Continuum theory for the mechanics of curved epithelial shells by coarse-graining an ensemble of active gel cellular surfaces

Epithelial tissues undergo complex morphogenetic transformations driven by cellular and cytoskeletal dynamics. To understand the emergent tissue mechanics resulting from sub-cellular mechanisms, we formulate a fully nonlinear continuum theory for epithelial shells that coarse-grains an underlying 3D vertex model, whose surfaces are in turn patches of active viscoelastic gel undergoing turnover. Our theory relies on two ingredients. First, we relate the deformation of apical, basal and lateral surfaces of cells to the continuum deformation of the tissue mid-surface and a thickness director field. We explore two variants of the theory, a Cosserat theory accommodating through-thickness tilt of cells, and a Kirchhoff theory assuming that lateral cell surfaces remain perpendicular to the mid-surface. Second, by adopting a variational formalism of irreversible thermodynamics, we construct an effective Rayleighian functional of the tissue constrained by the cellular-continuum kinematic relations, which therefore depends on continuum fields only. This functional allows us to obtain the governing equations of the continuum theory and is the basis for efficient finite element simulations. Verification against explicit 3D cellular model simulations demonstrates the accuracy of the proposed theory in capturing epithelial buckling dynamics. Furthermore, we show that the Cosserat theory is required to model tissues exhibiting apicobasal asymmetry of active tension. Our work provides a general frame-work for further studies integrating refined subcellular models into continuum descriptions of epithelial mechanobiology.