Variability is emerging as an integral part of development. It is therefore imperative to ask how to access the information contained in this variability. Yet most studies of development average their observations and, discarding the variability, seek to derive models, biological or physical, that explain these average observations. Here, we analyse this variability in a study of cell sheet folding in the green alga Volvox, whose spherical embryos turn themselves inside out in a process sharing invagination, expansion, involution, and peeling of a cell sheet with animal models of morphogenesis. We generalise our earlier, qualitative model of the initial stages of inversion by combining ideas from morphoelasticity and shell theory. Together with three-dimensional visualisations of inversion using light sheet microscopy, this yields a detailed, quantitative model of the entire inversion process. With this model, we show how the variability of inversion reveals that two separate, temporally uncoupled processes drive the initial invagination and subsequent expansion of the cell sheet. This implies a prototypical transition towards higher developmental complexity in the volvocine algae and provides proof of principle of analysing morphogenesis based on its variability.
‘The phenomena are always the same, and this is what matters to us, but their variations, for the greater or for the lesser, are beyond count. ’ Thus opined Xavier Bichat in the account of his investigations into life and death [1] and thereby spelt out how, to the present day, questions in developmental biology and cell sheet folding in particular are commonly approached: the vast majority of analyses average their experimental observations and seek to derive a model, biological or physical, that explains this average behaviour. In so doing, they discard the variability or deviations from average behaviour that are observed in experiments. A certain amount of noise is, however, unavoidable in biological systems; indeed, it may even be necessary for robust development, as demonstrated, for example, by Hong and colleagues [2], who showed that variability in cell growth is necessary for reproducible sepal size and shape in Arabidopsis. The natural question of how to use this variability to infer developmental mechanisms appears to lie in uncharted waters, however. This is the question that we explore in this paper to provide proof of principle of analysing cell sheet folding based on its variability. Cell sheet folding pervades multicellular development, and its general principles have been established in a large body of previous work: local cellular changes can produce forces that are transmitted via cell–cell connections along the cell sheet and drive its global deformations [3,4]. Simple events of cell sheet folding, such as ventral furrow formation in Drosophila [5,6], can be driven primarily by cell shape changes. In more complex metazoan developmental processes—such as gastrulation [7,8], optic cup formation [9,10], neurulation [11,12], and related processes in vivo [13] and in vitro [14]—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. Owing to this complexity, and in spite of significant progress in identifying the molecular components involved, the correspondence between local cellular changes and global deformations of cell sheets remains poorly understood. Biological analyses of morphogenesis are complemented, at a more physical level, by a whole host of mechanical models of morphogenesis. The first of these represented cells as discrete collections of springs and dashpots [15]; they were soon followed by elastic continuum models [16,17]. Notable among this early modelling of morphogenesis is, for example, the work of Davidson and colleagues [18,19], 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 [20], though the choice of model must ultimately be informed by the questions one seeks to answer [21]. More recent endeavours were directed at deriving models that can represent the chemical and mechanical contributions to morphogenesis and their interactions [22] and at establishing the continuum laws that govern these out-of-equilibrium processes [23]. However, all of this but emphasises a rather curious gap in the study of development: the importance of quantifying morphogenesis and its variability has been recognised [24,25], but analyses of the variability of development have been few and far between. What experimental data there are on the variability of the mechanical properties of cell sheets suggest a large amount of variability ([26] and references therein). The variability of the cell sheet deformations during development is even more unexplored, and accounts of this variability—e. g. , in the loach Misgurnus fossilis [27,28]—have often been merely descriptive. In this paper, we present the first comprehensive analysis of this variability in cell sheet folding and the lessons that can be drawn from it. The experimental system in which we explore these questions of variability is the multicellular green alga Volvox (Fig 1A), of which Julian Huxley said that ‘In some colony like Volvox, there once lay hidden the secret of the body and mind of [humans]’ [29]. Indeed, Volvox and the related volvocine algal genera have been recognised since the work of Weismann [30] as model organisms for the evolution of multicellularity [31–33], spawning more recent investigations of kindred questions in fluid dynamics and biological physics [34]. Being able to reproduce asexually, Volvox is perfectly suited for studying nongenetic sources of morphogenetic variation among genetically identical individuals. In the asexual life cycle, the cells of an adult Volvox spheroid (Fig 1B) are differentiated into biflagellate somatic cells and a small number of germ cells, or gonidia, that will form the next generation [31]. The somatic cells in the adult are embedded in a glycoprotein-rich extracellular matrix [35,36]. 