A Y-Junction is the insertion of a row and a column in the cone mosaic

To maintain approximately constant cell spacing as the retina grows by mitotic addition of cone photoreceptors at the rim, rows must be inserted (Fig 1C) [20,22]. It is straightforward to demonstrate that a simple row insertion causes a disruption, that is not limited to a point defect but extends along an entire line, in the cone mosaic (Fig 2A). To avoid this disruption along an entire line of the cone mosaic, it is necessary to consider more complex defects: the insertion of two rows (S1 Fig) or the insertion of a row and a column, neither of which disrupts formation of the cone mosaic. In the zebrafish cone mosaic, the most common topological defect is the insertion of a row and a column, i.e., a ‘Y-Junction’ (Fig 2B and 2C). To understand why the insertion of a row and a column is expected to be the most prevalent defect, we employ a tool used in analyzing defects in physical crystals: the Burgers vector [31,32].

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Fig 2. A Y-Junction, a topological defect in the cone mosaic, is an insertion of a row and a column.

(A) Schematic of simple row insertion in cone mosaic. As new cone photoreceptors are incorporated to the right of the defect, a series of improper cone contacts (black box) within columns form. White dashed lines: rows associated with defect. (B) Schematic of a Y-Junction, a topological defect in the zebrafish cone mosaic. A Y-Junction only disrupts the cone mosaic near the core rather than along an entire line of contacts. White (black) dashed lines: rows (columns) associated with the defect. (C) A Y-Junction in a flat-mount retinal preparation from an adult, double transgenic (Tg[sws1:GFP; sws2:mCherry]) line in which UV and Blue cones express distinct fluorescent reporters (pseudo-colored magenta and blue, respectively) under control of UV and Blue opsin promoters, respectively. Antibody staining labels Red and Green cones (both pseudo-colored green). White (black) dashed lines: rows (columns) associated with the defect. (D) Each UV cone from panel C is connected (white bonds) to its nearest UV cone neighbors. To the left and right of the defect, rows are counted. Seven-sided (five-sided) star: seven-coordinated (five-coordinated) UV cone. (E) A circuit of triangulation bonds around defect from panels C-D. Red arrow is the Burgers vector, the additional bond necessary to close the circuit containing a dislocation. (F) Y-Junction in the same flat-mount retinal preparation as in Fig 1B. The round cells with dim fluorescence are UV cones. The Red cones are pseudo-colored red. The Blue cones are pseudo-colored blue. The remaining cones (bright gray fluorescence) are Green cones. We connect (solid white lines) nearest neighbors in the Red cone sublattice and Blue cone sublattice.

https://doi.org/10.1371/journal.pcbi.1008437.g002

Y-Junction generates minimal lattice deformation, as quantified by Burgers vector

As discussed in the introduction, the unit cell of the cone mosaic is composed of one Blue cone, one UV cone, two Green cones, and two Red cones (Fig 1A and 1B). One can generate an infinite cone mosaic on a flat plane given the unit cell and two lattice vectors, which define the Bravais lattice [55]. The Bravais lattice defines which defects one expects to observe in the cone mosaic lattice, though the distribution of particles in the defect core may vary [31,32,55]. For the sake of clarity, we analyze the defects in the cone mosaic from the perspective of a cone subtype that appears only once in the unit cell: UV cones.

To define the Burgers vector, we build a triangulation for the UV cones in which we connect nearest neighbors of the same cone subtype (Fig 2D and 2E). Away from the core of the defect, every UV cone is surrounded by six nearest UV cone neighbors [31,32,55]. Near the defect core, as in physical crystals that are triangular, one UV cone is surrounded by seven nearest UV cone neighbors. Neighboring this seven-coordinated UV cone is a UV cone that has only five nearest UV cone neighbors. This pair of five- and seven-coordinated UV cones constitutes the core of the dislocation.

We, then, construct a circuit that surrounds the core of the defect. If there were no defect, the circuit would be a parallelogram (Fig 2E). The bottom side of the circuit would contain as many bonds as the top side of the circuit. The right side of the circuit would contain as many bonds as the left side of the circuit. If there is a dislocation inside of the circuit, to close the circuit, one must add a bond, called the Burgers vector [31,32]. The magnitude of the Burgers vector quantifies the amount of lattice deformation associated with the dislocation [31,32].

In physical crystals, where the elastic deformation associated with the dislocation is proportional to the magnitude of the Burgers vector squared, the defect that generates the least deformation is expected to be the most prevalent. Even though we have no reason a priori to treat this biological crystal as elastic, we expect this measure of deformation to be generally applicable. The mechanism that drives ordering in a non-physical crystal likely also resists large deformations due to defects in the lattice.

The Y-Junction in the cone mosaic lattice is the dislocation that introduces the smallest deformation. This can be seen by comparing the Burgers vector of a Y-Junction to the Burgers vector of a double row insertion. For a double row insertion, the length of the Burgers vector is equal to the spacing between UV cones along a column, which is approximately twelve microns, as compared to the Burgers vector of a Y-Junction, with a length of approximately eight microns (for quantification of spacings between UV cones in same column and in same row, see S2 Fig). For the sake of minimizing lattice deformations, we expect double row insertions to be less prevalent than Y-Junctions.

Distribution of Red and Green cones near the Y-Junction core

For Red and Green cones, that each appear twice in the unit cell, we connect nearest neighbors of the same subtype and analyze the spatial distribution in the Y-Junction core. Away from a Y-Junction core, the cells that form the lattice can be grouped into hexagons (Figs 1E and 2F; S3 Fig) [55–57], but this grouping breaks down near the defect core. The distribution of Red and Green cones in the Y-Junction core is variable but is often either a ‘glide’ dislocation or a ‘shuffle’ dislocation, which differ in the distributions of cones at the core.

A ‘glide’ dislocation has a heptagon and a pentagon in the core. For example, in S3A and S3B Fig, one heptagon of Red cones neighbors a pentagon of Red cones. A ‘shuffle’ dislocation has a single octagon in the core. For Red cones (S3E Fig), the octagon contains two UV cones, and for Green cones (S3D and S3F Fig), the octagon contains two Blue cones. Interestingly, both ‘glide’ and ‘shuffle’ dislocations are commonly observed in honeycomb crystals like graphene [56,57].

Y-Junctions form lines, called grain boundaries, from center of retina to its periphery

Having verified that individual Y-Junctions are the dislocations that generate minimal lattice deformation, we next study the spatial distribution of Y-Junctions on the retinal surface. On the retinal hemisphere, the row direction rotates by 2π about the pole of the hemisphere, similar to the convergence of longitudinal lines toward a pole on the globe (Fig 1C). For physical crystals in this orientation, the ground state contains lines of dislocations, called grain boundaries, from the center (pole) of the hemisphere to the edge (equator) [23,58]. In physical crystals, dislocations are usually mobile; therefore, in physical crystals, it is possible for defects to rearrange into grain boundaries after crystallization, regardless of the initial spatial distribution of defects [31,32,36,38,39,58].

In a biological crystal, it is not obvious that the Y-Junctions will form a spatial pattern that is equivalent to the ground state of a physical crystal. If Y-Junctions do form grain boundaries, however, we may be able to leverage that information to understand the mechanism by which the biological crystal forms. By manually tracing rows of UV cones over approximately fifty percent of the retinal area in eight retinae (see Row tracing of flat-mounted retinae, S1 Table), we identified the locations of approximately one thousand seven hundred Y-Junctions.

In flat-mounted retinae, a large fraction of Y-Junctions form grain boundaries

Y-Junctions do, indeed, form grain boundaries that run from the center of the retina to the periphery (Figs 3A, 3B and S4). These grain boundaries reconcile domains of differing crystallographic orientations (Fig 3C and 3D). The angle by which the local row direction rotates from one side of a grain boundary to the other side is determined by the linear density of Y-Junctions in the grain boundary and the length of the Burgers vector of an individual Y-Junction [31,32,58].

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Fig 3. Y-Junctions form lines, called grain boundaries, from the center of the retina to the periphery.

