Actomyosin-based tissue folding requires a multicellular myosin gradient

Actomyosin-associated folding of epithelial sheets is a common mode of morphogenesis during the development of complex organs and organisms. Defects in these folding events can lead to congenital birth defects, including spina bifida, which can be caused by failure of the neural plate to fold and fuse (Copp and Greene, 2010; Wallingford et al., 2013). Actomyosin-based epithelial folding is often associated with apical constriction, a process in which apical actomyosin contractility causes columnar epithelial cells to adopt a wedge shape by reducing their apical cell surface area (Martin and Goldstein, 2014). One unanswered question is how apical contractility must be spatially patterned in a tissue to generate three-dimensional (3D) form.

Drosophila gastrulation is a classic example of tissue folding in response to apical constriction. Cells on the ventral side of the embryo fold into the embryo as one of the first tissue rearrangements during development. The domain of invaginating cells is specified by two embryonic transcription factors, Twist and Snail (Leptin and Grunewald, 1990; Thisse et al., 1987). At the time of gastrulation, snail expression extends nine cells from the ventral midline (VM) (to form an 18-cell-wide domain) (Ip et al., 1992). twist expression extends a few cells further than snail (Leptin, 1991). Both genes are initially expressed in a narrower domain of cells that expands over time (Leptin, 1991). Expression of both twist and snail requires the maternal transcription factor Dorsal. twist is necessary for persistent apical constriction and non-muscle myosin 2 (myosin) accumulation (Mason et al., 2016; Xie and Martin, 2015). Two transcriptional targets of Twist appear to act in parallel to regulate actomyosin contractility in the ventral furrow: folded gastrulation (fog) and T48. Fog is a secreted protein that activates a maternally provided G_α12/13, concertina, through a G protein-coupled receptor (GPCR) pathway (Costa et al., 1994; Kerridge et al., 2016; Manning et al., 2013; Parks and Wieschaus, 1991; Xie et al., 2016). T48 is a transmembrane protein that contains a PDZ-binding consensus motif (Kolsch et al., 2007). T48 and G_α12/13 activation at the apical surface appear to recruit and activate RhoGEF2, a Rho guanine nucleotide exchange factor with a PDZ domain and a regulator of G protein signaling (RGS) domain (Barrett et al., 1997; Hacker and Perrimon, 1998). There, active RhoGEF2 can activate RhoA (Rho1 in flies), which, in turn, promotes apical actomyosin contractility (Kolsch et al., 2007; Mason et al., 2016). The effector of RhoA, Rho kinase (ROCK; Rok), is required for apical myosin accumulation in the ventral furrow (Dawes-Hoang et al., 2005).

As the tissue folds, there is significant variation in the extent and timing of apical constriction along the ventral-lateral axis of the furrow (Leptin and Grunewald, 1990; Oda et al., 1998; Sweeton et al., 1991) (Fig. 1A). Specifically, cells closer to the VM constrict earlier and to a greater extent than cells farther from the VM, despite the fact that all cells express twist prior to constriction (Leptin, 1991). The Twist target fog is transcribed in a subset of ventral cells that extends six cells from the VM (Costa et al., 1994); this region corresponds to the region of earliest constriction (Sweeton et al., 1991). Recently, it was shown that expression of the Twist transcriptional targets fog and T48 occurs in a graded manner along the ventral-lateral axis (Lim et al., 2017). The intensity profile of myosin during gastrulation has been illustrated at the tissue level, with highest myosin concentrations at the VM (Lim et al., 2017; Spahn and Reuter, 2013). However, whether there are cell-to-cell differences in transcription and active myosin levels and how patterns of transcription and contractility relate to each other is unknown. Most importantly, it is not known whether the variation in apical constriction/contractility is relevant to tissue folding.

