Hexagonal Patterning of the Insect Compound Eye: Facet Area Variation, Defects, and Disorder

Abstract

The regular hexagonal array morphology of facets (ommatidia) in the Drosophila compound eye is accomplished by regulation of cell differentiation and planar cell polarity during development. Mutations in certain genes disrupt regulation, causing a breakdown of this perfect symmetry, so that the ommatidial pattern shows onset of disorder in the form of packing defects. We analyze a variety of such mutants and compare them to normal (wild-type), finding that mutants show increased local variation in ommatidial area, which is sufficient to induce a significant number of defects. A model formalism based on Voronoi construction is developed to predict the observed correlation between ommatidium size variation and the number of defects, and to study the onset of disorder in this system with statistical tools. The model uncovers a previously unknown large-scale systematic size variation of the ommatidia across the eye of both wild-type and mutant animals. Such systematic variation of area, as well as its statistical fluctuations, are found to have distinct effects on eye disorder that can both be quantitatively modeled. Furthermore, the topological order is also influenced by the internal structure of the ommatidia, with cells of greater relative mechanical stiffness providing constraints to ommatidial deformation and thus to defect generation. Without free parameters, the simulation predicts the size-topology correlation for both wild-type and mutant eyes. This work develops formalisms of size-topology correlation that are very general and can be potentially applied to other cellular structures near the onset of disorder.

Introduction

Animal development is a progressive phenomenon in which cells within the body are organized into spatial patterns of increasing complexity. This is most obviously seen in the collective sheets of adherent cells known as epithelial tissues. During epithelial tissue growth, cells undergo division and have predictable effects on large-scale order, disrupting it to a certain degree (1). Cessation of proliferation, onset of differentiation, and cell morphogenesis are all processes that can then transform disordered epithelia into highly ordered patterns. These processes require coordination, and cell-cell signals organize differentiation while planar cell polarity (PCP) orients cell morphogenesis along prescribed tissue axes.

A classic system to study this transformation from a disordered to an ordered epithelium is the compound eye of Drosophila melanogaster. Like the eyes of many insects, it is an ordered array of hexagonal units called ommatidia, arranged in a crystalline pattern (Fig. 1 a). Each eye is a simple epithelium founded by 20 cells during embryogenesis, and the epithelium grows in size over the next four days to encompass 20,000 cells. Cells become postmitotic and progressively differentiate in a pattern where each periodic unit corresponds to a nascent ommatidium. Initially, neighboring ommatidia have variable numbers of differentiated cells, but this variation gradually diminishes over time until midway through animal pupation (Fig. 1 b). By this time, every ommatidium is composed of eight photoreceptor cells, four cone cells, two primary pigment cells, three secondary pigment cells, one tertiary pigment cell, and one bristle group of three cells. This 21-cell unit is repeated with virtually no variation in cell number and internal arrangement (>99.8% identical (2)). The apical-basal organization of each ommatidium is also highly reproducible. The cone cells and primary pigment cells occupy most of the cross section of the apical domain, while the other pigment cells and bristle group form a thin frame (Fig. 1 c). The basal domain is occupied by the photoreceptors and the secondary/tertiary pigment cells. The shape of an individual ommatidium rarely deviates from a regular hexagon, so that the overall eye pattern is a perfect honeycomb, a regular close-packed structure observed in many other biological and inanimate systems (3, 4, 5). At this stage of development, the eye epithelium is flat and has not yet acquired the curvature of the adult eye; therefore, potential effects of curvature on the regularity of the hexagonal pattern (6, 7) are not present.

Figure 1.