(A) Flat-mounted retina in which UV cones express a transgenic reporter (pseudo-colored magenta) under control of the UV cone opsin promoter. The dorsal side of the retina is left, ventral is right, temporal is down, and nasal is up. White lines: rows of UV cones. Yellow dots: Y-Junctions. Red dots: reverse Y-Junctions, generating row deletions. (B) Row tracing and identification of defects from retina in panel A. (C) Grain boundary from the retina in panel A. White squares indicate onset of patterning. Gray arrows indicate rotation of crystallographic orientation at the grain boundary. (D) Grain boundary presented in panel C with only the row tracing. White squares indicate onset of patterning. (E) Illustration of potential role of defect motion in generating the final spatial distribution of defects. Black region: photoreceptor epithelium. Gray region: retinal margin, where photoreceptor epithelium grows. Yellow circles: Y-Junctions. If defect motion does occur, it could allow defects to move together to form grain boundaries, as indicated by the white arrows. If defect motion is too slow, the patterning mechanism would instead have to generate grain boundaries during initial mosaic formation (not shown).

https://doi.org/10.1371/journal.pcbi.1008437.g003

Although these grain boundaries can be picked out by eye, we developed a less subjective way to identify them, which can be applied to biological data and, later, to simulation results of potential models of cone mosaic formation. Before we describe this method, however, it is worth noting that although the existence of a grain boundary in the idealized case of a single boundary in an infinite crystal is unambiguous, there is no completely sharp definition of a grain boundary in a finite-sized system like ours with a non-zero defect density. If the dislocations needed to accommodate the cone mosaic to the hemispherical surface are uniformly distributed across the retina, then we do not want to say that grain boundaries are present, whereas if these defects are arranged in perfectly straight lines, we certainly do. There is necessarily some measure of arbitrariness, however, in where exactly we draw the line between these two limiting cases. Importantly, we will use our method of identifying grain boundaries primarily to compare the fraction of Y-junctions in grain boundaries in experimental images to the corresponding fraction in simulation results. We will use the same cutoff to define a grain boundary in both sorts of data, and we are able to show that our central conclusions are independent of the precise choice of cutoff.

To determine whether a specific Y-Junction is in a grain boundary, we quantify how much the row orientation rotates in the vicinity of the Y-Junction; we, then, check whether that rotation of row orientation is above a fixed threshold (Fig 3C and 3D; see Detection of grain boundaries in flat-mounted, row-traced retinae). By applying this measure to the eight analyzed retinae, we found that a majority of the identified Y-Junctions are in grain boundaries (S4 Fig and Tables 1 and S2).

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Table 1. Quantification of grain boundaries in flat-mounted retinae.

From the row tracing of eight retinae, we show the percentage of retinal area analyzed, the number of Y-Junctions identified, and the percentage of Y-Junctions in grain boundaries (see Detection of grain boundaries in flat-mounted, row-traced retinae). Here we assume that the rotation of row orientation about a Y-Junction must be greater than twelve degrees for that Y-Junction to be in a grain boundary, but we systematically scan that (arbitrary) threshold in S2 Table.

https://doi.org/10.1371/journal.pcbi.1008437.t001

Defect motion is not responsible for grain boundary formation

To study whether defect motion is responsible for the existence of grain boundaries, we need to observe the dynamics of individual cones during cone mosaic formation in live fish rather than the fixed positions of cones in flat-mounted retinae. If defect motion is not responsible for the existence of grain boundaries, we will be able to test potential models of cone mosaic formation based on their ability to form grain boundaries at the boundary of an expanding cellular crystal.

To quantify the motion of individual UV cones during cone mosaic development, one must observe the same region of the retina in the same fish at two distinct time points as the fish remains alive. To make sure that one can locate the same set of UV cones at the two distinct time points, one must have some common point as a reference for UV cone positions in the two images.

To locate the same set of UV cones at two different time points in live fish, we use transgenic zebrafish in which the UV cones express a nuclear-localized, photoconvertible fluorescent protein under the control of the UV cone opsin promoter (see Generation of transgenic zebrafish with nuclear-localized photoconvertible (green-to-red) EOS protein expressed specifically in UV cones and nEOS photoconversion and imaging). We photoconvert and image a small patch at the retinal margin, near where newly generated UV cones are incorporated into the growing retina (Fig 4A and 4B). At the time of imaging, the cone mosaic is composed of approximately eighty columns and is growing by approximately three columns of cones per day [40]. Two, three, or four days later, we image both photoconverted and non-photoconverted UV cones in the same retinal area (Fig 4A and 4B). By comparing the two images, we can determine how the different UV cones have moved relative to each other (see Tracking UV cone positions in photoconverted regions and measuring glide motion).

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Fig 4. By photoconverting UV cones near the retinal margin, we track Y-Junction motion.

(A) Schematic of photoconverted UV cones in photoreceptor epithelium near the retinal margin. We photoconvert a patch of UV cones (purple box) near the margin, where new cone photoreceptors are incorporated by mitotic addition. After two, three or four days of retinal growth, we image the photo-converted region, which is now separated from the margin by newly added retinal tissue (black). (B) Example of patch of UV cones immediately after photoconversion and two days later. In this line (Tg[sws1:nEOS]), UV cones express a nuclear-localized, photoconvertible fluorescent protein under control of the UV cone opsin promoter. The non-photoconverted fluorescent protein is pseudo-colored yellow, and the photoconverted fluorescent protein is pseudo-colored magenta. Retinal margin is to the right of each image. Approximately eight columns of UV cones have been added in two days since photoconversion. (C) Glide motion involves subtle motion of individual UV cones near the Y-junction core. Magenta circles are UV cones with photoconverted fluorescent signal, and yellow circles are surrounding UV cones with non-photoconverted fluorescent signal. Every cone is connected to nearest neighbors. Five-sided and seven-sided stars: five-coordinated and seven-coordinated UV cones, respectively. Dashed black line: the “inserted” row. The two triangulations on the right describe positions of UV cones (from the left triangulation) after glide in the direction denoted by gray arrow. Note that the assignment of five- and seven-coordinated UV cones has shifted by one row.

https://doi.org/10.1371/journal.pcbi.1008437.g004

Eliminating the possibility of defect motion perpendicular to the Burgers vector

To understand what types of motion we expect to observe, it is useful to revisit the concept of a Burgers vector. When a dislocation moves along the direction defined by the Burgers vector, this motion requires only subtle rearrangements of individual particles, here UV cones, near the core of the dislocation. This motion is called glide motion [31,32]. To illustrate this motion, we focus on the inserted row associated with the Y-Junction as well as the five- and seven-coordinated UV cones. When the dislocation glides by one row, as shown in Fig 4C, the initial inserted row incorporates itself into a neighboring row, as a new inserted row is generated. The assignment of the five- and seven-coordinated UV cones shifts by one row in the direction of the glide motion. Glide by one row is the flipping of one bond in the UV cone triangulation near the Y-Junction core.

On the other hand, motion of dislocations perpendicular to the direction defined by the Burgers vector, called climb motion, requires the creation or annihilation of point defects, which are interstitials or vacancies in the crystal [31,32]. A vacancy in the cone mosaic corresponds to the absence of an entire unit cell—that is, six contiguous cones—which we never observe (S5 Fig). Therefore, in monitoring the motion of individual UV cones, we test specifically for glide motion rather than climb motion [38].

Quantifying glide motion

Based on the positions of UV cones, we connect nearest neighbors in the lattice (see Tracking nuclear positions in photoconverted regions and Tracking UV cone positions in photoconverted regions and measuring glide motion). We identify the location of the Y-Junction core based on the location of the inserted row (Fig 5A). To quantify glide motion, we search for bond flips (Figs 4C and 5A) along the glide line (see Tracking UV cone positions in photoconverted regions and measuring glide motion) near the Y-Junction core between the time of photoconversion and the time of subsequent imaging (two, three, or four days later).

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Fig 5. By estimating the timescale of Y-Junction motion, we conclude that Y-Junctions line up into grain boundaries during initial mosaic formation rather than by subsequent Y-Junction motion.

