The elongating body axis of Xenopus embryos is a long-standing and powerful model system for studying PCP-dependent vertebrate CE (Figure 1—figure supplement 1; Huebner and Wallingford, 2018). Xenopus CE can be considered most simply in terms of four-cell neighbor exchanges in which mediolaterally aligned cell-cell junctions (‘v-junctions’) shorten, followed by the elongation of new, perpendicularly aligned junctions (‘t-junctions’)(Figure 1A). To gain deeper insights into this process, we used high-speed super-resolution imaging to establish a quantitative physical description of the motion of tricellular vertices bounding v-junctions (Figure 1B).

First, we found that v-junction shortening was dominated by the movement of a single ‘active’ vertex, while the other ‘passive’ vertex moved comparatively less (Figure 1C,D), similar to the asymmetry observed previously during CE in Drosophila epithelial cells (Vanderleest et al., 2018). To quantify the asymmetric movement of vertices in Xenopus mesenchymal cells, we used a fixed coordinate system and defined an activity parameter, A, as the ratio of the net distance moved by the vertices to the initial junction length, (Figure 1C; Figure 1—figure supplement 2A) (Appendix, Section 1). This analysis demonstrated significant differences in the motion of active and passive vertices (Figure 1D).

We then explored the physical basis of asymmetric motion in active and passive vertices using mean squared displacement (MSD)(SI Section 2). Our analysis revealed that active vertices consistently displayed a highly fluidized movement (i.e. super-diffusive)(Figure 1E,F, red). By contrast, passive vertices displayed the hallmarks of more-constrained, glass-like motion (i.e. defined by sub-diffusive movement with an intermediate time slowdown, as observed in colloidal systems); (Kegel and van Blaaderen, 2000; Weeks et al., 2000; Figure 1E,F, blue). The juxtaposition of liquid- and glass-like motion along a single cell-cell junction was interesting, because while fluid-to-glass phase transitions are known features at tissue-level length scales (Angelini et al., 2011; Bi et al., 2015; Malmi-Kakkada et al., 2018; Sinha et al., 2020), such transitions have not been reported at sub-cellular length scales during morphogenesis.

Given these surprising results, we also controlled for the possibility that image drift may interfere with our quantification. To this end, we quantified the motion of active and passive vertices using two relative reference frames with respect to slowly moving landmarks within the tissue (Figure 1—figure supplement 2D–F). All three quantification strategies demonstrate that the asymmetry we observed was not a point-of-reference artifact (Figure 1—figure supplement 2D–G).

Finally, we confirmed the distinct patterns of motion in active and passive vertices using four additional physical metrics, the Van Hove function, the velocity auto-correlation function, the self-overlap parameter, and the fourth order susceptibility, ${\chi }_4\left(t\right)$ (SI sections 2-3). All four orthogonal approaches confirmed our finding that the active and passive vertices bounding individual v-junctions exhibit asymmetric dynamic behaviors, with one vertex displaying a fluid-like motion and the other, glass-like (Figure 1—figure supplement 3). Critically, this asymmetric behavior was specific to shortening dynamics of v-junctions, as the two vertices bounding non-shortening junctions in the same tissue were consistently symmetrical, both resembling passive vertices (Figure 1E,F, pink, black; Figure 1—figure supplement 3).

This physical analysis provided three important insights: First, glass-like dynamics previously observed only at tissue-length scales in morphogenesis also exist at the subcellular length scale of individual junctions. Second, the frequently invoked assumption of mechanical homogeneity along single cell-cell junctions, which underlies a wide swath of the biophysical work on morphogenesis, may not be valid. And finally, because only shortening junctions exhibited local mechanical heterogeneities, this phenomenon may be a specific and essential feature of convergent extension.

The possibility of mechanical heterogeneity along single cell-cell junctions has important implications, as many biophysical approaches and in silico tools for understanding morphogenesis (e.g. laser cutting, vertex models) assume that junctions are mechanically homogeneous along their length. We therefore developed a new theoretical framework for junction shortening that accommodates the possibility of local mechanical heterogeneity by independently modeling the movement of each vertex (Figure 2A,B; Appendix, Section 4–8).

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Our model involves (i) a local junction stiffness (or elasticity) modeled using a spring element, which is consistent with the pulsatile relaxation of v-junctions observed in Xenopus CE (Shindo and Wallingford, 2014) (ii) a dynamic rest length, recently shown to be important for modeling CE (Shindo et al., 2019) (iii) a viscoelastic parameter, $k/\gamma$, dictated by the spring stiffness, $k$, and the friction at the vertices,$\gamma$; and (iv) a rest length exponent,$\psi$, which describes the time dependence of plastic displacement of the vertices modeled with a piston (Figure 2A,B; Appendix, Section 4-8).

Using this model, we explored parameter space to find variables in elastic and viscous deformation that can support effective shortening of the junction (Appendix, Sections 7, 8). As shown in the heatmap in Figure 2C, for a given rest length exponent, junctions failed to shorten if the viscoelastic parameter was equal and small for both the vertices (Figure 2C, red box; Appendix, Sections 7, 8). When the viscoelastic parameter was asymmetric, junctions shortened effectively (Figure 2C, gold box). Thus, at the level of binary outcome (i.e. shorten versus fail-to-shorten), our model suggests that CE requires mechanical heterogeneity along single v-junctions.

For a more stringent test, we compared the temporal dynamics of junction shortening in our model to those quantified in vivo from high-speed super-resolution movies (Figure 2D,E). In time-lapse data, the relaxation behavior of v-junctions collapsed into a self-similar pattern when normalized; relaxation became progressively more efficient over time and could be described by a compressed exponential (Figure 2G, black lines; Figure 2—figure supplement 1A,B; Appendix, Sections 9). When the viscoelastic parameters in our model were asymmetric, the shortening dynamics closely recapitulated this compressed exponential relaxation (Figure 2G, gray lines, I; expanded view in Figure 2—figure supplement 1A,B; Appendix, Sections 8–10).

Finally, we also analyzed junction length dynamics in non-shortening junctions in vivo (Figure 2D,F), because unlike shortening junctions, these display symmetrical mechanics along their length (Figure 1E, black, pink). Defining non-shortening junctions as any that displayed no net reduction in length over the observation time scale of ~400 s, we found that the length dynamics of non-shortening junctions in vivo displayed wide fluctuations over time. Moreover, non-shortening junctions did not share a self-similar relaxation pattern and displayed large deviations from the compressed exponential (Figure 2H, black, I; Appendix, Sections 8–10). Likewise, when symmetric viscoelastic parameters ($k/\gamma$) were input into the model for both vertices bounding a single junction, the resulting junction length dynamics displayed wide fluctuations in length and deviated substantially from the normal relaxation pattern (Figure 2H, gray; I; Appendix, Sections 8–10).

Thus, by incorporating local mechanical heterogeneity, our new model not only recapitulates overall shortening/non-shortening outcomes, but also quantitatively recapitulates the dynamic patterns of length change observed in both shortening and non-shortening junctions in vivo. Because both modeling and observations suggest a key role for mechanical heterogeneity, we next sought to understand the contribution of such local mechanical regimes to cell movement during CE.

