deformations involving
groups of cells. Inspired by recent experiments, we also find that local,
subcellular curvature at cell-cell interfaces emerges in a group of cells in
response to the swelling of additional cells surrounding the group. Our
framework, therefore, can account for a wider array of multi-cellular responses
to constraints in the tissue environment.