(A) Expected motion of individual UV cones in the case of glide motion by one row in either direction. Left triangulation shows UV cones near the defect core; a UV cone sits on each site of triangulation. The center and right panels overlay the positions of UV cones before (black) and after (gray) glide in the direction denoted by the gray arrows. Note that the originally five- and seven-coordinated UV cones in the black triangulation both become six-coordinated. (B,B’) Example of Y-Junction in photoconverted region in which no bond flips in two days. The photoconverted fluorescent signal in UV cone nuclei is pseudo-colored magenta. For reference, the same five cones are numbered in both images. White lines: triangulation of UV cones. (C,C’) Y-Junction in photoconverted region from Fig 4B. One bond has flipped in the triangulation over two days, meaning that the Y-Junction has glided in the direction of the gray arrow by a row. (D,D’) Observation of grain boundary growth during initial mosaic formation. Immediately after photoconversion, one observes seven Y-Junctions (yellow dots), six within a grain boundary and an isolated Y-Junction nearby. White dashed lines: rows of UV cones. Two days later, one observes two additional Y-Junctions in the grain boundary. Based on the constraint that a Y-Junction does not glide faster than one row in two days, Y-Junctions must have initially formed within the regions indicated by red arrows (thus, within the grain boundary). White double-headed arrow: columns of cones incorporated since photoconversion.

https://doi.org/10.1371/journal.pcbi.1008437.g005

We observe non-negligible motion near the core; however, in our sample of sixteen Y-Junctions in thirteen distinct retinae (S3 Table), we never observe glide motion by more than one row per two days, where glide motion by one row is illustrated in Fig 4C. We show two examples of Y-Junctions within photoconverted regions: an example in which there is no glide motion (Fig 5B and 5B’) and an example in which the Y-Junction glides by one row in two days (Fig 5C and 5C’). These experiments thus provide an upper bound on the rate of glide motion of one row per two days. If we compare this constraint on the timescale of glide motion to the timescale of grain boundary formation, we can determine whether grain boundaries form via the coalescence of initially isolated dislocations that move together after initial crystallization.

Comparing the timescale of glide motion to the timescale of grain boundary formation

In many of our photoconversion experiments, we photoconvert a patch of UV cones near at least one existing grain boundary (Figs 5D, 5D’ and S6). At the time of subsequent imaging of the UV cones near the photoconverted region (two to four days later), approximately eight UV cone columns are newly incorporated into the cone mosaic. After identifying Y-Junctions in the newly incorporated columns of UV cones, we can ask whether their locations are correlated with the positions of existing (i.e., observed at the time of photoconversion) grain boundaries. If we do observe a correlation between the positions of new Y-Junctions and existing lines of Y-Junctions, we can ask whether that correlation could be explained by glide of Y-junctions from completely random initial positions during the 2–4 days of observation or whether it must imply that new Y-junctions form preferentially near existing grain boundaries.

Table 2 lists the lines of Y-junctions we observed in our live imaging at the time of photoconversion; for each example, Table 2 also lists how much the row orientation rotates at the retinal margin. Similar to our identification of grain boundaries in flat-mounted retinae, we classify a line of Y-Junctions as a grain boundary if the rotation of the row orientation at the retinal margin exceeds a fixed cutoff (see Statistical significance of growing grain boundaries in live fish). For a cutoff of twelve degrees as in Table 1, out of the eighteen samples, nine samples have grain boundaries. Since some samples have two grain boundaries, in total we observe eleven grain boundaries above the twelve-degree cutoff in live fish (S6 Fig; and Tables 2 and S4). Two to four days later, we identify the positions of newly incorporated Y-Junctions. To these samples, we apply the following null model: each new Y-Junction’s initial position is uncorrelated with existing grain boundaries, but after formation, each new defect moves one row closer to the closest grain boundary per two days (see Statistical significance of growing grain boundaries in live fish). We find that newly incorporated defects are more aligned with existing grain boundaries than can be explained by this null model (p<0.0004). (We explore the effect of changing the threshold in row rotation angle used to define a grain boundary in S4 Table.) Because glide motion is slow relative to the grain boundary growth, we conclude that grain boundaries form at the time of cone mosaic formation, not by subsequent defect motion.

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Table 2. Growth of grain boundaries in live fish.

For images immediately after photoconversion, we measure how much the row orientation rotates at the retinal margin (see Statistical significance of growing grain boundaries in live fish). Although we define a grain boundary based on a rotation of the row orientation by twelve degrees or more, for the sake of completeness, this table lists all cases with a rotation of the row orientation of ten degrees or more. (See S4 Table for the effect of changing this angle cutoff.) For each of these cases, we count the number of defects in the corresponding grain boundary at the time of photoconversion and the number of defects added to the grain boundary by the time of later imaging. Though we photoconverted one region per fish, that region sometimes neighbors two grain boundaries, allowing us to measure growth of two grain boundaries in the same fish (e.g., 4–1 and 4–2). Fig 5D and 5D’ show grain boundary 3.

https://doi.org/10.1371/journal.pcbi.1008437.t002

Testing computational models of cone mosaic formation

Near the rim of the retinal hemisphere (Fig 6A) is a region defined as the precolumn area, where newly generated but not fully differentiated cones are in approximately correct locations relative to each other based on cone subtype before the formation of mature columns [40]. Composed of differentiating post-mitotic cones, this precolumn area lies between two regions: a central region of mature columns and the rim, which contains proliferative cells. It remains unclear how proper positioning of cones by subtype in the precolumn area occurs, but importantly, this must occur within hours of the generation of post-mitotic cells by terminal divisions of neighboring proliferative cells [40].

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Fig 6. Cone mosaic formation is minimally disrupted in Red cone mutant but is significantly disrupted in UV cone mutant.

(A) Cross-section of wild-type retina in which immunostaining of Red cone opsin labels Red cones. White arrow indicates approximate location of precolumn area [40]. (B) Cross-section of trβ2 mutant in which immunostaining of Red cone opsin labels Red cones. Note absence of differentiated Red cones. White arrow indicates approximate location of precolumn area. (C) Apical plane of wild-type cone mosaic lattice in retinal flat-mount in which anti-ZO-1 stains cell profiles. UV cones are indicated in the inset. (D) Apical plane of cone mosaic in retinal flat-mount from trβ2 mutant, which lacks Red cones. Anti-ZO-1 stains cell profiles. UV cones (indicated in inset) identified based on large, rounded profiles. The triangular lattice of UV cones is minimally disrupted in absence of Red cones. (E) Apical plane of cones in retinal flat-mount from tbx2b mutant, which lacks UV cones. The cone mosaic is disrupted in this mutant.

https://doi.org/10.1371/journal.pcbi.1008437.g006

In this section, we ask whether our observations of grain boundary creation can provide information about the mechanism for proper positioning of cones by subtype into immature columns. In principle, one can imagine two extremal models for the creation of regular cell fate patterns in biological tissues. In the first model, cell fates are specified first, and motile cells with a clear identity then rearrange themselves into the final pattern; in the second model, cells instead first arrange themselves in space, then the correct fate pattern is imposed on this static cell packing by cell-cell signaling. Evidence from embryonic retinae suggests that post-mitotic cones are of fixed subtype and move relative to each other during integration into the retina [42].

This finding from embryonic retinae, together with other examples of cell-cell repulsion in neural cell mosaics [43–47], suggests that mosaic formation in zebrafish retinae falls closer to the first model. To test whether such a picture is consistent with the observed behavior of grain boundaries, we construct a computational model in which fate-specified cones repel each other during differentiation. After finding that this cell-cell repulsion model does indeed generate grain boundaries during the initial process of differentiation, we demonstrate that the alternative model, in which cell fate patterns arise through lateral inhibition in a static and disordered packing, is not likely to be responsible for cone mosaic formation. Before turning to the descriptions of the two models, we first gather some additional experimental data, on which cone subtypes are essential to establishing a crystalline mosaic and on lattice anisotropy in the zebrafish mosaic, to help in model formulation.

Absence of Red cones only minimally disrupts cone mosaic formation, but the absence of UV cones significantly disrupts mosaic formation

As noted above, crystals are described by identifying the smallest repeating unit that can be used to build the crystal, the unit cell–consisting of one Blue, one UV, two Red, and two Green cones (Fig 1A) for the zebrafish retina–and the way different unit cells are positioned relative to each other in space, the Bravais lattice–for the zebrafish retina, a slightly anisotropic triangular lattice [31,32,55]. It is well established that the defect core structures can depend on all the features of the unit cell but that elastic interactions between two defects are determined only by the Bravais lattice and the defects’ Burgers vectors [23,24,34,55,59]. Thus, we expect that most features of the spatial distribution of Y-junctions, which should depend primarily on defect-defect interactions [23,24,34,37,50,51], can be recapitulated by a model in which each unit cell is replaced by a single cone photoreceptor. In order to provide biological justification for this simplification of the cone mosaic and to determine which cone subtype to focus on, we consider mutants in which certain cone subtypes are absent.

