The emergence of two-dimensional stresses in the plane of the wing blade suggests that there are physical constraints on the movement of wing epithelial cells near the margin. We wondered whether there might be a matrix connecting the wing epithelium to the overlying pupal cuticle in this region. To investigate this, we used a laser to destroy the region between the margin of the E-Cadherin:GFP expressing wing epithelium and the cuticle after the two had separated as a consequence of molting. Although this treatment does not apparently damage either the wing or the cuticle, it causes the wing epithelium to rapidly retract away from the cuticle within seconds (Figure 1A–B′′, Video 1). Laser ablation causes epithelial retraction when performed at any region along the wing blade margin—anteriorly, posteriorly or distally. During tissue flows, the now disconnected margin moves even further away from the cuticle, producing abnormal wing shapes (Figure 1C–F). This shows that the wing is physically restrained by apical extracellular matrix connections to the overlying cuticle, and that these connections are required to shape the wing during tissue flows.

Figure 1

Download asset Open asset

Video 1

Download asset

We wondered whether the large apical extracellular matrix protein Dumpy might contribute to these connections. Dumpy is a 2.5 MDa protein that is predicted to form filaments at least 1 µm long (Wilkin et al., 2000). It forms an elastic matrix in the embryonic tracheal lumen, and provides mechanical resilience of tendon cell attachments to the overlying cuticle (Dong et al., 2014). While dumpy null mutations are lethal, some hypomorphs produce wings that are short and misshapen—a defect that arises during pupal development (Waddington, 1939, 1940). To ask whether shape defects in dumpy wings might arise during pupal tissue flows, we imaged dumpy^ov1 pupal wings that expressed E-Cadherin:GFP. The shape of dumpy^ov1 wings is normal at 16 hr after puparium formation (APF), before molting occurs (Figure 2A,B). Shortly afterwards, when hinge contraction begins, the shape of the dumpy^ov1 mutant wing blade begins to differ from wild type (WT). The wing blade epithelium retracts abnormally far from the distal cuticle and fails to elongate in the PD axis. By the time tissue flows have ended, the characteristic abnormal shape of the dumpy^ov1 wing is apparent (Video 2 and Figure 2A–B′′).

Figure 2 with 3 supplements see all

Download asset Open asset

Video 2

Download asset

To examine Dumpy distribution, we imaged wings from flies harboring a protein trap construct that expresses YFP:Dumpy from the endogenous chromosomal locus. YFP:Dumpy is present on the apical surface of epithelial cells throughout the wing, and within the overlying cuticle (Figure 2—figure supplement 1A, Video 3). Interestingly, Dumpy is also present in a fibrous-appearing matrix that connects the wing to the overlying cuticle in specific places. This matrix lies between the cuticle and the margin of the wing (Figure 2C–E,H), as well as in stripes that run on the dorsal surface of the wing between longitudinal veins L3 and L4, and over veins L2 and L5 (Figure 2F,G, Figure 2—figure supplement 1B). Dumpy-containing matrix also connects a subregion of the wing hinge to the overlying cuticle (Figure 2F,G, Figure 2—figure supplement 1B, Video 3).

Video 3

Download asset

To investigate the extent to which wing margin constraints had been relieved by the dumpy^ov1 mutation, we performed laser-severing experiments in the dumpy^ov1 mutant background. Cutting a dumpy^ov1 wing between the cuticle and the distal wing blade revealed almost undetectable retraction, suggesting that distal attachments of the wing blade to the cuticle are severely compromised. However retraction was still observed when severing was performed between the cuticle and the anterior or posterior margins (Figure 2—figure supplement 2). Furthermore, severing the matrix around the entire margin causes dumpy^ov1 mutant wings to develop even more dramatic wing shape abnormalities during tissue flows (compare Figure 2—figure supplement 2G to Figure 2A′′). This suggests that apical matrix connections to the cuticle are not completely abrogated in dumpy^ov1 wings.

To ask how the dumpy^ov1 mutation influenced PD tension in the wing blade, we performed circular laser cuts covering about 5–10 cells in different regions of WT and dumpy^ov1 wings (Figure 2I,I′, Figure 2—figure supplement 3). We observed a recoil of the ablated region, indicating that the blade epithelium is under tension. From the recoil, we can compare both the isotropic and the anisotropic components of epithelial stress in WT and dumpy^ov1 mutant wings (Figure 2I,I′ and Figure 2—figure supplement 3). These stress patterns differ between WT and dumpy^ov1 wings. The orientation of anisotropic stress in dumpy^ov1 is somewhat splayed and not as well aligned with the PD axis. Furthermore, anisotropic tension in dumpy^ov1 wings tends to be reduced in the central region and increased anteriorly and posteriorly.

Overall, Dumpy-dependent elastic connections are key to the emergence of the stress pattern during morphogenesis. This suggests that these stresses play an important role in guiding tissue flows.

What are the cellular events that shape the wing blade during tissue flows? To quantitatively address this question, we developed methods to quantify cell shape changes, cell divisions, cell rearrangements and cell extrusions during wing morphogenesis. We imaged three E-Cadherin:GFP-expressing wings at cellular resolution every 5 min between 16 and 32 hr APF. We then extracted and projected the planes containing the apical adherens junctions, automatically detected cell edges, and tracked each cell in the wing over the course of the videos (Figure 3A,A′ and Videos 4, 5). We designed a relational database (DB) to store information pertaining to all cells in a given video (‘Materials and methods’, Long-terms time-lapse imaging and Data handling and image processing). Querying these DBs provides information about individual cellular properties such as shape, area, and associated cell boundaries. It also provides information about neighbor and lineage relationships, identifying neighbor exchanges (T1 transitions), cell divisions and cell extrusions (T2 transitions).

Figure 3 with 2 supplements see all

Download asset Open asset

Video 4

Download asset

Video 5

Download asset

We showed previously that the area of the wing blade remains fairly constant during pupal morphogenesis, despite the fact that cells are dividing (Aigouy et al., 2010). To understand how the wing blade maintains area homeostasis, we quantified wing blade area, cell divisions, cell area changes and cell extrusions between 16 and 32 hAPF in three different WT wings (Figure 3A–E). We analyzed a large region of the wing blade in which all cells could be tracked during the whole video (green in Figure 3A,C).

The tissue area expansion rate is $v=\frac{1}{A}\frac{dA}{dt}$, where A is the tissue area and d/dt denotes the total time derivative. The tissue expansion rate can be decomposed into its cellular contributions as

where a is average cell area, k_d and k_e are cell division and extrusion rates, respectively. The cumulative area expansion rate of the whole wing blade is ln(A/A₀), where A₀ is the tissue area when recording starts (∼16 hAPF), can be obtained by integrating the area expansion rate v over time.

Figure 3D shows both the tissue area expansion rate (dark blue) and the contributions to this expansion rate from cell area changes, cell divisions and extrusions. These are averages over three WT wings. Figure 3E shows the corresponding cumulative quantities. The dynamics of wing area changes in the 3 WT wing blades are extremely similar—after contracting slightly during the first half of morphogenesis, blade area gradually returns to very close to its original value (dark blue lines in Figure 3D,E). This almost constant area reflects a balance between cell divisions (orange) on the one hand, and cell area changes (green) and extrusions (light blue) on the other. Interestingly, blade area in three analyzed wings is more reproducible than would be expected from the variation in each cellular contribution, if they were independent of each other (see shaded regions depicting standard deviations in Figure 3D,E). To quantify this observation we compared the variance of overall relative area change with the sum of the variances of the cumulative cellular contributions. We find that sum of variances is about 20 times larger than variance of the sum. This shows that cellular contributions are not independent and that normal variations in the rate of cell division can be compensated by changes in cell area and/or extrusion to maintain wing blade area (Figure 3—figure supplement 1).

