To measure the stress distribution on the actin cortex, we used laser nano-ablation, which has now become a standard method to assess forces exerted in cells, to sever the actin cytoskeleton and to observe the shape of the opening hole (Figure 2a). We visualized actin with a GFP- or mCherry-labelled actin-binding-domain protein (ABD) expressed in the epidermis (Gally et al., 2009) (Figure 1b–d). We adjusted the region of interest to cut within one cell, restricting our analysis to the early phase of elongation (≤1.7F; for staging, see Figure 1 legend).

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We observed two types of ablation responses (see Materials and methods). In the first (accounting for >80% of the cases), the opening hole within the actin cytoskeleton reached equilibrium in less than 10 s, and resealed within less than 2 min (Figure 2—figure supplement 1, Video 1). In such embryos, actin occasionally accumulated around the cut borders but not around cell borders (Video 1). Imaging calcium levels, which can rise after laser wounding (Xu and Chisholm, 2011; Razzell et al., 2013; Antunes et al., 2013), showed either no change or a localized increase (Video 2, Figure 2—figure supplement 2a,b). In the second ablation reponse, an actin ring accumulated around the cell borders during the repair process (Video 3) and a calcium wave propagated to nearby epidermal cells (Video 4; Figure 2—figure supplement 2c). Embryos showing the first response continued to develop and hatched, whereas those showing the second response arrested their development and eventually died. In all subsequent studies, we only took into account the first type of response, which should correspond to a local cortex disruption.

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To compare the response between different conditions, we detected the cut-opening shape, which we fitted with an ellipse to derive the shape parameters (see Materials and methods). The laser setup we used did not enable us to image the recoil dynamics within the first second after the cut, which other investigators previously used to assess the extent of mechanical stress (Rauzi and Lenne, 2015; Smutny et al., 2015; Saha et al., 2016). To circumvent this issue, we developed a novel analysis method to derive mechanical stress data, based on the equilibrium shape of a thin cut in an infinite elastic isotropic plane, subjected to biaxial loading (stress applied in two perpendicular directions) (Theocaris, 1986). The rationale for approximating the epidermis to such a plane is further outlined in the Appendices 2 and 3. In these conditions, a thin cut will open to form an elliptical hole at equilibrium (Figure 2a). The opening of the cut reflects mechanical stress in the direction perpendicular to the cut direction.

We cut the epidermal actin specifically in the AP and DV directions, which we found to correspond to the stress loading directions (Figure 2b; Appendix 2). For a cut in the AP direction, the minor axis of the ellipse at equilibrium, b_DV, will be proportional to the cut length, l, and to the ratio of stress in the DV direction, ${\sigma }_{DV},$ over the Young modulus E of the plane (Theocaris et al., 1986) (Figure 2b):

We will call the ratio $\frac{b_{DV}}{l}$ the opening in the DV direction of a cut made along the AP direction (Figure 2b), and similarly $\frac{b_{AP}}{l}$ the opening in the AP direction of a cut made in the DV direction. Thus, we used the opening of the hole in a given direction to derive the stress in that direction.

We compared the conclusions drawn from this method with methods relying on the recoil dynamics (Rauzi and Lenne, 2015; Smutny et al., 2015; Saha et al., 2016) (Appendix 3). The half-time of the cut border relaxation, which depends on the ratio of viscosity over stiffness, was similar in the AP and DV directions (Appendix 3), supporting the hypothesis that the seam cell cortex is isotropic. We found an agreement between both methods for the AP versus DV stress ratio, and similar trends for the stress magnitude.

To further examine the validity of this method, we performed two tests. First, the theory described above (Theocaris, 1986) predicts that the minor to major axis ratio of the opening ellipse is independent of the initial cut length. We found that it is the case when the cut length varied from 3 μm to 6 μm (Appendix 4). Second, to prove that the opening observed after laser cutting depends on myosin II activity, we performed cuts in embryos defective for the main myosin II regulator, LET-502/Rho-kinase (Gally et al., 2009). As shown in Figure 2c, the opening in the seam cell H1 at the 1.5F stage in let-502(sb118ts) embryos changed very little and was significantly smaller than that in WT embryos, consistent with a decrease in mechanical stress.

Thus, we feel confident that the method based on the opening shape measures actomyosin-dependent stress and can be used reliably to report on stress differences along the DV and AP directions.

We applied the method described above on three seam cells (head H1, body V3, tail V6; Figure 3a) because myosin II acts mainly in seam cells (Gally et al., 2009), and compared the response with embryonic morphological changes. We focused on the anisotropy of stress between the DV and AP directions (difference of stress along both directions) in a given cell (Figure 3b). Indeed, in other systems, such as Drosophila embryos (Rauzi et al., 2008) and C. elegans zygotes (Mayer et al., 2010), this parameter is critical.

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At the 1.3F stage in H1, there was no significant stress anisotropy; however, as the embryo elongated to the 1.5F and 1.7F stages, the stress became anisotropic (Figure 3b). In V3, the anisotropy of stress evolved in the opposite direction, with higher stress anisotropy at the 1.3F stage compared to the 1.5F stage (Figure 3b). In V6, the stress was slightly anisotropic at both the 1.3F and the 1.5F stage (Figure 3b). In all cells, whenever the stress became anisotropic, it was higher in the DV direction. Overall, the opening increased as the embryo elongated from the 1.3F to the 1.5F stage and from the 1.5F to the 1.7F stage for H1.

To correlate the stress anisotropy with the morphological changes in the embryo, we used markers labelling cortical actin (an ABD) and junctions (HMR-1/E-cadherin). We observed that the head, body and tail diameter (at the levels of H1, V3 and V6, respectively) decreased at different rates over time (Figure 3c), as also observed by Martin and colleagues (Martin et al., 2014). The head diameter did not diminish between the 1.3F and 1.5F stages when the stress was nearly isotropic, but decreased significantly between the 1.5F and 1.7F stages as the stress anisotropy increased. Conversely, the body diameter decreased most rapidly between the 1.3F and 1.5F stages, when the stress was highly anisotropic, then changed at a lower pace beyond the 1.5F stage, when the stress became less anisotropic. Finally, the tail diameter decreased nearly linearly between the 1.3F and 1.7F stages, at a lower rate than the body diameter, coinciding with a smaller anisotropic stress in V6. Thus, the local morphological changes within the embryo correlate with locally higher stress in the DV compared to the AP direction.

