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

TL;DR

The paper tackles how embryonic inversion in Volvox arises from glocal elasticity, where local changes in intrinsic curvature and stretch must overcome global geometric constraints. It develops an axisymmetric thin-elastic-shell model with locally varying intrinsic properties, then uses small-deformation asymptotics to separate bending from stretching and numerical bifurcation analysis to study invagination versus posterior contraction. Key findings show that a localized region of negative intrinsic curvature drives invagination while posterior contraction relieves global constraints, with an inner-layer asymptotic analysis revealing a preferred narrow invagination (global minimum at $\Lambda\approx2.44$) and contraction providing symmetry-breaking that lowers energy barriers. The resulting bifurcation picture—complete with a critical point and spinodal curves—explains the time-sequenced cell-shape changes in Volvox and offers design insights for programmable elastic systems exploiting glocal deformation pathways.

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. Höhn ${\it et~al}.$, ${\it Phys. Rev. Lett.}$ $\textbf{114}$, 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.

Figures (8)

• Figure 1: (color online). Volvox invagination and elastic model. (a) Adult Volvox, with somatic cells and one embryo labelled. (b) Volvox embryo at the start of inversion. (c) Mushroom-shaped invaginated Volvox embryo. (d) Cell shape changes to wedge shapes and motion of cytoplasmic bridges (CB) bend the cell sheet. Red line indicates position of cytoplasmic bridges. (e,f) Cross-sections of the stages shown in panels (b,c). Cell shape changes as in (d) occur in the marked regions. (g) Geometry of undeformed spherical shell of radius $R$ and thickness $h$. (h) Geometry of deformed shell. Scale bars: (a) $50\,\hbox{\textmu m}$, (e,f) $20\,\hbox{\textmu m}$. False color images obtained from light-sheet microscopy provided by Stephanie Höhn and Aurelia R. Honerkamp-Smith.
• Figure 2: (color online). Simple intrinsic deformations. (a) A sphere can be shrunk to smaller spheres of equal radii by both compatible and incompatible intrinsic deformations. (b) Contraction of a circular region of radius $R$ in a plane elastic sheet by a factor $f$. The boundary of this region is contracted to $s=FR$. (c) Numerical result for $F$ (+) agrees with analytical calculation (\ref{['eq:linear']}) (solid line).
• Figure 3: (color online). Asymptotic analysis of invagination and contraction. (a) Numerical shape resulting from contracting the posterior to a radius $r_{\mathrm{p}}<R$. (b) Numerical "hourglass" shape resulting from pure invagination. (c) Geometry of contraction with posterior radius $r_{\mathrm{p}}<R$, resulting in upward motion of the posterior by a distance $d$. (d) Geometry of pure invagination solution. (e) Asymptotic geometry: in the limit $h \ll R$, deformations are localized to an asymptotic inner layer of width $\delta$ about $x = X$, where $x = s/R$ is the angle that the undeformed normal makes with the vertical. In the deformed configuration, this angle has changed to $\beta(x)$. (f) Asymptotic invagination: upward motion of posterior by a distance $d$ requires inward deformations scaling as $(\delta d)^{1/2}$ in the inner layer of width $\delta$. (g) Relation between preferred curvature $k$ and width of invagination $\lambda$ for a given amount of upward posterior motion $d$, from asymptotic calculations. (h) Inward rotation of midpoint of invagination with, and without contraction, from asymptotic calculations.
• Figure 4: (color online). Setup for numerical calculations, following hohn14. (a) Geometrical setup: the intrinsic curvature $\kappa_s^0$ of a spherical shell of undeformed radius $R$ differs from the undeformed curvature in the range $\lambda_{\max} > s > \lambda_{\max}-\lambda$, where $s$ is arclength. Posterior contraction is taken into account by a reduced posterior radius $r_\mathrm{p}<R$. These intrinsic curvature and contraction result in deformations that move up the posterior pole by a distance $d$. (b) Corresponding functional form of $\kappa_s^0$; in the bend region, $\kappa_s^0=-k<0$. (c) Form of the intrinsic stretches $f_s^0,f_\phi^0$ for posterior contraction. (d) Functional form of $\kappa_\phi^0$ for posterior contraction.
• Figure 5: (color online). Bifurcation behaviour of invagination solutions. Solution space for $\lambda_{\max} = 1.1R$ and $r_{\mathrm{p}}=R$: each line shows the relation between $k$ and $d$ at some constant $\lambda$. A critical branch (at $\lambda=\lambda_*$) separates different types of branches. Branches with $\lambda>\lambda_\ast$ feature two extrema; the resulting spinodal curve (thick dashed line) defines a critical point. Insets illustrate representative solution shapes. See text for further explanation.