Elasticity and Glocality: Initiation of Embryonic Inversion in $Volvox$

Elastic objects across a wide range of scales deform under local changes of their intrinsic properties, yet the shapes are $glocal$, set by a complicated balance between local properties and global geometric constraints. Here, we explore this interplay during the inversion process of the green alga $Volvox$, whose embryos must turn themselves inside out to complete their development. This process has recently been shown [S. H"ohn $etal\mathrm{.}$, $Phys.Rev.Lett.$ $\mathbf{\text{114}}$, 178101 (2015)] to be well described by the deformations of an elastic shell under local variations of its intrinsic curvatures and stretches, although the detailed mechanics of the process have remained unclear. Through a combination of asymptotic analysis and numerical studies of the bifurcation behavior, we illustrate how appropriate local deformations can overcome global constraints to initiate inversion.