Drosophila stocks and genetic crossings<\/p>\n
To generate fluorescent cephalic furrow mutants, we performed genetic crosses using the loss-of-function alleles btdXA (FBal0030657), eve3 (FBal0003885), prd4 (FBal0013967), slp\u039434B (FBal0035631) and stg2 (FBal0247234); the membrane fluorescent marker Gap43-mCherry (FBal0258719; a gift from K. Skouloudaki); and the green fluorescent balancers FM7c, Kr-GFP (FBst0005193), CyO, twi-GFP (gift from A. Jain) and TM3, Kr-GFP (FBst0005195). We established stable lines balancing the loss-of-function alleles with fluorescent balancers and used the lack of GFP signal to identify homozygous embryos in our live-imaging recordings. For genes on chromosomes 1 and 2 (btd, eve and prd), we added the membrane marker on chromosome 3 ((btdXA\/FM7c, Kr-GFP;; Gap43-mCherry\/MKRS), (eve3\/CyO, twi-GFP; Gap43-mCherry\/MKRS) and (slp\u039434B\/CyO, twi-GFP; Gap43-mCherry\/TM6B)). For stg, which is located on chromosome 3, we recombined the allele with Gap (Gap43-mCherry, stg2\/TM3, Kr-GFP). As the btd\u2013stg double-mutant stable line is weak, we imaged the progeny of btdXA\/FM7c, Kr-GFP;; Gap43-mCherry, stg2\/Gap43-mCherry flies, identifying btd homozygosity by the GFP and stg homozygosity by the lack of cell divisions after gastrulation. For laser ablations, we used a moe-GFP line (a gift from E. Knust). The wild-type stocks contain the Gap43\u2013mCherry marker in the Oregon-R genetic background. We obtained the founder fly stocks from the Bloomington Drosophila Stock Center and the Kyoto Stock Center and deposited the lines in the MPI-CBG stock collection. The complete list of FlyBase35<\/a> accession numbers and genotypes is available in the data repository for the project36<\/a>.<\/p>\n Animal husbandry and embryo collection<\/p>\n We maintained the Drosophila stocks in 50-ml hard plastic vials containing standard fly food and enclosed with a foam lid to allow air exchange. They were kept in an incubator with a constant 25\u2009\u00b0C temperature and 65% humidity and a 12\u201312\u2009h light cycle. For imaging, we first amplified the stocks in larger 200-ml vials for a few weeks. We then narcotized the flies with CO2 and transferred them to a cage with a plate attached to one end containing a layer of apple juice agar and a slab of yeast paste on top. The flies were left to acclimatize in the cage for 2 days before the experiments. To guarantee that the embryos are at a similar developmental stage, we exchanged the agar plate once per hour at least twice (pre-lays) and let the flies lay the eggs on the agar for 1\u2009h before collecting the plate. After filling the plate with water, we used a brush to release the eggs from the agar and transferred them to a cell strainer with 100-\u00b5m nylon mesh (VWR). To remove the chorion, we immersed the embryos in 20% bleach (sodium hypochlorite solution; 1.05614.2500, Merck) for 90\u2009s, washed abundantly with water and proceeded to mounting for live imaging.<\/p>\n We maintained Clogmia flies in 9-cm-wide plastic Petri dishes with a layer of wet cotton at room temperature and fed them weekly with powdered parsley. To obtain embryos for fixation, we collected the adult flies in a 200-ml hard plastic vial with wet cotton and let them mate for 2\u20133 days. Then, we anaesthetized the flies with CO2, dissected the ovaries from ripe females and released the eggs using tweezers in deionized water, which activates embryonic development37<\/a>,38<\/a>. We let embryos develop in deionized water at room temperature until the desired stage. To remove the chorion, we transferred the embryos to a glass vial with 0.5\u00d7 PBS using a fine brush, exchanged the medium for 5% bleach in 0.5\u00d7 PBS for 2\u2009min and washed abundantly with 0.5\u00d7 PBS. Using the diluted PBS solution instead of water prevents the embryos from bursting after dechorionation.