A preferred curvature-based continuum mechanics framework for modeling embryogenesis with application to Drosophila mesoderm invagination

Mechanics plays a key role in the development of higher organisms. However, working towards an understanding of this relationship is complicated by the fact that it has proven difficult to model the link between local forces generated at the subcellular level, and tissue deformation at the whole-embryo level. Here we propose an approach first developed for lipid bilayers and cell membranes, in which force-generation at the cytoskeletal level only enters the shape mechanics calculation in the form of local changes in preferred tissue curvature. This allows us to formulate the continuum mechanics problem purely in terms of tissue strains. Relaxing the system by lowering its mechanical energy yields global morphogenetic predictions that accommodate the tendency towards this local preferred curvature, without explicitly modeling force-generating mechanisms at the molecular level. Our computational framework, which we call SPHARMMECH, extends a three-dimensional spherical harmonics parameterization known as SPHARM to combine this level of abstraction with a sparse shape representation. The integration of these two principles allows computer simulations to be performed in three dimensions, on highly complex shapes, gene expression patterns, and mechanical constraints. We demonstrate our approach by modeling mesoderm invagination in the fruit-fly embryo, where local forces generated by the acto-myosin meshwork in the region of the future mesoderm lead to formation of a ventral tissue fold. The process is accompanied by substantial changes in cell shape and long-range cell movements. Applying SPHARM-MECH to whole-embryo live imaging data acquired with light-sheet microscopy reveals significant correlation between calculated and observed tissue movements. Our analysis predicts the observed cell shape anisotropy on the ventral side of the embryo and suggests an active mechanical role of mesoderm invagination in supporting the onset of germ-band extension.