To define whether all cells contribute equally to the diameter change, we quantified the circumferential width of the epidermal cells H1, V3 and their adjacent DV cells (Figure 3d,e). At the level of V3, the decrease in body diameter came from both seam (V3) and DV cells, whereas in the head, it came mainly from DV cells (Figure 3e). Collectively, our results strongly suggest that stress anisotropy correlates with morphological changes. Furthermore, we found that both seam and DV epidermal cells contribute to the changes in embryo diameter, irrespective of their level of active myosin II.

Taking the H1 cell as an example, we considered some cellular factors that could contribute to the stress anisotropy in seam cells: (i) actin-anchoring proteins, and (ii) muscle-induced tension. To ease comparisons, we defined the anisotropy of stress (AS) as

which can be derived from the ratio of the opening along the DV and AP directions, see Equation (1).

First, we examined the actin-anchoring spectrin cytoskeleton, which is essential for embryonic elongation (Moorthy et al. 2000; Norman and Moerman, 2002; Praitis et al., 2005). In spc-1(RNAi) embryos, at a developmental timing equivalent to the 1.7F stage in control embryos, we found a smaller opening in both AP and DV directions and a decrease of AS compared to WT (Figure 3f,g). This may account for the slower elongation rate of spc-1(RNAi) embryos and their arrest at the 2F stage (Figure 3—figure supplement 1). Thus, spectrin partially contributes to the AS at the 1.7F stage.

Second, we wondered whether muscle contractions, which start after the 1.5F stage, could account for AS changes (Figure 3b). Compared to controls, embryos that are depleted in UNC-112/Kindlin, which mediates sarcomere assembly (Rogalski et al., 2000), showed a significantly larger opening in both the AP and DV directions at the 1.7F stage, but no change in stress anisotropy (Figure 3f,g). This is consistent with their wild-type elongation rate up to the 2F stage (Figure 3—figure supplement 1). Thus, AS establishment in the H1 cell after the 1.5F stage is independent of muscle contractions.

To define the possible causal relationship between the AS and embryonic shape changes, we aimed to simplify the shape of the embryo in order to allow the application of classical physical laws such as the Young-Laplace equation, which predicts the relationship between surface tension and the surface curvature. As illustrated in Figure 1, the embryo has a circular section and a cylindrical or conical shape depending on the stage, in which the epidermis is relatively thin (100 nm to 2 µm, depending on areas; www.wormatlas.org) when compared to the embryo diameter (25 µm). Within the embryo, the epidermis is subjected to hydrostatic pressure when the section decreases (Priess and Hirsh, 1986). We can thus model the C. elegans embryo as an isotropic thin-wall (the epidermis) vessel with capped ends under hydrostatic pressure, and can determine the relationship between the mechanical stress on the epidermis and the embryo shape.

First, we calculated the anisotropy of stress on the wall of such a vessel. For an axisymmetric vessel, the AS on the wall depends on the surface curvature and the radius (Appendix 5), which for simple geometrical configurations can be written as shown in Figure 4a–c. Typically, the AS factor, or the DV to AP stress ratio, is equal to one for a sphere, equal to two for a cylinder and takes an intermediate value between 1 and 2 for an ellipsoid. We can simplify the geometry of C. elegans embryos between the 1.3F and 1.5F stages as a curved cylinder (body), attached to a sphere (head) (Figure 4d,e). The head evolves into an ellipsoid between the 1.5F and 1.7F stages (Figure 4f). Thus, the AS of the head can be determined easily. We previously observed that the AP stress among the seam cells at a given stage differs by 20% (Figure 3b). Thus, if we approximate the AP stress as a constant at a given stage, the AS in the body will depend on the ratio of the body to head radius (Figure 4d,e, Appendix 5). Given the head and body diameter of the embryo (Figure 3c), we can compare the AS predicted by the thin-wall vessel model with those derived experimentally using laser ablation (Figure 4g). These values are nearly identical, showing that AS can be predicted on the basis of embryonic geometry.

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To examine whether the AS can dictate embryonic morphological changes, we related the deformation of the vessel wall with the forces applied using the Hooke’s law (Figure 4h, Appendix 6A) – for instance, Hooke’s law states that the one-dimensional deformation of a spring is equal to the ratio of the applied force to the spring stiffness. Similarly, in a two-dimensional system and for an isotropic material, the deformation is proportional to the mechanical stress (forces) and inversely proportional to the Young modulus (stiffness) along the different loading directions (Figure 4h). The resulting equations, which assume that seam cells have an isotropic cortex and are subjected to contractile stress, correctly predict that the seam cell dimension increases along the AP axis (ε_AP) with the AS (Figure 4h–j), and decreases along the DV axis (ε_DV). Indeed, consistent with the equations, the head evolves from a sphere to an ellipsoid between the 1.5F to the 1.7F stages as the AS becomes greater than 1 (Figure 3b,c).

In conclusion, our experimental and modelling data show that the AS induces morphological changes in embryonic seam cells and provide a basis for understanding how the embryo elongates from a mechanical standpoint.

As shown in Figure 3e, the head diameter reduction primarily involves changes in the circumferential width in the DV epidermis. As the RhoGAP RGA-2 maintains myosin II activation in these cells at a low level (Diogon et al., 2007), actomyosin contractility in DV cells cannot account for such changes. However, in contrast to seam cells, DV epidermal cells have circumferentially oriented actin bundles (Figure 1b–d), which based on recent observation could affect cell stiffness (Calzado-Martín et al., 2016; Salker et al., 2016). We thus hypothesized that the circumferential polarized actin distribution in DV epidermal cells could induce greater stiffness in that direction and thereby influences their deformation. To establish whether this is the case, we investigated both stress and stiffness distribution in the epidermal cells dorsal and ventral to the H1 seam cell using laser nano-ablation (Figure 5a). As these cells are the precursors of the HYP7 syncytium, we will denote them HYP7.

