Wing explants were grown ex vivo as described previously (Dye et al., 2017). Briefly, wing discs were dissected from larvae in growth media (Grace’s cell culture media + 5% fetal bovine serum ${\displaystyle +20nM}$ 20-hydroxyecdysone + Penicillin-Streptomycin) at room temperature. Then, they were transferred to a Mattek #1 glass bottom petri-dish to the center of a hole cut in a double-sided tape spacer (Tesa 5338) and covered with a porous filter. The dish was then filled with fresh growth media.

Data from movies 1 to 3 were used previously (Dye et al., 2017). Movie 4 was acquired after publication of the first manuscript, in the same exact way as movies 1 to 3. Briefly, Ecadherin-GFP-expressing wing discs were imaged in growth media using a Zeiss spinning-disc microscope to acquire ${\displaystyle 0.5\mu m}$ spaced Z-stacks at 5 min intervals with a Zeiss C-Apochromat 63X/1.2NA water immersion objective and ${\displaystyle 2\times 2}$ tiling (10% overlap). This microscope consisted of an AxioObserver inverted stand, motorized xyz stage, stage-top incubator with temperature control set to ${\displaystyle {25}^{\circ }C}$, a Yokogawa CSU-X1 scanhead, a Zeiss AxioCam MRm Monochrome CCD camera (set with ${\displaystyle 2\times 2}$ binning), and 488 laser illumination. We circulated growth media during imaging using a PHD Ultra pump (Harvard Apparatus) at a rate of ${\displaystyle 0.03ml/min}$. Time ${\displaystyle 0hr}$ of the movie was considered to be the start of imaging acquisition, typically ${\displaystyle 45-60min}$ from the start of dissection (time required for sample and microscope preparation).

Movie 5 was acquired on a newer microscope with a larger field of view that eliminated the need to tile across the wing pouch. This microscope has an Andor IX 83 inverted stand, motorized xyz stage with a Prior ProScan III NanoScanZ z-focus device, a Yokogawa CSU-W1 with Borealis upgrade, and a Pecon cage incubator for temperature control at ${\displaystyle {25}^{\circ }C}$. We used an Olympus 60x/1.3NA UPlanSApo Silicone-immersion objective with 488 laser illumination and an Andor iXon Ultra 888 Monochrome EMCCD camera. We acquired ${\displaystyle 0.5\mu m}$ spaced Z-stacks of a single tile at ${\displaystyle 5min}$ intervals, as for the other movies. Movie 5 was also acquired without the constant flow of new media.

For all movies, care was taken to limit light exposure, using laser power values of ${\displaystyle <0.08mW}$ and exposure times less than ${\displaystyle 350ms}$ per image.

Raw Z-stacks were denoised using a frequency bandpass filter and background subtraction tools available in FIJI (Schindelin et al., 2012). Then, we used a custom algorithm as described previously (Dye et al., 2017) to make 2D projections of the apical surface, marked by Ecadherin-GFP. This algorithm also outputs a height-map image, in which the value for each pixel corresponds to the level in the Z-stack of the identified apical surface. For those movies that were tiled, we used the Grid/Collection Stitching FIJI plugin (Preibisch et al., 2009) to stitch the 2D projections, and then used that calculated transformation to stitch the height map images, so that we could correct the cell area and elongation values for local curvature (see below). We focus in this work exclusively on the disc proper layer, which is the proliferating layer of the disc that goes on to produce the adult wing. We did not consider cell shape patterns in the peripodial layer, the non-proliferating layer of the wing disc that is largely destroyed at the onset of pupariation.

Samples for single timepoint imaging were acquired exactly as described for live imaging: dissected in growth media and imaged in Mattek dishes under a porous filter.

All imaging of single timepoint data was performed with the same microscope described above for Movie 5. We did not acquire timelapse data for these genetic perturbations, and thus we chose to image the entire disc using ${\displaystyle 2\times 2}$ tiling, ${\displaystyle 0.5\mu m}$ spaced Z-stacks. Approximately ${\displaystyle 6-10}$ discs were imaged per dish, within approximately ${\displaystyle 30-60min}$ .

Tiled images were stitched together as Z-stacks; then we obtained the apical surface projection and its corresponding height map as described above and in Dye et al., 2017.