<\/p>\n We obtained pupae of the EgyptII wild-type strain of Ceratitis from the Insect Pest Control Laboratory of the International Atomic Energy Agency. Adult flies were kept at 25\u2009\u00b0C, 65% humidity and 12\u201312\u2009h light cycle, inside 49\u2009\u00d7\u200930\u2009\u00d7\u200930\u2009cm plexiglass cages with the front and back ends covered by a nylon mesh. We provided water through a soaked towel and food as a 3:1 sugar:yeast mixture. As ripe females laid eggs through the nylon mesh, we placed a plastic container with water at the back end of the cage for several hours to collect eggs. We dechorionated Ceratitis embryos using the Drosophila protocol.<\/p>\n We performed the collection of Anopheles embryos at the Center for Integrative Infectious Diseases Research at Heidelberg University. To collect embryos, we placed a glass container with water and filter paper inside a cage with 300 mated females, which were fed a blood meal 72\u2009h before, and put the cage in the dark for 2\u2009h at 29\u2009\u00b0C. We then removed the container from the cage and let the embryos develop until the desired stage. To dechorionate, we collected the embryos on a cell strainer, incubated them in 5% bleach for 75\u2009s and washed them thoroughly with deionized water.<\/p>\n Embryo fixation and in situ hybridization<\/p>\n For Drosophila and Ceratitis, we transferred dechorionated embryos to a glass vial containing equal volumes of 4% formaldehyde in PBS and n-heptane and let the vial shake at 215\u2009rpm for 45\u2009min. For Clogmia, we diluted the fixative in 0.5\u00d7 PBS. After removing the fixative (lower phase) using a glass pipette, we added an equal volume of 100% methanol and shook the vial vigorously by hand for 1\u2009min. We then removed the n-heptane (upper phase) and collected the embryos on the bottom of an Eppendorf tube and washed several times with 100% methanol. For Anopheles, we followed a similar protocol that includes a longer 30-min wash in water after fixation, a 30-s boiling water step followed by 15\u2009min in ice-cold water, until the final methanol washes21<\/a>. All the samples were stored in 100% methanol at \u221220\u2009\u00b0C.<\/p>\n We performed the in situ hybridization of btd, eve, prd and slp genes using the Hybridization Chain Reaction (v3.0; HCR)39<\/a> reagents, except for the probe sets, which we designed using a custom script. The oligos were obtained from Sigma-Aldrich. We selected the HCR amplifiers to allow for triple (multiplexed) in situ combinations of btd\u2009+\u2009eve\u2009+\u2009slp or prd\u2009+\u2009eve\u2009+\u2009slp. Before starting, we manually devitellinized Anopheles embryos using fine tweezers. Then, we rehydrated the embryos in 100% methanol with a series of washes to 100% PBT. We permeabilized the embryos with 1:5,000 dilution of proteinase K (20\u2009mg\u2009ml\u22121) for 5\u2009min, except for Drosophila. All samples were re-fixed in 4% formaldehyde for 40\u2009min and washed thoroughly with PBT. We then followed the \u2018In situ HCR v3.0 protocol for whole-mount fruit fly embryos revision 4 (2019-02-21)\u2019 from Molecular Instruments. After the protocol, we stained the embryos with 1:1,000 DAPI in 5\u00d7 SSCT solution for 2\u2009h and mounted the embryos in 80% glycerol in 5\u00d7 SSCT for imaging.<\/p>\n Sample mounting for microscopy<\/p>\n For most of our live imaging, we used a Zeiss Lightsheet Z.1 microscope running ZEN 2014 SP1 (v9.2.10.54). To increase the throughput of samples imaged in one session, we optimized a mounting strategy previously developed in our laboratory40<\/a>. First, we cut a 22\u2009\u00d7\u200922\u2009mm glass coverslip (0.17\u2009mm thickness) into 6\u2009\u00d7\u200915\u2009mm strips using a diamond knife and attached a single strip to a custom sample holder using silicon glue, letting it harden for 15\u2009min. We then coated the coverslip strip with a thin layer of heptane glue and let it dry while preparing the embryos. Using a fine brush, we transferred the embryos collected in the cell strainer onto an agar pad and oriented them manually with a blunt cactus spine under a stereomicroscope. We aligned about 20 embryos in a single line (head to tail) along the main axis of the strip with the left or ventral sides up, depending on the experiment. To attach the embryos to the coverslip, we carefully lowered the sample holder over the agar pad until the glass coated with heptane