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 [37–39]. 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 embryonic development [31]. This process of inversion has become a model for the study of cell sheet deformations [40–42]. Inversion in Volvox [44,45] and in related species [46–49] 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 [46], here, we focus on the so-called type-B inversion arising, for example, in V. globator [44,46,50]. This shares features such as invagination and involution with developmental events in metazoans [51–53]. This inversion scenario is distinct from type-A inversion, in which four lips open at the anterior of the shell and peel back to achieve inversion [45]. This process is driven by a single wave of uniform cell shape changes moving from the anterior to the posterior pole of the embryo [45]. By contrast, type-B inversion involves different types of cell shape changes in different parts of the cell sheet [44], the coupling of which has remained unclear. This inversion begins with the appearance of a circular bend region at the equator of the embryo (Fig 1C and 1D, Fig 2A): cells there become wedge-shaped by developing narrow basal stalks [44]. At the same time, the cells move relative to the cytoplasmic bridges so as to be connected at their thin stalks, thus splaying the cells and bending the cell sheet [44]. Nishii and colleagues [54] showed that type-A inversion in V. carteri is arrested in the absence of analogous motion of cells relative to the cytoplasmic bridges. This relative motion is mediated by a motor protein, the kinesin InvA, associated to the microtubule cytoskeleton (S1 Fig); orthologues of InvA are found throughout the volvocine algae [32]. After invagination, the posterior hemisphere moves into the anterior (Fig 1E), the phialopore widens, and the anterior hemisphere moves over the subjacent posterior (Fig 1F) while' rolling' over a second circular bend region, the anterior fold [44]. Additional cell shape changes (Fig 1D–1F, Fig 2B–2D) in the anterior and posterior hemispheres are implicated in the relative contraction and expansion of either hemisphere with respect to the other [44]. This plethora of cell shape changes is possible as embryonic Volvox cells do not have a cell wall [31]. It is not yet known what triggers the initial cell shape changes, what determines their location, and what kind of signal drives the propagation of waves of cell shape changes in Volvox embryos. In the present study, we show that even without this knowledge, we are able to infer information on local changes from the variability of global dynamic embryo shapes. In a previous study [43], we combined light sheet microscopy and theory to analyse the early stages of inversion, showing that only a combination of active bending and active stretching (i. e. , expansion or contraction) can account for the cell sheet deformations observed during invagination. The crucial role of active stretching was also highlighted by Nishii and Ogihara [55], who showed that type-A inversion in V. carteri cannot complete if actomyosin-mediated contraction is inhibited chemically. We later analysed the mechanics of this competition between bending and stretching in more detail [56]. The general question of how the different parts of a morphogenetic process relate to each other, however, remained unanswered in this system, too: are the different deformations of either hemisphere during type-B inversion coupled? What drives the' peeling' of the anterior hemisphere? The present analysis addresses these questions and naturally divides into three parts: we begin by deriving an average sequence of Volvox inversion and quantifying its variability. This consensus inversion sequence serves as a template for the mathematical analysis in the second part of the paper: building on our earlier, qualitative model of the early stages of inversion and combining ideas from morphoelasticity [57] and shell theories [58,59], we derive a detailed, quantitative description of the entire process of inversion. In so doing, we show for the first time how detailed information on the underlying cellular changes can be deduced from deformations at the tissue level. In the third and final part of the paper, we compare the experimental distribution of variability to simulated distributions based on perturbations of the local active deformations in the model. We thus infer how the observed distribution of variability in the embryo shapes arises, and we find that inversion is driven by two separate, temporally uncoupled processes. This provides proof of principle of using developmental variability to infer developmental mechanisms and mechanics.