Wing discs from mid-third-instar larvae (${\displaystyle 96hr}$ AEL) were dissected and mounted exactly as was done for the live imaging timelapses. Due to constraints on the speed of ablation, we only cut regions of the wing disc pouch that were completely flat, so that we could cut in a single plane at the apical surface (rather than having to cut in each plane of a Z stack). Where this flat region lies depends on how the disc happens to fall on the coverslip during mounting. Because the anterior compartment is larger and higher, most of our ablations are in this compartment. To access other regions, we also mounted some wing discs on an agarose shelf: stripes of 1% agarose (in water) were dried onto the surface of the coverslip and wing discs were arranged their anterior half propped up on the agarose shelf prior to adding the porous filter cover.

Ablations were performed using ultraviolet laser microdissection as described in Grill et al., 2001 using a Zeiss 63X water objective. First, we took a full Z-stack of the sample prior to the cut. Then, we selected a ${\displaystyle 7\mu m}$ radius circle that would ablate in the flattest region of the tissue. No imaging is possible during ablation, but we acquired a ${\displaystyle 2min}$ timelapse immediately after the cut in a single Z-plane. This timelapse data was not used except to estimate whether or not the sample was fully cut. After, once the sample has finished expanding but not started to heal (${\displaystyle \sim 2min}$), we took another full Z-stack to image the endpoint. We excluded a small number of data points if any of the following were true: (1) the inner piece remaining after the ablation was no longer visible (sometimes it floats or is destroyed); (2) the cut appeared to expand highly asymmetrically (rare); (3) the wing discs were clearly too young to be considered ${\displaystyle 96hr}$ AEL (poor staging).

Due to a limited field of view on the microscope used for ablation, we performed immunofluorescence after the ablation in order to better estimate the position of the cut in the wing pouch. After all the discs in the dish were ablated (${\displaystyle <10}$ discs/dish), the entire dish was fixed through the filter by adding 4% PFA and incubating ${\displaystyle 20min}$ at room temperature. After, the dish was rinsed and kept in PBST (PBS + 0.5% Triton X-100) until all discs from that image acquisition day were completed (${\displaystyle 2-4hr}$). All samples were then blocked using 1% BSA in PBST + ${\displaystyle 250mM}$ NaCl for ${\displaystyle 45min-1hr}$, and then incubated in primary antibody overnight (diluted in BBX: 1% BSA in PBST). Initially, we labeled samples with SRF (Active Motif, 1:100 dilution), but later we switched to Patched and Wingless (DSHB, 1:100) to identify the compartment boundaries. Primary antibody toward GFP (recognizing the Ecadherin-GFP) was also included at 1:1000. After overnight incubation at ${\displaystyle 4^{\circ }C}$, we washed with BBX, followed by BBX+ 4% normal goat serum (NGS), for at least ${\displaystyle 1hr}$. Secondary antibodies were added for ${\displaystyle 2hr}$ at RT in BBX+NGS. Finally, samples were washed ${\displaystyle 4x10min}$ in PBST and imaged in this media. Imaging was performed on one of the two spinning disc microscopes described above for live imaging. We matched the stained samples with the ablation images by (1) keeping track of the position of each disc on the dish and where it was ablated during acquisition and (2) morphology of the disc before/after ablation. All antibody information are listed in the Key Resources Table.

To immunostain larval wing discs, late third instar larvae were partially dissected in ${\displaystyle 1\times }$ PBS, inverting the heads to expose the wing discs, and then fixed in 4% PFA for ${\displaystyle 20min}$. Samples were then washed twice for ${\displaystyle 15min}$ each with PBX2 (${\displaystyle 1\times }$ PBS + 0.05% Triton X-100), and blocked in BBX250 (PBX2 + ${\displaystyle 1mg/ml}$ BSA + ${\displaystyle 5mM}$ NaCl) for ${\displaystyle 45min}$. Incubation in primary antibody diluted 1:50 in BBX (PBX2 + ${\displaystyle 1mg/ml}$ BSA) was performed overnight at ${\displaystyle 4^{\circ }C}$. We used a mouse monoclonal antibody against Dachsous (Merkel et al., 2014). After, the sample was washed twice with BBX, ${\displaystyle 20min}$ for each wash, followed by one ${\displaystyle 45min}$ wash with BBX + ${\displaystyle 4^{\circ }C}$ normal goat serum (NGS). Incubation with secondary antibody was performed either for ${\displaystyle 2-3hr}$ at room temperature or overnight at ${\displaystyle 4^{\circ }C}$. We used a goat anti-mouse Alexa-Fluor 647 as the secondary antibody (ThermoFisher,Cat No. A28181) at 1:500 dilution in (BBX+4% NGS). The secondary antibody was removed by washing twice with PBX2, followed by washing twice with ${\displaystyle 1\times }$ PBS. The tissues were stored in PBS and dissected from the body wall just before mounting. Wing discs were mounted on a slide within a thin channel created by two strips of double-sided tape (Tesa 5338). Wings were transferred to the middle of the channel with a p200 pipette, the excess PBS was removed, and the tissues were arranged with apical sides up. Next, the sample was covered with a ${\displaystyle 22\times 22mm}$ #1 coverslip, adhering the coverslip to the double-sided tape. Vectashield mounting medium (Vector laboratories) was added to one side of the coverslip and allowed to seep in by capillary action. Excess Vectashield was removed, and the sides of the coverslip were sealed using transparent nail polish. The slides were then stored at ${\displaystyle 4^{\circ }C}$ until imaging.

