Mechanical strain focusing at topological defect sites in regenerating Hydra

Morphogenesis requires elaborate coordination between multiple biochemical and biophysical processes to generate robust, functional outcomes. To achieve this, the system must employ feedback spanning all levels of organization that can relate the emerging patterns with the mechanisms driving their formation (Braun and Keren, 2018; Collinet and Lecuit, 2021). There is a growing appreciation for the role mechanics plays in this process, and in particular the importance of mechanical feedback across scales in directing morphogenetic patterning (Dye et al., 2021; Hannezo and Heisenberg, 2019; Maroudas-Sacks and Keren, 2021). However, despite substantial progress in the field, the mechanisms by which such feedback operate and lead to the emergence of well-defined patterns remain largely elusive.

Hydra is a small fresh-water animal that provides an excellent model system for studying the integration between mechanics and other biophysical and biochemical processes in morphogenesis (Braun and Keren, 2018; Gierer, 2012). In particular, the capacity for full body regeneration in Hydra and its short regeneration time facilitate the study of an entire morphogenetic process in a holistic manner. The Hydra body is composed of a bi-layered epithelium that surrounds an internal, fluid-filled lumen. Due to the incompressibility of the fluid, the lumen acts as a hydrostatic skeleton that provides structural support (Kier, 2012). The osmolarity difference between the fresh-water environment and the lumen drives an inward osmotic flux (Ferenc et al., 2021; Fütterer et al., 2003; Kücken et al., 2008) that inflates the lumen and increases the hydrostatic pressure gradient across the bi-layered tissue.

The actomyosin cytoskeleton forms cortical networks and apical junctions in the bi-layered epithelium, as well as supracellular actomyosin fibers, known as myonemes, that organize into coherent, parallel arrays spanning the entire Hydra (Aufschnaiter et al., 2017; Livshits et al., 2017). These contractile fiber arrays run parallel to the body axis in the ectoderm, and perpendicular to the axis in the endoderm. All epithelial cells in both layers are excitable, capable of generating calcium-mediated action potentials that activate contraction of the actomyosin myonemes, as in muscles (Agam and Braun, 2023; Braun and Ori, 2019; Campbell et al., 1976; Szymanski and Yuste, 2019). This internal actomyosin-driven force generation, together with the hydrostatic pressure, are largely responsible for the deformations and movement of mature Hydra (Szymanski and Yuste, 2019; Wang et al., 2023a) and the shape changes during regeneration (Kücken et al., 2008; Livshits et al., 2017).

A crucial step in Hydra regeneration is the emergence of a head organizer that functions as the main signaling center in mature animals (Bode, 2012). The patterning of regenerating Hydra and the establishment of a new head organizer have mostly been attributed to biochemical morphogens, primarily those associated with the Wnt signaling pathway and its autoregulatory dynamics (Hobmayer et al., 2000; Holstein, 2022; Nakamura et al., 2011). Indeed, local activation of Wnt can promote head organizer formation (Wang et al., 2020) and global Wnt activation results in a multi-headed phenotype (Broun et al., 2005; Ferenc et al., 2021). Despite this progress, axial patterning in Hydra is still not well understood (Holstein, 2022). In particular, even though the significance of the head organizer was recognized over one century ago (Browne, 1909), the mechanisms involved in establishing a new organizer and specifying its location remain obscure (Cazet et al., 2021; Gufler et al., 2018; Suzuki et al., 2023 preprint; Tursch et al., 2022).

Regenerating Hydra tissues undergo cycles of osmotic swelling and collapse (Ferenc et al., 2021; Fütterer et al., 2003; Kücken et al., 2008). Previous experiments have shown that increasing the osmolarity of the external medium leads to diminished osmotic inflations (Fütterer et al., 2003; Kücken et al., 2008) and reduced activation of Wnt signaling (Ferenc et al., 2021). In particular, when the osmolarity of the external media and the lumen are comparable, Wnt expression falls off and head regeneration does not occur. While the mechanisms involved are unclear, the repression of the Wnt pathway and the failure to regenerate were attributed to the lack of tissue stretching under isotonic conditions (Ferenc et al., 2021).

Previously, we have found that sites of topological defects in the nematic order (i.e. in the parallel alignment) of the ectodermal actomyosin fibers emerge early in regenerating Hydra (<24 h from excision), and coincide with the sites of formation of morphological features (Maroudas-Sacks et al., 2021). Specifically, aster-shaped +1 defects emerge at the future head, and a pair of +½ defects come together in the regenerating foot, in accordance with a memory of body-axis polarity in the parent animal (Shani-Zerbib et al., 2022). Here, we study the dynamics of regenerating Hydra spheroids originating from rectangular tissue pieces at high spatiotemporal resolution. We find that tissue deformations are highly non-homogeneous in both space and time, and are strongly correlated with the nematic organization of the ectodermal actomyosin fibers. In particular, we find recurring tissue stretching concentrated at focus points of the actomyosin fiber pattern. We further find that rupture holes form exclusively at these actin foci, primarily at the future head site. Notably, we show that the actin foci can be identified in the folded spheroid from the onset of regeneration and coincide with the location where topological defects emerge and eventually the regenerated head and foot form. The mechanical strain focusing at actin foci is recapitulated by a biophysical model of Hydra tissue mechanics that considers transient activation of contraction in the ectodermal actomyosin fibers.

The colocalization of an aster-shaped +1 defect in the actin fiber organization, the emergence of a biochemical signaling center, and the focused tissue stretching at the future head site suggest a tight coupling between mechanics and signaling. Indeed, we find that disrupting the Wnt signaling pathway using iCRT14 (Cazet et al., 2021; Gufler et al., 2018), or placing tissues in elevated external osmolarity (Ferenc et al., 2021), prevents the formation and/or stabilization of aster-shaped defects and the tissues fail to regenerate. We propose a self-organization mechanism in which actomyosin fiber organization and tissue mechanics are coupled to the dynamics of a biochemical morphogen via a closed-loop feedback mechanism, involving strain-dependent morphogen production at defect sites. We hypothesize that this mechanochemical feedback loop underlies the robust formation and stabilization of the head organizer and use model simulations to demonstrate the viability of this idea, providing the basis for a putative mechanochemical framework for Hydra regeneration.

The dynamic organization of the ectodermal actomyosin fibers in regenerating Hydra is followed using live imaging of transgenic animals expressing Lifeact-GFP (see supplementary Materials and Methods). Rectangular tissue segments cut from the body of mature Hydra fold and seal into hollow spheroids within a couple of hours (Bode and Bode, 1984; Javois et al., 1988; Livshits et al., 2017) (Fig. 1A,B). This process is highly stereotypical: opposite ends of the tissue stretch to meet, such that the originally head-facing side of the excised tissue comes together with the originally foot-facing side of the tissue (Shani-Zerbib et al., 2022). The excised tissue inherits an array of aligned actomyosin fibers, but during the folding process, the fibers along the edges of the excised tissue disassemble (Livshits et al., 2017; Maroudas-Sacks et al., 2021). The resulting tissue spheroid contains a domain of well-ordered ectodermal fibers corresponding to the more central part of the original excised tissue piece, and a region that lacks supracellular fiber organization (Fig. 1A,B; Fig. S1). The ordered domain of parallel fibers spans roughly two-thirds of the tissue circumference in one direction. At either end of this ordered domain (along the direction of the fibers), we find disordered ‘caps’ connected by a disordered ‘bridge’. The two caps are mostly encircled by radially aligned fibers, whereas the connecting bridge is flanked on both sides by fibers aligned parallel to the domain boundary. Overall, the disordered region has a total nematic charge of +2, defined by the fiber organization surrounding it (Fig. 1A). This characteristic pattern allows us to identify two focus points of the actomyosin fiber organization, corresponding to the centers of the disordered caps. The location of these foci is determined based on the fiber organization in the surrounding ordered regions, where the fibers form a nearly-radial pattern around each cap. The center of the caps can be approximated by extending lines along the radial fibers from the surrounding regions to their intersection point within the disordered region (Fig. 1A,B; Fig. S1).

