Unified quantitative characterization of epithelial tissue development

1The advances in live imaging have beautifully illustrated how the growth and shaping of tissues and organs emerge from the collective dynamics of cells (for review see). In this context, the development of quantitative methods is essential to determine the role in tissue and organ development of each cell process, in particular cell divisions, cell rearrangements, cell size and shape changes and apoptoses. However, a general formalism valid in two and three dimensions and allowing to unambiguously decompose tissue growth and morphogenesis into the parts due to each cell process is still needed. Building such a formalism is critical to define the roles of each process and advance our understanding of how gene activities and mechanical forces cooperate in controlling cell dynamics to regulate the growth, shaping, repair or homeostasis of monolayered or three-dimensional cohesive tissues. In particular the lack of a general formalism has impeded a comprehensive characterization of the role of cell divisions and of their orientations during tissue development.

2The growth and morphogenesis of tissues typically involves cell divisions leading to the formation of organs of the correct size and shape. So far, two fundamental properties of cell division have been reported during the morphogenesis of proliferative tissues. First, the orientation of cell divisions has been shown to play a key role in tissue elongation, either by being an anisotropic source of force, or by reducing mechanical stress. Second, cell division orientation has been reported to be mainly oriented along the direction of mechanical stress. These conclusions were mainly drawn from the observation of tissues where cell divisions are the major contributor of tissue elongation or where cell divisions, tissue deformation and mechanical stress display a common preferred orientation. Yet, embryos, as well as many other tissues or organs, are heterogeneous: cell divisions display rates and orientations varying in space and time, and they are concomitant to other morphogenetic processes such as cell rearrangements, cell size and shape changes and apoptoses. One of the broad challenges in developmental biology is therefore to test and generalize these proposed functions of cell divisions in more heterogeneous contexts.

3We have previously proposed a statistical method to quantify cell rearrangements and cell size and shape changes during tissue development. The method took advantage of the links joining the centroids of a cell and its neighbors. Measuring the changes of position, size and direction of links, as well as link swapping, yielded a quantification of cell rearrangements and cell size and shape changes characterized by an amplitude, an anisotropy and an orientation. This enabled to separately measure cell rearrangements and cell size and shape changes during tissue development. Generalizing the method to incorporate the remaining biological cell processes, in particular cell divisions and apoptoses, is not straightforward since these cell processes can substantially modify the cell and link numbers. We therefore developed a novel formalism that takes into account the changes in link number and which disentangles the measurement of each cell process during tissue development (See Appendix for detailed information and comparison with previous approaches).

4In this novel formalism, valid in two and three dimensions, the four main cell processes (cell divisions, cell rearrangements, cell shape and size changes and apoptoses) are unambiguously distinguished and independently quantified by four measurements (,, and, respectively). These four measurements quantify the changes in link length or orientation as well as link appearances or disappearances due to their respective cell processes; they add up to the local tissue rate of deformation measuring the rate of tissue growth and morphogenesis due to these four cell processes (Figure 1a). Whenever needed, the subdivision of these main cell processes can be further refined, for instance in a mono-layered tissue to distinguish apoptoses from live cell extrusions, or to distinguish simple rearrangements through four-fold vertices from those with five-fold vertices or higher. Furthermore, introducing cell divisions and apoptoses naturally enables the addition of the other cell processes changing the number of cells such as: (i) the integration of new cells in epithelium sheets; (ii) the fusion (coalescence) of cells, and (iii) the in/outward cell flux, representing the cells entering and exiting the microscope field of view or the boundaries of the tissue of interest (Figure 1—figure supplement 1a). The formalism therefore yields a complete and unified quantitative characterization of tissue deformation and of all cell processes reported to occur during epithelial tissue development and homeostasis.

5In a tissue where tissue deformation is solely associated with cell divisions, cell rearrangements, cell size and shape changes and apoptoses, this unified characterization is expressed as a balance equation where the deformation rate of a region in the tissue is decomposed into the sum of the deformation rates associated with each cell process:

6Note that in the absence of divisions, rearrangements and apoptoses (i.e. ), our formalism therefore yields an exact equality between the rates of tissue deformation and of cell size and shape changes.

7Here, we explicitly describe its practical implementation and measurements in the context of the general interest in understanding the growth and morphogenesis of epithelial sheets. We first acquire a time-lapse movie in which cell apical contours have been labeled by E-Cadherin:GFP, images are segmented and cells tracked to determine their positions over time and their lineages. Then, the formalism is applied between successive images to separately measure the growth and morphogenesis associated to each cell process (,, and ) and of the tissue (Figure 1b and Figure 1—figure supplement 1b). Each measurement can be represented with a circle and a bar (Figure 1c and Figure 1—figure supplement 1c). The circle diameter represents the local rate of tissue isotropic dilation or tissue growth: it is positive for an increase in size (white filled circle) and negative for a decrease (grey filled circle). The bar, which has a length and orientation, represents the local rate of tissue anisotropic deformation or tissue morphogenesis: it quantifies the local elongation rate (and respective equal contraction rate in the orthogonal direction), thereafter named the contraction-elongation (CE) rate (Figure 1d). Finally, the analysis is multi-scale, in the sense that each statistical measurement can be averaged at any supra-cellular scale over space, over time, and over several animals, thereby linking the length and time scales associated with cells, groups of cells and the whole tissue (Figure 1e, Video 1).

8In order to confirm that each measurement quantifies unambiguously and accurately its associated biological process, we applied the formalism on computer simulations of cell patches undergoing known deformations. In each simulation, we imposed that the growth and morphogenesis of the cell patch was mainly driven by only one of the cell processes: cell divisions, cell rearrangements, cell size and shape changes or apoptoses. We first tested the measurements of and by imposing an isotropic dilation of the cell patch, followed by its CE along the horizontal axis, both patch deformations solely occurring via cell size and shape changes. We independently measured the imposed deformation rates for and with 0.3% of error, and obtained as expected (Figure 2a, Video 2a). Next, we tested the measurements of,, by allowing deformation of the cell patch by oriented cell divisions, oriented rearrangements and apoptoses, respectively. In each simulation, the balance equation shows that the tissue deformation rate was determined by the main process enabling the deformation of the cell patch (Figure 2b–d, Video 2b–d; see Figure 2—figure supplement 1 and Video 2e–i for the others processes). This confirmed that the formalism unambiguously measures the tissue deformation rate as well as the deformation rates associated with each individual cell process.

