Topological defects in epithelia govern cell death and extrusion...Topological defects in epithelia govern cell death and extrusion Thuan Beng Saw 1,2, Amin 3Doostmohammadi 3, Vincent

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LETTERdoi:10.1038/nature21718

Topological defects in epithelia govern cell death and extrusionThuan Beng Saw1,2*, Amin Doostmohammadi3*, Vincent Nier4, Leyla Kocgozlu1, Sumesh Thampi3,5, Yusuke Toyama1,6,7, Philippe Marcq4, Chwee Teck Lim1,2, Julia M. Yeomans3 & Benoit Ladoux1,8

Epithelial tissues (epithelia) remove excess cells through extrusion, preventing the accumulation of unnecessary or pathological cells. The extrusion process can be triggered by apoptotic signalling1, oncogenic transformation2,3 and overcrowding of cells4–6. Despite the important linkage of cell extrusion to developmental7, homeostatic5 and pathological processes2,8 such as cancer metastasis, its underlying mechanism and connections to the intrinsic mechanics of the epithelium are largely unexplored. We approach this problem by modelling the epithelium as an active nematic liquid crystal (that has a long range directional order), and comparing numerical simulations to strain rate and stress measurements within monolayers of MDCK (Madin Darby canine kidney) cells. Here we show that apoptotic cell extrusion is provoked by singularities in cell alignments9,10 in the form of comet-shaped topological defects. We find a universal correlation between extrusion sites and positions of nematic defects in the cell orientation field in different epithelium types. The results confirm the active nematic nature of epithelia, and demonstrate that defect-induced isotropic stresses are the primary precursors of mechanotransductive responses in cells, including YAP (Yes-associated protein) transcription factor activity11, caspase-3-mediated cell death, and extrusions. Importantly, the defect-driven extrusion mechanism depends on intercellular junctions, because the weakening of cell–cell interactions in an α-catenin knockdown monolayer reduces the defect size and increases both the number of defects and extrusion rates, as is also predicted by our model. We further demonstrate the ability to control extrusion hotspots by geometrically inducing defects through microcontact printing of patterned monolayers. On the basis of these results, we propose a mechanism for apoptotic cell extrusion: spontaneously formed topological defects in epithelia govern cell fate. This will be important in predicting extrusion hotspots and dynamics in vivo, with potential applications to tissue regeneration and the suppression of metastasis. Moreover, we anticipate that the analogy between the epithelium and active nematic liquid crystals will trigger further investigations of the link between cellular processes and the material properties of epithelia.

To understand the mechanisms that underlie apoptotic cell extru-sion (Fig. 1a), we investigated the relationship between extrusion and epithelial monolayer remodelling. By culturing Madin Darby canine kidney (MDCK) epithelial cells at confluency on micropatterned substrates coated with extracellular matrix proteins (Methods), we observed that extrusion events were preceded by a coordinated, long-range flow of cells towards the eventual location of the extrusion (Fig. 1b, − 160 min; Supplementary Movie 1). The group of cells that constituted these flows consistently formed a comet-like shape, with the head portion of the comet pointing towards the cell destined for

extrusion (Fig. 1c, d). Since the cells in the monolayer were aniso-tropic in shape over long periods of time (Extended Data Fig. 1a, b) and demonstrated supracellular orientational order in their alignment (Extended Data Fig. 1c, d), the comet shape can be identified as a sin-gularity in the cellular alignment9,10. We note that the singularity takes the form of a topological defect with + 1/2 topological charge, where the orientational ordering is destroyed in a nematic liquid crystal. There are predominantly two types of defects in nematic liquid crystals, that is, + 1/2 and − 1/2, and both were identified in the monolayer (Fig. 1e, Methods)12. Such nematic topological defects have been identified in a wide variety of biological systems including lipid vesicles13, fibroblast cell colonies9,10, suspensions of microtubule bundles14, motility assays of driven filaments15, and growing Escherichia coli colonies16.

We found that extrusion events were strongly correlated with the positions of a subset of + 1/2 defects (and less so with − 1/2 defects) (Fig. 1f, Extended Data Fig. 1e–h, Methods). We further found similar extrusion–defect links in different types of epithelium (Fig. 1f and Extended Data Fig. 1e–h), including a cell-division-inhibited (treated with mytomycin C) MDCK monolayer, a breast cell line (MCF10A) and human epithelial skin (HaCaT). In the last case, we found a correlation between extrusions and defects, but with a stronger correlation with − 1/2 defects, which may be attributed to the multi-stratified organiza-tion of HaCaT cells as well as the HaCaT cell layers being more elastic than the MDCK monolayer17. We then analysed the temporal correlation between nematic defects and cell extrusions within MDCK epithelial monolayers. Defects occurred well before cell extrusion and caspase activation (at about 100 min) (Extended Data Fig. 1i), consistent with spatio-temporal cellular flows observed in these regions. It suggests that singularities in cellular alignment are spontaneously generated in the epithelial monolayer in the form of nematic topological defects, and the defects in turn trigger cell apoptosis and extrusion.

To probe the first part of the hypothesis, we studied the properties of singular points of cellular alignment in wild-type (WT) MDCK to confirm their identification with topological defects in active nematic liquid crystals. We used particle image velocimetry18 (PIV; see Methods) to measure experimentally the velocity and strain rate fields around the singular points in cell alignment (Fig. 2a top row, Experiment), and compared them to numerical simulations (see Methods) of active nematic liquid crystals (Fig. 2a bottom row, Simulation)19. The close match between strain rate patterns and velocity fields around + 1/2 topological defects in experiments and simulations (Fig. 2a) revealed that the epithelium monolayer of cells indeed behaves as an extensile, active nematic. The extensile nature is manifest as flow along the elongated axes of the cells that moves towards the head region of the defect (Fig. 2a rightmost column, Average velocity field), in contrast to contractile flow, which is directed towards the tail region (Extended Data Fig. 2a).

1Mechanobiology Institute, T-Lab, Singapore 117411, Singapore. 2National University of Singapore Graduate School of Integrative Sciences and Engineering (NGS), Centre For Life Sciences (CeLS), Singapore 117456, Singapore. 3The Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, UK. 4Laboratoire Physico-Chimie Curie, Sorbonne Universités, UPMC Université Paris 6, Institut Curie, CNRS, UMR 168, Paris 75005, France. 5Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600036, India. 6Department of Biological Sciences, National University of Singapore, Singapore 117543, Singapore. 7Temasek Life Sciences Laboratory, National University of Singapore, Singapore 117604, Singapore. 8Institut Jacques Monod (IJM), Université Paris Diderot, CNRS, UMR 7592, Paris 75013, France.* These authors contributed equally to this work.

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Previous studies have shown that other active systems exhibited exten-sile behaviours, including reconstituted microtubule bundles14, E. coli bacteria20, sperm cells21, and cell division induced flow field in an epithelium19,22. We checked that cell division inhibition did not alter the extensile nature of MDCK (Extended Data Fig. 2b, Mytomycin C). Moreover, active nematic theory predicts that the defect density scales proportionally with the activity of the cells19 (Extended Data Fig. 2c, Supplementary Movie 2). Because active stresses in the cell monolayer

originate to a large extent from the actomyosin activity23, we tested this by introducing blebbistatin treatment. When blebbistatin was added to reduce activity at various concentrations (10 μ M and 50 μ M), there was a steady and considerable drop in the defect density with respect to the concentration we used (Extended Data Fig. 2d), and this trend was reversed during washout (Fig. 2b, Supplementary Movie 3). The rates of defect density decrease and increase were similar, as the defect density dropped to about 33% of its original value after approximately 10 h of blebbistatin treatment, and increased back to about 66% of the original density in approximately 6 h of washout. The blebbista-tin treated monolayer maintained an extensile flow field (Extended Data Fig. 2b, Blebbistatin 10 μ M). In addition, the extrusion–defect spatial correlation was slightly reduced under blebbistatin treatment, suggesting a stress regulation around the defects through actomyosin activity (Extended Data Fig. 2e). Altogether, these results confirm that the epithelium is behaving as an active nematic liquid crystal and that topological defects are spontaneously formed by active stresses in the monolayer. This is in contrast to a recent finding that fibroblast cells in a packed environment behaved as a non-active, nematic liquid crystal10, where the number of topological defects relaxed to the equilibrium state.

To understand why extrusions were related to defects, we first spec-ulated that defects might generate regions of high local cell density that induced extrusions, as tissue crowding is known to play a role in cell extrusion4–6. However, we did not find a clear spatial correla-tion between extrusions and the regions of highest local cell density (Extended Data Fig. 3a, Methods). The next clue came from liquid crystal theory12, in which spontaneously formed topological defects are expected to generate mechanical stress in their vicinity owing to large distortions in the cell orientation. We thus hypothesized that + 1/2 defects generated spatially localized, elevated compressive stress that is sufficient to provoke cell response, apoptosis and extrusion. This hypothesis is supported by simulations showing that the highest com-pressive isotropic stresses are strongly correlated with the locations of + 1/2 defects and less so with − 1/2 defects (Extended Data Fig. 3b), which is reminiscent of the stronger extrusion correlation with + 1/2 defects than with − 1/2 defects in experiments (Fig. 1f, Extended Data Fig. 1f). To investigate the impact of mechanical stress on cell extrusions, we measured the mechanical traction exerted by cells on the underlying substrate using traction force microscopy17 (TFM, see Methods), and converted the traction to two-dimensional stress in the monolayer (Fig. 2c) using Bayesian inversion stress microscopy24 (BISM, see Methods). Consistent with the velocity field results, the stress pattern around + 1/2 defects in the experimental measurements is again highly similar to that in extensile nematic simulations (Fig. 2d).

To prove directly that cells are more compressed at + 1/2 defects, we calculated the isotropic contribution to the monolayer stress measured experimentally around topological defects. The isotropic stress provides a clear distinction between + 1/2 and − 1/2 defects as it is only highly compressive (negative) at the head portion of a + 1/2 defect, while being more tensile at a − 1/2 defect (Fig. 3a, Experiment). These distinguish-ing isotropic stress distributions around the defects are again reflected in the simulations (Fig. 3a, Simulation). We then measured the time evolution of the average isotropic stress for cells that were going to extrude, and found that such cells were being increasingly compressed (increasingly negative isotropic stress with time, see Fig. 3b) in the time leading up to their extrusion. Notably, the time at which stress started to become more compressive (about − 110 min) was consistent with the start of the long-range flow and with the rise in the defect–extrusion correlation (Fig. 1b, Extended Data Fig. 1i). These results matched the observation that a + 1/2 defect is first created and propagated in the monolayer, and that a cell is then extruded at the head region of the defect where the compressive stress is concentrated (Fig. 3c). There was also a higher probability for a defect with higher compressive stress to induce an extrusion (Extended Data Fig. 3c, Methods). We observed that the + 1/2 defects tended to maintain their overall orientation as

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Figure 1 | Extrusion correlates with singularities in cell orientation in the epithelia. The singularities are defects with a + 1/2 topological charge. a, Left, diagrams of confluent monolayer (top) and extruding cell (bottom). Grey, cell body; blue, nucleus; orange, apoptotic extruding cell). Middle, side view confocal image of confluent MDCK monolayer (top) and extruding cell (bottom). Green, actin; blue, nucleus. Right, corresponding images of activation of caspase-3 signal (red): top, monolayer; bottom, extruding cell. b, Phase-contrast images showing monolayer dynamics before extrusion (yellow arrowhead) at t = 0 min, overlaid with velocity field vectors. Length of vectors is proportional to their magnitude. Colour code: bright (dark) green for high (low) speeds. c, d, Corresponding images overlaid with red lines (in c), and represented as black lines (in d) showing average local orientation of cells. The group of cells moving towards the extrusion forms a comet-like configuration (in d: blue dot, comet core; arrow, comet tail-to-head direction). e, Experimental (left) and schematic (right) images of + 1/2 defect (top, comet configuration) and − 1/2 defect (bottom, triangle configuration). Red lines, average cell orientations; blue dot and arrowmark, respectively defect core and tail-to-head direction of + 1/2 defect. Green triangle, − 1/2 defect core. f, Left, diagram showing determination of correlation between + 1/2 defects and extrusions: distance of each extrusion to its closest + 1/2 defect in the preceding frame is measured (re), and the number of these defects per unit area as function of re is normalized (right). See Methods. n = 50 (MDCK WT) extrusions from 4 independent movies in 3 independent experiments; n = 61 (MDCK mytomycin C treatment) extrusions from 3 independent movies in 2 independent experiments; n = 85 (MCF10A) extrusions in 2 independent movies; n = 79 (HaCaT) extrusions in 2 independent movies. Scale bars, 10 μ m.


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the neighbour cells elongated to compensate for the extruding cell (Extended Data Fig. 3d), leading to a slow relaxation (about 250 min) of the average compressive stress (Extended Data Fig. 3e).

We then investigated potential mechanotransductive effects of the stress localization observed at defects on biochemical signals within the monolayer. To this end, we first checked the YAP transcription factor distribution in cells (see Methods), as YAP is known to respond to compressive mechanical signals25 and act as a potent inhibitor of apoptosis by translocating between the nucleus and cytoplasm26—it is also implicated in extrusion27. We observed that there were substan-tially more cells at the head region of + 1/2 defects than in − 1/2 defect cores that had YAP sequestered in their cytoplasm (Fig. 3d, e). As YAP is known to translocate to the cytoplasm under compression25, this finding clearly indicates that certain cells at + 1/2 defects are subjected to high compressive stress that may lead to YAP inactivation and as such, further confirm cell apoptosis in these regions. We then observed that caspase-3 activation (a marker for cell death1) only occurred when

there was an extrusion (Extended Data Fig. 3f). Along these lines, we found that caspase-3 inhibition largely eliminated the occurrence of extrusions (Extended Data Fig. 3g), even though the defect density was maintained. These results showed that compressive stress localization at defects induced cell death signals which in turn triggered extrusion1.

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Figure 2 | MDCK WT epithelia behave as 2D, extensile, active nematic liquid crystals. a, Top, average (Avg.) yy- and xy-components of strain rate map around + 1/2 defect in experiments (left and middle respectively) and corresponding average velocity flow field (right: n = 2,142 defects from 4 independent movies in 3 independent experiments), compared with simulations of extensile, active nematic liquid crystal (bottom). Colour code is positive for stretching and negative for shrinkage. b, Time evolution of (total) defect areal density under 50 μ M blebbistatin treatment and washout (arrow). Data for each time point are binned over duration of 120 min (n = 6 different time frames), in n = 4 independent movies. A t-test is performed for each time point against time = 600 min. Data are represented as n = 24 scatter points for each time, and bars represent mean ± s.e.m of the scatter points. * * P < 0.01, * * * P < 0.001. c, Diagram of TFM set-up to measure traction (tx and ty) and to infer 2D stress in the monolayer (σxx, σyy and σxy). d, Left pair of plots, average (Avg.) yy- and xy-components of stress map around a + 1/2 defect in experiments (n = 1,339 defects in 2 independent experiments), compared with simulation of extensile, active nematic liquid crystal (right pair of plots). Colour code is positive for tensile stress and negative for compressive stress. Overlaid black lines in panels show representative nematic directors, grey circle denotes defect core.

