To perform these functions, most epithelia fold into three-dimensional structures that enclose a pressurized fluid-filled cavity called lumen. Epithelia are active materials that sculpt the early embryo, separate body compartments, protect against pathogenic and physicochemical attacks, and control fluid and biomolecular transport 1. The internal and external surfaces of the animal body are lined by thin cellular layers called epithelia. Our approach enables a systematic study of how geometry and stress influence epithelial fate and function in three-dimensions.
In epithelia with rectangular and ellipsoidal cross-section we find pronounced stress anisotropies that impact cell alignment. In epithelia with spherical geometry we show that stress weakly increases with areal strain in a size-independent manner. This method establishes a correspondence between epithelial shape and mechanical stress without assumptions of material properties. We develop a computational method, called curved monolayer stress microscopy, to map the stress tensor in these epithelia. We design pressurized epithelia with circular, rectangular and ellipsoidal footprints. Here we engineer curved epithelial monolayers of controlled size and shape and map their state of stress. To adopt shapes such as spheres, tubes and ellipsoids, epithelia generate mechanical stresses that are generally unknown. The function of organs such as lungs, kidneys and mammary glands relies on the three-dimensional geometry of their epithelium.
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