To ask whether connections to the cuticle were required to maintain wing blade area, we examined blade area changes and the underlying cellular contributions in dumpy^ov1 mutant wings (Figure 3F,G). As a complementary approach, we quantified the cellular contributions to area changes in laser-severed wings. We severed the wing either between the hinge and the blade, or at the very distal tip before hinge contraction occurred (Figure 3—figure supplement 2). We also severed connections between the wing margin and the cuticle (Figure 1D,F) at about 22 hAPF. In contrast to unperturbed WT wings, total wing area decreases dramatically when connections at the margin are weakened by dumpy^ov1 mutation or by laser severing (Figure 3G and Figure 3—figure supplement 2, see dark blue curve). Thus, connections to the cuticle are required to maintain wing blade area during morphogenesis. These connections provide mechanical linkages that permit the buildup of tensile stresses while maintaining wing blade area.

How does epithelial stress influence the cellular events contributing to area homeostasis? To answer this question, we first compared cellular contributions to area change during morphogenesis of WT and dumpy^ov1 mutant wings. Wing blade area decrease in dumpy^ov1 mutant wings is not a consequence of fewer cell divisions—cells actually divide more than in WT (Figure 3—figure supplement 2D–F,I). Cells in dumpy^ov1 mutant wings have a similar maximum division rate but divide over a longer period of time, resulting in more cells at the end of morphogenesis (Figure 3G and Figure 3—figure supplement 2D–F, yellow curve). The reduced wing blade area in dumpy^ov1 is quantitatively explained by reductions in cell area and by cell extrusions, which more than compensate the increased proliferation. Thus, reduced epithelial stresses in dumpy^ov1 wings perturb the balance between cell divisions, area changes and extrusions seen in WT.

All laser-severing perturbations decrease the final wing area, similar to dumpy^ov1 mutant wings (Figure 3—figure supplement 2). In these wings, the analysis of cellular contributions to wing area changes is complicated by the delay between cutting and the initiation of time-lapse imaging (about 45 min). During this intervening time, the wing responds acutely to reduced tension, and both wing and cell area decrease to values below those expected for WT wings of the same stage (Figure 3—figure supplement 2D–F,H). While we can estimate changes to cell area during this time, we cannot know the rates of cell division and extrusion. Nevertheless, several interesting conclusions can be drawn by analyzing final cell area, and the rates of division, area change and extrusion after time-lapse imaging begins. Wings that have been severed before hinge contraction (whether at the hinge-blade interface or at the distal tip) behave similarly to dumpy^ov1 mutant wings. After an initial delay, the rate of cell division increases and cells divide more than in unperturbed wings. However, cell extrusions and decreasing cell area more than compensate for increased cell division to produce smaller wings. When cuticle connections are severed later at 22 hAPF, most cell divisions have already occurred, and this treatment does not increase proliferation. In this case, the number of cell extrusions increases, and the final cell area is smaller than that of unperturbed wings. Taken together, analyzing dumpy^ov1 and laser-severed wings shows that epithelial stresses are required to balance cell divisions with cell extrusions and cell area changes to maintain area homeostasis during morphogenesis. This is consistent with observations in the thorax, where overcrowding drives delamination (Marinari et al., 2012).

In the preceding section, we discussed how cellular processes contributed to wing blade area changes. These area changes correspond to the isotropic component of a tensor characterizing the tissue strain (Figure 4A). Now, we discuss shape changes of the wing blade, which correspond to the anisotropic part of this tensor and characterize the process of elongation along an axis (i.e., pure shear). The rate of change of pure shear is described by the pure shear rate tensor $\stackrel{\sim }{v}$ (Figure 4B) with ${\stackrel{\sim }{v}}_{xx}$ characterizing the rate of elongation along the PD axis (Figure 1A). Note that pure shear, that is, convergence-extension flow, is different from so-called simple shear, which results from a superposition of pure shear and a rotation (Figure 4C). In the following, we use the term shear to refer to pure shear. We now discuss how tissue shear can be decomposed into contributions from cell shape changes and topological changes. These include cell divisions, cell neighbor exchanges (T1 transitions), and cell extrusions (T2 transitions) (Figure 4D–G).

Figure 4

Download asset Open asset

To understand the cellular contributions to the overall shear of the wing blade, we developed a method to distinguish and quantify shear caused by cell shape changes and shear caused by topological changes. In a piece of tissue where no topological changes occur within the cellular network, the deformation of the individual cells defines the deformation of the whole piece of tissue. However, when topological changes occur, deformation of individual cells no longer completely accounts for the overall shear (Figure 4D–H). The triangle method we outline below represents an exact geometrical formalism to decompose large-scale deformations of the wing blade into contributions by cell deformation and by each kind of topological change.

First, we tile the cellular network with triangles as follows. Each vertex of the cellular network that touches three cells (red dot in Figure 4J) gives rise to a single triangle (red), whose corners are defined by the centers of the three cells (green dots). Vertices that touch more than three cells are treated as described in Appendix 1, ‘Triangulation procedure’. The resulting set of triangles tiles the cellular network without gaps or overlaps (Figure 4K). We choose a tiling into triangles, because the deformation of a single triangle between two frames of the video can be uniquely characterized by a single 2 × 2 tensor describing a linear transformation (see Appendix 1, ‘Triangle deformation’). Note that such a characterization by a 2 × 2 tensor is in general not possible for polygons with more than three sides. For each triangle and time point, this tensor describes relative area changes, rotation, and shear of the triangle. The average shear rate of all triangles in the tissue corresponds to the overall tissue shear rate. To connect the tissue shear rate to cell elongation changes, we define a nematic tensor Q characterizing the state of triangle elongation (see Appendix 1, ‘Triangle elongation’). Then, the change of triangle elongation corresponds exactly to triangle shear. Cell elongation is obtained as the average of the elongation tensors Q of those triangles that belong to a given cell. Hence, in the absence of topological changes, we find an exact relation between cell elongation change and overall tissue shear (Appendix 1, ‘Large-scale shear in the absence of topological transitions’).

If topological changes occur, then average cell elongation change is not equal to tissue shear. To include the effect of topological transitions, we write (Appendix 1, ‘Contributions to shear by topological changes’):

where $\stackrel{\sim }{v}$ is the overall tissue shear rate tensor, DQ/Dt is a corotational rate of change in average triangle elongation, and R is the shear rate tensor due to topological changes. Contributions to R include T1 transitions (T), cell divisions (C), T2 transitions (E), as well as correlated cell shape changes and cell rotations (D): R = T + C + E + D (see Figure 4I).

How can we define the contributions by topological changes (T, C, and E) to tissue shear? During a topological transition, the triangulation changes and thus the average triangle elongation changes (Figure 4L–O). However, at the moment the topological change occurs there is no tissue shear. Therefore, tissue shear and triangle elongation are no longer the same. This can be compensated by introducing a contribution to tissue shear by topological transitions. This contribution corresponds to the negative change in the average triangle elongation caused by the change in the triangulation (Figure 4M and Appendix 1, ‘Intermediate network states’ and Appendix 1, ‘Contributions to shear by topological changes’).