Here, we demonstrate that there is a gradient in myosin contractility across the ventral furrow. This gradient starts two to three cells from the VM and extends to approximately six cells from the VM. In this region, two to six cells from the VM, each subsequent cell has lower levels of active myosin. This contractility gradient originates from the dorsal morphogen gradient, and perturbation of the dorsal morphogen gradient changes the spatial patterning of contractility. Our 3D model of the gastrulating embryo predicts the importance of contractility gradients in generating a tissue fold. Our experimental data validated a prediction of the model: tissue bending was associated with contractile gradients, but not absolute levels of contractility.

To determine how tissue-scale contractility is organized in the ventral furrow, we imaged embryos with labeled myosin (Sqh::GFP) and membrane (Gap43::mCherry) (Martin et al., 2010; Royou et al., 2002). We segmented all images from time-lapse movies of the folding process and partitioned cells into bins based on the initial distance of the cell centroid from the VM (see example in Fig. 1A). As previously observed (Jodoin and Martin, 2016), cells do not intercalate during furrow formation, and cell positions for bins at later time points show the same relative positions as at the initial reference time point (Fig. 1A). Thus, we were able to measure cell apical cross-sectional area over time as a function of relative position from the VM.

In agreement with a previous live-imaging study, which quantified groups of cells (Oda et al., 1998), we found that apical area reduction was not uniform along the ventral-lateral axis. Prior to the onset of constriction, all cells along the ventral-lateral axis had an apical area of ∼40 µm² (Fig. 1B, blue curves; Fig. S1A,B, blue curves). Over time, cells within four cells of the VM reduced their apical area and cells farther than five cells from the VM expanded their apical area (Fig. 1B, blue to yellow curves; Fig. S1A,B, blue to yellow curves). At late time points, the apical area distributions for the two cells adjacent to the VM were not statistically different, but each subsequent cell from the VM had significantly larger apical area until six cells from the VM (Fig. 1C,D; Fig. S1C,D). We refer to this pattern in the resulting apical area as a gradient in apical constriction. Differences in cell area were not seen when cells were binned based on their position along the orthogonal anterior-posterior axis of the embryo, demonstrating that the apical constriction gradient occurs mainly along the ventral-lateral axis (Fig. S2).

To measure the levels of ‘active’ myosin in the cell apex, we have used an image-processing method to capture activated, cortical myosin molecules and exclude cytoplasmic, ‘inactive’ myosin (see Materials and Methods). Several arguments support the fact that we have measured active myosin: (1) Drosophila myosin 2 can exist either as a single molecule in an inactive, folded conformation or as an active bipolar filament consisting of 12 molecules, depending on phosphorylation of its regulatory light chain (Vasquez et al., 2016), (2) inhibition of ROCK, a kinase that phosphorylates and activates myosin filament formation and motor activity, results in the immediate dissolution (within 10-15 s) of the intense cortical myosin structures that we measure (Coravos and Martin, 2016), and (3) the amount of myosin that we measured using this method correlates well with apical constriction in single cells (Martin et al., 2009; Xie and Martin, 2015). For all of these reasons, we conclude that the intensity of the cortical myosin structures measured in this study represents active myosin.

We found that there was a multicellular gradient in active myosin that accompanied ventral furrow formation. Specifically, active myosin intensity distributions were highest on average in the two cells adjacent to the VM (Fig. 1E-G; Fig. S1E-H). The average amount of active myosin in the third cell from the VM was often lower, but this was not always statistically significant (Fig. 1E-G; Fig. S1G-J). Beginning three cells from the VM, every subsequent cell position consistently exhibited a significantly lower active myosin distribution than the one before (Fig. 1F,G; Fig. S1G-J). Thus, there is a multicellular gradient in myosin activity that starts two to three cells from the VM and extends to six cells from the VM. We refer to this as a ventral-lateral contractility gradient.