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Figure 3 | Compressive stresses at +1/2 defects trigger both cell extrusion and YAP mechanosensitive response and can be modulated by cell–cell junction strength. a, Average isotropic stress around a + 1/2 and a − 1/2 defect in experiments (n = 1,339 (+ 1/2) defects in 2 independent experiments; n = 2,454 (− 1/2) defects in 2 independent experiments) (left pair of plots), and simulations (right pair of plots). Within each pair, left is + 1/2 defect, right is − 1/2 defect. Colour code, positive for tensile state, negative for compression. b, Average isotropic stress evolution for cells experiencing negative (compressive) stress values during extrusion initiation at t = 0 min (spatially averaged over 65 μ m × 65 μ m; n = 32 extrusions in 2 independent experiments). A t-test is performed for each time point against a normal distribution centred at zero. * P < 0.0001. c, Isotropic stress around a + 1/2 defect moving to top left corner of image, and extrusion. Top panel, nematic directors (red lines) overlaid on monolayer. Bottom panel, corresponding isotropic stress heat map. Colour code, positive for tensile stresses, negative for compressive stresses. d, Distribution of YAP and nuclei of cells at a + 1/2 defect, with composite image at right (red, nucleus; green, YAP). Green arrowheads, cells with YAP in cytoplasm. e, Percentage of cells with YAP in nucleus, in cytoplasm or uniformly distributed, at head of + 1/2 defects, at core of − 1/2 defects, or at random points. n = 78 (+ 1/2), n = 77 (− 1/2), n = 78 (random) from 17 independent movies in 2 independent experiments. See Methods. Data represented as minimum, first and third quartiles, median, maximum (lines) and mean (circle). ks-test, * * * P < 0.001. f, Average isotropic stress around + 1/2 defect (α -catKD experiments). n = 1,940 defects from 3 independent movies. Similar colour code as a. g, h, Average defect areal density (g) and extrusion rate (h) in WT and α -catKD MDCK. g, n = 685 frames from 3 independent movies in 2 independent experiments (WT), n = 360 frames from 3 independent movies (α -catKD). Two sample t-test, * * * P < 0.001. h, n = 6 independent movies in 4 independent experiments (WT), n = 6 independent movies in 2 independent experiments (α -catKD). ks-test, * * P < 0.01. All data, except in e, represented as mean ± s.e.m. Black lines show representative nematic directors, grey circle denotes defect core. Red lines show local cell orientation. Scale bars, 10 μ m.


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To further test the contribution of intercellular junctions to this mechanism, we perturbed collective cell movements and intercellu-lar forces by knocking down α -catenin17,18, which is known to be a core mechanosensor of force transmission at cell–cell junctions28. We observed that α -catenin knock down (α -catKD) MDCK cells main-tained similar levels of orientational order and similar extrusion–defect correlation patterns as WT MDCK at a similar cell density range (Extended Data Fig. 4a, b, Fig. 1f, Extended Data Fig. 1e–h). However, the defects were smaller in size and had more spatially focused stress patterns compared to WT MDCK (Fig. 3a, f, Supplementary Movie 4). From the nematic model, the size of the defect core is known to scale as R ∝ √K, where K is the orientational elasticity constant characterizing the resistance of nematic directors to a change in the orientation12, and thus the smaller defect size seen in α -catKD experiments can be interpreted as a reduction in K (Extended Data Fig. 4c). Collective cell bending may be favoured in epithelial α -catKD cells, as the alteration of

the mechanical connection between cell–cell adhesion proteins and the contractile actomyosin cytoskeleton facilitates the relative movements of adjacent cells17,28. More importantly, nematic theory predicts that the number of topological defects is inversely related to the orientational elasticity (Extended Data Fig. 4d), therefore a reduction in the orien-tational elasticity is expected to result in a larger number of topological defects. Indeed, experiments showed that there was an increase of about 40% in the number of defects on going from WT cells to α -catKD cells (Fig. 3g), which can explain a marked increase in the extrusion rate in α -catKD cells (Fig. 3h). These results suggest that the weakening of cell–cell junctions in α -catKD cells facilitated collective in-plane bending of multiple cells that decreased orientational elasticity, K, thus increasing defect formation and extrusion.

To further prove the causal role of defects in extrusions, we sought to control defect locations in the monolayer29, thus allowing the control of extrusion hotspots. Since MDCK cells preferentially align tangential to the boundary between monolayer-adherent and non-adherent substrates18, we microcontact printed (Methods) a star-shaped cell monolayer (Fig. 4a) to geometrically force comet-like defects to the four tips of the star. The length scale of the tip of the star (about 100–200 μ m) was chosen to match the size of the defects. We indeed found that the defect density increased at the corners of the star, and that extrusions predominantly happened close to the four tips (Fig. 4b, c, Extended Data Fig. 5a, Supplementary Movie 5). In contrast, the − 1/2 defect density became larger near the centre of the star, but there was no increase in extrusion events in this region. This biased distribution of extrusions was not found in a circle-shaped monolayer (Fig. 4d–f, Extended Data Fig. 5a). The extrusions were also more correlated with + 1/2 defects (more decorrelated with − 1/2 defects) in the star-shaped than in the circle-shaped monolayer (Extended Data Fig. 1e–h, Extended Data Fig. 5b, Fig. 1f). Thus, we demonstrated that extrusions could be controlled by artificially controlling the positions of + 1/2 defects in the monolayer.

These findings reinforce the idea that comet-like defects in epithe-lia can mechanically induce cell apoptosis and extrusions. However, the reverse question may be asked: can apoptotic cells destined for extrusion produce certain biochemical signals that can increase cell activity1,30 and hence generate new local defects to expedite their own extrusion? To investigate this possibility, we first checked defect-related properties in a caspase-3 inhibited monolayer, and found that the defect density was similar to that of a non-treated monolayer, and the flow field patterns at + 1/2 defects still showed an extensile flow field albeit having a reduced pattern size (Extended Data Fig. 5c). This suggested that cell death signals did not contribute to the extensile nature and activity of the epithelium. In another more direct experiment, we used an ultraviolet laser to induce a single cell apoptosis31 (Methods), and followed the time evolution of the number of + 1/2 defects imme-diately afterward (within a radius of 80 μ m, Fig. 4g) and up to several hours, until the first extrusion occurred. We did not observe any increase in the average number of defects after the laser induction of cell apoptosis (Fig. 4h), which confirmed that there was minimal influence of apoptotic signalling on the triggering of more defects.

Taken together, our results show that as cells collectively move in the epithelial monolayer, topological defects in cell alignments are formed spontaneously. The emergence of + 1/2 defects provides hotspots of compressive stress, which lead to a higher probability of cell apoptosis and extrusion (Fig. 4i). This newly identified mechanism appears to be the main pathway that triggers apoptotic extrusion in epithelia, as around 70% of cell extrusion events occurred at compressive stress regions (Extended Data Fig. 5d). Such regions increased in compres-sion as a function of time on average, leading to extrusion (Fig. 3b). Knowledge of this mechanism allows tuning of extrusion hotspots through the control of topological defects in the tissue29. Notably, the magnitude of compressive stress needed for extrusion is modest (up to 350 Pa μ m, or about 35–70 Pa taking the typical cell height to be approximately 5–10 μ m, Extended Data Fig. 3e) compared to other

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Figure 4 | Topologically induced +1/2 defects can control extrusion hotspots. a, d, Confluent MDCK monolayer confined on star and circle shapes (respectively a and d, Methods). Scale bar, 100 μ m. Red lines show local cell orientation. b, e, Heat map of normalized extrusion number per unit area. c, f, Normalized average areal density of extrusions, + 1/2 defects and − 1/2 defects as function of distance from confinement centre, rfc. Each point on the curve is averaged over the full 360° for each specific range of rfc. n = 145 extrusions, n = 6,738 (+ 1/2) defects and n = 5,083 (− 1/2) defects from 12 independent movies in 2 independent experiments (star). n = 361 extrusions, n = 5,389 (+ 1/2) defects and n = 4,858 (− 1/2) defects from 8 independent movies in 3 independent experiments (circle). g, Diagram of laser induction of single cell apoptosis. h, Time evolution of average number of + 1/2 defects within radius of 80 μ m around laser induced cell apoptosis (laser induction at t = 0 min, n = 9 independent apoptotic induction experiments). A ks-test is performed for each time point against t = 0 min, P-values for t = 50–300 min are respectively P = 0.86 (t = 50 min), 0.74 (t = 100 min), 0.81 (t = 150 min), 0.75 (t = 200 min), 0.30 (t = 250 min) and 0.73 (t = 300 min). NS, not significant. Data represented as mean ± s.e.m. i, Diagram of apoptotic cell extrusion induced by nematic defect. 1, 2, 3 denote event sequence in order. Red arrow, + 1/2 defect. Green arrow, − 1/2 defect. Blue, cell nuclei. Orange, apoptotic cell.


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measured stress values in two- and three-dimensional epithelium or cell aggregates32–34. This suggests that extreme physical environ-ments or niches (for example, extremely curved substrate surfaces, overcrowding4–6) are not the only places where mechanical activation of apoptosis, extrusions and other mechanosensitive cell activities could be triggered.

We conclude that an epithelial cell monolayer behaves as an active extensile nematic material, and that spontaneous formation of sin-gularities in cellular alignment in the form of nematic topological defects is a previously unidentified cause of cell apoptosis, suggesting that such defects govern cell fate. Hence it is anticipated that this defect-induced extrusion mechanism could be a common strategy for preserving homeostasis of a normal epithelium in vivo and for suppressing tumour invasion. It would be interesting to explore how differences in the intrinsic mechanical properties of different epithe-lia can influence the stress distributions around topological defects and thus alter the influence of defects on apoptosis and extrusion. Another interesting investigation would be to probe the role of top-ological defects in pathological conditions, including oncogenic cell extrusion.

Online Content Methods, along with any additional Extended Data display items and Source Data, are available in the online version of the paper; references unique to these sections appear only in the online paper.

Received 9 May 2016; accepted 21 February 2017.

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Supplementary Information is available in the online version of the paper.

Acknowledgements We thank G. Peyret, S. Jain, R.-M. Mège and group members from MBI for discussions, as well as I. Yow, A. P. Le, S. Begnaud and A. Kehren for experimental assistance, and S. Begnaud for providing the YAP nucleus-to-cytoplasm ratio quantification algorithm. We also thank the MBI Microfabrication core, the MBI Microscopy core and the MBI Science Communication core for support. W. J. Nelson and M. Sudol are thanked for their gifts of MDCK cell lines and YAP antibody, respectively. This work was supported by the European Research Council under the European Union’s Seventh Framework Program (FP7/2007-2013)/ERC grant agreements 617233 (B.L.) and 291234 (J.M.Y.), and by the Mechanobiology Institute. T.B.S. was supported by an NGS scholarship.

Author Contributions T.B.S., A.D., J.M.Y. and B.L. designed research, T.B.S., L.K. and Y.T. performed experiments, A.D. and S.T. implemented the numerical method and did the in silico simulations, T.B.S., A.D., V.N. and P.M. contributed new reagents, modelling and computational tools, T.B.S., A.D., J.M.Y. and B.L. wrote the paper, and C.T.L., J.M.Y. and B.L. supervised the project. All authors read the manuscript and commented on it.

Author Information Reprints and permissions information is available at www.nature.com/reprints. The authors declare no competing financial interests. Readers are welcome to comment on the online version of the paper. Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Correspondence and requests for materials should be addressed to C.T.L. ([email protected]), J.M.Y. ([email protected]) and B.L. ([email protected]).

Reviewer Information Nature thanks A. Bausch, D. Discher and the other anonymous reviewer(s) for their contribution to the peer review of this work.




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METHODSExperimental techniques. Cell culture and reagents. MDCK strain II WT (MDCK WT, gift from J. W. Nelson, Stanford Univ) and HaCaT (Cell Lines Service) cells were maintained in a culture medium composed of high glucose DMEM 1X medium (Invitrogen), 100 μ g/ml penicillin, 100 μ g/ml streptomycin (Invitrogen), and 10% fetal bovine serum (Invitrogen). Culture medium for MDCK with stable GFP-actin and stable α -catenin knockdown MDCK (α -catKD), gifted by James W. Nelson, Stanford University, was additionally supplemented with 250 μ g/ml geneticin (Invitrogen). MCF10A cells (ATCC) were maintained in MEGM (Lonza) medium supplemented with cholera toxin (100 ng/ml). All cells were grown at 37 °C and 5% CO2. For microscopy imaging, DMEM was replaced with low glucose Leibovitz (Sigma-Aldrich). MCF10A imaging was done in its culture medium. To reduce MDCK WT actomyosin activity, blebbistatin (Selleckchem) was added at 10 μ M for 1 h before imaging. To observe the defect density temporal evolution as a function of activity, 50 μ M blebbistatin was added and washout was performed after a certain amount of time. To inhibit MDCK proliferation, mytomycin-C (Sigma) was added at 10 μ g/ml for 1 h, and rinsed before imaging. For live imag-ing of fluorescent cell nuclei, the confluent monolayer was incubated with Hoechst 33342 (ThermoFisher) at 1 μ g/ml for 5 min and rinsed. To monitor cell viability in experiments, caspase-3 indicator NucView (Abcam) was added to the medium at 1:1,000 dilution for ~ 30 min before imaging. Cell apoptosis was inhibited by adding Z-VAD-FMK (Promega) to the medium at 50 μ M. Cell lines tested for mycoplasma but not authenticated are MDCK WT, MDCK GFP-actin, MDCK α -catKD, MCF10A, and HaCaT.Microscopy imaging. A mature and confluent monolayer was allowed to develop overnight before experiments started. Confluency is reached when all space pre-sented to the epithelium is filled. Typical experiments were run for 1–2 days. Phase contrast and fluorescence time-lapse imaging for monolayer on a glass-bottom Petri dish, PDMS spin-coated dish, and TFM soft gel were performed using a Biostation (Nikon) or Olympus 1X2-UCB inverted microscope. Confocal time-lapse images were obtained on a Nikon A1R MP laser scanning microscope or a Zeiss upright Z2 microscope.Microcontact printing. Silicon wafers with desired patterns were made using SU-8 photoresist for soft lithography35. To form stamps, PDMS (Sylgard 184, Dow Corning) at 1:10 mixture ratio (curing agent: silicone elastomer) was moulded onto silanized wafers and cured at 80 °C for 2 h. A mixture of 50 μ g/ml pure fibronectin (Roche) and 25 μ g/ml conjugated fibronectin (Cy5.5 or Atto dye, GE Healthcare and Sigma) was incubated on the stamps for 1 h and dried before stamping. Patterns were stamped onto a UV-activated, PDMS spin-coated Petri dish and the unstamped area was passivated by 1% Pluronics (F127, Sigma) for 1 h. Samples were rinsed with PBS before cell seeding.Traction Force Microscopy (TFM). Soft silicone gel with attached fluorescent beads was used as the substrate for TFM so that in-plane cell traction forces on the sub-strate could be measured36. To prepare the gel with a stiffness of 10–20 kPa, CyA and CyB components (Dow Corning) were mixed at 1:1 ratio and spin-coated on a Petri dish to achieve a flat substrate of height ~ 60–100 μ m. After curing at 80 °C for 2 h, the substrate was silanized with 5% 3-aminopropyl trimethoxysilane (Sigma) in ethanol for 5 min. Carboxylated fluorescent beads (100 nm, Invitrogen) were functionalized on the substrate at 1:500 dilution in deionized water. The beads were passivated with 1X Tris (Sigma) for 10 min and pure fibronectin (50 μ g/ml) was incubated on the substrate for 1 h before cell seeding. Between each step, the samples were rinsed 3 times with 1X PBS. After the experiment, the cells were completely removed by adding SDS, so that the resting state of the gel could be measured. Bead displacements (with respect to its resting state) acquired during experiments were measured and converted to cell traction forces with an ImageJ plugin37.Cell apoptosis induction. Laser induction of cell apoptosis31 was done on a Nikon A1R MP laser scanning confocal microscope with Nikon Apo 60× /1.40 oil-immersion objective. An ultraviolet laser (355 nm, 300 ps pulse duration, 1 kHz repetition rate, PowerChip PNV-0150-100, Teem Photonics) was focused on the nucleus of a target cell for 10 s at a laser power of 25 nW at the back aperture of the objective.YAP antibody staining. Cells were seeded until confluency in circularly stamped regions and then imaged using 10× phase contrast, before being fixed and stained with YAP. The monolayer was fixed with 4% PFA in 1X PBS for 15 min, then incubated with blocking buffer—that is, 5% horse serum (GIBCO) and 0.3% Triton X-100 in 1X PBS—for 1 h. To stain for YAP localization, the sample was first incubated with a YAP antibody (1:500) gifted by Marius Sudol (MBI, Singapore) for 1 h then incubated with a secondary antibody (1:1,000, goat anti-rabbit IgG (H+ L), Alexa Fluor 568, Invitrogen) for 30 min, both diluted in the blocking buffer. Between each step, the samples were rinsed 3 times with 1X PBS.Image analysis. Cell extrusion and apoptosis determination. The typical geometry of an extruded cell as a protrusion out of the relatively flat epithelial monolayer