(a) Scanning electron micrograph of an adult Drosophila compound eye. (b) Eye epithelia dissected from midpupal stage and imaged for Discs Large protein. The left sample is wild-type and the right sample is a Fz mutant. (Arrows) Body axis orientation relative to the eye epithelia. (c) Magnified image of a single ommatidium in which the optical section is taken through the apical domain of the epithelium. C, cone cells; P, primary pigment cells; S, secondary pigment cells; T, tertiary pigment cells; B, bristle group. (d) Scanning electron micrograph of an Fz mutant adult compound eye. (e) Example of topological defects in a Fz mutant, color-coded by number of neighbors n. (f) Topology diagram of the entire eye from a Fz mutant (left) and its Voronoi reconstruction based on ommatidial centroids from image analysis (right); the latter faithfully reproduces most defects in the original experimental image. (g) Probability distribution of ommatidial area for wild-type and Fz mutant; the latter shows a considerably greater width (larger $c_A$ value). The bar in (a) provides a scale for all photographic images and is 35 μm for (a) and (d), 40 μm for (b) and (f), 1.5 μm for (c), and 5.5 μm for (e). To see this figure in color, go online.

One hundred years of Drosophila research has generated a collection of mutants that affect the hexagonal pattern of the compound eye. These mutants exhibit defects in the pattern that vary in extent, and are visually detectable even under a low power microscope. Such mutants have been given the term “rough” (Fig. 1 d). The primary causes of such roughness vary greatly. For example, the number of cells per ommatidium might become variable in a mutant. When a dominant-negative form of the Ras1 protein (Ras) is synthesized in R7 precursor cells, proper differentiation is inhibited which causes sporadic loss of this cell from certain ommatidia (8). Other mutants have the same number of cells per ommatidium, but their internal arrangement becomes variable. Mutation of the sevenless (Sev) gene generates ommatidia with 20 cells per ommatidium, but the photoreceptors in an ommatidium can adopt one of three different arrangements (9). Still other mutants affect both cell number and orientation of cells in ommatidia, such as the PCP mutant frizzled (Fz) (10). An Fz mutant has a much more disordered pattern at the midpupal stage (Fig. 1 b). In particular, the broken regularity of the hexagonal lattice can be quantified by the identification of topological defects, i.e., ommatidia whose number of neighbors is different from six (Fig. 1, e and f).

It is well known in solid-state physics that two-dimensional packings and tilings can lose translational (crystalline) order without losing orientational order (hexatic phases (3, 11)). Indeed, we have shown in previous work that the occurrence of defects in a two-dimensional pattern is intimately correlated with a disruption of the uniformity of cross-sectional areas of the individual units: a sufficiently large area variation (polydispersity) will give rise to defects, whose density then follows a well-defined function of polydispersity (12, 13, 14, 15). We reasoned that perhaps the mutations that cause roughness do not directly disrupt long-range order, but perturb the uniformity of areas of the ommatidia, so that the presence of defects is merely a consequence of local area disorder. To test this hypothesis, we quantify the relation between area variation and topological disorder in Ras, Sev, and Fz mutant eyes. Our analysis has led to discovering a normal long-range area variation in the eye that, when taken into account together with local statistical variations, explains the experimental results without reference to long-range orientational disorder. We also found that the different mechanical properties of distinct classes of cells in the ommatidia influence the topological disorder significantly.

Materials and Methods give the experimental data acquisition, as well as the protocols for image analysis and data extraction. In Translational and Orientational Order, we quantify the translational and orientational order of our samples using tools from statistical mechanics. In Statistical Area Variation, we then show by comparison with simulations that local statistical variation alone cannot explain the size-topology correlations observed in the retinal epithelium. We quantify and model the role of the systematic area variation, then the role of varying cell-mechanical properties is discussed in Cone Cell Exclusion Volume. In Order Measures in the Model, the statistical order measures from the model are compared with experimental data. Conclusions are presented in the Discussion.

Materials and Methods

Experimental data acquisition

Genetic strains of Drosophila melanogaster were obtained from the Bloomington Drosophila Stock Center. Analysis was performed on the strain genotypes listed in the following table. We will use the abbreviations in the first column of Table 1 as shorthand notations to address the different strains.

Table 1.