In the definition of R in Equation 2, we have also introduced the contribution D to tissue shear, which accounts for collective cellular events that combine to increase average triangle elongation in the absence of tissue shear and topological transitions. This occurs when several triangles have fluctuating shapes, such that the instantaneous elongation and the rotation rate or area expansion rate of triangles are correlated. Note that this effect does not occur when several triangles undergo equal deformations and rotations. One example of a cellular network deformation that produces the contribution D to tissue shear is shown in Figure 4H. Here, cells in neighboring rows slide relative to each other in alternating directions, such that no net pure shear occurs. However, there are alternating rows of simple shear and a net change in triangle elongation. We introduce the contribution D in the definition of R in order to compensate for the increase in the average triangle elongation DQ/Dt stemming from such correlations, should they exist in the wing blade.

We first used the triangle method to calculate the patterns of tissue shear and cellular contributions to this tissue shear in WT (Figure 5, Video 6, and Appendix 1, ‘Spatially resolved shear patterns’) and dumpy^ov1 mutant wings (Figure 5—figure supplement 1 and Video 7). To visualize these patterns we averaged all quantities within squares of about 26 × 26 µm². Figure 5 shows shear patterns in WT at early, intermediate, and late time points during pupal wing morphogenesis. Shear is indicated by a line whose orientation represents the shear axis and whose magnitude corresponds to the shear rate.

Figure 5 with 1 supplement see all

Download asset Open asset

Video 6

Download asset

Video 7

Download asset

Tissue shear in the WT wing blade is oriented in a fan-shaped pattern with a strong PD component (Figure 5A–A′′). At about 21 hAPF, the shear pattern develops a sharp reorientation between veins L3 and L4, where shear is oriented along the AP axis. This region corresponds to a stripe of Dumpy-containing matrix that attaches the blade to the cuticle (Figure 2—figure supplement 1B). Shear decreases over time, beginning distally and proceeding proximally, and finishes by about 32 hAPF.

What are the patterns of cellular contributions to the tissue shear? These patterns reveal a surprising complexity that changes with time. Shear caused by cell elongation and cell rearrangements (T1 transitions) display significant contributions that are antagonistic. Unexpectedly, T1 transitions cause shear along the AP axis early in the process (Figure 5B–B′′). This shear is opposed by increasing cell elongation along the PD axis (Figure 5C–C′′). At intermediate and late stages, T1 transitions shift their average orientation to cause PD shear—at the same time, cells reduce their elongation along the PD axis, causing AP shear. The contribution of cell extrusions to tissue shear is negligible (not shown) and cell divisions result in significant shear only during the early stages of wing morphogenesis (Figure 5D–D′′). Finally, plotting the contribution D to shear due to correlation effects reveals that these effects do exist in the wing (Figure 5E–E′′). Thus, patterns of cell elongation, cell rearrangement, cell division and correlation effects make dynamically changing contributions to tissue shear that are sometimes antagonistic.

To investigate the cause of altered wing shape in dumpy^ov1, we performed a similar analysis. The patterns of cellular contributions to tissue shear in a dumpy^ov1 mutant wing display subtle abnormalities (Figure 5—figure supplement 1). However, a more quantitative analysis is required to understand the origin of the altered dumpy^ov1 mutant wing shape.

To better understand the quantitative relationships between the cellular processes contributing to tissue shear, we studied spatially averaged shear in the wing blade projected on the PD axis. We quantified average tissue shear, and shear caused by each cellular contribution over time in three different WT videos. These averages were taken over the tracked region shown in Figure 3A and the resulting quantities were further averaged over the three videos. These averages and the standard deviations between the videos are shown in Figure 6A,B. Positive values indicate shear along the PD axis and negative values indicate shear orthogonal to the PD axis (i.e., shear in the AP axis). Adding the contributions of shear caused by cell divisions, cell rearrangements, cell shape changes, and correlation effects (light-pink line in Figure 6—figure supplement 1A) reproduces the independently calculated total shear curve (blue line in Figure 6A and Figure 6—figure supplement 1A). Small differences between these two curves (about 3%) stem from small inaccuracies (see Appendix 1, ‘Decomposition of the large-scale tissue shear rate’). Thus, we can decompose tissue shear into its individual cellular contributions.

Figure 6 with 1 supplement see all

Download asset Open asset

Does the cumulative tissue shear we calculate account for the shape change of the wing blade? To verify this, we characterize the blade shape by a nematic determined by the outline of the tracked region (see Appendix 1, ‘Characterization of wing blade anisotropy’). The change of this nematic with respect to its initial value over time is shown together with the cumulative tissue shear obtained by the triangle method (Figure 6—figure supplement 1B). These quantities agree well, indicating that the shear projected on the PD axis accounts for the main features of tissue shape changes. Thus, we can now use the shear decomposition method to discuss how different cellular events contribute to shape change of the wing over time.

On average, the wing shears smoothly along its PD axis between 17 and 32 hAPF as the hinge contracts. In contrast, the cellular processes that combine to produce tissue shear change over time. During the first 6 hr of our videos, shear caused by cell elongation in the PD axis (green curves in Figure 6A,B) is even larger than PD tissue shear (blue curves). This shows that cell elongation increases more than the tissue elongates suggesting that active cellular processes also contribute to PD cell elongation. Subsequently, starting at about 22 hAPF, cell elongation decreases although the blade continues to elongate. These discrepancies between cell shape changes and tissue shape changes require topological changes in the cell network. To more clearly discuss these events, we define two distinct phases of wing morphogenesis (phases I and II) that are separated by the peak of cell elongation occurring at about 22.5 hAPF.

Quantifying shear caused by T1 transitions and by cell elongation reveals that they change dynamically throughout morphogenesis with a striking reciprocal relationship (green and red curves in Figure 6A,B). This reciprocal relationship accounts to a large extent for the discrepancies between cell elongation and tissue shear. It further suggests that the active contribution to cell elongation (i.e., the amount of cell elongation that exceeds tissue shear) may be linked to AP-oriented T1 transitions—the orientation of these T1s, which work against the observed tissue shear, suggests that they are autonomously controlled. Active AP-oriented T1 transitions could produce PD cell elongation if mechanical constraints prevent the wing from shearing. In principle, it is also possible that active PD-oriented cell shape changes could produce AP-oriented T1 transitions under the same constraints. Cell divisions also contribute to PD shear in the wing blade. Although cell divisions initially cause a small amount of AP shear, their direction changes during phase I such that their net contribution shears the wing in the PD axis. In addition, correlation effects produce significant shear in the AP direction and contribute most strongly at the time that T1 transitions are changing their orientation.

In summary, the continuous large-scale deformation of the wing blade emerges from complex patterns of cell dynamics on small scales. During phase I, cells undergo AP-oriented T1 transitions while elongating in the PD axis. Cell divisions during phase I contribute shear along the PD axis. During phase II, the orientation of T1 transitions shifts to the PD axis and cells relax their shape. Correlation effects contribute AP shear, and peak roughly at the time that T1's shift their orientation.

To ask whether the cellular contributions to tissue shear were independent of each other, we compared the sum of the variances of the final cellular contributions to tissue shear with the variance of final tissue shear itself. The ratio of these values is about 25, indicating that the cellular contributions to tissue shear—like the contributions to area change—are not independent of each other. Thus, the overall tissue shear is more reproducible than would be expected from the variations of the cellular contributions.