allowed simple and clear detection of the extrusion under phase contrast or bright field imaging (the extrusion appears as a bright spot). In fluorescence microscopy, extrusion is marked by the disappearance of the nucleus. The first frame where the extruding cell started to adopt a distinct morphology from its neighbour cells was defined as the time of initiation of extrusion. All extruding cells checked for viability displayed clear caspase-3 activation simultaneously with the initiation of extrusion, showing that the extrusions were apoptotic extrusions (Fig. 1c).Automated nematic characterization of experimental images. To obtain the nematic director field in an epithelium, a clear epithelial monolayer image (which can be phase contrast or fluorescence image) was obtained, with individual cell boundaries visible (Extended Data Fig. 6a, Step 1). The image was smoothed using Bandpass Filter in ImageJ to remove unnecessary details (Extended Data Fig. 6a, Step 2). The filter size of small structures was set to roughly one-third the size of a single cell. The ImageJ plugin OrientationJ was used to detect the direction of the largest eigenvector of the structure tensor of the image38 for each pixel (for a window size of roughly one-quarter the size of single cell), output as grey values from − 90° to + 90° (Extended Data Fig. 6a, Step 3). The local nematic order parameter tensor, Q, (ref. 12) was calculated (the nematic window size averages over pixel directions in a fixed-size region that contained 3–5 cells) for each point on a grid that discretized the image, using an in-house Matlab code. The grid distance was 75% of the nematic window size. Only pixels that resided in the region of the cell body were taken into account for this calculation (white regions obtained by Auto Local Threshold function in ImageJ, Extended Data Fig. 6a, Step 4) as cell boundary regions could have orientations that are perpendicular to the cell body. The largest eigenvector of Q was taken to be the local orientation of 3–5 cells, and plotted (red lines) over the original image for inspection (Extended Data Fig. 6a, Step 5). Once the nematic director field was established, automatic nematic defect detection was done based on calculation of winding number39. Only two types of defects (+ 1/2 and − 1/2) were predominantly found. To reduce false positives, only stably detected defects (that is, defects found in at least two consecutive frames at the same location) were considered for further analysis. See below for detailed discussion regarding the robustness and optimality of this method, with respect to small variations to the chosen algorithm parameters. To determine a measure for the global ordering of the epithelium, the order parameter, S (ref. 12) (Extended Data Fig. 1c), for all local regions (which can incorporate ~ 16 nematic directors each) throughout the epithelium was averaged.Cell aspect ratio, area and density calculation. The outlines of cells in typical phase contrast images were tracked in MATLAB using Fogbank software40. The cell outlines in several images were manually traced to verify the results. These outlines were fitted with ellipses using ImageJ, and the aspect ratio taken as the ratio of the long axis to the short axis. The area of these outlines and the number of cells were also calculated.Particle Image Velocimetry (PIV). The velocity field in 10× , phase contrast epithelium time-lapse images was obtained using an open source MATLAB code (PIVlab41), with three passes (64 × 64, 32 × 32 and 16 × 16 pixel size interrogation window with 50% overlap each). The interrogation window sizes were scaled accordingly for higher magnification images.YAP spatial localization in the cells at +1/2 and −1/2 defects. Using an in-house ImageJ script, the nucleus-to-cytoplasmic YAP intensity ratio (NCYap-ratio) of cells in the monolayer was measured, where cells were classified as having YAP in the nucleus (NCYap-ratio > Min_threshold), in the cytoplasm (NCYap-ratio < Max_threshold) or uniformly distributed across the whole cell (other cells not falling in the other two categories). Parameter values were varied with a com-bination of pairs of Min_threshold ∈ (1.02, 1.05, 1.1, 1.15, 1.2) and Max_threshold ∈ (0.95, 0.96, 0.97, 0.98, 0.99) and were all considered for the statistics. The cor-responding phase contrast movie taken earlier for each circular shaped, fixed monolayer was used for identification of the defects. Cells falling in a circular region of radius r ≈ 10 μ m centred at the head region of + 1/2 defects, ~ 22–25 μ m away from the + 1/2 defect cores, were considered as cells at + 1/2 defects, while cells falling in an ROI (r ≈ 10 μ m) centred at the − 1/2 defect cores were consid-ered as cells at − 1/2 defects. For comparison, cells at randomly generated points were considered. For each pair of Min_threshold and Max_threshold values, the percentage of cells having YAP in the nucleus, in the cytoplasm and uniformly distributed were calculated.Calculation of physical quantities. Determination of high local cell density spots. Local cell density was calculated as the number of cells divided by area in a square area of side length ~ 50 μ m, centred on points on a grid that discretized the entire monolayer. High local cell density spots were defined as the grid points with the highest density value in each time frame.Extrusion correlation with defects and temporal evolution. If extrusions correlated with defects spatially, there would be a high probability of finding a defect close to an extrusion. To quantify this, the distance, re, between each extrusion and


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its closest defect in the preceding frame was determined. The number of closest defects (+ 1/2 or − 1/2 done separately) within a given distance range from their corresponding extrusions was converted into the areal probability of the defects at the specific distance, and the areal probabilities were normalized such that they sum to 1 (to facilitate comparison between different experimental conditions). A similar closest defect areal probability as a function of distance was also obtained for random points for comparison. For each extrusion, a random point was gen-erated in the same frame as the extrusion, but at a random location (uniformly spaced in the monolayer). For the closest defect probability curve with random points, n = 30 (different sets of random points). A defect probability curve which shows a higher spatial correlation between extrusion and defect has larger positive values at re close to 0 μ m. The ratio of the defect probability at re = 10 μ m (closest region to extrusions, between re = 0–20 μ m) to re = 120 μ m (furthest point, between re = 110–130 μ m) is calculated as a measure of the strength of the extrusion–defect correlation. For example, since this ratio for the + 1/2 MDCK WT case is ~ 7 (Extended Data Fig. 1h), this tells us that there is an approximately sevenfold higher probability to find the closest defect for an extrusion situated at ~ 10 μ m compared to ~ 120 μ m from it. To quantify when the corresponding defects start to spatially approach the extrusions, this defect probability ratio is calculated as a function of time (t < 0 min) leading to extrusion at t = 0 min, comparing at each time frame between the defects in that frame and the eventual extrusion spots at t = 0 min.Strain rate and stress measurements. Strain rate was calculated using the formula Eij = (∂iuj + ∂jui)/2, with i, j ∈ (x, y) and velocity field, ui, obtained from PIV meas-urements. Stress was estimated from traction force data by inverting the force bal-ance equation24. This underdetermined problem was solved by Bayesian inversion from the BISM method, independently of the epithelial rheology (details below). The stress estimate was defined as the mode of the posterior stress probability dis-tribution function (maximal a posteriori estimate, see below for details). Isotropic stress was taken as half the trace of the stress tensor, (σxx + σyy)/2 in the tissue. Average strain rate and stress maps for the defects were calculated by rotating and aligning the defects to the y axis at the origin, and averaging the values of the rotated vectors and matrices (in their new basis) for the corresponding pixel loca-tions in all defects. Smoothing was done by linear interpolation of maps.+1/2 defect stress magnitude as indicator of the ability of defect to induce extrusion. For each + 1/2 defect, the distance to the closest extrusion within the next 40 min after the time of the defect was determined. These distances were pooled into groups based on the values of the compressive isotropic stress at the head regions (area of 60 μ m × 60 μ m) of their corresponding defect. The number of extrusions within a given distance range was converted into the areal density of the defects at the specific distance, and the areal densities were normalized such that they sum to 1. The peak in each areal density curve increases near r = 0 μ m, as the compressive stress categorizing the group increases. This shows that as the defect compressive stress magnitude increases, the probability of the defect inducing an extrusion increases.Simulations, models and robustness studies of techniques. Robustness of the nematic director and defect detection method. To verify the effectiveness of the winding angle approach in detecting half-integer topological defects, a different algorithm based on diffusive topological charge16,42 to detect the defects was used and compared to the original approach. The results were found to be identical in both cases. Extended Data Fig. 6b shows an example of topological defects detected using the winding angle approach and the diffusive charge approach.

In addition, errors could occur in the identification of the nematic director field from the pattern of cells in the images, rather than in identifying the defects from the director field. As a check, we have performed a systematic variation of the relevant parameters in the orientation analysis and confirmed the robustness of our approach against small variations in the parameter values. Extended Data Fig. 6c shows the effect of varying the coarse-grained window size (for each nematic director, we average over the orientation of the cells within the corresponding window) on the number of topological defects detected. To reduce the number of false defects detected: (1) each nematic director represents the average orientation of 3–5 cells within the window (window size of ~ 50 μ m). This reduces the pos-sible errors due to fluctuations in the orientation of each cell. (2) Only the stably detected defects were kept (defects that appear in at least 2 consecutive frames). (3) The final window size was chosen as ~ 51 μ m (80 pixels) (after checking a range of 60–110 pixels). The minimum window size checked was ~ 38 μ m (60 pixels), corresponding to 2–4 cells, which clearly showed detection of ‘noise defects’. The maximum window size checked was ~ 71 μ m (110 pixels), which was too big so that the orientation field was too smooth and certain clear defects were clearly missed. 51 μ m was chosen as a compromise between these values (Extended Data Fig. 6c).Active nematic model for epithelial monolayer. Individual cells in a confluent epi-thelial monolayer have an in-plane anisotropic shape, which can be approximated by an ellipse, and are constantly moving in the epithelium, corresponding to a non-zero velocity field. Cell alignments show local orientational order of the cells.

The orientational order is destroyed at singular points called topological defects. The strength of a defect is determined by the change in the orientations of cells in a closed curve around the defect core. Integer (± 1) and half-integer (± 1/2) top-ological defects correspond to ± 2π and ± π rotation of the cell alignment along a closed curve around a singular point12. The strength of topological defects in cell alignment is an important determinant of the type of orientational order of cells, with ± 1 specific to polar materials and ± 1/2 specific to nematic materials. Therefore, the emergence of ± 1/2 defects in the experiments indicates that despite their individual polarity, MDCK cells behave as nematic materials (the orientation vector is ‘head-tail’ symmetric) at the level of the epithelium.

In order to capture the dynamics of cell motion and orientation, we use a continuum model of nematohydrodynamics for the monolayer43,44 to account for the combined effects of cell velocity and orientation. This nematohydrodynamic approach has proven successful in describing active systems19,45–48. The orienta-tional order of cells is characterized by the nematic tensor Q = 3S(nn − I/3)/2, valid for a 3D or quasi-2D simulation used here, where n is the cell orientation, S is the magnitude of the order and I is the identity matrix. The nematic tensor is evolved according to the Beris–Edwards equation12:

Γ∂ + ∂ − =u Q S H( ) (1)t k k ij ij ij

where (∂ t + uk∂ k) is the total derivative with u denoting the velocity field and

λ Ω δ δ λ Ω λ

δ

= + + / + + / − −

+ / ∂

S E Q Q E

Q Q u

( )( 3) ( 3)( ) 2

( 3)( )(2)

ij ik ik kj kj ik ik kj kj

ij ij kl k l

is the co-rotation term accounting for the response of cell orientation to the veloc-ity gradients, where δij is the identity matrix or the Kronecker delta. Here, velocity gradients are characterized by the strain rate tensor Eij = (∂ iuj + ∂ jui)/2 and the vorticity tensor, Ωij = (∂ iuj − ∂ jui)/2, corresponding to extensional and rotational flows, respectively. The relative strength of extensional and rotational flows is deter-mined by the alignment parameter λ. Therefore, the alignment parameter accounts for the different responses of particles of different shapes to the symmetric and asymmetric parts of the velocity gradient tensor49. Mapping the alignment parameter to the Leslie–Ericksen equation for liquid crystal dynamics gives λ= β

β+ −

+S

S3 4

911

2

2 (refs 12, 50), where β = a/b is the ratio of the length of the cell along its axis of symmetry, a, to its length perpendicular to this axis, b. Therefore, for prolate ellipsoids β > 1, while for oblate ellipsoids β < 1 and for spherical particles β = 1, which correspond to λ > 0, λ < 0, and λ = 0, respectively. The experiments show that MDCK cells exhibit an in-plane anisotropic shape in the form of prolate ellipses and therefore are characterized by λ > 0. The molecular field in equation (1),

δδ

δ δδ

=− +

H F

QF

Q3Tr (3)ij

ij

ij

kl

describes the relaxation of the orientational order to the minimum of the free energy, F = Fb + Fel. The bulk free energy Fb is calculated from the Landau– De Gennes expansion,

= + +FA Q Q B Q Q Q C Q Q( )

2( )

3( )

4(4)ij ji ij jk ki ij ji

bQ Q Q

2

where AQ, BQ and CQ are material constants. In addition, the free energy corre-sponding to spatial inhomogeneities in the orientation field is described by the Oseen–Frank expansion using a single elastic constant approximation12,

=∂

FK Q( )

2(5)k ij

el

2

where K is the elastic constant. There are in general two elastic constants, cor-responding to bend, Kb, and splay, Ks, in 2D systems. However, setting different values of Kb and Ks gives only small qualitative changes in the flow fields around topological defects in the simulations, as these are predominantly controlled by active stresses (introduced below).

The velocity field in the monolayer is evolved according to the incompressible Navier–Stokes equation:

∂ =u 0 (6)i i

ρ σ∂ + ∂ =∂u u( ) (7)t k k i j ij

which reduces to the force balance equation ∂ jσij = 0 in the low Reynolds number limit relevant to monolayer mechanics. The assumption of an incompressible monolayer in equation (6) is supported by experimental measurements of the


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divergence of the flow field in the epithelium. Although divergence hotspots (larger values) do arise at singular points in the epithelium, the temporal average of this divergence field shows only small deviations from zero (< 4% every 10 min, Extended Data Fig. 7a, b).

Equation (7) describes how the rate of change of linear momentum is driven by stress gradients in the monolayer. The total stress, σ, consists of four contributions: isotropic pressure, − Pδij, viscous stress, σ η= E2ij ij

viscous , nematic elastic stress, σijel,

and active stress, σijactive. The elastic stress

σ λ δ λ δ λ δ

δδ

= + / − + / − + /

−∂

∂

+ −

Q Q H H Q Q H

Q FQ

Q H H Q

2 ( 3)( ) ( 3) ( 3)

(8)ij ij ij kl lk ik kj kj ik ik kj

i klj lk

ik kj ik kj

el

corresponds to distortions in cell alignments. Finally, the active stress

σ ζ=− Q (9)ij ijactive

accounts for the local stresses generated by active processes in the cells, including actomyosin polymerization and cell contractility22,47,51–53. The activity coefficient, ζ, determines the strength of the activity, with positive and negative values for extensile and contractile stresses. The extensile stress corresponds to the flow generated by the cell activity outwards along the elongated axis of the cell, while contractile stress characterizes the flow outwards along its shorter axis. The meas-urements of strain rates around defects in the experiments and comparison with the simulations show that the cells produce extensile stresses in the epithelium and that the epithelium behaves as an extensile active nematic.