To evaluate the role of Red cones in establishing a crystalline mosaic, we generated a targeted mutation in a single gene that resulted in a loss of Red cones (Fig 6B; see CRISPR-Cas9 mediated mutation in the thrb gene). This gene, trß2, is an early fate marker of Red cones and is expressed in proliferating progenitors and mature Red cones, but not other cone subtypes [40,42]. All other cone subtypes are still present in the outer retinal layer (S7A Fig). Strikingly, in the trß2 mutant, we find that cone mosaic formation, including ordering of UV cones, is only minimally disrupted by the absence of Red cones (Figs 6D and S7B).

The robustness of cone mosaic formation to the absence of Red cones is in marked contrast with previous experiments with tbx2b mutants in which UV cones are largely absent [60]. In tbx2b mutants, it is difficult to discern long-ranged crystalline order in the cone positions (Fig 6E). The spatial distribution of cones and absence of long-ranged order are similar in other zebrafish mutants in which cell-cell adhesion is perturbed [21,41]. Given this evidence from both trß2 mutants and tbx2b mutants, we simplify the cone mosaic to a lattice of UV cones.

Measuring lattice anisotropy of the cone mosaic in live fish

Previous studies have established the importance of anisotropy of the mosaic lattice both in its formation and refinement [21,40–42]. In modeling the cone mosaic lattice as a lattice of UV cones, we need to make sure that we produce a lattice with the same anisotropic spacing as the cone mosaic. For this reason, we first measure the lattice vectors of the UV cone sublattice in live fish. We use images of photoconverted regions, immediately after photoconversion, in which there are no Y-Junctions (S2 Fig).

In an isotropic triangular lattice, the ratio of the distance between UV cones in the column direction to the distance between UV cones in the row direction is equal to the square root of three. We find that this ratio is approximately six-fifths in our live images of the UV cone sublattice (S2 Fig). This means that the spacing along the row direction is longer than would be expected in an isotropic triangular lattice. We use this degree of anisotropy as an input for modelling the cone mosaic formation.

Model of repulsive interactions between fate-committed cells generates grain boundaries during initial cone mosaic formation

In building a model of cone mosaic formation, we hypothesize that cell motion is generated by repulsive interactions between fate-committed cones of the same subtype as they differentiate in a mechanically anisotropic medium. This model is motivated by cell mosaics in other vertebrate retinal layers [43,45–47] and in the nervous system in Drosophila larvae [44]. For example, in mice, specific neuron subtypes disperse after fate commitment and during morphological differentiation [45,47]. By cell ablation, previous investigators have established that for retinal horizontal cells in mice, the cells are fate-committed and homotypically interact via transient neurites (i.e., cell processes) [45]. Similarly, in Drosophila, certain classes of peripheral sensory neurons tile the body wall. By ablation of dendritic processes, previous investigators have established that neurons of a specific class (class IV) interact homotypically to establish a tiling of the body wall by non-overlapping cell territories [44].

To model repulsive interactions between fate-committed cones of the same subtype, leading to a preferred cell spacing in the row and column directions, we employ a phase-field model of crystallization (Fig 7A and 7B). These models are widely employed to describe various phenomena in physical crystals, where they are known to give qualitatively, and in some cases quantitatively, accurate descriptions of processes including nucleation of crystalline domains in a supercooled fluid and the mechanical hardness of a solid based on the microstructure of crystalline domains, while having vastly improved computational efficiency compared to particle-based models [61,62]. Importantly, the basic form of a phase-field crystal model can be determined from system symmetries and similar phenomenological considerations, so that these models in some sense provide a generic description of crystallization, independent of many specifics of the microscopic interactions between individual particles.

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Fig 7. Phase-field crystal model of cone mosaic formation.

(A) Schematic of contact-interaction model in which fate-committed cones interact homotypically and form an anisotropic lattice. UV cones (magenta circles) interact with nearest neighbors of same subtype. White arrows: contact interactions. New fate-committed cones are incorporated to right of the ordered region. (B) In the phase-field crystal model, a continuum field describes the positions of UV cones. UV cones are most likely to be found around peaks in the density (white regions) and least likely to be found in troughs (dark regions). (C) Flat-mounted retina (same as Fig 3A and 3B) in which UV cones express a transgenic reporter under control of the UV cone opsin promoter. Yellow dots: Y-Junctions. Red dots: reverse Y-Junctions. (D) Simulation of phase-field crystal model on the surface of a cone. The number of initial rows and total number of columns are comparable to those in retinae, and this leads to a defect density similar to that measured in experiments. The degree of anisotropy of the triangular lattice is constrained by the anisotropy in unit cell dimensions measured in our live-imaging experiments. Yellow dots: Y-Junctions. Red dots: reverse Y-Junctions.

https://doi.org/10.1371/journal.pcbi.1008437.g007

Instead of representing a crystal as composed of discrete particles, phase-field crystal models employ a continuum field, which corresponds to modulations in the particle density (Fig 7B) [61–68]. Particles are most likely to be found around peaks in the density and least likely to be found in troughs. Based on our simplification of the cone mosaic to a lattice of UV cones, the particles in our case can be identified with UV cones. If the field is homogeneous in space, the UV cones form a “liquid,” with no clear periodicity. If the field is periodic in space, they form an ordered crystal.

The phase-field crystal model assumes that the density field evolves to minimize a free energy functional while conserving the total number of particles. This dynamical law can be derived from a microscopic model of individual particles whose pairwise interactions include a short-ranged repulsion and that are subject to frictional damping and thermal noise; the approximations involved in the derivation amount in essence to assuming that higher-order correlations relax quickly compared to two-body correlations and that the crystal structure can be described by keeping only Fourier modes in the particle density of the dominant wavelength [62]. The existence of a dominant wavelength, in turn, arises from the fact that the particles repel each other at short range and thus have a preferred minimum spacing. (Although there is no a priori reason that the dynamics of a biological system should be governed by a free energy, we expect that many models with symmetric, pairwise, repulsive interactions between unpolarized cells can be mapped onto an effective free energy of the phase-field crystal form.)

To model the growth of the cone mosaic, we initialize the phase-field to be entirely uniform except near the center of the simulation domain (S1–S6 Videos; see Numerical solutions of anisotropic phase-field crystal model on cone). The crystalline region grows outward from the center, as fate-committed cones in the disordered region are incorporated (Fig 7A and 7B).

For a fixed anisotropy of the lattice spacing (S2 Fig), the simulation results depend on two free parameters in the model free energy. The first parameter specifies the degree of undercooling. When the temperature is below the melting temperature (i.e., the liquid is “undercooled”), the density field is unstable to the formation of periodic structures; otherwise, only the liquid phase is stable. As there is no clear analogue to temperature in our system, we interpret this parameter as quantifying the strength of the interactions relative to the random noise in cone cell motion. The second parameter is the mean of the cone density field, which is conserved by the dynamics.

The values of these two free parameters, the interaction strength and the mean of the density modulation field, determine which phase (or phases) are stable. Depending on these parameters, this model can generate three phases: a constant (“liquid”) phase, a striped phase, and a triangular phase [61,64]. We can constrain these two free parameters to the region of the phase diagram in which the only stable phase is the triangular phase. To ensure that any conclusions from the model do not depend on fine-tuning of parameters (S8 Fig; see Numerical solutions of anisotropic phase-field crystal model on cone), we scan the parameters over a one-dimensional cut of this region of the phase diagram, while also varying the magnitude of noise in the initial conditions.

By simulating the phase-field crystal model in a geometry of comparable size to the retina and with comparable defect densities, we find that this model generically produces spatial distributions of Y-Junctions that are similar to those observed in the flat-mounted retinae (Figs 3, 7C, 7D, S4 and S8 and S5 Table and S6 Video). As we scan over simulation parameters, the fraction of Y-Junctions in grain boundaries, as quantified by the same measure that we applied to the flat-mounted retinae with the same cutoff as in Table 1, is slightly greater than fifty percent on average (S8 and S9 Figs and S5 Table). For our simulations with the smallest magnitude of undercooling, the fraction of Y-Junctions in grain boundaries reaches approximately seventy percent. In the flat-mounted retinae, approximately seventy-five percent of Y-Junctions are in grain boundaries (Tables 1 and S2; see Detection of grain boundaries in simulated cone mosaic for a discussion of sources of error that could lead to a mild overcount of Y-Junctions in grain boundaries in flat-mounted retinae and see Alternative method for detection of grain boundaries for an alternative grain boundary detection method that avoids these issues at the expense of introducing additional ad hoc parameters). Thus, this model, for which there is supportive evidence in embryonic retinae [42], tends to group a majority of Y-Junctions into grain boundaries, consistent with observations of flat-mounted retinae (Figs 7C, 7D, S8 and S9 and Tables S2 and S5).