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In the HYP7 cell, the opening in the DV direction was larger than that in the AP direction (Figure 5b; dorsal and ventral cells behaved similarly after laser cutting) at the 1.5F and 1.7F stages, as was the case in the H1 cell (Figure 3b). However, the ratio of DV/AP opening in HYP7 was greater than that in H1 (Figure 5c). Assuming that the HYP7 cell cortex has isotropic material properties like those of H1, our model (Figure 4; Appendix 5) would predict that the DV/AP opening ratio in HYP7 depends only on the head axisymmetric shape and is equal to that of H1, and would thus contradicts our observations. Hence, this suggests that the HYP7 cell has anisotropic cortical material properties.

To model the DV epidermal cell deformation, we examined two classes of anisotropic stiffness materials that have been described previously: orthotropic materials such as bones (Miller et al., 2002; Helwig et al., 2009), and fiber-reinforced materials such as arteries (Gasser et al., 2006), articular cartilage (Federico and Gasser, 2010) or fibrous connective tissues (Ben Amar et al., 2015). Orthotropic materials have different stiffnesses along orthogonal directions, and thus respond differently to the same stress magnitude along these directions. Fiber-reinforced materials also have different stiffnesses in the directions along and transverse to the fibers; in addition, such materials can respond differently to extensive or compressive stress (Bert, 1977). To define which model best applies to the DV epidermis, we used continuum linear elastic analysis (Muskkhelishvili, 1975; Suo, 1990; Theocaris, 1986; Yoffe, 1951) (Appendix 7) to interpret the laser-cutting data from the DV epidermis. We discarded the orthotropic model, as it did not adequately describe our data (Appendix 8), and focused on the fiber-reinforced plane model, which better accounts for the presence of well aligned actin fibers in DV cells.

In a fiber-reinforced material that is composed of a matrix$$ superimposed with fibers, the contribution of the fibers to the stiffness of the material depends on their orientation. Along the direction parallel to the fibers, the Young modulus of the composite is much increased due to fiber reinforcement, whereas along the direction perpendicular to the fibers, the contribution of the fibers to the composite stiffness is small. According to our modelling, the Young modulus along the fiber direction increases linearly with a factor K related to the fiber stiffness and density; whereas the stiffness along the direction transverse to the fibers varies as a hyperbolic function of K and reaches a plateau (Appendix 7). For fiber-reinforcement in the DV direction, the change in Young modulus along the DV and AP directions predicted by modelling is given in Figure 6a,b. Cuts perpendicular to the fibers opened similarly to an isotropic material with the matrix Young modulus, because they locally destroyed the fibers (Figure 6c; see Equation (1) above). By contrast, cuts along the fibers opened with an equilibrium value that depends on the fiber stiffness and distribution through the factor K defined above (Figure 6d, Appendix 7).

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Since the H1 seam and the head HYP7 cells are adjacent along the circumference (Figure 3d), they should be under the same DV stress due to tension continuity across cell-cell junctions. According to Equation (1), if the stress in two cells is the same, their opening should vary inversely with their respective Young moduli. As the DV opening of HYP7 was about 1.5 times smaller than that of H1 (Figure 7a), we infer that the Young modulus of the HYP7 matrix without fibers was about 1.5 times stiffer than that of H1 (Appendix 9), suggesting that these cells have distinct material properties. Comparing the DV and AP opening for the HYP7 cell, we found that the factor K increased during early elongation (Figure 7b; Appendix 10). More importantly, the calculated ratio of DV/AP Young moduli also increased, and was greater than the DV/AP stress anisotropy (Figure 7c).

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To understand how a change in stiffness affects head HYP7 deformation, we again applied Hooke’s law to these cells (Appendix 6B; Figure 7d). As myosin II activity in DV cells is low, their cortex should be exposed to tensile stress induced by actomyosin contractility in the seam cells. The cell length along the AP direction increased when the stiffness anisotropy (DV/AP stiffness ratio) increased (Figure 7d,e), whereas the trend was opposite in the DV direction (Figure 7d,f). Thus, the stiffness anisotropy helps the HYP7 cell to extend along the AP direction and to shrink along the DV direction. Interestingly, the equations predict that increasing stress anisotropy has an opposite effect on HYP7 cell deformation, as it prevents these cells from extending antero-posteriorly (Figure 7e). Altogether, our model strongly suggests that when the DV/AP stiffness anisotropy increases and is greater than the DV/AP stress anisotropy, as observed in the head HYP7 (Figure 7c), elongation along the AP direction is favored. Furthermore, our data demonstrate that the distinct mechanical properties of the cells composing a complex tissue enables its morphogenesis , which does not require all cells to be contractile.

Myosin II is not polarized (Figure 1e,f), but to find out whether actin distribution accounts for the stress and stiffness anisotropies, we carried out an analysis of actin filament alignment in the seam cells H1 and V3, as well as in the head HYP7 cell. We found that the polarization of actin filaments in seam cells correlated with the observed pattern of stress anisotropy. Indeed, in H1 at the 1.3F stage, actin filaments had a nearly isotropic angular distribution correlating with the isotropic stress (Figure 8a, Figure 3b), whereas they became increasingly aligned along the DV direction from the 1.3F to the 1.7F stage (Figure 8a), mirroring the increasing stress anisotropy from 1.3F to 1.7F (Figure 3b). Likewise, actin alignment decreased along the DV direction in V3 from the 1.3F to the 1.5F stage, in parallel to the decrease of stress anisotropy between those stages (Figures 8b and 3b). The changes in H1 (Figure 8c), but not in V3 (Figure 8d), were statistically significant. In the HYP7 cell, actin filaments already acquired a preferential DV alignment at the 1.3F stage (Figure 8c), but became increasingly organized along the DV direction as the embryo elongated to the 2F stage, with a highly significant difference between the 1.5F and 1.7F stages (Figure 8c and f). These changes correlated with the increased stiffness anisotropy observed in the HYP7 cell (Figure 7c).

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We have attempted to test functionally how actin organization could affect stress and stiffness by manipulating actin polymerization through two different strategies to express cofilin during early elongation. However, we could not obtain meaningful results. Altogether, we conclude that the pattern of actin distribution showed a good correlation with the observed stress and stiffness anisotropy. It will remain important to define the mechanisms that mediate changes in actin distribution, and ultimately to determine whether this distribution is a cause or a consequence of anisotropy.

