A Preferred Curvature-Based Continuum Mechanics Framework for Modeling Embryogenesis

Mechanics plays a key role in the development of higher organisms. However, understanding this relationship is complicated by the difficulty of modeling the link between local forces generated at the subcellular level and deformations observed at the tissue and whole-embryo levels. Here we propose an approach first developed for lipid bilayers and cell membranes, in which force-generation by cytoskeletal elements enters a continuum mechanics formulation for the full system in the form of local changes in preferred curvature. This allows us to express and solve the system using only tissue strains. Locations of preferred curvature are simply related to products of gene expression. A solution, in that context, means relaxing the system's mechanical energy to yield global morphogenetic predictions that accommodate a tendency toward the local preferred curvature, without a need to explicitly model force-generation mechanisms at the molecular level. Our computational framework, which we call SPHARM-MECH, extends a 3D 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.