Within this framework, the isotropic contribution to the total stress reduces to just the pressure and an isotropic contribution from the nematic elastic stress, as the trace of σij

active is zero because Qij is traceless, and the trace of σijviscous is also zero

due to the incompressibility condition, equation (6). Thus, the specific spatial patterns of the isotropic part of the total stress in the simulations are determined only by the elastic stress.

The hybrid Lattice Boltzmann algorithm is used to solve the equations of motion (equations (1), (6) and (7)). Details of the algorithm can be found elsewhere45,54,55. The simulation parameters used are Γ = 0.34, AQ = 0.0, BQ = − 0.3, CQ = 0.3, K = 0.08, ζ = 0.006, λ = 0.7 and η = 2/3, in lattice units, unless otherwise is stated.Bayesian Inversion Stress Microscopy (BISM). According to Newton’s laws, the stress in a cell monolayer is balanced by the traction forces the monolayer exerts on its substrate everywhere in space and time. In 2D and neglecting inertia, the force balance reads

σ∇⋅ = t (10)

where σ is the stress tensor field (units Pa μ m in 2D) and t is the 2D in-plane cell–substrate traction force field. In Cartesian component form, the equations

σ σ∂∂+∂∂=

x yt (11a)xx xyx

σ σ∂∂+∂∂=

x yt (11b)yx yy

y

can be discretized on a square grid of spatial resolution L and rewritten in a matrix format,

σ=A T (12)where the vectors σ and T respectively consist of all stress and traction components over the whole of space (see ref. 24 for the exact forms of A, σ and T). Since t is readily measured by Traction Force Microscopy (TFM), the stress components could be determined by inverting the matrix A. However, the equation is under-determined as σ has three components (σxx, σyy, σxy) at each space position, while t has only two components (tx and ty), and thus cannot be solved for σ even if T is fully known. Bayesian inference provides a statistical framework for the integration of current data, prior knowledge and reasonable assumptions about the system to extract information (in the form of a probabilistic distribution) from such ill- conditioned situations56,57. Two distributions, that is, the likelihood function, L(T| σ), and prior density, π(σ), need to be constructed to calculate the desired out-put distribution that describes σ. L(T| σ) accounts for the physical relation between measured data, T and σ through equation (10) up to an additive noise. Assuming the noise has a Gaussian profile with zero mean, the likelihood can be written as

σσ

| ∝

−| − |

T TL A

s( ) exp

2(13)2

2

where s2, the variance of the noise, allows the definition of a diagonal covariance matrix S = s2I, with I the identity matrix. π(σ) accounts for any additional con-straints and assumptions on the stress components. Specifically, the off-diagonal components of the stress are enforced to be equal, that is, σxy = σyx (with an extra hyperparameter, α) and stress is also assumed to be Gaussian with zero mean and covariance Is0

2 . The prior is written as

σσ σ σ

σ σπα

∝

−| | + | − |

= − −S

s( ) exp

2exp[ ] (14)xy yx

2 2

02

2t

01

with =S Bs0 02 the covariance matrix of the prior, embedding the two quadratic

terms in B.Bayes’ theorem links the posterior distribution Π(σ| T) on σ (conditional

probability of σ given data and assumptions), with L(T| σ) and π(σ) by a product rule

σ σΠ π σ| ∝ | ×T TL( ) ( ) ( ) (15)

Since the likelihood and prior are both Gaussian, σΠ |T( ) is also Gaussian with covariance matrix SΠ and mean σΠ written as,

= +Π− − −S S A S A( ) (16)0

1 t 1 1

and

σ =Π Π−S A S T (17)t 1

where At is the transpose of A. σΠ is the inference of monolayer stress that we need. Recall that whereas actual stress, σ, cannot be obtained (because the matrix A itself cannot be inverted), σΠ can be readily calculated since the inversion of +− −S A S A( )0

1 t 1 is possible. The Maximum A-Posteriori (MAP) solution to the inference problem is the posterior mode, identical to the mean for a Gaussian: σσ= Πˆ (see Extended Data Fig. 8a for a schematic of the BISM algorithm). The

solution depends on the dimensionless parameter Λ= ,I ss

2 2

02 which is found using

the L-curve method58. Note that error bars on the inferred stress are provided by the posterior covariance if needed.

In this Letter, the hyperparameter α is fixed to a value (103) large enough to enforce the equality of off-diagonal components of the stress tensor. With a spatial resolution I = 5.12 μ m, the L-curve method yields a value of Λ = 10−6 that we use for all stress inferences.Robustness study. Validation of BISM with unknown boundary conditions. BISM has been validated primarily for confined systems where the boundary conditions are known (free boundary conditions σ·n = 0, where n is the vector normal to the boundary)24, but not yet in the case of unknown boundary conditions (applicable for our current study, as explained below). For the former case, the boundary conditions are incorporated in the prior, with a supplementary term (compare with equation (14)):

σσ σ σ σ

σ σπα β

∝

−| | + | − | + | |

= − −S

s( ) exp

2exp[ ] (18)xy yx

2 2 2 2BC

2

02

tBC

1

with a modified covariance matrix =S BsBC 02

BC, where the three quadratic terms are embedded in BBC. The inferred stress is then obtained from equations (16) and (17) using SBC instead of S0. BISM has been validated using numerical simulations that yield stress and traction force data sets, as well as using experimental data in a quasi-1D geometry24. In particular, we showed that the absolute value of the stress could be inferred in confined systems. Further, the robustness of the method for different likelihood and prior distribution functions has also been tested. In par-ticular, we checked that a non-Gaussian prior, a non-Gaussian likelihood, or a smoothness prior all lead to the same level of accuracy as the Gaussian model presented here.

Since the focus in this Letter is on extrusions and defects occurring in the bulk of the monolayer, we needed to perform stress inference away from physical boundaries, without knowledge of the correct boundary conditions applying to the epithelium patch that surrounds the extrusion. We now present an additional numerical validation for this case. Briefly, we used BISM on a larger and confined area using free boundary conditions in the prior (equation (18)), and compared the stress thus inferred in a smaller, central region to that obtained without imposing the boundary conditions in the prior (equation (14)), see Extended Data Fig. 8b–j. Good agreement between the two inferences showed that, except in the vicinity of the boundaries, BISM also infers the absolute value of the stress in the case where the boundary conditions are unknown.


LETTERRESEARCH

Following previous work, we used a simplified, viscous rheology, neglecting orientational degrees of freedom, with the constitutive equation

σ η η∇⋅ ∇⋅ ∇⋅= + + ′u u u I( ( ) ) ( ) (19)t

and drove the fluid layer with active moving force dipoles. Including a dissipative interaction with the substrate through an effective fluid friction force, the traction force reads:

ξ= −t u f (20)u act

The active force fact derives from the sum of nd force dipoles, =∑ ∂=x xf p( ) ( )i nn

j ijn

act, 1d ,

with a density of 10−2 dipoles per μ m2, such that for each dipole n, xp ( )ijn is

defined as:

δ δ= − −x x xp p p D( ) ( ) (21)ij

n nij

nij ntr dev

with deviatoric angular matrices θ θθ θ

=

−

D cos 2 sin 2

sin 2 cos 2n n

n n, and amplitudes

= μp 10 kPa mntr

0 and = μp 10 kPa mndev

1 for the trace and deviator respectively (numerical values taken from experiments59). The dipole positions xn and orientations θn are random variables uniformly distributed over the spatial domain and [0, 2π ] respectively. The delta functions are implemented as finite-size Gaussian distributions, with a spatial extension 2d = 10 μ m of the order of a typical cell size. We chose material parameter values typical for cell monolayers60–62: friction coefficient ξu = 100 kPa μ m−1 s, shear viscosity η = 103 kPa μ m s and compression viscosity η′ = η.

Using the finite element software FreeFem+ + 63, we solved the equations (equa-tions (10), (19), (20)) for u on a 300 × 300 μ m2 square with boundary conditions σ·n = 0. The stress σnum (Extended Data Fig. 8e, h) and traction force fields are derived from the velocity field and sampled over a regular Cartesian grid of spatial resolution I = 2 μ m, giving N = 50 points for 100 μ m. To account for the measure-ment error, we add a white noise to the traction force field with an amplitude sexp equal to 5% of the maximum traction force amplitude. We apply our algorithm to these numerical traction forces in two cases. (1) Whole monolayer of area 300 × 300 μ m2 with zero stress boundary conditions and prior (equation (18)). We obtain the stress σwhole with 3N × 3N values for each component (Extended Data Fig. 8f, i). (2) Central part of the monolayer of area 100 × 100 μ m2: without boundary conditions and prior (equation (14)). We obtain the stress σcentral with N × N values for each component (Extended Data Fig. 8g, j).

We compare the outcomes of two inferences with the exact stress field σnum, and find excellent agreement between σnum and both σwhole (blue dots) and σcentral (red dots, Extended Data Fig. 8b–d). Note that the inference is less accurate close to the border of the central part (black circles), yet allows us to obtain the correct absolute values of the stress in the bulk of the central domain with an accuracy equivalent to that of the whole system inversion.Statistical analysis. No statistical methods were used to predetermine sample size. The sample size was chosen to see a statistical difference between data sets. In the case where no differences were observed (Fig. 4h), the sample size chosen was at least as big as those where differences were observed. Blinding was achieved when comparing results between experimental analysis, active nematic numerical simulations and Bayesian stress inference as these are done independently in three different institutes. All experimental data were tested with the Anderson–Darling test to check for normality of the distribution. In the case when we were comparing data which were normally distributed and have similar variance, the two-sided t-test was used (Figs. 2b, 3b, g, and Extended Data Figs 2d, 3e, 4a). Otherwise, the non-parametric Kolmogorov–Smirnov test (ks-test) was used. Throughout, * P < 0.05, * * P < 0.01, * * * P < 0.001, apart from data in Fig. 3b and Extended Data Fig. 3e, where * P < 0.0001. All relevant statistics are reported in the corresponding legends.

35. Vedula, S. R. K. et al. Microfabricated environments to study collective cell behaviors. Methods Cell Biol. 120, 235–252 (2014).

36. Brugués, A. et al. Forces driving epithelial wound healing. Nat. Phys. 10, 683–690 (2014).

37. Martiel, J.-L. et al. Measurement of cell traction forces with ImageJ. Methods Cell Biol. 125, 269–287 (2015).

38. Rezakhaniha, R. et al. Experimental investigation of collagen waviness and orientation in the arterial adventitia using confocal laser scanning microscopy. Biomech. Model. Mechanobiol. 11, 461–473 (2012).

39. Huterer, D. & Vachaspati, T. Distribution of singularities in the cosmic microwave background polarization. Phys. Rev. D 72, 043004 (2005).

40. Chalfoun, J. et al. FogBank: a single cell segmentation across multiple cell lines and image modalities. BMC Bioinformatics 15, 431 (2014).

41. Thielicke, W. & Stamhuis, E. J. PIVlab–Towards user-friendly, affordable and accurate digital particle image velocimetry in MATLAB. J. Open Res. Softw. 2, e30 (2014).

42. Blow, M. L., Thampi, S. P. & Yeomans, J. M. Biphasic, lyotropic, active nematics. Phys. Rev. Lett. 113, 248303 (2014).

43. Giomi, L., Mahadevan, L., Chakraborty, B. & Hagan, M. Excitable patterns in active nematics. Phys. Rev. Lett. 106, 218101 (2011).

44. Thampi, S. P., Golestanian, R. & Yeomans, J. M. Instabilities and topological defects in active nematics. Europhys. Lett. 105, 18001 (2014).

45. Marenduzzo, D., Orlandini, E., Cates, M. E. & Yeomans, J. M. Steady-state hydrodynamic instabilities of active liquid crystals: hybrid lattice Boltzmann simulations. Phys. Rev. E 76, 031921 (2007).

46. Thampi, S. P., Golestanian, R. & Yeomans, J. M. Vorticity, defects and correlations in active turbulence. Phil. Trans. R. Soc. A 372, 20130366 (2014).

47. Bittig, T., Wartlick, O., Kicheva, A., González-Gaitán, M. & Jülicher, F. Dynamics of anisotropic tissue growth. New J. Phys. 10, 063001 (2008).

48. Volfson, D., Cookson, S., Hasty, J. & Tsimring, L. S. Biomechanical ordering of dense cell populations. Proc. Natl Acad. Sci. USA 105, 15346–15351 (2008).

49. Larson, R. G. The Structure and Rheology of Complex Fluids Vol. 33 (Oxford Univ. Press, 1999).

50. Edwards, S. A. & Yeomans, J. M. Spontaneous flow states in active nematics: a unified picture. Europhys. Lett. 85, 18008 (2009).

51. Bittig, T., Wartlick, O., González-Gaitán, M. & Jülicher, F. Quantification of growth asymmetries in developing epithelia. Eur. Phys. J. E 30, 93–99 (2009).

52. Basan, M., Joanny, J.-F., Prost, J. & Risler, T. Undulation instability of epithelial tissues. Phys. Rev. Lett. 106, 158101 (2011).

53. Delarue, M. et al. Mechanical control of cell flow in multicellular spheroids. Phys. Rev. Lett. 110, 138103 (2013).

54. Denniston, C., Marenduzzo, D., Orlandini, E. & Yeomans, J. M. Lattice Boltzmann algorithm for three-dimensional liquid–crystal hydrodynamics. Phil. Trans. R. Soc. A 362, 1745–1754 (2004).

55. Fielding, S., Marenduzzo, D. & Cates, M. E. Nonlinear dynamics and rheology of active fluids: simulations in two dimensions. Phys. Rev. E 83, 041910 (2011).

56. von Toussaint, U. Bayesian inference in physics. Rev. Mod. Phys. 83, 943–999 (2011).

57. Kaipio, J. & Somersalo, E. Statistical and Computational Inverse Problems Vol. 160 (Springer Science & Business Media, 2006).

58. Hansen, P. C. Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev. 34, 561–580 (1992).

59. du Roure, O. et al. Force mapping in epithelial cell migration. Proc. Natl Acad. Sci. USA 102, 2390–2395 (2005); correction 102, 14122 (2005).

60. Cochet-Escartin, O., Ranft, J., Silberzan, P. & Marcq, P. Border forces and friction control epithelial closure dynamics. Biophys. J. 106, 65–73 (2014).

61. Harris, A. R. et al. Characterizing the mechanics of cultured cell monolayers. Proc. Natl Acad. Sci. USA 109, 16449–16454 (2012).

62. Jenkins, A. D. & Dysthe, K. B. The effective film viscosity coefficients of a thin floating fluid layer. J. Fluid Mech. 344, 335–337 (1997).

63. Hecht, F. New development in FreeFem+ + . J. Numer. Math. 20, 251–266 (2012).

Data and code availability. Source data and codes for cell orientation detection, active nematic simulations and Bayesian inference of tissue stress are available upon request.