glue touched the embryos. We placed the sample holder into the microscope chamber filled with water and rotated it so that the samples were facing the detection objective directly and the coverslip was orthogonal to the detection objective; this is important to prevent the light sheet from hitting the glass edges. With the embryos oriented vertically along the coverslip, the light sheet generated from the illumination objectives coming from the sides only needed to pass through the width of the embryo (about 200\u2009\u00b5m). This approach gives the best results for recording lateral and dorsal views and is ideal for live-imaging homozygote embryos as they are only about one-fourth of the total number of imaged embryos. For imaging\u00a0fixed in situ samples, we used an inverted Zeiss LSM 700 confocal microscope running ZEN 2012 SP5 FP3 (v14.0.25.201). We mounted the samples immersed in 80% glycerol between a slide and a glass coverslip supported by tape.<\/p>\n Microscopy acquisition parameters<\/p>\n For the light-sheet lateral datasets, we used a Zeiss \u00d720\/1 NA Plan-Apochromat water immersion objective to acquire stacks with 0.28-\u00b5m xy resolution and 3-\u00b5m z resolution covering half of the volume of the embryo in a single view. This z resolution was restored to 1\u2009\u00b5m during image processing (see below). For the dorsal datasets, we used a Zeiss \u00d740\/1 NA Plan-Apochromat water immersion objective to acquire stacks with 0.14-\u00b5m xy resolution and 3-\u00b5m z resolution covering a volume around in the middle section of the anterior end of the embryo. We adjusted the time resolution between 45\u2009s and 60\u2009s per frame to maximize the number of embryos acquired in one session. To visualize both the membrane signal (mCherry) and the green balancer signal (GFP), we acquired two channels simultaneously using the 488-nm and 561-nm lasers at 3% power with an image splitter cube containing a LP560 dichromatic mirror with SP550 and LP585 emission filters. All live-imaging recordings were performed at 25\u2009\u00b0C. For the confocal datasets, we used a \u00d720\/0.8 Plan-Apochromat Zeiss air objective to acquire four-channels using three tracks (405, 488 and 639, and 555\u2009nm, respectively) with a BP575-640 emission filter and about 0.4-\u00b5m xy resolution and 2-\u00b5m z resolution covering about half the volume of the embryo.<\/p>\n Image processing and visualization<\/p>\n We converted the raw light-sheet imaging datasets into individual TIFF stacks for downstream processing using a custom macro (ProcessZ1Coverslip.ijm) in Fiji\/ImageJ (v2.16.0\/1.54p) with Java (v1.8.0_172)41<\/a>,42<\/a>. To visualize the presence and dynamics of ectopic folds, we generated 3D renderings of the surface of embryos in lateral recordings using a custom animation (3D_animation.txt) in the Fiji plugin 3Dscript (v0.2.1)43<\/a>. For analysing the entire epithelial surface, we first improved the signal-to-noise ratio and z resolution of lateral datasets from 3\u2009\u00b5m to 1\u2009\u00b5m by training a deep learning upsampling model using CARE CSBDeep (v0.3.0)44<\/a>. Then, we created cartographic projections of the lateral recordings using the ImSAnE toolbox (v3a7be24)45<\/a> by loading the restored data in MATLAB (R2015b)46<\/a>, segmenting the epithelial surface using ilastik (v1.3.3b2)47<\/a>, and generating 3D cartographic projections of lateral views following a workflow established for fly embryos48<\/a>. As the pixel size varies across the projection, the provided scale bars represent approximate values at the central portion of the image. To visualize in situ hybridization data, we performed maximum intensity projections or extracted single slices from the raw volumes. For all microscopy images, we only performed minimal linear intensity adjustments to improve their contrast and brightness49<\/a>. The imaging data for the light-sheet and in situ hybridization experiments analysed in this study are available on Zenodo50<\/a>.