What exactly are cells doing that results in correlation effects? We found that shear due to correlation effects was mainly generated by correlations between local elongation and rotations (Figure 6C–I). To investigate this further, we determined the magnitude and orientation of shear and the rotation rate associated with each triangle for each frame of the video. We observed that triangles rotated and sheared in striking spatial patterns that rapidly fluctuate in time (Figure 6C,F,G). These patterns correspond to rows of correlated shear and rotation that are distributed throughout the wing blade. To characterize these correlated patterns, we calculated the spatial power spectrum of the local tissue rotation rate (Figure 6D, Appendix 1, ‘Power spectrum of local tissue rotation’). This revealed that shear and rotation are correlated in regions corresponding to PD-oriented rows that were about 3–7 triangles long. These rows consist of triangles that all rotate and shear in the same direction. The rows are interspersed with other rows of similar length with mirrored patterns of shear and rotation (note blue and red rows in Figure 6F,G). Such a pattern of rotation and pure shear is characteristic of neighboring rows of triangles and cells undergoing simple shear in alternating directions (Figure 6H). This would occur if PD-oriented rows of cells slide with respect to each other. As discussed above, such rearrangements can indeed contribute to correlation terms (see cartoon in Figure 4H).

If rows of cells slide past each other, cells typically engage in T1 transitions. Since the peak of AP correlation effects coincided with a shift in the net orientation of T1 transitions (i.e., when the red curve in Figure 6A crosses 0), we wondered whether correlation effects could be associated with T1 transitions at this time (see also Figure 6—figure supplement 1C). We therefore examined whether correlation effects were associated with a particular type of topological change. Indeed, correlation effects are mostly accounted for by those cells that are about to undergo a T1 transition within the next 9 frames, although they cover only a small fraction of the total area (Figure 6I, Appendix 1, ‘Role of T1 transitions in the correlation-induced shear’).

For rows of cells to slide past each other, cells would have to undergo a peculiar type of T1 transition in which the orientation of the boundaries gained and lost are similar. Boundary orientations around T1 transitions are difficult to measure, because boundary length is small. The triangulation method provides us with a more robust measure of a bond orientation. As a proxy for cell boundary orientation, we use the orientation of the line connecting the centers of the two corresponding triangles (Figure 6—figure supplement 1D). We calculated the proportion of connection losses or acquisitions occurring in different directions over the course of morphogenesis, and compared them with the distribution of connection angles in general. We observe that at early times, when T1-induced shear is mainly along the AP axis, connections are typically lost along the PD axis and gained along the AP axis (Figure 7A–D). At intermediate times, when the correlation term is maximal, both lost and gained connections were oriented along the PD axis (Figure 7B). This is consistent with the unusual T1 transitions associated with sliding cell rows. Finally, as T1 transitions begin to cause net PD shear, AP connections are preferentially lost and PD connections are gained (Figure 7D). Thus, as T1 transitions shift from an AP- to a PD-oriented shear, they pass through an intermediate state where connection gains and losses are still oriented but do not cause shear. Interestingly, at the same time, the correlation term has maximal magnitude. This suggests that the correlation effects are related to these unusual patterns of T1 transitions at intermediate times.

Figure 7

Download asset Open asset

To investigate which cellular events depended on epithelial stresses, we quantified shape changes and cellular contributions to tissue shear in the dumpy^ov1 mutant wing blade (Figure 8A,B). As a complementary approach, we also studied these events in wings that had been subjected to laser severing (Figure 8C–F and Figure 8—figure supplement 1). In dumpy^ov1 wings, cells experience hinge contraction but the counterforces exerted by cuticle connections seem to be reduced (Figure 2A–A′′). Tissue shear is dramatically altered in dumpy^ov1 mutant wings as compared to WT—instead of shearing in the PD axis, these wings shear on average along the AP axis (Figure 8A,B). Examining the different contributions to tissue shear in dumpy^ov1 wings shows that the rates of AP shear caused by T1 transitions and by correlation effects are similar to WT and persist for longer times. Thus, these processes are likely to be autonomously driven. By the end of the video, they cause more accumulated AP-oriented shear than in WT. Analogously, cell divisions cause more cumulative PD shear than in WT—consistent with the increased number of cell divisions in the dumpy^ov1 mutant wing. In contrast, cell elongation during phase I causes less PD shear than in WT. Thus, PD-oriented epithelial stresses must contribute to PD cell elongation. Interestingly, the increase of cell elongation in the PD axis still exceeds the increase of elongation of the blade in the dumpy^ov1 mutant wing. This suggests that autonomous cellular processes cause the residual PD cell elongation in dumpy^ov1 mutant wings. Finally, PD shear by T1 transitions in phase II is smaller than in WT. This is not due to a premature cessation of T1's—indeed quantifying the rate of T1 transitions (regardless of orientation) shows that they occur at a higher rate and for a longer time than in WT wings (Figure 8—figure supplement 2). Rather, T1 transitions fail to orient as effectively with the PD axis in the dumpy^ov1 mutant wing (see shear patterns in Figure 5—figure supplement 1B–B′′).

Figure 8 with 3 supplements see all

Download asset Open asset

As a complementary approach, we asked how tissue shear and cell dynamics were altered in wings that had been subjected to laser severing. We first examined wings that had undergone laser severing before hinge contraction at the distal tip of the wing blade (Figure 8C,D and Figure 8—figure supplement 1A,B). These wings experience mechanical conditions similar to those in dumpy^ov1 wings in that they undergo hinge contraction but cannot attach properly to the distal cuticle. Consistent with this, the final shape of distally severed wings resembles that of dumpy^ov1 wings. Furthermore, distally severed wings show similar tissue shear and cellular contributions to shear as dumpy^ov1 mutant wings. These observations suggest that weakened connections to the distal cuticle are key to the altered wing shape of dumpy^ov1 mutants.

To perturb PD stresses even more strongly, we severed the region between the hinge and the blade before hinge contraction occurred (Figure 8E,F). These wings are not subjected to externally generated PD stresses at all, although they may still experience autonomously-induced stresses. We expected that cellular events that depended on externally generated stresses would be even more strongly perturbed in proximally severed wings than in dumpy^ov1 or distally-severed wings. The proximally severed wing undergoes significant but short-lived PD shear during the molting process until 18 hAPF. Subsequently, PD shear stops and becomes even negative by 20 hAPF. Cells do not elongate in the PD axis as much as in WT wings, although PD cell elongation still exceeds PD tissue shear. Again, this suggests that only a fraction of PD cell elongation is normally caused by external stresses, and that autonomous cellular events must produce the residual PD cell elongation in proximally severed wings. PD shear due to cell division is larger than in WT (as it is in dumpy^ov1 and distally severed wings) confirming that these divisions do not depend on external stresses. Furthermore, like dumpy^ov1 and distally severed wings, proximally severed wings undergo greater AP shear resulting from correlation effects. Thus the cellular events underlying correlation effects produce even more shear when stresses are reduced. However later, T1 transitions fail to generate significant PD shear in proximally severed wings. The reduction in T1-dependent PD shear is much stronger than in either dumpy^ov1 or distally severed wings, confirming that reorientation of T1 transitions in phase II is dependent on externally generated PD stress. At the very beginning of the video, PD-oriented connections are preferentially lost and AP-oriented connections are preferentially gained, consistent with the AP shear caused by T1 transitions at this time (Figure 8—figure supplement 2C–F). However, unlike WT, the preferential loss of connections never shifts towards the AP axis. Loss of connections remains biased towards the PD axis throughout the video—despite the fact that net shear caused by T1 transitions becomes very small. Shear caused by T1 transitions becomes small because the preferred orientation of gained connections gradually shifts from the AP to the PD axis. By the end of the video, both the assembly and disassembly of cell boundaries are preferentially oriented along the PD axis. These observations suggest that PD-oriented cell boundaries have a greater tendency to disassemble than those oriented at other angles, and that this is an autonomous, planar polarized feature of wing epithelial cells.