LETTER RESEARCH

Extended Data Figure 1 | Further characterization of cell, monolayer and extrusion–defect correlation properties. a, b, Time evolution of cell aspect ratio (a) and cell area (b) in a confluent MDCK epithelium. Data for each time point are binned over a duration of 120 min. From lowest to highest time points, n = 5,101, 5,537, 5,772, 6,549, 6,572, 6,876, 6,593 and 6,831 cells. c, Time evolution of nematic measure (averaged local order parameter, S) of corresponding epithelium (Methods), n = 294 data points for each bar. d, MDCK monolayer in circular confinement (left). Red lines (represented again as black lines in middle) show local cell orientation, colour coded at right. Scale bar, 100 μ m. e, Number of closest + 1/2 defects per unit area as function of re normalized such that area underneath the density curve sums to 1. n = 30 different random point sets. See Methods. f, Left, diagram of determination of correlation between − 1/2 defects and extrusions: distance, re, of each extrusion to

its closest − 1/2 defect in preceding frame is measured, and the number of these defects per unit area as function of re is normalized (right). See Methods. n = 50 (MDCK WT) extrusions from 4 independent movies in 3 independent experiments; n = 61 (MDCK, mytomycin-c) extrusions from 3 independent movies in 2 independent experiments; n = 85 (MCF10A) extrusions in 2 independent movies; n = 79 (HaCaT) extrusions in 2 independent movies. g, Similar to e, but is the normalized density curve between random points and − 1/2 defects. h, A measure of correlation strength between extrusions and defects: ratio of density curve values at re = 10 μ m to that at re = 120 μ m (first and last points in respective curves, Fig. 1f and Extended Data Fig. 1f). i, Measure of correlation strength for eventual extrusion points and + 1/2 defects as function of + 1/2 defect distributions at each time point (WT-MDCK). All data represented as mean ± s.e.m.

b

d

−90°

−45°

0°

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0.50.55

0.60.65

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vera

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spec

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e f

g h i

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MDCK−MytoC

MCF10A

HaCaT0

2

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y r e=

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MD

CK

-WT,

Rat

io o

f +1/

2 d

efec

t pro

babi

lity

r e=10

µm

to r e=

120

µm

120 360 600 840Time (min)

120 360 600 840Time (min)

Extrusion

Distance with closest

-1/2 defect in preceding

frame

re

0 20 40 60 80 100 1200

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MDCK−WTMDCK−MytoCMCF10AHaCaT

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LETTERRESEARCH

Extended Data Figure 2 | Further examination of active nematic properties and the correlation of extrusion with defects in the epithelium under different conditions. a, Velocity field around + 1/2 defect in contractile, active nematic liquid crystal simulation. Director configuration around the defect is same as Fig. 2a b, Average velocity field around + 1/2 defect for mytomycin C and blebbistatin treated MDCK. n = 2,003 (mytomycin C) defects from 3 independent movies in 2 independent experiments; n = 3,061 (blebbistatin) defects from 3 independent movies. + 1/2 defect has same orientation and position as in Fig. 2a. c, Total defect areal density evolution in simulation, activity parameter decreased at simulation time t = 0, then increased at t = 5. d, Average total defect density for WT MDCK (‘Non-treated’) and blebbistatin (10 μ M and 50 μ M) treated MDCK. n = 314 frames from

4 independent movies in 3 independent experiments (WT); n = 155 frames from 3 independent movies (blebbistatin 10 μ M); n = 26 frames from 4 independent movies (blebbistatin, 50 μ M). t-test, * * * P < 0.001. e, Left, number of closest + 1/2 and − 1/2 defects per unit area as function of distance, re, measured from respective extrusions (Ext, + 1/2) and (Ext, − 1/2), in 10 μ M blebbistatin treated monolayer. Similar curves also plotted for + 1/2 and − 1/2 defects measured from random points (Random, + 1/2) and (Random, − 1/2). Each density curve is normalized such that area under each curve sums to 1. n = 78 extrusions in 3 independent movies; n = 30 different random point sets. Right, ratio of density curve values at re = 10 μ m to re = 120 μ m for the curves (Ext, + 1/2) and (Ext, − 1/2) on the left. All data represented as mean ± s.e.m.

−80 −40 0 40 80−80

−40

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y (µ

m)

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a

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m)

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LETTER RESEARCH

Extended Data Figure 3 | Further investigation of the relation between local cell density, compressive stress, defects and extrusions, and the role of caspase-3 activation. a, Number of closest high cell density spots per unit area as function of radius, r, measured from extrusion points in monolayer (Ext, High cell density). Similar curve plotted for high cell density spots measured from random points (Random, High cell density). n = 50 (MDCK WT) extrusions from 4 independent movies in 3 independent experiments; n = 30 different random point sets. b, Normalized number of closest defects per unit area as function of distance, re, against spots with top 5% of all compressive stresses in simulation domain. See Methods for calculation of simulation stress. c, Normalized number of closest extrusions per unit area as a function of distance against + 1/2 defects, grouped by magnitude of compressive stress at head regions of defects. Extrusions are within 40 min after the frame of defect. From lowest (least negative) to highest compressive isotropic stress (most negative), n = 331, 215, 180 and 72 defects in 2 independent experiments. All density curves in a, b and c are normalized such that area

under the curve sums to 1. d, Typical example of extrusion event, showing configuration of + 1/2 defect before and after extrusion. The same neighbour cell before (t < 0 min) and after (t > 0 min) extrusion is outlined with same colour. e, Left, same data set as Fig. 3b, showing average isotropic stress evolution around cells experiencing negative (compressive) stress values during extrusion initiation at t = 0 min, before and after extrusion (n = 32 extrusions in 2 independent experiments). A t-test was performed for each time point against a normal distribution centred at zero, * P < 0.0001. Right, scatter plot of same data. f, Typical example of extrusion event with caspase-3 activation (yellow arrowhead) (t = 0 min). Top panel, nematic directors (red lines) overlaid on monolayer. Bottom panel, corresponding caspase-3 signalling. g, Average extrusion rate in non-drug-treated (ND) and caspase-3 inhibited MDCK. n = 8 (ND) independent movies in 3 independent experiments; n = 7 (caspase-3 inhibited) independent movies in 2 independent experiments. ks-test, * * * P < 0.001. All data represented as mean ± s.e.m.


LETTERRESEARCH

WT α-catKD0

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Extended Data Figure 4 | Order parameter and extrusion–defect correlation for α-catKD MDCK, and simulations comparing small and large nematic bending elasticity. a, Average local order parameter, S, for WT and α -catKD MDCK epithelium. n = 3 (WT) independent movies in 2 independent experiments; n = 3 (α -catKD) independent movies. t-test, * * * P < 0.001. b, Left, number of closest + 1/2 and − 1/2 defects per unit area as a function of distance, re, measured from respective extrusions (Ext, + 1/2) and (Ext, − 1/2) in α -catKD MDCK. Similar curves plotted for + 1/2 and − 1/2 defects measured from random points (Random, + 1/2)

and (Random, − 1/2). Each density curve is normalized such that area under each curve sums to 1. n = 56 extrusions in 3 independent movies; n = 30 different random point sets. Right, ratio of density curve values at re = 10 μ m to re = 120 μ m, for the curves (Ext, + 1/2) and (Ext, − 1/2) on the left. c, d, Simulations comparing size of defects (c) and total defect areal density (d) for small and large nematic bending elasticity, K. K is 0.02 and 0.08 for c, and 0.04 and 0.08 for d. All data represented as mean ± s.e.m.


LETTER RESEARCH

Extended Data Figure 5 | Further analysis of topologically induced defects and extrusions. a, Maps of normalized number of + 1/2 (left column) and − 1/2 (right column) defects per unit area in star (top row) and circle (bottom row) epithelium confinements (Methods). n = 6,738 (+ 1/2) defects and n = 5,083 (− 1/2) defects from 12 independent movies in 2 independent experiments (star); n = 5,389 (+ 1/2) defects and n = 4,858 (− 1/2) defects from 8 independent movies in 3 independent experiments (circle). b, Top, number of closest + 1/2 and − 1/2 defects per unit area as function of distance, re, measured from respective extrusions (Ext, + 1/2) and (Ext, − 1/2) in star-shaped monolayer. Similar curves plotted for + 1/2 and − 1/2 defects measured from random points (Random, + 1/2) and (Random, − 1/2). Each density curve is normalized

such that area under each curve sums to 1. n = 145 extrusions from 12 independent movies in 2 independent experiments; n = 30 different random point sets. Bottom, ratio of density curve values at re = 10 μ m to re = 120 μ m, for the curves (Ext, + 1/2) and (Ext, − 1/2) on the top. c, Average velocity field around + 1/2 defect in caspase-3 inhibited monolayer. n = 2,993 defects from 7 independent movies in 2 independent experiments. Defect has same orientation and position as in Fig. 2a. d, Isotropic stress measured around cells just before extrusion (t = 0 min is time of extrusion). > 70% (< 30%) of cells experienced negative (positive respectively) stress, denoted as Group 1 (Group 2 respectively). n = 44 total number of extrusions in 2 independent movies. ks-test, * * * P < 0.001. All data represented as mean ± s.e.m.

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Extended Data Figure 6 | Automated nematic director detection and robustness study. a, 5 step process of automated nematic director detection. Step 1, phase contrast image of monolayer is obtained. Step 2, details of image are smoothed. Step 3, local orientation of cells are obtained using OrientationJ. Step 4, local contrast is applied to identify cell body regions. Step 5, nematic directors are obtained. b, Example of defect detection in a given nematic director field using the winding

number approach (left, red for + 1/2 defect, blue for − 1/2 defect) and the diffusive charge approach (right, yellow for + 1/2 defect, blue for − 1/2 defect). c, Number of stable defects detected as a function of window size. A bigger window size allows the orientation of more cells to be averaged for the direction of a nematic. Averaging is done over n = 50 frames of a monolayer movie, for each window size analysis. ks-test, * * P < 0.01, * * * P < 0.001. Data are represented as mean ± s.e.m.


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Extended Data Figure 7 | Epithelium can be modelled as a 2D incompressible material. a, Instantaneous (for every 10 min frame interval) velocity divergence field (colour coded) in circularly confined epithelium for one time point. b, Temporal average of velocity divergence field in circularly confined epithelium (averaged over about 20 h or 128 images consecutively).


LETTERRESEARCH

Extended Data Figure 8 | Bayesian inference stress method (BISM) and robustness study. a, Diagram of inference algorithm. b–d, Plots of inferred stress versus simulated stress for each component, in kPa μ m. Red line, bisector y = x; blue dots, 3N × 3N stress, σwhole for the whole system; red dots, N × N stress, σcentral for the central region; black circles, stresses obtained less than 2 μ m from the boundary of the central region,

N = 50 points. e–g, Pressure, and h–j, shear stress fields in kPa μ m for the whole system: from left to right are shown exact values, σnum; inferred values obtained for the whole monolayer, σwhole; and inferred values obtained for the central region, σcentral. Black dashed box represents the central region.


1

Topological defects in the nematic order of actin fibers as

organization centers of Hydra morphogenesis

Yonit Maroudas-Sacks1, Liora Garion1, Lital Shani-Zerbib1, Anton Livshits1, Erez Braun1,2,

Kinneret Keren * 1,2,3

1 Department of Physics, 2 Network Biology Research Laboratories and 3 The Russell Berrie

Nanotechnology Institute, Technion – Israel Institute of Technology, Haifa 32000, Israel

Animal morphogenesis arises from the interaction of multiple biochemical and mechanical

processes, spanning several orders of magnitude in space and time, from local dynamics at

the molecular level to global, organism-scale morphology. How these numerous processes

are coordinated and integrated across scales to form robust, functional outcomes remains

an outstanding question 1-4. Developing an effective, coarse-grained description of

morphogenesis can provide essential insight towards addressing this important challenge.

Here we show that the nematic order of the supra-cellular actin fibers in regenerating

Hydra 5, 6 defines a slowly-varying field, whose dynamics provide an effective description of

the morphogenesis process. The nematic orientation field necessarily contains defects

constrained by the topology of the regenerating tissue. These nematic topological defects

are long-lived, yet display rich dynamics that can be related to the major morphological

events during regeneration. In particular, we show that these defects act as organization

centers, with the main functional morphological features developing at defect sites. To our

knowledge, this provides the first demonstration of the significance of topological defects

for establishing the body plan of a living animal. Importantly, the early identification of

topological defect sites as precursors of morphological features, suggests that the nematic

orientation field can be considered a “mechanical morphogen” whose dynamics, in

conjugation with various biochemical signaling processes, result in the robust emergence of

functional patterns during morphogenesis.

Hydra is a classic model system for morphogenesis, thanks to its simple body plan and

remarkable regeneration capabilities. Historically, research on Hydra regeneration inspired the

development of many of the fundamental concepts on the biochemical basis of morphogenesis,

including the role of an “organizer” 7, the idea of pattern formation by reaction-diffusion

author/funder. All rights reserved. No reuse allowed without permission. The copyright holder for this preprint (which was not peer-reviewed) is the. https://doi.org/10.1101/2020.03.02.972539doi: bioRxiv preprint

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dynamics of morphogens 8, 9, and the concept of positional information 10. However, in these

studies, as well as in the majority of subsequent works, the role of mechanics in Hydra

morphogenesis was largely overlooked. Here we revisit this classic model system, focusing on

the cytoskeletal dynamics during the regeneration process from the point of view of nematic

physics, and reveal a previously unappreciated relation between cytoskeletal organization and the

morphogenesis process.

A mature Hydra has a simple uniaxial body plan, with a head on one end and a foot on the other.

Its tubular body consists of a double layer of epithelial cells, which contain highly organized

arrays of parallel supra-cellular actin fibers (Fig. S1). The fibers are globally aligned along the

body axis in the outer (ectoderm) layer, and perpendicular to the body axis in the inner

(endoderm) layer 5. The actin fibers are decorated with myosin motors, forming contractile

bundles called myonemes, which are akin to muscular structures in other organisms. The fibers

lie along the basal surfaces of each epithelial layer and are connected through cell-cell junctions

to form supra-cellular bundles, which exhibit long range directional order over scales

comparable to the size of the animal, ranging from ~300 μm in small regenerated Hydra to

several mm in fully grown Hydra. The myonemes are coupled via adhesion complexes to a thin

viscoelastic extracellular matrix layer (the mesoglea) sandwiched between the two cell layers

(Fig. S1) 11.

Large-scale supra-cellular arrays of contractile actin-myosin fibers are a common organizational

theme found in many animal tissues 12-14. While the mechanism for the development of these

parallel fiber arrays is not entirely clear, mechanical feedback has been shown to play a central

role in their formation and alignment 14, 15. Such parallel alignment is characteristic of systems

with nematic order, i.e. systems in which the microscopic constituents tend to locally align

parallel to each other exhibiting long-range orientation order 16. Nematic systems can be

described by a director field which denotes the local orientation, and its spatio-temporal

evolution reflects the dynamics of the system. We focus here on the nematic organization of the

supra-cellular actin fibers in the ectoderm of regenerating Hydra, which are the most prominent

cytoskeletal feature affecting Hydra mechanics 5, 6.

The director field describing the alignment of the ectodermal actin fibers in Hydra defines a

slowly-varying field that persists over extended spatial and temporal scales compared to


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molecular and cellular scales. In particular, unlike other tissue characteristics (e.g. cell shape

anisotropy, local curvature), the fiber orientation is maintained even under extensive tissue

deformations. As such, the fiber orientation field provides a valuable coarse-grained description

of the system. In that respect, the aligned fibers serve a dual role: an integrated probe reporting

on a host of underlying biochemical and mechanical processes, and the major force generator in

the system. Importantly, the contractile actin fibers simultaneously contribute and respond to the

stresses and deformations in the tissue, consuming energy and maintaining the system far from

thermal equilibrium. Such coupling between a nematic field and surface deformations in non-

equilibrium systems introduces rich physics, which are only beginning to be explored

experimentally 17, 18 and theoretically 19, 20. Note that the activity of the nematic actin-fiber

system arises from local force generation within the deformable tissue (as in e.g. muscles),

unlike other recently studied active nematic systems in which the activity is due to the

underlying self-driven constituents characterized by a velocity field that is coupled to the

nematic orientation field 21.