Additional insights generated by model of cell-cell repulsive interactions

Armed with this cell-cell repulsion model, we now generate insights into cone mosaic formation. First, we address how the specific orientation of the cone mosaic lattice is selected and how it is maintained as the crystal grows. With isotropic interactions in an isotropic medium, at the onset of ordering, crystallites form with random orientations (S4 Video). Additionally, as the crystal grows outward, the crystallites tend to rotate into an orientation that is misaligned with the orientation observed in the retinae by thirty degrees (i.e., maximally misaligned) (S10A Fig and S3 and S4 Videos). For an anisotropic crystal, as in the retinae, the correct crystallographic orientation is selected even without spatial ordering in the original cone positions, and that crystallographic orientation is maintained as the crystal grows (S10B Fig and S2 and S5 Videos).

This model also suggests a mechanism by which grain boundaries form during initial mosaic formation. We consistently observe in our phase-field crystal simulations that the density profiles near a grain boundary remain poorly resolved even as neighboring regions of the crystal grow outward, leading to a characteristic V-shape of the crystal surface (S10C Fig). Our model predicts that there should be a lag in proper positioning of cones near a grain boundary.

Implausibility of lateral inhibition mechanism for cone mosaic formation

Before concluding, we briefly discuss a possible alternative model of cone mosaic formation and consider its shortcomings in generating a crystalline mosaic. The alternative model is motivated by cell fate lattices, like those formed by sensory bristles or R8 photoreceptors in Drosophila [3,5–7,27,48,49], where neural cells are selected through a process of lateral inhibition mediated by the Notch signaling system. In these examples, cell motion is largely absent during pattern formation. Mathematical models of motionless tissues in which cells differentiate, inhibiting neighboring cells within some range from committing to the same fate, have had some success in reproducing the patterns observed in these systems [3,5,6,27,69–76].

Specifically, we adapt a lateral inhibition model originally developed to describe sensory bristle patterning in Drosophila [27] (see Numerical solutions of lateral inhibition on disordered cell packing). An important difference between sensory bristle patterning in Drosophila and cone mosaic formation in zebrafish is that the lattice vectors in the cone mosaic are much shorter, in units of cell diameters, than the lattice vectors of the sensory bristle lattice. There are five to six cells between nearest-neighbor sensory bristle cells [27], but only one to two cells between nearest-neighbor UV cones. Therefore, we expect a cone mosaic lattice generated by lateral inhibition to be more sensitive to disorder in cell packing than is the sensory bristle lattice.

In S11 Fig, we provide illustrative examples of lateral inhibition with different signaling ranges (see Numerical solutions of lateral inhibition on disordered cell packing). As a wave of differentiation moves from the left side of the cell packing to the right, individual cells differentiate into the inhibiting, neural cell fate (dark cells, u≈1), and these cells signal to their neighbors, causing them to adopt the alternative (non-neural, u≈0) fate. For the shortest inhibition range, appropriate for the UV cone mosaic, the resulting fate pattern has many defects, disrupting the long-ranged crystalline order expected in the cone mosaic [21,40]. Thus, in this model, the underlying disordered cell packing prevents the formation of a precise triangular lattice, even before we consider the effects of patterning on a curved surface. This contrasts with the results of the model discussed in previous sections in which fate specification occurs first and cells of known spectral fate then move into the correct lattice position.

Since the basic impediment to pattern formation by lateral inhibition in this model is the disorder in the cell packing, one might be tempted to consider a model in which lateral inhibition instead acts to impose a fate pattern on an ordered packing of equipotent cells. In such a model, however, there would have to be defects in the ordered packing for it to fit onto the retina’s curved surface, and the pattern of defects in the eventual cone mosaic might be expected to follow the pattern of defects in the underlying packing. Thus, the problem would again be reduced to that of arranging cells (necessarily with some short-ranged repulsion, as they cannot overlap) into a crystal on the surface of a growing hemisphere–that is, essentially to the same problem solved by our earlier phase-field model of cone mosaic formation.

Zebrafish

Fish were maintained at approximately 28°C on a 14/10 h light/ dark cycle with standard husbandry procedures [79]. Zebrafish lines, Tg(-5.5sws1: EGFP)kj9 [80], Tg(-3.2sws2: mCherry)mi2007 [21], Tg(trβ2: tdTomato) [42], Tg(-3.2sws2: EGFP) [81], Tg(gnat2:H2A-CFP), and pigment mutant ruby carrying albino (slc45a)^b4/b4 and roy^a9/a9 were used. All animal procedures were approved by the Institutional Animal Care and Use Committee at the University of Michigan.

Histology

Retinal dissection, fixation and immunocytochemistry were performed as previously described [40,82,83]. Briefly, the isolated retina was fixed in 4% paraformaldehyde with 5% sucrose in 0.1M phosphate buffer, pH 7.4, at 4°C overnight. After antigen retrieval with 10 mM sodium citrate in 0.05% tween 20 (pH 6.0), the retina was incubated in blocking buffer for 2 hours followed by primary antibody incubation, mouse anti-Zonula Occludens (ZO1-1A1, 1:200, ThermoFisher Scientific, Waltham, MA) and rabbit anti-GFP (1:200, ThermoFisher Scientific) at room temperature overnight. Incubation with secondary antibodies (Alexa Fluor 555 and 649, ThermoFisher) was performed at room temperature overnight, and the retina was flat-mounted on a glass slide. For retinal cross sections, affinity-purified rabbit polyclonal opsin antibodies, a gift from Dr. David R. Hyde [84], were used. Images were acquired with a Zeiss AxioImage ZI Epifluorescent Microscope (Carl Zeiss Microimaging, Thornwood, NY) equipped with an ApoTome attachment for optical sectioning structured illumination, Leica DM6000 Upright Microscope System (Leica Microsystems, Werzlarm Germany) and a Leica TCS SP5 confocal microscope equipped with Leica 40X HCX PL APO CS Oil Immersion lens.

Generation of transgenic zebrafish with nuclear-localized photoconvertible (green-to-red) EOS protein expressed specifically in UV cones

Multi-Gateway-based tol2 kit system was used to generate expression vectors [85]. In brief, the 5’ entry clone, p5E- 5.5sws1 [80], middle entry clone, pME-nEOS (gift from Dr. David Raible), and 3’ Entry clone, p3E-polyA, were assembled into a destination vector, pDestTol2pA [85] using LR Clonase II Plus enzyme (Thermo Fisher Scientific). Embryos of the transparent ruby genetic background [40] at the 1-cell stage were injected with 1 nL of solution containing 25 pg plasmid DNA and 25 pg tol2 transposase mRNA [86]. Founders (F0) with germline transmission of the transgene were identified by outcrossing with wildtype animals, and their F1 progenies were screened for nEOS expression at 4 days post fertilization.

CRISPR-Cas9 mediated mutation in the thrb gene

A genetic mutation targeting the type 2 isoform of the thrb gene (synonym, trß2) was generated by CRISPR-Cas9 gene editing methods [87]. Briefly, pT7 gRNA vector (Addgene #46759) was used as a template to construct the thrb2 gRNA [87]. PCR based method was performed using specific primers, 5’-GGGGTAATACGACTCACTATAGGCAACACAGCCAACCCTATGTTTTAGAGCTAGAAATAGCAAG-3’; 5’-AAAAAGCACCGACTCGGTGCC-3’. The MEGAscript T7 Kit (Ambion Inc., Austin, TX) was used to transcribe the gRNA. For the nlsCas9nl mRNA synthesis and purification, mMESSAGE mMACHINE T7 Transcription Kit (Ambion) and Qiagen RNeasy mini kit (Qiagen, Hilden, Germany) were used. For genotyping of the thrb mutation, PCR fragments of the thrb gene, amplified using specific primer set, 5’-CATGGTGTAAGTGGCGGATATG -3’; 5’-TCCACTGCATCTGAGAGAAATCC-3’, were subjected to restriction with BstXI (New England Biolabs, Ipswich, MA).

nEOS photoconversion and imaging

Photoconversion of nEOS protein was performed on ruby; Tg(sws1:nEos) fish [88]. Juvenile zebrafish (0.7 to 0.88 cm standard body length) were anesthetized with 0.672 mg/ml Tricaine S/ MS-222 (Western Chemical Inc., Ferndale, WA) and placed dorsal side down on a 50 mm glass bottom petri dish with a No. 1.5 coverslip (MarTek Corporation, Ashlan MA, see [40]) and held in place with damped Kimwipes. Imaging and photoconversion were performed with a Leica TCS SP8 LSCM (Leica Microsystems, Werzlar, Germany) equipped with Leica 40X PL APO CS2 Water Immersion lens, 1.1 NA with 650 μm working distance. Green to red photoconversion of nEOS protein was performed by a 405 Diode laser at 400 Hz scan speed with a resolution of 512 x 512 pixels in the xy dimension at a single optical plane. Pre and post photoconversion images were captured with the White Light Laser tuned to 506 nm for nEOS (green) and 573 nm for nEOS (red). Leica HyD hybrid detectors were tuned to 516–525 nm for nEOS (green) and 620–761 nm nEOS (red).