To define an average inversion sequence and analyse its mechanics, we compare the local geometry of the traced curves. The rather philosophical question of how to define an appropriate metric for this kind of comparison goes back at least to the work of D' Arcy Thompson [61] and has no unique answer. Thompson showed, for example, how the outlines of fish of different species could be mapped onto one another by dilations, shears, and compositions thereof. Our averaging approach must allow for the different types of variability that arise in Volvox inversion (as discussed in the previous paragraph), while recognising that the posterior poles and the rims of the phialopores of the different embryos must correspond to each other. Our approach is therefore based on minimising the euclidean distance between individual embryo shapes and their averages, with alignments obtained using dynamic warping of shapes (Materials and methods). Results are shown in Fig 4. Averaging approaches that do not consider both stretching in time of individual inversions and local stretching of corresponding points of individual shapes tend to give unsatisfactory results: the simplest averaging approach is to align the inversion sequences by a single time point—for example, when the posterior-to-bend distance reaches half of its initial value (Materials and methods and S2 Fig). The absence of time stretching, however, means that large variations arise at later stages of inversion. (Given the dramatic embryonic shape changes during inversion, it is not surprising that there should be no single parameter that could be used to align inversions of different embryos.) A better alignment is obtained if we allow stretching in time (Materials and methods and S3 Fig), but this method, without local stretching of individual shapes relative to each other, produces unrepresentative kinks in the bend region of the average shapes (S3 Fig). To quantify the time course of individual inversions further, we shift the time coordinate of each embryo half so that t = 0 is the time when it reaches the first fitted stage. We then define the average time ⟨t⟩ to be the average, over all embryo halves, of these shifted times. Plotting the time course of individual inversions in the resulting (⟨t⟩, t) diagram (Fig 5A), we observe that different stages of inversion take different times in different embryos, with some embryos seeming to linger in certain stages. Nonetheless, despite this' stop-and-go' behaviour, each time course is well approximated by a straight line in the (⟨t⟩, t) diagram (Fig 5B), which signifies that inversion proceeds at constant speed in all embryos. To analyse the local variations of the embryo shapes, we define, at each point of the average shapes, a covariance ellipse. The curves that are parallel to the average shape and tangent to the covariance ellipse define what we shall term the standard deviation shape. These standard deviation shapes measure the variability of the average shapes and are shown in Fig 4. The variations they represent naturally divide into two components: first, those variations that are parallel to the average shape and, second, those perpendicular to the average shape. The former represent mere local stretches of the average shapes, while the latter correspond to actual variations of the shapes; we shall therefore refer to the thickness of the standard deviation shapes as' shape variation' in what follows. We report the mean shape variation and its standard error in Fig 5C. This plot shows that the mean shape variation reaches a maximal value around the stages in Fig 4G–4I: different embryos start from the same shape and reach the same inverted shape after inversion (up to a scaling) but may take different inversion paths. Plotting the mean shape variation for different averaging methods (S4 Fig), we confirm that the present averaging method yields a better alignment than the alternative methods discussed earlier. It is intriguing, however, to note the spatial structure of the local shape variations. In particular, during the early stages of posterior inversion (Fig 4D–4F), the shape variation is smaller in the active bend region than in the adjacent anterior fold (Fig 1E, the second bend region of increased positive curvature). As the phialopore opens, and the anterior begins to peel back over the partially inverted posterior (Fig 4H), the relative shape variation becomes smaller in the anterior fold. The initially small variation in the bend region is especially intriguing, since this is where cells become wedge-shaped to drive invagination, while the anterior fold bends passively [43]. In other words, the shape variation is reduced in the part of the cell sheet where the active cell shape changes that drive invagination arise. If there were no variability in the cell shape changes, then inversion could not fail. This correspondence therefore characterises what one might term, from a teleological point of view, a' good' inversion. We shall focus on a less exalted question, the answer