We quantified neural tube (NT) morphologies in wild type, cadherin 2 mutants (cdh2^-/-), maternal zygotic integrin α5 mutants (MZ itgα5^-/-) and cdh2^-/-; MZ itgα5^-/- double mutants. We fixed embryos at 12–14 somite-stage and imaged them for Fibronectin and cell nuclei. cdh2 mutants exhibited a wider neural tube than wild-type embryos consistent with its known convergence defect (Figure 1B–C; Lele et al., 2002). In contrast, MZ itgα5 mutants exhibit a narrower neural tube (Figure 1D) suggesting precocious neural tube convergence after reduction of cell-Fibronectin matrix adhesion. Surprisingly, the convergence defect of cdh2 mutants is rescued in cdh2^-/-; MZ itgα5^-/- double mutants (Figure 1E).

We quantified convergence of the neural tube along the anterior-posterior axis across genotypes. We measured the medial-lateral width of the neural tube (cartoon in Figure 1F) on transverse sections every 20 μm along the anterior-posterior axis starting from the last somite boundary until the posterior end of the PSM (Figure 1—figure supplement 1). In wild-type embryos, the width of the neural tube progressively narrows, comparing posterior to anterior, reflecting convergence over time (Figure 1F and Video 1). This change in the medial-lateral width of the neural tube is shallower in mutants. In cdh2 mutants, the neural tube is wider than wild-type embryos along the full anterior-posterior length of the PSM (Figure 1F). In MZ itgα5 mutants and cdh2; MZ itgα5 mutants, the neural tube has a narrower medial-lateral width in the posterior PSM compared to wild type (Figure 1F). Similarly, wild-type embryos exhibited a posterior to anterior decrease in the dorsal-ventral length and the cross-sectional area of the neural tube, while the posterior to anterior decreases in these two metrics were shallower in other genotypes (Figure 1—figure supplement 2A and B). These data suggest that inter-tissue adhesion via cell-Fibronectin matrix adhesion constrains neural tube convergence.

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We examined the effect of loss of inter-tissue adhesion on the PSM and observed an abrogation of bilaterally symmetric morphogenesis. We measured the left-right differences in PSM cross-sectional areas (Figure 1G) and found that both MZ itgα5^-/- and cdh2^-/-; MZ itgα5^-/- double mutants exhibited loss of bilateral symmetry. By contrast, cdh2 mutants did not have symmetry defects. Local asymmetries were also observed in MZ itgα5^-/- and cdh2^-/-; MZ itgα5^-/- double mutants in the length of the interface between PSM and neural tube (PSM|NT interface) and the angle formed at the lateral edge of this interface (Figure 1—figure supplement 2C and D). An explanation of these asymmetries is that reduction of cell-matrix interactions creates local tissue detachments (arrowheads in Figure 1D and E, Figure 1—figure supplement 2E and F) which in turn leads to disequilibrium in inter-tissue adhesion along the left and right PSM|NT interfaces.

To understand how variable levels of cell-cell adhesion and inter-tissue adhesion underlie the differences in tissue shapes across genotypes, we designed a computational 2D model with variable cell-cell adhesion and inter-tissue adhesion and tested whether the model can predict tissue shapes. We modeled a 2D transverse section of four connected tissues (NT, left and right PSM and notochord [NC]) (see Materials and methods for details). In this coarse-grained model, we do not directly consider individual cells, rather the tissues are considered as soft units with an internal pressure (blue arrows in Figure 2A) and an elastic surface made of movable points interlinked by springs (green surface springs in Figure 2A). These connected surface springs model tissue surface tension. This tissue surface tension resists the internal pressure and was shown to be proportional to the cell-cell adhesion strength inside a tissue (David et al., 2014; Manning et al., 2010). The tissues are attached to each other by another set of springs (red springs in Figure 2A), which model inter-tissue adhesion via cell-Fibronectin interactions. Overall, there are three parameters in the model: the surface stiffness (spring constant K_S for surface springs), the adhesion stiffness (spring constant K_adh for adhesive springs), and the internal pressure (P). The surface stiffness and the adhesion stiffness positively correlate with the levels of Cadherin-dependent cell-cell adhesion and Fibronectin-mediated inter-tissue adhesion, respectively.

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To determine how the model parameters affect tissue morphology, we analyzed the simulated tissue shapes. The internal tissue pressure should depend on the net fluid content inside a tissue, and there is no obvious reason to vary this quantity across genotypes. Hence, we fixed the internal pressure (p=5) depending on an earlier model of soft grains (Åström and Karttunen, 2006) in such a way that the simulated shapes qualitatively resemble in vivo tissue shapes in 2D transverse sections (Figure 2B). Irrespective of choices of K_s and K_adh, we found that the PSM|NT and the PSM and epidermis interface (PSM|E) always have very different curvatures (Figure 2C). The PSM|NT interface is straight and has much higher radius of curvature than the PSM|E interface which is curved (Figure 2C). This prediction was confirmed experimentally in vivo (Figure 2C).

The parameters K_s and K_adh were systematically varied in simulations to assess their influence on the shape of the interface between NT and PSM (Figure 2—figure supplement 1). We measured two shape metrics: the length of the interface between the PSM and neural tube (L PSM|NT, blue boxes in Figure 2B) and the angle at the lateral edge of this interface (angle Θ in Figure 2B). We found that increasing surface stiffness decreases the interfacial length, while increasing inter-tissue adhesion increases interfacial length (Figure 2—figure supplement 1A and B). Thus, inter-tissue adhesion acts like a ‘zipper’ between two tissues. In fact, low adhesion stiffness decouples the tissues, which is evident from the local detachments seen in our simulations (Figure 2B top right, arrowhead) and similar to detachments observed in MZ itgα5 mutants and cdh2; MZ itgα5 mutants (Figure 1D and E).

This simple computational model can account for the gradual narrowing of the neural tube along the medial lateral axis as it develops (Figure 2—figure supplement 1C). We can assume that the NT-PSM interface is locally at steady-state along the anterior-posterior axis, and the observed tissue shape change along the anterior-posterior axis may be caused by a progressive increase in K_s along that axis. Indeed, the zebrafish PSM progressively solidifies from posterior to anterior in a cadherin2-dependent manner (Mongera et al., 2018). We found that the length of the medial to lateral interface between the PSM and NT (L PSM|NT) steadily decreases with increasing surface stiffness when other parameters are fixed (Figure 2—figure supplement 1A).