One of the key features of nematic systems is the possible existence of singularities in the

director field (locally disordered regions), which are called topological defects, and are classified

by the number of rotations of the nematic director around the singular point (called topological

charge; see Methods) 16. The total charge of defects in a closed system is constrained by topology

16. In particular, for a nematic on the surface of a closed hollow shell, which is the relevant case

for Hydra, the total charge has to equal +2. Recently, the framework of active nematics has been

found to be informative in understanding various phenomena in a variety of biophysical systems

21, 22 including in vitro cytoskeletal networks 17, 18, 23-25, cell monolayers 26-28 and bacterial

cultures 29. Interestingly, topological defects in the nematic order have been shown to induce

local changes in the dynamics of cell monolayers in vitro 26, 28, that were related to mechanical

feedback whereby alteration of the local stress field near defects triggered changes in cellular

behavior (e.g. cell death and extrusion in epithelial sheets 26).

Here we follow the nematic dynamics of the actin fibers during Hydra regeneration from excised

tissue pieces by live microscopy. The inherited fiber organization in excised tissues is only

partially maintained as the tissues fold into spheroids 6. During regeneration, the regions with

aligned fibers induce order in neighboring regions, yet nematic topological defects in the actin


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fiber alignment must appear due to the change in topology. We show that these topological

defects are long-lived, and display rich dynamics, including motility, as well as merging and

annihilation of defect pairs. Importantly, we find that the dynamics of defects during

regeneration are closely related to the formation of functional morphological features such as the

head and foot, with the fiber alignment ultimately determining the regenerated body axis. The

observed relation between defect dynamics and the regeneration process offers a complementary

biophysical viewpoint on morphogenesis, with the nematic orientation field serving as a

“mechanical morphogen” and sites of defects in this field acting as organization centers of the

developing body plan.

Nematic organization of the supra-cellular actin fibers in mature Hydra

Mature Hydra have a parallel array of actin fibers in the ectoderm which is aligned with their

body axis and can be described by a nematic director field (Fig. 1A,B). Since the fibers reside

on a closed 2D hollow shell, nematic defects are unavoidable. Importantly, the topological

defects in mature Hydra coincide with the morphological features of the animal. Specifically,

two +1 point defects are found at the apex of the head and foot regions; a +1 defect at the tip of

the mouth (hyptosome; Fig. 1C), and a +1 defect at the base of the foot, surrounding the basal

disc (Fig. 1D). The body axis of the animal is aligned between these two defects, which have a

total charge of +2, equal to the topologically required charge. Additionally, mature Hydra have a

variable number of tentacles that contain further topological defects; each tentacle has a +1

defect at its tip (Fig. 1E) that is accompanied by two -1/2 defects at opposite sides of its base

(Fig. 1F). The total defect charge of a tentacle is thus equal to 0, and hence does not contribute to

the net defect charge in the animal. Similarly, multi-axis Hydra (Fig. S2) or Hydra with an

emerging bud 5, contain an additional +1 defect at the extra head/foot, which is balanced by

two -1/2 defects on either side of the junction between the two body axes (Fig. S2C). In general,

+1 defects are localized in regions of high positive surface curvature, whereas -1/2 defects reside

in saddle regions with negative curvature. Interestingly, mature Hydra contain only +1 defects

(which are “aster”, rather than “vortex” shaped 22) and -1/2 defects, whereas regenerating Hydra

tissues exhibit +1/2 defects, in addition to +1 and -1/2 defects (Fig. 1G-I).


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Nematic organization of the supra-cellular actin fibers in regenerating Hydra tissues

How does the nematic configuration found in mature Hydra arise during regeneration from a

tissue segment? To address this question, we followed actin dynamics in regenerating tissue

fragments excised from transgenic Hydra expressing lifeact-GFP in the ectoderm (Fig. 2).

Initially, an excised tissue fragment is characterized by a perfect nematic array of actin fibers

with no defects (Fig. S3A). The excised tissue folds within a couple of hours, in an active

actomyosin-dependent process, to form a closed spheroid 6. The change in surface topology,

from a tissue fragment with open boundaries to a closed, hollow spheroid, necessarily implies

that the nematic order cannot be maintained throughout the system. Indeed, the folding process is

accompanied by a dramatic reorganization during which actin fibers near the center of the

fragment are stably maintained, whereas fibers near the closure region of the spheroid undergo

extensive remodeling and the nematic organization is essentially lost 6. We find that the ordered

nematic fiber array occupies roughly half of the spheroid’s surface area, while the remaining

surface is essentially devoid of aligned fibers (Fig. 2B, left).

The folded spheroid with partial nematic order regenerates into a mature Hydra within 2-3 days

(Fig. 2; Movie 1). This process involves extensive tissue deformations that are primarily driven

by internal forces, generated by the actomyosin cytoskeleton, as well as by osmotic forces that

cause changes in fluid volume in the internal gastric cavity, leading to cycles of osmotic swelling

and ruptures 30. During the first ~24 hours after excision, nematic arrays of actin fibers gradually

form in the previously disordered regions (Fig. S3), and the fraction of the spheroids’ surface

area that contains ordered fiber arrays increases to span nearly the entire surface (Fig. 2B,C).

The order appears to “invade” the disordered regions, as the orientation of newly formed fibers is

influenced by the alignment of neighboring regions (Fig. S3). As previously shown, this

induction of order confers a structural memory of the original body axis orientation 6.

Due to the topological constraint of the closed surface, fiber alignment cannot be continuous

everywhere and topological defects in the nematic order must emerge (Fig. 2A). As the extended

disordered regions progressively become ordered, the topological defects become spatially-

localized and develop into well-defined point defects with charges of +1/2, -1/2 or +1 (Figs. 1G-

I, 2B,D). The defect configuration that emerges depends on the geometry of the excised tissue,

and is generally different from the final defect configuration in regenerated Hydra. In particular,


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the common defect type in regenerating square tissue fragments in the first 24 hours after

excision is a +1/2 defect (Fig. 2D), which is entirely absent in mature Hydra (Fig. 1). Over time,

+1 defects become more prevalent and the +1/2 defects become less frequent, until eventually

they are completely eliminated (Fig. 2D). In contrast to square tissue fragments, tissue rings

excised from the tubular body of a mature Hydra can form a spheroid with two +1 defects from

the onset, as the top and bottom faces of the ring seal 6. Nevertheless, the different defect

configurations that initially emerge in regenerating fragments and rings evolve and typically

stabilize into the same stereotypic defect configuration of the mature Hydra (Fig. 1).

Defect dynamics and the emergence of morphological features in regenerating Hydra tissues

To understand how the defect configuration evolves, we next sought to follow actin fiber

dynamics in regenerating Hydra and track defect positions over time. This goal is challenging

due the extensive tissue movements and deformations that naturally occur during Hydra

regeneration. To tackle this problem, we developed a method to label specific tissue regions by

introducing a cytosolic photoactivatable dye into the ectodermal layer of regenerating tissues by

electroporation and using a confocal microscope to locally uncage the dye at well-defined

regions (Methods). Using this technique, we examined the movement of defects relative to

labeled regions throughout the regeneration process, allowing us to characterize the dynamics

displayed by defects of different types and relate them to the emergence of morphological

features and body plan formation (Figs. 3,4).

We find that +1/2 defects, which are prevalent early in regeneration from tissue fragments (Fig.

2D), are inherently unstable and mobile (Fig. 3). We observe +1/2 defects moving relative to the

underlying tissue, in the direction of their rounded end (Fig. 3A; Movie 2). Moreover, +1/2

defects that appear during the initial 24 hours after excision, are eliminated over time (Fig. 3B).

Their elimination occurs most commonly by merging of two +1/2 defects into a +1 defect (Fig.

3C), or alternatively by annihilation with a -1/2 defect (Fig. 3D; Movie 3). The motility of +1/2

defects, as observed in many other active nematic systems 17, 23-28 and their two accessible modes

of elimination, explain their inherent instability in the tissue and their absence from the mature

Hydra morphology.


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In contrast, we find that +1 defects are stable and can be related to the emergence of the head and

foot in regenerated animals (Fig. 4). The +1 defects originate either from newly-aligned fibers in

the initially disordered regions, or from merging of two previously-formed +1/2 defects (Fig.

3C). Typically, a +1 defect forms in the previously disordered regions within the first 24 hours,

sometimes as early as 3-4 hours after excision (Fig. 4D inset). These +1 defects remain

stationary relative to the underlying tissue and can be followed throughout the regeneration

process, at the end of which they will predominantly be located at the mouth of the regenerated

animal (Fig. 4A,D; Movie 4). Alternatively, +1 defects can arise through merging, when two

+1/2 defects aligned with their curved ends facing each other, move closer together as the tissue

elongates until coalescing, most typically to form the foot (Fig. 4B,E; Movie 5). The final

merging often occurs late in the regeneration process, after the formation of tentacles and the

establishment of the body axis. Together, these findings indicate that it is possible to discern the

sites where +1 defects will emerge, already at very early stages of the regeneration process.

These sites remain stationary with respect to the tissue and hence identify the region of cells that

will eventually form the functional morphological features of the regenerated animal (Movies 4,

5). Remarkably, the asymmetry between the head and foot formation process with respect to

their initial defect configuration (Fig. 4D,E), implies that the sites of emergence of the head and

the foot in regenerating fragments and the body axis polarity can be predicted long before the

physical appearance of these features, solely based on the distribution of actin fiber alignment

shortly after excision.

Although -1/2 defects are sometimes observed early in regenerating spheroids, they are relatively

rare (Fig. 2D), and can be eliminated through annihilation with +1/2 defects (Fig. 3D). Early -1/2

defects that survive can end up defining the junction of an additional body axis in multi-axis

animals (Fig. S2). Most commonly, however, -1/2 defects emerge much later in regeneration,

after the regenerated body axis is well-defined, during the formation of tentacles. De novo defect

formation typically involves the coupled appearance of two -1/2 defects and a +1 defect (with net

zero charge), in conjunction with an increase in surface curvature and the formation of a “bump”

that develops into the tentacle (Fig. S4). This process is analogous to that observed during

budding, where two -1/2 defects emerge at the base of the early bud together with a +1 defect at

its tip, as the bud begins to protrude 5. These de novo defect formation processes in Hydra differ

from other active nematic systems, where new defects emerge exclusively as +1/2 and -1/2


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defect pairs 17, 23-26. Apart from the protrusion-coupled emergence of defects in the tentacles (Fig.

S4), we do not observe spontaneous generation of defects in ordered regions during regeneration.

As such, the configuration of defects that establishes the regenerating animal’s body plan, and

defines its head and foot regions, is the result of the evolution and interaction of defects

emerging from the disordered regions early in regeneration following the induction of order there

(Fig. S3).

Interestingly, we have also found an intriguing relation between the nematic fiber organization,

the morphological outcome of the region, and the appearance of transient rupture holes which

allow the release of pressure and fluid from the spheroid cavity 30, 31. The formation of holes in

regenerating spheroids is never observed within regions containing ordered nematic fiber

alignment (Fig. 4F). Rather, during the earlier stages of regeneration, before localized point

defects develop, the ruptures occur in disordered regions (Fig. 4F). Most commonly, the holes

appear in disordered regions that are surrounded by ordered fibers that create an effective, spread

out +1 “aster”-shaped defect, and proceed to evolve into localized +1 defects as the ordered

nematic array of fibers forms (Fig. S5). At later times, after localized defects appear, hole

formation occurs almost exclusively at +1 defect sites (Fig. 4C,F), indicating that the mechanical

environment at these sites is significantly altered relative to the surrounding ordered regions. The

unique mechanical environment at +1defect sites is inherent to their structure as the focal point

of an aster of contractile fibers. Since the same defects that facilitate tissue rupture are also those

giving rise to the formation of the mouth, we suggest that the altered mechanical environment at

these locations plays an important role in supporting the biological specialization required for

mouth formation, with +1 defect sites acting as precursors for mouth formation.

Discussion

Our main finding here is that topological defects in the nematic actin fiber alignment that are

identified early in regeneration, act as organization centers of the body plan in the

morphogenesis process (Fig. 4). The nematic actin fiber orientation stands out as a relevant

observable, i.e. a coarse-grained field whose dynamics provide a valuable phenomenological

description of the morphogenesis process. This nematic orientation field contains only a small


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number of topological defects that evolve over the course of regeneration and change at rates that

are slow compared to the underlying molecular and cellular processes (Figs. 2-4). The unique

mechanical environment at +1 defects can provide localized mechanical cues at these sites,

which we identify as the regions where the head and foot eventually develop. The formation of

these morphological features also defines the polarity of the animal, which in regenerating tissue

fragments cannot be trivially inherited from the parent animal due to the initial tissue folding

(Fig. 2). Our observation that different defect configurations identify and differentiate between

the distinct sites of head and foot formation (Fig. 4D,E), imply that the defect distribution and

dynamics are coupled to the definition of the body-axis and its polarity, and thus likely play a

role in the establishment of the body plan.

The coincidence between defect locations and the emergence of morphological features with

specialized biological functions, demonstrates the innate coupling between biochemical and

mechanical fields in the tissue, rather than implying a direct causal sequence of events driven by

the presence of defects. For example, tentacle or bud formation are not initiated by preexisting

defects, but rather involve de novo defect formation, likely triggered by biochemical signals 5.

Nevertheless, the interaction between the mechanical field and biochemical signaling pathways

provides a possibility for reinforcement of specific processes at defect sites, with their unique

mechanical environment, towards the establishment of the required biological functions there.

Moreover, beyond their local effects, topological defects in nematic systems have long-range

interactions and are coupled to the geometry of the surface, in addition to the topological

constraint on the total defect charge 16. As such, topological defects naturally relate global

characteristics of the animal’s emerging body plan with local properties of the regions in which

morphological features develop, and can thus contribute to the large-scale coordination and

robustness of the morphogenesis process.

Based on our results, we suggest a complementary view of morphogenesis, in which a physical

field (here, the nematic actin fiber orientation) can be considered as a “mechanical morphogen”

field that provides relevant cues for directing the morphogenesis process. This view extends

Turing’s framework for the chemical basis of morphogenesis that has been extremely influential

in developmental biology 8, 9, and suggests a physical field as a mechanical counterpart to the

chemical morphogen field. The physical field reflects an integration of multiple non-trivial


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biochemical and biophysical factors including the mechanical properties, geometry and stress in

the tissue, and the complex mechanosensitive response to these features within the tissue. We

hypothesize that the interplay between various biochemical signaling processes and this

mechanical morphogen field with its associated pattern of topological defects, leads to the

establishment of a robust body plan in regenerated Hydra. Moreover, the identification of defect

sites as precursors for morphological features, draws attention to these regions as focal points of

the patterning process, offering the opportunity for focused investigation of the synergistic

interplay between biochemical signaling pathways and mechanical cues at these locations. The

development of an integrated framework for morphogenesis, which incorporates this dynamic

interplay between self-organized physical fields and signaling processes, and extends it to other

tissues and developing organisms, presents an exciting challenge for future research.

Acknowledgements

We thank Gidi Ben Yoseph for superb technical assistance. We thank Nitsan Dahan from the

LS&E Imaging and Microscopy Unit for help with confocal and light-sheet microscopy. We

thank Prof. Bert Hobmayer for generously providing transgenic Hydra expressing lifeact-GFP.