The early stages of the regeneration process are characterized by an induction of order process, during which the inherited fibers guide the formation of aligned fibers throughout the spheroid, and point topological defects emerge (Maroudas-Sacks et al., 2021). We find that the location of the emerging defects is highly stereotypical, forming at the center of the caps at the two foci of the fiber organization (Fig. 1; Fig. S1). Using laser-induced uncaging of a caged dye (Abberior CAGE 552) (Maroudas-Sacks et al., 2021), we label groups of cells at the cap regions and follow them over time (Fig. 1C,D; Movie 1). The cap closer to the head-facing side of the excised tissue develops a +1 aster-shaped defect at the future head site, whereas in the other cap, a pair of +½ defects form at the future foot region. Meanwhile, the ‘bridge’ between the two caps develops a complete, ordered array of fibers aligned parallel to the inherited fibers along its boundary.

The tissue dynamics during regeneration are followed in conjunction with the supracellular actomyosin fiber organization at high spatiotemporal resolution. The Lifeact-GFP probe that labels the actomyosin fibers at the ectodermal basal surface also binds filamentous actin at cell-cell junctions at the apical surface (Aufschnaiter et al., 2017). We developed an image-processing pipeline to separate the fluorescent signal from the apical and basal sides of the ectoderm, allowing us to identify individual cells at the apical surface, while simultaneously following the actomyosin fiber organization. The regenerating tissue is extremely dynamic, with continuous movement and deformations. The most pronounced deformations are seen when the tissue experiences large-scale, coordinated contractions that are reflected in extensive, yet transient, distortion of the tissue, primarily along the direction of the ectodermal actomyosin fibers (Fig. 2). During these events, we observe stretching of cells at the center of the disordered cap regions (Fig. 2A), and at +1 defect sites after the induction of order at later stages of regeneration (Fig. 2B). Occasionally we observe ‘doming’, where the stretched tissue exhibits an abrupt curvature change between the core stretched region and the rest of the tissue (Fig. S2), suggestive of inhomogeneous or non-linear material properties (Latorre et al., 2018).

To quantify the frequency and localization of stretching events, we use spinning-disk confocal microscopy in a custom up-and-under setup, which enables simultaneous imaging of multiple samples from two opposite sides (see supplementary Materials and Methods). We imaged at a time resolution of 2-2.5 min, which is shorter than the typical event duration (7±6 min, mean±s.d., n=375 stretching events; Fig. 2F,I), and manually recorded tissue stretching events during the first 24 h after excision (Fig. 2D-I). Large stretching events, defined as incidents in which stretched cells reach a greater than twofold increase in apical cell area, are observed only at actin foci, mostly at the disordered cap regions early on, or later at sites of +1 defects (Fig. 2A,B,D; Movie 2). These large stretching events occur at the future head site, at a rate of ∼1 event/h (Fig. 2E). Notably, we never observe stretching events in ordered regions, where the fibers are organized in a parallel array, and observe only smaller stretching events (defined as a less than twofold increase in apical cell area) at regions containing +½ defect pairs (Fig. 2C,G). As the regenerating tissue becomes elongated, it often aligns parallel to the imaging plane making it harder to visualize the defect regions, yet pronounced tissue deformations continue. Similarly, in mature animals, contraction of the ectodermal fibers leads to stretching at the +1 defect region and mouth opening (Carter et al., 2016).

The tissue stretching events typically occur simultaneously at both foci, with the future head region consistently exhibiting larger amplitude events compared to the future foot region (Fig. 2E,H; Fig. S3). This is true even at early stages of the regeneration process, where both cap regions lack ordered fibers and we are unable to distinguish between them based on the fiber organization. We have previously shown that the future head and foot form in accordance with a memory of body-axis polarity (Javois et al., 1988; Shani-Zerbib et al., 2022). Thus, while the details of how body-axis polarity memory is encoded in the tissue remain unknown, our results indicate that there is a mechanical manifestation of this memory in the tissue deformations, which is apparent already at the earliest stages of the regeneration process (Fig. 2E,H).

The patterns of tissue deformations at the future head and foot regions become more distinct as the fibers organize and form the characteristic defect configuration, with a +1 defect at the future head and a +½ defect pair at the future foot (Fig. 2B,C; Fig. S3). Events at the future head region remain radially symmetric around the +1 defect and are similar in amplitude to earlier events (Fig. 2A,B). However, events localized between two +½ defects exhibit more moderate cell area stretching (Fig. 2C). These differences in the pattern and extent of stretching, can be attributed to the different contractile fiber organization, and particularly the different defect configuration (+1 vs +½ pair, respectively) at the future head and foot regions.

To quantify the pattern of ectodermal tissue deformations during stretching events, we segment individual cells based on the Lifeact-GFP signal at the apical cell-cell junctions, correcting for the geometrical distortions due to the curved tissue surface (Fig. 3; see supplementary Materials and Methods). As in other epithelial tissues, cell segmentation provides a local measure of tissue deformations (e.g. Blanchard et al., 2009; Etournay et al., 2015; Guirao et al., 2015; Hashimoto et al., 2015; Priya et al., 2020). Tracking individual cells in regenerating Hydra is difficult due to the highly dynamic nature of the tissue. However, the actomyosin fiber pattern, which remains stable during stretching events (∼several minutes; Fig. 2F,I), provides a useful frame of reference. We thus use the actin foci as reference points, and quantify the pattern of cellular deformations as a function of distance from these points. Since geometric distances depend on the varying tissue strain, we use graph distance (i.e. the degree of minimal neighbor separation between cells) to quantify the distance from the actin foci. We measure cell area strain using logarithmic strain, which is defined as the natural logarithm of the ratio of cell area during the peak of a stretching event to the cell area immediately before the event started.

We find that area strain during stretching events is highest at the actin focus at the future head region, located early on at the center of the cap region in the disordered domain and later at the +1 defect site (Fig. 3A,B). The area strain decays within a distance of ∼3-5 cells from the foci (Fig. 3D,E). The strain amplitude at the core varies between events, with an average of ∼ln(2), i.e. a twofold increase in cell area (Fig. 3E). At the same time, cells away from the focus deform anisotropically, contracting along the direction of the ectodermal fibers (Fig. 3C,F). We quantify the degree of cell anisotropy using the cell shape tensor Q (Merkel et al., 2017). The deformation pattern at the future foot site that contains a +½ defect pair exhibits only moderate local area stretching at the core (Fig. 3H-N). The cells in the region between the two defects, transiently contract in the direction of the fibers, similar to what is observed in the ordered fiber arrays in the future gastric region. The observed tissue deformations are thus correlated with the nematic field describing the actin fiber orientations (Fig. S4). The relationship between the nematic field and the strain field connects the overall pattern of these two fields throughout the tissue, rather than being a local relation.

The observed tissue stretching is not only localized in space, but also in time, with rapid stretching events typically separated by considerably longer time intervals (Fig. 3D,K). Notably, the correlation between cell area deformations and fiber organization (Fig. 3E,F,L,M) is not apparent during most of the regeneration process. In particular, the time-averaged changes in cell area strain around defect sites are substantially smaller than the changes observed during stretching events, and within the typical range of variation in cell area (Fig. 3G). Thus, although the area strain experienced by regenerating tissues during stretching events generates a clear mechanical signature at defect sites, these enhanced tissue deformations are transient and are not apparent upon time averaging. This suggests that if these tissue deformations provide relevant mechanical cues at the future head site, the response to these cues must be non-linear, accentuating the large, yet transient, strains encountered at defect sites.