Given the above correspondence between in vivo and in silico observations, we next assigned reasonable values of surface stiffness (K_S) and inter-tissue adhesion stiffness (K_adh) that reproduce morphometrics of wild-type embryos. We note that in vivo the value of the interfacial length (L PSM|NT) decreases to 0.6 of the maximum value at the anterior end of PSM relative to the posterior end (Figure 2—figure supplement 1C). We also found in silico that L PSM|NT roughly falls to 0.6 of the maximum value at K_S ≈ 100 for a fixed K_adh (Figure 2—figure supplement 1A). Since we are interested in steady-state shapes near the anterior PSM, we may take this value to represent the wild type. Next, we consider the value of K_adh. In simulations, we varied K_adh for different fixed values of K_S, and measured the medial-lateral length of NT relative to the interfacial length between NT and PSM (Figure 2—figure supplement 1G). In vivo, this ratio (ML length of NT/L PSM|NT) is about two on average for the anterior 140 μm of PSM. In silico, we found that irrespective of the K_S value, this ratio saturates around a value of 2.7 in the range of K_adh = 8 to 12 (marked in red in Figure 2—figure supplement 1G). Based on the above analysis, we then represented the wild-type embryos by a pair of values: (K_S,K_adh)=(100,10).

After fixing the wild-type parameters, we then chose the parameter values corresponding to cdh2 mutants and MZ itgα5 mutants by matching the in silico and in vivo lengths of PSM|NT interface in these mutants relative to its wild-type values (Figure 2E). For cdh2 mutants, the mean length of PSM|NT is around 1.5 times higher than wild type. To reproduce the experimentally observed increase of L PSM|NT relative to wild-type, we assigned reasonable parameter values to cdh2 mutants depending on biological expectations. First, we lowered the surface stiffness relative to wild type, since low cell-cell adhesion is known to reduce tissue surface tension (David et al., 2014; Manning et al., 2010). Second, Cadherin 2 was shown to inhibit Integrin α5 activation and Fibronectin matrix assembly in the PSM (Jülich et al., 2015; McMillen et al., 2016). Therefore, we may assign a higher adhesion stiffness value to cdh2 mutants compared to wild type. After systematic exploration of the parameter space in simulations, we found a pair of parameter values, (K_S,K_adh)=(55,12), that reproduce the in vivo increase of L PSM|NT relative to wild-type (1.55 times).

Following the same procedure for MZ itgα5 mutants, the in vivo data indicate that MZ itgα5 mutants exhibit a mean PSM|NT interface length 0.77 times that of the wild-type value. Since cell-matrix interaction is reduced in MZ itgα5 mutants, it is logical to assume a lower inter-tissue adhesion for this genotype relative to wild type, but the surface stiffness was kept same as wild type. We found that a pair of parameter values, (K_S,K_adh)=(100,6.5), reproduce the experimentally observed decrease of L PSM|NT relative to wild-type (0.78 times).

Using the parameter values corresponding to cdh2 mutants and MZ itgα5 mutants, we can then predict the interfacial length for cdh2; MZ itgα5 mutants simply by combining these parameter values. The double mutants are represented using the value of inter-tissue adhesion in MZ itgα5 mutants, and the same value of surface stiffness as cdh2 mutants (Figure 2D). Hence, cdh2; MZ itgα5 mutants are represented by the values: (K_S,K_adh)=(55,6.5). These parameter choices for cdh2 mutants and MZ itgα5 mutants accurately predicted the in vivo interfacial length of cdh2; MZ itgα5 mutants, (Figure 2E). Importantly, although we assigned the parameter values depending on the relative interfacial lengths of cdh2 mutants and MZ itgα5 mutants, these parameter choices also reproduced the experimentally observed trends in the variation of interfacial angle Θ across genotypes (Figure 2F).

The observation that the PSM|NT and PSM|E interfaces have very different curvatures intuitively suggest that they are under different levels of tension. Therefore, we analyzed the tension distributions at tissue surfaces in our simulations (Figure 3A). Interestingly, the model predicted a gradient of tension at the neural tube side of PSM|NT interface with a higher tension on the lateral side of the interface and a lower tension on the medial side (see dashed box, Figure 3A). In contrast, the medial-lateral tension distribution at the PSM|E surface is predicted to be homogeneous.

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To test these predictions in vivo, we analyzed F-actin intensity as an indicator of cortical tension. Consistent with the model prediction, we observed an increasing medial to lateral gradient of F-actin along the PSM|NT interface (Figure 3B, F and L). A parallel gradient of Myosin-II localization was also observed (Figure 3—figure supplement 1A–1C). Fibronectin matrix assembly is a force-induced process dependent upon cell actomyosin contractility (Zhong et al., 1998; Lemmon et al., 2009; Dzamba et al., 2009). We therefore analyzed the Fibronectin matrix to determine whether there was a correlation between levels of F-actin and Fibronectin matrix assembly at the tissue interfaces (Figure 3C and D). Since Fibronectin matrix is assembled from soluble and small matrix aggregates into large fibers, we quantified the topology of the Fibronectin matrix by sorting into small and large matrix elements, color coded in cyan and magenta, respectively (Figure 3H, K, M and O). In correlation with the F-actin gradient, we observed a medial to lateral gradient of Fibronectin matrix assembly as large assembled elements were enriched at lateral edge of the interface, while small matrix elements were evenly dispersed. Lastly, we used a monoclonal antibody that recognizes an epitope in human Fibronectin that is exposed when Fibronectin is under tension (Cao et al., 2017). In embryos expressing a chimeric zebrafish/human Fibronectin, we observed a medial to lateral gradient of Fibronectin under tension along the PSM|NT interface (Figure 3—figure supplement 1D–1F, J and K). All together, these data support the model prediction that there is an increasing medial to lateral gradient of tension along the interface between the PSM and neural tube.

These medial-lateral gradients of F-actin and Fibronectin matrix were only observed at the PSM|NT interface. Both F-actin, Fibronectin matrix assembly and Fibronectin under tension were homogenously distributed medial-laterally at the PSM|E interface (Figure 3B–O and Figure 3—figure supplement 1G–1I and L–M), which is again consistent with the model predictions. Instead, this interface was characterized by a progressive posterior to anterior increase in F-actin density (Figure 3I and P). Similarly, we observed a posterior to anterior increase in the density of large matrix elements and decrease in the density of small matrix elements (Figure 3K and Q). These opposing gradients suggest that the matrix is progressively assembled from posterior to anterior by crosslinking small Fibronectin fibrils to form large fiber networks (Figure 3J). This posterior-to-anterior increase in matrix density and F-actin at the PSM|E interface also correlates with the progressive epithelialization of PSM surface cells and a posterior to anterior increase of F-actin signal in both internal cells and surface cells of the PSM (Figure 3—figure supplement 2).