Our theory makes a prediction: that the more fluid-like motion of the active vertex occurs in the context of increased local stiffness (i.e. higher viscoelastic parameter), while the more glass-like motion of the passive vertex occurs in a relatively decreased stiffness regime. Given that vertex movement, while highly directional, is not entirely directed along the line joining two vertices (in-line movement) (Figure 3A), we reasoned that a stiffer mechanical regime might limit the movement of the active vertices perpendicular to the in-line direction; the perpendicular motion is referred to as transverse movement (Figure 3A, green). Transverse movement is indeed limited for active vertices thereby resulting in more smoothly processive, fluid-like motion in the line of shortening as compared to passive vertices (Figure 3A, orange).

Figure 3

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To test our model’s prediction in vivo, we used our time-lapse data to quantify the transverse fluctuations of vertices (Figure 3B, green arrows; Appendix, Section 12). Consistent with our model’s prediction, active vertices displayed significantly less transverse fluctuation than did passive vertices at the same junctions (Figure 3B,C,E), indicating a higher local stiffness at active vertices (Marmottant et al., 2009). Analysis of the straightness index, quantifying how straight vertices move along the in-line direction, independently validated this conclusion (Figure 3E,F; Appendix, Section 12).

This analysis of in vivo imaging data validates our physical model’s prediction of an increased stiffness regime near active junctions and suggests that the lower stiffness regime of passive vertices allows more transverse fluctuation, resulting in less-directed, more glass-like movement. As such, multiple independent lines of observation and theory suggest that local mechanical heterogeneity along cell-cell junctions is a fundamental feature of CE. We next sought to understand the molecular underpinnings of this feature, asking if patterns of protein localization during CE might reflect the local mechanical patterns we identified here.

We first considered that the observed mechanical asymmetry along shortening v-junctions might result from asymmetric distribution of actomyosin, for example from asymmetric cellular protrusions from neighboring cells or asymmetric junction contraction events along v-junctions. We therefore measured actin intensity in the region abutting active and passive vertices, but we observed no such asymmetry (Figure 4—figure supplement 1A–C).

We next turned our attention to cadherin cell adhesion proteins, which have been shown to tune the very local mechanics of individual cell membranes in culture (Strale et al., 2015). We specifically examined Cdh3, as it is essential for CE in Xenopus (Brieher and Gumbiner, 1994; Fagotto et al., 2013; Figure 4—figure supplement 1D) and was recently implicated in CE cell movements in the mouse skin (Cetera et al., 2018). Like all classical cadherins, Cdh3 forms both intercellular trans-dimers and also cis-clusters mediated by intracellular interactions (Figure 4A; Yap et al., 1997). Such cis-clustering is a key regulatory nexus for cadherin function (Yap et al., 2015), so it is interesting that while the mechanisms governing formation of cadherin cis-clusters during CE has been studied, cis-cluster function during CE remains unknown (e.g. Levayer and Lecuit, 2013; Truong Quang et al., 2013).

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We used high-speed super-resolution microscopy to image a functional GFP-fusion to Cdh3 and used the spatial autocorrelation function for an unbiased quantification of Cdh3-GFP cluster size (Figure 4B)(Appendix, Section 13). Using this function, an exponential decay in spatial correlation is expected for clusters that are regularly ordered, and this pattern was observed for Cdh3-GFP (Figure 4C). Moreover, this decay reached zero at ~1 μm (Figure 4C), consistent with the size reported for cis-clusters of cadherins in vertebrate cell culture (Yap et al., 2015).

This analysis revealed that together with pulsatile junction shortening (Figure 4D; Shindo and Wallingford, 2014) Cdh3 clusters undergo dynamic fluctuations in size (Figure 4E). Moreover, fluctuations in mean Cdh3 cluster size significantly cross-correlated with shortening pulses (Figure 4F). Mean cluster size peaked ~20 s prior to the onset of junction shortening pulses (Figure 4E,F)(Appendix, Section 14,15), suggesting a functional relationship between Cdh3 clustering and junction remodeling.

We then reasoned that mechanical heterogeneity observed along cell-cell junctions during CE might be driven by local patterns of Cdh3 clustering, since cadherins can tune the local mechanics of free cell membranes in single cultured cells (Strale et al., 2015). This led us to measure Cdh3 cluster size specifically in the ~3 micron region abutting vertices of shortening v-junctions during shortening pulses. Patterns of Cdh3 clustering were complex and highly heterogeneous, consistent with the mechanical heterogeneities we report here (Figure 4—figure supplement 2). Nonetheless, the mean size of Cdh3 clusters near active junctions was significantly larger than that for clusters near passive vertices (Figure 4G; Appendix, Section 16). We confirmed this important result using an alternative quantification of cluster size involving fits to the exponential decay of the spatial autocorrelation (Figure 4—figure supplement 1E–G; Appendix, Section 16).

Importantly, asymmetric Cdh3 clustering was specific to shortening v-junctions and was not observed along non-shortening junctions in the same tissue. Rather, all vertices bounding non-shortening junctions displayed clustering similar to that near passive vertices in shortening junctions (Figure 4G). Symmetrical clustering in non-shortening reflects the symmetrical dynamics of vertices bounding these junctions, described above (Figure 1E). Accordingly, these results demonstrate that asymmetric cis-clustering of Cdh3 is a specific property of shortening v-junctions during CE and suggests that such clustering may drive the asymmetric mechanics of active and passive vertices that we observed in vivo and predicted in silico.

The patterned, asymmetric cis-clustering of Cdh3 during CE is a significant finding, because as mentioned above the function of cis-clustering remains undefined not only for CE, but indeed in any in vivo context. We therefore took advantage of point mutations in Cdh3 that specifically disrupt the hydrophobic pocket that mediates cis clustering, without affecting trans dimerization (cisMut-Cdh3; Figure 5A; Harrison et al., 2011; Strale et al., 2015). To test this mutant in vivo, we depleted endogenous Cdh3 as previously described (Figure 5—figure supplement 1; Ninomiya et al., 2012), and then re-expressed either wild-type Cdh3-GFP or cisMutant-Cdh3-GFP.

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We first confirmed the cis mutant’s impact on clustering in vivo. Re-expressed wild-type Cdh3-GFP clustered normally and displayed the expected exponential decay in spatial autocorrelation that indicates regular spatial order and a mean cluster size ~1 μm (Figure 5B,D) (Appendix, Section 13,17). By contrast, when cisMut-Cdh3-GFP was re-expressed, clusters were clearly absent, and the signal was diffuse along cell-cell junctions (Figure 5C). Moreover, the spatial autocorrelation of cisMut-Cdh3-GFP did not decay exponentially (Figure 5D), consistent with a lack of spatial order (Appendix, Section 13,17). We confirmed this result using fits to the exponential decay of the spatial autocorrelation (Figure 7—figure supplement 1) (Appendix, Section 13,17).

We next used the same replacement strategy to directly test the function of cis-clustering in Xenopus CE. At neurulation stages, embryos depleted of Cdh3 display severe defects in axis elongation (Figure 5E,F,I, green) (Brieher and Gumbiner, 1994; Lee and Gumbiner, 1995). At later stages, these embryos disassociate to individual cells due to the widespread requirement for Cdh3 in cell cohesion (Ninomiya et al., 2012; Figure 5J, green). We found that re-expression of wild-type Cdh3-GFP rescued both axis elongation and embryo integrity, as expected (Figure 5G,I,J, purple).