9Having validated the formalism in silico, we illustrate its relevance to study tissue development by undertaking an analysis of the role of cell division orientation and its relationship to other cell processes and to mechanical stress during the development of a heterogeneous epithelial tissue. During pupal metamorphosis, the Drosophila dorsal thorax (notum, yellow dashed box in Figure 3a,b) is a monolayered cuboidal epithelial tissue. From 10 h after pupa formation (hAPF), it undergoes several morphogenetic movements associated with cell divisions, cell rearrangements and cell size and shape changes as well as delaminations, which can be due to live cell extrusions or apoptoses. An important feature of this tissue is its heterogeneity, which enables to simultaneously investigate the various mechanisms driving morphogenesis and their interplays. Furthermore, applying our formalism on this tissue will provide a valuable resource since it is a general model to uncover conserved mechanisms that regulate planar cell polarization, tissue morphogenesis, tissue homeostasis and tissue mechanics, and to perform genome-wide RNAi screen (see for example).

10We imaged the development of this tissue by labeling cell adherens junctions with E-Cadherin:GFP and followed ~10 103 cells over several cell cycles with 5 min resolution from at least 14 to 28 hAPF. We segmented and tracked the cells of the whole movie (~3 106 cell contours with a relative error below 10-4, Figure 3c, Video 3a). The display of cell displacements, as well as the tracking of cell patches deforming over time enable to visualize the heterogeneity of tissue growth and morphogenesis between 14 and 28 hAPF (Figure 3d, Video 3b,c). Directly measuring the rate and orientation of cell divisions, we found that ~17 103 divisions take place during the development of the tissue, and that both the cell division rates and orientations display major variations in space and time (Figure 3c,e). Cell division rate is higher in the posterior part of the tissue (Figure 3c) and many regions harbor oriented cell divisions (Figure 3e). Division orientation is represented by a bar (, Appendix C.3.2), the length and orientation of which represent respectively the cell division orientation anisotropy and main orientation in each region. In particular, in the central posterior part of the tissue the orientation of cell divisions is medial-lateral, while in a more anterior and lateral domain cell divisions are oriented at roughly 45° relative to the anterior-posterior axis (Figure 3e, boxes). While these descriptions of tissue development by following patches of cells, cell division rate and orientation are essential, we now explain how the formalism enables to rigorously tackle three major steps to quantitatively study the morphogenesis of an epithelial tissue: (i) measure the local CE rates associated with each process for one animal; (ii) determine the average and the variability of cell dynamics over several animals; and (iii) measure the components of each cell process CE rate along tissue morphogenesis.

11In order to quantify development over the whole tissue from cell to tissue level, we applied our formalism to measure,,,, using 2 h sliding window averages in cell patches of initially 4040 μm, typically encompassing tens of cells that were tracked between 14 and 28 hAPF (Video 4a-e). Dilation and CE rate maps vary smoothly in both space and time, while remaining symmetric relative to the midline axis. This indicates that the averaging length scale (4040 μm2) and time-scale (2 h) are appropriate to describe the dynamics of the tissue; moreover, the symmetry indicates that each process is regulated. The results were also plotted as 14 h average maps on the last image analyzed to quantify the growth and morphogenesis of each patch between 14 and 28 hAPF (Figure 3f–j and Figure 3—figure supplements 1, 2). We find that tissue dilation mainly occurs through divisions, cell size changes and apoptoses, the dilation rates of the other processes being negligible (Figure 3—figure supplement 1). While the dilation rates are critical to study important processes such as apical constriction or local tissue growth, we focus here on the CE rates to illustrate how the formalism enables to better understand the role of cell processes in morphogenesis, with a focus on oriented cell divisions. The tissue CE rate map demonstrates that the tissue undergoes various CE movements, in particular in its posterior medial and lateral domains (Figure 3f, boxes). As expected, the cell division orientation and division CE rate maps (Figure 3e,g) show strong similarities (alignment = 0.94). Importantly, the formalism now enables to compare directly the division CE rate to the other CE rates. First, the and CE rates are roughly aligned in the medial posterior region, while they have clearly different orientations in the lateral domains (Figure 3f,g). Second, both cell rearrangements and cell shape changes have strong CE rates in many domains where cell divisions are also oriented (Figure 3g–i). Third, while cell delaminations are spatially regulated and numerous (~2.5 103), their CE rates are negligible, and their effect is mostly isotropic (Figure 3j and Figure 3—figure supplements 1e, 2e). Therefore, tissue CE mostly occurs through oriented divisions, rearrangements and cell shape changes. Together this shows how our formalism enables to express quantitatively the relevant information extracted from millions of cell contours over space and time and to separately measure and represent the CE rates of the tissue and of each cell process at the scale of groups of cells in a whole epithelial tissue.

12In order to determine the average and the variability of cell dynamics of a tissue, we developed a general method to register in time and space, and to rescale movies obtained from different animals. This method uses reliable landmarks (for space registration) and prominent, stereotyped rotational movements (for time registration); and it can be applied to mutant conditions and to other tissues (see Materials and methods and below for mutant conditions). Upon space-time registration, each deformation rate measured in each cell patch in 5 hemi-notum movies can be averaged to yield average maps of an archetype hemi-notum representing the dynamics of ~7 106 cell contours, i.e. ~23 106 links, 40 103 divisions, 5.4 103 delaminations (Figure 4a,b and Figure 4—figure supplement 1a). For each process and in each region of the archetype hemi-notum, the corresponding biological variability is measured by quantifying its variations between hemi-nota in each region. This was used to determine whether each CE rate was significant in a given region (plotted in color or grey, respectively). Collectively, these approaches yield average maps of the tissue CE rate and of the main cell process CE rates (,, and ) that can be superimposed for comparison (Figure 4a,b). These maps confirmed that division CE rates are parallel to the tissue deformation rates in some regions but not all (compare regions 1, 2 and 3), suggesting that oriented divisions may not have a single and simple effect in morphogenesis.

13Studying the morphogenesis of epithelial tissues requires the analysis of both the alignment and the magnitude of each cell process CE rate with respect to the local tissue CE rate. This can be achieved by projecting each CE rate (,, and ) onto the direction of the local tissue CE rate (noted ) to determine their components along the direction of, thereby yielding,, and components respectively (Figure 4c). The projection of along its own direction yields the local magnitude of tissue morphogenesis. Each of these components along can therefore be interpreted as the effective participation of each process in tissue morphogenesis of the wild-type tissue; it should not be confused with a functional role of a cell process that can only be studied using loss and gain of function approaches and modeling. A CE rate aligned with the tissue CE rate has a positive component along tissue morphogenesis, whereas a CE rate displaying a bar orthogonal to the tissue CE rate has a negative component along tissue morphogenesis (Figure 4c). In a tissue where tissue deformation is solely associated with cell divisions, cell rearrangements, cell size and shape changes and apoptoses, is equal to the sum of the CE rate of divisions, rearrangements and cell shape changes, as well as apoptoses. The magnitude of local tissue morphogenesis can therefore also be expressed as the sum of the local components of each cell process along, namely:

14Maps of the components associated with each cell process can be then plotted using circles that are hollow or filled for positive or negative components, respectively, and the areas of which scale with the magnitude of each components (Figure 4d–g and Figure 4—figure supplement 1b,c).

