A mechanical bifurcation constrains the evolution of cell sheet folding in the family Volvocaceae

Abstract

The processes of morphogenesis that give rise to the shapes of organs and organisms during development are often driven by mechanical instabilities. Can such mechanical bifurcations also drive or constrain the evolution of these processes in the first place? We discover an instance of these constraints in the green algae of the family Volvocaceae. During their development, their bowl-shaped embryonic cell sheet turns itself inside out. This inversion is driven by a simple wave of cell wedging in the genus Pleodorina (16-128 cells) and more complex programmes of cell shape changes in Volvox (~400-50000 cells). However, no species with intermediate cell numbers (256 cells) have been described. Here, we relate this gap to a mechanical bifurcation: Focusing on the inversion of Pleodorina californica (64 cells), we develop a continuum model, in which the cell shape changes driving inversion appear as changes of the intrinsic curvature of an elastic surface. A mechanical bifurcation in this model predicts that inversion is only possible in a subset of its parameter space. Strikingly, parameters estimated for P. californica fall into this possible subset, but those that we extrapolate to 256 or more cells using allometric observations and a model of cell cleavage in Volvocaceae do not. Our work thus suggests that the more complex inversion strategies of Volvox are an evolutionary necessity to obviate this bifurcation and indicates more broadly how mechanical bifurcations can drive the evolution of morphogenesis.

Figures (10)

• Figure 1: Inversion in Pleodorina californica: mechanical model and bifurcation. (a) Cell numbers $N$ of the genera Pleodorina and Volvox in the family Volvocaceae. Inset: Schematics of type-A and type-B inversion in different species of Volvox, reproduced from Ref. hohn2015inversion. (b) Inversion of P. californica, reproduced from Ref. hohn2016distinct. Top row: schematics of a section of the axisymmetric bowl-shaped cell sheet at different stages of inversion; the length of the cross-section before inversion is $L$, and arclength along the cross-section is $s$. Red line: position of cytoplasmic bridges; post: posterior pole. Bottom row: corresponding light micrographs of stained semi-thin sections. Scale bar: $20\,\text{\textmu m}$. (c) Mechanical model of inversion in P. californica. The cell sheet is represented as an elastic spherical cap of radius $R=1$. The slow cell shape changes of inversion are encoded in quasistatic variations of its intrinsic curvature $\kappa^0(s;t)$, where $s\in [0,L]$ is undeformed arclength, and $t$ is the time of inversion. The front of the wave of cell shape changes at $s=W(t)$ separates the uninverted part of the cell sheet (where cells are spindle-shaped, $\kappa^0=1$) from the inverted part (where cells are more wedge-shaped, $\kappa^0=-k<0$). As inversion progresses, $W(t)$ decreases from $s=L$ towards $s=0$ (arrow). Insets: cell shapes on either side of the front, from Ref. hohn2016distinct. (d) Mechanical bifurcation: If $k>k_\ast$ is large enough (top), the edges of the cap flip over as the bend region progresses and the cell sheet inverts (shape insets at different times $t$); if $k<k_\ast$ is too small, the edges do not flip over and inversion fails.
• Figure 2: Bifurcation diagrams for inversion in P. californica. Plots of the elastic energy $\mathcal{E}$ against the position $W$ of the front of the wave of cell shape changes, for different values of the absolute value $k$ of the inverted intrinsic curvature. Insets: corresponding plots of the signed distance $\mathcal{Z}$ [defined in panel (a)]; $\mathcal{Z}>0$ ($\mathcal{Z}<0$) indicates inverted (uninverted) configurations of the cell sheet. Bifurcations at $k=k_1$ and $k=k_2$ separate three qualitatively different bifurcation diagrams. (a) For $k<k_1$, the cell sheet does not invert on the branch of lowest energy (red line) connected to the undeformed configuration. The cell sheet does invert on another branch (dashed line) of lower energy, but this branch is not connected to the undeformed configuration. (b) For $k_1<k<k_2$, the cell sheet inverts on the branch of lowest energy (red line). Fold points on this branch imply a discontinuous snapthrough as $W$ decreases (arrows). The branch on which the lips do not flip over (dashed line) is no longer connected to the undeformed configuration. (c) For $k>k_2$, the cell sheet inverts without snapthrough on the branch of lowest energy. Parameter values: $L=1.75$, $h=0.15$, (a) $k=1.8$, (b) $k=2$, (c) $k=3.1$.
• Figure 3: Quantitative model of inversion in P. californica. (a) Plot, in $(h,L,k)$ space, of the bifurcation boundaries $k_1(h,L),k_2(h,L)$. Inversion is possible (arrow) for $k>k_1(h,L)$ and continuous for ${k>k_2(h,L)}$. Inset: definition of $h,L$. Marker and error bars: parameter values for P. californica estimated in Table \ref{['table1']}, falling within the region in which the model allows continuous inversion. (b) Cross-section through this plot at $L=2.0$ (Table \ref{['table1']}). The hatched area ${k>8/(3h)}$ is forbidden geometrically because the intrinsically deformed cell shapes would self-intersect (Appendix \ref{['appA']}). The shading indicates the region of strong cell constriction (Appendix \ref{['appA']}). (c) Analogous cross section at $h=0.4$ (Table \ref{['table1']}).
• Figure 4: Plot of the ranges reported for the numbers of cells of individuals for different species in the families Volvocaceae and Goniaceae, grouped according to genus, based on the Algaebase database algaebase and data from Refs. grove1915pleodorinachodat1931quelquespocock1955studiesstarr1962newberg1970structurestarr1970volvoxkarn1972sexualakiyama1977illustrationsmarchant1977colonymainireland1981inversionnozaki1982morphnozaki1983morphologybatko1989goniumnozaki1989morphologicalnozaki1989pleodorinanozaki1990volvulinanozaki1991pandorinanozaki1992ultrastructurenozaki1993asexualnozaki2001morphologyhallmann2006morphogenesisnozakimorphologyyamada2008taxonomicsolari2008volvoxhayama2010morphologyiida2011embryogenesishohn11nozaki2011newisaka2012descriptioniida2013cleavagenozaki2014newnozaki2015morphologyhohn2016distinctnozaki2017rediscoverynozaki2018morphologykimbara2019morphologicalnozaki2019morphologynozaki2019morphologybnozaki2022morphology collected in Appendix \ref{['appD']}. Each line corresponds to one species. The grey bar highlights the lack of species with individuals of 256 cells.
• Figure 5: Allometry of Volvocaceae and Goniaceae. Illustration of the empirical relation $H\propto N^{-1/4}$ (dashed line) between the non-dimensional cell sheet thickness $h$ and the number $N$ of cells, fitted from micrographs hallmann2006morphogenesishohn2016distinctmarchant1977colonymain of Gonium pectorale, Pandorina morum, Eudorina unicocca, Eudorina elegans, Pleodorina californica ($N=\text{8--64}$), as shown in the inset and described in Appendix \ref{['appC']}. The allometric relation is also consistent with estimates from micrographs nozaki2018morphologyhohn11 of Volvox carteri and Volvox globator ($N=\text{1400--6000}$).