Overall, the intriguing observation here is that the homogeneous medial-lateral distribution in F-actin and Fibronectin density observed at the PSM|E interface is lost at the PSM|NT interface, which instead displays a medial-lateral gradation both in F-actin and the Fibronectin fibers. Our simplified model suggests that this behavior may result from the particular distribution of mechanical stress at the PSM|NT interface. This interface between two converging tissues mechanically resembles an ‘adhesive lap joint’ that is commonly used in engineering and is comprised of partially overlapping components bound via an adhesive. The structure formed by the apposition of neural tube and PSM tissues can be idealized as a particular type of lap joint, a single-sided strapped joint (Ghoddous, 2017), where the neural tube on top of the PSM acts as a single sided strap bridging the two PSMs via a Fibronectin adhesive (Figure 3R). Shear stress at the overlapping interface of a lap joint is highest at the lateral edges of the overlap and lowest in the middle (Goland and Reissner, 1944). This nonhomogeneous stress distribution can be compensated for by grading the adhesive in the overlap (Carbas et al., 2014). In accordance to this engineering insight, we observe a graded Fibronectin ‘adhesive’ at the PSM|NT interface. An important distinction is that shear stress is generated by an externally applied load in an engineered lap joint, while at the PSM|NT tissue interface, shear stress is generated via tissue motion relative to one another due to convergence extension. This shear stress at the adhesive interface resists the convergence of neural tube (green and black arrows in Figure 3R, right). In engineering, excess adhesive can be added to the edges of the overlap and sculpted into an arc to strengthen the joint by distributing the forces over a larger area (Figure 3R; Lang and Mallick, 1998). These arced adhesive ‘spew fillets’ are also observed in our system at the lateral edges of the PSM|NT interface (Figure 3S). In addition, the geometry of the adherent materials impacts the strength of a lap joint. A rounded end (rather than an end with sharp corners) of adherent materials eliminates singularities in the stress field and strengthens the joint by more evenly distributing the forces (Adams and Harris, 1987). Here, the tissue edges are rounded at PSM|NT interface. Thus, the PSM|NT interface behaves like an optimally engineered adhesive lap joint with rounded edges, a graded adhesive and an adhesive spew fillet.

In our computational model, we hypothesized a greater inter-tissue adhesion between the neural tube and the PSM in cdh2 mutants and a lower inter-tissue adhesion in itgα5 mutants compared to wild-type embryos. These assumptions were sufficient to accurately predict the relative in vivo values of both the interfacial angle and length of PSM|NT interface. To further test these model assumptions, we quantified Fibronectin matrix assembly and F-actin at PSM|NT interface in cdh2 mutants and MZ itgα5 mutants. We observed relatively high average levels of Fibronectin at the interface in the cdh2 mutants and the gradients of both Fibronectin matrix assembly and F-actin were lost (Figure 4A–C and G–I). In contrast, MZ itgα5 mutants have lower levels of Fibronectin, and the medial-lateral gradients of F-actin and Fibronectin matrix assembly were maintained (Figure 4D–K).

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Lap joints exhibit graded shear stress at the interface of two overlapping materials (Goland and Reissner, 1944). In our system, the presence of a shear stress at the interface should depend on the relative motion between the attached converging tissues, and this relative motion can be reduced in the absence of neural tube convergence. If the medial-lateral gradation of Fibronectin matrix assembly and F-actin are shear-driven, we can expect a positive correlation between neural tube convergence and presence of these gradients. Our data for cdh2 mutants strengthen this idea as these mutants exhibit a reduction in neural tube convergence as well as loss of the gradients. On the other hand, itgα5 mutants do not have a defect in convergence and they retain the potential to establish this graded assembly, though we observe an overall reduction in Fibronectin matrix production.

To directly examine whether Fibronectin matrix is required for inter-tissue adhesion and restriction of neural tube convergence, we generated double mutants for the two zebrafish fibronectin genes, fn1a and fn1b, using CRISPR/Cas9 (Figure 5—figure supplement 1B). Double homozygous fibronectin mutant embryos (fn1a^-/-;fn1b^-/-) completely lacked Fibronectin matrix as detected by immunohistochemistry (IHC) (Figure 5D). fn1a^-/-;fn1b^-/- embryos exhibited a precociously converged neural tube, local tissue detachments, and displayed local disruptions of bilateral symmetry consistent with the MZ itgα5^-/- phenotype (Figure 5A and B, Figure 5—figure supplement 1C–1E).

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We created a photoconvertible Fibronectin transgenic to enable us to ‘paint’ the extracellular matrix and quantify deformation of the ‘painted spots’ in live embryos. The transgenic line expresses Fibronectin 1a tagged with the green-to-red photoconvertible protein mKikumeGR under the control of the heat-shock promoter (Tg hsp70:fn1a-mKIKGR, Figure 5—figure supplement 1A). We tested the functionality of the transgene by creating fn1a^-/-;fn1b^-/-; hsp70:fn1a-mKIK embryos. fn1a^-/-;fn1b^-/- embryos are not viable, and we observed somite border defects consistent with published loss of function studies in zebrafish (Figure 5G; Jülich et al., 2005; Koshida et al., 2005). Heat-shocked fn1a^-/-;fn1b^-/-; hsp70:fn1a-mKIK embryos exhibited normal Fibronectin assembly and somite boundary morphologies, demonstrating the functionality of the transgene (Figure 5C–H and Tables 1 and 2). Using this transgene, we compared the dynamics of matrix remodeling at different tissue interfaces in wild-type and mutant embryos.

We assayed Fibronectin matrix dynamics at tissue interfaces in vivo via confocal timelapse microscopy. When transgenic embryos were heat-shocked at the 2–3 somite stage, we observed a fluorescently labeled Fibronectin matrix at the 12-somite stage (Figure 6B and C). The topology of the Fn1a-mKikGR matrix was consistent with the matrix topology observed at each interface in fixed wild-type samples subjected to Fibronectin IHC. We photoconverted spots of matrix 25–35 μm in diameter either at the PSM|NT or PSM|E interface (Figure 6A–C) at a distance of 150–200 μm from the last somite, and imaged every 15 min. One hour after the photoconversion, the spots at the PSM|NT interface showed significant shrinking along the medial-lateral direction, while the spots at the PSM|E interface did not shrink (Figure 6B and C and Video 2).

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Video 2

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We quantified the spatiotemporal dynamics of photoconverted Fibronectin matrix spots after thresholding to account for photobleaching. The spots were converted into binary images and, we used the standard deviation of the white pixel distribution along the medial-lateral and anterior-posterior axes as measures of the width and height, respectively. This metric quantifies the general deformation of the photoconverted spot over time. At the PSM|NT interface, the medial-lateral width of the photoconverted region decreased by 40%, while the anterior-posterior height was unchanged (Figure 6D and Figure 6—figure supplement 1A). In contrast, the matrix at the PSM|E interface did not exhibit medial-lateral shrinking (Figure 6D and Figure 6—figure supplement 1A). Further analysis of the directionality of matrix remodeling revealed that there is medial to lateral bias in the displacement fields of the PSM|NT interface but no anisotropy in the displacement fields of the PSM|E interface (Figure 6—figure supplement 1E–1G).