to which will be falsifiable, however: how is this spatial structure of the variability related to the mechanics of inversion? Before we can address this question, we need to analyse the mechanics of inversion in some more detail. The second step of our analysis is to derive a quantitative theoretical model of inversion. We have previously described the early stages of inversion using a qualitative mathematical model [43] in which cell shape changes appear as local variations of the intrinsic (meridional and circumferential) curvatures κs0, κϕ0 and stretches fs0, fϕ0 of an elastic shell (Fig 6A). Open, one-dimensional elastic filaments can simply adopt a shape in which the curvature and stretch are everywhere equal to their intrinsic values, but two-dimensional elastic shells cannot, in general, do this: the intrinsic curvatures and stretches may not be compatible with the global geometry, causing the shell to deform elastically and adopt actual (meridional and circumferential) curvatures κs, κϕ and stretches fs, fϕ different from the imposed intrinsic curvatures and stretches (Fig 6A). A more technical discussion of these issues, couched in the language of differential geometry, is provided in [57]. Our previous model [43] revealed that active bending, active contraction, and active expansion are necessary for the early stages of inversion (Fig 6C). The relation between these processes remained unclear, however, and the model could not describe the large deformations during later stages of inversion (Fig 6D). Indeed, that model was derived under the assumption of small strains. While the elastic strains are small indeed (since the metric tensor, which describes the deformed shape, is close to the intrinsic tensor defined by the cell shape changes), the geometric strains are large: both the metric tensor of the deformed shell and the intrinsic tensor differ considerably from the metric tensor of the undeformed sphere. We must therefore generalise our previous mathematical model by combining ideas from morphoelasticity and shell theories (Materials and methods) in order to obtain a quantitative description of the entire inversion process. The cell shape changes (Fig 6B and 6C; see also Fig 1C–1F and Fig 2) observed previously in [44] suggest simple functional forms of the intrinsic stretches and curvatures defined in terms of 15 parameters (Fig 6D–6F) that vary over the course of inversion: the parameters f1, …, f5 and κ1, …, κ5 encode the magnitudes of the intrinsic stretches and curvatures of the different cell types that arise in different positions of the cell sheet at different times during inversion, while the parameters s1 <…< s5 encode the arclength positions of the transitions between cell types. These 15 parameters allow for a minimal representation of the cell shape changes [44] and ensuing variations of the intrinsic stretches and curvatures: We proceed to fit the generalised elastic model to the experimental average embryo shapes (Materials and methods). The fitting algorithm compares experimental and numerical embryo shapes to obtain values of the 15 parameters described above for each stage of inversion. In the model, we impose a larger extent of the phialopore than in the biological system, in which the phialopore is initially very small (Fig 4A). This is an important simplification to deal with the discrete nature of the few cells that meet up at the phialopore. In spite of this simplification, the model captures the various stages of inversion (Fig 7); fitted numerical values of the 15 parameters are given in S1 Data. This supports our interpretation of the observed cell shape changes (Fig 6B and 6C, Fig 2) and their functions. At this stage, we are finally set up to relate the spatial structure of the shape variations to the mechanisms and mechanics of inversion. This structure of the local shape variations results from variations of the underlying cell shape changes, via the mechanics of inversion, and from geometric effects associated with averaging the shapes. Some of the structure observed in Fig 4 is clearly geometric: since the shapes are aligned so that the positions of their centres of mass along the axis coincide (Materials and methods), the shape variations accumulate and are thus expected to, e. g. , increase in the anterior hemisphere, towards the phialopore, as at the stage in Fig 4C. At the same stage, however, the shape variation is smaller in the bend region than in the adjacent anterior fold. Both of these regions are, however, close to the centre of mass, and so we do not expect this difference to arise from mere geometric accumulation of shape variations. We must therefore ask: can this global structure arise purely mechanically (i. e. , from a uniform variability of the local parameters implementing the changes of intrinsic curvatures and stretches so that each parameter varies by the same relative amount), but possibly as a statistical fluke, or must there be some regulation (i. e. , nonuniform variability of these local intrinsic parameters)?
