The noisy basis of morphogenesis: Mechanisms and mechanics of cell sheet folding inferred from developmental variability

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.

duration of inversion, but also the relative timing of parts of it. We additionally note that there is no correlation between the size of an embryo and the duration of its inversion (Fig. A1d), not even between embryos from the same parent spheroid. It is natural to ask to what extent the different deformations of inversion must arise in a particular order: while invagination occurs before phialopore opening in all our samples, analysis of characteristic 'checkpoints' of inversion ( Fig. A2) reveals that there is still considerable leeway in the timing of posterior inversion and phialopore opening. This is the same non-linearity that we inferred more generally in Fig. 5a from the global shapes, with some embryos lingering in some stages of inversion.
In our previous work [43], we discussed in detail three geometric descriptors of the traced embryo outlines, which we have reproduced for the present dataset: 1. the distance e (  Fig. A3, normalised by the total duration of inversion from appearance of the bend region until closure of the phialopore. a: first measurement of posterior-to-bend distance e (when the tangent at the point of the most negative curvature κ * is horizontal); b: maximal negative curvature κ * ; c: maximal surface area A; d: e reaches half of its initial value; e: phialopore has widened to 20% of its maximal diameter, f: e reaches 10% of its initial value; g: phialopore reaches its maximal diameter; h: phialopore has shrunk to 20% of its maximal diameter. Posterior inversion (characteristic points a,d,f) is shown in red, and anterior inversion (characteristic points e,g,h) is shown in blue. The purple regions indicate an overlap of posterior and anterior inversion. See S1 Data for numerical values.
2. the embryonic surface area A (Fig. A3b), which was computed by determining a surface of revolution from each half of the midsagittal slice and averaging the two values for each timepoint; 3. the minimal (most negative) value κ * of the meridional curvature in the bend region (Fig. A3c).
We have computed three additional descriptors associated with the progress of later of inversion: 4. the diameter d of the phialopore (Fig. A3d) as an indicator of progress of inversion of the anterior hemisphere; 5. the width w of the bend region (Fig. A3e), where the bend region is defined as the region of negative curvature; 6. the position of the bend region (Fig. A3f), measured along the arclength of the deformed shell from the posterior pole to the midpoint of the bend region.
From the aligned shapes, these geometric descriptors were computed as follows: the posterior-to-bend distance e was computed as the distance from the apex line to the posterior pole. The maximal surface area A max and the most negative value of curvature κ * in the bend region were computed as described previously [43]; traces were smoothed before computing the curvature. The phialopore width d was computed as the absolute distance between the two ends of a complete embryo trace. The bend region was defined as the region of negative curvature; the distance between the first and last points of negative curvature defined the bend region width. The bend region position is defined by While the posterior hemisphere moves into the anterior hemisphere (Fig. A3a), the surface area increases considerably due to stretching in the anterior hemisphere [43], in most embryos before the phialopore begins to widen (Fig. A3b,d). While the negative curvature in the bend region increases, the meridional width of the bend region widens (Fig. A3c,e) and the distance of its midpoint to the posterior pole decreases (Fig. A3f). Comparing these geometric descriptors for the experimental averages and the fitted shapes of Fig. 7, we find a good agreement (Fig. A4), although we notice that the fitted shapes underestimate the width of the bend region. Because curvature is a second derivative of shape, it is not surprising that larger differences arise in the minimal bend region S1 Text -4 curvature of the average and fitted shapes (Fig. A4).