15We briefly illustrate here how the component measurements can further be analyzed to uncover novel interactions between tissue morphogenesis, cell divisions and other cell processes. The measurements show that the cell division component (Figure 4e) can either be strongly positive in some regions (regions 1 and 2) or strongly negative in others (region 3). In contrast, cell rearrangements and cell shape changes mainly have positive components along tissue morphogenesis (Figure 4f,g). In regions 1 and 2, the positive component of cell divisions corresponds to the one described in the literature, namely that cell divisions and tissue elongation are oriented in the same direction. Conversely, our measurements in region 3 show that cell divisions have a negative component corresponding to divisions taking place mostly orthogonally to the direction of tissue elongation. The measurements of cell rearrangements and cell shape components show that the elongation of the tissue in this region results from the strongly positive components of cell shape changes or of cell rearrangements (Figure 4f,g). Finally in region 4, although most cells divide once, the cell division component is small relative to that of cell shape changes that almost completely accounts for tissue deformation in that region (Figure 4d–g).

16The existence of these distinct relationships between process CE rates and total CE rate can be further analyzed by plotting each relative component versus time to characterize its temporal evolution throughout tissue development (Figure 4h–k). In addition to the spatial heterogeneity of morphogenesis, these plots reveal that morphogenesis also strongly varies over time in a given region of the tissue and that it can be further decomposed in different periods (see Figure 4h–k legends for details). More generally, such analyses will provide important insight about the respective roles of each cell process in tissue morphogenesis and their time evolution.

17To illustrate the generality of our approach and to determine whether cell divisions can be oriented perpendicularly to the tissue CE rate in another tissue, we performed a similar characterization of the morphogenesis in another epithelium, the pupal wing (Figure 5, Figure 5—figure supplement 1, Video 5a). The important deformations of cell patches over time show that wing morphogenesis is heterogeneous as well, displaying strong tissue deformations in the anterior and posterior domains of the wing, and milder deformations in the medial domain (Figure 5a, Video 5b). To characterize quantitatively these deformations of the tissue and relate them to cellular processes occurring in this wing, we applied our formalism to this pupal wing between 15 and 32 hAPF and determined maps of averaged tissue CE rates and cell processes CE rates,, and (Figure 5b–e), as well as their projections onto direction,,,, and (Figure 5f–i). Even on a single wing, all CE rate maps display heterogeneous but smooth patterns over time, showing that this scale of space-time averaging (4040 μm2, 2 h) is also suitable for the wing (Video 5c–f). These measurements confirmed the observed heterogeneity in tissue morphogenesis by evidencing major tissue CE rates in anterior and posterior domains of the wing, with a tilt of about ±45° with respect to the horizontal axis, respectively, and smaller CE rates in the medial domain (Figure 5b,f). The CE rates of cell processes,, display smooth and heterogeneous maps while displaying different patterns in space (Figure 5c–e), while the CE rate of is negligible (Figure 5—figure supplement 1b). Like in the dorsal thorax, rearrangements and cell shape changes mainly have positive components along tissue CE rate and they make up most of tissue morphogenesis (Figure 5f,h,i). Like in the dorsal thorax, oriented cell divisions in the wing display more variety in their component along tissue morphogenesis than and: has positive components along in anterior and posterior wing domains, where tissue morphogenesis is the strongest, while it has negative components in the medial wing domain, where morphogenesis is milder (Figure 5c,g box). Together our analyses of morphogenesis in the notum and the wing illustrate how our formalism enables the quantitative characterization of morphogenesis in various epithelial tissues.

18We found that in the wild-type tissue cell divisions display various orientations with respect to the direction of tissue elongation and thus can have negative and positive components along tissue morphogenesis. These observations raise important questions regarding the role of cell divisions per se in tissue development, namely the role of proliferation and division orientation, as well as its interplay with the other cell processes during tissue morphogenesis. We illustrate here how the formalism can help analyze these central questions by allowing for a rigorous quantification of different experimental conditions.

19To experimentally study the role of cell divisions in morphogenesis, we overexpressed the tribbles gene (trblup), an inhibitor of G2/M transition using the Gal4/Gal80ts system to inhibit proliferation specifically at pupal stage. We segmented and tracked five trblup hemi-thoraxes over time (corresponding to ~3.7 106 cell contours). Both visual inspection of the movie and cell tracking revealed that a trblup hemi-notum hardly displays any division as compared to wild-type tissue: ~1.7 103, i.e. only 4.3% of the number of wild-type divisions (Figure 6a,b). We then registered and rescaled in space the five hemi-thoraxes, synchronized them in time, and applied the formalism to determine tissue morphogenesis and the respective CE rates of each cell process (Figure 6d,f and Figure 6—figure supplement 1b,d). As expected, the measured division CE rate nearly vanishes in accordance with the nearly complete disappearance of cell proliferation (Figure 6c,d). Furthermore, we find that tissue CE rate is disrupted in trblup tissue, both in direction and amplitude, suggesting that the absence of proliferation impacts tissue morphogenesis (Figure 6e,f).

20Two complementary maps can be used to quantitatively study the effects of trbl overexpression: the difference between the tissue CE rates in trblup tissue and in wild-type tissue, namely (Figure 6g), and its projection onto the wild-type tissue CE rate, namely (Figure 6h). represents the change brought to wild-type tissue morphogenesis by the trbl overexpression, and measures the effective contribution of this change to wild-type tissue morphogenesis. A region where is positive means that wild-type tissue morphogenesis has been increased by trbl overexpression, and conversely, is negative means that it has been decreased in this region. Therefore, the map provides a visual representation of where the tissue morphogenesis is increased or reduced and can be interpreted as the role of cell divisions (proliferation and oriented divisions) during tissue morphogenesis. This map reveals that in almost all regions of the tissue, and regardless of the orientation of cell divisions relative to tissue elongation in wild-type, overexpressing trbl reduces wild-type tissue elongation (full circles in Figure 6h).

21A similar approach can be applied to each cell process to determine how it is impacted by the trbl overexpression (Figure 6h–j and Figure 6—figure supplement 1). Thus, the respective changes in each cell process due to trbl overexpression can be measured using,, and (, not shown). As expected from the nearly complete absence of division in trblup tissue, the map representing the changes in tissue morphogenesis due to the loss of cell division is almost the opposite of the map (compare Figure 4e and Figure 6—figure supplement 1e). More interestingly, the, and maps directly demonstrate that both cell rearrangements and cell shape changes are significantly modified in trblup tissue and contribute to the overall changes in tissue morphogenesis due to trbl overexpression (Figure 6i–j). This indicates that suppressing proliferation not only makes oriented division CE rate vanish, but also has an indirect impact on both cell rearrangements and cell shape changes. In conjunction with our results in the wild-type tissue, this suggests that both cell proliferation and the orientation of divisions determine the morphogenesis of the tissue, and that a complex interplay exists between cell divisions and the other processes such as cell shape changes and rearrangements.