To determine the source of the active myosin gradient, we examined an upstream component of the signaling pathway that regulates ventral furrow formation, the integral membrane protein T48 (Fig. 2A). We engineered a T48 allele with an internal GFP tag. The GFP::T48 fusion allele was expressed from the endogenous T48 promoter using bacterial artificial chromosome-mediated recombineering (Venken et al., 2006). GFP::T48 was visualized in fixed embryos stained with a GFP antibody (Fig. 2B). We segmented the image based on phalloidin staining and binned cells with respect to their position from the VM (Fig. 2C; Fig. S3A). We found that average signal density of GFP::T48 is highest in cells closest to the VM (Fig. 2D; Fig. S3B). Similar to the distribution of active myosin, a difference in T48 levels first appears between cells 3 and 4 from the VM. The gradient in T48 seems to extend slightly further than the gradient in myosin (encompassing seven to eight cells on either side of the midline) (Fig. 2D,E; Fig. S3B,C). The fact that there is a T48 gradient is consistent with the myosin gradient resulting from upstream signals.

The measured gradient in T48::GFP signal density (a.u. per µm²) could reflect the gradient in apical constriction, rather than a gradient in T48 gene expression. As cells apically constrict, T48::GFP on apical membranes could be concentrated. To test this, we measured the total amount of T48 at the apical surface of a cell. We found that total signal is strongly correlated with cell area, possibly due to a low signal-to-noise ratio (Figs 2F and S3D). However, for cells of a given area, cells closest to the VM have a higher level of total T48::GFP signal, suggesting that differential expression of T48 accounts for at least some of the difference in signal density (Figs 2F and S3D, blue versus yellow points). This is supported by recent work measuring T48 expression in the ventral domain that found that T48 and fog, another Twist target, are expressed earlier in cells closest to the VM (Lim et al., 2017). Our data demonstrates that between three and seven cells from the VM there are differences in both T48 density and overall levels, in a pattern that mirrors active myosin.

It has been previously shown that Dorsal, a maternal morphogen that is required for twist transcription, is active in a gradient along the ventral-lateral axis with highest nuclear levels in the most ventral cells (Kanodia et al., 2009; Reeves et al., 2012). twist expression levels have previously been described as being uniform across the central region of the ventral furrow with a gradient of twist at the edge of the furrow, where cells stretch (Leptin, 1991). We measured twist mRNA levels at the beginning of nuclear cycle 14, about an hour before gastrulation, using quantitative fluorescence in situ hybridization (FISH). We found that twist mRNA was detected as small puncta in the cytoplasm of ventral cells (Fig. 3A). We measured twist mRNA levels by quantifying the mean intensity of the twist FISH signal in the cytoplasm adjacent to each nucleus (McHale et al., 2011). At early nuclear cycle 14, twist is expressed in a gradient around the VM. The gradient begins two to three cells from the VM and extends to about 12 cells from the VM (Fig. 3B,C). Early differences in twist levels are consistent with, and could explain, the later differences in transcription and downstream signaling measured in this paper and previously (Costa et al., 1994). Time delays due to transcription of target genes and the translation of Twist and target genes could explain the delay between the measurement of twist mRNA and the downstream gradients. Thus, there is an early, transient gradient in twist mRNA levels that could explain the subsequent gradient in apical T48 levels, myosin activity, and contractility.

To understand the mechanisms underlying furrow formation by contractility gradients, we studied a 3D continuum model of the Drosophila embryo during gastrulation. At this developmental stage, the embryo consists of a thin, single-layer epithelial sheet of approximately ellipsoidal shape. Its mechanics can be described mathematically by an elastic shell (Koiter shell model; Ciarlet, 2000), represented by its middle surface (Fig. 4A); for further explanation see Materials and Methods. Passive mechanical stresses appear whenever the middle-surface geometry deviates from its original shape by local stretching or bending deformations. Active stresses from apical constriction were incorporated into this model by modifying the local preferred curvature of the middle surface (mimicking constriction at the apical surface); these cause active bending moments that drive furrow formation (Fig. 4B). This is analogous to the bending of bimetallic strips, which assume a curved state when one of the two adjacent layers expands less strongly than the other in the presence of a thermal stimulus (Timoshenko, 1925). Similarly, apical contraction is expected to create a preferred tissue curvature. To relate the local myosin activity to active stress generation, we analyzed how apical constriction affects preferred curvature in a vertex-based cell model (Fig. 4B). This allowed us to relate active stresses to a dimensionless parameter M. M measures the relative strength of active and passive stresses and fully determines the induced shape deformation. The parameter M can thus be understood as a dimensionless contractility coefficient that is proportional to the local myosin activity. This theoretical framework allowed us to efficiently study spatial patterns of contractility and compare the predicted 3D shell shapes against our experimental data for wild-type and mutant embryos.