Bristol N2 was used as the wild-type (WT) strain and animals were maintained as described in Brenner (1974). The strain ML1540: mcIs50[lin-26p::vab-10(abd)::gfp; myo-2p::gfp] LGI carrying the actin-binding domain (ABD) of the protein VAB-10 under the epidermal promoter lin-26 has been described elsewhere (Gally et al., 2009). The endogenous NMY-1::GFP reporter strain was built by CRISPR knock-in (ML2540: nmy-1(mc82)[nmy-1::gfp] LGX; the NMY-2::GFP reporter strain LP162 nmy-2(cp13)[nmy-2::gfp + LoxP] LGI was a generous gift from Daniel Dickinson.

For parallel calcium and actin imaging during ablation, we used the strain ML2142: mcIs43 [lin26p:: vab-10::mCherry; myo-2p::gfp]; juIs307[dpy-7p::GCaMP3] carrying a calcium sensor under the epidermal promoter dpy-7p and mCherry-labeled VAB-10(ABD) under the lin-26 promoter. The thermosensitive Rho kinase mutation let-502(sb118ts) was crossed with ML1540 to give the strain ML2216: let-502(sb118ts); mcIs50[lin-26p::vab-10(abd)::gfp; myo-2p::gfp] LGI.

For determining morphological changes, we used the strain ML2386: mcIs50[lin-26p::vab-10(abd)::gfp; myo-2p::gfp] I; xnIs97[hmr-1::gfp] III, which expresses both a junctional marker (HMR-1/E-cadherin) and an actin marker (VAB-10(ABD)).

For actin alignment analysis, we used the ML1966 unc-119(ed3) mcIs67 [dpy7p::LifeAct::GFP; unc-119(+)] strain, expressing the actin reporter LIFEACT under the dpy-7 epidermal promoter.

RNAi experiments were done using injection of double-stranded RNA synthesized from PCR-amplified genomic fragments using a T3 or T7 mMESSAGE mMACHINE Kit (Ambion, Austin, TX, USA). The embryos were analyzed from 24 hr to 48 hr post-injection.

Freshly laid embryos or embryos from dissected hermaphrodites were mounted on 5% agarose pads in M9 buffer and the coverslip was sealed with paraffin oil. DIC time-lapse movies were recorded at 20°C using a Leica DM6000 upright microscope with a 40X oil immersion objective. For each embryo, a Z-stack of 7–8 focal planes with 4 μm step size was acquired. The length of embryos was estimated by tracing the embryo body axis (through the middle of the embryo). Fluorescence time-lapse movies were recorded at 20°C using a spinning-disk Zeiss microscope Axio Observer.Z1 using a 63X oil immersion objective. Other fluorescence images were acquired with the same microscope using a 100X oil immersion objective. To determine the morphological changes in the embryo, sections of the embryo imaged with junctional and actin markers at the level of H1, V3 and V6 were reconstructed to determine the radius, seam and DV cell width along the circumferential direction. All images were analyzed using the ImageJ (FiJi) software (NIH, Bethesda, Maryland, USA; http://rsb.info.nih.gov/ij/) and MATLAB R2014b (The MathWorks Inc., Natick, MA).

Z-stack images of LIFEACT::GFP fluorescence expression in the epidermis were acquired using a confocal Leica SP5 microscope with a 63X oil immersion objective and zoom factor 8. We used a step size of 0.08 μm, a pinhole opening of 0.6 Airy Unit and projected 2 μm around the actin cortex. The embryos were rotated on the scan field to have the same antero-posterior orientation. The acquired images were deconvoluted using the Huygens Essential software from Scientific Volume Imaging (Hilversum, Netherlands). We chose a region of interest (ROI) of 4 × 4 μm² within the seam cell H1 or dorso-ventral epidermal cell HYP7, and of 3 × 3 μm² within the seam cell V3 to perform Fast Fourier Transform (FFT). We used a high-pass filter to remove the low frequencies then did inverse FFT. We found that the high pass filter removed changes in intensity due to unequal labelling or out of focus signals but retained the actin texture. Finally, we used an ImageJ plugin, ‘Spectral Texture Analysis’, written by Julien Pontabry to derive the angle distribution of actin texture. This plugin performed the Fast Fourier Transform (FFT) of the given ROI and computing coefficients in Fourier space, such as the angle distribution of the given structure, as detailed in Gonzalez and Woods (2008). Digital image processing, Nueva Jersey, chapter 11, section 3.3 (Gonzalez, 2008).

For ablations, we compared embryos of the same developmental timing. To do so, we recorded the elongation curve of different genetics background (Figure 3—figure supplement 1) and took embryos at the corresponding developmental time from the beginning of elongation. Thus, unc-112(RNAi) embryos elongating up to 1.7F similarly to WT have the same length as WT at 1.7F stage. By contrast, spc-1(RNAi) embryos elongated slower than WT (Figure 3—figure supplement 1) and thus at a time corresponding to 1.7F in a control embryos were shorter than wild-type embryos at 1.7F stage. For comparison between let-502(sb118ts) embryos, measurements were carried out at 25.5°C and the embryos were taken when muscles started to twitch (at around 1.5F in control embryos).

Laser ablation was performed using a Leica TCS SP8 Confocal Laser Scanning microscope, with a femtosecond near-infrared Coherent Chameleon Vision II, Ti:Sapphire 680–1080 nm laser, 80 MHz. To make a line cut, a region of interest with a length varying from 3 μm to 6 μm and a width of 0.08 μm (1 pixel width) was drawn. We used a laser wavelength varying from 800 nm to 900 nm, which gave consistent ablation responses. The laser power was tuned before each imaging section to obtain local disruption of the cortex response (>80% of the cases, visible opening, no actin accumulation around cell borders in the repair process and the ablated embryos developed normally). Typically, the power of the laser was 2000 mW, and we used 50% power at 100% gain. Wounding response (actin accumulation around cell borders in the repair process, embryo died afterwards) was rarely observed at the power used for local disruption, but more often when the power was increased to 60–65%. The first time point was recorded 1.44 s after cutting, which corresponded to the time needed to reset the microscope from a two-photon to a regular imaging configuration. The image scanning time recorded by the software was usually less than 400 ms, so the total exposure time of the chosen ROI to multiphoton laser was less than 1 ms. The cuts were oriented either in the antero-posterior (AP) or dorso-ventral (DV) directions relative to the global orientation of the embryos. After ablation, the embryos were monitored to see whether they continued to develop normally or whether they expressed the desired phenotype. More precisely, we verified whether embryos ablated at 1.3F and 1.5F developed past the 2F stage, whether embryos ablated at 1.7F developed past the 2.5F stage, and whether unc-112(RNAi) embryos and spc-1(RNAi) embryos arrested at 2F stage.