To disturb connections to the overlying cuticle without damaging the wing epithelium we disrupted apical extracellular matrix between the wing margin and the cuticle shortly before the onset of phase II, when this region becomes accessible (see Videos 8, 9). When anterior connections are severed, the wing blade shears even more in the PD axis while the area decreases slightly (Figure 8—figure supplement 3). This suggests that these connections restrain the narrowing of the wing blade in the AP axis. Increased PD tissue shear is mainly a consequence of T1 transitions. Since severing anterior connections reduces AP stress while PD stress does not change, this supports the idea that T1 events during phase II are oriented by anisotropic stress.

Video 8

Download asset

Video 9

Download asset

When distal connections are severed, cells rapidly reduce their elongation and area (Figure 8—figure supplement 3). Surprisingly, the rate of PD shear caused by T1 transitions continues to increase as it does in WT at this stage. However, after about 4 hr the PD shear rate due to T1 transitions decreases prematurely. Since the analysis of dumpy^ov1 and proximally severed wing shows that PD T1 transitions are stress dependent, this suggests that there is a time delay between the change in tissue stress and the resulting T1 transitions.

Taken together, the analyses of dumpy^ov1 wing blades and laser-severed wing blades distinguish autonomously driven cellular processes from those induced by tissue stresses. PD cell elongation and PD-oriented T1 transitions clearly depend on tissue stresses, whereas PD-oriented cell divisions are autonomously driven. AP-oriented T1 transitions and a corresponding fraction of PD-oriented cell elongation are also driven autonomously, as are the cellular events underlying correlation effects.

The overall picture that emerges is that changes of wing blade shape arise due to force balances that involve stresses exerted at the boundary of the tissue, and internal tissue stresses. Boundary stresses are due to hinge contraction and to the resistance of extracellular matrix attachments to the cuticle. Internal tissue stresses are generated by cell autonomous processes, like T1 transitions, and by elastic cell deformations. Tissue mechanics depends strongly on elastic connections of the wing to the cuticle. We must now understand how these isotropic and anisotropic mechanical stresses in the tissue, combined with boundary stresses, lead to cell and tissue remodeling. The interplay between boundary stresses and forces generated in the tissue is complex and requires a physical approach. We now present a continuum mechanical theory to understand these force balances and to calculate both tissue and cell shape changes.

We first define tissue material properties, starting with elastic properties, adding cell autonomous stresses and tissue shear due to topological changes. We then introduce elastic linkers to the surrounding cuticle. Finally, we compare predictions of this theory to the experimentally measured cell and tissue shape changes and determine key biophysical parameters characterizing tissue material properties.

We first investigated stresses present in the tissue. In an elastic tissue, tissue deformations stem from cell shape changes and cell elasticity is responsible for tissue elasticity. For small deformations, Hooke's law states that the mechanical stress in the material is proportional to its deformation. We write the isotropic part of the stress

where P denotes two-dimensional tissue pressure, a and a₀ are cell area and preferred cell area respectively, and $\overline{K}$ is the area compressibility. The preferred cell area a₀ changes when cells divide. For small (a − a₀)/a₀, Equation 3 corresponds to Hooke's law. Equation 3 implicitly contains the active contribution to pressure, which influences the preferred cell area a₀ (see Appendix 2, ‘Constitutive equation for the tissue stress’).

We now focus on the anisotropic part of the stress $\stackrel{\sim }{\sigma }$, also called the shear stress. For simplicity, we write the elastic anisotropic stress in the form of Hooke's law ${\stackrel{\sim }{\mathit{\sigma }}}_e=2K\mathit{Q}$, where K is a shear elastic modulus and Q is the cell elongation. In this expression, the cell shape is isotropic in the absence of anisotropic stresses. However, in tissues, planar polarized cells may spontaneously elongate. Therefore, we postulate the following constitutive equation for the anisotropic tissue stress

where ζ is a tensor that can be interpreted as a cell autonomous active stress related to spontaneous cell elongation.

To test whether Equation 4 accounts for anisotropic stresses present in the tissue, we further analyzed the circular laser ablations performed in different regions of WT and dumpy^ov1 wings at 22 hAPF (Figure 2I,I′). We defined a shear rate ${\stackrel{\sim }{v}}^{\mathrm{cut}}$ that characterizes the anisotropic recoil of the circular cut boundary into an elliptic shape (see ‘Materials and methods’, Analysis of circular laser ablations). We plotted the projection of this shear rate on the PD axis ${\stackrel{\sim }{v}}_{xx}^{\mathrm{cut}}$ as a function of the projected average cell elongation Q_xx (Figure 9A). Cell elongation was determined as the average cell elongation within the corresponding region in unperturbed wings. We found that the anisotropic part of the shear rate varied linearly with cell elongation (Figure 9A). The positive slope of this linear relation indicates that the shear modulus K is positive. In this argument, we use the recoil shear rate ${\stackrel{\sim }{v}}^{\mathrm{cut}}$ as a measure proportional to tissue stress. A linear fit to this data also has a positive intercept, corresponding to a positive ζ_xx in Equation 4. This implies that wing blade cells would spontaneously elongate along the AP axis in the absence of stress. Equivalently, in the absence of cell deformation (Q = 0) wing blade cells are exerting higher contractile stress in the PD axis than in the AP axis. The relative contributions of elastic stress and stress associated with spontaneous cell elongation can be quantified from this linear fit. The ratio of the intercept and the slope of the linear fit equals ζ_xx/2K, which leads to the estimate ζ_xx/K = 0.333 ± 0.003 in WT and ζ_xx/K = 0.316 ± 0.004 in dumpy^ov1 wing. Experimental data obtained in WT and dumpy^ov1 wings fall on similar lines (Figure 9A), suggesting that internal mechanical properties of the tissue are not perturbed in dumpy^ov1 mutant wings. Since dumpy mutant cells are less elongated than those of WT (Figure 9—figure supplement 1A), their similar mechanical properties imply that they are under less anisotropic stress consistent with the loosened connections to the overlying cuticle.

Figure 9 with 2 supplements see all

Download asset Open asset

If we perform the same analysis for the isotropic part of the tissue stress, we use the area expansion rate of the cut circle during recoil as a proxy for negative tissue pressure −P. Plotting this rate as a function of average cell area in a given region, we find that a₀ is smaller than cell area a, which reveals that the tissue is under contractile tension or negative P (Figure 9B). This is consistent with the observed tissue area contraction in dumpy^ov1 mutant and severed wings. However, parameter values a₀ and $\overline{K}/K$ cannot be reliably estimated by this method because the average cell area a varies too little, and because cell divisions may introduce heterogeneity in the preferred cell area a₀.

Overall, laser ablation experiments indicate that there are two contributions to anisotropic stress in the PD axis. First, blade cells are elastically deformed along the PD axis, in response to hinge contraction and margin attachments. Second, these cells would tend to spontaneously shorten in the PD axis even in the absence of external stresses.

