Beyond 2D Tissue Modeling: A 3D Continuum Shell Theory of Epithelial Cells*

Two-dimensional (2D) models for confluent tissues correlate the mechanical state of monolayers to morphological indicators such as the shape of their cellular components. We show that such an approach, relying on perimeter elasticity and adhesion of the cells, proves inadequate when applied to in-vitro cultured MDCK epithelia, and that this discrepancy is a consequence of the response of epithelial cells to localized stresses exerted by actin fiber bundles at their basal faces and by a diffuse actin cortex at their apical face. We model the 3D shape of the cell boundary by treating it as an elastic shell subject to appropriate boundary conditions. The analytical solutions agree with our experimental observations of changing cross-sectional shapes along the apical-basal axis and explain the discrepancy in 2D modeling. Furthermore, an analysis of the contributions to deformation energy suggests that the main effects are attributable to bending, as opposed to stretching, and to deformation gradients in the axial, rather than the perimeter, direction, both of which contrast with 2D modeling assumptions. By considering important biologically motivated effects, this 3D continuum shell model not only provides a more realistic representation of epithelial cell mechanics but also serves as a valuable platform for extracting effective morphological indicators relevant to mechanically diagnosing tissues.