The Different Drosophila Genotypes Used in This Investigation

Name	Genotype	Reference
Wild-type	W1118	(32)
Ras	W1118; $sev>Ras{1}^{T17N}/+$	(8)
Sev	w1118$se{v}^{d2}$	(33)
Fz	Df(3L)fz-GF3b/Df(3L)Exel6122	(34, 35)

Animals were raised on standard cornmeal molasses food at 23°C. Compound eyes were dissected from animals 48 h after pupariation, and were fixed for 1 h in 40 mg/mL paraformaldehyde in PBS. After washing and permeabilization in PBST (PBS + $0.1\text{\%}$ (v/v) Triton-X100), eyes were incubated overnight at 4°C with MAb 4F3 anti-Discs Large (Developmental Studies Hybridoma Bank, University of Iowa, Iowa City, IA) diluted 1:10 (v/v) in PBST. Eyes were washed in PBST and incubated for 1 h at 23°C with goat anti-mouse IgG conjugated to the fluorophore Alexa488 (Thermo Fisher Scientific, Waltham, MA). Eyes were washed in PBST, followed by clearing in Vectashield (Vector Labs, Burlingame, CA) and mounting for microscopy.

Confocal laser-scanning microscopy was performed on a model No. SP5 system using a Plan Apo 20×/0.8 objective and 72 μm pinhole (Leica, Wetzlar, Germany). A 636 × 636 μm field of view was imaged in 2048 × 2048 pixels by resonance scanning. Each optical section was 1.95 μm in depth, and a stack of sections was collected for each eye sample.

In total, we have acquired data from 76 eyes, each with a number of ommatidia ranging from 368 to 745. In a small percentage of the samples, the retinal epithelium was torn during preparation, with damage ranging from small fissures to missing retinal areas. As described below, ommatidia next to such lesions were excluded from the analysis, resulting in smaller ommatidial counts.

Image analysis

A reliable image analysis protocol is crucial to extract accurate statistics from experimental images. Raw images are first converted to segmented images in which ommatidial interfaces (and for larger magnification, also cell-cell interfaces within ommatidia) are identified as edges of single-pixel thickness. For the segmentation, a MATLAB (The MathWorks, Natick, MA) watershed algorithm is implemented after a series of preprocessing and manual corrections are applied to the images (Fig. 1 f). All measurements exclude boundary ommatidia and ommatidia abutting torn regions. The cross-sectional areas A and number of neighbors n of the individual segmented ommatidia can now be determined automatically. As shown in Fig. 1 g, the Fz mutant shows a much wider probability distribution $P\left(A\right)$ than wild-type. We quantify polydispersities of the $P\left(A\right)$ distributions by their coefficients of variation $c_A$ (the quotient of standard deviation and mean). Likewise, one way of quantifying the number of defects is the coefficient of variation $c_n$ of the (discrete) probability distribution of neighbor numbers; it was shown that the quantities $c_A$ and $c_n$ are strongly correlated in systems in mechanical equilibrium (12, 13, 14, 15, 16, 17).

Wild-type, as well as the mutants Ras and Sev, display a small number of defects statistically consistent with zero, while defects in Fz mutants are much more frequent (Table 2). Furthermore, most defects in wild-type, Sev, and Ras eyes are located near the boundary and are generally artifacts of the segmentation process (typically, at most one such boundary artifact is detected per sample, and manually corrected). After resolving these cases, the entire domain becomes an almost perfectly regular honeycomb. By contrast, defects in Fz mutants exist throughout the bulk of the domain. Thus, Fz mutants yield a larger $c_n$ significantly different from zero, in qualitative agreement with the larger $c_A$ seen in Fig. 1 g.

Table 2.