Strikingly however, while re-expression of cisMut-Cdh3-GFP significantly rescued overall embryo integrity (Figure 5J, red), it failed to rescue axis elongation (Figure 5H,I, red). These data provide the first experimental test of the role of cadherin cis-clustering in vivo, and moreover, provide an experimental entry point for testing the role of cis-clustering in the generation of local mechanical patterns along cell-cell junctions.

To understand the relationship between Cdh3 clustering (Figure 4) and the asymmetric mechanics and vertex dynamics of shortening v-junctions (Figures 1–3), we applied our battery of physical methods to quantify the motion of vertices in cells with disrupted Cdh3 cis-clustering (i.e. Cdh3 knockdown +cisMut-Cdh3 re-expression). We found first that defects in axis elongation in cisMut-Cdh3 expressing cells were accompanied by defects in cell polarization (Figure 6A–C), reflecting the phenotype seen when PCP signaling is disrupted (Wallingford et al., 2000). Second, v-junctions in cells with disrupted Cdh3 clustering displayed large fluctuations in length that deviated significantly from the compressed exponential relaxation pattern observed for normal v-junctions (Figure 6D,E). The aberrant length dynamics of cisMut-Cdh3 expressing junctions resembled those of junctions that lack mechanical heterogeneity (i.e. non-shortening junctions in normal embryos in vivo or those modeled in silico (compare Figure 6D with Figure 2H)).

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We then asked if cisMut-Cdh3 expression also disrupted the normal mechanical heterogeneity of v-junctions by quantifying transverse fluctuations of vertices. We found that all vertices in cells with defective cis-clustering of Cdh3 displayed the elevated transverse fluctuations observed only in passive vertices of normal cells (Figure 6F,G). These results provide direct experimental evidence that Cdh3 cis-clustering restricts transverse movement of vertices, thereby facilitating fluid-like shortening of the junction.

A key challenge in animal morphogenesis is to understand how ubiquitous cellular machinery such as cadherin adhesion is directed by tissue-specific developmental control mechanisms. PCP signaling is a central regulator of vertebrate CE and PCP proteins localize to shortening v-junctions during Xenopus CE (Figure 7A), where they control actomyosin contractility (Butler and Wallingford, 2018; Shindo et al., 2019), but how these systems interface with cadherin adhesion during CE is poorly defined. Because cells with disrupted Cdh3 cis-clustering superficially resemble those with defective PCP (Figure 6B,C), we asked if Cdh3 clustering may be under the control of PCP signaling.

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We disrupted PCP with the well-characterized dominant-negative version of Dvl2, Xdd1, which severely disrupted cell intercalation behaviors as expected (Wallingford et al., 2000; Figure 6C). Strikingly, expression of Xdd1 also elicited a significant disruption of Cdh3 clustering that was apparent in both images and in the lack of exponential decay in spatial autocorrelation data (Figure 7B,C). Finally, Xdd1 expressing junctions also displayed exaggerated length fluctuations, significant deviation from the compressed exponential relaxation behavior, and symmetrical, elevated transverse fluctuations (Figure 7D–F), all features associated only with junctions lacking local mechanical heterogeneity in vivo or in silico.

These data not only provide an independent experimental confirmation of the link between Cdh3 cis-clustering, local mechanical heterogeneity of junctions, and asymmetric vertex dynamics (Figure 7G), but also provide a novel mechanistic link between a conserved and essential developmental regulatory module (PCP), and the ubiquitous machinery of Cadherin adhesion.

Ovulation was induced by injection of adult female Xenopus with 600 units of human chorionic gonadotropin and animals were kept at 16°C overnight. Eggs were acquired the following day by squeezing the ovulating females and eggs were fertilized in vitro. Eggs were dejellied in 3% cysteine (pH 8) 1.5 hr after fertilization and embryos were reared in 1/3X Marc’s modified Ringer’s (MMR) solution. For microinjection, embryos were placed in 2% ficoll in 1/3X MMR and then washed in 1/3X MMR after injection. Embryos were injected using a Parker’s Picospritizer III with an MK1 manipulator. Embryos were injected in the dorsal blastomeres at the four cells stage targeting the presumptive dorsal marginal zone. Keller explants were excised at stage 10.25 in Steinberg’s solution using eyelash hair tools.

The Cdh3 morpholino had been previously described (Ninomiya et al., 2012) and was ordered from Gene Tools. Cdh3-GFP, (Pfister et al., 2016) lifeact-RFP, and membrane-BFP were made in pCS105 and Xdd1 was made in CS2myc (Sokol, 1996). Cdh3 antibody was ordered from Developmental Studies Hybridoma Bank (catalog number 6B6). The Cdh3-cis-mutant was generated using the Q5 Site-Directed Mutagenesis Kit (NEB, catalog number A13282) and here we changed valine 259 to aspartic acid and isoleucine 353 to aspartic acid.

Capped mRNA was generated using the ThermoFisher SP6 mMessage mMachine kit (catalog number AM1340). mRNAs were injected at the following concentrations per blastomere, Membrane-BFP (100 pg), Cdh3-GFP for imaging (50 pg), Cdh3-GFP for rescue (300 pg), Cdh3-cis-mutant (300 pg), lifeact-RFP (100 pg), and Xdd1 (1 ng). Cdh3 morpholino was injected at a concentration of 10 ng per blastomere.

Explants were mounted on fibronectin coated glass coverslips in either Steinberg’s solution or Danilchik’s for Amy solution. Experiments were repeated in the absence of fibronectin to ensure fibronectin did not confound results. Explants were incubated at room temperature for 4 hr or at 16°C overnight before imaging. Standard confocal images were acquired with either a Nikon A1R or a Zeiss LSM 700. Super-resolution images were acquired with a commercially available instantaneous structured illumination microscope (BioVision Technologies). Standard confocal time-lapse movies were acquired with a 20 s time interval and super resolution images were acquired with a 2 s time interval. All images were acquired at a z-depth of 5 μm above the coverslip to insure similar z-depth sampling between images.

All image analyses were performed using the open-source image analysis software Fiji (Schindelin et al., 2012). Images were first processed with 50-pixel rolling ball radius background subtraction and smoothed with a 3 × 3 averaging filter, which allowed better distinction of individual cadherin clusters. The segmented line tool, with width set to the thickness of the junction (~16 pixels), was used to set a line of interest (LOI) across the length of the cell junction. Next the multi-plot tool was used to extract cdh3 intensity values across the length of the cell junction and the measure tool was used to collect data such as junction length and mean intensity values. The Fiji Time Lapse plugin Line Interpolator Tool was used to make successive measurements for movies. Here a segmented line LOI was drawn every 10–30 frames, the line interpolator tool was then used to fill in the LOIs between the manually drawn LOIs allowing rapid semi-manual segmentation. The multi-plot tool and measure tool were then used to extract data for each time-point of the movie. Source data for all imaging experiments can be found in the Dryad Server (doi: 10.5061/dryad.pg4f4qrph).

Samples were prepared by micro-dissection as described above and incubated at room temperature for 4 hr or overnight at 16°C. Samples were then fixed in 1x MEMFA for 1 hr at room temperature and washed three times with PBS to remove fixative. Next samples were permeabilized with 0.05% Triton X-100 in PBS for 30 min and then blocked in 1% normal goat serum (NGS) in PBS for 2 hr at room temperature. The primary antibody was then diluted 1:100 in fresh 0.1% NGS/PBS and samples were incubated with primary antibody at 4°C overnight. Samples were then blocked a second time at room temperature for 1 hr and then washed twice with fresh blocking solution. Secondary antibody (goat anti-Mouse 488, #A32723) was diluted 1:500 and samples were incubated at 4°C overnight. Finally, samples were washed three times in 1X PBS and imaged.