Regenerating Hydra spheroids exhibit cycles of swelling and collapse driven by osmotic influx of fluid into the lumen followed by tissue rupture and fluid release (Ferenc et al., 2021; Fütterer et al., 2003; Kücken et al., 2008; Wang et al., 2019). Tissue ruptures typically involve both layers of the epithelium. This can be shown directly by imaging the ectoderm and endoderm simultaneously (Carter et al., 2016), as well as indirectly by observing fluid efflux accompanied by a reduction in lumen volume (Ferenc et al., 2021; Fütterer et al., 2003; Kücken et al., 2008; Maroudas-Sacks et al., 2021; Wang et al., 2019). Tissue ruptures could have various physiological implications, including, for example, release of internal pressure (Stokkermans et al., 2022), changes in the electric potential gradient across the tissue (Macklin and Josephson, 1971), and induction of a wound-healing response in the tissue (Tursch et al., 2022).

Our high-resolution imaging allows us to visualize the formation of rupture holes and to characterize their location and the associated tissue deformations. We find that osmotic inflations involve a gradual, roughly homogeneous increase in cell area throughout the ectodermal shell (Fig. 4A). However, rupture holes form exclusively at actin foci, and their formation is preceded by focused tissue stretching of the future hole site (Fig. 4; Movies 5 and 6). The rupture holes recur at the actin foci sites, and never appear in regions with ordered fibers (Fig. 4A,F). This is true from the very first ruptures that occur in the folded spheroid, contrary to what has been reported previously based on lower-resolution imaging (Wang et al., 2019). We further observe that the formation of rupture holes is biased toward the future head region (Fig. 4G), which also exhibits more extensive tissue stretching (Fig. 3N).

The mechanical strain focusing at future rupture sites and compression of cells in the ordered regions far from these sites (Fig. 4A), is analogous to the cellular deformation patterns observed during stretching events (Figs 2 and 3). As such, our observations indicate that the tissue ruptures are not the result of random local failure after homogeneous osmotic inflations, as previously thought (Ferenc et al., 2021; Kücken et al., 2008; Wang et al., 2019). Rather, the ruptures form when large-scale actomyosin fiber contractions occur in an osmotically inflated tissue spheroid. In this case, the contractions further increase the isotropic pressure in the lumen (Stokkermans et al., 2022) and, more importantly, generate an additional inhomogeneous in-plane stress that is strongest at the actin foci (see also modeling section below). Rupture hole formation occurs at the site of highest in-plane stress, and hence is localized at the actin foci (Fig. 4F).

Rupture events typically involve disruptions of the epithelial bi-layer along cell-cell contacts, generating fractures along cell boundaries (Fig. 4B), as observed in other tissues (Bonfanti et al., 2022). Immediately after their formation, rupture holes can appear as elongated cracks along the junctions connecting adjacent cells (Fig. 4B). These cracks round up within a short time (<1 min) and acquire a smooth boundary, spanning multiple cells, with an enriched actomyosin ring lining the edge of the hole (Movie 5). The rupture holes typically reseal within ∼10 min (Fig. 4D). The formation of rupture holes is reminiscent of the mechanism of mouth opening in adult Hydra, which similarly involves tearing the epithelium along cell-cell junctions at a well-defined location that is a focal point of the ectodermal actomyosin fibers (Carter et al., 2016; Wang et al., 2019).

An interesting phenomenon that is observed in about half of the regenerating tissues (8/13 samples observed) is the opening of extremely large rupture holes in the ectoderm, which can even reach the entire circumference of the spheroid (Fig. 4C; Movie 6) and remain open for nearly 1 h (Fig. 4E). Remarkably, despite the considerable deformation, the spheroids reseal and appear to recover their pre-rupture organization and eventually regenerate successfully. We suspect that these extremely large ruptures may occur when the ectodermal layer partly detaches from the mesoglea due to the strong shear forces generated by the contraction of the ectodermal fibers. The large opening in the ectoderm in this scenario reflects transient sliding of the ectoderm relative to the endoderm, which subsequently recovers.

The observed pattern of cellular deformations during large stretching events can be understood intuitively by assuming that the Hydra tissue is elastic (Perros et al., 2024) and that contraction of the ectodermal actomyosin fibers is the primary source of force generation during these events. Upon activation, the aster-shaped +1 defect regions will focus stress (and hence strain) at the defect site, due to contraction of the surrounding radially oriented fibers. Similar stress focusing is expected at the center of the cap regions, which are encircled by radially oriented fibers from nearly all directions (apart from the disordered bridge). The fiber organization in the vicinity of +½ defect pairs is expected to be less efficient at focusing stress, and no strain focusing is expected in ordered regions with a parallel fiber array. While the perpendicularly oriented endodermal fibers also generate forces, we neglect their contribution during stretching events. This assumption is justified by our observations that the tissue rapidly contracts primarily along the direction of the ectodermal fibers during these events (Fig. 3F,M). This is also consistent with recent studies of the behavior of mature Hydra showing that the ectodermal fibers are dominant during large contractions (Wang et al., 2023a).

To corroborate our intuitive understanding of mechanical strain focusing during stretching events, we develop a biophysical model of Hydra tissue mechanics (Fig. 5A; see supplementary Materials and Methods). We describe the regenerating Hydra as a deformable shell with an embedded nematic field describing the actomyosin fiber orientation, which encapsulates a fluid-filled lumen. The deformable shell is realized as a collection of cells specified by the positions of their vertices. As such, our model is a generalization of 2D vertex models (Farhadifar et al., 2007; Honda and Eguchi, 1980) to curved closed surfaces. The tissue mechanics are described by an energy function E, accounting for cell area elasticity, cell-cell adhesion, and cell perimeter elasticity, as in standard vertex models (reviewed by Alt et al., 2017; Fletcher et al., 2013). The tissue deformations conserve the volume of the fluid-filled lumen. Furthermore, since we do not explicitly model the tissue thickness, we include a bending energy term that penalizes tissue curvature. We describe the embedded nematic field by assigning a two-dimensional tensor to each cell in the cell-tangent plane. Finally, to emulate stresses generated by fiber contraction, we introduce active cellular stresses, oriented along the nematic field, with a tunable magnitude ζ (Comelles et al., 2021). The model parameters are chosen to reproduce the elastic behavior of regenerating Hydra and the magnitude of deformations observed experimentally (see supplementary Materials and Methods).

We use this model to study strain patterns induced by individual tissue-stretching events (Fig. 5; Movies 7 and 8). Starting from an initially relaxed state, we emulate global contraction activation by introducing a pulse of active stress with magnitude of ζ_M (Fig. S5; see supplementary Materials and Methods), and duration of ΔT_ζ. Since the time scale for stretching events (minutes; Fig. 2F,I) is shorter than the time scale for fiber reorganization (hours) (Maroudas-Sacks et al., 2021), we assume that the nematic field describing the fiber orientation does not evolve during these events (apart from being conveyed by the cells as the tissue deforms). We consider two particular configurations of the nematic field that recapitulate the observed fiber patterns during early and later stages of regeneration, respectively. For early stretching events, we consider a spheroid with a large disordered domain of net charge +2 surrounded by a fully ordered region (Fig. 5B). For later events, following the induction of order, we consider a spheroid with fully ordered fibers except for a +1 defect on one end and a +½ defect pair at the opposite end (Fig. 5C).

Our simulations recapitulate the observed correlation between the deformation field and the nematic fiber patterns (Fig. S4, Movies 3 and 4). Specifically, the simulations exhibit cell stretching concentrated at the two actin foci, and cell compression in the direction of the fibers in the ordered regions between the foci (Fig. 5B-D; Movies 7 and 8). This pattern of mechanical strain focusing at the foci arises from global activation of contraction of the nematic actin fibers throughout the tissue. Since the fiber organization at early stages of the regeneration is similar around the two foci, the simulated mechanical strain focusing has comparable magnitudes in both foci (Fig. 5E). At later stages of regeneration, when the nematic pattern becomes asymmetric, with a +1 defect at the future head region and a +½ defect pair at the future foot, the model accounts for the asymmetric deformation pattern observed, exhibiting pronounced stretching at the +1 defect site (Fig. 5C,F; Movie 8), and more moderate stretching at the +½ defect pair (Fig. 5D). Overall, these results demonstrate that considering regenerating Hydra as elastic shells with an incompressible lumen that experience global fiber contraction is sufficient to account for the tissue deformations that occur during stretching events.