Large tile scans of flat-mounted retinae

Large tile scans of entire flat-mounted retinae from adult Tg(sws1:EGFP) zebrafish immunostained for ZO1 were acquired with a Leica TCS SP8 LSCM (Leica Microsystems) equipped with Leica 20X PL APO Dry lens. The GFP signal was recovered by immunostaining with anti-GFP antibody. The White Light Laser was tuned to 555 nm for Alexa Fluor 555 and 649 nm for Alexa Fluor 649. The Leica HyD hybrid detectors were tuned to 600–641 nm for Alexa Fluor 555 and 701–751 nm for Alexa Fluor 649. Images were acquired at 700 Hz scan speed with a resolution of 2048 x 2048 pixels in the xy dimension with a 2.0 μm interval between optical sections in the z-dimension.

Row tracing of flat-mounted retinae

We manually traced rows of UV cones, starting near the region coinciding with the disorder-to-order transition (Figs 3A, 3B and S4 and Tables 1 and S1). The row tracing extends over approximately one hundred columns of UV cones from the larval remnant to the periphery, avoiding regions of the retinae that were damaged during flat mounting. Based on the row tracing, we identified where rows are inserted (i.e., Y-Junctions) and where rows are removed (i.e., reverse Y-Junctions, see S1 Table).

Detection of grain boundaries in flat-mounted, row-traced retinae

In this section, we describe our method for identifying grain boundaries in images from experiments or simulations. In an infinite crystal, the rotation of the crystal axes in its vicinity is a clear signature of a grain boundary: The direction of these axes is unperturbed far from an isolated dislocation, but a finite linear density of dislocations must induce a rotation in the grain orientation. The situation is somewhat more ambiguous in our finite-sized, flat-mounted retinae (Figs 3A, 3B and S4). In particular, it is certainly possible to distribute Y-junctions roughly uniformly across the retina, such that the orientation of the rows in the crystalline cone mosaic rotates smoothly through 360 degrees as one goes around the pole of the retinal hemisphere. In such a situation, we would not say that grain boundaries are present. Instead, we would like to define a grain boundary as a region where the Y-junctions tend to coalesce into radially directed lines, or, equivalently, as one where the row orientation undergoes a rapid change. There is, however, no unique answer to the question of how large a jump in orientation over how short a distance is required for a grain boundary to be present. Instead, we will be forced to introduce cutoffs that, while corresponding reasonably well to our intuitive picture of what constitutes a grain boundary, are nonetheless somewhat arbitrary.

Specifically, in order to facilitate comparisons between experimental and simulation results, we would like to analyze our data in a way that tells us whether or not a given Y-Junction is part of a grain boundary. In the flat-mounted retinae, we have the positions of Y-Junctions, which generate row insertions, in the traced regions. In addition, individual rows are traced (meaning that each row has a distinct curve running along it). To define whether a given Y-junction belongs to a grain boundary, we check whether the row orientation of the surrounding domain rotates appreciably in the vicinity of the Y-Junction.

This method requires two arbitrary parameters:

1. the length-scale over which we average the domain rotation (which we will call Δr)
2. how large the domain rotation must be for a defect to be “in a grain boundary”

We describe positions on the flattened retinae via polar coordinates (where the approximate center of the retina lies at the origin). For each Y-junction, we average the domain orientation in five distinct boxes (which we will use to compute how much the row orientation changes near the Y-Junction):

1. The first box is centered on the Y-junction of interest (at radial coordinate r = r₀ and angular coordinate θ = θ₀). It runs from to and from to .
2. The second box runs from to and from to .
3. The third box runs from to and from to .
4. The fourth box runs from to and from to .
5. The fifth box runs from to and from to .

To calculate the orientation within each box, we first identify all row lines that run through the box. For each row line (a skeletonized line composed of individual pixels), we calculate its orientation via principal component analysis:

1. computing the covariance matrix of the row’s pixels’ positions within the box of interest (cov; MATLAB 2016B, MathWorks, Natick, MA)
2. subsequently identifying the direction along which the row’s pixels’ positions vary the most (eig; MATLAB 2016B, MathWorks, Natick, MA)

By convention, row lines always run from the center of the retina to its periphery, so we encode each row’s orientation with a unit vector that runs along the row from center to periphery (rather than from periphery to center). Then, within the box of interest, we do a weighted average over the unit vectors (one for each row line within the box), where the weight of each row is simply the number of pixels of that row line that fall within the box. Then, we convert this weighted average of row vectors to a unit vector, which represent the row orientation within the box.

Now that we have the row orientation for each of the five boxes (unit vector for box 1, …, unit vector for box 5). We calculate how much the row orientation changes about the Y-Junction along the θ direction () as well as along the r direction (). If the grain boundaries were purely radial, we could focus on Δϕ₄₅ and ignore Δϕ₂₃. In flat-mounted retina, because of retinal deformations due to cutting and flattening, grain boundaries are frequently not well-aligned with the nominal radial direction. Because of this, we compute takes into account the change in row orientation in the vicinity of the Y-Junction in both directions.

To determine whether a Y-Junction is within a grain boundary or not, we compare its Δϕ_rms to some cutoff (for example, ten degrees or twelve degrees or fourteen degrees). If Δϕ_rms is greater than the chosen cutoff, the Y-Junction is in a grain boundary; if not, the Y-Junction is not in a grain boundary. Because by its very nature this cutoff is arbitrary, we vary this parameter (as well as Δr) to see how the fraction of defects in grain boundaries changes with these parameters. We find that the fraction of Y-Junctions in grain boundaries depends very weakly on Δr, but strongly on the cutoff in rotation of row orientation. To see if our simulations and experimental images have similar fractions of Y-Junctions in grain boundaries, we fix the values of these two parameters when comparing flat-mounted retinae to simulations.

We conclude this section by noting a possible pitfall of this method as applied to flat-mounted retinae: In regions where the row orientation is slightly warped due to deformation of the retina from the flattening process, Y-junctions may be designated as “in a grain boundary” (even if they are isolated) because of this artifactual deformation of row orientations. Thus, it is possible that this approach somewhat overestimates the fraction of Y-junctions in grain boundaries in flat-mount images compared to the fraction in simulations. (See below for an alternative method of defining grain boundaries that is expected to be less sensitive to distortion upon flattening.)

Detection of grain boundaries in simulated cone mosaic

The basic method for detection of grain boundaries in simulated retinae is identical to that for flat-mounted (experimental) retinae with a couple of exceptions:

First, unlike for the experimental images, we do not trace rows by hand. Instead, we compute a Delaunay triangulation that connects peaks in the simulated particle density. Then, we identify “row” bonds based on which bonds are well-aligned with the local radial direction (that runs from the cone periphery to the cone center). More specifically, for each peak in the particle density, we:

1. compute the negative of the local radial direction ()
2. compute a vector from this peak to each of its nearest neighbors in the Delaunay triangulation
3. compute the dot product of with each of the vectors to nearest neighbors

Then, whichever bond has the largest (positive) value of this dot product is a “row” bond. Note that based on this definition, a particle density peak associated with a Y-Junction ends up with three row bonds because it has two row bonds emanating from it (towards the periphery of the simulation domain) and one row bond flowing into it (from the center of the simulation domain). On the other hand, a reverse Y-Junction only has one row bond because no row bonds emanate from it to the periphery of the simulation domain.

After identifying row bonds in the Delaunay triangulation, to calculate the row orientation within a given box, we find row bonds that fall entirely within the box. We compute a weighted average over these row bonds, represented as vectors that run (mostly) from center of simulated retina to its periphery; each weight is equal to the length of the corresponding row bond.