Embryos were injected in the dorsal blastomeres with Cdh3-MO, Cdh3-MO + Cdh3 GFP (rescue), Cdh3-MO + Cdh3 cis-mutant (mutant),or left as un-injected controls. Live embryos were kept at room temperature for 26 hr post fertilization (~stage 33). Embryos were then fixed with MEMFA in glass vials on and rotated for 1 hr at room temperature. Post fixation samples were washed three times in 0.1% Tween-20 in 1X PBS and then images of embryos were acquired using a Zeiss AXIO Zoom stereoscope. The embryos anterior-posterior length and dorsal-ventral width were then measured using Fiji.

Embryos were injected in the same manner as done for the length to width measurement and the number of embryos injected for each condition was recorded. Embryos were then kept at room temperature for 20 hr (~stage 20) and the number of surviving embryos was recorded. The percentage of embryos surviving (embryo integrity) was reported.

Embryos were injected at the four-cell stage in a single dorsal blastomere with Cdh3-MO + membrane BFP generating embryos with mosaic knockdown of Cdh3 on the dorsal side of the embryo. Explants were next dissected from embryos, immuno-stained for Cdh3, and images were acquired as described above. The mosaic labeling allowed us to compare wild type and Cdh3-KD cells (marked by membrane-BFP) within a single explant. First, we used Fiji to measure endogenous Cdh3 intensity at cell junctions in wild type and Cdh3-KD cells and used a t-test to statistically compare these conditions. Next cellular polarity was assessed for each condition by measuring the ratio of the mediolateral length to the anterior-posterior width of individual cells.

We used the Manual Tracking plugin in FIJI to obtain the trajectories of vertex pairs. Individual vertex positions were tracked for a time interval of 400 s every 2 s. By obtaining the time-dependent two-dimensional (2D) vertex co-ordinates $(x_L,y_L)$ and $(x_R,y_R)$ for the left (L) and right (R) vertices respectively, the net distance travelled by the left(L) vertex is,

where ${x_L(t}_f)$, ${x_L(t}_0)$ are the vertex positions at the final ($t_f$) and initial time ($t_o$) of measurement respectively. A similar equation with $x_R,y_R$ applies for the right vertex. The length of the junction is,

To determine the weight of the contribution of each vertex to junction shortening, we define an activity parameter, $A$, as the ratio of net vertex distance moved to the initial junction length that is $A_L=\frac{\Delta r_L}{L\left(t_0\right)}$. Similarly, $A_R=\frac{\Delta r_R}{L\left(t_0\right)}$, for the right vertex. If ${\displaystyle A_L>A_R}$, the left vertex is labelled as the ‘active’ vertex while the right vertex is the ‘passive’ one, and vice versa if ${\displaystyle A_R>A_L}$. Over the time frames that we have analyzed the vertex movement, the median value of $L\left(t_f\right)/L\left(t_0\right)~$ 0.30, implying that the junctions have shortened by $\sim 70\%$ as compared to the initial junction length. Both high time resolution (2s per frame) and low time resolution (20s per frame) imaging data show the same trend that one of the vertices tend to be active, contributing more to junction shortening (Figure 1B–C, Main Text). We confirm that this observation is not due to the overall motion of the cells as detailed below (Figure 2—figure supplement 2).

The characteristics of vertex dynamics could provide clues as to the active mechanisms that promote or impede vertex movement. An important parameter to quantify vertex dynamics is the Mean Square Displacement (MSD), as a function of the lag time t. Time averaged MSD, $\stackrel{-}{\Delta }{\left(t\right)}_i,$ is calculated using the vertex positions ${\overrightarrow{r}}_i\left(t\text{'}\right)$,

where $T=400s$ and subscript $L$ stands for the left vertex. Taking the average over N independent vertex trajectories, labelled by the index $i$, we obtain the ensemble averaged MSD,$\Delta {\left(t\right)}_L=\frac{1}{N}\sum _{i=1}^N{\stackrel{-}{\Delta }\left(t\right)}_{L,i}$. The same procedure is used to calculate the MSD for the right vertex (see Figure 1E, Main Text). In many physical systems, the MSD increases with a power law, that is. $\Delta \left(t\right)~t^{\alpha }$. When the vertex motion is uncorrelated in time and along random directions, the dynamics is described as Brownian, and the MSD exponent is unity, α=1. Sub-diffusive, α<1, movement occurs when there is a hindrance to motion or the dynamics is highly correlated. For example, when a particle in caged by its immediate neighbors, sub-diffusive motion results. Super-diffusive MSD, α>1, is seen when the motion is highly directed.

We found substantial heterogeneity in the individual vertex MSD as seen from the plot of $\stackrel{-}{\Delta }{\left(t\right)}_i$ (Figure 14—figure supplement 2C). Active and passive vertex MSDs span 3 orders of magnitude of time lag. Two distinct time regimes are observed for both active and passive vertex movements: (i) at short time lags, ${\displaystyle t<30s}$, active and passive vertex movements are random, characterized by MSD exponent $\alpha ~1.$ (ii) For ${\displaystyle t>40s}$, active vertices show strong superdiffusive movement while passive vertices undergo a slowdown followed by a recovery toward superdiffusive motion (see Figure 1E,F Main Text). These distinct differences between active versus passive vertices are observed in the ensemble averaged MSD, $\Delta \left(t\right)$ for 20 vertices from 10 different embryos.

To eliminate the effect of motion of the entire tissue, we tracked vertex positions with respect to the center of an egg yolk particle (Figure 1—figure supplement 2D, E) typically present within cells as well as nearby stationary vertices (Figure 1—figure supplement 2F, G). In this manner, we analyzed the relative vertex positions, ${\overrightarrow{r}}_{rel}$, with respect to a frame of reference within the tissue being imaged. By extracting the co-ordinates of the center of an egg yolk within a cell or nearby vertices, ${\overrightarrow{r}}_c,$ we obtain the relative vertex positions, ${\overrightarrow{r}}_{rel,L}={\overrightarrow{r}}_L-{\overrightarrow{r}}_c.$ We then evaluated the mean square relative displacements (MSRD) for the left and right vertex pairs using Equation (3) above (Figure 1—figure supplement 2E, G). The distinct differences between active versus passive vertex dynamics is conserved in this relative co-ordinate system, indicating that the asymmetry in active versus passive vertex movement is not due to motion of the whole tissue (see Figure 1—figure supplement 2D, G).

To quantify the highly asymmetric vertex movement that underlies CE, we measured the fractional change in vertex positions over a time interval t using the self-overlap order parameter, defined as:

where $w_i=1$ if ${\displaystyle \left|{\overrightarrow{r}}_i\left(t^{\prime }+t\right)-{\overrightarrow{r}}_i\left(t^{\prime }\right)\right|<L_c}$ and $w_i=0$ otherwise. The self-overlap parameter is dependent on the length scale that is probed by $L_c$ and represents the probability that vertices have moved by a specified length scale over a time interval, t. We chose $L_c=1.3\mu m,$ as this is the distance scale over which movement of active and passive vertex become distinct. This is evident from the plot of MSD (Figure 1E,F, Main Text) for active and passive vertices where the dynamics begins to differ at a length scale of ${\displaystyle >1\mu m}$. If a vertex moves less than $L_c=1.3\mu m$ over the time interval t, the vertex is considered to have 100% overlap with its previous position, and hence assigned a value 1. However, if the vertex has moved more than $1.3\mu m$ within the time interval $t$, we consider this as 0% overlap. The self-overlap function, $⟨Q\left(t\right)⟩,$ is calculated by averaging over a range of initial times, $t\text{'}$, followed by ensemble averaging over individual vertices (Figure 1—figure supplement 3G-H). The active vertex self-overlap function decays rapidly and can be fit to a single exponential decay function, indicating liquid like dynamics. However, passive vertex overlap function shows a two-step decay, a signature of glass-like dynamics (Figure 1—figure supplement 3G).