Guided by our myosin activity measurements in wild-type embryos, we approximated the contractility profile as ⁠. Here, φ is the distance from the VM, M₀ denotes the maximum value of M (assumed at φ=0), k is a steepness coefficient characterizing the profile decay, and w is the width of the profile (Fig. 4C). We adjusted k, M₀ and w to fit the distribution of myosin and apical constriction that we observed in wild-type embryos.

To test whether the gradient in contractility was important for folding, we compared furrow formation in the model for increasing values of w (Fig. 4C). For low values of w, we observed furrows that were morphologically similar to those of wild-type embryos (Fig. 4C, left and middle). Increasing w increased the width of the region of contractility and resulted in a region around φ=0 where the contractility profile is flat (Fig. 4C, right). Simulated equilibrium shapes showed that, as the distance between regions of graded contractility increased, the furrow width also increased, leaving a relatively flat surface in the region of uniform contractility (Fig. 4C). Importantly, changes in shell curvature were associated with the position of the gradient in M (Fig. 4D). This was the case regardless of the width of the myosin profile. Changes in shell curvature always occurred near the boundary where there was a gradient in M. In contrast, where there was an extended region of uniform M, curvature was low (Fig. 4D,E).

We next explored how our model responded to changes in M₀ and k, i.e. the level of contractility and steepness of the contractility gradient, respectively. We found that with low contractility M₀, but normal steepness k, our model does not form a furrow (Fig. 5A,B). Additionally, in our model, having high contractility M₀, but a low steepness k for the gradient, also fails to form a furrow and instead results in an hour-glass shaped flattened depression (Fig. 5A,D). Only when both M₀ and k are above a certain threshold will the model form a furrow (Fig. 5A,C). Thus, our model predicts that furrow formation in embryos will fail if either the total activity of active myosin or the steepness of the gradient is too small.

Our 3D model simulations predicted that tissue folding occurs where there is a gradient in contractility. To show that uniform levels of active myosin were insufficient to cause tissue bending, we attempted to flatten the active myosin gradient in vivo using a genetic perturbation. Spn27A is a negative regulator of ventral cell fate in the Drosophila embryo (Fig. 2A). Hypomorphic alleles of Spn27A and depletion of Spn27A using RNA interference (RNAi) have been shown to increase the width of the ventral domain (Ligoxygakis et al., 2003; Chanet et al., 2017). We hypothesized that increasing Dorsal activity could lead to an extended region of uniform Dorsal activity around the VM. We first tested the effect of Spn27A depletion on the upstream gradient in twist mRNA and found that, as expected, twist expression levels were uniform in an expanded region around the VM in Spn27A-RNAi embryos (Fig. S4). We next examined the effect of Spn27A depletion on the contractility gradient, and found that the region of uniform active myosin was broadened (Fig. 6A). Embryos depleted of a control gene (Rh3) exhibited a gradient in active myosin similar to that of wild-type embryos (Fig. S5A-F). In contrast, Spn27A depleted embryos did not exhibit a normal gradient; instead they exhibited uniform active myosin within six cells of the VM (Fig. 6C-E; Fig. S6A-F). Furthermore, the uniform levels of active myosin were associated with uniform apical constriction compared with the normal apical area gradient seen in the control RNAi embryos (compare both Fig. 6F-H and Fig. S6G-J with Fig. S5G-J). Because Spn27A attenuates Dorsal activity, our data suggest that the gradient in twist, active myosin, and apical constriction is the result of the gradient in Dorsal transcriptional activity.