So far, we have characterized the elastic properties of the tissue. On long time scales, elastic stresses can be relaxed by topological changes of the cellular network generating tissue shear viscosity. We therefore now develop a model describing the average rate and orientation of tissue shear R due to such topological changes. Although R includes the shear caused by T1 and T2 transitions, cell divisions and correlation effects, it is dominated by T1 transitions (Figure 6A,B). We asked how the shear rate R due to topological changes is regulated in the wing blade. If cells in the tissue are elongated, this elongation could drive topological changes that give rise to the shear rate R = Q/τ_r. Here, τ_r can be interpreted as the time during which cell elongation relaxes. Indeed, τ_r is also the time beyond which the tissue behaves as a viscous fluid with viscosity Kτ_r, due to elastic stress relaxation by topological changes including cell rearrangements and cell divisions. In addition, our observations of laser-severed wings suggest that the response of R to cell elongation is not instantaneous, but follows with a time delay τ_d of a few hours (see previous discussion and Figure 8—figure supplement 3), which also must be incorporated in the theory.

In addition to elongation-induced rearrangements, planar polarized tissues may undergo oriented topological changes even in the absence of cell elongation. For instance, topological changes could be driven autonomously by increased contractility of certain bonds. Taking into account all these contributions, we postulate the following relation between the shear rate R and cell elongation (Figure 9C):

Here, the delayed response to cell elongation is introduced via the time derivative of the shear R. This way of implementing a delay implies that the current value of cell elongation has the strongest impact on shear due to topological changes and the effect of recent values of cell elongation fades exponentially over time, disappearing beyond the time τ_d. Shear driven by polarity dependent processes is characterized by the tensor λ. Thus, even if cells are not elongated (Q = 0), λ drives shear due to topological changes. For shear along the AP axis λ_xx < 0, while for shear along the PD axis λ_xx > 0.

In order to verify whether Equation 5 captures the dynamics of the shear created by topological changes during pupal wing morphogenesis, we plotted R_xx vs the cell deformation component Q_xx for different times (Figure 9D). For an instantaneous response of R (no delay, τ_d = 0) these points would fall on a straight line. For non-zero delay, the history of the process matters and these points follow a curve spiraling towards a fixed point (see Appendix 2, ‘Effect of delay in topological changes’). We find indeed that experimental data for each WT wing follow a spiral, confirming the existence of a delay. A fit of the theory to the data allows us to estimate the coefficients in Equation 5 (τ_d = (3.7 ± 0.9) h, τ_r = (1.8 ± 0.6) h and λ_xx = (−0.10 ± 0.04) h⁻¹). Interestingly, λ_xx is negative, indicating that polarity driven topological changes create AP shear—consistent with conclusions from laser ablation experiments. Polarity driven AP-oriented topological changes may be related to increased tension of PD-oriented as compared to AP-oriented cell bonds. This anisotropy of cell bond tension could account for the positive value of ζ in Equation 4 because it would tend to elongate cells in the AP-axis.

The fact that a constant negative value of λ_xx accounts for the experimental data suggests that the tendency to undergo polarity-driven topological changes exists during both phase I and II. The transition from phase I to phase II occurs when cell elongation-driven topological changes begin to exceed the effect of polarity-driven topological changes. During phase I topological changes are largely polarity driven along the AP axis. Equation 2 then implies that these topological changes, together with hinge contraction, contribute to the buildup of cell elongation in the PD axis.

Once flows have stopped, the value of λ_xx determines the final value of cell elongation. Indeed, Equations 2, 5 predict that in a steady state without shear flows (R = 0, dQ/dt = 0), the cell elongation is given by Q = −τ_r λ. We can therefore use the cell elongation at the end of the videos, where the tissue is almost stationary, to estimate τ_r λ_xx. In other words, the final cell elongation should be independent of external stresses. Instead final cell elongation is governed only by the internal dynamics of the tissue, that is, the tendency to undergo AP-oriented topological changes λ_xx as well as the stress relaxation time-scale τ_r. Interestingly, examining different perturbed conditions, we find that the final cell elongation and hence λ_xx depend on cell area (Figure 9—figure supplement 1B and Appendix 4, ‘Area dependence of λ_xx’).

Overall, we find that the shear created by topological changes is driven in part by cell elongation and in part by cell polarity-dependent processes. Furthermore, topological changes respond to cell elongation with a delay of several hours.

Equations 1–5 constitute a full theory for tissue mechanics taking into account cell shape changes. We now want to test whether this tissue description quantitatively accounts for cell and tissue shape changes during wing morphogenesis.

We ask here whether the tissue and cell properties described by our equations can quantitatively account for the observed changes in the shapes of hinge and blade as well as the elongation and topological changes of their constituent cells. We use a simplified description of the wing where the hinge and blade are represented by two rectangles that are attached to each other. These rectangles have the same areas as the hinge and blade and they undergo pure shear that is the average shear in the respective parts of the wing. To describe tissue flow, Equations 1–5, which characterize local tissue properties, have to be complemented by the condition of force balance. At the boundary of the tissue, elastic linkers connected to the cuticle impose external forces that have to balance tissue stresses. In our model the rectangles are therefore connected to an external frame by elastic elements (Figure 10A and Figure 10—figure supplement 1). These external elements correspond to the extracellular matrix and the frame corresponds to the cuticle. The elastic elements provide resistance to extension of blade and hinge along the PD and AP axes (see Appendix 3, ‘Boundary stresses’).

Figure 10 with 2 supplements see all

Download asset Open asset

In each rectangle, anisotropic stresses and shear are described by Equations 4, 5, with different tissue parameters in hinge and blade. Isotropic internal stresses in the blade are described by Equation 3. Preferred cell area a₀ changes rapidly when cells divide and is modulated by the cell contractility $\overline{\zeta }$ (see Appendix 2, ‘Constitutive equation for the tissue stress’). In the hinge, we do not solve the full mechanical problem but rather impose the observed hinge area contraction. We do so by adjusting the pressure in the hinge to match the observed hinge area.

We start with rectangles whose areas and aspect ratio are consistent with those of hinge and blade at 16 hAPF (see Appendix 3). The initial conditions for Q and R are the observed average cell elongation and the initial shear rate due to topological changes. To trigger cell flows in the model, we turn on active stresses ζ, $\overline{\zeta }$ and polarity dependent topological changes λ. We solved Equations 1–5 in two rectangles that correspond to hinge and blade. Note that the pressure in the hinge is not determined by Equation 3, but by imposing hinge area. In the blade, we used the measured rates of cell division and T2 transitions to calculate the blade area changes (Equation 1).

We normalized all elastic moduli and friction coefficients to the blade shear modulus K. The solutions Q(t), a(t), v(t), and $\stackrel{\sim }{\mathit{v}}\left(t\right)$ of Equations 1–5 depend on a set of parameters characterizing the tissue both in hinge and blade (see Tables 1, 2). In addition, elastic coefficients describe constraints imposed by linkers at the boundary. Friction due to motion with respect to the cuticle is captured by a friction coefficient γ. Furthermore, to fully account for the observed flows, we found that we needed to add a contribution of tissue area viscosity $\overline{\eta }$ to Equation 3 (see Appendix 2, ‘Constitutive equation for the tissue stress’, Equation 52). We include this term in all subsequent calculations. Laser ablation of the extracellular matrix is introduced in our model by removing the elastic linkers at the tissue boundary at the side of ablation.

We first analyzed WT wings in both unperturbed and mechanically perturbed conditions where only the extracellular matrix was ablated. Because blade tissue was not damaged by matrix ablation we expected that most parameters characterizing the hinge and blade would be identical to those in unperturbed wings. However, λ_xx can differ in wings characterized by different cell areas, as noted above. We therefore used the values of τ_r, τ_d and λ_xx determined by jointly fitting Equation 5 to the data (see Table 1 and Figure 9—figure supplement 2). In addition, we used experimentally determined values of ζ/K estimated from circular laser ablation experiments in WT wing blades (Figure 9A). A similar fit is performed for the hinge and corresponding parameters are superscripted with an index H.