Alternative method for detection of grain boundaries

The definition of grain boundaries described in the preceding sections and used in the main text has the advantages that it connects directly to the basic definition of a grain boundary as the interface between two domains with different crystallographic orientations and that it depends on only two parameters, both with clear biophysical interpretations. As we noted above, however, when it is applied to images of flat-mounted retinae, it is potentially susceptible to systematic biases caused by distortions introduced by the mounting and fixation process. To verify that these biases do not dominate our results, in this section we consider an alternative method of defining grain boundaries that views the boundaries as collections of Y-junctions forming a (nearly) straight line. Because it does not depend as strongly on the local row orientation, this method is more robust against artifacts like deformation of flat-mounted tissue. This robustness, however, comes with the tradeoff that the method requires setting several ad hoc parameters that lack clear and appealing interpretations.

To define grain boundaries, we search for approximately linear chains of five Y-Junctions, which are nearest neighbors. To the image of all Y-Junctions in the traced regions, we apply the following algorithm:

1. Loop through all Y-Junctions one-by-one. We will build a chain of nearest neighbors, of five Y-Junctions, for each Y-Junction.
   1. Look for the Y-Junction’s nearest neighboring Y-Junction, using a k-nearest neighbors search (knnsearch; MATLAB 2016B, MathWorks, Natick, MA). Add the nearest neighbor to the chain.
   2. For that nearest neighbor, add its nearest neighbor, excluding any Y-Junctions that already belong to the chain.
   3. Repeat b until you have a chain of five Y-Junctions, including the Y-Junction that initialized the chain.
2. Now, based on the calculation in step 1, every Y-Junction, indexed by i below, has a chain of nearest neighbors, including the Y-Junction itself. We index the five defects in the chain by j. The position of the j^th defect in the i^th chain is . For each Y-Junction, we compute the following sum:
3. Now, based on the calculations in steps 1 and 2, every Y-Junction, indexed by i, has a chain of nearest neighbors, which is assigned a score a_i. If a_i = 1, the chain of five Y-Junctions is perfectly linear. If a_i>a_gb, we call the i^th chain a grain boundary. We set the cutoff a_gb equal to seven-eighths.
4. We initialize an empty array, in which we will store Y-Junctions that belong to grain boundaries. Loop through all Y-Junctions, indexed by i, one-by-one.
   1. If a_i>a_gb, store (i.e., in the array of all Y-Junctions in grain boundaries) the five Y-Junctions which belong to this chain.

We also perform this computation (steps 1–4) on the Y-Junctions in simulations of the phase-field crystal model. In that case, we map the cone frustum to the flat plane. This does not generate distortions because cones are isometric to the plane. We, then, calculate which dislocations form grain boundaries, respecting the periodic boundary conditions of the flattened cone frustum. To compare the experimental results to simulations, see S6 Table and S12 Fig.

The fraction of defects in grain boundaries, as quantified by this alternative method that we applied to the flat-mounted retinae, is approximately sixty percent (S12 Fig). In the eight flat-mounted retinae, approximately fifty percent of Y-Junctions are in grain boundaries (S6 Table).

Tracking nuclear positions in photoconverted regions

In the photoconversion experiments, we observe the same region of the same retina at two different times in live fish. Given a nucleus at one time point, we want to find the same nucleus in the image at the other time point. One image of the region is taken immediately after photoconversion, which we call day 0. Across fish, we vary the time between photoconversion and the time of the second observation (i.e., two days after photoconversion at the earliest and four days after photoconversion at the latest). We call the second time point day 2–4.

At both times of observation for each fish, we have an image with two channels. One channel corresponds to the color of the photoconverted fluorescent protein. The other channel corresponds to the color of the non-photoconverted fluorescent protein. For the image analysis below, we use the photoconverted channel at both times. The image is three-dimensional, and the plane which contains the UV cone nuclei (i.e., where the fluorescent protein is localized) is mostly parallel to the x-y plane. This fact allows us to perform most of the computations, for tracking each nucleus from one image to the other, based on two-dimensional projections.

For each z-stack, we compute a two-dimensional wiener filter (wiener2; MATLAB 2016B Image Processing Toolbox, MathWorks) with a filter size of eight pixels, which is approximately a micron. This filter removes noisy specks (i.e., spikes in intensity at small length scales). We, then, compute a two-dimensional projection by summing over z-stacks. The photoconverted UV cones are in the middle of the image. The intensity in the photoconverted channel is significantly weaker for UV cones near the edge of the image. This provides us the reference boundary by which we can identify common nuclei (i.e., which nucleus in the day 2–4 image corresponds to a specific nucleus in the day 0 image).

We perform an image registration, computing the combination of rotation and translation that optimizes the normalized cross-correlation between the two images (normxcorr2; MATLAB 2016B Image Processing Toolbox, MathWorks). Then, we segment nuclei in the two images. Because the intensity of UV cone nuclei varies significantly across the image, we use both adaptive thresholding (adaptthresh; MATLAB 2016B Image Processing Toolbox, MathWorks) and a low absolute threshold. We morphologically open the thresholded image, followed by morphological closing. We fill holes in the image (imfill; MATLAB 2016B Image Processing Toolbox, MathWorks) and clear the border of the image (imclearborder; MATLAB 2016B Image Processing Toolbox, MathWorks). We perform minimal manual correction of these segmentations. Given that we have aligned the two images and segmented the nuclei, we track each nucleus from one image to the other by computing for each nucleus in the day 0 image its nearest neighbor in the day 2–4 image (knnsearch; MATLAB 2016B, MathWorks). As a sanity check, for each nucleus in the day 2–4 image we compute its nearest neighbor in the day 0 image to make sure that this calculation returns, for each nucleus, an answer consistent with the reverse calculation. We manually correct any errors.

Following this segmentation and identification of common nuclei between the two images, we want to estimate the three-dimensional position of each nucleus based on the raw z-stacks rather than on a post-processed version. We identify a circular region, of radius two and a half microns, in the xy-plane centered on each of the segmented nuclei. This radius is larger in the xy-plane than the nuclear radius but small enough not to encompass other nuclei. This circular region corresponds to a pillar in the z-direction. To estimate the three-dimensional position of each nucleus in both images, we use the raw z-stacks, computing the center of intensity of each pillar (i.e., weighted average of voxel positions in each pillar where the weights are the voxel intensities). At the end of this entire procedure, for each nucleus common to both images, we know its position at both time points.

Tracking UV cone positions in photoconverted regions and measuring glide motion

To identify the location of the Y-Junction, we need to calculate a triangulation over the nuclear positions. At both day 0 and at day 2–4, the UV cone nuclei positions in each experiment are well fit by a plane, which we fit by simple least-squares minimization (see RMSE information in S3 Table). For calculating the triangulation, we project the UV cone positions onto the plane of best fit. We, then, calculate the triangulation in that plane (delaunayTriangulation; MATLAB 2016B, MathWorks).

We want to track movement of UV cones near the Y-Junction core along the direction of glide motion. We systematically search for bond flips (i.e., any change in nearest neighbor assignments as in Fig 5C and 5C’) between day 0 and day 2–4 for any bonds that could be flipped in glide motion (see Fig 4C). Which UV cone bonds lie along the glide line is always unambiguous based on the triangulation. We never observe glide motion by more than one row, as illustrated in Figs 4C and 5A. We show an experimental example of glide motion by one row in Fig 5C and 5C’ (see S3 Table).

Statistical significance of growing grain boundaries in live fish

In S6 Fig, we show examples of images in which we can identify newly incorporated Y-Junctions lining up into grain boundaries. These are images of UV cone nuclei near the retinal margin (i.e., where the layer grows by addition of post-mitotic cells). These images are oriented such that the margin is parallel to the y-axis. Our field of view in these images contains approximately forty rows of UV cones and forty columns of UV cones.

To identify grain boundaries that are already visible immediately after photoconversion, we trace rows of UV cones at the retinal margin. If immediately after photoconversion the row direction rotates about a group of defects by ten degrees or more at the margin, we call the group of Y-Junctions in-between the rotated rows a grain boundary. (S4 Table also lists results for analyses where this threshold is instead taken to be twelve or fourteen degrees.) As a justification for this use of row rotation to identify grain boundaries, see Figs 3C, 3D, 5D, 5D’ and S6. For a cutoff of twelve degrees, out of the eighteen samples, nine samples have grain boundaries. Since some samples have two grain boundaries, in total we observe eleven grain boundaries above the twelve-degree cutoff in live fish (Table 2).