In accordance with the prediction of our 3D continuum mechanical model (Fig. 4C, right), Spn27A-depleted embryos exhibited reduced cross-sectional curvature in the region surrounding the VM, whereas the tissue does curve where there is a gradient of apical myosin activity at the edge of the broadened ventral domain (Figs 7A and S7A,B). This failure to form a sharp furrow was associated with defective/reduced apical constriction, specifically along the ventral-lateral axis (Fig. S7C). This phenotype is consistent with the prediction of our model that increasing the width (w) of the region of contractility results in an expanded furrow, which is flattened at the VM where there is no myosin gradient.

In many cases, epithelial folding is thought to result from myosin accumulating at the site of the fold (Lecuit et al., 2011; Leptin, 2005; Martin and Goldstein, 2014). Here, we show that active myosin accumulation fails to effectively change tissue curvature, except in regions in which neighboring cells or tissue regions are accumulating different amounts of apical, active myosin. This conclusion is based on several pieces of evidence: (1) there is a gradient in mean levels of active myosin, which originates two to three cells from the VM and extends to six cells from the VM (Fig. 7B), (2) there are graded levels of both twist and its transcriptional target GFP::T48 that originate three cells from the VM and extend slightly farther than mean levels of active myosin (Fig. 7B), (3) mechanical modeling of a 3D elastic shell requires a contractility gradient to form a fold that resembles a ventral furrow, and (4) experimentally expanding and flattening the myosin gradient in vivo results in an abnormal tissue shape that is predicted by the mechanical model. Thus, changes in tissue curvature and folding occur at sites of active myosin gradients. In the case of the Drosophila ventral furrow, the gradient in active myosin is the result of a gradient in the activity or levels of upstream transcription factors.

Gradients in the level of contractility, in our case apical contractile myosin, promote curvature by promoting a contractile force imbalance between neighboring regions of cells (Fig. 7C). In the case of the wild-type ventral furrow, this imbalance occurs along the ventral-lateral axis. Cells within two to three rows from the VM, accumulate similar levels of active myosin and constrict the same amount. Despite uniform active myosin in these central four to six cells, it appears that these cells are able to pull the surrounding tissue and effectively constrict. Beyond two to three cells from the VM, each cell exhibits progressively lower apical, active myosin, T48 and twist levels and constricts later and to a lesser extent (Fig. 7B,C).

Furrow formation requires asymmetric constriction (i.e. a wedge shape), such that the cells' apical surface remains stretched along the anterior-posterior axis and constricted along the orthogonal (ventral-lateral) axis (Martin et al., 2010). When mechanically isolated, ventral furrow cells constrict symmetrically, suggesting that the elongated shape of the cell apical surface is the result of differences in forces generated by the surrounding tissue (Martin et al., 2010). We propose that the contractility gradient we measured in this paper allows cells to constrict more along the ventral-lateral axis because of less resistance from the surrounding tissue. This results in wedge-shaped cells and promotes curvature/folding specifically along the ventral-lateral axis (Fig. 7C).

In contrast to wild-type embryos, uniform contractility levels (i.e. in Spn27A-RNAi) increase the resistance to apical constriction from neighboring cells; thus, cells do not constrict enough to generate high curvature (Fig. 7A,D). Consistent with this interpretation, Spn27A-RNAi exhibited a defect in constriction along the ventral-lateral axis (Fig. S7C). Spn27A-RNAi embryos could occasionally fold and internalize the mesoderm, but these folds consistently exhibited a flattened depression and the invagination was often uneven and not temporally smooth.

The in vivo experimental outcomes described above are consistent with our mechanical model. We found that having a contractile gradient around the circumference of an ellipsoid resulted in a sharp, narrow furrow. In contrast, increasing the width of the contractile zone between opposing contractile gradients resulted in a flattening of the tissue at the VM, similar to observations in Spn27A-RNAi embryos. In a systematic parameter scan of our model, the ventral furrow fails to develop if either contractility or the steepness of the gradient is too small. Thus, it is not simply absolute apical myosin activity that is required for shape, but active myosin differentials between neighboring regions or cells.

