Elasticity and Glocality: Initiation of Embryonic Inversion in ${\it Volvox}$

TL;DR

This paper investigates how local elastic deformations in the embryonic shell of Volvox initiate the inversion process, balancing local properties and global constraints through analysis and numerical modeling.

Contribution

It provides a detailed analysis of the mechanics behind Volvox inversion, highlighting how local curvature changes can overcome global constraints to trigger shape transformation.

Findings

Local deformations can initiate inversion despite global constraints.

Asymptotic analysis reveals the bifurcation behavior of the elastic shell.

Numerical studies support the role of local curvature variations in inversion initiation.

Abstract

Elastic objects across a wide range of scales deform under local changes of their intrinsic properties, yet the shapes are ${\it glocal}$, 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 ${\it Volvox}$, whose embryos must turn themselves inside out to complete their development. This process has recently been shown [S. H\"ohn ${\it et~al}.$, ${\it Phys. Rev. Lett.}$ $\textbf{114}$, 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.