The shape of the cut opening was detected using the Active Contour plugin ABsnake (Boudier, 1997). A starting ROI was drawn around the opening as the initiation ROI for ABsnake. After running the plugin, the results were checked and corrected for detection errors. The detected shape was fitted with an ellipse to derive the minor axis, major axis and the angle formed by the major axis with the initial cut direction. The average opening of the five last time points before the repair process began (Figure 2—figure supplement 1, from around 8 to 10 s after cutting) was taken as the opening at equilibrium. The standard error of the mean is reported.

The curve fit was performed on the average value of the cut opening (defined as the minor axis/initial cut length) using GraphPad Prism 5.00 (San Diego, California, USA) and the equation of one-phase association:

where y₀ is the initial width of the cut opening, Plateau is the minor axis of the opening at equilibrium and γ is the relaxation rate. The standard error of the mean given by the software is reported.

The two-tailed t-test was performed on the average of the last five time points (from about 8 s to 10 s) of the cut opening using MATLAB R2014b (The MathWorks Inc., Natick, MA). Z-test was performed using QuickCalcs of GraphPad Prism (San Diego, California, USA) to compare the anisotropy of stress (AS), the relaxation half-time and the initial recoil speed of the cut opening.

Soon after the ventral enclosure has completed, C. elegans embryos elongate from a lima-bean shape to the characteristic cylindrical shape, resulting in a 4-fold increase in length and approximately a 2-fold decrease in diameter (Priess and Hirsh, 1986) (Figure 1a). The elongation is thought to be driven by cell shape changes, as can be most easily observed among seam cells (Figure 1a). Muscle contractions, starting about midway through the process, are essential: muscle-defective embryos are paralyzed and arrest at the 2F stage. In the following paragraphs, we mostly discuss the early phase of elongation that occurs prior to the onset of muscle contraction.

Pharmacological and genetic studies have proved the critical role of actomyosin contractility during the early elongation. Inhibition of actin polymerization with cytochalasin-D prior to the 1.5F stage blocks elongation, whereas application at later stage causes embryos to retract to their pre-elongation state (Priess and Hirsh, 1986). Non-muscle myosin II activation is regulated through phosphorylation and dephosphorylation of the regulatory light chain MLC-4 by the LET-502/Rho-binding kinase and the MEL-11/Myosin phosphatase, respectively (Wissmann et al., 1997, 1999; Gally et al., 2009). LET-502, the effector of the Rho GTPase RHO-1, can be activated by the C. elegans RhoGEF (Guanine Exchange Factors) RHGF-2 and inactivated by the RhoGAP (GTPase Activating Protein) RGA-2 (Chan et al., 2015; Diogon et al., 2007). Mutations affecting myosin II or its activation, such as MLC-4 and NMY-1/Non-muscle Myosin II heavy chain, RHGF-2/RhoGEF, and LET-502/ROCK, lead to hypo-elongation, causing embryos to arrest earlier than or at the 2F stage (Wissmann et al., 1997; Gally et al., 2009; Chan et al., 2015; Shelton et al., 1999; Piekny et al., 2003). By contrast, mutations affecting negative regulators of myosin II, such as mel-11 or rga-2, cause embryos to burst during elongation due to increased tension exerted on adherens junctions (Wissmann et al., 1999; Diogon et al., 2007). The two myosin heavy chains, NMY-1 and NMY-2, work redundantly to regulate actomyosin contractility (Piekny et al., 2003). Although essential during early embryonic development, our data show that NMY-2 is not required during elongation. Indeed, embryos that are homozygous for the strong thermosensitive mutant nmy-2(ne3409) (Liu et al., 2010) still elongated normally after shifting to the restrictive temperature (25.5^o C data not shown).

Several lines of evidence suggest that seam cells generate most of the actomyosin forces, while the DV cells may remain passive. First, the myosin II regulatory light chain MLC-4 is mainly required in seam cells (Gally et al., 2009). Second, MLC-4, MLC-5/myosin essential light chain and NMY-1 are expressed at higher levels in seam cells (Gally et al., 2009; Shelton et al., 1999; Piekny et al., 2003). Third, rescue experiments have shown that the positive regulator of contractility RHGF-2/RhoGEF is required only in seam cells, whereas the negative regulator RGA-2/RhoGAP acts specifically in DV cells (Shelton et al., 1999; Piekny et al., 2003). Thus, all the players of acto-myosin regulation pathway promote a high contractility in seam cells and keep a low contractility in DV cells.

Although the myosin II activity is crucial in seam cells, the phenotypes of several mutants affecting junctional proteins, which are thought to anchor actin bundles (Costa et al., 1998), show an important role of actin bundles in DV epidermal cells during elongation. In particular, zygotic hmp-1/$\alpha$-catenin mutants, in which actin bundles detach from the junctional belt, show bulges and cannot elongate. Similarly, loss of ZOO-1/ZO-1 and VAB-9/Claudin homologues affects actin bundle organization in DV cells, leading to deformities and an incomplete elongation (Lockwood et al., 2008; Simske et al., 2003).

In summary, the epidermal actomyosin network is essential for the early elongation phase of the C. elegans embryo.

Biological materials are in general viscoelastic (Kasza et al., 2007), but the elastic aspect of C. elegans embryos seems to be more important. Indeed, inhibition of actin polymerization with cytochalasin-D induces a retraction of the embryo to nearly its original length (Priess and Hirsh, 1986), like a spring after force release. Thus, we used an elastic model to describe C. elegans embryonic deformation. In particular, we considered the epidermal cell cortex as an elastic plane. In this section, we examine, for the analysis of laser ablation responses, whether the epidermal cell cortex is subjected to biaxial stress loading (stress along two orthogonal directions) along the AP and DV directions.

