Modeling the mechanics of growing epithelia with a bilayer plate theory

Epithelia, which consists of cell sheets lying on a substrate, are prevalent structures of multicellular organisms. The physical basis of epithelial morphogenesis has been intensely investigated in recent years. However, as 2D mechanics focused most attention, we still lack a rigorous description of how the mechanical interactions between the cell layer and its substrate can lead to 3D distortions. This work provides a complete description of epithelial mechanics using the most straightforward model of an epithelium: a thin elastic bilayer. We first provide experimental evidence in Drosophila tissues that localized alterations of the cell substrate (the extracellular matrix) can lead to profound 3D shape changes in epithelia. We then develop an analytical model modifying the Foppl-von Karman equation with growth for bilayers. We provide a complete description of all contributions from biophysical characteristics of epithelia. We show how any localized inhomogeneity of stiffness or thickness drastically changes the bending process when the two layers grow differently. Comparison with finite element simulations and experiments performed on Drosophila wing imaginal discs validates this approach for thin epithelia