The coincidence of a +1 defect with the biochemical signaling center at the head organizer in mature Hydra (Fig. 6A) suggests an intimate coupling between mechanics and biochemical signaling. To explore this interplay, it is useful to perturb the Wnt signaling pathway, which is considered to be the main activator associated with organizer formation, and characterize the actin fiber organization, tissue dynamics and regeneration outcome. Previous work has shown that local upregulation of Wnt leads to rearrangements of the ectodermal actomyosin fibers toward regions of high Wnt and the formation of aster-shaped +1 defects, both during budding (Aufschnaiter et al., 2017) and after grafting of an excised organizer or a piece of tissue overexpressing Wnt onto a host tissue (Wang et al., 2020).

To investigate the influence of downregulation of Wnt on actin organization and tissue mechanics, we use iCRT14, which inhibits the β-catenin-TCF interaction and has been shown to suppress Wnt signaling and disrupt regeneration of bisected Hydra (Cazet et al., 2021; Gufler et al., 2018). We follow the dynamics of excised tissue fragments subject to iCRT14 treatment, and examine its effect on actin fiber organization and tissue mechanics. Animals are pretreated with 5 μM iCRT14 for 2 h before excision (Cazet et al., 2021), and the excised tissues are subsequently maintained at the same iCRT14 concentration. Excised fragments fold and seal as untreated samples, with a similar pattern of inherited ordered fibers and a large disordered region (Fig. 6B, top). However, while the fibers in the ordered region remain stable, there is no induction of order in the disordered region (Fig. 6B, bottom; Movie 9). Thus, unlike control tissues, the treated samples remain essentially in their initial configuration, with an inherited ordered region and a large disordered domain. Notably, under these conditions, aster-shaped defects fail to form and the samples do not regenerate.

Recent experiments showed that placing regenerating Hydra in a medium with elevated osmolarity, among various possible effects, also leads to suppression of Wnt signaling (Ferenc et al., 2021). These experiments corroborated previous findings, showing that, when regenerating tissue fragments are placed in media that is isotonic with their lumen (∼70 mOsm), osmotic inflations cease (Kücken et al., 2008). They further showed that, under these conditions, Wnt is initially upregulated in response to the wound, but its expression is not sustained and subsequently declines, and the excised tissues fail to regenerate. Here, we follow the tissue deformations and actin fiber dynamics in spheroids placed in Hydra medium supplemented with 70 mM sucrose. Initially, the folded spheroids have an ordered region and a disordered domain (Fig. 6E, left), which organizes into an ordered nematic fiber array (Movie 10), as in untreated samples. However, under isotonic conditions, the spheroids develop an unusual defect configuration with four +½ defects (Fig. 6C,D; Fig. S6H), rather than the characteristic defect configuration with a stable +1 defect at the future head site and a +½ defect pair at the future foot (Maroudas-Sacks et al., 2021). While we occasionally observe the formation of a +1 defect under isotonic conditions, these defects are unstable and dissolve into a +½ defect pair (Fig. 6E; Movie 10).

The treated tissues still undergo some stretching events with mechanical strain focusing at actin foci, yet these are diminished compared to control samples (Fig. S6, Movie 10). In particular, we observe fewer tissue ruptures under isotonic conditions and large stretching events are not observed once the four +½ defect configuration develops. In this configuration, the strain pattern in both foci is similar to the pattern seen at the future foot region in control samples, which contains a pair of +½ defects (Fig. S6C,G). Interestingly, even though the treated tissues fail to regenerate, they still elongate along the fiber direction, such that the two +½ defect pairs localize at either end of the cylindrical tissue, rotated 90° from each other (Fig. 6E, Movie 10). Overall, these results highlight the inherent coupling between actin fiber organization and Wnt signaling, and demonstrate that the formation of aster-shaped +1 defects can be disrupted when Wnt expression is perturbed.

Our results show that the actomyosin fiber organization leads to mechanical strain focusing at defect sites. Notably, the primary site of strain focusing can be identified from the onset of regeneration and coincides with the location of the emergence of a head organizer at the tip of the regenerating head. Motivated by these observations, we propose a self-organized mechanism in which mechanical strain focusing at actin foci and the localized biochemical signaling required for establishing a new head organizer reinforce each other (Fig. 7A). Specifically, we suggest a closed-loop feedback in which the strain localized at the actin foci induces morphogen production. Concurrently, regions of high morphogen concentration align fibers along morphogen gradients and stabilize aster-shaped +1 defects, further enhancing the local strain focusing at these sites.

While the mechanistic details involved are unknown, recent results by us and others support different parts of the proposed feedback loop. The relation between mechanical strain and Wnt production was suggested by Ferenec et al., who correlated reduced tissue stretching under isotonic conditions with the inability to stabilize Wnt expression and regenerate a head (Ferenc et al., 2021). In their work, the mechanical strain was assumed to be spatially homogenous, whereas our high-resolution imaging reveals strain focusing at the future organizer site. The idea that morphogen gradients can direct the orientation of actin fibers was recently suggested theoretically (Wang et al., 2023c), and is supported by observations showing that local Wnt expression can induce the formation of +1 defects in grafting experiments or during budding (Aufschnaiter et al., 2017; Wang et al., 2020). This notion is further supported by our current results showing the absence of stable aster-shaped +1 defects after perturbations of the Wnt pathway (Fig. 6).

We examine the feasibility of the proposed self-organization mechanism by developing a biophysical model of regenerating Hydra (Fig. 7; see supplementary Materials and Methods). This model extends the description of Hydra tissue mechanics (Fig. 5), by including the coupled dynamics of the nematic field and a biochemical morphogen throughout the regeneration process. The morphogen field is defined by specifying the number of molecules N_c in each cell (Fig. 7B). We introduce a strain-dependent morphogen source term as a sigmoid function of cell area strain, characterized by a threshold strain value and a saturation level r₊ (Fig. 7B; see supplementary Materials and Methods). The morphogen molecules can diffuse between cells and are degraded at a constant rate r₋. The dynamics of the nematic field are governed by an interaction term that favors alignment of fibers in neighboring cells, as well as a coupling of fiber orientation to local morphogen gradients (Fig. 7B; see supplementary Materials and Methods). The recurring transient fiber contractions are emulated by periodically turning on the active stress in all cells with a period T_ζ and amplitude ζ_M using a Gaussian profile in time of width ΔT_ζ.

Using this model, we simulate regeneration from tissue spheroids with an initial fiber configuration containing a large disordered region (as in Fig. 5B) and no morphogen. We evolve the full dynamics of the model through a series of global contraction cycles, and examine whether the disordered region can reliably resolve to form a ‘head organizer’, i.e. a region containing a +1 nematic defect colocalized with a morphogen peak. The outcome defect configuration depends on the availability of morphogen, controlled by ⁠, and on the coupling between the nematic field and the morphogen, controlled by α (Fig. S7). In particular, configurations with stabilized +1 defects appear only for sufficiently low values of and for high enough values of α (see supplementary Materials and Methods). We find model parameters in this range that reproduce the emergence of a single organizer and a +½ defect pair on the opposite side (Fig. 7C; Movie 11), as observed during Hydra regeneration. For the same set of model parameters, different initializations of the random nematic orientation in the disordered region can also yield a spheroid with two organizers (Fig. 7C; Movie 12). We find that the appearance of these different defect configurations in our simulations is consistent with a scenario in which each actin focus develops independently into a +1 defect or a pair of +1/2 defects (see supplementary Materials and Methods). Thus, while the suggested feedback loop can robustly produce at least a single ‘head organizer’, it does not prevent the formation of two organizer sites. We provide a detailed analysis of the parameter space spanned by and α (Fig. S7). As expected, +1 defects are completely absent in simulations with no morphogen production (⁠→∞) or no coupling to morphogen (α=0).