22To better understand this last point, and more generally the effect of oriented cell divisions, we used computer simulations. We compared our previous simulation of cell divisions oriented along the horizontal axis of tissue elongation (Figure 2b and 6k and Video 6a) with simulations where only the pattern of divisions has been modified in two distinct ways: (i) we aligned all cell divisions along the vertical axis, namely orthogonally to tissue elongation, thereby leading to a negative component of ( < 0) (Figure 6l and Video 6b), thus mimicking our observation in region 3 of the wild-type tissue (Figure 4e); (ii) we suppressed divisions ( = 0, Figure 6m and Video 6c), mimicking our observation in trblup tissue (Figure 6d). In both cases, we found that modifying the pattern of divisions impacts simultaneously, and in addition to (Figure 6l,m). When divisions are orthogonal to tissue elongation, cell rearrangements, and to a lesser extent cell shape changes, are greatly increased along the direction of deformation, but they only partly compensate the CE rate of horizontally oriented divisions in the initial simulation, thereby resulting in reduced tissue elongation (Figure 6l). When divisions are suppressed, cell rearrangements and cell shape changes are moderately increased along the direction of deformation, and compensate even less horizontally oriented divisions, thereby resulting in further reduced tissue elongation (Figure 6m). These two simulations therefore recapitulate two aspects of our experimental observations: (i) how divisions orthogonal to the tissue CE rate in wild-type have a negative component along tissue morphogenesis, as found in some regions of the wild-type tissue (Figure 4e, region 3); (ii) how divisions, regardless of their orientation, can facilitate tissue elongation by indirectly impacting cells rearrangements and cell shape changes, as observed in trblup tissue where proliferation is severely reduced and tissue deformations are globally decreased (Figure 6h).

23Altogether, our formalism reveals the extent of the heterogeneity of division orientation in a tissue, and our analyses of simulations and trblup experimental condition show that both cell proliferation and oriented divisions can influence tissue morphogenesis. Lastly, our formalism provides a unified approach to independently quantify each cell process, thus revealing a complex interplay between cell divisions, cell rearrangements and cell shape changes and providing a rigorous framework for its future characterization using both mutant conditions and modeling.

24Epithelial tissue growth and morphogenesis is regulated by mechanical stress. To provide a complete set of methods to study tissue development, we therefore aimed to combine our formalism with the measurement of mechanical stress due to tension in adherens junctions. This ‘junctional stress' gathers all forces (regardless of their biological origin, including cortical and cytoplasmic forces) transmitted between cells via adherens junctions. The relevance of junctional stress quantification to understand tissue development has been demonstrated by methods such as laser ablation (for review see) or optical trapping of cell junction. However, with these methods, it is difficult to obtain spatial and temporal stress maps at the scale of the whole tissue.

25Others and we have previously developed force inference approaches to quantify junction stress from segmented images independently of possible external forces such as a friction of the epithelium on an outer layer. We improved our method to make it numerically more robust and efficient, thereby enabling the determination of cell pressures, junction tension and junctional stress over the whole tissue (see Materials and methods, Figure 7—figure supplement 1). The stress has an isotropic part related to the pressure represented by a circle (Figure 7—figure supplement 1c). Its anisotropic part has an amplitude and a direction of traction represented by a bar, and a direction of compression (of equal magnitude and perpendicular, the display of which is redundant). Even on a single animal, the junctional stress maps vary smoothly over time and space, and are symmetric with respect to the midline, revealing the quality of the signal-to-noise ratio (Figure 7a, Figure 7—figure supplement 1c and Video 4f). We then performed their ensemble average over several animals (Figure 7b) and compared the anisotropic part of the junctional stress maps and of the CE rate maps of the different processes measured by the formalism. Focusing here on divisions, the analysis confirms that on average cell division orientation aligns well with junctional stress orientation, even in such a heterogeneous tissue (Figure 7—figure supplement 2, alignment = 0.87). Moreover, the division CE rate, which is more relevant to tissue morphogenesis, is also well correlated with junctional stress orientation (Figure 7b, alignment = 0.73).

26Taking further advantage of averaged maps of division CE rate on the one hand, and of tissue and cell process component maps on the other hand, enables to explore more finely the alignment between cell divisions and stress. In particular, we can exclude that a positive or negative component of cell divisions would be due to distinct relationships between division CE rate and stress orientations. Indeed, cell divisions have a positive component in region 1 and 2, while cell division CE rate is either poorly aligned (region 1, alignment = 0.16) or well aligned (region 2, alignment = 0.97) with junctional stress orientation (Figure 7b). In addition to regions where stress, division CE rate and tissue elongation are well aligned (region 2, Figure 7b, Figure 4e), we also find regions where, although cell divisions and junctional stress remain well aligned (region 3, alignment = 0.94, Figure 7b), the tissue CE rate is almost orthogonal to divisions and stress (alignment = -0.88, Figure 4e), mostly occurring through cell rearrangements and cell shape changes (Figure 4f,g). Altogether our results illustrate how the combination of the formalism and a stress inference method enables to uncover additional interplays between cell divisions, stress and tissue elongation. This sets the stage for in-depth spatial and temporal investigations of the interactions between each cell process and mechanical stress during tissue development.

27We have developed a unified multiscale formalism that relates cell and tissue behaviors to characterize the growth and morphogenesis of epithelial tissues in two and three dimensions. The formalism is free from assumptions regarding biological mechanisms, modeling or external forces and it has numerous advantages. Its unified and separate measurements of the contributions of each cell process to tissue growth and morphogenesis significantly help describe and quantify the mechanisms governing tissue development. These measurements have been validated with computer simulations. They can be easily represented on spatial and temporal maps or graphs to describe the interplay between divisions, cell rearrangements, cell shape and size changes and apoptoses, as well as the interplays between cell processes and junctional stress, thus facilitating their comparison in wild-type and mutant conditions. In combination with the recent advances in light microscopy, genetics and physical approaches, our unified framework and methods provide a basis for comprehensive analyses of the mechanisms driving tissue development.

28ubi-E-Cad:GFP and E-Cad:GFP were used for live imaging of apical cell contours in notum and wing pupal tissues. In all experiments the white pupa stage was set to 0 h after pupa formation (hAPF), determined with 1 h precision. For notum tissues, pupae were prepared for live imaging as described in. Pupae were imaged for a period of 15–26 h, starting at 11–12 hAPF, with an inverted confocal spinning disk microscope (Nikon) equipped with a CoolSNAP HQ2 camera (Photometrics) and temperature control chamber, using Metamorph 7.5.6.0 (Molecular Devices) with autofocus. Movies were acquired at 25±1°C. We took Z-stacks with 18 to 28 slices (0.5 μm/slice) to make sure we included the adherens junctions. Maximum projections of Z-stacks were obtained using a custom ImageJ routine (publicly available as the ‘Smart Projector' plugin) and have been used for tissue flow analysis. Multiple position movies were stitched using a customized version of the ‘StackInserter' ImageJ plugin. Filming 10 to 12 positions with 0.32 μm spatial resolution (pixel size) every 5 min yielded a tiling of a half dorsal thorax (3 movies), 24 positions yielded a tiling of the whole dorsal thorax (1 large movie, available as Supp. Video in), resulting in 5 hemi-notum movies.