Lap joint theory suggests that inter-tissue adhesion is required for shear forces at the PSM|NT interface. We examined the effect of inter-tissue adhesion on shear driven Fibronectin matrix remodeling utilizing the variable inter-tissue detachment phenotype of MZ itgα5 mutants in three groups of photoconversion experiments (Figure 6E and F). In Group 1 embryos, the neural tube is attached on both sides to the PSM at the level of the photoconverted region (Figure 6G). In Group 2 embryos, tissues are detached ipsilateral to the photoconverted side but attached on the contralateral side (Figure 6H). In Group 3 embryos, tissues are attached ipsilateral to the photoconverted side but detached on the contralateral side (Figure 6I). In the absence of tissue detachment, the photoconverted matrix was remodeled in the medial-lateral direction similar to wild type (Figure 6G and J). In contrast, where tissues were ipsilaterally detached the medial-lateral remodeling was lost (Figure 6H and J). This result demonstrates that inter-tissue adhesion is necessary for the medial-lateral matrix remodeling. Strikingly, tissue detachment contralateral to the photoconverted region also affected matrix remodeling (Figure 6I and J). This contralateral phenotype illustrates a long-range effect of mechanical coupling, namely that ECM remodeling on one side of the embryo is responsive to morphogenic variability on the opposite side of the embryo.

We next asked whether the medial-lateral remodeling of the Fibronectin matrix at the PSM|NT interface was dependent on neural tube convergence by performing the photoconversion experiment in cdh2 mutants. These mutants exhibit variable neural tube convergence as measured by the change in medial-lateral width of the neural tube over time. We also observed bilaterally asymmetric convergence of the neural tube in some embryos, which we quantified as the change in medial-lateral width of the left and right halves (Figure 6M). We divided cdh2 mutants into three groups based the variable neural tube convergence phenotype. In Group 1, the neural tube is symmetric and slowly converges at the level of the photoconverted region (Figure 6K and N). In Group 2, the neural tube is symmetric at the level of the photoconverted region but convergence is severely retarded (Figure 6L and N). In Group 3, the neural tube converges asymmetrically at the level of the photoconverted region (Figure 6M and P). When the neural tube converges symmetrically, the matrix exhibits normal medial-lateral remodeling (Figure 6K and O). However, when the neural tube does not converge, there is no medial-lateral matrix remodeling (Figure 6L and O). Thus, anisotropic matrix remodeling at the PSM|NT interface depends on the neural tube convergence.

An interesting phenotype was observed when the neural tube converges asymmetrically. The side that exhibited a strong neural tube convergence defect displayed normal medial-lateral matrix remodeling, while anisotropic matrix remodeling was lost on the contralateral side where the neural tube converged (Figure 6M and Q). This result indicates that neural tube convergence on one side of the body impacts the Fibronectin matrix remodeling on the opposite side of the body.

We next performed the photoconversion experiments in cdh2; MZ itgα5 double mutants. Mutant embryos were sorted into two groups based on the degree of neural tube convergence. We observed medial-lateral Fibronectin matrix remodeling when the neural tube converged, whereas medial-lateral matrix remodeling was lost with diminished neural tube convergence (Figure 6R–U). Thus, these experiments confirmed that the medial-lateral remodeling of the matrix positively correlates with the neural tube convergence. Lastly, we did not observe any change in the anterior-posterior length of the photoconverted spots for any mutants, similar to wild type (Figure 6—figure supplement 1B–1D).

We revisited these contralateral effects with our in silico model using different parameter values for the left and right sides. Uniform parameters produce symmetric gradients in tension in the neural tube at the left and right PSM|NT interfaces (Figure 7A). When the adhesion stiffness (representing inter-tissue adhesion) is reduced along the left interface relative to the right interface, the left and right PSM become asymmetric in size and the contralateral tension gradient becomes shallower (Figure 7B). This result mimics both the morphological phenotype as well as the contralateral effects on Fibronectin matrix remodeling observed in MZ itgα5^-/- embryos. We computationally implemented asymmetric neural tube morphogenesis observed in some cdh2 mutants by reducing the surface stiffness of one half of the neural tube (Figure 7C). Here, we observed a contralateral reduction in the tension gradient which mimics the contralateral effects on Fibronectin matrix remodeling observed in cdh2^-/- embryos.

Figure 7

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Zebrafish were raised according to standard protocols (Nüsslein-Volhard and Dahm, 2002) and approved by the Yale Institutional Animal Care and Use Committee. The TLAB and TLF wild-type strain were used. The tm101B allele of cdh2 was used (Lele et al., 2002). The MZ itgα5^-/- mutant line is a maternal zygotic mutant line using the thl30 allele (Jülich et al., 2005). The single mutant lines and double mutant lines were crossed to the Tg (hsp70:fn1a-mKIKGR) line to the generate the mutants lines in transgenic background used in photoconversion experiments. Embryos were collected from pair-wise natural matings. Sex-specific data were not collected as zebrafish do not have a strictly genetic sex determination mechanism, and sex is determined after the first 36 hr of development studied here (Wilson et al., 2014).

To generate a double mutant line for fn1a and fn1b genes, we first created single mutant lines for fn1a and fn1b genes separately using CRISPR. Single cell stage embryos were injected with a mix containing: Cas9 mRNA, a single guide RNA (sgRNA) specifically designed to target a chosen sequence in the exon 4 of fn1a gene or in the exon 5 of fn1b gene (see Figure 5—figure supplement 1 for detailed target sequences, and a stop codon cassette oligonucleotide Gagnon et al., 2014). Sequences for the fn1a^-/- line: sgRNA(5’ATTTAGGTGACACTATAGGAGGGCACTCCTACAAGATGTTTTAGAGCTAGAAATAGCAAG3’); stop codon cassette oligonucleotide (5’GAGGGAGGGCACTCCTACAAGTCATGGCGTTTAAACCTTAATTAAGCTGTTGTAGGATTGGAGACACATGGCAGA3’). Sequences for the fn1b^-/- line: sgRNA(5’ATTTAGGTGACACTATAGGACTGCACATGTTTGGGAGGTTTTAGAGCTAGAAATAGCAAG3’); stop codon cassette oligonucleotide (5’CGTGGACTGCACATGTTTGGGTCATGGCGTTTAAACCTTAATTAAGCTGTTGTAGGAGAGGGAAACGGACGCATC3’).

General details regarding the choice of target sequence, the design of the stop codon cassette oligonucleotide, the design and synthesis of the sgRNA and conditions of injection can be found in the detailed supplemental protocol described in Gagnon et al. (2014). Individuals positive for the insertion of the stop codon cassette were identified by PCR on genomic DNA using fn1a DiagA1/stopA primers or fn1b DiagA1/stopA primers: fn1a Diag A1(5’GACTGTACTTGCATTGGCTCTG3’) fn1b DiagA1 (5’GAGCGTTGCTATGATGACTCAC3’) stop A (5’GCTTAATTAAGGTTTAAACGCC3’).