Our answers to the questions of developmental regulation that we have raised here in the context of Volvox inversion have so far been either negative (i. e. , ruling out certain mechanisms of regulation) or of what one might term the Occam' s razor variety (i. e. , invoking the law of parsimony to find the simplest modification of the model that can explain the observations). This approach of testing falsifiable hypotheses [68] has the advantage of mitigating the risk of drawing teleological conclusions. However, a fuller answer to the questions above requires estimation of the variability of all the model parameters from the experimental data. Solving this full inverse problem would provide a firmer grip on the relatively large number of fitting parameters required to reproduce the experimental observations, yet that endeavour entails significant statistical, computational, and experimental difficulties: to quantify the full range of variability, one would need a very much larger number of experimental samples to estimate the experimental distribution; additionally, for each step of the optimisation algorithm used to estimate the large number of variability parameters, a large number of computational samples would have to be computed to estimate the distribution under the model. Similar difficulties arise when estimating the variability allowed mechanically. While we have previously noted [56] that the dynamic data for type-B inversion suggest that invagination proceeds without a' snap-through' bifurcation, there is no general requirement for individual developmental paths to lie on one and the same side of a mechanical bifurcation boundary. This poses an additional challenge for modelling approaches. Coupling the deformations described by the elastic model to the signalling processes that underlie its intrinsic deformations is a further challenge for these models. Feedback loops relating the diffusion of contractility-inducing' mechanogens' that are degraded by the resulting strain have been studied theoretically [69] and were coupled to a differential-tension model of a discrete epithelium [70] in more recent work [71]. After this discussion of general challenges for a mechanobiological analysis of morphogenesis and its regulation, we mention some of the remaining questions specific to Volvox inversion: it remains unclear what triggers the initial cell shape changes, what determines their location, and what kind of signal drives the propagation of waves of cell shape changes. It seems likely that the cytoplasmic bridges play a role in chemical or mechanical signal transduction. It is curious that inversion starts at the equator in type-B inversion but starts at the phialopore in type-A inversion. It is not known whether there are patterning mechanisms in Volvox that predetermine the spatial distribution of specific cell shape changes. It is unlikely that morphogens known from animals are conserved in Volvox, but plant hormones have been suggested to act as morphogens in photosynthetic organisms [72]. Alternatively, the position of the bend region could be determined by mechanical and/or chemical cues right at the start of inversion. Interestingly, inversion is preceded by temporary local dents in both embryo hemispheres [44]. One could speculate that this' denting' plays a role in determining the location of the bend region. Once triggered, a wave of cell shape changes could be propagated either by mechano-sensing and/or a chemical signal. Calcium waves, for example, are known to play a role in plant development and can be triggered by mechanical stimuli [73]. In Chlamydomonas reinhardtii, a close relative of Volvox, calcium signalling plays a role, for instance, in the flagellar response [74]. Moreover, in the type-A inverter V. carteri, cells mechanically released from preinversion embryos undergo shape changes prematurely [38,45], which could either indicate the absence of a chemical repressor [31] or that the act of isolating the cells serves itself as a mechanical cue. Combined molecular and physical approaches will be needed to address these questions. Moreover, our model does not resolve the details of the phialopore and hence does not describe the closure of the phialopore at the end of inversion, which remains a combined challenge for experiment and theory: as discussed above, the cytoplasmic bridges elongate drastically at the phialopore [44], and confocal imaging has revealed the possibility of rearrangements within the cell sheet at the phialopore. Do some cytoplasmic bridges rend to make such rearrangements possible, or are some cells next to phialopore not connected to all of their neighbour, as in type-A inversion? Understanding the details of the opening of the phialopore may also require answering more fundamental questions, the answer to which has remained elusive [38,54]: what subcellular structures apart from endoplasmic reticulum [75] are located within the cytoplasmic bridges? How is it possible for them to stretch to such an extent? At the theoretical level, rearrangements of cells near the phialopore raise more fundamental questions of morphoelasticity [57]: in particular, how does one describe the evolution of the boundary of the manifold underlying the elastic description? Cytoplasmic bridges rending next to the phialopore would lead to the formation of lips similar to those seen in type-A inversion [45,46]. Is there a simple theory to describe the elasticity of this nonaxisymmetric setup? We note in passing that the curling of the membranes of red blood cells upon malaria parasite egress [76] leads to shape changes qualitatively similar to the curling of the lips during type-A inversion (albeit at very different scales). These shape changes have been described theoretically by intrinsic membrane curvature [77,78]. At the close of this discussion, it is meet to briefly dwell on questions of more evolutionary flavour: all genera of Volvocaceae and its sister group Goniaceae—with the exception of the single genus Astrephomene [79]—display some form of inversion [42]. There is a general trend among these genera for complexity of inversion to increase with cell number, enabling comparative studies of the evolution of this complexity [49]. The simplest inversion occurs in Gonium [48]: as cells uniformly change their shape, the initially bowl-shaped, convex embryos become concave. Increases of this complexity may appear in different guises: certain cell shape changes may arise only in part of the cell sheet, as in Pleodorina [49], or cell shape changes may proceed in a wave, as exemplified by type-A inversion in Volvox [45]. The separate regulation of different processes and heterochrony in type-B inversion described here may be a prototype of an additional trait of the evolution of multicellularity that can be studied in the volvocine algae: the transition between cell sheet deformations driven by a single process and those resulting from two separate processes. This complements the similarly prototypical transition from organisms with one cell type to organisms with two cell types associated with germ-soma differentiation in the volvocine algae [80]. The question how the different species of the polyphyletic genus Volvox [81] evolved different ways of turning themselves inside out remains, however. Phylogenetic studies of the volvocine algae show that different inversion types evolved several times independently in different lineages [46,82]. Additionally, Pocock [83] reported that in V. rousseletii and V. capensis, inversion type depends on the (sexual or asexual) reproduction mode. This may be a manifestation of the poorly understood role of environmental and evolutionary cues in morphogenesis [84], but such cues remain subject to the mechanical constraints on the respective tissue.