To image the adult wing morphology, male flies of the desired genotype were collected and stored in isopropanol. Wings were dissected and collected in isopropanol. To mount, wings were transferred to a glass slide using a p200 pipet. In a centrifuge tube, ${\displaystyle 500\mu l}$ of isopropanol was mixed with ${\displaystyle 500\mu l}$ of Euperal, and about ${\displaystyle 50\mu l}$ of this mix was added to the slide containing the wings. The wings were then aligned on the slide, adding more isopropanol/Euperal mix when necessary to avoid drying. Once the wings were all aligned, more isopropanol-Euperal mix was added and allowed to partially dry. Once the mix was almost dry, a clean ${\displaystyle 22\times 22}$ coverslip containing about ${\displaystyle 20-50\mu l}$ of Euperal was inverted on top of the wings. The slides was allowed to cure for ${\displaystyle 24hr}$ before imaging.

Immunostained samples were imaged on an Olympus IX81 microscope equipped with a spinning disk module (Yokogawa) and back illuminated EMCCD (Andor Technology, iXon Ultra 888) with an exposure time of ${\displaystyle 100ms}$ and EM gain of 250. A confocal z-stack of immunostained samples was acquired using a ${\displaystyle 60\times }$ silicon immersion objective with ${\displaystyle 0.47\mu m}$ z-spacing.

Adult wings were imaged on an inverted wide field microscope (Zeiss) using a ${\displaystyle 5\times }$ objective. The images were analyzed using a custom code written in MATLAB (Mathworks, USA) to measure wing area.

Using the 2D projections, we performed cell segmentation and tracking using the FIJI plugin, TissueAnalyzer (Aigouy et al., 2016). We manually corrected errors in the automated segmentation and tracking as much as possible and then generated a relational database using TissueMiner (Etournay et al., 2016). Images were rotated to a common orientation (Anterior left, dorsal up). We then manually identified three regions of interest at the last point in the timelapse, using the FIJI macros included with TissueMiner: the ‘blade’ was defined roughly as an elliptical region surrounded by the most distal folds; and the Anterior-Posterior and Dorsal-Ventral boundaries were estimated using the brightness of Ecadherin-GFP and apical cell size (Jaiswal et al., 2006).

For the timelapse data, we used only the cells within these manually defined regions that were trackable during the entire course of the movies. An example of this region is presented for Movie 1 in Figure 1A. Furthermore, we also excluded from all of our analysis the first ${\displaystyle 2hr}$ of imaging, the so-called adaption phase, where cells uniformly shrink in response to culture (Dye et al., 2017).

For the RNAi data (Figures 5 and 7), we did not acquire timelapses, rather a single timepoint at late stages of development. We nevertheless chose to analyze these data using the same TissueMiner workflow for simplification. Because TissueMiner was developed for timelapse data, however, it requires at least two timepoints. Thus, we duplicated the data and labeled the two images as if they were timepoints 0 and 1 of a timelapse. We then only analyzed timepoint 0. The regions of interest in these data, therefore, are manually defined and not simply the region that is trackable (since it is static data).

To average across all movies of timelapse data, or all discs of the same genotype of the static RNAi data, we generated a disc-coordinate system by normalizing the position of each cell to the AP and DV boundaries for that disc. To do so, we averaged the absolute xy positions of all the cells in the AP or DV boundaries over all time after the adaption phase. We then calculated the distance of each cell to the new X axis (average position of the AP boundary) and Y axis (average position of the DV boundary).

The cell elongation tensor $\mathit{Q}$ is a traceless symmetric tensor that quantifies the anisotropy of cell shapes in a region of the tissue. We define cell elongation using a triangulation of the tissue obtained by connecting centroids of connected cells (Etournay et al., 2015; Merkel et al., 2017). The state of each triangle is described by a tensor $\mathit{s}$, defined by a mapping an equilateral reference triangle to the triangles in the tissue. The state tensor contains information about the triangle elongation tensor ${\tilde{\mathit{q}}}^t$, orientation angle $\theta$, and area ${\displaystyle a}$:

Here, $a_0^t$ is the area of the reference triangle, and $R$ is a 2-dimensional rotation matrix. Cell elongation in the tissue region is defined as an area weighted average of triangle elongation. For details about the method see Merkel et al., 2017.