This leaves 10 parameters unspecified. These characterize the isotropic stresses in the blade, the anisotropic stresses in the hinge, the stiffness of the elastic linkers at the boundary, and the coefficient describing external friction (Table 2). We estimated these parameters by performing joint fits of Equations 1–5 to the quantified time dependence of tissue area changes and shear in unperturbed and mechanically perturbed WT wings. We found that anisotropic stresses in the hinge are small compared to isotropic stresses and therefore we could set ζ^H and K^H to zero. A single set of the remaining 8 parameters accounted well for all time dependent data (see Figure 10B and Figure 10—figure supplement 2 and Table 2). It is remarkable that this very simplified model can account for the main features of the complex shape changes that occur during pupal wing morphogenesis. The choice of rectangles indeed captures the average shear and area change that determine tissue shape change. Displaying the time evolution of the rectangles within a time-lapse video of the pupal wing reveals the close agreement between the calculated and observed shape changes of hinge and blade (see Videos 10–13).

Video 10

Download asset

Video 11

Download asset

Video 12

Download asset

Video 13

Download asset

Our experimental data suggest that distal elastic connections to the cuticle are weakened in the dumpy^ov1 mutant wing. We therefore asked whether we could fit the rectangle model described above using WT tissue parameters but allowing parameters describing connections to the cuticle to change. We removed distal boundary connections to the blade (Figure 10—figure supplement 1) as suggested by laser ablation experiments and allowed other parameters characterizing external linkers to change. We used these values and the measured value of ζ/K (which is almost the same as in WT, see Figure 9A) in fits of the full dynamics of the rectangle model to the dumpy^ov1 mutant data (see Figure 10B and Figure 10—figure supplement 2). These calculations show that distal and lateral connections to the wing blade margin are weakened, as expected (see Table 2).

Overall, the continuum theory of epithelia outlined in Equations 1–5 together with appropriate boundary conditions recapitulates the dynamics of cell and tissue shape over 16 hr of morphogenesis. By fitting the theory to experimental measurements, we determine the values of tissue parameters characterizing intrinsic tissue time-scales, cell autonomous stresses, tissue elastic moduli and elastic moduli of external linkers. Using this theory to investigate mechanical parameters in a dumpy^ov1 mutant wing, we confirm the existence of Dumpy-dependent elastic connections between the wing and cuticle. Thus, this theoretical approach is a powerful tool for studying how mutations influence specific aspects of cell and tissue mechanics.

Flies were raised at 25°C under standard conditions unless stated otherwise. Pupae were collected for imaging as described previously (Classen et al., 2008). Ecad::GFP flies (Huang et al., 2009) were used as control for all live imaging experiments. The dumpy^ov1 mutation (Sturtevant et al., 1929) was recombined with Ecad::GFP for imaging, and for quantifying tissue flows and cell behaviors in a dumpy^ov1 mutant (Bloomington, reference number 276). The Dumpy::YFP protein trap line (DGRC) was used to describe Dumpy distribution in the pupal wing. To inhibit cell divisions in the pupal wing while imaging it, we used the thermosensitive allele cdc2^E1-E24 of cdk1 that we recombined with Ecad::GFP (Stern et al., 1993). Cells expressing two copies of cdc2^E1-E24 and shifted to 30°C arrest in G2 phase just prior to entering mitosis (Weigmann et al., 1997).

Pupae were prepared for live imaging as previously described (Classen et al., 2008). Long-term time-lapses were acquired with a Zeiss spinning disk microscope driven by the Axiovision software and equipped with an inverted Axio Observer stand, a motorized xyz stage, and a Zeiss LCI Plan-Neofluar 63× 1.3 NA Imm Corr lens associated to an objective heater set to 25°C. The fly pupa was placed in a temperature-controlled chamber set to 25°C and equipped with a humidifier to prevent desiccation. Images were recorded with an AxioCamMR3 camera (2 × 2 binning). Laser power was measured through a 10×/0.45 NA lens using a power-meter (PT9610, Gigahertz-Optik). A power of 0.980 mW was found to be optimal to prevent noticeable bleaching during 24 hr of continuous acquisition with an exposure time of 265 ms. Briefly, the dorsal cell layer of the pupal wing was scanned within 5 min in the (x, y, z) dimensions, over 24 overlapping positions of about 30 z-sections each. This scan procedure was continuously repeated about 260 times to cover more than 20 hr of development. Custom Fiji macros helped keeping the tissue in focus in a semi-automated way.

We benefited from the flexible architecture provided by our computer department to handle several TB of data using custom unix bash scripts to archive, compress and store data on tape and retrieve them easily. Image pre-processing steps were mostly performed using custom Fiji macros called from a master bash script to enable parallelization using GNU parallel (Tange, 2011). A low-pass filter was first applied to images to remove high frequency noise. For each z-stack, the signal of highest contrast was projected using a C++ algorithm as previously described (Merkel et al., 2014). Tiles were stitched using Fiji (Preibisch, 2009; Schindelin et al., 2012). Stitched images were then loaded in Packing Analyzer v8.5 (PA8.5) for cell edge detection by using a seeded-watershed algorithm. To facilitate cell tracking, cross-correlation of subsequent images was performed in PA8.5 to calculate local tissue displacement beforehand. Resulting cell-tracking masks were parsed using a custom C++ parser that extracted cell shape properties, cell lineages and cell neighbor relationships for further storage in a SQLite relational DB. Topology was kept in the DB by ordering cell neighbors counter-clockwisely around each cell, namely by following a directed path of cell–cell junctions around each cell. One DB file was created per video. All DB queries were carried out using R (R Development Core Team, 2014) or Python (www.python.org; Hunter, 2007; Pérez and Granger, 2007; van der Walt et al., 2011).

We restricted our analyses to a subset of cells that were trackable throughout the entire course of the video, excluding cells that became visible only at later stages due to the apposition of the two cell layers. At 32 hAPF, most cells of the dorsal cell layer are visible. Therefore, we manually drew regions of interest (ROIs) on 32 hAPF wings using the veins and hinge–blade interface as landmarks, where the Ecad::GFP signal is more intense. These ROIs include the blade region delimited by the hinge–blade interface and the wing margin, but also the hinge–blade interface itself. In addition, we defined a triangular portion of the hinge delimited by the hinge–blade interface and the most anterior sensory organ located in the bulk of the hinge. In other hinge regions, cells were too small and elongated to be consistently tracked. Starting from these ROIs at 32 hAPF, we developed a backward-tracking algorithm that uses information about cell divisions and cell extrusions to reconstitute the corresponding group of cells at start of recording (usually ∼16 hAPF), discarding margin cells that were not present through the course of the video. In the resulting set of cells, cells in contact with the wing margin at start of recording (about 16 hAPF) were not perfectly segmented and therefore further discarded. This gave rise to a group of cells that is fully consistent in time.

Since part of the hinge could not be analyzed at cellular resolution, we complemented our analysis of the small portion of the hinge by using particle image velocimetry (PIV) (Adrian and Westerweel, 2011) to extract information about the entire hinge deformation as described thereafter. PIV was implemented by Benoit Aigouy and described elsewhere (Merkel et al., 2014).