<\/p>\n Ectopic fold analyses<\/p>\n To characterize the relative timing of ectopic folding, we annotated the position of the germ band and the number of frames after the onset of gastrulation at the initial buckling, when the first cells disappear from the surface in the lateral 3D renderings. We defined the onset of gastrulation (T\u2009=\u20090) as the moment immediately after the end of cellularization and immediately before the beginning of the ventral furrow invagination. To visualize the variability of ectopic folding, we manually traced the fold outlines in lateral recordings using Fiji. Because embryos have different sizes, we first used the plugin bUnwarpJ (v2.6.13)51<\/a> (https:\/\/imagej.net\/plugins\/bunwarpj<\/a>) to register individual frames and then applied the same transformation to the fold traces for a standardized comparison. We analysed the dynamics of ectopic folds by measuring the relative angle and tortuosity of the segmented line traces over time and to visualize the kinetics, we generated colour-coded temporal projections using the script Temporal Color Code (v101122; https:\/\/imagej.net\/plugins\/temporal-color-code<\/a>) with the perceptually uniform mpl-viridis colour map (https:\/\/bids.github.io\/colormap<\/a>) bundled in Fiji.<\/p>\n To estimate the folded area in the cephalic furrow and ectopic folds, we annotated the region of the blastoderm before gastrulation that infolded in the cartographic projections using Fiji and calculated the area, correcting the pixel dimensions according to the coordinates in the projection. For the fold depth, we measured the distance between the vitelline envelope to the tip of the fold at the moment of maximum depth in the dorsal recordings. For the analysis of the epithelial surface, we used the plugin MorphoLibJ (v1.6.0)52<\/a> (https:\/\/imagej.net\/plugins\/morpholibj<\/a>) to segment, measure and colour-code the cell apical areas, and the plugin Linear Stack Alignment with SIFT (v1.5.0)53<\/a> (https:\/\/imagej.net\/plugins\/linear-stack-alignment-with-sift<\/a>) to register cells between timepoints.<\/p>\n Laser cauterization experiments<\/p>\n We performed laser cauterization experiments in two microscope setups, a light-sheet Luxendo MuVi SPIM with a photomanipulation module and a confocal Zeiss LSM 780 NLO with multiphoton excitation running ZEN Black (v14.024.201). For the MuVi SPIM, we embedded dechorionated embryos in 2% low-melting agarose and mounted the samples in glass capillaries to obtain in toto recordings. We used a pulsed infrared laser at 1,030\u20131,040\u2009nm with a 200-fs pulse duration and 1.5\u2009W power to cauterize the posterior region of the dorsal embryonic surface, attaching the blastoderm to the vitelline envelope. Using an Olympus \u00d720\/1.0 NA water immersion objective, we acquired stacks with 0.29-\u00b5m xy resolution and 1-\u00b5m z resolution of four different angles every 1\u2009min. For the Zeiss microscope, we attached the embryos with the dorsal side down onto coverslips using heptane glue and immersed them in halocarbon oil. We cauterized the embryos sequentially using a near-infrared 800-nm laser (Chameleon Vision II) through a single pixel line (210\u2009nm per pixel and 100\u2009\u00b5s per pixel) around the same dorsal region to block the germ band extension. We used a Zeiss \u00d725\/0.8 NA LD LCI Plan-Apochromat glycerol immersion objective to acquire every 2:38\u2009min two different planes of the blastoderm: (1) the surface to monitor the germ band extension, and (2) 40\u2009\u00b5m deep in the equatorial region to monitor the occurrence of ectopic folding. The stacks had 0.21-\u00b5m xy resolution and 1-min time resolution. To obtain a quantitative measure of ectopic folding, we analysed the degree to which the tissues deform between non-cauterized and cauterized mutants, using as a proxy the tortuosity of the epithelium outline. For that, we took the profile slices from dorsal recordings and transformed the curved vitelline envelope into a straight line using the Straighten tool of ImageJ (Supplementary Fig. 4a<\/a>). We then cropped a 200\u2009\u00d7\u200925\u2009\u00b5m region along the head\u2013trunk interface and applied Gaussian blur, thresholding and edge detection to obtain the epithelium outline for individual timepoints covering about 50\u2009min after gastrulation (Supplementary Fig. 4a,b<\/a>). We extracted measurements from the epithelium outlines using the ImageJ plugin Analyze Skeleton (v3.4.2)54<\/a> (https:\/\/imagej.net\/plugins\/analyze-skeleton<\/a>) and generated the colour-coded temporal projections as described above. The imaging data for the laser cauterization experiments are available on Zenodo55<\/a>.