Then single mutant lines were then crossed together to obtain the double mutant line.

Information concerning the sequence of the transgene is described in Figure 5—figure supplement 1A. The plasmid construct containing the transgene was generated following the previously described method (Jülich et al., 2015) with the mKikGR plasmid provided by the Atsushi Miyawaki laboratory (Habuchi et al., 2008). 75 ng/μl of plasmid were co-injected with 120 ng/μl of Tol2 mRNA at 1 cell stage. For both in vivo tracking of Fibronectin matrix dynamics and rescue experiments, embryos were heat-shocked by incubation 30 min at 38°C.

12–14 somite stage embryos were fixed overnight (4°C) in 4% PFA (in 1X PBS). After dechorionation in 1X PBS, embryos were rinsed 3 × 5 min in PBSDT (1% DMSO, 0.1% Triton in 1X PBS), permeabilized 3 min with Proteinase K (5 μg/ml in PBSDT), rinsed 2 × 5 min in PBSDT, post-fixed 20 min at RT in 4% PFA and finally rinsed 3 × 5 min in PBSDT. Blocking was done 2–3 hr at RT in blocking solution (1% Blocking reagent from Roche in PBSDT). Embryos were incubated O/N at 4°C with the primary antibody against Fibronectin (rabbit polyclonal SIGMA, F3648) (1/100 dilution in blocking solution). Embryos were rinsed 2 × 15 min in blocking solution, 2 × 15 min and 3 × 1 hr in PBSDT. Embryos were incubated O/N at 4°C with both secondary antibody (Alexa fluor 555 donkey anti-rabbit IgG, Invitrogen A31572) and Alexa Fluor 488 Phalloidin (Life technologies, A12379) respectively diluted 1/200 and 1/100 in blocking solution. Embryos were incubated 15 min with DAPI solution (100 pg/ml in PBSDT), and then rinsed 2 × 15 min and 3 × 1 hr in PBSDT.

For morphometric analysis, whole embryos were mounted in glass capillaries (1.4 mm diameter) to preserve tissue morphologies (1% agarose in 1X PBS) and imaged with a Lightsheet Z.1 (ZEISS, 20x objective). For imaging of fibronectin matrix and F-actin at high resolution, only the tails of the embryos were mounted between standard slides and coverslips (Fisher Finest Premium Microscope Slides) in 50% glycerol in 1X PBS and imaged with an inverted confocal microscope (ZEISS LSM880, 40x oil objective). For the photoconversion assay, 12 s stage embryos were mounted in glass bottom dishes in drops of 1% agarose in 1X E2 covered with E2 medium. An inverted confocal microscope ZEISS LSM880, 20x objective) was used for both photoconversion and subsequent imaging of the photoconverted region. Regions of interest were photoconverted using a 405 nm laser (30–45 cycles, speed 9 average 1, 10% of maximum laser power).

We model the 2D cross-section of the posterior tail (see Figure 2A in main text) based on an earlier model of soft grains (Åström and Karttunen, 2006). The tissue surfaces are modeled by closed loops of mass-spring chains. An isolated tissue cross-section is made of 50 mass-points connected by elastic springs with homogeneous stiffness constant $K_S$. This parameter ($K_S$) is named 'surface stiffness'. Thus, the tension force along the i-th spring is: ${\overrightarrow{T}}_i=K_S(l_i-l_0){\widehat{h}}_i$, where $l_i$ and $l_0$ are the instantaneous length and the equilibrium (unstretched) length of the i-th spring respectively, and ${\widehat{h}}_i$ is a unit vector along the i-th spring. The sum of these tension forces along all the surface springs represents the surface tension of the tissue. An internal pressure, P also acts in the bulk of each tissue and it is directed perpendicularly outward at the tissue surface. Therefore, the net force acting on the i-th mass point of an isolated tissue (${\overrightarrow{F}}_i^{tissue}$) is given by:

Here, ${\widehat{n}}_i$ denotes an outward directed unit vector perpendicular to the i-th spring. We next modeled the interaction forces between adjacent tissues. Two mass-points belonging to two different tissues are considered to be interacting only if they are within a distance $R_{adh}$ from each other. If this distance between the mass-points is between $R_{rep}$ and $R_{adh}$, there exists a spring-like adhesive force that represents 'inter-tissue adhesion'. In addition, to prevent tissues from penetrating into each other, we implemented 'volume exclusion' by assuming spring-like repulsion forces between the mass-points, when the distance between the mass-points is below $R_{rep}$. Therefore, the interaction forces (${\overrightarrow{F}}_{ij}^{int}$) between i-th and j-th mass-points belonging to adjacent tissues can be summarized as:

Here, $r_{ij}$ is the metric distance between the mass-points (i.e. $r_{ij}=|{\overrightarrow{r}}_i-{\overrightarrow{r}}_j|$) and ${\widehat{r}}_{ij}$ is a unit vector pointing from the j-th to the i-th point. $K_{adh}$ and $K_{rep}$ represent the strength of adhesive and repulsive interactions respectively. In general, $K_{rep}$ is much higher than $K_{adh}$, and it is kept constant in all simulations. On the hand, a variation in $K_{adh}$ represents a variation in the level of inter-tissue adhesion. This parameter, $K_{adh}$ is referred as 'adhesion stiffness' that represents the strength of cell-ECM interaction.

We finally assumed an over-damped dynamics (neglecting inertia) of the mass-points to evolve their positions over time as below:

Here, ${\overrightarrow{v}}_i$ is the velocity of i-th mass-point and $c$ is the viscous coefficient representing a drag force. By evolving the above dynamics in computer, we produced steady final shapes of the tissues at the limit of very long time. We used the 'Explicit Euler' method to simulate the dynamics with a time step set at $\Delta t=0.001$.

In all simulations, the equilibrium spring lengths ($l_0$), the repulsion strength ($K_{rep}$), the viscous coefficient ($c$), the range of forces between tissues ($R_{rep}$ and $R_{adh}$) and the internal pressure (P) are kept constant. These fixed parameters are: $l_0=0.1$,$K_{rep}=30000$, $c=10$, $R_{rep}=0.1$, $R_{adh}=0.2$, and $P=5$. To theoretically explore the impact of variations of the surface stiffness (K_s) and adhesion stiffness (K_adh) in Figure 2B, we used K_adh = 4 and K _adh= 12, while keeping the surface stiffness fixed at K_s = 50. We also used K_s = 35 and K_s = 100, keeping the adhesion stiffness fixed at K_adh = 8. We next systematically varied the surface stiffness (K_s) and adhesion stiffness (K_adh) to mimic the conditions of different genotypes in Figure 2D, E and F. The chosen values for these parameters are: for wild-type K_s = 100, K_adh = 10; for cdh2^-/- mutant K_s = 55, K_adh = 12; for itgα5^-/- mutants K_s = 100, K_adh = 6.5; and for cdh2^-/-;itgα5^-/- mutants K_s = 55, K_adh = 6.5 (also see the parameter space plot in Figure 2D). In Figures 3A and 7A, we used K_s = 200 and K_adh = 10 to generate the symmetric tension distributions along the PSM|NT interfaces.

