Mechanics and Variability of Cell Sheet Folding in the Embryonic Inversion of $Volvox$

Many embryonic deformations during development are the global result of local cell shape changes and other local active cell sheet deformations. Morphogenesis does not only therefore rely on the ability of the tissue to produce these active deformations, but also on the ability to regulate them in such a way as to overcome the intrinsic variability of and geometric constraints on the tissue. Here, we explore the interplay of regulation and variability in the green alga Volvox, whose spherical embryos turn themselves inside out to enable motility. Through a combination of light sheet microscopy and theoretical analysis, we quantify the variability of this inversion and analyse its mechanics in detail to show how shape variability arises from a combination of geometry, mechanics, and active regulation. Introduction Julian Huxley’s pronouncement, “In some colony like [the green alga] Volvox, there once lay hidden the secret to the body and shape of [humans]” (Huxley, 1912), emphasises that morphogenesis across kingdoms relies on the fundamental ability of organisms to, firstly, produce active forces that drive the deformations of cell sheets underlying the development of many organs and tissues and, secondly, regulate these active deformations in such a way to complete morphogenesis. Unravelling the biomechanics of these processes is therefore of crucial importance to understand pathological errors and foster bioengineering to address these errors (Sasai et al., 2012). Local cellular changes can produce forces that are transmitted along the cell sheet to drive its global deformations (Lecuit and Lenne, 2007; Lecuit et al., 2011). Simple events of cell sheet folding such as ventral furrow formation in Drosophila can be driven primarily by cell shape changes (Sweeton et al., 1991). In more complex metazoan developmental processes such as gastrulation (Leptin, 2005;Wang and Steinbeisser, 2009), optic cup formation (Fuhrmann, 2010; Chauhan et al., 2015), neurulation (Lowery and Sive, 2004; Vijayraghavan and Davidson, 2017) and related processes (Sherrard et al., 2010), the effect of such cell shape changes is overlaid by that of other cellular changes such as cell migration, cell intercalation, cell differentiation, and cell division. In all of these processes however, these local cellular changes occur in specific regions of the cell sheet and at specific stages of morphogenesis. On the one hand, the spatio-temporal distribution of these local cellular changes affects the global tissue shape. On the other hand, a certain amount of noise is unavoidable in biological systems; indeed, it may even be necessary for robust development, as demonstrated for example by Hong et al. (2016), who showed that variability in cell growth is necessary for reproducible sepal size and shape in Arabidopsis. While 1 of 33 ar X iv :1 70 8. 07 76 5v 1 [ co nd -m at .s of t] 2 5 A ug 2 01 7 Mechanics and Variability of Cell Sheet Folding in the Embryonic Inversion of Volvox some processes may be subject to less intrinsic variability than others, one must therefore ask: how are these processes orchestrated so that development can complete despite the intrinsic biological variability? Differences in the observed shapes of organisms at certain stages of development (i.e. what one might term their geometric variability) stem from a combination of mechanical variability (i.e. differences in mechanical properties or mechanical state) and active variability (i.e. differences in the active forces generated by individual cells). What experimental data there are suggest that the mechanical properties are subject to a large amount of variability (von Dassow and Davidson, 2007, and references therein). Finally, differences in the mechanical stress state of the tissue are another facet of mechanical variability that is induced by active variability. The first mechanical models of morphogenesis (Odell et al., 1981) represented cells as discrete collections of springs and dashpots; they were soon followed by elastic continuum models (Hardin and Cheng, 1986;Hardin and Keller, 1988). Notable among this early modelling ofmorphogenesis is for example the work of Davidson et al. (1995, 1999), who combined models of several mechanisms of sea urchin gastrulation with measurements of mechanical properties to test the plausibility of these different mechanisms. These models heralded the emergence of a veritable plethora of mechanical modelling approaches over the subsequent decades (Fletcher et al., 2017), though the choice of model must ultimately be informed by the questions one seeks to answer (Rauzi et al., 2013). More recent endeavours were directed at deriving models that can represent the chemical and mechanical contributions to morphogenesis and their interactions (Howard et al., 2011) and at establishing the continuum laws that govern these out-of-equilibrium processes (Prost et al., 2015). There is, however, a rather curious gap in the study of the variability of development: the importance of quantifying the morphogenesis and its variability has been recognised (Cooper and Albertson, 2008; Oates et al., 2009), yet accounts of the variability of development, e.g. in the loach (Cherdantsev and Tsvetkova, 2005; Cherdantsev and Korvin-Pavlovskaya, 2016), have often been merely descriptive. For this reason, the interplay between mechanics and active variability has seemingly received little attention and hence a question we believe to be fundamental appears to lie in uncharted waters: how does active variability lead to geometric variability? Conversely, what does geometric variability tell us about active variability? This is the question that we explore in this paper in the context of the development of the multicellular green alga Volvox (Fig. 1a). Volvox and the related Volvocine algal genera have been recognised since the work ofWeismann (1892) as model organisms for the evolution of multicellularity (Kirk, 1998, 2005; Herron, 2016), spawning more recent investigations of kindred questions in fluid dynamics and biological physics (Goldstein, 2015). The cells of Volvox (Fig. 1b) are differentiated into biflagellated somatic cells and a small number of germ cells, or gonidia, that will form daughter colonies (Kirk, 1998). The somatic cells in the adult are embedded in a glycoprotein-rich extracellular matrix (Kirk et al., 1986; Hallmann, 2003). The germ cells undergo several rounds of cell division, after which each embryo consists of several thousand cells arrayed to form a thin spherical sheet confined to a fluid-filled vesicle. Cells are connected to their neighbours by cytoplasmic bridges (Fig. 1b), thin membrane tubes resulting from incomplete cell division (Green and Kirk, 1981; Green et al., 1981; Hoops et al., 2006). Those cell poles whence will emanate the flagella however point into the sphere at this stage, and so the embryos must turn themselves inside out through an opening at the anterior pole of the cell sheet (the phialopore), to enable motility and thus complete their development (Kirk, 1998). Because of this process of inversion, Volvox has become a model organism for the study of cell sheet deformations, too (Kirk et al., 1982; Kirk and Nishii, 2001;Matt and Umen, 2016). Inversion in Volvox (Viamontes and Kirk, 1977; Höhn and Hallmann, 2011) and in related species (Hallmann, 2006; Iida et al., 2011, 2013; Höhn and Hallmann, 2016) results from cell shape changes only, without the complicating additional processes found in metazoan development discussed above. This simplification facilitates the study of morphogenesis. While different species of Volvox have developed different ways of turning themselves inside out (Hallmann, 2006), here, we focus on the so-called type-B inversion arising, for example, in Volvox globator (Zimmermann,