<\/p>\n Laser ablation experiments<\/p>\n We performed laser ablations in a Yokogawa CSU-X1 spinning disk confocal with an EMCCD camera (Andor iXon DU-888) and the software AndorIQ for image acquisition. We attached dechorionated embryos laterally to a MatTek glass-bottom Petri dish and covered the samples with water. Then, we performed the ablations using a Titanium Sapphire Chameleon Ultra II (Coherent) laser at 800\u2009nm tuned down from 80\u2009MHz to 20\u2009kHz with a pulse-picker. The laser power measured before the microscope port was 6\u2009mW, and the pixel dwell time for scanning was 2\u2009\u00b5s. To ensure the cut, we repeated the scan ten consecutive times along a single cell, acquiring a single slice with a \u00d760\/1.2 NA water immersion objective with 0.18-\u00b5m xy resolution and 200-ms time steps. We ablated each embryo just once. The temperature was maintained at 28\u2009\u00b0C. To analyse the ablation data, we created a line crossing the edges of the ablated cell perpendicular to the cut and generated a kymograph using the Fiji plugin Multi Kymograph (v3.0.1; Supplementary Fig. 5<\/a>). We then binarized the kymographs, measured the distance between cell edges over the first 30\u2009s after the cut and performed a linear fit of the data to obtain the recoil velocity (Supplementary Fig. 5<\/a>). We performed additional laser ablations in an inverted Zeiss Axio Observer.Z1 spinning disk confocal microscope running ZEN Blue (v3.2) with a Rapp OptoElectronic setup for photo-manipulation running SysCon2. The imaging data for the laser ablation experiments are available on Zenodo55<\/a>.<\/p>\n Strain rate analysis<\/p>\n To estimate the strain rates, we first performed particle image velocimetry on cartographic projections using the ImageJ plugin iterativePIV (v2.0)56<\/a> (https:\/\/sites.google.com\/site\/qingzongtseng\/piv<\/a>). Then, we used the equation<\/p>\n $$E=\\left|\\frac{1}{2}(\\overrightarrow{\\nabla }.\\overrightarrow{v})+\\frac{1}{2}({\\partial }_{x}{v}_{y}+{\\partial }_{y}{v}_{x})\\right|$$<\/p>\n to define and calculate the magnitude of the strain rate, where \\(v\\) is the displacement obtained in the particle image velocimetry analysis divided by the time in minutes. The measurements combine isotropic and anisotropic strain rates. We used these values to create a colour-coded overlay for the strain rate (Supplementary Fig. 2b<\/a>). To generate the line plots, we averaged the strain rate along the dorsoventral axis in two predefined regions, the head\u2013trunk (canonical cephalic furrow position) and the trunk\u2013germ (posterior to the mitotic domain 6; Supplementary Fig. 2b<\/a>).<\/p>\n Model and simulations<\/p>\n Our model follows an approach similar to a previously published model of epithelial buckling under confinement17<\/a>. It represents the monolayer epithelium of the early Drosophila embryo in a cross-section as a single line through the apicobasal midline of the epithelial cells. The tissue is modelled as an elastic rod with a stretching energy per unit length \\({W}_{s}\\) and bending energy per unit length \\({W}_{b}\\) so that the total energy per unit length is \\({W}_{T}={W}_{s}+{W}_{b}\\). In full,<\/p>\n $${W}_{T}=\\mathop{\\int }\\limits_{{L}_{o}}\\frac{1}{2}{K}_{s}{\\left(\\frac{{ds}}{d{s}_{o}}-1\\right)}^{2}d{s}_{o}+\\mathop{\\int }\\limits_{{L}_{o}}\\frac{1}{2}{K}_{b}{(\\kappa -{\\kappa }_{o})}^{2}d{s}_{o}$$<\/p>\n where \\({K}_{s}\\) is the stretching rigidity and \\({K}_{b}\\) is the bending rigidity of the tissue; \\(d{s}_{o}\\) and \\({ds}\\) are the preferred and current lengths of the curve, respectively; and \\(\\kappa \\) is the curvature of the rod. \\({L}_{o}\\) is the total length of the tissue in a stress-free condition. To