By using Imaris, we made transverse sections of 3D reconstructs of the posterior tail in every 20 um starting from the last somite boundary. Each transverse section was exported into image-J and rotated such that the medial-lateral axis of the notochord is exactly parallel to the horizontal axis of the image. We then traced the contours of neural tube and PSM to define regions of interests (ROIs). We applied the ‘bounding rectangle’ tool to these ROIs and the medial-lateral and dorsal-ventral lengths of the tissues are given by the width and heights of the rectangle. The same ROIs were also used to determine the cross-sectional area of the tissues (Path in image-J: Analyze - > Set measurements - > select ‘area’ and ‘bounding rectangle’). The interfacial length and angle were measured with standard ‘segmented line tool’ and ‘angle tool’ from image J. The radius of curvature was measured using a online ‘Fit circle’ Image J macro adapted from Pratt V., 1987 (see the website: http://www.math.uab.edu/~chernov/cl/MATLABcircle.html, http://imagej.1557.x6.nabble.com/Re-bending-and-radius-of-curvature-td3686117.html). We provide the image-J macro at the end of the methods section.

To get the cell aspect ratio, we first prepared cross-sections of the PSM prepared by the Imaris software. In these cross-sections, each cell was manually traced and aspect ratio (major axis/minor axis) was extracted using the ‘Shape descriptor’ plugin of ImageJ (path: analyze - > Set measurements - > click ‘shape descriptor’).

To quantify the mean F-actin intensity of the PSM surface cells or PSM internal cells, we first traced a region of interest (ROI) encompassing either surface or internal cells. Then the mean grey value within the ROI was extracted after setting a threshold. The threshold was set such that we remove the cytoplasmic background without removing the low membrane signals in the posterior of the PSM.

We first oriented the images of photoconverted spot in Imaris to get a view perpendicular to the spot in each time frame, and cropped around the spot. These images were then manually thresholded by Image J (path: Image - > Adjust - > Threshold) to account for photobleaching and then converted into binary images. From these binarized images, we used the standard deviation of the white pixel distribution along the medial-lateral and anterior-posterior axes as measures of the medial-lateral width and anterior-posterior height of the photoconverted spot respectively. These metrics were measured by a custom Matlab code provided below:

for k = 1:8 
filename = strcat('SLICE', num2str(k), '.tif'); 
IMECM = imread(filename); 
IMECM = im2bw(IMECM); 
nwhite = nnz(IMECM); 
[rows,cols]=find(IMECM == 1); 
Time(k)=(k-1)*15; 
area(k)=nwhite*0.2*0.2; 
width(k)=std(cols); %./mean(cols); 
height(k)=std(rows); %./mean(rows); 
%aspect(k)=allaspect(biggest); 
%NoElement = CC.NumObjects 
end 
output=[Time', width', height', width'/width(1), height'/height(1)]; 
plot(Time',height'/height(1),'o k-'); hold on 
plot(Time', width'/width(1),'o r-') 
xlswrite('RedSpot_WT-4_PSML.xls',output)

To determine if there is any directional bias in the medial-lateral remodeling of matrix at the PSM|NT interface, we analyzed the displacement field of the photoconverted signal using a Particle Image Velocimetry (PIV) plugin available in Image J (downloadable versions and tutorials at https://sites.google.com/site/qingzongtseng/piv#tuto). For this analysis, images of photoconverted spot were prepared in Imaris to get a view perpendicular to the spot in each time frame. We used the raw grey scale images with full intensity distribution (without any thresholding). In Image-J, we selected the ‘iterative PIV (advanced)’ method, which determines the correlation between two images at a time and requires three user-defined parameters: the ‘interrogation window size’ (IW), the ‘search window size’ (SW), and the ‘vector spacing’ (VS). The set of two images is analyzed by PIV through three passes with a progressive decrease in parameter values so that a fine-tuned vector field of pixel displacement is finally produced. For our analysis, the set of two images corresponds to two consecutive time frames of the spot and we chose the following parameter values: for 1^st pass, IW = 60, SW = 120, VS = 15; for 2^nd pass, IW = 50, SW = 120, VS = 12; and for 3^rd pass, IW = 40, SW = 80, VS = 10. The displacement was analyzed over the course of 45 min (using a frame to frame correlation for the first four consecutive time frames) as most of the remodeling of the photoconverted region happened in the first hour. All the vectors from the Frame 1–2, Frame 2–3 and Frame 3–4 correlations were plotted together in the same rose plot (see Figure 6—figure supplement 1E).

mRNA coding for myl12.1-eGFP was synthesized using mMessage mMachine SP6 kit (Thermofisher, AM1340) according to standard manufacturer instructions using a PCS2+ myl12.1-eGFP plasmid as a template (a gift from Carl-Philipp Heisenberg). Embryos were injected at the one-cell stage with 100 pg of myl12.1-eGFP mRNA (25 ng/μl with a 4 nl droplet). Live embryos were imaged at the 12–14 somites stage on a ZEISS LSM880 (40x water objective, zoom 2, with airyscan detector set in a fast mode acquisition). Each image stack was then automatically sliced transversally every 5 μm using the ‘Reslice tool’ in Fiji. To quantify medio-lateral differences in myl12.1-eGFP localization along the NT border on each transverse slice, we used a custom Fiji code (see code below) to divide the NT border into 3 regions of 3–5 μm in width and measured the mean intensity value in each region.

roiManager("Select", newArray(0,1,6,7)); 
roiManager("Combine"); 
getSelectionCoordinates(xpoints, ypoints); 
makeSelection("polygon", xpoints, ypoints); 
run("Measure"); 
roiManager("Deselect"); 
roiManager("Select", newArray(1,2,5,6)); 
roiManager("Combine"); 
getSelectionCoordinates(xpoints, ypoints); 
makeSelection("polygon", xpoints, ypoints); 
run("Measure"); 
roiManager("Deselect"); 
roiManager("Select", newArray(2,3,4,5)); 
roiManager("Combine"); 
getSelectionCoordinates(xpoints, ypoints); 
makeSelection("polygon", xpoints, ypoints); 
run("Measure"); 
roiManager("Deselect"); 
roiManager("Select", newArray(2,3,4,5));
roiManager("Combine");
getSelectionCoordinates(xpoints, ypoints);
makeSelection("polygon", xpoints, ypoints);
run("Measure");
roiManager("Deselect");
roiManager("Save", "/image path/ROIn.zip"); 
roiManager("Deselect"); 
roiManager("Delete"); 
selectWindow("Results"); 
String.copyResults(); 
Table.deleteRows(0, 2);

