A mathematical theory for the mechanics of three-dimensional cellular aggregates reveals the mechanical atlas for Ascidian embryogenesis

The intricate three-dimensional (3D) structures of multicellular organisms emerge through genetically encoded spatio-temporal patterns of mechanical stress. Cell atlases of gene expression during embryogenesis are now available for many organisms, but connecting these to the mechanical drivers of embryonic shape requires physical models of multicellular tissues that identify the relevant mechanical and geometric constraints, and an ability to measure mechanical stresses at single-cell resolution over time. Here we report significant steps towards both these goals. We describe a new mathematical theory for the mechanics of 3D multicellular aggregates involving the quasi-static balance of cellular pressures, surface tensions, and line tensions. Our theory yields a quantitatively accurate low-dimensional description for the time-varying geometric dynamics of 3D multicellular aggregates and, through the solution of a mechanical inverse problem, an image-based strategy for constructing spatio-temporal maps of the mechanical stresses driving morphogenesis in 3D. Using synthetic image data, we confirm the accuracy and robustness of our geometric and mechanical approaches. We then apply these approaches to segmented light sheet data, representing cellular membranes with isotropic resolution, to construct a 3D mechanical atlas for ascidian gastrulation. The atlas captures a surprisingly accurate low-dimensional description of ascidian gastrulation, revealing the adiabatic nature of the underlying mechanical dynamics. Mapping the inferred forces onto the invariant embryonic lineage reveals a rich correspondence between dynamically evolving cell states, patterns of cell division, and local regulation of cellular pressure and contractile stress. Thus, our mechanical atlas reveals a new view of ascidian gastrulation in which lineage-specific control over a complex heterogenous pattern of cellular pressure and contractile stress, integrated globally, governs the emergent dynamics of ascidian gastrulation.