The Interplay of Curvature, Geometry, and Topology Shapes Tissue Organisation in Epithelial Shells.

During morphogenesis, multicellular patterns emerge from coordinated processes at the single-cell level. This is characterised in flat epithelial layers, where average neighbour distributions follow conserved arrangements. However, less is known about how curvature defines tissue architecture. Epithelial shells constitute a critical structural stage in early mammalian embryonic development. Our analysis reveals how the closed, curved geometry of epithelial shells limits their 3D organisation. Here, we designed computational models that predict shell surface topology across different cell numbers. To validate the model, MDCK cysts with precise cell-level segmentation were analysed. Our results show that more pentagons are required to accommodate higher curvature per cell patch, a restriction also observed in early mouse embryos. Small cysts and embryos exhibited a restricted subset of polygon configurations, predominantly those maximising the area-to-perimeter ratio. We conclude that epithelial shells are organised under general rules mirroring other natural cages, revealing that geometric and curvature constraints guide early epithelial organisation.