If a thin cut is introduced in an infinite isotropic elastic plane under biaxial stress loading (Appendix 2—figure1a), Theocaris et al. (1986) have shown that the opening is an ellipse. These authors have shown that the rotation angle $\theta$ between the direction of the cut and the major axis of the opening ellipse is in general different from zero (Appendix 2—figure1b). $\theta$ is equal to zero when the cuts are parallel to the directions of stress loading or when there is equal tension-tension loading (${\displaystyle {\sigma }_{xx}={\sigma }_{yy}>0}$) (Theocaris et al., 1986).

To test whether the AP and DV directions are indeed the directions of stress loading for different seam cells (H1, V3 and V6; Figure 3a) during early elongation, we performed laser cuts in the AP and DV directions and measured the rotation angle $\theta$ at equilibrium from the 1.3F to the 1.7F stages. Appendix 2—figure 1c shows that $\theta$ was not significantly different from zero for different magnitudes of stress (Figure 3b), consistent with the hypothesis that AP and DV are the principal directions of stress loading. The difference of $\theta$ compared to 0 was more important for V6, as we had difficulties in determining unambiguously the AP and DV direction for V6. In conclusion, the epidermal cell cortex can be considered as an elastic plane under biaxial stress loading along the AP and DV directions.

Appendix 2—figure 1

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In order to compare the two analysis methods of laser ablation using the equilibrium shape of the cut opening or the recoil dynamics (Rauzi and Lenne, 2015; Smutny et al., 2015), we estimated the initial recoil speed and the relaxation half-time by fitting the relaxation of the cut borders (the minor axis of the cut opening) with the equation:

where $y_0$ is the initial width of the cut opening, $Plateau$ is the minor axis of the cut opening at equilibrium and $\gamma$ is the relaxation rate. The recoil speed can be obtained by taking the derivation of the previous equation versus time:

Thus, the initial recoil speed is:

The half-time, defined as the time interval needed to reach half of the distance between the initial opening and the $Plateau$, is given by:

According to the model described in Rauzi and Lenne (2015), Smutny et al. (2015)and Mayer et al. (2010), the initial recoil speed is proportional to the ratio of the cortical stress $\sigma$ to the viscosity of the cellular medium $\eta$:

and the relaxation half-time depends on the ratio of $\eta$ to the cortex stiffness $k$:

The fitting of the Equation (1) depends on the initial width of the cut opening $y_0$. We used three methods to estimate $y_0$. First, since the cut opening depends on the actomyosin contractility (Figure 2c), we reasoned that $y_0$ should be close to the opening observed in a mutant where actomyosin contractility is strongly inhibited. Indeed, in the let-502(sb118ts) mutant, the opening changed very little (Figure 2c), which corresponded to a nearly complete inhibition of the early elongation. The smallest opening in the let-502(sb118ts) mutant was around 0.6 $\mu m$ (Figure 2c). Second, our evaluation of the initial width of the opening using the same setup to photobleach a thin fluorescent layer also gave an opening of around 0.6 $\mu m$. Finally, we tried to fit the recoil of cut borders using Equation (1) with values of $y_0$ decreasing from 0.6 $\mu m$ to 0.2 $\mu m$ (in 0.1 $\mu m$ steps) and found a decrease of goodness of fit $R^2$ (Supplementary table 2). The goodness of fit $R^2$ is a fraction between 0 and 1, and higher values indicate better fits (GraphPad Prism 5, Goodness of fit of nonlinear regression). Thus, all three approaches indicated that 0.6 $\mu m$ was a good estimation of the initial width of the cut opening (an example of fit is shown in Figure 2—figure supplement 1). Subsequently, we used $y_0=0.6\mu m$ μm to derive the initial recoil speed and the relaxation half-time, and the results are shown in Supplementary table 3.

We found that the relaxation half-time in the AP direction was similar with that in the DV direction in most of the seam cells examined at different stages (Appendix 3—figure 1a). As the relaxation time is proportional to the ratio of viscosity over the stiffness of the cortex, and the cytoplasmic viscosity is likely to be homogeneous within a given cell, the cortex in seam cells is likely to be isotropic. However, it can vary from one cell to another as indicated in Appendix 3—figure 1a.

Next, we wanted to know whether the two methods to analyze the ablation response (based on the cut recoil dynamics and the cut opening at equilibrium) gave consistent results for the stress magnitude and anisotropy. As we compared different seam cells (H1, V3 and V6) with potentially different material properties (viscosity and stiffness), we multiplied the initial recoil speed by the relaxation half-time to have the stress to stiffness ratio (Equations [5, 6]). We normalized the previous ratio to the different cut lengths used in the different seam cells (5 $\mu m$ in H1, 4 $\mu m$ in V3 and V6), then plotted the resulting values against the cut opening at equilibrium, which also reports the stress over stiffness ratio (Figure 2b, Appendix 3—figure 1a,b,d). The linear regression strongly suggests that the two methods are in good agreement on the stress magnitude. Moreover, the anisotropy of stress obtained by the two methods showed a linear correlation (Appendix 3—figure 1e).

In summary, the two methods to analyze laser ablation responses gave results consistent with each other on the magnitude and the anisotropy of stress. The relaxation half-time obtained from recoil dynamics analysis indicated that the seam cell cortex is isotropic.

Appendix 3—figure 1

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As predicted by Theocaris et al. (1986), the minor to major axis ratio of the cut opening should be independent of the cut length. Indeed, the equation for the major axis $a$ over the cut length $l$ of the opening at equilibrium is given as:

where ${\sigma }_{xx}$ and ${\sigma }_{yy}$ are the stresses in the principal loading directions $x$ and $y$, respectively, which are also the laser cut directions. $E$ is the Young modulus of the plane (Appendix 2—figure 1a,b). Thus, according to the ratio of minor axis to cut length given in Figure 2b, the minor to major axis ratio of the ellipse at equilibrium is:

and does not depend on the cut length $l$. Our experimental data fitted well with this prediction as shown in Appendix 3—figure 1f,g. Indeed, when we plotted the minor versus major axes of the opening at equilibrium, we observed a linear relationship (Appendix 3—figure 1f) when the cut length varied from 3 to 6 $\mu m$ in the seam cell H1, for cuts in both the AP and DV directions. Thus, the minor to major axis ratio is nearly a constant and independent of the cut length (Appendix 3—figure 1g), consistent with the theory of Theocaris et al. (1986).