Our mechanical model differs from previous models of Drosophila gastrulation. It represents the first 3D continuum model of ventral furrow formation in which apical constriction is the only active force input into the system (Conte et al., 2009; Conte et al., 2008; Odell et al., 1981; Polyakov et al., 2014; Spahn and Reuter, 2013). In particular, our theoretical analysis implies the role of active myosin gradients in tissue bending. Spahn and Reuter explored the role of a gradient in contractility on the pattern of cellular constriction, but their planar model did not reveal the importance of the gradient to folding (Spahn and Reuter, 2013). Conte and colleagues presented a 3D model of Drosophila gastrulation, but needed to introduce ectodermal pushing to generate folding (Conte et al., 2008). Our results suggest that the embryo does not require ectodermal pushing if there is a contractility gradient.

We note that contractility conditions for successful furrow formation are intrinsic 3D effects not captured by 2D cross-section models. In particular, 2D models assume spatial invariance along the anterior-posterior axis and thus can be thought of as a model of furrow formation on an infinite cylinder. A cylinder has vanishing Gaussian curvature and axially invariant deformations introduce bending, but no stretching deformations. By contrast, furrow formation on an ellipsoid requires stretching of the membrane due to positive Gaussian curvature. The high energetic cost of stretching thus presents an effective energy barrier against such deformations, an important mechanical effect neglected in 2D models. Therefore, 2D models might significantly underestimate the conditions required for the furrow to form. The present 3D analysis underscores the mechanical importance of contractility gradients for the robust formation of the ventral furrow.

There are some limitations to our particular modeling approach. The most important is that the height of the elastic shell in our model is constant. This is not the case in vivo, where apical constriction initially results in cells lengthening along their apical-basal axis, followed by shortening along same axis (Sweeton et al., 1991). This final shortening step is associated with the generation of the highest curvature (Polyakov et al., 2014). Given that our model successfully predicts several experimental observations, the concept that a given magnitude of apical contractility generates a local preferred curvature of the surface of a shell provides a useful framework with which to understand tissue folding. The final tissue curvature will also be governed by the extent to which epithelial cells lengthen (which minimizes curvature) versus expanding their basal domain (which promotes curvature). A future experimental challenge will be to understand the cell biological and physical mechanisms that control epithelial cell height during this process. Regardless of the specific mechanisms that control cell height, the spatial pattern of contractility will play a central role in governing final tissue shape.

Because of our ability to resolve cell shape and active myosin at cellular resolution, we have been able to show that there is a gradient in apical myosin that starts three cells from the VM and extends six cells distal to the VM. Thus, a four-cell wide domain exhibits a contractility gradient, with differences in contractility existing between individual rows of cells with given ventral-lateral positions (Fig. 7B). Interestingly, the domain with uniform active myosin levels is about the width of a hinge point associated with mouse neural tube closure (Smith et al., 1994). The gradient we measured depends on a gradient in twist and probably also T48. This was the basis for our mechanical model of folding. Importantly, we show that flattening of this gradient leads to defects in tissue shape (i.e. reduced tissue curvature at the VM).

Other work also supports the importance of an active myosin gradient. Mutation of a possible attenuator of Fog signaling, Gprk2, results in a broader activation of myosin (Fuse et al., 2013). Interestingly, the Gprk2 mutant exhibited delayed folding at the VM and instead only displayed changes in curvature at the edges of the myosin domain (Fuse et al., 2013). The Gprk2 mutant phenotype resembles that of Spn27A-RNAi. Thus, multiple mutant phenotypes are consistent with the requirement of an active myosin gradient to promote tissue curvature, which reinforces our conclusions.

Fly lines and crosses are listed in Table S1. Embryos were collected from cages at 25°C except where noted.

