Epithelia are ubiquitous tissues that display a large diversity of functions and forms, from totally flat to highly curved. Various morphogenetic events, such as gastrulation or branching morphogenesis, correlate to changes in the curvature of epithelia. Building a physical framework to account for the shape of cells in epithelia is thus an important challenge to understand various normal and pathological biological processes, such as epithelial morphogenesis or cancer metastasis. It is widely recognized that the shape of epithelial cells is determined by the tension generated by the actomyosin cortex and the adhesion of cells to the substrate and to each other. These tensions and adhesions are not homogeneously distributed on the cell surface, which makes a 3D view of the problem valuable. To account for these biological and structural contributions to cell shape, different physical models have been proposed, which include surface energies, adhesions, line tensions, volume compressibility, or elasticity terms. However, an experimental procedure that would allow a validation of a minimal physical model for the shape of epithelial cells in 3D has not yet been proposed. In this study, we first made a quantitative analysis of the correlation between cell thickness and curvature during the formation of the ventral furrow in the early Drosophila embryo. We then cultured Madin-Darby Canine Kidney (MDCK) epithelial cells on substrates with a sinusoidal profile, allowing us to measure the shape of the cells on various positive and negative curvatures. We found that both in the early Drosophila ventral furrow and in MDCK epithelia cells are thicker when positively curved (on valleys of sinusoidal substrates) than when negatively curved (on the crests). The influence of curvature on the shape of epithelial cells could not be understood with a model using only differential apical, basal, and lateral surface energies. However, the addition of an apical line tension was sufficient to quantitatively account for the experimental measurements. The model also accounts for the shape of MDCK cells that overexpress E-cadherin. On the other hand, when reducing myosin II activity with blebbistatin, we measured a saturation of the difference in cell thickness between valleys and crests, suggesting the need for a term limiting large cell deformations. Our results show that a minimal model that accounts for epithelial cell shape needs to include an apical line tension in addition to differential surface energies, highlighting the importance of structures that produce anisotropic tension in epithelial cells, such as the actin belt linking adherens junctions. In the future, the model could be used to account for the shape of epithelial cells in different contexts, such as branching morphogenesis. Furthermore, our experimental procedure could be used to test a wider range of physical models for the shape of epithelia in curved environments, including, for example, continuous models.
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