The rationale for modeling the C. elegans embryo as a thin-wall pressured vessel is given in the main text. To do so, first, we calculated the stress anisotropy for an axisymmetric vessel. We derived the stress anisotropy for the head from this calculation because the head was considered axisymmetric. Second, we calculated the stress anisotropy in the embryo body at the position of the seam cell V3. The body was not axisymmetric due to the important folding of the embryo in the eggshell, but the stress anisotropy can be obtained using a similar method.

Anisotropy of stress for an axisymmetric thin-wall pressured vessel

We consider an axisymmetric thin-wall pressured vessel with two ends capped. To make a parallel with C. elegans embryos, we call the axis of the vessel AP and the circumferential axis DV (Appendix 5—figure 1a). We calculate the anisotropy of stress $\frac{{\sigma }_{DV}}{{\sigma }_{AP}}$, where ${\sigma }_{AP}$ is the longitudinal (AP) stress and ${\sigma }_{DV}$ is the circumferential stress (DV) on the wall.

Appendix 5—figure 1

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Let us consider a point Q on the wall for which we calculate $\frac{{\sigma }_{DV}}{{\sigma }_{AP}}$. The tangent at Q in the plane going through Q and the AP axis makes an angle $\alpha$ with AP.

Imagine that we cut the vessel in two parts by a plane going through Q and perpendicular to the AP axis (Appendix 5—figure 1a). The vessel is divided into ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$. The forces applied by ${\mathrm{\Omega }}_2$ on ${\mathrm{\Omega }}_1$ have two components $F_1$ and $F_2$ (Appendix 5—figure1b): $F_1$ is the force applied by the wall of ${\mathrm{\Omega }}_2$ on ${\mathrm{\Omega }}_1$, whereas $F_2$ is the force exerted by the hydrostatic pressure from ${\mathrm{\Omega }}_2$. We have

(9) $F_1={\sigma }_{AP}2\pi Rh$

where $R$ is the radius of the vessel at Q; $h$ is the thickness of the wall (epidermis), considered as a constant. $F_{1AP}$ is the component in the AP direction of $F_1$ and is written as:

(10) $F_{1AP}=F_{1}cos\alpha ={\sigma }_{AP}2\pi Rhcos\alpha$

$F_{1AP}$ is balanced by the hydrostatic force $F_2$ because ${\mathrm{\Omega }}_1$ is at equilibrium

(11) $F_2=p\pi R^2$

where $p$ is the hydrostatic pressure. By combining the Equations (10, 11), we have:

(12) ${\sigma }_{AP}2\pi Rhcos\alpha =p\pi R^2$

(13) $\Rightarrow {\sigma }_{AP}=\frac{pR}{2hcos\alpha }$

Now let’s consider a volume element ${\mathrm{\Omega }}_4$ of length $\mathrm{\Delta }l$ with two limiting sections perpendicular to the AP axis, so that ${\mathrm{\Omega }}_4$ is between ${\mathrm{\Omega }}_3$ and ${\mathrm{\Omega }}_5$ (Appendix 5—figure 1c). Imagine that we cut ${\mathrm{\Omega }}_4$ into two halfs ${\mathrm{\Omega }}_{41}$ and ${\mathrm{\Omega }}_{42}$ (Appendix 5—figure1d,e).

Let’s examine an element $\mathrm{\Delta }s$ on the wall of ${\mathrm{\Omega }}_{41}$, at the interface between ${\mathrm{\Omega }}_3$ and ${\mathrm{\Omega }}_{41}$ (Appendix 5—figure 1f). The force applied by the wall of ${\mathrm{\Omega }}_3$ to this element is:

(14) $F_3={\sigma }_{AP}(R_3)h\mathrm{\Delta }s={\sigma }_{AP}(R_3)h{R}_{3}d\beta$

where ${\sigma }_{AP}(R_3)$ means that ${\sigma }_{AP}$ is a function of $R_3$, $\beta$ is the angle formed by the position of $\mathrm{\Delta }s$ with the ${\mathrm{\Omega }}_{41}$-${\mathrm{\Omega }}_{42}$ interface as shown in Appendix 5—figure 1f. The force applied by ${\mathrm{\Omega }}_3$ on ${\mathrm{\Omega }}_{41}$ in the radial direction (Appendix 5—figure 1e,f) is:

(15) $F_{3R}=F_{3}sin\alpha (R_3)={\sigma }_{AP}(R_3)h{R}_{3}d\beta sin\alpha (R_3)$

where $\alpha (R_3)$ means that $\alpha$ is a function of $R_3$. The force applied by ${\mathrm{\Omega }}_3$ in the direction $\overrightarrow{n}$ perpendicular to the ${\mathrm{\Omega }}_{41}$–${\mathrm{\Omega }}_{42}$ interface (Appendix 5—figure 1f) is:

(16) $F_{3n}={\int }_0^{\pi }{F}_{3R}sin\beta d\beta ={\int }_0^{\pi }{\sigma }_{AP}(R_3)h{R}_{3}sin\alpha (R_3)sin\beta d\beta$

(17) $=2{\sigma }_{AP}(R_3)h{R}_{3}sin\alpha (R_3)$

If we replace ${\sigma }_{AP}$ obtained from the Equation (13), we have:

(18) $F_{3n}=2{\sigma }_{AP}(R_3)h{R}_{3}sin\alpha (R_3)=\frac{p{R}_3^{2}sin\alpha (R_3)}{cos\alpha (R_3)}$

The force $F_{5n}$ exerted by ${\mathrm{\Omega }}_5$ to ${\mathrm{\Omega }}_3$ can be calculated in the same manner. We obtain:

(19) $F_{5n}=-\frac{p{R}_5^{2}sin\alpha (R_5)}{cos\alpha (R_5)}$

The resulting force applied by ${\mathrm{\Omega }}_3$ and ${\mathrm{\Omega }}_5$ to ${\mathrm{\Omega }}_{41}$ in the $\overrightarrow{n}$ direction can be expressed as :