Although the MSD and the self-overlap function $⟨Q\left(t\right)⟩$ are useful to quantitatively characterize vertex movement, other metrics are needed to gather further insights into the dynamic heterogeneity and correlations in vertex movement that emerge temporally during CE. In systems approaching the glass transition, the cooperativity of motion increases such that the length and time scales characterizing the dynamic heterogeneity are expected to grow sharply. In supercooled liquids, the fourth order susceptibility, ${\chi }_4\left(t\right),$ provides a unique way to distinguish the dynamic fluctuations between liquid and frozen states (Kirkpatrick and Thirumalai, 1988). Therefore, we compute the fourth order susceptibility from the variance of the self-overlap parameter,

Similar to structural glasses, the dynamic heterogeneity, quantified by ${\chi }_4\left(t\right)$ increases with time, peaks at a maximum time interval, $t_M$ and then decays (Figure 1—figure supplement 3I). The dynamic heterogeneity is manifested as dramatic variations between individual vertex trajectories in both active and passive vertex movements. For active vertices, ${\chi }_4\left(t\right)$ peaks at $t_M~120s$ while for passive vertices heterogeneity peaks at a longer time interval $t_M~170s$ (Figure 1—figure supplement 3l). The time scale associated with the peak in dynamic heterogeneity is consistent with the viscoelastic relaxation time (further discussed below), known to be the characteristic relaxation time for vertices connected by the cell cortex under tension (Solon et al., 2009). For non-shortening junctions, ${\chi }_4\left(t\right)$, does not show a peak (Figure 1—figure supplement 3J). We anticipate the peak to be at a much longer time scale for vertices of non-shortening junctions.

Vertex based models are important for studying the dynamics of confluent cell layers (Fletcher et al., 2014). The junction between three or more cells (vertices) are represented as point particles. The connecting edge between vertices represent cell-cell interfaces. We developed a theoretical model for junction shortening to understand the asymmetric dynamics of vertices. Our model, shown in (Figure 2A,B Main Text), is a coarse-grained representation of a collection of cells intercalating mediolaterally. Each vertex, bounding the v-junction, are connected to Maxwell-like components with viscous and elastic elements. Elastic properties are modeled by springs with stiffness, $k$, and actuators characterize the viscous motion of cell vertices (see Figure 2A Main Text; $\mathit{\gamma }$ is the viscosity). For the purposes of visualization, we depict the spring-actuator element as being in the direction away from the cell-cell interface, exerting a compressive force on the vertices. This need not be the case as the forces and mechanical factors contributing to junction shortening can also be localized within the cell-cell junction. For the purposes of simplicity in visualization, we picked a direction for the spring-actuation element.

We assume that the position of the left vertex, ${\overrightarrow{r}}_L(\stackrel{\scriptscriptstyle\mathrm{def}}{=}{x}_L,y_L),$ evolves according to the equation of motion:

where ${\overrightarrow{k}}_L$ is the elasticity of the left ($L$) vertex, ${\overrightarrow{F}}_L$ is the contractile force responsible for viscous deformation of the vertex and ${\gamma }_L$ is viscosity coefficient of the vertex. Replacing the subscript $L$ with $R$ above gives the equation of motion for the right vertex. The local elasticity near the vertices are accounted for by a connected harmonic spring with strength ${\overrightarrow{k}}_L$. The spring is connected in series with an actuator that supplies the contractile force, ${\overrightarrow{F}}_L.$ It is likely that the noise in a physical or biological system is correlated in time. Consistent with our observation that fluctuations in junction length are correlated in time, we model ${\zeta }_L$ as the colored noise experienced by the vertices. The noise, ${\zeta }_L$, represents the coupling of the vertices to their immediate local environment, satisfying $⟨{\zeta }_L\left(t\right){\zeta }_L\left(s\right)⟩=A{e}^{-|t-s|/{\tau }_n}$ with the mean $⟨{\zeta }_L\left(t\right)⟩=0$. The coefficient, $A$, is the noise strength. For large noise strength, vertex positions show large amplitude deviations from the position dictated by the minimum of the elastic force, as constrained by the spring. For small persistence time of the correlated noise, ${\tau }_n$, the vertex dynamics is highly uncorrelated in time. At large persistence times, however, the noise induced fluctuations in the vertex positions are correlated over the timescale ${\tau }_n$. We set the noise correlation time to be the persistence time of junction length fluctuations. The colored noise satisfies, $\frac{d{\zeta }_L}{dt}=-\frac{{\zeta }_L}{{\tau }_n}+\frac{1}{{\tau }_n}\eta \left(t\right)$, where $\eta \left(t\right)$ is the Gaussian white noise source characterized by delta correlation $⟨\eta \left(t\right)\eta \left(s\right)⟩=\delta (t-s)$ and mean $⟨\eta ⟩=0.$

Since the movement of vertices along the medio-lateral direction is much more persistent as opposed to the perpendicular direction, as evident from the closure of junctions, we simplify the model to consider only one-dimensional (1D) motion. Henceforth, we drop the vector notation and focus on the vertex dynamics along the x-axis.

By considering the basic vertex equations in the Langevin picture,

we model the vertex equations of motion in analogy to particles moving in a translating optical trap. The minimum of the left elastic ‘trap’ changes dynamically due to the term $a_L{t}^{{\psi }_L}$ in Equation 10 (modeled by the left actuator). Similarly, the right elastic ‘trap’ is translated from its initial position $L_0$ by $a_R{t}^{{\psi }_R}$ in Equation 11 (modeled by the right actuator). These terms serve as a proxy for active contractile forces which viscously deform the cell edges. Hence, we refer to the exponents, ${\psi }_L$ and ${\psi }_R$, as the rest length exponents. The physical implication of the rest length exponent is that the rest length of the junction varies dynamically. The contractile force is, ${\displaystyle F_{L,total}=-V^{\prime }\left(x_L\right)}$ in Equation 9, where $V\left(x_L\right)$ is the time-dependent ‘trap’ potential of the form $V_L={0.5k}_L{\left(x_L-a_L{t}^{{\psi }_L}\right)}^2$ and $V_R={0.5k}_R{\left(x_R-\left(L_0-a_R{t}^{{\psi }_R}\right)\right)}^2$. The stochastic movement of the vertices in a translating potential leads to a ratchet-like effect where the vertex dynamics has a specified direction. This directionality in the motion of the vertices does not arise, however, from the asymmetry in the potential but rather from the asymmetric translation of the potential well minimum or the dynamic rest length.