GFP was inserted in the T48 coding sequence at amino acid residue 85, with a standard linker (amino acids GGSG) added before and after the GFP. The T48::GFP construct was inserted in the bacterial artificial chromosome (BAC) (CH321-58L17) using standard recombineering techniques (Venken et al., 2006; Warming et al., 2005). We injected nanos-integrase; attp40; vk33 embryos with the purified BAC and selected for transgenic flies using white expression as a marker. Primers for T48 transgenic were as follows: 5′ upstream homology arm: 5′-ATGCGGTCGGTGGAAGTGGC-3′; 3′ upstream homology arm: 5′-GCTCACGCCCGAGCCGCCCATGCCGGGCAGCACCAC-3′; 5′ GFP with linker: 5′-GGCGGCTCGGGCGTGAGCAAGGGCGAGGAGCTGTTCACCGG-3′; 3′ GFP with linker: 5′-GCCCGAGCCGCCCTTGTACAGCTCGTCCATGCCGAGAGTG-3′; 5′ downstream homology arm: 5′-GTACAAGGGCGGCTCGGGCCTGCAAAATTCAGGTGGG-3′; 3′ downstream homology arm: 5′-CACACGCTTTATTTGGGGCTC-3′.

Embryos were prepared for live imaging as previously described (Martin et al., 2009). Images were acquired on a Zeiss 710 microscope with an Apochromat 40×/1.2 numerical aperture W Korr M27 objective. Pinhole size: 2.2 or 2.4 airy units. A 488/561 beam splitter was used. Channels were excited (argon 488 laser, 2.6% or 4.5%; DPSS 561-10, 2% or 2.8%) and acquired (detected range: 493-561 nm and 599-696 nm) concurrently.

Immuno- and phalloidin staining was performed using standard methods (Martin et al., 2009). GFP antibody (rabbit, 1:500; Abcam, ab290) and AlexaFluor 647 phalloidin (Invitrogen) were used. All imaging was carried out on a Zeiss 710 confocal microscope with a Plan-Apochromat 63×/1.4 numerical aperture Oil DIC M27 objective. Pinhole size: 1 airy unit. Channel 1: DPSS 651-10 laser (1%), detector range: 574-712 nm. Channel 2: argon 488 laser (2%), detector range: 493-574 nm.

For quantitative FISH, embryos were fixed (8% paraformaldehyde, 45 mins) and stained as described by Little et al. (2011). Probes complementary to the twist open reading frame were conjugated to AlexaFluor 594 (Life Technologies). DAPI was used to detect nuclei. Embryos were mounted in Aqua Poly/Mount (Polyscience). All imaging was carried out on a Zeiss 710 confocal microscope with a Plan-Apochromat 63×/1.4 numerical aperture Oil DIC M27 objective. Channels were acquired on separate tracks. Pinhole size: 1 airy unit. Track 1: Diode 405-30 (0.5%) detector range: 410-582 nm. Track 2: HeNe594 (15%) detector range 602-734 nm. Each line was scanned four times and the sum recorded.

All image analysis was performed in Fiji (http://fiji.sc) (Schindelin et al., 2012) and MATLAB (MathWorks). Custom software for image processing is available upon request.

First, cytoplasmic background signal (defined as the mean cytoplasmic signal plus 2.5 standard deviations) was subtracted from the myosin channel (Martin et al., 2009; Vasquez et al., 2014). This subtraction left only the concentrated apical signal that evidence suggests is active myosin. The maximum myosin signal intensity in the z-plane was used to generate a rough map of the embryo surface. A Fourier transform was used to generate a smooth continuous surface. Myosin signal was averaged over the 4 µm above the surface and membrane signal was the sum of the signal from 1-2 µm below the surface. (Membrane signal was only used for segmentation, not quantification.)

T48 signal was quantified in a single apical slice from a 3D image. Fixed embryos were flattened during mounting so the surface projections used in live embryos were not necessary. The T48 signal was not background subtracted.

All live and immunostained images were then segmented using an existing MATLAB package, Embryo Development Geometry Explorer (EDGE) (Gelbart et al., 2012). Membrane signal (Gap43::mCherry) or cortical actin (phalloidin) were used to detect cell boundaries and track cells in time for live images. Errors in segmentation were corrected manually. Our segmentation algorithm was used to determine centroid position, cell diameter, cell area and total myosin or T48 signal intensity within a cell.

