Dynamics of a $Volvox$ Embryo Turning Itself Inside Out

Spherical embryos of the algal genus $Volvox$ must turn themselves inside out to complete their embryogenesis. This `inversion', which shares important features with morphological events such as gastrulation in animals, is perhaps the simplest example of a topological transition in developmental biology. Waves of cell shape changes are believed to play a major role in the process, but quantification of the dynamics and formulation of a mathematical description of the process have been lacking. Here, we use selective plane illumination microscopy on $V. globator$ to obtain the first quantitative three-dimensional visualizations of inversion $in~vivo$. A theory is formulated for inversion based on local variations of intrinsic curvature and stretching of an elastic shell, and for the mechanics of an elastic snap-through resisted by the surrounding fluid.