Wild-type strain V. globator Linné (SAG 199. 80) was obtained from the Culture Collection of Algae at the University of Göttingen, Germany [85], and cultured as previously described [86] with a cycle of 16 h light at 24°C and 8 h dark at 22°C. From the traced cell sheet outlines, anterior–posterior axes of the embryos were determined as follows: for shapes for which the bend region was visible on either side of the cross-section, the embryo axis was defined to be the line through the centre of mass of the shape that is perpendicular to the common tangent to the two bend regions (the apex line). Shapes were then rotated and translated manually so that their axes coincided. Since embryos do not rotate much before the flagella grow, the orientation of the axes of the earliest traces (for which the bend regions are not apparent) were taken to be the same as that of the earliest timepoint for which two bend regions were visible. The intersection of the embryo trace and axis defines the posterior pole. After manually recentring some embryos with more pronounced asymmetry, embryos were halved to obtain N = 22 embryo halves. We consider a spherical shell of radius R and uniform thickness h ≪ R (Fig 16A), characterised by its arclength s and distance from the axis of revolution ρ (s), to which correspond arclength S (s) and distance from the axis of revolution r (s) in the axisymmetric deformed configuration (Fig 16B). We define the meridional and circumferential stretches fs (s) =dSds, fϕ (s) =r (s) ρ (s). (6) The position vector of a point on the midsurface of the deformed shell is thus r (s, ϕ) =r (s) ur (ϕ) +z (s) uz, (7) in a right-handed set of axes (ur, uϕ, uz), and so the tangent vectors to the deformed midsurface are es=r′ur+z′uz, eϕ=ruϕ, (8) where dashes denote differentiation with respect to s. By definition, r′2+z′2=fs2, and so we may write r′=fscosβ, z′=fssinβ. (9) Hence, the normal to the deformed midsurface is n=r′uz−z′urfs=cosβuz−sinβur. (10) We now make the Kirchhoff' hypothesis' [58] that the normals to the undeformed midsurface remain normal to the deformed midsurface (Fig 16C). Taking a coordinate ζ across the thickness h of the undeformed shell, the position vector of a general point in the shell is r (s, ϕ, ζ) =rur+zuz+ζn= (r−ζsinβ) ur+ (z+ζcosβ) uz. (11) The tangent vectors to the shell are thus es=fs (1−κsζ) (cosβur+sinβuz), eϕ=ρfϕ (1−κϕζ) uϕ, (12) where κs = β′/fs and κϕ = sin β/r are the curvatures of the deformed midsurface. The metric of the deformed shell under the Kirchhoff hypothesis accordingly takes the form dr2=fs2 (1−κsζ) 2ds2+fϕ2 (1−κϕζ) 2ρ2dϕ2. (13) The geometric and intrinsic deformation gradient tensors are thus Fg= (fs (1−κsζ) 00fϕ (1−κϕζ) ), F0= (fs0 (1−κs0ζ) 00fϕ0 (1−κϕ0ζ) ), (14) where fs0, fϕ0 and κs0, κϕ0 are the intrinsic stretches and curvatures of the shell. Thence, invoking the standard multiplicative decomposition of morphoelasticity [57], the elastic deformation gradient tensor is F=Fg (F0) −1= (fs (1−κsζ) fs0 (1−κs0ζ) 00fϕ (1−κϕζ) fϕ0 (1−κϕ0ζ) ). (15) While we do not make any assumption about the geometric or intrinsic strains derived from Fg and F0, respectively, we assume that the elastic strains derived from F remain small; we may thus approximate εss≈fs (1−κsζ) fs0 (1−κs0ζ) −1, εϕϕ≈fs (1−κsζ) fs0 (1−κs0ζ) −1, (16) with the off-diagonal elements vanishing, εsϕ = εϕs = 0. For a hookean material with elastic modulus E and Poisson' s ratio ν [58,59], the elastic energy density (per unit extent in the meridional direction) is found by integrating across the thickness of the shell: E2πρ=E2 (1−ν2) ∫−h/2h/2 (εss2+εϕϕ2+2νεssεϕϕ) dζ=Eh2 (1−ν2) { (1+h24 (κs0) 2) Es2+ (1+h24 (κϕ0) 2) Eϕ2+2ν (1+h212 ( (κs0) 