FISH signal was performed using a previously published software package, which partitions the cytoplasm into ‘cells’ based on nuclear position and subtracts the nuclear volume from measurements of average cytoplasmic signal per ‘cell’ (McHale et al., 2011). This package was used to determine cell centroid position for each cell, average cytoplasmic signal per pixel for each cell, and average cell diameter for the image.

For all image quantifications, data was aggregated into ‘cell bins’. Cells were assigned to bins based on the ventral-lateral position of the cell centroid. We first identified the approximate position of the VM in all images. In live images, the VM was defined as the position at which the furrow closes over (Figs 1 and 6; Figs S1, S5 and S6). In fixed images, the signal was first confirmed to be symmetric, and the position of the VM was determined by symmetry of the signal (Figs 2 and 3; Figs S3 and S4). We were unable to determine the precise VM position in Spn27A-RNAi embryos stained for twist mRNA because the boundaries of the domain expressing twist mRNA extended beyond the field of view. In this case (Fig. S3), we did not assume symmetry of the signal, and simply determined the distribution of twist mRNA in a ventral region. When assigning cells to bins based on centroid position, live images were binned based on initial position of the cell centroid before tissue contraction and folding and the boundaries of the bins were set by the average cell diameter along the ventral-lateral axis (Figs 1 and 6; Figs S1, S5 and S6). For example, cell bin 1 includes all cells with a cell centroid that fell within one average cell diameter of the VM. For fixed images in which cells had already begun to constrict, the width of each bin was set manually to approximate the width of cells at that ventral-lateral position (Fig. 2; Fig. S3). For fixed images in which cells had yet to constrict, the boundaries of each bin were determined by an average cell diameter estimate (Fig. 3; Fig. S4). We used MATLAB to generate box-and-whiskers plots depicting the distribution of data.

In some time series, image acquisition began slightly after myosin accumulation, in which case t=0 s is the first acquired image in the time series. In other time series, the image acquisition began long before furrow formation, in which case we truncated the time series so that the initial time point is ∼1 min before cells begin to constrict.

Cell length was measured as the projected length along either the ventral-lateral or anterior-posterior axes. Embryos were aligned in time based on the onset of apical constriction. Measurements were made for cells within approximately two cell diameters of the VM.

Testing for statistical significance was carried out using the MATLAB statistical toolbox. We used the non-parametric two sample Kolmogorov–Smirnov to generate all reported P-values.

Here, Y is the Young's modulus, ν the Poisson Ratio (we set corresponding to an incompressible material), and the integral extends over the surface ⁠. The parameter S accounts for the spatially dependent stiffness variations and is typically set to S=1 for passive equilibrium materials. However, when including apical constriction stresses, S becomes myosin dependent, as outlined below. The first term in the integral corresponds to a stretching energy, and the second term captures the energetic cost of bending the surface. Tr denotes a generalized trace, similar to the usual matrix trace (see supplementary Materials and Methods for details). and are the 2×2 metric tensors of the deformed surface ω and reference surface ⁠. They describe the local geometry of the surface by measuring the lengths of and angles between tangent vectors at any point of the surface (Fig. 4A). Specifically, the components of are given by with · the usual (vector) dot product. The first term in Eqn 1 thus measures the deviations of lengths and angles between the reference and deformed configurations, thereby capturing the energetic cost of stretching and shearing the surface. Similarly, the curvature tensors and measure the local curvature of the deformed and the reference surface, respectively; see supplementary Materials and Methods for their precise definitions. Thus, the integrand in the second line assigns an energetic cost whenever there is a mismatch between the local and preferred curvatures and ⁠, respectively.

We thank N. Perrimon, L. Perkins and the Transgenic RNAi Project at Harvard Medical School (National Institutes of Health/National Institutes of General Medical Sciences R01-GM084947) for providing transgenic RNAi stocks. We also thank A. McHale for the software used to analyze of FISH images. We thank S. Little for his FISH protocol and C. Engbert for help implementing the protocol. We thank R. Horvitz, R. Gupta, and current and former Martin lab members for helpful discussions and feedback.