The wing disc pouch has a slightly domed shape (Figure 1—figure supplement 1A). After projection onto a 2D surface, the cell shapes and areas will be distorted. To ensure that the radial profiles we measure in a 2D projection (Figure 1B,C,E and F and Figure 3F,G) are not a result of tissue curvature, we account for the distortion caused by projection. We use the height maps generated by our projection algorithm, which identifies the apical surface of the pouch within the 3D Z-stacks. We smooth this height map with a gaussian filter of width $\sigma =2\mu m$ to find the height field $h(x,y)$, and then we calculate the height gradient field. We then smooth the result again with the same gaussian filter to find $\overrightarrow{\nabla }h=({\partial }_{x}h,{\partial }_{y}h)$. The deprojected cell or triangle area $a_0$ is obtained from the measured area $a$ as $a_0=a\sqrt{1+\left|\overrightarrow{\nabla }h\right|}$, where $\overrightarrow{\nabla }h$ is evaluated at the cell center. To find the deprojected cell elongation of a triangle we evaluate $\overrightarrow{\nabla }h$ at the triangle center. Then, we find the angle of steepest ascent $\alpha =\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}({\partial }_{y}h/{\partial }_{x}h)$ in the projected plane and define the tilt transformation:

We apply this transformation to the triangle shape tensor $\mathit{s}$, as defined in Merkel et al., 2017, to determine the deprojected triangle shape tensor ${\mathit{s}}_0$:

The cell elongation tensor $\mathit{Q}$ is obtained from the triangle shape tensor ${\mathit{s}}_0$ as the corresponding area weighted average using the deprojected cell area, as described in Merkel et al., 2017.

To generate the color-coded smoothed plots of area and elongation (Figure 1B–C), we divided the aligned wings into a grid with boxsize ${\displaystyle =10}$ pixels (${\displaystyle \sim 2\mu m}$). For each position, we averaged cell area or performed an area-weighted averaged of triangle elongation in a neighborhood box = 20 pixels (${\displaystyle \sim 4\mu m}$). To create the nematic elongation pattern, we similarly averaged elongation of triangles whose centers lie within each grid box, with grid box size ${\displaystyle =30}$ pixels (${\displaystyle \sim 6\mu m}$). Tables containing the corrected cell area and triangle elongation values for each movie at each timepoint have be uploaded to Dryad: doi:10.5061/dryad.jsxksn06b.

We define the center of the wing pouch to be on the DV boundary. To determine the location of the center along the DV boundary $x_c$ we divide the pouch into four regions defined by the DV boundary and a line perpendicular to it located at some position $x$, as shown in Figure 1—figure supplement 1E. Then, we define a function:

for all values of $x$ along the DV boundary. Here, $A^{I-IV}$ are the areas of the four regions and $⟨Q_{xy}{⟩}_a^{I-IV}$ are the area weighted averages of the cell elongation component $Q_{xy}$ calculated in the four regions. Finally, $x_c$ is the value that minimizes $F\left(x\right)$ (Figure 1—figure supplement 1F). We find that the center point lies just anterior to the intersection with the anterior-posterior (AP) boundary (Figure 1, Figure 1—figure supplement 1, consistent with Legoff et al., 2013).

In time-lapse experiments, cell centers were determined using the last 20 timepoints. In single timepoint image experiments (Figures 5 and 7), all images of a single genotype were used together after alignment on the DV and AP boundaries.

In contrast to the other regions (blade, AP and DV boundaries), the definition of the band around the DV boundary (Figure 1—figure supplement 2) and the region of the blade that excludes this region (Figure 1D) were not defined manually using the FIJI scripts of TissueMiner. Rather, we defined them after the TissueMiner databases were generated, using the position of cells relative to the DV boundary in the last frame. Using the plots of the radial elongation on each disc, we estimated the width of the stripe region and then filtered for cells included in and excluded from this region. Once these cells in the last frame were identified, we used the UserRoiTracking.R of TissueMiner to backtrack these two regions, producing a list of all cells belonging to this lineage that are traceable forward and backward in the movie.

For the static RNAi data, we used the average width of the band around the DV boundary from timelapse data as an estimate and excluded this region from the static images to analyze the radial gradients in cell area and elongation (Figures 5 and 7).