Hence, the active time-dependent forces contributing to junction shortening were modeled in silico as,

where $a_L$ and $a_R$ are the ‘acceleration’ of the left and right vertices respectively, and the exponents ${\psi }_L$ and ${\psi }_R$ determine the temporal dynamics of the contractile force. We include the acceleration term to account for the experimentally observed increase in the persistence of junction shortening as a function of time (see Figure 2G Main Text; See also Sec. 9 below). The initial condition is set as $x_L\left(t=0\right)=0$, and $x_R\left(t=0\right)=L_0,$ with $L_0$ being the initial cell-cell junction length. We arbitrarily assign the left side to be active, with the time dependent active force rising in proportion to ${\displaystyle t^{{\psi }_L}\left({\psi }_L>{\psi }_R\right)}$. The right side is assigned to be passive, with force increasing with time as $\propto t^{{\psi }_R}$. The difference in the rest length exponents, ${\psi }_L$ versus ${\psi }_R$, determines which vertex is active.

The equations of motion then become:

Defining ${\stackrel{-}{x}}_L=\frac{x_L}{x_0}$, $\stackrel{-}{t}=\frac{t}{\tau }$ and ${\stackrel{-}{a}}_L=\frac{a_L}{a_L^0}$, where $x_0=10\mu m$, $\tau ={10}^{2}sec$ and $a_L^0{\tau }^{{\psi }_L}=x_0$, we recast the equations of motions into dimensionless forms. Similar normalization with L replaced by R applies for the right vertex. The system of equations is scaled with the characteristic length and time, $x_0$ and $\tau$, physiologically relevant for cells undergoing convergent extension. In terms of the normalized quantities, the equation of motion is,

where the parameter ${\displaystyle \frac{k_L}{{\gamma }_L}=\frac{1}{{\tau }_L}}$ has the dimension of inverse time $\frac{1}{s}$. When normalized by the characteristic timescale $\tau$, $\frac{\tau }{{\tau }_L}=\tau \times \left(\frac{k_L}{{\gamma }_L}\right)=\frac{1}{{\stackrel{-}{\tau }}_L},$ we obtain a dimensionless parameter which we refer to as the viscoelastic ratio.

In vertex-based models for plant cells, the cell-cell interface length is modeled with a spring having a characteristic rest length (Merks et al., 2011). Any deviation in the length of the cell-cell interface from the rest length is energetically unfavorable. In vertex models for animal cells, such a rest length is typically not included (Fletcher et al., 2014). In our coarse-grained vertex model, we include a spring term with dynamic rest lengths. We show that this model accounts for the asymmetric vertex dynamics and quantitative experimental features of the junction shortening behavior. By studying actomyosin contractility in combination with theoretical modeling, it has recently been shown that epithelial junctions exhibit both elastic and viscous remodeling behavior (Staddon et al., 2019).

The existence of memory effects in junction shortening necessitates the addition of the colored noise term. Previous vertex-based models have considered random white noise indicating no memory effect. However, by experimentally quantifying the junction length fluctuations, we would like to point out that colored noise may be important to consider in modeling biological systems.

The viscoelastic ratios, $1/{\stackrel{-}{\tau }}_L$ and $1/{\stackrel{-}{\tau }}_R,$ were varied from 0.05 to 5 equivalent to ${5\times 10}^{-4}{s}^{-1}-0.05s^{-1}$ in dimensional units. Therefore, the viscoelastic relaxation time is in the range of $20s-2000s.$ Spring stiffness, $k,$ in the range between $100pN/\mu m$ and $1nN/\mu m$ (Bittig et al., 2008; Girard et al., 2007) and the viscosity, $\gamma$$~100nN.s/\mu m$ (Forgacs et al., 1998), accounts for the elastic and viscous properties of tissues previously reported in the literature. For these values, one obtains the viscoelastic relaxation time in the range of $1s-100s.$ Therefore, the viscoelastic ratio used in our model is within an order of magnitude of the physiological values for both tissue stiffness and viscosity.

We consider a wide range of values for both the viscoelastic ratio and the rest length exponent for the active vertex, ${\psi }_L$. The time step in the simulation is $\Delta t=0.0022=0.22s$, chosen to be smaller than the characteristic viscoelastic relaxation time (of order $10s$). We evolve the simulation for a total of n=20, 000 steps (4, 400s in real units). The equations of motion are solved using the Euler method for each vertex. If at any point during the simulation, the left and right vertex positions approach one another to a distance less than $0.5\mu m$, we label the junction as having successfully completed the shortening. The initial junction length was set to be $L_0=2$, equal to $20\mu m$ in real units. The range of rest length exponents we consider is limited by the need to ensure that the minima of the potentials do not overlap during a given simulation run. To generate the phase diagram for the probability of junction shortening as a function of the rest length exponent and the viscoelastic ratio, we consider for the left active vertex $0.05\le \frac{1}{{\stackrel{-}{\tau }}_L}\le 5$ at intervals of 0.5. Rest length exponents in the range, 1.7 <${\displaystyle {\psi }_L<2}$, were simulated at intervals of 0.25 for the active vertex. ${\psi }_R=1.3$ is fixed for the passive right vertex. The acceleration of the potential minima, is set to be ${\stackrel{-}{a}}_L={\stackrel{-}{a}}_R=0.001$. The viscoelastic ratio for the right passive vertex is fixed at $\frac{1}{{\stackrel{-}{\tau }}_R}=0.1$. We simulated 100 junction shortening events at each value of the parameters $\frac{k_L}{{\gamma }_L}$ and ${\psi }_L.$ By monitoring the percent of successful junction shortening events, we generate the phase diagram (Figure 2C, Main Text).

We observe in the phase diagram (Figure 2C, Main Text) that at a fixed value of the rest length exponent, modulating the asymmetry of the viscoelastic parameter (($\frac{k_L}{{\gamma }_L})/(\frac{k_R}{{\gamma }_R})$) leads to a transition from non-shortening (failure to shorten) to junction shortening (successful shortening) regime. At constant ${\psi }_L=1.95$, for low values of the active vertex viscoelastic parameter ${\displaystyle ({\overline{\tau }}_R/{\overline{\tau }}_L)<6.9,}$ less than 40% of the junctions shorten. However, at higher values of the viscoelastic parameter, ${\displaystyle {\stackrel{-}{\tau }}_R/{\stackrel{-}{\tau }}_L>15,}$ more than 80% of the junctions successfully execute shortening. Therefore, the theory predicts that the asymmetry in local viscoelasticity is critical for cells to intercalate medially and effect convergent extension. We calculate the normalized length for non-shortening junctions and found that the self-similarity in junction length dynamics is broken, in agreement with experimental results Figure (2H,Main Text). $L_n$ for non-shortening junctions is characterized by large fluctuations away from the expected compressed exponential behavior, as quantified by the residue (see details below). Simulated junction length dynamics for the non-shortening case(gray curves in Figure 2H Main Text) is obtained for parameter values $1/{\stackrel{-}{\tau }}_L=0.05$, $1/{\stackrel{-}{\tau }}_R=0.05$ and ${\displaystyle {\psi }_L=2,{\psi }_R=1.3}$. Meanwhile, for the shortening phase (gray curves in Figure 2G Main Text), $1/{\stackrel{-}{\tau }}_L=5$, $1/{\stackrel{-}{\tau }}_R=0.05$ and ${\displaystyle {\psi }_L=2,{\psi }_R=1.3}$. Therefore, asymmetry in viscoelasticity is critical for junctions to execute shortening. Our model points out that the persistent dynamics of active vertices, enabling the efficient shortening of the cell-cell interfaces, is a direct consequence of the faster viscoelastic relaxation time.