(20) $F_{35n}=F_{3n}+F_{5n}=\frac{d}{dR}(\frac{p{R}^{2}sin\alpha (R)}{cos\alpha (R)})\mathrm{\Delta }R=2pRtan\alpha (R)\mathrm{\Delta }R+\frac{p{R}^2}{co{s}^2\alpha (R)}\frac{d\alpha }{dR}\mathrm{\Delta }R$

The force applied by ${\mathrm{\Omega }}_{42}$ to ${\mathrm{\Omega }}_{41}$ (Appendix 5—figure 1e,f) in the $\overrightarrow{n}$ direction is :

(21) $F_{12}=p\mathrm{\Delta }lcos\alpha (R)2R-{\sigma }_{DV}2\mathrm{\Delta }lh$

Since ${\mathrm{\Omega }}_4$ is at equilibrium, we have :

(22) $F_{35n}+F_{12}=0$

Note that

(23) $\mathrm{\Delta }R=\mathrm{\Delta }lsin\alpha$

From the Equations (20, 21, 22, 23) we have

(24) $2pRtan\alpha \mathrm{\Delta }lsin\alpha +\frac{p{R}^2}{co{s}^2\alpha }\frac{d\alpha }{dR}\mathrm{\Delta }lsin\alpha +p\mathrm{\Delta }lcos\alpha 2R-{\sigma }_{DV}2\mathrm{\Delta }lh=0$

(25) $\Rightarrow {\sigma }_{DV}=\frac{pRsi{n}^2\alpha }{hcos\alpha }+\frac{pRcos\alpha }{h}+\frac{p{R}^{2}sin\alpha }{2hco{s}^2\alpha }\frac{d\alpha }{dR}=\frac{pR}{hcos\alpha }+\frac{p{R}^{2}sin\alpha }{2hco{s}^2\alpha }\frac{d\alpha }{dR}$

Thus we obtain the anisotropy of stress on the wall:

(26) $AS=\frac{{\sigma }_{DV}}{{\sigma }_{AP}}=\frac{\frac{1}{cos\alpha }+\frac{Rsin\alpha }{2co{s}^2\alpha }\frac{d\alpha }{dR}}{\frac{1}{2cos\alpha }}=2+R\frac{d\alpha }{dR}tan\alpha$

We can now calculate the $AS$ for vessels with a particular shape : a sphere, a cylinder and an ellipsoid.

-—For a sphere of radius $R_0$ :

(27) $R=R_{0}cos\alpha$

According to the Equation (13), the AP stress is:

(28) ${\sigma }_{AP}=\frac{p{R}_0}{2h}$

If we take the derivation of the Equation (27) with respect to R, we have

(29) $-R_{0}sin\alpha \frac{d\alpha }{dR}=1$

(30) $\Rightarrow R_{0}cos\alpha tan\alpha \frac{d\alpha }{dR}=-1$

(31) $\Rightarrow Rtan\alpha \frac{d\alpha }{dR}=-1$

Combing the previous equation with the Equation (26), we have:

(32) $\Rightarrow AS=\frac{{\sigma }_{DV}}{{\sigma }_{AP}}=1$

—For a cylinder : $\alpha =const=0$

(33) $AS=\frac{{\sigma }_{DV}}{{\sigma }_{AP}}=2$

According to the Equation (13), the AP stress is:

(34) ${\sigma }_{AP}=\frac{pR}{2h}$

where $R$ is the radius of the cylinder.

—For an ellipsoid with the major axis $a_1$ and minor axis $a_2$ (Appendix 5—figure 1g), we can write the coordinates of point Q in the plane going through Q and AP axis as:

(35) ${\displaystyle \Rightarrow \{\begin{array}{l}x=a_{1}cost\\ y=a_{2}sint\end{array}}$

thus

(36) $tan\alpha =\frac{dy}{dx}=\frac{a_{2}cost}{a_1(-sint)}=-\frac{a_2}{a_1}cotant$

(37) $\frac{d(tan\alpha )}{dt}=(1+ta{n}^2\alpha )\frac{d\alpha }{dt}=\frac{a_2}{a_1}(1+cota{n}^{2}t)$

Thus

(38) ${\displaystyle \frac{d\alpha }{dt}=\frac{\frac{a_2}{a_1}(1+cota{n}^{2}t)}{1+\frac{a_2^2}{a_1^2}cota{n}^{2}t}=\frac{a_1{a}_2}{a_1^{2}si{n}^{2}t+a_2^{2}co{s}^{2}t}}$

Due to the symmetry of the system, we examine only $t\in [0,\pi ]$. We have:

(39) $R=y=a_{2}sint$

(40) $\frac{d\alpha }{dR}=\frac{d\alpha }{dt}\frac{dt}{dR}=\frac{a_1{a}_2}{(a_1^{2}si{n}^{2}t+a_2^{2}co{s}^{2}t)a_{2}cost}=\frac{a_1}{(a_1^{2}si{n}^{2}t+a_2^{2}co{s}^{2}t)cost}$

From Equations (36, 39, 40) we have

(41) $R\frac{d\alpha }{dR}tan\alpha =\frac{-a_2^2}{a_1^{2}si{n}^{2}t+a_2^{2}co{s}^{2}t}$

Thus:

(42) $AS=\frac{{\sigma }_{DV}}{{\sigma }_{AP}}=2-\frac{a_2^2}{a_1^{2}si{n}^{2}t+a_2^{2}co{s}^{2}t}$

For the middle of the ellipsoid, $y=a_2$ and $x=0$, $t=\frac{\pi }{2}$, thus

(43) $AS=\frac{{\sigma }_{DV}}{{\sigma }_{AP}}=2-{\left(\frac{a_2}{a_1}\right)}^2$

According to the Equation (13), the AP stress at the middle of the ellipsoid is:

(44) ${\sigma }_{AP}=\frac{p{a}_2}{2h}$

Note that the radial stress on the wall is $-p$. As the wall is thin, i.e $h\ll R$, we expect that the radial stress is much smaller than the AP and DV stress on the wall for a sphere, an ellipsoid or a cylinder.

In this section, we further detail how we used the Hooke’s law to describe the deformation of the seam and DV epidermal cells.