29For trblup notum tissues, temporal control of gene function was achieved by using the temperature sensitive Gal4/GAL80ts. Tribbles was overexpressed using an UAS-Trbl transgene during pupal stage. Embryos and larvae were raised at 18°C. Late third instar larvae were switched to 29°C. After 24 to 30 h, pupae were examined. Those which were timed as 111 hAPF were mounted for live imaging at 29°C. Five hemi-notum movies were acquired.

30For wing tissue, E-Cad:GFP pupa was prepared for live imaging as described in. Immersol W 2010 (Zeiss) was used instead of voltalef oil. Pupal wing was imaged for a period of 17 h at 5 min interval, starting at 15 hAPF with an inverted confocal spinning disk microscope (Olympus IX83 combined with Yokogawa CSU-W1) equipped with iXon3 888 EMCCD camera (Andor), an Olympus 60X/NA1.2 SPlanApo water-immersion objective, and temperature control chamber (TOKAI HIT), using IQ 2.9.1 (Andor). Other details of treatment and quantification of wing data will be published elsewhere.

31The local tissue flow within notum tissues was first estimated by image cross-correlation velocimetry along sequential images using customized Matlab routines based on the particle image velocimetry toolbox, matpiv (http://folk.uio.no/jks/matpiv). Each image correlation window was a 3232 pixels square (~1010 µm2, pixel size 0.32 μm), had a 50% overlap with each of its neighbors. The resulting estimate of the velocity field was used to initiate the tracking procedure, and also for time registration of different movies.

32Images were denoised with the Safir software. Cell contours were determined and individual cells were identified using a standard watershed algorithm. Errors were corrected through several iterations between manual and automatic rounds. Automatic rounds consisted of a custom automatic software (Matlab) which tracked segmented cells through all images and detected each event where two cells fuse. The tracking relied on the comparison between cells in one image and the following, based on overlap of cells as well as centroid positions. Divisions were detected and we checked the sizes and elongation of daughter cells. Delaminations were characterized by cell size decrease before cell disappearance, and distinguished from fusions. A fusion between times and is an artifact which almost always reveals an error of segmentation which is either a false positive cell junction at time, or a cell junction missing at time. Both by manual tests on small subsamples, and by automatic tracking of false cell appearances or disappearances, we estimated the final relative error rate to be below, which was sufficient for the analyses presented here. The whole notum movie contains ~8.8 103 cells at the beginning. After one to three divisions per cell, several delaminations, and a flow of cells out of the field of view, the final image contains ~18.4 103 cells. Altogether, on the five hemi-notum movies, ~7.7 106 cell contours were segmented and tracked. For trblup tissues, there were ~3.7 106 cell contours. Cells were larger and easier to segment, resulting in errors even smaller than in wild-type tissues (data not shown). Cells in the anteriormost part get out of focus due to tissue flow and elongation, so that we cannot track them throughout the whole movie; similarly, on lateral sides, some cells become visible during the movie (purple cells in Video 3). We observed a total of ~40 103 divisions and ~5 103 delaminations in the wild-type tissues, ~1.7 103 divisions and ~7 102 delaminations in trblup tissues. This yielded maps of cell apical surface area and shape, cell centroid displacements (Videos 1 and 3), cell lineages, and neighbour changes.

33For wing tissue, ~2 106 cell contours were segmented and tracked. The segmentation was less manually corrected than in the notum. Errors in segmentation or tracking resulted in cell fusions, cell integrations and fluxes; their measured values were small enough that the measurement for other cell processes were not significantly affected. Interestingly, this shows the robustness of our formalism with respect to errors in input data. Altogether, ~13 106 cell contours (~40 106 cell-cell junctions) were analyzed.

34In epithelial tissues where cells are confluent, assuming mechanical equilibrium, cell shapes are then determined by cell junction tensions and cell pressures, and information on force balance can be inferred from image observation. For instance, if three cell junctions which have the same tension end at a common meeting point, their respective angles should be equal by symmetry, and thus be 120° each. Reciprocally, any observed deviation from 120° yields a determination of their ratios. The connectivity of the junction network adds redundancy to the system of equations, since the same cell junction tension plays a role at both ends of the junction.

35Mathematically speaking, there is a set of linear equations, which involve all cell junction tensions, and cell pressure differences across junctions. We simultaneously estimate tensions and pressures by using Bayesian statistics formulated by maximizing marginal likelihood, or equivalently, by minimizing the Akaike Bayesian information criterion (ABIC). Our expectation that junction tensions are distributed around a positive value is incorporated as a prior. This inverse problem requires to invert large matrices, whose typical size is a few. Our custom program takes advantage that these matrices are sparse (http://faculty.cse.tamu.edu/davis/suitesparse.html), which not only increases the speed of resolution, but also minimizes the ABIC more robustly. It infers all junction tensions up to only one unknown constant which is the tension scale factor, and an additional unknown constant which is the average cell pressure. By integrating tensions and pressures, their contributions to tissue stress can be calculated in any given cell patch, again up to these two constants, with the Batchelor formula

36where is the two-dimensional identity matrix, is the cell patch area, the area of cell, the sum is over each cell in the patch and its neighbours, is the vector representing the chord of the junction between cells and (it links two vertices, its orientation being unimportant). Only the junction tensions contribute to the deviator of (hence called ‘junctional stress' in the main text), and thus determine the main direction and the difference of two eigenvalues of.

37Regarding this junctional stress, we recall three points:

38Junctions contribute to transmit stress between cells even if the biological origin of this stress lies in non-junctional cytoskeletal elements and other structures in the body of the cell. Forces created in the cell body or at the cell boundary are transmitted to its neighbours by contact at the cell-cell interface. These forces transmitted by contact can be decomposed in components parallel and perpendicular to the junction, namely tension and pressure, which we both estimate. Changes of cell volume lead to measurable changes in pressure and thus to detectable changes in our stress estimation.

39We focus on junctional stress because of the growing consensus that it is a important determinant of tissue morphogenesis, as measured by laser ablations of (and outside of) adherens junctions (for review see), or directly probed by experimental manipulation of adherens junctions and compared with simulations. Note that we cannot measure other contributions to the total stress, for instance through cell membrane parts far from adherens junctions.