Summary of Experimental Data

Genotype	Retina Samples	Ommatidia per Sample	Defects	cn	cA	cA,sy	cA,st
Wild-type	21	607 ± 42	1.3 ± 1.8	0.005 ± 0.006	0.102 ± 0.022	0.077 ± 0.019	0.067 ± 0.015
Sev	15	516 ± 62	4.5 ± 3.2	0.014 ± 0.008	0.104 ± 0.013	0.072 ± 0.012	0.075 ± 0.010
Ras	10	598 ± 104	0.7 ± 2.2	0.002 ± 0.005	0.119 ± 0.019	0.095 ± 0.022	0.069 ± 0.007
Fz	30	648 ± 47	124 ± 39	0.074 ± 0.012	0.151 ± 0.018	0.101 ± 0.019	0.111 ± 0.015

“Defects” is the number of ommatidia with a number of neighbors different than six. Coefficients of area variation are given for the total, systematic, and statistical variation. Error margins are standard deviations.

We now investigate whether this correlation between area variation and topological disorder can be understood from quantitative modeling. We first verify that constructing Voronoi domains (using the centroids of ommatidia as seed points (18), as determined from the watershed algorithm) gives an accurate representation of the ommatidial structure (Fig. 1 f). The area variation, the defect density, and the location of individual defects are found to be in good agreement with experiment, so that we conclude that Voronoi construction based on centroid information is a suitable modeling framework to study size-topology correlations in the Drosophila eye.

Results

Translational and orientational order

All tissue samples analyzed in this work show a high to moderate degree of order by visual inspection. While our main objective is to explain the density of defects, a wide variety of other measures can be used to quantify the degree of order in a pattern (19, 20), and statistical mechanics concepts such as the following have been used in the context of confluent biological tissues recently, e.g., in analyzing photoreceptor patterns in avian retinas (20).

As the main input for assessing the short- and long-range ommatidial order in the mutant and wild-type eyes, we use the coordinate positions ${\mathbf{r}}_k$ of the ommatidial centroids as determined from the image analysis and their nearest-neighbor connectors (dual lattice, Fig. 2 a). Direct information about the regularity of the centroid pattern is obtained from the spatial pair correlation function $g_2\left(r\right)$, a measure of local density for centroid pairs at relative distance r (3).

Figure 2.

(a) Dual lattice of centroid points with nearest-neighbor bonds, extracted from an experimental image of an Fz mutant eye. (b and c) The pair correlation function $g_2\left(r\right)$ of a wild-type sample (b) and an Fz mutant sample (c), showing stronger loss of correlation in the latter. Distances are normalized by the mean distance in the dual lattice $\overline{s}$. (d) The sample average of the translational correlation function ${\langle g_T\left(r\right)\rangle }_s$, for all genotypes discussed in this work (symbols). The wild-type, Ras, and Sev retinas display a slow decay superimposed over a large-scale oscillation, while the Fz mutant shows rapid (exponential) loss of correlation over a scale of only a few ommatidial lengths. (Dashed line) Modeling $g_T$ from a lattice with appropriate systematic area variation predicts the decay of the wild-type, Ras, and Sev results, but cannot explain Fz (see Systematic Variation of Ommatidial Area for more detail). (e) The orientational correlation function $g_6\left(r\right)$ is sensitive to hexagonal order. While Fz mutants display significantly smaller values, the correlation does not decay to zero over the size of the eye and the decay is not exponential. To see this figure in color, go online.

Fig. 2, b and c, shows examples of $g_2\left(r\right)$ for wild-type and Fz-mutant eye samples, respectively, with r scaled by the mean centroid-to-centroid distance obtained from the mean area $\overline{A}$ of the ommatidia of a sample as

$$\overline{s}={\left(2\overline{A}/\sqrt{3}\right)}^{1/2}.$$ (1)

The oscillations of period $\mathcal{O}\left(1\right)$ indicate the periodicity of the lattice, and their rate of decay is codetermined by the longer-range disorder. It is apparent that Fz eyes show a more rapid loss of translational correlation than wild-types, whose centroid pattern is closer to a regular triangular lattice (Sev and Ras mutants show $g_2\left(r\right)$ behavior very similar to wild-types).