All subsequent analysis is based on the later image (i.e., two, three, or four days later). We trace rows in the later image. We identify newly inserted rows (i.e., newly incorporated Y-Junctions) in the later image, and we again identify the old defects within each grain boundary (i.e., those not newly incorporated). We calculate a one-dimensional coordinate for the location of each grain boundary in the later image. This one-dimensional coordinate is the average of y-coordinates (i.e., axis approximately parallel to the margin in the image) of all defects (i.e., not newly incorporated) within each grain boundary. We are interested in how close, along the y-direction, newly incorporated Y-Junctions are to the nearest grain boundary in the image.

Suppose that in the image there is only one grain boundary that is identifiable at the time of photoconversion and later imaging. Suppose this grain boundary is located at coordinate y_gb. The image spans from y = 0 to y = y_max. For each new Y-Junction, we generate one hundred thousand random Y-Junction positions, uniformly distributed from y = 0 to y = y_max (rand; MATLAB 2016B, MathWorks). We calculate the distance between each of these one hundred thousand random Y-Junction positions and the grain boundary (at y_gb). We call the array of distances between each random Y-Junction position and the grain boundary . We also store the actual distance, which we call δ_actual, between each observed newly incorporated Y-Junction in the image and the nearest grain boundary in the image.

Suppose that in the image there are two grain boundaries that are identifiable at the time of photoconversion and later imaging. Suppose their coordinates are y_gb,1 and y_gb,2. The image spans from y = 0 to y = y_max. For each new Y-Junction, we generate one hundred thousand random Y-Junction positions, uniformly distributed from y = 0 to y = y_max (rand; MATLAB 2016B, MathWorks). We calculate the distance between each of these one hundred thousand random Y-Junction positions and the nearest grain boundary (at either y_gb,1 or y_gb,2). We call the array of distances between each random Y-Junction position and the nearest grain boundary . We also store the actual distance, δ_actual, between each newly incorporated Y-Junction and its nearest grain boundary.

Based on the procedure outlined above, for each new Y-Junction, we have a vector of length one hundred thousand and a scalar, and δ_actual, respectively (see S4 Table). If after random incorporation with respect to the grain boundaries in the image, a newly incorporated Y-Junction moves at a speed of one row per two days closer to the nearest grain boundary (with spacing between rows approximately equal to six microns as shown in S2 Fig), the distribution of distances with respect to the nearest grain boundary becomes , where Δt is the time between photoconversion and later imaging (i.e., two, three, or four days).

Depending on which cutoff we use to define a grain boundary (S4 Table), we have as many as thirty-seven new Y-Junctions across twelve samples (with at least one grain boundary at the retinal margin). We would like to compare these scalar values of δ_actual to a concatenated vector of across the newly incorporated defects. This concatenated vector has a length that is one hundred thousand times as long as δ_actual. We test whether the distribution of δ_actual has the same median as the concatenated vector of . We assign a p-value to that comparison via Mann-Whitney U-test (ranksum; MATLAB 2016B Statistics and Machine Learning Toolbox, MathWorks).

Numerical solutions of lateral inhibition on disordered cell packing

Starting with a Voronoi tessellation of uniformly (randomly) distributed points, we generated large, disordered, periodic cell packings (e.g., 20,000 total cells in S11 Fig) via vertex model simulations with equal tensions on all edges as described in [89,90]. We model dynamics of individual cell fates on the static cell packing according to the model described in [27], but do not include noise in the dynamics (D = 0). Since we changed some aspects of the model, including the external signaling gradient and the noise in fate, we describe the model in [27] below for the sake of clarity. The fate of cell i, u_i, evolves as: where s_i is the signal each cell receives from other cells as well as from any external gradients. We interpret the u = 1 fate as the UV cone spectral subtype and the u = 0 fate to be other spectral subtypes. Also, f(u, s) is sigmoidal: .

The signal that cell i receives, s_i, includes an external, time-dependent signal s₀(x, t) as well as signals from neighboring cells in a distance-dependent manner.

The external signal provided to the cells has the following form: where S₀ = 1 and and . τ is the timescale for cell fate dynamics, and l is the characteristic cell-cell signaling range. The distance-dependent coupling constant c_ij between cell i and cell j is of the form: , where d_ij is the distance between the centroids of cell i and cell j. No cell signals to itself directly: c_ii = 0.

A cell of fate u_j produces signal D*(u_j) = a(u_j)D(u_j). The ligand level of cell j, D(u_j), is directly proportional to the fate u_j. The ligand activity of cell j, called a(u_j), is of the form: . The constants a₀ and a₁ are 0.05 and 0.95, respectively.

To explore the effects of cell-cell signaling range on the final fate pattern, we systematically change the signaling range l from to to , where A₀ is the mean cell area. All cells are initially in the u = 0 state. The sigmoidal signaling front, sharper than the characteristic cell size, starts at left side of the packing (x = 0 at t = 0) and moves to the right. In the wake of the front, individual cells differentiate into the u≈1 fate, inhibiting their neighbors from adopting the u≈1 fate within the specified cell-cell signaling range. We solve the differential equations for cell fates using ode45 (MATLAB 2016B, MathWorks).

Numerical solutions of anisotropic phase-field crystal model on cone

The free energy functional F for an anisotropic phase-field ψ, with undercooling parameter R, is an integral over the simulation domain [61,68,91]: (stretched along the x direction) (stretched along the r direction in polar coordinates)

The particle-density-modulation field ψ is evolved to minimize the free energy functional F while conserving the mean of the field ():

For solving this equation on the cone, we first map the cone to a flat plane, which does not generate distortions because the cone is isometric to the plane. We, then, use the Laplacian for polar coordinates, respecting the periodic boundary condition of the cone. We set up the problem in terms of the variables u, v, ψ as defined in [92]. For computational efficiency, we take the Fourier transform along any direction which is periodic (e.g., along the θ direction on the cone). We use first-order implicit-explicit methods, as in [93], treating the non-linear term in ψ explicitly. We implement all derivatives by finite differences [94]. We use no-flux boundary conditions at each non-periodic boundary.

We evolve the system with a fixed step size in time (Δt = 0.075). The computational grid is such that there are approximately 25 grid points per lattice spacing along the circumferential direction in the initial column, and approximately 10 grid points per lattice spacing along the radial direction.

We systematically vary the parameters of the phase-field crystal model as done in [61]; note that stretching the crystal does not change the phase diagram as discussed in [64]. In short, we take a 1D cut of the phase diagram, setting as we vary the undercooling parameter R. For this cut, we also vary the strength of the noise in the initial conditions. These three parameters are the parameters on which we do not have any quantitative handle (relative to experiments); therefore, we perform this robustness analysis on these three parameters. The results of this analysis are shown in S8 Fig.

Geometry for cone mosaic growth

The retina is approximately hemispheric. A hemisphere might, thus, seem like the most obvious choice of geometry in which to test cone mosaic growth. It is important to note, however, that the retina is not a hemisphere of a fixed radius during development. Its radius increases as new retinal cells are incorporated. As the hemisphere dilates, the existing cone photoreceptor layer must be deformed. The exact way in which the existing pattern deforms, and how that affects subsequent cone mosaic formation, is beyond the scope of this paper. Our aim is to choose a minimal geometry which allows us to test the phase-field crystal model’s ability to form grain boundaries.

We choose a geometry in which we can easily tune the defect density. We choose a cone frustum, which is constructed by slicing off the top of a cone with a plane that is parallel to its base. By changing the level at which we slice the cone, we tune the number of UV cones in the initial column. We choose the top level such that there are approximately two hundred initial rows, which is consistent with the number of initial rows identified in flat-mounted retinae. By changing the opening angle of the cone, we can tune the number of Y-Junctions required to maintain constant cell-cell spacing per added column. We choose an opening angle such that two row insertions are required per added column to maintain approximately constant cell-cell spacing. The number of Y-Junctions necessary to maintain constant cell-cell spacing is comparable to the number of Y-Junctions observed in the retinae (Table 1 and S4 Fig).

Initial conditions for cone mosaic growth

At the very top level of the cone frustum, we lay down one column of cones (see one-mode approximation in [61]). We add a white-noise mask to this initial column of cones. Because we do not have any quantitative handle on the noise in cell positions at the onset of patterning in the zebrafish retina, we vary the noise strength, exploring its effect on the subsequent pattern of defects (S8 Fig and S5 Table).