To analyze the tension within the Fibronectin matrix, we used the H5 monoclonal antibody which recognizes a tension dependent epitope on the human Fibronectin 10^th FN type III repeat (Cao et al., 2017). We generated a chimeric FN1a-mKIKGR13.2-hsFNIII10-11 protein in which the zebrafish 10^th and 11^th FN type III repeats have been replaced by the human ortholog’s 10^th and 11^th FN type III repeats. (The 10^th FN type III repeat contains the PSHRN motif and was formerly identified as the 9^th FN type III repeat. The 11^th FN type III repeat contains the RGD motif and was formerly identified as the 10^th FN type III repeat.) The PCS2+ FN1a-mKIKGR13.2-hsFNIII10-11 construct was generated by first subcloning the FN1a-mKIKGR13.2 CDS into the PCS2+ plasmid. The zebrafish FNIII10-11 repeats were replaced with a zebrafish codon optimized CDS of the human ortholog FNIII10-11 repeats using Gibson assembly. To synthetize the mRNA, we followed the protocol of Prince and Jessen (2019). Briefly, 30 μg of plasmid was linearized with Not I-HF (200 μl total volume reaction with 8 μl of enzyme) and purified via phenol chloroform extraction, overnight sodium acetate/ethanol precipitation and eluted in 10 μl RNAse free water. mRNA was synthesized using the mMessage Machine SP6 kit (Thermofisher, AM1340) following manufacturer’s instructions for long mRNA synthesis (1 μl of GTP was added to the reaction mix (total volume 21 μl) with 1 μg of DNA template at high concentration (1–3 μl of DNA in the final reaction mix).

One-cell stage embryos were injected with 600–800 pg of FN1a-mKIKGR13.2-hsFNIII10-11 mRNA. Embryos were sorted for bright fluorescence at the 12–14 somite stage, fixed 40 min in 4% PFA (gentle lateral shaking) at RT, washed twice for 5 min in PBST (0.1% Tween in 1X PBS), dechorionated, washed twice for 5 min in PBST, treated for 3 min with 5 μg/ml Proteinase K diluted in PBST, washed 3 × 5 min in PBST, post-fixed 20 min in 4% PFA at RT(gentle lateral shaking), washed 2 × 5 min in PBST, blocked for 3.5 hr at RT (gentle lateral shaking) in blocking solution (2% blocking reagent from Roche, 1X maleic acid buffer (2X maleic acid buffer: 200 mM maleic acid 300 mM NaCl pH = 7.4)), incubated 20 hr at 4°C with H5-V5-tag antibody (a gift from Thomas Barker) diluted at 50 μg/ml in blocking solution, washed 4 × 15 min in 1X maleic acid buffer (gentle lateral shaking at RT), incubated 20H at 4°C with goat anti-V5 antibody (Abcam, Ab 9137 diluted 1/400) in blocking solution, washed 4 × 15 min in 1X maleic acid buffer (gentle lateral shaking at RT), incubated 20H at 4°C with donkey anti-goat Alexa 555 antibodies (Thermofisher scientific A32816) diluted 1/200 in blocking solution, washed 3 × 10 min in 1X maleic acid buffer (gentle lateral shaking at RT), checked for fluorescence, incubated in 25% and 50% glycerol (10 min each) and then mounted in 50% glycerol for confocal imaging (Zeiss LSM880, 40x water objective). All antibodies incubations were done with 50 μl antibody solution without shaking.

To quantify H5 and mKIKGR levels and the H5/mKIKGR ratio, H5 and mKIKGR signals were imaged in conditions minimizing pixel saturation. Tissue interfaces were then segmented using Imaris software and exported for analysis with display adjustments preventing any distortion of the raw pixel values (gamma 1, full dynamic range 0–255). From this step onward, further image treatment and quantification were performed using a custom MATLAB code (see code provided below). To summarize, the H5 and mKIK images were individually thresholded (based on visual assessment) to define background pixels corresponding to areas between matrix fibrils and to discard any saturated pixels (value = 255, which represented than 0.8% of pixels per image, on average). We then generated the H5/mKIK ratiometric image by dividing, pixel by pixel, the H5 image by the mKIKGR image, and displayed it with a scaled color based on pixel value (imagesc MATLAB function). For the quantification of the medial-lateral distributions of each signal, images were divided in 10 sections of equal size along the medial-lateral axis, and the average signal in each section was quantified and then normalized to the average signal in the most lateral section. For the H5 and mKIKGR levels analysis, the calculation of the average signal in each section included the background pixels, which were attributed a value of zero. Thus, the tissue scale average incorporates both matrix brightness as well as matrix density. By contrast, for the H5/mKIKGR ratio analysis, the calculation of the average signal in each section excluded the background pixels that were attributed a NaN value. Thus, this metric reflects the average signal within the matrix elements themselves and represents the local level of tension within the Fibronectin matrix.

MATLAB code for quantification of medial-lateral distribution of H5/mKIK ratio and H5, mKIK signals

clr = 'rgbk'; 
lw = 3; 
fs = 20; 
th1 = 33; 
th2 = 30; 
% th1 and th2 are manually determined to exclude zones in between matrix elements 
imgs = {'H5.tif','mKIK.tif','ratio.tif'}; 
img1 = double(imread(imgs{1})); 
img2 = double(imread(imgs{2})); 
img1_255 = img1(img1 == 255); 
img2_255 = img2(img2 == 255); 
percent255_1 = numel(img1_255)/numel(img1) 
percent255_2 = numel(img2_255)/numel(img2) 
img1(img1 == 0)=NaN; 
img2(img2 == 0)=NaN; 
% 2 previous lines exclude the zones of picture outside of the tissue 
img1(img1 == 255)=NaN; 
img2(img2 == 255)=NaN; 
% 2 previous lines exclude the saturated pixels 
img1(img1 <th1)=NaN; 
img2(img2 <th2)=NaN; 
% Values in the 2 previous lines are NaN for an averaging calulating the brightness of the matrix 
% content itself. Change them to 0 for an avering taking into account the 
% total surface of the tissue 
imgRatio = (img1./img2); 
imgRatio(imgRatio == Inf)=NaN; 
imgRatio(imgRatio == 0)=NaN; 
fig = figure('Renderer', 'painters', 'Position', [0 0 size(imgRatio,2) size(imgRatio,1)]); 
imagesc(imgRatio); 
caxis([0 4]); %change the upper value to tune the colorbar and colormap 
colorbar; 
box off; 
axis off; 
saveas(fig,'ratio.tif','tif'); 
nbin = 10; 
img = {img1, img2, imgRatio}; 
fig = figure; 
figure(2); 
hold on; 
for i = 1:size(imgs,2) 
[imgmean, imgstd, imgmeannorm, position,scale]=mlGradient(img{i}, nbin); 
plot(position, imgmeannorm, clr(i), 'LineWidth', lw, 'DisplayName', imgs{i}); 
set(gca, 'xminorgrid', 'off', 'yminorgrid', 'off', 'fontsize', fs, 'xminortick', 'off',. .. 
'yminortick', 'off', 'linewidth', lw); 
set(gcf, 'color', 'w'); 
legend('Location','bestoutside','FontWeight','Normal','FontSize',12); 
set(gca,'FontSize',fs,'linewidth',lw); 
box on; 
ylabel('Average pixel intensity'); 
xlabel('Medio-lateral position (relative)'); 
end 
Function mlGradient 
function [imgmean,imgstd,imgmeannorm,position,scale]=mlGradient(img, nbin) 
img(isinf(img))=NaN; 
width = size(img, 2) 
height = size(img, 1) 
binsize = floor(width/nbin); 
imgmean = zeros(nbin, 1); 
imgstd = zeros(nbin,1); 
for k = 1:nbin 
if k ~= nbin 
imgmean(k)=nanmean(img(:,(k-1)*binsize+1:k*binsize),'all'); 
imgstd(k)=std(double(img(:,(k-1)*binsize+1:k*binsize)),0,'all', 'omitnan'); 
else 
imgmean(k)=nanmean((img(:,(k-1)*binsize+1:width)),'all'); 
imgstd(k)=std(double(img(:,(k-1)*binsize+1:width)),0,'all', 'omitnan'); 
end 
end 
scale = [0:1/nbin:1]'; 
position = [0+(1/(nbin*2)):1/nbin:1-(1/(nbin*2))]'; 
imgmeannorm = imgmean/imgmean(nbin); 
end