We calculated the normalized cell-cell junction contact lengths to characterize the self-similarity in the length change underlying cell neighbor exchanges during convergent extension. We selected all cell-cell contacts that shorten over time intervals > 100 s, and normalized the change in length as,

where ${L(t}_f)$, ${L(t}_0)$ are the junction lengths at the final and initial time points respectively. The normalized junction length dynamics, $L_n\left(t\right)$, provides insight into the active processes that underlie vertex movement driving CE. Since junction lengths are highly heterogeneous (Figure 2—figure supplement 1A) relative to, ${L(t}_0),$ and the time to closure, $t_f-t_0$, the normalization in Equation 18 allows us to rescale all the length changes to values between 1 and 0. The normalized length curve was smoothed (over 10-time frame windows = 20s) to remove higher frequency noise. To determine if junction shortening exhibits a self-similar behavior across multiple embryos, we rescaled the time axis in $L_n\left(t\right)$ by the relaxation time ${\tau }_f$, defined as the time at which $L_n(t={\tau }_f)=$ 0.3. This corresponds to a 70% reduction in the junction length. Rescaling the time axis by $t/{\tau }_f$ collapses the normalized lengths onto the functional form,

which is a single compressed exponential (Figure 2—figure supplement 1B). The extent of the self-similarity is striking in comparison to both non-shortening (Figure 2H, Main Text) and cis-mutant normalized junction lengths (Figure 6D, Main Text). Notice that for ${\displaystyle t<{\tau }_f}$, change in normalized junction length is slower than exponential decay. However, for ${\displaystyle t>{\tau }_f}$, normalized junction length shortens significantly faster than would be predicted based on exponential decay. Therefore, the compressed exponential behavior for $L_n$ provides evidence that the persistence of junction shortening increases with time.

We quantify the deviation of the normalized junction shortening from the expected compressed exponential behavior by calculating the rescaled time, $t_r=\frac{t}{{\tau }_f}$, and ${\displaystyle \omega =|e^{-1.5{\left(t_r\right)}^{3.8}}-L_n|}$, where $\omega$ is the residue. ${\tau }_f$, is defined as the time at which $L_n(t={\tau }_f)=$ 0.3. In (Figure 2I, Main Text), non-shortening junctions show strong deviations from the expected compressed exponential behavior while shortening junctions closely follow the compressed exponential form.

To test the robustness of the conclusions obtained using our model, we consider an alternative form of the contractile force experienced by the vertices. We model the actuators contributing to viscous junction shortening as moving with constant velocities - ${\stackrel{-}{v}}_L$ and ${\stackrel{-}{v}}_R$- for the left and right vertices respectively:

The ‘trap’ potential in this scenario is of the form, $V_L=k_L{\left(x_L-v_{L}t\right)}^2$ and $V_R=k_R{\left(x_R-\left(L_0-v_{R}t\right)\right)}^2$, moving with constant velocities. Left vertex is defined to be active with ${\displaystyle {\stackrel{-}{v}}_L>}$ ${\stackrel{-}{v}}_R.$ The velocity is normalized as ${\stackrel{-}{v}}_L=v_L/\left(\frac{x_0}{\tau }\right)$. The passive vertex velocity is fixed at ${\stackrel{-}{v}}_R=0.011$, which in dimensional units correspond to $0.0011\mu m/s$. The active vertex velocity is varied in the range of $0.03\le {\stackrel{-}{v}}_L\le 0.034$, which in dimensional units is between $0.003\mu m/s$ - $0.0034\mu m/s$. Experimental vertex shortening velocities in the range of $0.001\mu m/s$ to $0.021\mu m/s$ was reported by some of us in a previous work (Shindo and Wallingford, 2014). Fixing the passive viscoelastic ratio at, $1/{\stackrel{-}{\tau }}_R=0.1$, we varied $0.05\le 1/{\stackrel{-}{\tau }}_L\le 5$ for the active vertex. Keeping all the other parameters the same, we arrive at the same conclusion that local junction viscoelastic response is critical to effect junction shortening (Figure 2—figure supplement 1D). Therefore, our conclusions are not affected by the specific form of the vertex dynamics. A crucial aspect is that the two potential well minima should move asymmetrically in time.

We quantify the intermittent movement (See Figure 3A, Main Text) of the vertices perpendicular to the motion that contributes to the junction shortening by calculating the transverse fluctuations, $R_T$. The transverse step size is given by, ${\delta r}_T\left(t\right)=\left|{\delta \overrightarrow{r}}_L\left(t\right)\right|sin\left(\theta \right),$ where ${\delta \overrightarrow{r}}_L\left(t\right)={\overrightarrow{r}}_L\left(t\right)-{\overrightarrow{r}}_L\left(t-\delta t\right)$ and the angle $\theta$ is the obtained from the dot product, ${\delta \overrightarrow{r}}_T.\Delta {\overrightarrow{r}}_L=\left|{\delta \overrightarrow{r}}_T\right|\left|\Delta {\overrightarrow{r}}_L\right|cos\left(\theta \right)$. Here, the net displacement of the Left(L) vertex is given by, ${\displaystyle \mathrm{\Delta }{\overrightarrow{r}}_L=\left(x_L\left(t_f\right)-x_L\left(t_0\right)\right)\hat{x}+\left(y_L\left(t_f\right)-y_L\left(t_0\right)\right)\hat{y}.}$ Similar equation applies for the right vertex with $x_L,y_L$ replaced by $x_R,y_R$. To better quantify the intermittent dynamics, we compute the transverse 'hop' function,

The angular bracket above ${⟨..⟩}_B$ denote the average over the time window $B\stackrel{\scriptscriptstyle\mathrm{def}}{=}\left[t-\delta t,t+\delta t\right].$ We chose for the hop duration parameter, $\delta t=4s,$ to probe short time transverse fluctuations. The probability distribution of all $R_T\left(t\right)$ values are shown in (Figure 3D, Main Text). By averaging the transverse fluctuations over all vertices, ${⟨R}_T\left(t\right)⟩=\frac{1}{N}\sum _{i=1}^N{R}_T{\left(t\right)}_i$, we obtain the mean transverse fluctuation for active and passive vertices (Figure 3C, Main Text). ${⟨R}_T\left(t\right)⟩$ for Cis-mutant and Xdd1 vertices are shown in Figure 6F and Figure 7F of the Main Text, respectively.