Laser ablations were performed using an ultraviolet laser microdissection apparatus as described elsewhere (Grill et al., 2001). For tissue severing, a C-apochromat-40×/1.2 water immersion lens was used to focus the beam along AP-oriented line segments to ablate both dorsal and ventral cell layers. Subsequent long-term imaging of the whole wing was then carried out on our dedicated Zeiss spinning disk. Extracellular-matrix ablations were done using a 63×/1.4 oil lens to focus the beam between the tissue and the cuticle and to cut over 10 μm in depth. Circular cuts were also conducted with the 63×/1.4 oil lens but over a depth of 5 μm. Each circular cut experiment was repeated on five distinct pupae.

Fly wings expressing Ecad::GFP were used in all circular cut experiments. Circular cuts were performed to disconnect a small subset of cells present inside the circle from the rest of the tissue. The interface of the tissue with the ablated circular region was always visible and well spatially defined after laser ablation. Therefore we quantified its displacement to calculate the initial velocity gradient that describes the immediate tissue deformation in response to the ablation. To do so, we first fitted an ellipse to this manually segmented interface, at 50 s after ablation. The major and minor axes of the ellipse define the orientation of two orthogonal kymographs, each intersecting the ellipse in two points corresponding to two opposite sides of the interface between the tissue and the ablated region. Each kymograph depicts the displacement as a function of time of the two opposite sides of this interface, which were manually segmented using sub-pixel resolution ROI. Thus, we quantified the displacement of four points corresponding to the intersections of the minor and major axes with the ellipse (see Figure 2—figure supplement 3). This procedure was semi-automated in a custom Fiji macro. The initial response of the tissue reflects the orientation and amplitude of tissue stresses. This can be captured by the initial velocity gradient within a few seconds after the 4 s ablation. Indeed, the initial displacement normal to the cut boundary was approximately linear as a function of time within the first 5 s after ablation. A linear fit to the data provided the initial normal velocities V_|| and V_⊥ along the major and minor axis of the ellipse, respectively. We then define the velocity gradient tensor in the coordinate system of the ellipse and rotate it into the image coordinate system:

where θ is the angle between the major axis of the ellipse and the PD axis (x axis). The radii r_|| and r_⊥ are the half lengths of major and minor axes of the ellipsoidal shape at the time the velocities are determined. This velocity gradient can be decomposed into trace and a traceless part as:

where $v_{kk}^{\text{cut}}=2C$ is the isotropic expansion rate, ${\stackrel{\sim }{v}}_{xx}^{\text{cut}}$ is the shear rate projected onto the PD-axis. While in the main text, tensors are denoted as bold face symbols, in the supplement we often use an explicit index or matrix notation for tensors. Latin indices i, j denote the x and y coordinates of a cartesian coordinate system of the tissue.

Circular cuts were performed in nine different regions of the wing where cell elongation differed (Figure 2I,I′). Due to the fast imaging settings optimized to catch the initial velocity gradient, the image quality was not sufficient for cell segmentation. Therefore we used wings of different animals at the same stage (22.5 hAPF) to estimate cell elongation Q_xx for each of the nine ablation regions that were easily located using morphological landmarks such sensory organs and veins (see Figure 9A). Average cell area a_cell was estimated using a similar approach (see Figure 9B).

We find that the shear rate projected onto the PD axis ${\stackrel{\sim }{v}}_{xx}^{\text{cut}}$ increases with cell elongation. We find that data of ${\stackrel{\sim }{v}}_{xx}^{\text{cut}}$ as a function of ${\stackrel{\sim }{v}}_{xx}^{\text{cut}}$ can be accurately fitted by a linear function both in WT and dumpy^ov1 wing. Fit parameters are given by

• WT: ${\stackrel{\sim }{v}}_{xx}\left[h^{-1}\right]=\left(0.018\pm 0.008\right)Q_{xx}+\left(0.003\pm 0.001\right)$

• Dp: ${\stackrel{\sim }{v}}_{xx}\left[h^{-1}\right]=\left(0.019\pm 0.005\right)Q_{xx}+\left(0.003\pm 0.002\right)$

The correlation between the isotropic expansion rate $v_{kk}^{\text{cut}}$ and the logarithm of cell area is less apparent. However, following Equation 8, we performed a linear fit to the data and found the following best fit parameters:

• WT: $\frac{1}{2}{v}_{kk}\left[h^{-1}\right]=\left(0.008\pm 0.007\right)\text{ln}\left(a_{cell}/a_{ref}\right)+\left(0.017\pm 0.002\right)$

• Dp: $\frac{1}{2}{v}_{kk}\left[h^{-1}\right]=\left(0.005\pm 0.008\right)\text{ln}\left(a_{cell}/a_{ref}\right)+\left(0.015\pm 0.002\right)$

Parameters obtained in isotropic expansion rate fits have high uncertainties and we do not use them in the rest of the paper.

In order to compare theory to the experimental data we need to approximate the observed hinge and blade shapes by rectangles. We first show how a characteristic height and length can be associated to an arbitrary two-dimensional shape. We then specify how hinge and blade regions were selected to obtain the associated height and length. Note that for the purpose of estimating the wing dimensions, we do not used tracked regions of a subpart of the wing as in ‘Data handling and image processing’, but we use all available segmented data. Finally, we introduce corrections that account for changes of the visible part of the tissue in the field of view.

Here we define the average pure shear rate of a tissue region and its decomposition into cellular contributions (Merkel, 2014). The cellular network of the tissue region is triangulated (‘Triangulated cellular networks’). Averaging triangle deformations over all triangles within the region yields expressions for the large-scale shear rate and the shear rate contributions by different cellular processes (‘Decomposition of the large-scale tissue shear rate’). The determination of spatial patterns of shear (Figure 5, Figure 5—figure supplement 1) and rotation (Figure 6C,D,F,G) are discussed in ‘Spatially resolved shear patterns’ and ‘Power spectrum of local tissue rotation’.

In this section, we discuss a coarse-grained hydrodynamic theory that describes the mechanics of a two dimensional epithelium, the cells of which can rearrange through topological changes. The hydrodynamic theory describes the evolution of two main physical quantities: the velocity field v_i within the tissue, and the average cell elongation Q_ij, treated as an internal variable of the tissue. In addition, we take into account a polarity field q_ij, which gives a preferred axis in the tissue and allows for internal anisotropic processes to occur.

To obtain a full description of the epithelium mechanics, it is necessary to specify tissue processes that drive the shear created by topological changes. The shear rate R_ij created by topological changes is a tensorial quantity. In the spirit of hydrodynamic theories, we propose a constitutive relation for this tensor by writing all possible terms compatible with its symmetry, at lowest order in the driving fields. In the following we describe the full hydrodynamic theory.

In this section, we use the theoretical framework described in the previous section to obtain a simplified description of the morphogenesis of the pupal wing. We represent the hinge and blade by rectangles, connected by elastic springs to an external fixed frame, corresponding to the cuticle (Figure 10A and Figure 10—figure supplement 1).

In this section, we describe the fitting procedure to experimental measurements. The fitting procedure was performed in two stages:

• First, the constitutive equation for topological changes, Equation 5 in the main text, was fitted to experimental data.

• Using the parameters obtained for the fit to Equation 18, the dynamics of cell shape and tissue shape predicted by the rectangle model of section Appendix 3 was then fitted to experimental data by adjusting the remaining parameters.

We analysed six wings, for which the full blade and a fraction of the hinge were segmented. In this section, tissue flows were measured by PIV analysis rather than by cell tracking (see ‘Materials and methods’, Data handling and image processing). The six analysed wings were 3 WTs, one dumpy^ov1 mutant (Dp), one wing where the extracellular matrix was distally ablated (ECMDistCut) and one wing where the extracellular matrix was anteriorly ablated (ECMAntCut).