Our results show that the nematic fiber organization in folded spheroids obtained from excised tissue fragments contains two discrete actin foci. These foci develop into a characteristic set of topological defects that eventually coincide with the sites of head and foot formation in the regenerated animal. We further show that these actin foci experience recurring mechanical strain that arises from transient contraction of the actomyosin fibers, converging at the defect core. This localized strain pattern provides a clear mechanical signature at the future head site, despite the presence of extensive disorder and fluctuations in the tissue.

Hydra has been a classic model system for studying axial patterning during animal morphogenesis for over one century, since the pioneering work of Ethel Brown, who discovered the head organizer (Bode, 2012; Browne, 1909). Subsequent work established the presence of head-inducing activation and inhibition activities that are graded along the body axis (Shimizu, 2012). These observations led to the development of the famous Gierer-Meinhardt model that describes axial patterning in Hydra as a reaction-diffusion process (Meinhardt, 2012b). This model successfully integrated a large body of observations into a coherent picture that has dominated the field, in which tissue-scale gradients generate short-range activation and long-range inhibition of head formation. Despite its popularity, the mechanistic basis for this model is still unclear and, in particular, the nature of the source term, a crucial ingredient of the model, has remained obscure (Meinhardt, 2012a; Wang et al., 2023b). In the context of our work, it is also important to note that the Gierer-Meinhardt model does not consider the possible contribution of mechanics and the nematic fiber organization in the patterning process.

The characteristic defect pattern found in regenerating Hydra fragments, containing a +1 defect at the future head and a +½ defect pair at the future foot (Fig. 1), differs from the expected stable defect configuration for a nematic on a sphere with four +½ defects (Shin et al., 2008). Interestingly, we observe that spheroids placed in elevated osmolarity develop such a four +½ defect configuration (Fig. 6C,D; Fig. S6H). These results demonstrate that +1 defects are not a necessity of the nematic organization in Hydra, but rather require some additional stabilizing interaction. As previously suggested (Wang et al., 2023c), +1 defects can be stabilized through a coupling of the nematic field to morphogen gradients (Fig. 7). We hypothesize that this additional coupling is part of a mechanochemical feedback involved in head organizer formation.

Ferenc et al. recently suggested that tissue stretching due to osmotic inflations are essential for regeneration because of their role in stabilizing Wnt expression (Ferenc et al., 2021). While we believe that their results are valuable in highlighting the importance of mechanochemical coupling in Hydra regeneration, we disagree with some of their interpretations. In particular, while cycles of large osmotic inflations and ruptures typically occur during regeneration (Ferenc et al., 2021; Kücken et al., 2008), they are not essential (Fig. S8). Moreover, the localized Wnt expression at the future head was attributed to random symmetry breaking under the assumption of homogenous tissue inflation. However, the observed tissue stretching is inhomogeneous, and the location of the future head coincides with the location of the actin foci at the onset of regeneration.

We suggest a closed-loop feedback that couples topological defects in the nematic field with localized morphogen production, via the spatiotemporal strain patterns generated during actomyosin fiber contraction (Fig. 7). Within our model, the emergence of a head organizer at the defect site is not a coincidence or a corollary of biochemical patterning. Rather, we suggest that head organizer formation is a dynamic self-organization process that involves an interplay between morphogen dynamics and the nematic field. This hypothesis is appealing, since it naturally provides a mechanism for inducing local organizer activity at defect sites. The nematic field robustly generates discrete foci that produce a localized strain, and the slow dynamics of the morphogen source field in this scenario arise from the slowly varying dynamics of the actin fiber pattern (Maroudas-Sacks et al., 2021).

Obviously the formation of a new organizer is not solely determined by this mechanochemical coupling. The Wnt signaling pathway has an autocatalytic regulatory network and exhibits non-linear dynamics that can generate localized production on their own (Nakamura et al., 2011). Moreover, the memory of polarity in regenerating tissues implies the presence of additional relevant fields (Javois et al., 1988; Shani-Zerbib et al., 2022). While our current model of morphogen field dynamics is too simplistic, we believe that the proposed mechanochemical coupling between Wnt signaling and the nematic field, plays a role in facilitating the robust yet highly flexible regeneration capabilities in Hydra. Interestingly, similar mechanochemical coupling has been shown to be important in regenerating jellyfish fragments, where reorganization of the muscle fibers into aster-shaped defects was shown to define structural landmarks for the formation of the Wnt signaling center at the future manubrium (Sinigaglia et al., 2020).

The proposed mechanochemical feedback seems particularly important in regeneration from excised tissue pieces. Unlike organizer grafting experiments or bud formation, where defect formation initiates in a region containing a pre-established morphogen peak (Hobmayer et al., 2000), regenerating fragments are not expected to possess a well-defined initial biochemical pattern. The excised tissue spans only a small fraction of the body length of the parent animal and folds so that its originally head- and foot-facing sides adhere to each other, making it unlikely that the folded spheroids contain a localized morphogen peak emanating from gradients in the parent animal. Similarly, the possibility that the head formation site is specified by a localized wound response (as could be the case in bisected animals; Cazet et al., 2021; Gufler et al., 2018) appears unlikely, since the excised tissue fragment is wounded from all directions and folds into a spheroid in which the closure region spans nearly half of the regenerating spheroid (Fig. 1). Nevertheless, our observations show that head formation occurs in a well-defined location that can be identified from the actin fiber pattern. Our ability to specify this location and identify a unique mechanical signature there highlights the strong coupling between mechanics and biochemical signaling associated with organizer formation.

In the context of regenerating fragments, we suggest that mechanical fields can play an instructive role in specifying the future head site (Fig. 7). However, importantly, the relation between the presence of nematic defects and the establishment of a head organizer is not a simple causal relationship. The proposed mechanism involves a feedback loop coupling the morphogen field and the nematic field, where neither is upstream or downstream from the latter (Fig. 7A). We suggest that, depending on the context, different aspects of the initial conditions can be more or less instructional in guiding the highly flexible yet robust regeneration process. The memory of polarity, even in small excised tissues (Javois et al., 1988; Shani-Zerbib et al., 2022), in this view, does not arise from a well-defined pre-pattern, but rather emerges dynamically from the amplification of initial biases that are present in the excised tissue. Future work is needed to identify what mediates the memory of polarity in Hydra, and how it guides the formation of a polar body axis in a robust manner.

This work is only an initial step in deciphering the mechanochemical coupling underlying axial patterning in Hydra. While our proposed mechanochemical model is consistent with various experimental observations and has attractive features, it is nonetheless still speculative. Future work is required to establish this model from a phenomenological point of view, by proving that it can predict the behavior of regenerating tissues from various initial conditions (Bailles et al., 2024 preprint; Livshits et al., 2017) and under different perturbations (Maroudas-Sacks et al., 2024; Ravichandran et al., 2025; Wang et al., 2020). From a mechanistic point of view, activation of contraction in the nematic actomyosin fibers can nicely account for the observed relationship between the nematic field and the tissue strain field (Fig. 5), yet the relationship between the strain field and morphogen production is still unclear. Tissue stretching has been shown in other systems to activate biochemical signaling (Martino et al., 2018). In particular, the Hippo-YAP pathway is known to respond to a diverse set of mechanical cues (Panciera et al., 2017), and is coupled to the Wnt signaling pathway (Azzolin et al., 2014). Similarly, while the Wnt pathway has been shown to influence actin dynamics (Stamatakou et al., 2015), how this could lead to nucleation and alignment of myonemes along morphogen gradients remains unknown.

