Tissue shape from cell-scale active tensions

Connecting cell behavior to tissue shape and mechanics is a key challenge in the physics of morphogenesis. Cytoskeletal turnover precludes a fixed reference state, and tensions are actively generated independently of strain; so conventional elasticity theory is not applicable. Here, we study epithelia governed by quasi-static force balance between intracellular pressure and internal, active tensions. This makes the tissue a distributed hydrostatic skeleton. Our theory starts from a set of prescribed active tensions. It treats cell interfaces as force dipoles whose embedding in physical space - the physical cell configuration - is constrained by force balance. To solve this constraint problem geometrically, we represent the tensions as a triangulation dual to the cell tiling. This allows us to use (and extend) the mathematics of discrete conformal geometry to link tensions to cell and tissue shape. Adiabatic changes of tensions cause changes in the physical configuration. Thus, rather than fluidizing, tissues can deform - or "morph" - while resisting external forces like a solid. This constitutes a form of emergent elasticity, mediated by two geometric soft modes. Importantly, tissue-scale stress depends on cell shape, but is independent of microscopic tension anisotropy, with consequences for experimental stress measurements and modeling mechanosensitive feedback loops. Discrete conformal geometry also allows us to analyze how cellular tension dynamics drive cell rearrangement, required for large plastic deformation. The unified description of emergent elasticity of epithelial tissues and their plastic morphing, driven by adiabatic tension dynamics and cell rearrangement, provides a foundation to better understand the role of mechanics in morphogenesis. Furthermore, we highlight connections to the mechanics of other amorphous materials, such as granular media.