For the timelapse data, we defined the radial elongation center (described above) for each movie and then calculated the distance away from this center for each cell. After excluding the band of cells around the DV boundary, we binned cells by radius by rounding to the nearest ${\displaystyle 10\mu m}$. We also binned the movie across five equal time windows (${\displaystyle \sim 2hr}$), excluding the adaption phase. Average cell area and area-weighted average of cell elongation were calculated for each of these bins in each time group for each movie. Including data from all five movies, we report the global average and its standard deviation (Figure 1E–F). For Figure 3, we averaged over the last ${\displaystyle \sim 5hr}$. For the band of cells around the DV boundary (Figure 1—figure supplement 2), we binned cells in $x$, rather than in radius, and report the gradient along the $x$ axis, defined as the DV boundary.

The static RNAi data were analyzed similarly. We report the global average of all discs in the genotype and the standard deviation (Figures 5D,I and 7D). The number of discs analyzed in each genotype is listed in the figure legend.

To analyze the pattern of isotropic deformation, we locally averaged cell behavior by dividing the tissue into a grid centered upon the crossing of the AP and DV boundaries. First, we determined the average position of cells in the AP and DV boundaries to generate a common frame of reference across all movies. Second, we divided the tissue visible in the last frame into a grid centered on these compartment boundary positions, with grid size = ${\displaystyle 8\mu m}$. Grid boxes that were incompletely filled (less than 33% of the area of the box filled by cells) were discarded to eliminate noise along the tissue border. Third, each grid box was considered an 'ROI' and then tracked through the entire movie using TissueMiner’s ROI tracking code. Fourth, the rate of deformation by each type of cellular contribution was calculated as a moving average (kernel ${\displaystyle =11}$) for each grid position for each timestep. Last, we averaged these rates over all time points post-adaption period and plotted in space (Figure 1H,I,K,L).

To show tissue deformation as a function of distance to the center, we first calculated the distance to the center of symmetry for each cell. Second, we divided the tissue visible in the last frame into radial bins, rather than a grid, by rounding the radial position of each cell to its nearest ${\displaystyle 10\mu m}$. Third, as we did for the grid, we defined each radial bin to be an 'ROI' and then tracked the region forward and backward using TissueMiner’s ROI tracking code. Fourth, the rate of deformation by each type of cellular contribution was calculated as a moving average (kernel ${\displaystyle =11}$) for each radial bin ROI at each timestep. Last, we averaged these rates over all time points post-adaption period and plotted this rate as a function of distance to the center. Note that we also show the spatial profiles of tissue growth in the band around the DV boundary in Figure 1—figure supplement 1, but here we binned along $x$, not $r$.

We previously published patterns of cellular contributions to anisotropic tissue shear from movies 1 to 3 (Dye et al., 2017); however here, we calculated these patterns in a more accurate way and report averages over multiple movies (Figure 2A). Previously, we assigned a grid at each timepoint as in Etournay et al., 2015; Etournay et al., 2016. While this method provides a simple first approximation of the pattern, it is not completely accurate because we are not tracking the box in time but reassigning it at each timepoint; thus, cell movement in/out of the box is not counted. Here, we perform grid box tracking, as described above for the calculation of cellular contributions to isotropic tissue deformation but with grid box size ${\displaystyle =15\mu m}$. Further, we present a global average of not just movies 1 to 3, but also the two new movies. We calculated for each movie the rate of tissue deformation by each cellular contribution as a moving average (kernel ${\displaystyle =11}$) in time. Then, we averaged across all five movies at each timepoint and presented the pattern as a cumulative sum, starting from the end of the adaption phase (first ${\displaystyle 2hr}$) and continuing through the end of the timelapse. All correlation terms were added together (see Merkel et al., 2017). The contribution to tissue shear from cell extrusion is very small and thus not shown.

We calculated a moving average (kernel size ${\displaystyle =11}$) total value for each cellular contribution to radial tissue shear, averaged across the blade (excluding the band around the DV boundary). Then, we accumulated the contributions after the adaption phase (first ${\displaystyle 2hr}$) and plotted over time (Figure 2B). In addition to the previously described types of correlations (Merkel et al., 2017), the radial decomposition also involves a correlation between cell area and shear (see Appendix 3). We added all correlation terms together in Figure 2B.

For the ablation data, we first projected the Z-stack images of the wing discs before and after ablation using our custom surface projection (Dye et al., 2017).