2+κs0κϕ0+ (κϕ0) 2) ) EsEϕ}+Eh324 (1−ν2) {Ks2+Kϕ2+2νKsKϕ−4κs0EsKs−4κϕ0EϕKϕ−2ν (κs0+κϕ0) (EϕKs+EsKϕ) }, (17) where we have expanded the energy up to third order in the thickness and where we have defined the shell strains and curvature strains Es=fs−fs0fs0, Eϕ=fϕ−fϕ0fϕ0, Ks=fsκs−fs0κs0fs0, Kϕ=fϕκϕ−fϕ0κϕ0fϕ0. (18) As in our previous work [43,56], the elastic modulus is an overall constant that ensures that E has units of energy but does not otherwise affect the shapes. This property of the model enables us to neglect global variations of the elastic modulus between different embryos. We make the additional assumption that the elastic modulus does not vary locally within embryos. We shall also assume that ν = 1/2 for incompressible biological material; the cell size measurements of [62] for type-A inversion in V. carteri support this assumption qualitatively. (These considerations also explain why we do not perturb these mechanical parameters in our analysis of the shape variations.) We finally set h/R = 0. 15 as in our previous work. For the purpose of fitting the model to the observed average shapes, we fit values of the 15 parameters f1, …, f5, κ1, …, κ5, s1, …, s5 defined in Fig 6D–6F. The other geometrical parameter of the shell, the angular extent P of the phialopore, is not fitted for. We arbitrarily set P = 0. 3. The reasons for this simplification are discussed in the main text. We do not fit either for the distance Δs over which we regularise the functional form of κϕ0 (Fig 6F), since we lack information about the cell shape changes that define it. We arbitrarily set Δs = 0. 05. Numerical shapes were fitted to the average shapes by distributing M = 100 points uniformly along the arclength of the numerical and average shapes and minimising a euclidean distance between them using the Matlab (The MathWorks) routine fminsearch, modified to incorporate the variant of the Nelder–Mead algorithm of [92] as well as a modified shrinking step of the Nelder–Mead simplex. A custom-written adaptive stepper was used to move about in parameter space and select the initial guess for the Nelder–Mead simplex. For each shape, the fit for the previous stage of inversion was used as the initial guess for the optimisation. Sample code is given in S2 Code. To define perturbations for the F = 15 fitted model parameters P0 ∈ ℝF at noise level δ, we draw independent N uniform random samples X∼U[0,1]F on the unit interval and define the perturbed parameters P = P0 (1 − δ + 2δX) by pointwise multiplication.
Biological noise is unavoidable in—and even necessary for—development. Here, we ask whether this variability can teach us something about the process that underlies it. We show how to access the information hidden in the variability in an analysis of the variability of cell sheet folding in the green alga Volvox globator. Through a combination of light sheet microscopy and mathematical modelling, we show how the inversion process, by which the spherical embryos of Volvox turn themselves inside out, results from two separate mechanisms of bending and stretching (expansion and subsequent contraction). Our analysis therefore uncovers a prototypical transition of developmental complexity in Volvox and the related volvocine algae, from a morphogenetic process driven by a single mechanism to one driven by two separate mechanisms. This complements the similarly prototypical transition from one cell type to two cell types that has made the volvocine algae a model system for the evolution of multicellularity.
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The noisy basis of morphogenesis: Mechanisms and mechanics of cell sheet folding inferred from developmental variability