perform numerics, we discretize the curve into \\(N\\) particles indexed by \\(i\\). The total energy per unit length for this discretized model is given by<\/p>\n $${W}_{T}^{* }=\\frac{1}{2}{K}_{s}\\mathop{\\sum }\\limits_{i=2}^{N-3}{\\left(\\frac{\\varDelta {r}_{i}}{\\varDelta {r}_{o}}-1\\right)}^{2}\\varDelta {r}_{o}+\\frac{1}{2}{K}_{b}\\mathop{\\sum }\\limits_{i=2}^{N-3}{({\\kappa }_{i}-{\\kappa }_{o,i})}^{2}\\varDelta {r}_{o}$$<\/p>\n where \\(\\varDelta {r}_{o}\\) is the preferred length of springs connecting consecutive points (equal for all springs); \\(\\varDelta {r}_{i}\\) is the current length between \\(i\\) and \\(i+1\\); \\({\\kappa }_{i}\\) is the discretized curvature at point \\(i\\); and \\({\\kappa }_{o,i}\\) is the preferred curvature at point \\(i\\) (equal to 0, except when specified). The first and last two points of the curve are fixed in space. To obtain a physically meaningful dimensionless bending rigidity, we divided the bending rigidity by the factor \\({K}_{s}{L}^{2}\\) as<\/p>\n $${K}_{b}^{* }=\\frac{{K}_{b}}{{K}_{s}{L}^{2}}$$<\/p>\n where \\(L\\) is the semi-major axis of the embryo. To minimize the total energy, we added a ground level of noise to the particles and let the particles move in the direction of the forces. The motion of the particles is governed by<\/p>\n $$\\frac{\\varDelta {\\overrightarrow{r}}_{i}}{\\varDelta t}=-\\frac{L}{{K}_{s}\\tau }\\frac{{\\rm{\\partial }}{W}^{\\ast }}{{\\rm{\\partial }}{\\overrightarrow{r}}_{i}}+{\\overrightarrow{\\zeta }}_{i}$$<\/p>\n where \\({\\overrightarrow{r}}_{i}\\) is the current position of the ith particle; \\(\\tau \\) represents an arbitrary timescale introduced here to balance dimensions (set to 1); \\(\\varDelta t\\) are the timesteps (set to \\({10}^{-5}\\times \\tau {K}_{s}\/L\\)); and \\({\\overrightarrow{\\zeta }}_{i}\\) is the noise chosen from a Gaussian distribution with mean 0 and standard distribution \\({10}^{-5}\\times L\\).<\/p>\n In our model, the position of the germ band corresponds to the position of the last particle in the curve on the semi-ellipse that represents the embryonic blastoderm. The extent of the germ band is given by \\(g\\), which is the projection of the germ band arc length onto the mid-axis of the embryo normalized by the embryo length (\\(2L\\)). When \\(g=0\\), the tissue is free of stretching stress, but at any other \\(0 , the blastoderm will be compressed. The preferred lengths of the individual springs are obtained by dividing the elliptical arc length into \\(N\\) equal segments. The length of each segment is given by \\(\\varDelta {r}_{\\text{o}}=\\frac{1}{N}\\left(L{\\int }_{0}^{\\pi }\\sqrt{1-{e}^{2}{\\cos }^{2}(u)}{du}\\right)\\). To find the initial lengths of the springs, we used<\/p>\n $$\\varDelta r(t=0)=\\frac{1}{N}\\left(L\\underset{u{\\prime} }{\\overset{\\pi }{\\int }}\\sqrt{1-{e}^{2}{\\cos }^{2}(u)}{du}\\right)$$<\/p>\n where \\(e=\\sqrt{1-{(0.4)}^{2}}\\) and the angle \\({u}^{{\\prime} }\\) corresponds to the position of the blastoderm end. \\({u}^{{\\prime} }\\) is obtained for a given value of \\(g\\) by \\({u}^{{\\prime} }={\\cos }^{-1}(1-2g)\\). Here we obtained the initial lengths by dividing the compressed blastoderm into \\(N\\) equal segments. For any simulation, the value of \\(g\\) is constant (the blastoderm end is static in position). To model mitotic domains, we introduced new particles and springs on the midpoints between two particles in specific regions of length \\(0.5L\\). The new springs were given the same \\(\\varDelta {r}_{o}\\) as the rest of the springs in the tissue. The blastoderm is confined by a rigid boundary in the shape of a semi-ellipse. Any particle that lands outside this boundary at any timestep was repositioned onto the rigid boundary. This new position was prescribed by taking the intersection point of the rigid boundary curve and the line segment that connects the position before this iteration (which was inside or on the vitelline envelope) and the position outside the vitelline envelope. Finally, we defined and counted a fold when we found that the distance of a particle from the rigid boundary is greater than a threshold value. To calculate this threshold, we measured the maximum distance that particles can achieve when the tissue is in a stress-free state. This threshold was calculated to be \\(0.035L\\). The code for the model and the simulation data are available in the theory repository on Zenodo57<\/a>.