To quantify cell elongation and area, we used the images taken before the cut. Cells were segmented using the FIJI plugin TissueAnalyzer and then processed with TissueMiner to rotate the disc to a common orientation (anterior to the left; dorsal up) and to create a triangle network and a database structure. The area-weighted average of triangle elongation and the average cell area was calculated for those cells included in the center of a circle of a radius ${\displaystyle r=1.3\times r_{cut}}$ (${\displaystyle 4.55\mu m}$) centered at the center of the cut. Varying the size of this region only slightly affects the result: ${\displaystyle <1.0\times r_{cut}}$, the distribution becomes more noisy because we are averaging a smaller region with less cells, but regions that are too big (${\displaystyle \sim 1.8\times r_{cut}}$) may begin to include cells that span different regions of the wing (i.e. the band around the DV boundary). In Figure 4C and Figure 4—figure supplement 1B,E, we plot cell elongation projected onto the stress axis.

To measure the shape of the tissue after ablation, we fit ellipses to the inner and outer piece left by the cut using the projected after-cut images. These images were first rotated using the transformation performed on the before-cut images to orient the tissue. We used Ilastik (v. 1.2.2) (Berg et al., 2019) to segment the cut region: we trained the classifier using all the data from a single day’s acquisition, delineating three regions: cells, membranes, and dark regions. Using the trained classifier, we then generated a thresholded binary image of the cut tissue’s shape and cropped the image around the cut. We then fit inner and outer ellipses to these images.

To quantify the position of the cut in the wing pouch, the immunofluorescence post-cut images were projected using maximum intensity. In FIJI, we manually measured (in ${\displaystyle \mu m}$) the distance from the center of the cut to the DV boundary and AP boundary. We noted that fixation causes the tissue to shrink. We estimated the extent of shrinkage using a subset of the samples in which the compartment boundaries were visible in the Z-stack taken of the live sample before the cut. We used FIJI to measure distances from the center of the cut to the compartment boundaries in both the pre-fix live images and in the post-fix immunofluorescence images for this subset. We measure a discrepancy indicating that the tissue shrinks by ∼15%. Thus, we compensate for this shrinkage when measuring distances to the compartment boundaries in our analysis. We also noted the compartment (dorsal/ventral/anterior/posterior) in post-fix images and added a sign to the distances to the boundaries (ventral and posterior getting negative 'distances') to create the spatial map shown in Figure 4B.

To classify the cuts by position, we first estimated the width of the band around the DV boundary from the timelapse data to be ${\displaystyle \sim 22\mu m}$ (centered on the DV boundary). We then classified a cut as in the DV boundary if its cut boundary extended no more than ${\displaystyle 1.4\mu m}$ (20% of the ${\displaystyle 7\mu m}$ radius) outside this ${\displaystyle 22\mu m}$ horizontal stripe region. Likewise, cuts were classified as outside this region if it did not extend more than ${\displaystyle 1.4\mu m}$ into the ${\displaystyle 22\mu m}$ horizontal stripe. We excluded all cuts that appear to straddle the border between these regions (gray in Figure 4B). We also excluded those cuts with centers lying within ${\displaystyle 7\mu m}$ of the AP boundary, in case material properties at this boundary region are also different.

To obtain the normalized shear stress $\tilde{\sigma }/\left(2K\right)$, normalized isotropic stress ${\displaystyle \sigma /\overline{K}}$ and the ratio of elastic constant ${\displaystyle 2K/\overline{K}}$, we fit ellipses to the inner and outer tissue outlines simultaneously. These three parameters determine the large and small semi-axes of the two ellipses. Other fitting parameters are the center positions of the two ellipses and the angles of major axes. Stress is calculated by considering how the ellipses were generated by a circular laser ablation, as described in detail in the Appendix 2. When calculating the deformation from initial to final positions, we considered that the cells that were directly hit by the laser would die and thus not contribute to the evolution of the tissue after the cut. To account for this loss of cells, we first calculated average cell area for those cells that would be hit by the laser. Then, we adjusted the radius of the initial position by this amount.

Fits were performed on all of the laser ablations in the same region (either within or outside of the band around the DV boundary) for a range of fixed values of the ratio ${\displaystyle 2K/\overline{K}}$. The optimal value was considered to be the one that minimizes the sum of fit residuals (Figure 4—figure supplement 1C,F). We defined a threshold of fit residual, as shown in Figure 4—figure supplement 1A, to eliminate cuts with non-elliptical outlines. The uncertainty of ${\displaystyle 2K/\overline{K}}$ was estimated by bootstraping a subset of 7 cuts in each region 100 times; we report the standard deviation of the results.