A different measure, the translational correlation function (21) $g_T$, uses explicit deviations from an underlying regular lattice, and is thus particularly suited to detecting and quantifying onset of disorder. The regular triangular reference lattice has lattice constant $\overline{s}$, and its overall orientation is determined by the best fit of all experimental centroid positions; it is then described by a single reciprocal lattice vector $\mathbf{G}$ (the second vector follows from the regularity of the lattice).

The Fourier coefficients ${\text{Ψ}}_G\left(\mathbf{r}\right)=\text{exp}\left(i\mathbf{G}\cdot \mathbf{r}\right)$ can be evaluated at every centroid ${\mathbf{r}}_k$, and the translational correlation function $g_T\left(r\right)$ is determined from a binned correlation average of all centroid pairs within a distance r from each other,

$$g_T\left(r\right)={\langle {\text{Ψ}}_G^{\text{∗}}\left({\mathbf{r}}_j\right){\text{Ψ}}_G\left({\mathbf{r}}_k\right)\rangle }_{r-\Delta r/2\le \left|{\mathbf{r}}_j-{\mathbf{r}}_k\right|\le r+\Delta r/2}.$$ (2)

The most natural choice of bin is $\Delta r=1$ in this scaling. Fig. 2 d shows a further average of $g_T\left(r\right)$ over all samples of the same genotype, denoted by ${\langle \cdot \rangle }_s$. The rapid loss of translational correlation in Fz mutants is now very clear and quantifiable. The larger-scale oscillations in the other samples hint at systematic variations, which are explored in detail in Systematic Variation of Ommatidial Area. By contrast, the average Fz mutant with its significant number of defects shows a strong exponential decay without noticeable oscillations.

Analogously, orientational order relative to the reference lattice can be quantified by the orientational correlation function

$$g_6\left(r\right)={\langle {\text{Ψ}}_6^{\text{∗}}\left({\mathbf{r}}_j\right){\text{Ψ}}_6\left({\mathbf{r}}_k\right)\rangle }_{r-\Delta r/2\le \left|{\mathbf{r}}_j-{\mathbf{r}}_k\right|\le r+\Delta r/2},$$ (3)

with ${\text{Ψ}}_6\left({\mathbf{r}}_k\right)=\left(1/n_k\right){\sum }_{m=1}^{n_k}\text{exp}\left(6i{\theta }_m\left({\mathbf{r}}_k\right)\right)$. This expression uses the angular orientations ${\theta }_m\left({\mathbf{r}}_k\right)$ of the connectors between centroid k and its $n_k$ nearest-neighbor centroids m. Like $g_T$, this function approaches 1 for a perfect triangular lattice. In Fig. 2 e, we present sample averages showing that orientational order is almost perfectly conserved over distances comparable to the eye diameter for wild-type, Ras, and Sev. The Fz mutants display significant decay, but also show a stabilization of $g_6$ over long distances, and in particular do not exhibit an exponential decay to zero which would indicate a loss of correlation on length scales smaller than the system size. That the long-range orientation remains intact in Fz can also be appreciated visually in images such as Fig. 1 f. This result indicates that, in the language of solid state physics (3), the Fz eye epithelium is a hexatic phase, i.e., its translational order is strongly disturbed but orientational correlations persist over long distances. These findings encourage us to perform a detailed analysis of the density and pattern of defects in the eye samples and determine what combination of local and/or global variation gives rise to the observed disorder.

Statistical area variation

Even the relatively large defect density of the Fz mutant is moderate compared to that of a fully disordered domain pattern (14, 15). Therefore, a careful model capturing the onset of nonzero defect densities accurately is necessary for comparison with experiment. We expand here on previous work by Lucarini (22), in which it was shown how the symmetry of a perfectly regular lattice is broken as the centroids of domains are displaced randomly to varying degree. Here we start with a regular triangular lattice of centroid points, apply random Gaussian perturbations to the location of the centroids in the plane (at a given standard deviation α), and construct the pattern of Voronoi cells given by the perturbed points. Above a finite α the hexagonal symmetry is broken, and the first defective Voronoi cells appear.