Each experiment includes a minimum of three to five biological replicates. A biological replicate is equal one embryo whereas technical replicates represent separate experiments (e.g. staining reactions or timelapses). In Figure 6, the number of biological and technical replicates are equal. In other figures, we provide the number of biological replicates which derive from at least two technical replicates. In our experience, this sample size combined with detailed quantitative analysis of each sample will reveal any significant differences between experiment samples and controls. Samples were excluded from analysis if the signal to noise in the image data were too low to perform a quantitative analysis. To establish if there is any significant difference between samples in a metric, we used a two-sample T-test provided by the Matlab function’ ttest2’ or by standard ‘ttest’ function in Excel. Significance was defined as: *p<0.05, **p<0.005, and ***p<0.0005.

requires("1.43k");
if (nImages == 0)
exit("No image is open.");
showMessageWithCancel("Circle Fit Macro.", "This macro will reset your ROI Manager.\nDo you
want to proceed?");
roiManager("reset");
}
setTool("multipoint");
waitForUser("Circle Fit Macro.", "Multipoint tool selected.\nSelect points in your image, then click OK.");
roiManager("Add"); roiManager("Select", 0); roiManager("Rename", "points");
getSelectionCoordinates(x, y);
n = x.length;
if(n < 3)
exit("At least 3 points are required to fit a circle.");
sumx = 0; sumy = 0;
for(i = 0;i < n;i++) {
sumx = sumx + x[i]; 
sumy = sumy + y[i];
}
meanx = sumx/n;
meany = sumy/n;
X = newArray(n); Y = newArray(n);
Mxx = 0; Myy = 0; Mxy = 0; Mxz = 0; Myz = 0; Mzz = 0;
for(i = 0;i < n;i++) {
X[i]=x[i] - meanx;
Y[i]=y[i] - meany;
Zi = X[i]*X[i] + Y[i]*Y[i];
Mxy = Mxy + X[i]*Y[i];
Mxx = Mxx + X[i]*X[i];
Myy = Myy + Y[i]*Y[i];
Mxz = Mxz + X[i]*Zi;
Myz = Myz + Y[i]*Zi;
Mzz = Mzz + Zi*Zi;
}
Mxx = Mxx/n;
Myy = Myy/n;
Mxy = Mxy/n;
Mxz = Mxz/n;
Myz = Myz/n;
Mzz = Mzz/n;
Mz = Mxx + Myy;
Cov_xy = Mxx*Myy - Mxy*Mxy;
Mxz2 = Mxz*Mxz;
Myz2 = Myz*Myz;
A2 = 4*Cov_xy - 3*Mz*Mz - Mzz;
A1 = Mzz*Mz + 4*Cov_xy*Mz - Mxz2 - Myz2 - Mz*Mz*Mz;
A0 = Mxz2*Myy + Myz2*Mxx - Mzz*Cov_xy - 2*Mxz*Myz*Mxy + Mz*Mz*Cov_xy;
A22 = A2 + A2;
epsilon = 1e-12;
ynew = 1e+20;
IterMax = 20;
xnew = 0;
//Newton's method starting at x = 0for(iter = 1;i <= IterMax;i++) {
yold = ynew;
ynew = A0 + xnew*(A1 + xnew*(A2 + 4.*xnew*xnew));
if (abs(ynew)>abs(yold)) {
print(‘Newton-Pratt goes wrong direction: |ynew| > |yold|");
xnew = 0;
i = IterMax+1;
}
else {
Dy = A1 + xnew*(A22 + 16*xnew*xnew);
xold = xnew;
xnew = xold ynew/Dy;
if (abs((xnew-xold)/xnew)<epsilon)
i = IterMax+1;
else {
if (iter >= IterMax) {
print(‘Newton-Pratt will not converge’);
xnew = 0;
}
if (xnew <0) {
//print(‘Newton-Pratt negative root: x = "+xnew);
xnew = 0;
}
}
}
}
DET = xnew*xnew - xnew*Mz + Cov_xy;
CenterX = (Mxz*(Myy-xnew)-Myz*Mxy)/(2*DET);
CenterY = (Myz*(Mxx-xnew)-Mxz*Mxy)/(2*DET);
radius = sqrt(CenterX*CenterX + CenterY*CenterY + Mz + 2*xnew);
if(isNaN(radius))
exit("Points selected are collinear.");
CenterX = CenterX + meanx;
CenterY = CenterY + meany;
print("Radius = "+d2s(radius,2));
print("Center coordinates: ("+d2s(CenterX,2)+", "+d2s(CenterY,2)+")");
print("(all units in pixel)");
makeOval(CenterX-radius, CenterY-radius, 2*radius, 2*radius);
roiManager("Add"); roiManager("Select", 1); roiManager("Rename", "circle fit");
setTool("point");
makePoint(CenterX, CenterY);
roiManager("Add"); roiManager("Select", 2); roiManager("Rename", "center");
roiManager("Show All without labels");

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

We thank Madhusudhan Venkadesan for insightful discussions and members of the Holley lab for critical comments on the manuscript. Research support from the NIH initially provided by R01GM107385 and later by R01HD092361 to SAH.

Animal experimentation: Zebrafish were raised according to standard protocols (Nüsslein-Volhard and Dahm, 2002) and approved by the Yale Institutional Animal Care and Use Committee.

© 2020, Guillon et al.

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.