It is interesting to notice that from Equation A11 and the value for $2K/\overline{K}=3.4\pm 0.4$ obtained from the laser ablation experiments (main text and Materials and methods), we can estimate

This value is significantly higher than the one obtained by a direct fit to the laser ablation data (Equation A14). This apparent discrepancy stems in part from the fact that the elastic constant $\overline{K}$ in Equation A5 is a non-linear elastic modulus. It can only be related to the one used in the laser ablation method by linearizing it around a typical value of $a$ in the data. When this value is not equal to ${\displaystyle a_0}$ the linearized elastic constant will differ from the original $\overline{K}$ in Equation A5. Furthermore, Equation A5 does not take into account any adaptation of cell area to pressure or nematic cell polarity, which could influence the measured value $K^{*}/K$ when fitting the cell area profile. However interesting, these effects are beyond the scope of our current work.

We propose a dynamical equation of the polarity tensor $q_{ij}$

where the parameter ${\displaystyle \mu }$ represents the mechanosensitive response of nematic cell polarity to shear stress. We use Equations A1, A2, A3 and A19 to obtain a dynamic equation for shear stress

The system of linear Equations A19 and A20 becomes unstable when

This instability will result in a finite value of nematic cell polarity in the tissue. To account for this polarized state we have to include a stabilising higher order term in Equation A19

In the steady state, the magnitude squared of the nematic cell polarity is

Here, we have introduced the strength of polarity ${\displaystyle q_0}$, corresponding to the magnitude of polarity in a homogeneous system. Note that the definition of $q_0^2$ allows for negative values, which would correspond to no spontaneous polarization in the tissue.

To account for spatial variations of nematic cell polarity, we introduce a Laplacian term in Equation A22

To fit the radial profile of $Q_{rr}$ observed in time-lapse experiments we account for non-vanishing $R_{rr}$ as well. Consistent with the observation that during the experiments $R_{rr}$ does not vary significantly in time (Figure 2B main text) we use approximation $\tau {\partial }_t{R}_{rr}\ll R_{rr}$. Then $q_{rr}$ satisfies

where the left hand side is the radial component of the Laplacian of $q_{ij}$ in the polar coordinate system. From the solution of this equation, the $Q_{rr}$ profile is obtained as $Q_{rr}\approx -\tau \lambda q_{rr}$. This quantity is fitted to the experimentally measured tangential cell elongation profile by optimizing four tissue parameters, as well as two boundary conditions $q_{rr}(r_{in})$ and ${\partial }_r{q}_{rr}(r_{in})$ required to solve Equation A25. Here $r_{in}=10\mu m$ is the smallest radius at which we measure the tangential cell elongation profile in Figure 6C of the main text. Parameters obtained by the fit are reported in Table 1 of the main text, and the fitted $Q_{rr}$ profile is shown in Figure 6 of the main text. Parameter uncertainties were estimated as the 10^th and 90^th percentile intervals obtained by fitting uniformly sampled profiles of $Q_{rr}$ from the interval $Q_{rr}(r)+\mathrm{\Delta }{Q}_{rr}(r)$ and $Q_{rr}(r)-\mathrm{\Delta }{Q}_{rr}(r)$, see Figure 3 of the main text, with 101 realisations. Obtained uncertainty intervals are reported in the Table 1 of the main text.

Spontaneous cell polarity in our model depends on mechanosensitive feedback characterized by the feedback strength ${\displaystyle \mu }$. The characteristic polarity strength ${\displaystyle q_0}$ grows monotonically with ${\displaystyle \mu }$. A prediction from our model is therefore that when ${\displaystyle \mu }$ is reduced, cell elongation will be reduced because it is mediated by cell polarity. In order to test this prediction, we performed MyoVI knockdown experiments, which indeed show a decrease in cell elongation (Figure 7).

Our qualitative prediction is rather robust and does not depend on details of the system, such as the influence of compartment boundaries that disrupt the radial symmetry. However, boundary conditions, which we do not know, could have a significant influence on the system. Therefore, a full quantitative prediction of experiments is difficult. To show the self-consistency of our prediction, we test if the value of ${\displaystyle q_0}$ that we use to describe the data is consistent with the actual cell elongations observed. Indeed we have $q_0\approx 0.1$, see Table 1 which predicts a typical strength of cell elongation $Q_0=-\lambda \tau q_0\approx -0.1$, consistent with experimentally measured values, see Figure 6C of the main text. This suggests that our qualitative prediction is robust given the uncertainties in the system.