40The force inference method used to estimate the junctional stress is independent of other contributions to stress or of external force contributions; it thus remains valid even in the presence of cell-substrate interactions that can cause an additional contribution to the total mechanical interactions.

41Each of the five wild-type hemi-notum movies was oriented with the midline along the top horizontal side. The whole notum movie was cut into two hemi-notum movies; the left one was flipped to be oriented like the others movies. As spatial landmarks, we use the macrochaetae, which we identify manually at ∼16h30 APF 10 min, when the sensory organ precursor (SOP) cells divide, and the following results were independent of the choice of this reference time. For each movie, each macrochaeta is labelled, with for the closest to the midline, and the largest or 8 according to the movie. For each of the five movies, labelled to 5, we call axis the midline oriented from posterior to anterior, axis the perpendicular axis oriented from medial to lateral, and as origin of axes the barycenter of the five macrochaetae closest to the midline, to 5. A box is defined on the first image as the set of cells which barycenter is within a 128128 pixels square (~4040 μm2). A box contains several tens of cells, and has a 50% overlap with each of its neighbors. Each box is then labelled by two integer numbers which define our two-dimensional space coordinates on a fixed square lattice (Eulerian representation). Averaging over space is implemented by averaging over. When the whole notum movie is analyzed alone as a whole, the midline is chosen as axis.

42Data near the sample boundary are less reliable because of poorer statistics. In order to improve the quality of all quantitative calculations and visual representations, a cell patch at location at time near the boundary of the sample was assigned a weight, as follows.

43We defined the ‘bulk’ of the tissue in Eulerian description as boxes containing only core cells, and in Lagrangian description as patches placed at least three patches away from the boundary.

44In Eulerian description, we defined the ‘relative area’ of a box as the sum of the surface area actually occupied by recognizable cells in this box, divided by the surface area of the box. It was close to 1 in all boxes of the bulk, possibly slightly larger than one due to large cells with their barycenter in the box but some of their area outside of the box, and possibly lower than one due to junction pixels (junctions are one pixel wide, they belong to the box but are not assigned to a cell). In Lagrangian description, was the number of cells in the box, divided by its maximum value in the tissue: in the first images of the movie it was slightly less than 1 over the whole bulk, while it decreased by a factor at least 2 towards the end of the movie.

45Its minimum value in the bulk at time was used to normalized all values as

46which can occur only for boxes out of the bulk, and

47which occur for all boxes in the bulk. This way, is equal to 1 all over the bulk, and gradually vanishes when approaching the tissue boundary as cell patches contain fewer cells.

48Since the results presented below relied on the inversion of the texture matrix (Section C.1.1), we measured the condition number of the texture matrix in this box. The condition number of a function with respect to an argument measures how much the output value of the function can change for a small change in the input argument. For a symmetric 22 matrix such as, the condition number is, always larger than 1. We used its inverse, Matlab’s ‘reciprocal condition number’, which is 1 when is isotropic for instance and vanishes when is not invertible (i.e. for a box which contains only one half-link, or half-links which are all parallel to each others). Its minimum value in the bulk at time, typically 0.5, was used to normalized all values as

49This way, is equal to 1 all over the bulk, and vanishes when approaching the tissue boundary. The weight of the box was the square of the normalized relative area times normalized reciprocal condition number:

50Note that since the stress estimation does not use the inversion of the texture matrix, the weights used for stress measurements are simply.

51In this work, we mostly consider quantities that have been averaged over some time period (2h or 14h) rather than their instantaneous values that can be noisy. For any quantity at, calling its instantaneous value, its weighted average over time is and is defined as follows:

52the sum being made between and. The corresponding mean weights averaged over the same period are:

53with is the number of frames during, the time between two frames being. Weights decrease according to the number of data in the box, from 1 for a box fully in the bulk of the tissue between, to 0 for a box outside of the tissue boundary between. These weights were used in all calculations, averages and graphs. It was even used in maps as an opacity, so that data become more transparent near the boundary where they are less reliable (Figure 3, Figure 3—figure supplements 1, 2, Figure 5, Figure 7 and Videos 4, 5).

54We define the scalar product of two second-order tensors in dimension as follows:

55The scalar product of with itself is the square of its norm

56Note that with this definition, the identity is unitary:

57In dimension 2, for two deviatoric tensors represented as bars, where is the difference in their bar directions. is positive if the bar eigendirections are aligned (close to 0°), negative if these directions are perpendicular (close to 90°), and vanishes in between (close to 45°). The alignement coefficient between two tensors is:

58It is close to 1 if both tensors are aligned everywhere, close to if if both tensors are perpendicular everywhere, and vanishes if the tensor directions are independent. Equation 8 is an average of with weights combining and the bar lengths.

59When the data were averaged over larger time or length scales, the left-right symmetry was visually better (we have checked it quantitatively, data not shown), the reproducibility from one animal to another was increased, and the data maps appeared smoother. Unless stated otherwise, all results presented are computed in boxes of size 128128 pixels2 (at the onset of the movie), namely 4040 μm2, with 50% overlap. Time averages are over 2 h for movies or 14 h for still images. Whole notum images are measurements over one animal; archetype refers to average over 5 hemi-notum movies (1 whole animal and 3 half animals). This yielded good statistics while preserving the fine spatial variations of the data maps. Averaging different movies was made possible by defining their common space and time coordinates. We developed a general method to rescale and register movies from different animals and genotypes in time and space, as follows.

60We translated the five movies in order to match their origins. The position of macrochaetae is reproducible from one animal to the other: their standard deviation is 5.5 μm along and 6.3 μm along. We define as an archetype of species (e.g. the wild-type) a virtual animal which macrochaeta would be the barycenter of the actual macrochaetae in the five movies:

61We then rescale each actual movie, separately along and axes. This is necessary at least for movies taken at different resolutions, or for mutants. The procedure is as follows. Along, the multiplicative factor which minimizes the dispersion of macrochaetae of from the archetype is:

62In the wild-type tissues, we find ranging from 0.96 to 1.07, with a standard deviation of 0.03. Along axis, we perform the same analysis, and also include the position of the midline as an independent information; we find rescaling factors of 1 0.04. We thus observe that for these five hemi-notum movies the rescaling is not crucial. The variability in tissue scale for these wild-type tissue movies is lower than the variability in macrochaetae positions (not shown). After rescaling, the residual dispersion is slightly lower than the initial one: the standard deviation of macrochaetae position becomes 4.5 μm along and 5.5 μm along. The referential in the archetype defines the grid, whereby (0,0) is centered around one box. In the trblup tissues, standard deviations of macrochaetae positions were 6.2 μm along and 6.3 μm along. With respect to the archetype, the standard deviations were 10.2 and 7.9 μm, respectively. With the same procedure as the wild-type tissue, they were rescaled; after rescaling, the standard deviations were 6.2 and 5.7 μm, respectively. Note that the change in temperature from 18°C to 29°C has apparently no effect on the tissue shape, according to tests performed on the posterior part of the wild-type tissue.

