Spontaneous patterning of cell size on curved surfaces

Tissue surfaces exhibit complex curvature during embryogenesis and oncogenesis.

Evidence shows that cells can actively sense curvature to regulate behavior and fate, yet the underlying mechanism remains unclear.

Here, we develop a vertex model for arbitrary curved surfaces and uncover spontaneous cell size patterning on ellipsoidal surfaces: cells in high-curvature regions are consistently larger than those in low-curvature regions.

This non-uniformity arises from a mechanical competition encoded in Riemannian geometry: positive Gaussian curvature reduces the perimeter-to-area ratio of polygonal cells, relaxing cell-edge tension in high-curvature regions, which is compensated by area expansion to maintain global force balance.

This area pattern is robust against variations in model parameters and matches observations in biological systems.

The perimeter pattern, in contrast, is governed by competition between the intrinsic geometric tendency and the deformation required by force balance, and undergoes reversal beyond a critical shape index.

Together, these findings establish self-organized spatial variations in cell size as a potential physical mechanism for curvature sensing.