Our suggested mechanochemical model for Hydra morphogenesis bears similarities with emerging views of plant morphogenesis, where mechanical forces have been shown to play an instructional part in patterning through mechanochemical feedback that couples cytoskeletal alignment, mechanical stress/strain fields and the distribution of morphogens such as auxin (Trinh et al., 2021). We believe that similar self-organized dynamics involving mechanochemical feedback between stresses generated by the actomyosin cytoskeleton and signaling pathways are broadly relevant for animal morphogenesis. More generally, the presence of feedback from the emerging structure into the patterning process itself naturally provides developing systems with immense flexibility, while simultaneously promoting the emergence of robust outcomes with well-defined patterns.

All the experiments were performed using a transgenic Hydra strain expressing Lifeact-GFP in the ectoderm, generously provided by Prof. Bert Hobmayer (University of Innsbruck, Austria). Additionally, we used a Wnt expression reporter strain, HyWnt3:GFP-HyAct:dsRED transgenic strain (Nakamura et al., 2011; Vogg et al., 2019), kindly provided to us by Prof. Brigitte Galliot's lab (Geneva University, Switzerland), for studying Wnt expression in fixed Hydra (Fig. 6A). Animals were cultured in standard Hydra culture medium (HM; 1 mM NaHCO₃, 1 mM CaCl₂, 0.1 mM MgCl₂, 0.1 mM KCl and 1 mM Tris-HCl at pH 7.7) at 18°C. The animals were fed three times per week with live Artemia nauplii and rinsed after 4-8 h. Tissue fragments were excised from the middle body section of mature Hydra, ∼24 h after feeding, using a scalpel equipped with a #15 blade. Tissue rings were excised by performing two nearby transverse cuts, and were subsequently cut into two to four parts by additional longitudinal cuts to obtain rectangular tissue pieces.

To label specific tissue regions, we used laser-induced activation of a caged dye (Abberior CAGE 552 NHS ester) that was electroporated uniformly into mature Hydra and subsequently uncaged in the desired region (Maroudas-Sacks et al., 2021; Shani-Zerbib et al., 2022). Electroporation of the probe into live Hydra was performed using a home-made electroporation setup. The electroporation chamber consisted of a small Teflon well, with two perpendicular platinum electrodes, spaced 2.5 mm apart, on both sides of the well. A single Hydra was placed in the chamber in 10 μl of HM supplemented with 6-12 mM of the caged dye. A 75 V electric pulse was applied for 35 ms. The animal was then washed in HM and allowed to recover for several hours to 1 day before tissue excision. After excision, the specific region of interest was activated by a UV laser in a LSM 710 laser scanning confocal microscope (Zeiss), using a 20× air objective (NA=0.8). The samples were first briefly imaged with a 30 mW 488 nm multiline argon laser at up to 1% power to visualize the Lifeact-GFP signal and identify the region of interest for activation. Photoactivation of the caged dye was carried out using a 10 mW 405 nm laser at 100%. The activation of the Abberior CAGE 552 was verified by imaging with a 10 mW 543 nm laser at 1%. Subsequent imaging of the Lifeact-GFP signal and the uncaged cytosolic label was carried out by spinning-disk confocal microscopy, as described below.

Intact Hydra from the HyWnt3:GFP-HyAct:dsRED transgenic strain were relaxed in 2% urethane in HM for 1 min and fixed in 4% formaldehyde in HM for 1 h at room temperature. Samples were permeabilized with 0.1% Triton-X100 in PBS (PBT; three washes×5 min) and then incubated for 2 h with 0.1% Triton-X100, 3% bovine serum albumin (BSA, w/v) in PBS (PBSBT). For staining, samples were incubated with an AlexaFluor 647-conjugated anti-GFP, rabbit polyclonal antibody (2 mg/ml; Invitrogen) and AlexaFluor 488-conjugated phalloidin (200 u/ml; Invitrogen) diluted 1:1:100 in PBSBT, and incubated overnight at room temperature with gentle shaking. Subsequently, samples were washed three times in PBT (three washes×15 min) and then in PBS (3 washes×5 min). Each sample was then placed in a cube of 2% low gelling agarose (Sigma) and the cube was placed on a coverslip with its head facing the objective in order to image the mouth region.

Time-lapse live imaging was performed either in HM or in soft gel (0.5% low gelling point agarose, Sigma) prepared in HM to reduce tissue movement during imaging. The general characteristics of the regeneration process in soft gel are similar to regeneration in aqueous media. Samples were made in 50 mm glass-bottom Petri dishes (Fluorodish), polymer coverslip 24-well plates (Ibidi µ-plate 24 well, tissue culture treated) or in custom-made sample holders with a coverslip bottom. The samples were placed in wells cast of 2% agarose (Sigma) prepared with HM. For experiments in gels, the regenerating tissues were placed in liquefied 0.5% low gelling agarose gel that was cool enough not to damage the samples (∼35°C) at ∼3-6 h after excision (to allow the tissue pieces to first fold into spheroids). The gel subsequently solidified around the tissue. We added a few ml of HM above the gel to ensure it did not dry out during imaging.

To image samples from all directions by spinning-disk confocal microscopy, regenerating tissues were loaded within soft gel into FEP tubes (which have a similar refractive index to the surrounding solution) with a square cross-section and an internal diameter of 2 mm (Adtech). As above, the samples were inserted into liquefied 0.5% low gelling agarose gel and positioned within the FEP tubes. During imaging, the tubes were manually rotated to each of the four flat facets of the square tube and secured using a homemade Teflon holder to keep them stationary at each orientation. Images from four directions were acquired at the specified time points.

Samples for the up-and-under spinning-disk confocal microscope were prepared as follows. When using two air objectives, the samples were sandwiched between two glass coverslips at a set distance apart and were sealed from the environment to prevent leaks and fluid evaporation. The samples were placed within 2% agarose wells, which were filled with 0.5% low gelling agarose gel around the samples to maintain an aqueous environment between the coverslips. When using an air objective from below and a dipping lens from above, the samples were placed in a 50 mm glass-bottomed Petri dish (Fluorodish) equipped with a homemade Teflon ring with tubing to allow perfusion of media to prevent evaporation. The samples were placed in wells made from 2% agarose that were filled with 0.5% low gelling agarose gel and the samples were layered with 3-4 ml of media on top of the gel. On the microscope stage, the small tubes from the Teflon ring were connected to a peristaltic pump (Ismatec) and media was allowed to flow slowly over the samples.

Samples for the light-sheet microscope were loaded in liquefied 0.5% low gelling agarose gel into a ∼1 cm long cylindrical FEP tube with an internal diameter of 2.15 mm (Zeiss Z1 sample preparation kit) and positioned along the central axis of the tube. The imaging was carried out through the FEP tubing, which has a similar refractive index to the surrounding solution.

Pharmacological inhibition of the Wnt pathway with iCRT14 was carried out as follows (Cazet et al., 2021). Parent animals were preincubated in 5 µM iCRT14 (Sigma, SML0203) in HM for 2 h before excision. Tissue fragments were cut and left to seal for ∼3 h in the same solution (5 µM of iCRT14 in HM). Tissue spheroids were then placed in wells prepared from 2% agarose containing 5 µM iCRT14, embedded in 0.5% low gelling agarose gel containing 5 µM of iCRT14, and further layered with 5 µM of iCRT14 in HM. The medium with iCRT14 was refreshed every 24 h.

Perturbations with isotonic media (Ferenc et al., 2021) were carried out as follows. Tissue fragments were excised and allowed to seal for 3 h in normal HM. Tissue spheroids were then placed in wells prepared from 2% agarose containing 70 mM sucrose (Sigma, 84097), embedded in 0.5% low gelling agarose gel containing 70 mM sucrose, and further layered with HM containing 70 mM sucrose with constant perfusion.