63While the hAPF was determined with 1 h absolute precision, the tissue rotation rate analysis provided a better relative precision that we used to synchronize the different movies, as described in detail in. In the region of the tissue located near the origin of axes, we observed that the rotation rate systematically passed through a maximum during the development: this rotation peak could be used as a biological reference time. For that purpose, the rotation rate was measured and spatially integrated over a rectangular reference window. Plotting this average versus frame number yielded a bell-shaped curve (shown in Figure S4 of). We applied a time translation to each movie so that these curves overlapped. We matched the portion of the curve which had the steepest slope: we thus used the time corresponding to 3/4 of the peak value, in the ascent (rather than the maximum itself, which by definition had a vanishing slope). Its average value was 18:40 hAPF. Hereafter, ‘hAPF' indicates the time after this temporal translation has been applied. For instance, after synchronization, the maximum of the contraction-elongation and rotation rates were consistently found at 19:20 and 19:40 hAPF, respectively. This determination reached a 1 interframe (i.e. 5 min) relative precision. After this synchronization, we determined that the period of time included in all movies was 13:55-27:55 hAPF (hereafter and in the main text, rounded to ‘14-28 hAPF’), corresponding to 169 frames, 68 interframes analyzed in what follows. Global time averages were performed over all these frames. Sliding window averages were performed over 2 h every 5 min. The macrochaetae were again manually determined at the time 16h30 APF determined with this synchronization. To improve the precision, the time and space registration was iterated: it changed only slightly, evidencing the robustness of this double registration. The trblup tissues were synchronized with the same procedure as the wild-type tissue, up to a 10 min precision. For experiments performed at 29°C, the development is accelerated. This was taken into account by dividing time intervals by 0.9 (as determined with precision by the widening of the wild-type tissue rotation peak).

64These spatial and temporal adjustments allowed us to assign a system of space-time coordinates common to the 5 hemi-notum movies, from pupae with a given species (wild-type or mutant). We again checked, now with spatial details, the reproducibility from one pupa to another. To estimate the averages and standard deviation among animals at each space-time point, for each movie labelled to 5, we used the time-average weights defined in Equation 4. For any given time-average quantity, we defined its ensemble average over the movies of the species (here, the wild-type genotype, ), local in space and time, as

65The corresponding mean weights averaged over the same set of animals are:

66These weights were used in maps as an opacity so that measurements become more transparent near the boundary where data come from fewer animals and are less reliable (see Figure 4, Figure 6, Figure 6—figure supplement 1, Figure 7).

67The biological variability of the quantity was determined by measuring the weighted standard deviation for the species

68The fraction in Equation 12 generalizes the usual term for unbiased standard deviation,, which is recovered when all weights are equal to 1. This unbiased weighted standard deviation can be calculated for each of the different independent components of a symmetric tensor. In general, these components are, and. When we consider only the deviator of, the components are and. We compare them to the biological variability, and define that a measurement at a given position and time is significant if it satisfies:

69namely if:

70Using the significance criterion of Equation 14, measurements larger than the biological variability are plotted in color while smaller ones are shown in grey. Note that if the direction of the anisotropic part of the tensor varies a lot between animals, this variation decreases the amplitude of the anisotropic part of the mean tensor, or equivalently the amplitude of, and therefore the measurement is considered as non significant based on Equation 14.

71We numerically simulated the different processes using the cellular Potts model, an algorithm relevant in biology to describe variable cell shape, size, packing and irregular fluctuating interfaces of cells in 2 or 3 dimensions. Each cell is defined as a set of pixels, here on a 2D square lattice; their number defines cell area. The pixelisation of the calculation lattice can be chosen to match the resolution of experimental images. A cell shape changes when one of its pixels is attributed to another cell instead. We used periodic boundary conditions, with external medium surrounding cells (a state without adhesion or area and perimeter constraints). We use here a simple version where cells minimise their surface energy. Energy minimisation uses Monte Carlo sampling and the Metropolis algorithm, as follows. We randomly draw (without replacement) a lattice pixel and one of its eight neighboring pixels. If both pixels belong to different cells, we try to copy the state of the neighboring pixel to the first one. If the copying decreases the total energy, we accept it, and if it increases the total energy, we accept it with a probability <1 known as ‘fluctuation allowance’.

72Divisions were implemented as follows. Cells were growing with an initial asynchrony in their cycle ranging from 0 to 40% (to avoid divisions all occurring at the same time). As they grew, the group of cells underwent a force gradient along the axis which tended to stretch each individual cell shape as well as the cell patch shape along the axis. Cells divided only once when their target area had doubled, along their long axis; they did not regrow after division, thereby recovering their initial size. Cells divided along their long axis, which resulted in divisions oriented along. Cells tended to relax their elongated shapes before or after divisions, resulting in some rearrangements. Each image was 800800 pixels2 and was cropped to yield the final movie. There were 50 Monte-Carlo steps between two successive images.

73Other processes were implemented in a similar way (except when stated otherwise), as follows. Stretching the pattern and allowing for cell shape relaxation led to oriented rearrangements. Affinely stretching the pattern by direct image manipulation using an image treatment software instead of Potts simulations (hence not followed by any cell shape relaxation) led to strong cell shape changes. Delaminations were obtained by gradually decreasing the cell target area. Cell integration movie was produced by reversing the order of images of delamination simulation movie. Fusions were forced on cells by random removal of cell-cell junctions (the same image was gradually Gaussian-blurred then thresholded), and cells were then let to relax their shapes. Boundary flux was implemented by gradual removal of successive external cell layers. Rotation of the cell patch was achieved by direct image manipulation. A simulation movie is made with 241 frames, 240 interframes. It would be analogous to a experimental movie of 240 5 min, or 20 h. Hence an experimental scale of 10-2 h-1 would be equivalent to 8.3 10-4 interframe-1 in simulations.

74The approach which leads to the formalism and to the decomposition into separate cellular processes is described in details in the Appendix. We describe here its practical implementation.

75In practice, when studying a monolayer of cells, we are mainly interested in morphogenetic movements within the plane of the monolayer: we focus on in-plane components and actually implement the formalism in two dimensions, as follows. We use two successive images of the segmented and tracked movie. In both images, for each cell the list of neighbour cells is identified. A half-link is the link between a cell center and its neighbour center, and is listed independently from the half-link oriented from to. In both images, we list all half-links. The rare cases where four cells meet (along a vertex, in 2D, or along a line, in 3D) are listed separately: in what follows, it is possible to treat them separately if needed. This is what we do, and at the end we lump their contribution with that of other cells.