We thank Vincenzo Vitelli, Christina Marchetti and Julia Yeomans for valuable discussions. We

thank Natalie Dye and Meghan Driscoll for advice on image analysis. We thank Pascal

Silberzan, Alex Mogilner, Jacque Prost, Natalie Dye, Yariv Kafri, Tom Schultheiss, Guy Bunin,

Anna Frishman and Niv Ierushalmi for comments on the manuscript.

This work was supported by a grant from the European Research Council (ERC-2018-COG

grant 819174) to K.K., a grant from the Israel Science Foundation (grant No. 228/17) to E.B.,

and a Miriam and Aaron Gutwirth Memorial Fellowship to Y.M.S.


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Figures

Figure 1. Topological defects in Hydra. (A) Image showing the nematic actin fiber organization

in the ectoderm of a small mature Hydra. (B) Schematic illustration of the nematic actin fiber

organization in a mature Hydra. The topological defects in the nematic organization coincide

with the morphological features of the animal, with defects localized at the mouth (+1), foot (+1)

and tentacles (+1 at the tip, and two -1/2 at the base). The sum of defect charge is constrained by

topology to be equal to +2. (C-F) Images of the actin fiber organization containing topological

defects localized at the functional morphological features of a mature Hydra: the tip of the head

(C), the apex of the foot (D), and the tip (E) and base (F), respectively, of a tentacle. The zoomed

maps in (C,F) depict the actin fiber orientation extracted from the image intensity gradients 32


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(black lines) and the local order parameter (color; see Methods). (G-I) Images showing the

different types of nematic defects found in regenerating Hydra tissue spheroids: +1 defect (G),

-1/2 defect (H) and +1/2 defect (I). Inset: zoomed maps of the corresponding actin fiber

orientation and the local order parameter. All images shown are 2D projections extracted from

3D spinning-disk confocal z-stacks of transgenic Hydra expressing lifeact-GFP in the ectoderm.


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Figure 2. Nematic actin organization in regenerating tissue fragments. (A) Schematic

illustration of the nematic organization of the actin fibers at different stages during the

regeneration of an excised square tissue fragment. Immediately after excision (left), the tissue

fragment contains an ordered array of fibers. After ~2 hours (middle-left), the tissue is folded

into a closed spheroid with partial nematic order and an extended disordered region. Induction of

order over time, leads to the formation of localized topological defects (middle-right), which


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evolve over the course of a couple of days as the tissue regenerates into a mature Hydra (right).

(B) Images from a time-lapse light-sheet movie of a regenerating tissue fragment (Movie 1). At

each time-point, projections from two opposite sides depict the actin fiber organization in the

regenerating tissue. At T=5 hr, half of the spheroid is disordered (top), whereas a nematic array

of fibers is observed on the other half (bottom). At T=34 hr, ordered fibers are apparent

everywhere except for localized point defects. Three +1/2 defects are visible in the images. At

T=71 hr, the defect configuration of a mature Hydra is apparent with two +1 defects at opposite

sides of the tissue marking the tips of the head and the foot, respectively, and additional defects

in the emerging tentacles. (C) The fraction of area containing ordered fibers in regenerating

fragments as a function of time (see Methods). The graph depicts the ordered area fraction for 5

different regenerating fragments that were imaged by light-sheet microscopy from 4 directions

(thin lines), together with the average trend (thick line – mean; shaded region – standard

deviation). (D) The average number of localized point defects of each type (+1, -1/2 and +1/2) at

different stages of fragment regeneration (shaded region – standard deviation). The configuration

of point defects develops over time, with their net charge reaching +2 (the topologically required

value) only at ~24 hours, after the ordered regions expand and defects become localized. The

defect distribution was determined for a population of regenerating fragments that were imaged

using a spinning-disk confocal microscope from 4 directions within square FEP tubes at the

specified time-points (T=0, 2, 6, 24, 48, 72 hr with N=16, 25, 56, 44, 43, 41 samples at each time

point, respectively; see Methods). The data refers to defects that arise from the disordered

regions and their evolution, and does not account for the additional defects associated with

tentacle formation.


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Figure 3. +1/2 defects in regenerating Hydra are mobile and unstable. (A) Projected images

from a spinning-disk confocal time-lapse movie depicting the movement of a +1/2 defect relative

to the underlying tissue (Movie 2). The lifeact-GFP signal (gray scale) is overlaid with the


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fluorescent tissue label (green). The +1/2 defect (cyan arrow) moves away from the labeled

tissue landmark (green arrow) in the direction of its rounded end. (B) Bar plot showing the

number of +1/2 defects in regenerating tissue fragments as a function of time. The number of

defects which are part of a +1/2, +1/2 pair or a +1/2, -1/2 pair, and hence expected to undergo

merging or annihilation, respectively, is indicated. The data was extracted from regenerating

fragments that were imaged using a spinning-disk confocal microscope from 4 directions within

square FEP tubes at the specified time-points (T=2, 6, 24, 48, 72 hr, with N= 25, 56, 44, 43, 41

samples at each time point, respectively; see Methods). Two defects were considered a pair when

the distance between their cores was smaller than 40 μm. (C) Projected images from a spinning-

disk confocal time-lapse movie showing the merging of two +1/2 defects into a +1 defect. (D)

Projected images from a spinning-disk confocal time-lapse movie showing the annihilation of a

+1/2 defect and a -1/2 defect (Movie 3). Insets in (A), (C) and (D) depict maps of the

corresponding actin fiber orientation (black lines) and the local order parameter (color). The

tissue outlines are indicated with dashed lines.


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Figure 4. +1 topological defects as organization centers of head and foot regeneration in

Hydra. (A) Projected images from a spinning-disk confocal time-lapse movie depicting a +1

defect site that becomes the tip of the head of a regenerated Hydra (Movie 4). The lifeact-GFP

signal (gray scale) is overlaid with the fluorescent tissue label (green). The +1 defect remains

localized adjacent to the labeled tissue throughout the regeneration process, until the region

eventually develops into the mouth of the regenerated animal. The position of the defect and the

labeled region are indicated. (B) Projected images from a spinning-disk confocal time-lapse

movie depicting two +1/2 defects, facing each other (left), that eventually merge to form a +1

defect and become the apex of the foot of the regenerated animal (Movie 5). The lifeact-GFP


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signal (gray scale) is overlaid with the fluorescent tissue label (green). The position of the defects

and a landmark in the labeled region are indicated. (C) Projected images from a spinning-disk

confocal time-lapse movie depicting the formation of a rupture hole at a +1 defect site. The

image sequence shows the labeled +1 defect site prior (top), during (middle) and after (bottom)

hole formation. The lifeact-GFP signal (green) is overlaid with the fluorescent tissue label

(magenta). (D,E) The time evolution of defect sites was followed in 44 regenerating Hydra

samples, labeled by uncaging of a photoactivatable dye or using adhered beads to generate tissue

landmarks that could be followed over time (Methods). The outcome of the tissue regions

surrounding +1 defects that were formed either directly from the disordered region as in (A), or

from merging of two +1/2 defects facing each other as in (B), are depicted in (D) and (E),

respectively. Typically, since the samples are imaged from one side, we could only study one of

the two +1 defect sites in each sample. The inset in (D) shows the distribution of time after

excision when localized +1 defects become apparent. (F) The fiber organization surrounding

sites of hole formation was determined from time-lapse spinning-disk confocal movies of 27

regenerating tissues. For each sample the time after excision when localized point defects

become apparent was determined. The fiber organization surrounding rupture sites was then

analyzed for holes that were observed before (N=31; light grey) and after the appearance of

localized point defects (N=39; dark grey), all within the first 24 hours after excision.


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Methods

Hydra strains, culture and sample preparation

All the experiments are performed using transgenic Hydra strains expressing lifeact-GFP in the

ectodermal cells, generously provided by Prof. Bert Hobmayer from the University of Innsbruck,

Austria. Animals are cultured in standard Hydra culture medium (HM) (1mM NaHCO3, 1mM

CaCl2, 0.1mM MgCl2, 0.1mM KCl, 1mM Tris-HCl pH 7.7) at 18o C. The animals are fed 3-4

times a week with live Artemia Nauplii and rinsed after ~4 hours. Tissue segments are excised

from the middle body section of mature Hydra, ~ 24 hours after feeding, using a scalpel

equipped with a #15 blade. Tissue rings are excised by performing two nearby transverse cuts.

To obtain square tissue fragments, a ring is cut into ~4 parts by additional longitudinal cuts.

Sample Preparation

To reduce tissue movements and rotations, samples are either embedded in a soft gel (0.5% low

melting point agarose (Sigma) prepared in HM), or placed in HM supplemented with 1% Methyl

Cellulose. The general characteristics of the regeneration process in these environments are

similar to regeneration in normal aqueous media. For experiments in gels, the regenerating

tissues are placed in liquefied gel ~2-6 hours after excision (to allow the tissue pieces to first fold

into spheroids), and the gel subsequently solidifies around the tissue.

Time-lapse spinning-disk confocal imaging is done on samples placed in 35mm glass-bottom

petri dishes (Fluorodish), or polymer coverslip 24-well plates (Ibidi µ-plate 24 well, tissue

culture treated), embedded in 2-3 mm of gel that is layered with ~3-4 mm of HM from above.

Light-sheet microscopy samples are loaded in liquefied gel into a ~1cm long cylindrical FEP

tube with an internal diameter of 2.15 mm (Zeiss Z1 sample preparation kit). Once the gel has

set, the tube is mounted on one end onto a glass capillary which is held by the microscope

sample holder. The imaging is done through the FEP tubing, which has a similar refractive index

to the surrounding solution. Movies from 4 different angles are acquired by rotating the

specimen.

To image a large number of samples from all directions by spinning-disk confocal microscopy,

regenerating tissues are similarly loaded into FEP tubes with a square cross-section and an

internal diameter of 2 mm (Adtech). The tubes are rotated manually to each of the 4 flat facets of


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the square tube and secured using a homemade Teflon holder to keep them stationary at each

orientation. Images from 4 directions are acquired at the specified time points.

Microscopy

Spinning-disk confocal z-stacks are acquired on a spinning-disk confocal microscope (Intelligent

Imaging Innovations) running Slidebook software. The lifeact-GFP is excited using a 50mW

488nm laser and the activated Abberior CAGE 552 is excited using a 50mW 561nm laser.

Imaging is done using appropriate emission filters at room temperature, and acquired with an

EM-CCD (QuantEM; Photometrix). Long time lapse movies of regenerating Hydra are taken

using a 10× air objective (NA=0.5). Short time lapse movies and images in FEP tubes are taken

using a 10× water objective (NA=0.45).

Light-sheet microscopy is done on a Lightsheet Z.1 microscope (Zeiss). The light-sheet is

generated by a pair of 10× air objectives (NA=0.2), imaged through 20× water objectives

(NA=1), and acquired using a pair of CMOS cameras (PCO.edge). The lifeact-GFP is excited

using a 50mW 488nm laser and the activated Abberior CAGE 552 is excited using a 50mW

561nm laser. The imaging is performed using the “pivot scan” setting to minimize imaging

artefacts that introduce streaking in the direction of illumination, yet some remnants of the

streaking artefacts are still apparent in the images.

Tissue labeling using photoactivation of caged dyes

To label specific tissue regions we use laser photoactivation of a caged dye (Abberior CAGE 552

NHS ester) that is electroporated uniformly into mature Hydra and subsequently activated in the

desired region. Electroporation of the probe into live Hydra is performed using a homemade

electroporation setup. The electroporation chamber consists of a small Teflon well, with 2

perpendicular Platinum electrodes, spaced 2.5 mm apart, on both sides of the well. A single

Hydra is placed in the chamber in 10μl of HM supplemented with 2mM of the caged dye. A 75

Volts electric pulse is applied for 35ms. The animal is then washed in HM and allowed to

recover for several hours to 1 day prior to tissue excision.

Following excision, the specific region of interest is activated by a UV laser in a LSM 710 laser

scanning confocal microscope (Zeiss), using a 20× air objective (NA=0.8). The samples are first

briefly imaged with a 30 mW 488 nm multiline argon laser at 0.4-0.8% power to visualize the


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lifeact-GFP signal and identify the region of interest for activation. Photoactivation of the caged

dye is done using a 10 mW 405nm laser at 100 %. The activation of the Abberior CAGE 552 is

verified by imaging with a 10 mW 543 nm laser at 1%. Subsequent imaging of the lifeact-GFP

signal and the uncaged cytosolic label is done by spinning-disk confocal microscopy or light-

sheet microscopy as described above.

Experiments on hole formation

To follow the formation of holes in regenerating Hydra tissue and visualize the expulsion of

fluid from the holes, we incubate tissue segments immediately after excision in a solution of

fluorescent beads (0.2 µm Fluorospheres, Carboxylate-modified Microspheres, dark red 660/680,

Invitrogen), so that as the tissue folds and seals into a hollow spheroid, the internal fluid will

contain beads. The beads are first blocked by incubation in a 3% solution of Bovine Serum

Albumin (Sigma) in HM, whilst shaken for one hour, and then sonicated for 2 minutes. The bead

solution is then further diluted 1:10 in HM. Tissue segments are added to the bead solution

immediately after excision and incubated for 3 hours to allow tissue folding and sealing. Tissue

spheroids are then rinsed 3-6 times in HM, and imaged in wells (diameter 1 mm) prepared from

2% Agarose gel (Sigma). Imaging was done in a solution of HM containing 1% Methyl

Cellulose (Sigma) in order to increase the viscosity of the solution and allow more precise

visualization of tissue and bead movement. Some of the beads adhere to the tissue (especially

around the closure region), providing labeled landmarks on the regenerating tissue.

Image analysis

The image analysis tools used for extracting the 2D surface geometry of the tissue, the nematic

director field and the localization of the activated tissue label are based on custom-written code,

as well as adaptation and application of existing algorithms, written in Matlab and ImageJ as

detailed below.

Layer analysis. The apical and basal surfaces of the ectoderm are computationally identified in

3D fluorescence z-stacks of the lifeact-GFP signal in regenerating Hydra acquired with a z-

interval of 3-5 μm using the “Minimum Cost Z surface Projection” plugin in ImageJ

(https://imagej.net/Minimum_Cost_Z_surface_Projection). The cost images are generated by

processing the original z-stacks using custom-code written in Matlab. First, the signal from the


https://imagej.net/Minimum_Cost_Z_surface_Projection

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ectoderm layer was manipulated to make it more homogenous within the layer without

increasing its thickness, by applying the built-in Matlab anisotropic diffusion filter.

Subsequently, we employ a gradient filter to highlight the apical and basal surfaces as the top

and bottom boundaries of the ectoderm layer. The apical and basal surfaces are determined using

the minCost algorithm (Parameters used: Rescale xy: 0.5, rescale z: 1, min distance: 15µm, max

distance: 45µm, max delta z: 1, two surfaces). The surfaces given by the minCost algorithm are

then smoothed by applying an isotropic 3D Gaussian filter of width 1 pixel (after rescaling to an

isotropic grid matching the resolution in the z-direction) and selecting the iso-surface with value

0.5.

Image Projection. 2D projected images of the ectodermal actin fibers (that reside on the basal

surface of the ectodermal layer, Fig. S1) in spinning-disk confocal images are generated by

extracting the relevant fluorescence signal from the 3D image stacks based on the smoothed

basal surface determined above. To obtain the projected 2D image value for each x-y position,

we employ a Gaussian weight function in the z-direction with a sigma of 3 μm, which is centered

at the z-value determined from the smoothed basal surface with a small (2-3 pixels) fixed offset

found to optimally define the desired surface. 2D projected images of the ectodermal actin fibers

in light-sheet movies are generated by taking a maximum intensity projection of the 3D image

stacks, which result in better or similar quality projection images compared to method described

for spinning-disk images. The resulting 2D projected images are further subject to contrast

limited adaptive histogram equalization (CLAHE) in Matlab as described below.