Spinning-disk confocal z-stacks were acquired on a spinning-disk confocal microscope (Intelligent Imaging Innovations) running Slidebook software. The Lifeact-GFP was excited using a 50 mW 488 nm laser and the activated Abberior CAGE 552 was excited using a 50 mW 561 nm laser. Images were acquired with an EM-CCD (QuantEM; Photometrix). Time-lapse movies of regenerating Hydra were taken using either a 10× air objective (NA=0.5), a 10× water objective (NA=0.45) or a 20× air objective (NA=0.75).

The up-and-under setup was a custom, double spinning-disk confocal microscope (Intelligent Imaging Innovations) running Slidebook software, which enables imaging of the sample from two opposing angles – above and below. The Lifeact-GFP was excited using a 50 mW 488 nm laser and the activated Abberior CAGE 552 was excited using a 50 mW 561 nm laser. Images were acquired with two sCMOS cameras (Andor Zyla 4.1). Time lapse movies of regenerating Hydra were taken using a 10× air objective (NA=0.5) or a 20×air objective (NA=0.75) from below, and a 10× air objective (NA=0.3) or a 20× dipping lens (NA=0.5) from above.

Light-sheet microscopy was carried out using a Lightsheet Z.1 microscope (Zeiss). The light-sheet was 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 was excited using a 50 mW 488 nm laser and the activated Abberior CAGE 552 was excited using a 50 mW 561 nm laser. Four views from different angles were acquired for the same sample by rotating the specimen. The imaging was 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 were still apparent in the images.

All 3D stacks in the spinning-disk microscopy systems and light sheet microscope were acquired at room temperature, typically taken at 3 μm z-intervals. The time resolution of the movies ranged from 20 s to 30 min, depending on the experiment. Imaging was carried out using appropriate emission filters at room temperature.

Time-lapse epifluorescence and bright-field movies of regenerating Hydra were recorded on a Zeiss Axio-Observer microscope with a 5× air objective (NA=0.25) at room temperature. Images were acquired on a charge-coupled device (CCD) camera (CoolSnap, Photometrix) and illuminated with a Xenon lamp. Time lapse imaging begins ∼4.5 h after excision and continues for 3 days at a time interval of 10 min.

The tools used for image processing and analysis are based on a combination of custom-written code with adaptation and application of existing algorithms, written in MATLAB, Python and ImageJ, as detailed below. The codes are available in Github (https://github.com/KinneretLab).

In order to define the image region for analysis, masks were generated for every image based on the maximum intensity projections of the Lifeact-GFP signal, using automatic thresholding in ImageJ (‘Li’ method), and performing morphological operations (erosion, dilation and hole-filling) in MATLAB to obtain masks that accurately mark the tissue region in the image. All subsequent analysis was performed for the regions within the tissue masks.

The regenerating Hydra tissue spheroids consisted of a bi-layered epithelium surrounding a hollow cavity. The 2D apical and basal surfaces of the ectoderm were computationally identified in the 3D fluorescence z-stacks of the Lifeact-GFP signal in the ectoderm. The supracellular actin fibers reside on the basal surface of the ectoderm, while the apical junctions marking the cell outlines are visible on the apical surface. 2D projected images of the basal and apical surfaces of the ectoderm were automatically extracted from the 3D spinning-disk z-stack acquired with a 3 μm z-interval using the ‘Minimum Cost Z surface Projection’ plug-in in ImageJ (https://imagej.net/Minimum_Cost_Z_surface_Projection). The cost images were generated by pre-processing the original z-stacks using custom-code written in MATLAB. First, the signal from the ectoderm layer was manipulated to make it more homogeneous within the layer without increasing its thickness, by applying the built-in MATLAB anisotropic diffusion filter. Subsequently, we employed 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 were determined using the minCost algorithm (Parameters used: Rescale xy, 0.5; rescale z, 1; min distance, 15 μm; max distance, 45 μm; max delta z, 1; two surfaces). The surfaces given by the minCost algorithm were then smoothed by applying an isotropic 3D Gaussian filter of 1-3 pixels in width (after rescaling to an isotropic grid matching the resolution in the z-direction) and selecting the iso-surface with value 0.5.

2D projected images of the ectodermal actin fibers that reside on the basal surface of the ectodermal layer and the apical junctions, which are visible on the apical surface, were generated by extracting the relevant fluorescence signal from the 3D image stacks based on the smoothed basal and apical surfaces determined above, respectively. For each x-y position, we employed a Gaussian weight function in the z-direction with a sigma of 3 μm, which was centered at the z-value determined from the smoothed surface. This procedure was repeated for surfaces positioned 1-3 z-intervals in each direction along the z-axis to obtain additional 2D projected images at a small offset from the detected surface. We then manually selected the offset range that best showed the signal for the basal/apical surface and created a maximum intensity projection of these projected images. This last step was beneficial for improving the surface projection in regions closer to the edges of the tissue, where the tissue was curved at a steep angle relative to the imaging plane, and the surface detection could be less reliable. The resulting 2D projected images were further subject to contrast limited adaptive histogram equalization (CLAHE) with MATLAB's ‘adapthisteq’ function with Rayleigh distribution and a tile size of 26 μm. Finally, the binary mask was applied again to the adjusted images.

2D projections of the photoactivated dye were 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, within the masked region.

The local orientation of the supracellular ectodermal actin fibers is 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 is determined from the signal intensity gradients in the 2D projected images of the ectodermal actin fibers, as described by Maroudas-Sacks et al. (2021). In brief, we first calculated a raw orientation field θ_ij (where i,j refer to the indices of the original image) from the image intensity gradients in a box of size ∼6 μm around each pixel. The raw orientation field was then smoothed using a Gaussian filter with σ=4 μm. The coherence was calculated from the raw orientation field before smoothing as |cos(θ−θ_ij)|, where the averaging was carried out over a square window of ∼15 μm, providing a measure of the local variation in the orientation field. We used this measure to distinguish between regions with low fiber alignment (whether due to lack of parallel supracellular fibers or due to limitations in the imaging), and regions with high coherence having a clear array of nematic fibers. The nematic order parameter was defined as ⁠, and was calculated from the smoothed orientation field and averaged over a larger window of ∼25 μm. The order parameter provides a measure of the orientational order of the nematic array, reaching maximum values in regions with parallel alignment and minimum values at defect cores.

Analysis of defect sites and their outcome in the regenerated animal was performed manually by identifying and following defects over time in 2D projections on the basal ectodermal surface 3D from fluorescent time-lapse movies of LifAact-GFP in regenerating Hydra. The defect sites included in the analysis were those that can be reliably tracked throughout the regeneration process from their formation until the final morphological outcome.

The contrast-enhanced apical surface images were segmented using EPySeg, a machine-learning based segmentation tool (Aigouy et al., 2020), applying a custom-trained model using hand-segmented images of cell junctions as the ground truth for training. The resulting output is a binary segmentation mask for each image marking all cell outlines. The automatic segmentation was manually corrected using ‘Tissue Analyzer’ (Aigouy et al., 2020), also used for manual segmentation of the training set. Based on the segmentation masks, two additional output images were created for each original image using custom-written python code: an image marking all cell vertices and an image marking all cell bonds, with each bond uniquely numbered (matching the output format of ‘Tissue Analyzer’).

The segmentation masks, vertices and bond images hold the information on cell geometries and neighbor relations in the projected 2D image. This information is extracted from the images using custom-written MATLAB code, and stored in a relational database format matching that of TissueMiner software (Etournay et al., 2016), with some modifications, as detailed here. We apply a geometrical correction to the projected data to account for the curvature of the surface on which the cells reside, using the height map of the apical surface (see above). Since cell size is small relative to surface curvature, we assume each cell to be roughly planar in the plane tangent to the surface at the location of the cell. To determine the tangent plane for each cell, we use the mean normal to the surface over a window of width 20 μm around the cell center. For each cell, we project the full cell outline from the segmented images onto the plane normal to the mean normal of the cell and subsequently smooth the resulting outline using a five-point moving average before calculating bond lengths and cell areas to avoid artefacts due to discrete pixel resolution. 3D coordinates for cell centers, vertices and bonds (both original and smoothed), as well as neighbor relations, are stored in the database.