To determine the characteristic spatial correlation of cadherin intensity fluctuations, we analyze the pixel-by-pixel Cadherin3 (Cdh3) intensity data, $I\left({\overrightarrow{r}}_i\right)$, along the medio-lateral cell-cell interface (v-junction). Here, $\overrightarrow{r_i}$ is the position of the i-th pixel in the iSIM image. The spatial autocorrelation function of the cadherin intensity fluctuations as a function of distance, r, along the cell-cell interface is,

where $\theta \left(z\right)=1$ if $z=0$, $\theta \left(z\right)=0$ for any other value of $z$. $⟨I⟩$ is the mean cadherin intensity over all the pixels along the cell-cell junction. $C\left(r\right)$ is normalized such that $C\left(r=0\right)=1.$ The cadherin correlation length is defined as the distance, $\xi$, at which $C\left(r=\xi \right)=0$. This provides a measure of the distance scale at which the correlation in cadherin intensity fluctuations is lost. Equivalently, $\xi$, sets the spatial persistence of cadherin fluctuations along the cell-cell junction, providing a quantitative measure of lateral cadherin clustering. We analyzed cadherin clustering patterns along individual cell-cell junctions separately at time intervals of 2s and obtained the spatial correlation behavior for individual junctions from 10 embryos. The mean of the cadherin spatial correlation (over 100s of time points) for wild type embryos is reported in Figure 4C, Main Text with the error bar denoting the standard deviation. To analyze the dynamic variation in cadherin cluster size as a function of time, $C\left(r\right)$, was calculated over a time interval of 320s at 2s resolution. The fluctuation in cluster size is given by, $\delta \xi \left(t\right)=\xi \left(t\right)-{⟨\xi ⟩}_t$, where ${⟨\xi ⟩}_t$ is the mean cluster size over the analyzed time interval (, 4E Main Text). The cluster size fluctuation, $\delta \xi \left(t\right)$, was smoothed (over 10-time frame windows = 20s) in order to remove high frequency noise.

To analyze the instantaneous change in the junction length, we calculated, $\delta L\left(t\right)=(L\left(t\right)-L\left(t+\delta t\right))$, where $\delta t=2s$ and $t$ is the time. The unit of the length fluctuations is μm. When the junction shortens, ${\displaystyle \delta L\left(t\right)>0}$, while extension implies ${\displaystyle \delta L\left(t\right)<0}$ (Figure 4E, Main Text).

The normalized cross-correlation between junction length fluctuations, $\delta L\left(t\right),$ and cadherin cluster size fluctuations, $\delta \xi \left(t\right),$ was calculated in MATLAB using,

where T is the total time of analysis and $\tau$ is the lagtime. We analyzed the cross-correlation for 18 junction shortening events and show the correlation coefficient as a heatmap in Figure 4F, Main Text

To quantify the asymmetry in Cdh3 clustering in the spatial region near the left and right vertices, we calculated the spatial correlation in cadherin intensity fluctuations, $C\left(r\right)$ (see Equation 24), in a region spanning $3.25\mu m$ adjacent to left and right vertices. The spatial region is chosen such that on average it is $3X$ larger than typical cadherin cluster size of order $1\mu m$. The localized cadherin clustering behavior adjacent to active and passive vertices, quantified by the spatial correlation in cadherin intensity fluctuations, is shown in Figure 4G Main Text. Main Text also compares the local cadherin clustering behavior in non-shortening junctions to shortening junctions.

We then wondered if spatial correlation in cadherin expression is stable in time across a single cell-cell junction. We computed the spatial cadherin correlation at 2 second time intervals along single cell-cell junctions from multiple embryos. Our results indicate that the spatial correlation length of cadherin expression is highly heterogeneous in time, with the correlation length varying from 0.2 $\mu m$ to 1.6 $\mu m$, as shown in Figure 4—figure supplement 2. To decipher how the spatial correlation in cadherin expression along the cell-cell junction varies with time near active and passive vertices, we present the data for spatial autocorrelation of cadherin fluctuations in Figure 4—figure supplement 2A-J. In each panel, individual blue (red) lines correspond to the spatial correlation in cadherin fluctuations near passive (active) vertices. The difference in spatial cadherin correlation between active and passive vertices in shortening junctions are shown in Figure 4—figure supplement 2A-J whereas the same data for non-shortening junctions are shown in Figure 4—figure supplement 2K-Q. By extracting the length of the spatial correlation as discussed in Section 6 above, the summary of the cadherin spatial correlation data is presented in the Main Text Figure 4G.

We used an alternative definition of C-cadherin cluster size to confirm our results. By fitting the decay in the cadherin spatial autocorrelation function to zero by an exponential function, we can extract the cluster size. We find that the asymmetry in the local cadherin clustering behavior is independent of the definition of the cluster size (Figure 4—figure supplement 1E–G). Hence, we conclude that cadherin clustering is enhanced near active vertices as opposed to passive vertices in shortening junctions (Figure 4G Main Text, Figure 4—figure supplement 1F) while it is symmetric near left and right vertices in non-shortening junctions ( Figure 4—figure supplement 1G).

We calculate the spatial autocorrelation of the cadherin intensity fluctuations ($\mathit{C}\left(\mathit{r}\right)$ see Section 13,Equation (24)) for four different embryo development scenarios, (i) wild type Cdh3 Figure 4C,(ii) Cdh3 rescue (Cdh3-GFP) (see Figure 7—figure supplement 1), (iii) Cdh3 cis-mutant (cisMut-Cdh3-GFP) Figure (5D,Main Text), and (iv) Xdd1 Figure (7C,Main Text). The mean spatial correlation in cadherin fluctuations for wild type Cdh3 and Cdh3-GFP junctions show similar behavior with the decay to zero characterized by an exponential form (Figure 7—figure supplement 1B). The exponential spatial dependence is evidence for the existence of a characteristic spatial scale for correlations in cadherin spatial distribution. The local peak in the cadherin autocorrelation function is identified using the findpeaks algorithm in MATLAB. Local peak in a data array is identified when a data point is larger than its two neighboring data points or equal to infinity. The prominence of the peak is set to 0.1, identifying the peak amplitudes that stands out relative to other peaks. Wild type and rescue embryo cell-cell junctions are characterized by well-defined spatial periodicity in cadherin clustering, as observed from the secondary peaks in the spatial correlation (see black triangles, Figure 7—figure supplement 1C). Therefore, cadherin spatial organization in wild-type Cdh3 and Cdh3-cis-mutant rescue embryos is in a crystal-like phase ( Figure 7—figure supplement 1C), with regularly repeating spatial patterning. Cadherin spatial correlation for individual frames (with no averaging) is shown for control (solid lines in ; Figure 7—figure supplement 1D) and cadherin rescue embryos (dashed lines, Figure 7—figure supplement 1D). However, cadherin spatial correlation in Xdd1 and cisMut-Cdh3 embryos show diffuse spatial organization, with little to no secondary peak structures visible in the spatial autocorrelation (see Figure 7—figure supplement 1E). This is indicative of disrupted periodicity in cadherin clustering. Therefore, cadherin spatial organization in cisMut-Cdh3 and Xdd1 embryos is in a gas-like phase. Individual frame cadherin spatial correlation for cisMut-Cdh3 (solid lines Figure 7—figure supplement 1F) and Xdd1 is shown as dashed lines in Figure 7—figure supplement 1F. For Xdd1 and cisMut-Cdh3-GFP junctions, the decay in the spatial correlation is better fit by a power law, indicating the lack of existence of a coherent length scale associated with fluctuations in cadherin expression ( Figure 7—figure supplement 1G).

The statistical test used and other relevant details such as the number of embryos/image frames analyzed are described in the figure legends.

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

We thank Dan Dickinson for use of the iSIM microscope and for critical reading and helpful discussions. We thank Andy Ewald for critical reading of the manuscript. This work was supported by grants from the NICHD (R21HD084072) and the NIGMS (R01GM104853) to JBW and from the NSF (Phys 17–08128) and the Collie-Welch Chair through the Welch Foundation (F-0059) to DT.

Animal experimentation: Animal work described here was performed in accordance with the UT Austin Institutional Animal Care and Use Committee protocol #AUP-2018-00225.

© 2021, Huebner et al.

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