76The tracking enables to classify each half-link into one and only one category (which ensures the completeness of the formalism): appeared or disappeared through one of the processes, or conserved between both images. For instance, when a cell divides, all its half-link disappearances and the appearances of half-links for its daughters are included in the division process (conversely, the ‘division orientation' includes only the link between the newly created daughter cells) (Figure 1a,b). Similarly, when cell undergoes a delamination, some of its former neighbors enter in contact: their new half-links are also counted in the delamination contribution (Figure 1a,b). The half-link can be in a different category from half-link, for instance if in this interframe cell undergoes a division while cell undergoes a delamination, rearranges with other neighbours, or exits at the sample boundary. The result is a classification of all half-links in each image.

77All individual movies were analyzed using Lagrangian measurements, thanks to the tracking of cells and patches illustrated in Figure 3d, Figure 5a, Videos 1 and 3. To coarse-grain the measurements, the whole image is subdivided into cell patches. In the Lagrangian description chosen here, each patch is tracked from one frame to the next. After a division, daughter cells are assigned to the patch of their mother. All cells and links belonging to each patch are used to calculate tissue deformation and the deformations associated with each cell processes occurring between two images, and we then turn them into rates and average them over time (see Appendix, sections B and C). Those deformation rates are related by the balance equation (see Equation 77):

78We measure separately each term of both sides of Equation 15, by assigning each link geometric change to, and each link topological change to a specific cell process, taking advantage of their completeness. Finally, we systematically check that Equation 15 is satisfied.

79We consider any of the cell process (e.g. divisions, rearrangements, cell shape and size changes, delaminations, … ), which we note, measured by the tensor (e.g.,,,, … ). While the isotropic part of is a scalar, which directly quantifies the growth rate, the CE part of is a tensor characterized by an amplitude and a direction that differ from the amplitude and direction of tissue CE rate. In order to determine the effective contribution of process to local tissue deformation, we project it onto. To do so, in each patch from the tissue, we first define the unitary tensor that is aligned with (Figure 4c):

80which trivially ensures. Then we project process contribution along using the scalar product defined in Equation 5 (Figure 4c):

81We call this projection the component of parallel to tissue morphogenesis (or ‘along ', for short). It is expressed as a rate of change per unit time, i.e. hour-1. It represents the effective contribution of to tissue morphogenesis. The additivity (Equation 15) also applies separately to these components:

82In 2D,, where is the difference in directions of and bars. Thus, if a process CE rate is exactly parallel to the tissue CE rate, it has a positive component on, which is exactly its own amplitude:. If the bar is rather parallel to, it has a positive component on. If the bar is at 45° angle to, its component on is zero. If the bar is rather perpendicular to, its CE rate has a negative component on. If the bar is exactly perpendicular to, its CE rate has a negative component on, which is exactly minus its own amplitude:. As a consequence, the component of tissue CE rate along itself,, is the amplitude of, and is positive by construction. Figure 4c shows the sign of.

83In this section, we list the various sources of uncertainties that we encounter, present our methods to decrease them as much as possible, and discuss why their impact on our analyses remains limited. They fall into two main categories: those related to the acquisition of data which will be used as input for the formalism; and those related to the formalism itself.

84Errors due to image analysis affect the input of our formalism. Segmentation errors can result in false identification of cells, while tracking errors can result in false identification of cells and their lineages. It is thus necessary to choose an image acquisition time sufficient to ensure an image contrast and quality which enable a segmentation with a low error rate. This sets a minimal value to the acceptable time difference between two successive images. In the notum and wing tissues we observe here, fusion and integration events are used as markers of segmentation and/or tracking errors. In the notum, we use them as feedback to correct the segmentation until the contributions of fusions and integrations are negligible with respect to the contributions we want to measure.

85The tissue is a three-dimensional (‘cuboïdal’) monolayer and we do not assume it to be 2D. However, we perform its study in 2D. More precisely, we image it using the E-Cadherin:GFP marker, which labels the apical adherens junctions. Moreover, several quantites we measure are actually 2D, such as the velocity field, or its gradient which represents the morphogenesis. Other quantities like cell division orientation or stress field are 3D but can be studied in 2D independently from what occurs in the third dimension. Finally, some 3D processes like delamination in which apical area shrinks have an effect on tissue morphogenesis which is similar to that of an apoptosis where a cell entirely shrinks, and is thus not necessary to distinguish within the scope of the present study.

86The 3D structure of the notum also plays a role because the tissue is curved. It is imaged in 3D using a confocal spinning disc microscope; the height of its surface is recorded and we can entirely reconstruct its curvature. We can then obtain the local angle of the tissue surface with the horizontal plane: in the notum, this is mainly along the direction (medio-lateral), since along the axis the tissue curvature is much smaller. We obtain the projection factor cos which should in principle be applied as a correction in the corresponding direction either to the raw image before any treatment, or a posteriori to results of the formalism. We measure is at most 0.01 radian over the tissue and reaches at most 0.02 radian in the most lateral part of the whole thorax image. The absolute correction factor is the same for all tensors, and it is negligible with respect to 1 in the results presented here (1-cos ~10-4), so that it does not affect significantly the results presented here. For simplicity, we choose not to perform any correction to our results, knowing that we can perform the correction a posteriori if it appears necessary in the future. All these effects are completely negligible in the wing, which is much flatter than the notum.

87For a given set of data used as input to our formalism, the formalism itself does have uncertainties which affect its output. For instance, the time difference between two successive images should be small enough so that the fraction of conserved links remains larger or comparable to the fraction of non-conserved links, and that our measurement of tissue growth and morphogenesis (which is based on conserved links) accurately quantifies the actual morphogenesis. Since should be sufficiently large to keep a good image quality, the question is how to choose an optimal. We find that in our case, min is optimal, and for these value both constraints are satisfied: the image is good enough to be automatically segmented for a large part, and the morphogenesis is well quantified by our measurement. Finally, in order to validate our workflow, we have checked that with the data of the preceding paper our new formalism recovers consistent results. In conclusion, our pipeline which links the image acquisition to the data representation (Figure 1e) is valid and its accumulated uncertainties are lower than the biological variability within an animal or between different animals.

88In mutant tissues, the space and time registration are performed as in the wild-type tissues. Mutant tissues can exhibit a different variability than wild-type tissues; in practice, we observe a larger variability in trblup mutant tissues than in wild-type tissues. Moreover, wild-type and mutant tissues can be registered together. The total morphogenesis as well as the measurement of each cell-level process is computed similarly in a mutant tissue. We compare them term by term with the corresponding values in wild-type tissues. We assay at which time and position the mutation has a significant effect with an inter-genotype variability larger than the intra-genotype one. It is also possible to subtract term by term each measurements performed in the wild-type tissues and the measurements performed in the mutant tissues. We can plot the measurement difference defined as this difference represents the part of that has been added by the mutant condition to. It represents the effect of the overexpressed gene in the process of wt morphogenesis, and its projection along the wild-type tissue CE rate represents its effective contribution to wild-type morphogenesis.