<\/p>\n Data visualization and figure assembly<\/p>\n We created illustrations and assembled the final figure plates using Inkscape (v1.2.2)58<\/a>. For microscopy videos, we exported the original stacks as AVI without compression at 10\u201315\u2009fps using Fiji and post-processed them to MPEG-4 format 1,080p resolution using the H.264 encoding at a constant bitrate quality factor of 18 for visualization using HandBrake (v1.6.1)59<\/a>. The high-resolution figures and videos are available in a Zenodo repository60<\/a>. We performed the data wrangling, statistical analyses and plotting in R (v4.2.1)61<\/a> using R Markdown notebooks in RStudio (v2022.7.2.576)62<\/a>, and in Python (v3.10.7) using Jupyter notebooks (v6.5.4)63<\/a>. The source files and analysis pipelines are available in the main repository on Zenodo36<\/a>.<\/p>\n Statistics and reproducibility<\/p>\n The phenotypes that we report in this study were reproducible across multiple independent experiments. For the live imaging, we performed 7 experiments in btd mutants (total of 50 embryos), 5 experiments in eve mutants (total of 36 embryos) and 3 experiments in prd mutants (total of 41 embryos; Fig. 1b,c,f,g<\/a>, Extended Data Fig. 3c<\/a>, Extended Data Fig. 1a,b<\/a>, Extended Data Fig. 2a,b,h,k,l<\/a> and Fig. 2a\u2013c,f<\/a>, respectively). The phenotypes were also consistent across 6 experiments in slp mutants (total of 39 embryos; Extended Data Fig. 5a,g<\/a>), 3 experiments in stg mutants (total of 46 embryos; Extended Data Fig. 3a,b<\/a>), 6 experiments in btd\u2013eve double mutants (total of 35 embryos; Extended Data Fig. 3d<\/a>) and 2 experiments in wild-type embryos (total of 36 embryos; Fig. 1c<\/a>, Extended Data Fig. 2k,l,m<\/a> and Fig. 2a<\/a>). For the germ band cauterization, we performed 6 experiments in btd mutants (total of 10 embryos; Extended Data Fig. 3g<\/a>), 5 experiments in eve mutants (total of 10 embryos; Fig. 2i\u2013k<\/a>) and 8 experiments in wild-type embryos (total of 12 embryos; Extended Data Fig. 3e,f<\/a>). For the gene expression, the wild-type patterns of btd, eve and slp were highly consistent across 3 experiments in Drosophila (total of 26 embryos; Fig. 4a\u2013d,g<\/a> and Extended Data Fig. 6a<\/a>), 3 experiments in Ceratitis (total of 38 embryos), 4 experiments in Anopheles (total of 43 embryos) and 4 experiments in Clogmia (total of 44 embryos; Fig. 4f\u2013h<\/a> and Extended Data Fig. 8a\u2013c<\/a>). We also obtained consistent patterns of prd expression among 4 experiments in Drosophila (total of 10 embryos; Fig. 4c<\/a> and Extended Data Fig. 6b,c<\/a>) and 1 experiment in Clogmia (total of 20 embryos; Extended Data Fig. 8d,e<\/a>). The expression patterns in mutant embryos were repeatable across 4 independent experiments in slp mutants (total of 30 embryos; Extended Data Fig. 5c\u2013f,h<\/a>), 5 experiments in btd mutants (total of 20 embryos), 2 experiments in eve mutants (total of 12 embryos) and 2 experiments in prd mutants (total of 12 embryos; Extended Data Fig. 7a\u2013c<\/a>).<\/p>\n We performed no previous estimation for sample size and no randomization or blinding strategy for experiments. Following previous studies in the field, we determined the number of experiments and sample size based on the repeatability of the observed phenotypes. Sample numbers refer to biological replicates. In all boxplots, the centre represents the median, the lower and upper hinges correspond to the first and third quartiles (25th and 75th percentiles) and the whiskers extend from the hinges until 1.5 times the interquartile range. The asterisks in plots indicate P\u2009U-test contrasting the condition against wild type; exceptions are described in figure legends. We report P values rounded to three decimal places and show values lower than 0.001 as P\u200936<\/a>.<\/p>\n Reporting summary<\/p>\n