To query the database, quantify and visualize the data, we used a custom-written object-oriented MATLAB platform. The full documentation can be found on the lab Github repository (https://github.com/KinneretLab). All geometrical measures for cells were calculated using the cell outlines on the tangent planes. Cell geometrical measures that were calculated include area (inside the smoothed outline of the cell), perimeter (length of the smoothed outline), and cell shape anisotropy Q, defined by Merkel et al. (2017). Briefly, the cell shape anisotropy tensor Q was calculated by dividing each cell into triangles that connect each pair of adjacent vertices with the cell center, and then calculating for each triangle how it differed from an equilateral triangle aligned along a set reference axis to obtain the anisotropy tensor for each triangle. For each cell, Q was then calculated by taking an area-weighted average over the triangles from which it is comprised. To calculate cell shape anisotropy along the fiber orientation, the cell shape tensor Q was then projected along the direction of fiber orientation of the cell. To calculate the cell shape anisotropy relative to the direction to the defect, we drew a line between the cell and defect in the 2D projected image, projected this orientation to the cell plane, and projected Q along this direction. Fiber orientation and coherence per cell were assigned by taking the mean smoothed orientation and coherence field of all pixels within the cell outline. The mean fiber orientation was projected to the tangent plane of the cell. A coherence threshold of 0.88 was used to discriminate between cells with low coherence, indicating a lack of clearly visible, aligned, supracellular fibers and cells with well-defined nematic fibers. This coherence threshold was used for visualization purposes only, whereby the cellular nematic was depicted in gray for cells with an average coherence below the threshold and in black for cells with higher coherence in Fig. 3 and Movies 3 and 4. The distance of cells from defects was determined based on the graph distance, which is the minimal number of cells connecting a given cell and the cell at the center of the defect.

To calculate the logarithmic area strain at a given distance from the defect, we subtracted the mean logarithm of the cell areas of all cells at that distance before the event, from the mean logarithm of the cell areas of all cells at that distance during the peak of the event. This is mathematically equivalent to calculating the mean of the logarithm of the ratio of cell area during the peak and before the event for each cell. The last frame before the deformation was determined manually, and the peak of an event was determined as the frame in which the total area of the cells within a distance of three cells from the defect was largest.

Analysis of local tissue deformation and rupture events was performed manually by identifying frames within time-lapse movies in which these events occurred, using the projected images of both the apical and basal surface of the ectoderm. Events were defined as coherent tissue deformation involving dozens of cells. We recorded all observed events and classified them according to the pattern of deformation and its localization. Since local stretching and ruptures typically occurred simultaneously at both actin foci, more than one event was often recorded in different tissue regions at the same time. The categories used to classify events were as follows: small local stretching (cell area increase of less than twofold), large local stretching (cell area change of twofold or more), small ruptures, large ruptures (hole diameter∼tissue diameter), global contraction (large cell deformation in the direction perpendicular to fibers in the ordered region between the foci), and other cell shape deformations not involving area change (characterizes cell shape changes at 2×½ defect sites during global contractions). The criteria for defining a hole included the following: a transient, visible gap in fluorescence in both the Lifeact-GFP signal and in an additional tissue label, if present; a change in tissue volume before and after the hole appearance; and/or visible expulsion of tissue debris from the hole. Timing and location of the observed deformation and rupture events were recorded, and the sites were followed over time throughout the regeneration process (using fiduciary landmarks in the form of local photoactivated fluorescent tissue labels) to determine the fiber organization and morphological outcomes at these sites. To obtain statistics for the cumulative event distribution over time, time-lapse movies were taken in the up-and-under system to simultaneously image the samples from above and below with a time resolution of 2-2.5 min, which is shorter than the characteristic duration of an event (Fig. 2F,I).

In our custom vertex model of Hydra tissue mechanics, individual cells are defined by the position of their vertices. Since these vertices are, in general, not all in the same plane, we need to define the area and local tangent plane for each cell (Fig. S9A). To this end, we first determined the cell center, ⁠, as the mean position of all the cell vertices. We then defined a collection of cell triangles that each contained two neighboring vertices and the cell center. The cell area is defined as the sum of all cell triangle areas. The normal vector to the cell tangent plane is defined as the area weighted average of all the cell triangle normal vectors (see supplementary Materials and Methods for more details).

The cellular network evolves according to the forces acting on the vertices. If at any time-step of the simulation, the length of any bond becomes smaller than a threshold value ⁠, the two vertices of that bond are merged into a single vertex and the bond disappears. In this way, fourfold vertices are formed from merging of two threefold vertices. Such fourfold vertices can resolve back into pairs of threefold vertices in one of two ways: reverting back to the original cellular configuration or rearranging the cells. A cell rearrangement event, also called a T1 transition, consists of the disappearance of one cell bond by the transient formation of a fourfold vertex, which subsequently resolves into a new bond shared by cells that were previously not in contact. The stability of a fourfold vertex is tested with respect to each of these two possible resolutions (see supplementary Materials and Methods). Note that our model supports vertices with an arbitrary large number of bonds, but in practice we observe only fourfold vertices in our simulations.

To simulate the focusing of isotropic strain during stretching events (Fig. 5), we used a fixed nematic pattern that emulates the experimentally observed patterns of fiber alignment (Fig. 5B,C, left panels), and followed the vertex dynamics through global fiber contraction activation. Since we do not observe cell rearrangements in the Hydra ectoderm during stretching events, we chose parameters of the vertex model for which the cellular network has a negative two-dimensional Poisson ratio (see supplementary Materials and Methods). This allows us to generate large isotropic strains without inducing cell rearrangements. Furthermore, we take the cell active stress amplitude to be equal to ⁠, such that the isotropic strain profile we obtain in the simulation of a single event is comparable to the one experimentally observed in Hydra (Fig. 3; Figs S4D, S5).

To describe the alignment of the nematic field to the local morphogen gradient, we first had to calculate the gradient on the curved surface. As mentioned above, we assumed that the intra-cellular diffusion was fast compared to inter-cellular diffusion. To estimate the overall morphogen gradient across a cell, we had to take into account the morphogen concentrations in neighboring cells (Fig. S9D). For each cell, we aimed to find a gradient vector that satisfies relations for all neighboring cells c′. However, in general, these relations cannot all be satisfied with a single vector and, so we defined the morphogen gradient to be the vector ⁠, which produces the least-squared error of these relations (see supplementary Materials and Methods for technical details). Finally, due to the nematic symmetry, the nematic will align only to the axis of the morphogen gradient, independent of the polarity of the gradient along this axis. For this reason, the nematic aligns to a tensor constructed as the traceless-symmetric component of the outer product ⁠. The parameter α characterizes the strength of alignment of the nematic to the morphogen gradient.

The choice for all the model parameter values used in the simulations and the rationale behind these choices are presented in the supplementary Materials and Methods.

We thank Erez Braun for many valuable discussions and comments on the manuscript. We thank Gidi Ben-Yoseph for excellent technical help and Lital Shani-Zerbib for her contributions to the development of the image analysis pipeline. We thank Alexandra Schauer, Jana Fuhrmann, Alex Mogilner, Aurelien Roux and Ram Adar for their comments on the manuscript. We thank Ben Atkinson from Intelligent Imaging Innovations for his help in designing the up-and-under spinning disk microscopy system, Nitzan Dahan from the LS&E Imaging Center for advice and help in imaging, Eran Kafri for help in 3D visualization, and the participants of the UCSB 2023 QBio course and the KITP program for discussions and providing a stimulating environment. K.K. dedicates this work to the memory of Naomi Lindenstrauss.

The peer review history is available online at https://journals.biologists.com/dev/lookup/doi/10.1242/dev.204514.reviewer-comments.pdf