The deformation of an elastic object is described by the displacement vector of each point on the sheet

where ${\displaystyle x_i}$ is the intial position of a point on the elastic sheet and $x_i^{\prime }$ is the position of that point after the deformation. Shape changes are described by the symmetric traceless gradient of displacement

which can be decomposed into the isotropic part $u=u_{kk}$ and the shear part ${\tilde{u}}_{ij}=u_{ij}-{\delta }_{ij}u/2$. Note that we use summation convention over repeated indices. In polar coordinates displacement gradient is given by

We are considering the linear elastic constitutive relation

where ${\tilde{\sigma }}_{ij}$ is the traceless symmetric component of shear stress, with shear elastic modulus $K$ and bulk elastic modulus $\overline{K}$. Here, we allow for the presence of active stresses due to a nematic field $q_{ij}$, corresponding to the nematic cell polarity in the wing disc. However, a constant active stress does not influence our results in the small deformation limit, since it simply redefines the reference configuration. We can therefore write this constitutive equation as

Force balance imposes

In two dimensions, force balance is satisfied by all stress fields whose components are derived from the Airy stress function $\mathrm{\Phi }$

which can be expressed in polar coordinates

Now, a compatibility condition, which follows from a requirement of continuity of the displacement field, can be written as

The general solution of this equation in polar coordinates, also called the Michell solution, is given by

Inferring the stresses from laser ablation experiments consists of solving two elastic problems using the Michell solution as we now show.

We consider a piece of elastic sheet stretched by stress ${\sigma }_{xx}={\sigma }_{xx}^0$, ${\sigma }_{yy}={\sigma }_{yy}^0$, ${\sigma }_{xy}=0$. After a circular laser ablation, the inner piece is under no stress. To infer the original stresses in the sheet, we determine the shape of the sheet boundary assuming the stresses are known. Then, given the boundary shape, we will be able to infer the stresses.

Stress in a sheet that is in mechanical equilibrium, under no external body force, is uniform throughout the sheet. Therefore, using Equation B3 we can integrate Equation B7 to find the components of the displacement vector ${\displaystyle u_i}$

where we have introduced

To relate the displacement to the shape of the inner piece boundary, we have to solve Equation B1 for $r(\phi )$, in the configuration illustrated in Appendix 2—figure 1

Using the fact that

we obtain

To simplify the notation, we introduce

and we write the solution in the form

We denote semiaxes of this ellipse along the $x$ and $y$ axes, as defined in Appendix 2—figure 1, by $a_{\text{in}}$ and $b_{\text{in}}$, respectively. We find that

Appendix 2—figure 1

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As in the case of the inner piece, we need to find the boundary shape as a function of the original stresses in the sheet. In this case, the normal stress at the ablation boundary is zero, but the sheet is not stress-free. To simplify the problem, we consider an initially stress-free sheet with a circular hole to which boundary stresses $-{\sigma }_{ij}^0$ are applied. For small deformations, where the elastic problem is linear and solutions of the problems commute, the boundary shape is equivalent to the one created by the laser ablation of a sheet under stress ${\sigma }_{ij}^0$.

Now, we have to solve the full elastic problem, since stresses are not uniform. We keep the terms of the Airy stress function $\mathrm{\Phi }$, which obey nematic symmetry of the problem and relax sufficiently fast at infinity. We also assume that the polar axis is oriented along one of the principal axes of stress tensor, such that the off-diagonal component of stress vanishes. We obtain

and for stress components, using Equation B10

The stress boundary condition reads

allows us to determine coefficients in Equation B29

Now we can calculate and then integrate the deformation gradient components to obtain

Finally, as for the inner cut, we have to solve

for $r^{\prime }({\phi }^{\prime })$. We express the solution using the angle $\phi$ as a parameter

Note the ratio of elastic constants appears as a parameter of shape.

Now we show that the shape defined by Equations B40 and B41 is an ellipse. To this end, we first define

This allows us to write Equations B40 and B41 as

Now we show that $r^{\prime }({\phi }^{\prime })$ is an ellpise. An ellipse equation can be written as

where $a_{\text{out}}$ and $b_{\text{out}}$ are semiaxes of the ellipse oriented along $x$ and $y$ axes, respectively. Starting from Equation B46 and expressing ${\phi }^{\prime }$ from Equation B44 we will show that we can recover Equation B45 for a particular choice of parameters $a$ and $b$.

We first calculate

Inserting these identities into Equation B46 we find

Now, we only need to find $a_{\text{out}}$ and $b_{\text{out}}$ for which

for all $\phi$. The solutions are

or in terms of the original parameters