2D projection of spinning-disk confocal images of the photoactivated tissue label are generated

by taking a maximum intensity projection of the 3D image stacks in the z-region between the

smoothed apical and basal surface determined above.

Fiber orientation analysis. A system with nematic orientation is a system in which the

constituents tend to align parallel to each other, with no distinction between opposite ends (as

opposed to a polar system). The average orientation of the constituents can be described by a

director field, which is oriented along the mean orientation determined in a small region

surrounding every point in space. The nematic director field that describes the local orientation

of the supra-cellular ectodermal actin fibers is determined from the 2D projected images of the

ectodermal actin fibers. The local fiber orientation is determined from the smoothed signal


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intensity gradients in the image as described in 32 and implemented in Matlab according to

“FingerPrint Matching: A Simple Approach” toolbox by Vahid K. Alilou from the Matlab File

Exchange Central (https://www.mathworks.com/matlabcentral/fileexchange/44369-fingerprint-

matching-a-simple-approach) with small modifications. All the relevant length scales below are

given in pixels, assuming a calibration of 1.28 µm/pixel (as in our typical spinning-disk images).

For images with different calibration, the number of pixels is adjusted to maintain the same

distances in real spatial units

The signal intensities in the images are first normalized employing CLAHE with Matlab’s

“adapthisteq” function with Rayleigh distribution and a tile size of 20 pixels. The signal intensity

gradients x

y

G

G

in the normalized images are calculated by convolving the adjusted image,

G(x,y), with the gradient of a Gaussian with a width of 0.5 pixel. The covariance matrix of the

gradient is then calculated from the signal intensity gradient (after averaging using a Gaussian

filter with σ=5 pixels), as:

2

2

x x yxx xy

G

yx yyy x y

G G GG GC

G G G G G

The raw orientation angle is calculated from the covariance matrix as:

1

,22

xx yy xyG G G

Where ,x y is defined as:

1

1

1

tan / 0

, tan / for 0 0

tan / 0 0

y x x

x y y x x y

y x x y

The coherence field, used to determine the extent of nematic order, is calculated from the raw

orientation field according to: cos ij , with 𝜃𝑖𝑗 the orientation of a given pixel, and the

average is taken over angles within a square window of 20 pixels. The orientation field is further

smoothed by applying a Gaussian filter (σ=3 pixels) on the raw orientation field.


https://www.mathworks.com/matlabcentral/fileexchange/44369-fingerprint-matching-a-simple-approach

https://www.mathworks.com/matlabcentral/fileexchange/44369-fingerprint-matching-a-simple-approach

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The orientation and coherence fields are defined within a masked area that excludes the regions

close to the tissue edges, as the sharp signal gradients near the edges introduce artefacts into the

orientation field.

The nematic local order parameter, which is used to identify the sites of topological defects, is

defined as:

2 2

cos 2 sin 2Q

Where is the orientation field and the averaging is done over a window size of 32 pixels.

Defining regions with ordered fibers. To automatically identify regions containing ordered actin

fibers in our images, we define a threshold value for the coherence field (defined in the previous

section), above which the region is considered ordered. The coherence threshold (taken as 0.95)

is used to create a binary image. The thresholded region in the binary image is subsequently

simplified by smoothing with a kernel of size 32 pixels and selection of the contour with value

0.5. The fraction of area (after smoothing) out of the total area of the coherence field is taken as

the fraction of the tissue surface with ordered fibers (Fig. 2C).

Definition and identification of topological defects in the fiber orientation field. A nematic

system can contain singularities in the orientation field, which are points at which the alignment

is locally disordered. These singularities are called nematic topological defects. These

topological defects form specific patterns around the defect, and can be characterized by their

“topological charge”, or winding number, which is the number of times the director orientation

rotates around the center of the defect. For example, the orientation field surrounding a defect

with topological charge of +1/2 rotates exactly half a rotation (180 degrees) around the defect

center. In a nematic system, the topological charge can be any integer multiple of +1/2, because a

rotation of 180 degrees brings the director back to its original orientation. A positive sign refers

to counter-clockwise rotation, whereas a negative sign refers to clockwise rotation.

The local order parameter Q (see definition above) is a measure of how much the local

orientation in a certain region varies. At the center of a topological defect, the local order

parameter will receive a minimal value, since the orientation field surrounding this point contains

all possible orientations.


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In the Hydra tissue, we identify topological defects as points around which there is well-defined

alignment of fibers, and the orientation rotates around the center of the defect, forming the

specific patterns characteristic of nematic topological defects. The identification is based on the

calculated fiber orientation field, and the associated local order parameter (Q; see definition

above). Defects are first manually identified by specifying a rectangular region around the

defects in every time frame, using the Matlab Ground Truth Labeler application. The defects are

then precisely localized by taking the position of the minima in the local order parameter within

the specified rectangular regions and tracked over time.

Analysis of hole formation. Analysis of hole formation is performed manually by identifying

frames within time-lapse movies in which a hole opens in the tissue. The criteria for defining a

hole include a transient, visible gap in fluorescence in both the lifeact-GFP signal and additional

fluorescent signal if exists, as well as change in tissue volume before and after the hole

appearance and/or visible expulsion of inserted beads or other material from the hole. Times and

locations of observed hole opening events are recorded, and hole opening sites are followed over

time (using tissue landmarks) to determine the fiber organization and morphological outcomes at

these sites.

Analysis of +1 defects and outcome. Analysis of +1 defects and outcome is performed manually

by identifying and following defects over time in fluorescent time-lapse movies of lifeact-GFP in

regenerating Hydra with local fluorescent marking using photo-activated dyes (see above). The

defects included in the analysis are those that can be reliably tracked (with the aid of the

introduced landmarks inserted in the tissue) throughout the regeneration process from their

formation until the final morphological outcome. The time defined for defect appearance is the

time at which ordered fibers are completely visible in the full region encircling the defect.


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Supplementary Figures

Figure S1. The structure of a mature Hydra. (A) Schematic illustration of a mature Hydra

showing the organization of the supra-cellular actin fibers in the outer ectoderm layer (green) and

in the inner endoderm layer (purple). (B) Images of the ectodermal (top) and endodermal

(bottom) supra-cellular actin fibers in the body of transgenic Hydra expressing lifeact-GFP in the

ectoderm or the endoderm, respectively. The fibers in the ectoderm are aligned along the animal

axis, whereas the fibers in the endoderm are aligned in a perpendicular, circumferential

orientation. We focus in this work on the more prominent ectodermal fibers, which are thicker

and appear continuous over supra-cellular scales. (C) Image showing the supra-cellular

ectodermal actin fiber organization in a small mature Hydra expressing lifeact-GFP. (D)

Schematic illustration of a perpendicular cross-section of the tubular Hydra body. Part of the ring

cross-section is shown, depicting the external ectoderm cell layer, the internal endoderm cell

layer, and the extra-cellular matrix (mesoglea) sandwiched between the two layers. The cells in

each layer form a polarized epithelial sheet, with their basal surfaces facing the mesoglea, and

their apical surfaces facing either the external medium in the ectoderm, or the internal gastric

cavity in the endoderm. The ectodermal and endodermal arrays of supra-cellular fibers lie along

the basal layer of each epithelial sheet, on a pair of nearly parallel 2D curved surfaces.


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Figure S2. Topological defects in multi-axis Hydra. (A) Schematic illustration of the

configuration of topological defects in a multi-axis Hydra that has two heads and a foot. The two

heads and the foot each have a +1 defect at their tip, whereas the junction between the two axes

has two -1/2 defects (one visible; the second -1/2 defect is on the opposite side). (B,C) Images of

a two-headed transgenic Hydra expressing lifeact-GFP in the ectoderm. (B) Bright-field (left)

and spinning-disk confocal (right) images showing the entire Hydra body with two heads and a

foot. The arrow indicates the junction between the two body axes. (C) Zoomed images showing

the actin fiber organization near the junction. Left: Projected spinning-disk confocal images

showing the actin fibers. Right: Images of the actin fiber orientation extracted from the image

intensity gradients (black lines) and the local order parameter (color; see Methods). The -1/2

defect at the junction is clearly evident.


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Figure S3. Induction of nematic order in a regenerating tissue fragment. (A) Image of the

actin fiber organization in a tissue fragment immediately after excision. The excised fragment is

characterized by an open boundary and a complete nematic array of actin fibers inherited from

the parent animal. (B) Images from a time-lapse movie of a regenerating tissue spheroid that

contains a region labeled by uncaging a photoactivatable dye (Methods). The ectodermal actin

organization (green) and photoactivated tissue label (magenta) are shown at two time points from

the movie. (C) Zoomed images showing the actin organization (top) and the photoactivated label

marking the tissue (bottom) in the same region over time. Initially, the actin in the folded tissue

fragment exhibits partial nematic order, with a large disordered region in the center of the

imaged region (left). Over time, parallel actin fibers emerge in the disordered region, with an

orientation that is aligned with existing fibers in the surrounding tissue. Since the fiber

orientation of the surrounding regions is not homogenous, localized topological defects appear in

the previously disordered region (right). The image sequence depicts steps in this process, as the

labeled tissue region transitions from a disordered state to an ordered one, disrupted by localized


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point defects. All images shown are 2D projections extracted from 3D spinning-disk confocal z-

stacks of transgenic Hydra expressing lifeact-GFP in the ectoderm.

Figure S4. De novo formation of topological defects during Hydra regeneration. (A)

Schematics of de novo defect formation: a +1 defect appears together with two -1/2 defects

simultaneously with the formation of a local protrusion, e.g. during tentacle formation. The +1

defect localizes to the region of highest positive curvature at the tip of the protrusion, whereas

the two -1/2 defects localize at saddle regions of negative curvature at the base of the protrusion.

(B) Images from a time-lapse movie showing the ectodermal actin organization during the

formation of a tentacle in a regenerating Hydra. The process involves the appearance of a +1

defect at the tip of an emerging protrusion together with two -1/2 defects at opposite sides of its

base, which subsequently elongates to become a tentacle in the regenerated animal. 2D

projection images extracted from 3D spinning-disk confocal z-stacks of a transgenic Hydra

expressing lifeact-GFP in the ectoderm at different time points during this process are shown.

Top right: zoomed image of the protrusion showing the actin fiber orientation extracted from the

image intensity gradients (black lines) and the local order parameter (color; see Methods). The

+1 defect at the tip of the protrusion and the -1/2 defect on the visible side of its base are

apparent.


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Figure S5. Hole formation in a disordered region prior to the appearance of a +1 defect.

Images from a time-lapse movie of a regenerating fragment that was incubated with fluorescent

beads (Methods). Some of the beads adhere to the tissue providing labeled landmarks on the

regenerating tissue. 2D projection images extracted from 3D spinning-disk confocal z-stacks of

the ectodermal lifeact-GFP (green) and the beads (magenta) are shown. Images show the sample

before (at 7:05 hr:min), during (7:15), and after (7:45) the formation of a hole in a disordered

region that is surrounded by regions with ordered fibers that generate a very spread out +1

defect. The beads that were trapped in the internal cavity during folding and were ejected from

the hole during its rupture are visible. After several hours (10:30), following the induction of

order in the region surrounding the rupture site, a localized +1 point defect develops at that site.


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Supplementary Movies

Movie 1: Light-sheet movie of a regenerating fragment

Time-lapse light-sheet movie showing projected images of lifeact-GFP in a regenerating tissue

fragment (top) and the corresponding nematic orientation field and local order parameter

(bottom; Fig. 2B). Images are shown from 4 different views, each rotated by a 90-degree angle,

as indicated. The movie shows the regeneration process beginning at the early tissue spheroid

stage, with regions of well-aligned fibers inherited from the original Hydra tissue as well as

disordered regions. Over the first 24-hours, the disordered regions become aligned through

induction of order from neighboring ordered regions, and point defects in the nematic order

develop in the previously disordered regions. The evolution of the defects can be followed to the

formation of the mouth and foot defining the body axis of the regenerated animal, in addition to

the later formation of tentacles. The tissue dynamics characteristic of the regeneration process,

including contractions, expansions, and changes in shape are apparent. The tissue outlines are

indicated with dashed lines. The images were centered to correct for movements of the whole

tissue. The elapsed time from excision is displayed in hours, and the scale bar is 100 µm. The

horizontal streaking of the fluorescent signal apparent in some frames is an artefact of the light-

sheet illumination.

Movie 2: Movement of a +1/2 defect relative to the underlying tissue

Time-lapse, spinning-disk confocal movie depicting the movement of a +1/2 defect relative to

the underlying tissue (Fig. 3A). The +1/2 defect moves away from the labeled tissue in the

direction of its rounded end. Right: images of the lifeact-GFP signal (gray scale) overlaid with

the fluorescent tissue label (green). Left: corresponding nematic orientation field and local order

parameter. The defect position (cyan arrow) and labeled landmark (green arrow) are indicated.

The tissue outlines are indicated with dashed lines. The images were centered and rotated to

correct for movements of the whole tissue. The elapsed time from excision is displayed in hours,

and the scale bar is 100 µm.

Movie 3: Annihilation of a +1/2 and a -1/2 defect

Time-lapse, spinning-disk confocal movie showing the annihilation of a +1/2 and a -1/2 defect

(Fig. 3D). Near the end of the movie, the two defects (that have a net charge of 0) merge and the


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region becomes ordered with well-aligned parallel fibers. Right: images of the lifeact-GFP

signal. Left: corresponding nematic orientation field and local order parameter. The position of

the +1/2 and -1/2 defects are indicated (cyan and magenta arrows, respectively). The tissue

outlines are indicated with dashed lines. The images were centered and rotated to correct for

movements of the whole tissue. The elapsed time from excision is displayed in hours, and the

scale bar is 100 µm.

Movie 4: A stationary +1 defect becomes the site of mouth formation.

Time-lapse, spinning-disk confocal movie showing a +1 defect that remains stationary with

respect to the underlying tissue. The defect site can be followed through time and coincides with

the site of head formation in the regenerated Hydra (Fig. 4A). The labeled cells located near the

center of the +1 defect can be followed directly to the mouth in the regenerated animal. Right:

images of the lifeact-GFP signal (gray scale) overlaid with the fluorescent tissue label (green).

Left: corresponding nematic orientation field and local order parameter. The defect position

(box) and labeled landmark (green arrow) are indicated. The tissue outlines are indicated with

dashed lines. The images were centered to correct for movements of the whole tissue. The

elapsed time from excision is displayed in hours, and the scale bar is 100 µm.

Movie 5: The region between two +1/2 defects becomes the site of foot formation.

Time-lapse, spinning-disk confocal movie showing two +1/2 defects with their rounded ends

facing each other moving closer together. The labeled cells located between the two +1/2 defects

remain centered between the defects as they move closer, and the regenerated foot forms at this

site (Fig. 4B). In this case, the final merging of the two +1/2 defects into a +1 defect at the foot

occurs late in the regeneration, after the establishment of the body axis. Right: images of the

lifeact-GFP signal (gray scale) overlaid with the fluorescent tissue label (green). Left:

corresponding nematic orientation field and local order parameter. The defect positions (cyan

arrows) and labeled landmark (green arrow) are indicated when present on the visible side of the

tissue. The tissue outlines are indicated with dashed lines. The images were centered to correct

for movements of the whole tissue. The elapsed time from excision is displayed in hours, and the

scale bar is 100 µm.


https://doi.org/10.1101/2020.03.02.972539

33

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