As the eye epithelium shape is well approximated by an ellipse with a ratio of major to minor axis of $a_m\approx 1.3$, independent of the genotype, we choose the domain of the Voronoi simulation to be such an ellipse. Setting the distance between points in the unperturbed triangular lattice to $s=1$, the vertices of this regular triangular lattice, ${\left(x_i,y_j\right)}_r$, are generated first in a rectangular domain of dimensions $\left[\mathrm{0,2}L\right]\times \left[\mathrm{0,2}L{a}_m\right]$. Here, L is chosen to be 15 for the simulated domain to have a similar number of cells as the eye samples. The vertex positions are

$${\left(x_i,y_j\right)}_r=\{\begin{array}{cc}\left(i,\frac{\sqrt{3}j}{2}\right),& \text{if}j\text{is}\text{even,}\\ \left(i+\frac{1}{2},\frac{\sqrt{3}j}{2}\right),& \text{if}j\text{is}\text{odd}.\end{array}$$ (4)

A statistical variation is applied to the positions of the vertices by generating random displacements $\left({\xi }_{x_i},{\xi }_{y_j}\right)$ from a Gaussian distribution with standard deviation α, $N\left(0,\alpha \right)$; the new set of vertices ${\left(x_i,y_j\right)}_{st}$ is obtained as

$${\left(x_i,y_j\right)}_{st}={\left(x_i,y_j\right)}_r+\left({\xi }_{x_i},{\xi }_{y_j}\right).$$ (5)

Voronoi cells are now constructed based on these seed points; by construction, the average area of these polygons must be $\overline{A}=\sqrt{3}/2$. Finally, Voronoi cells lying outside of an ellipse defined by ${\left(\left(x-L\right)/L\right)}^2+{\left(\left(y-L{a}_m\right)/L{a}_m\right)}^2=1$ are excluded from the statistics to approximate the elliptical domain shape (Fig. 3 b).

Figure 3.

(a) Schematic figure of the regular triangular lattice and a displacement of one of its points. (b) Voronoi diagram for different strength α of Gaussian perturbation or centroid points. As the perturbation becomes larger, the domain gets more disordered (color code is as in Fig. 1e). (c) Correlation of $c_n$ and $c_A$ obtained from the Voronoi simulation, compared with the experimental values for wild-type equivalents and mutants. Error bars are standard deviations. To see this figure in color, go online.

The simulation generates a well-defined correlation between $c_n$ and $c_A$. In agreement with other, local algorithms of assessing neighbor statistics (12, 15), the Voronoi diagram stays topologically perfectly ordered (six neighbors for every domain) for small but finite α (thus, $c_n=0$). For very large α, on the other hand, the lattice structure is completely disrupted and the Voronoi diagram asymptotically approaches that of a random Poisson Voronoi tessellation (22, 23). The onset of symmetry breaking is expected at $\alpha \approx 0.11$, where the first defects are generated; this corresponds directly to a critical polydispersity $c_{A,c}\approx 0.086$. However, the size-topology correlation of wild-type, Ras, and Sev eyes show significant discrepancies compared to this simulation: given their area variation $c_A$, they should exhibit significant defect densities (above onset of disorder), while the actually observed $c_n$ are much smaller and generally consistent with zero (Fig. 3 c and Table 2). Even entirely defectless experimental samples exhibit $c_A$ values larger than the critical $c_{A,c}$. Therefore, size-topology correlations for ommatidia must have properties distinct from the Voronoi domains of the model.

Systematic variation of ommatidial area

Experimental evidence for systematic area variation

We analyzed the spatial variation of the area distribution in the data (see Table 2 for numbers of samples and ommatidia). Surprisingly, we found that the area of an ommatidium varies along a gradient that spans the entire eye epithelium. The area gradient, which we refer to here as a systematic area variation, is quantified by performing a least-squares fit of the experimental data to a linear function, capturing the main trend of ommatidium area variation across the eye (Fig. 4 a). By itself, this gradient represents an area variation equivalent to a coefficient of variation $c_{A,sy}$; Table 2 shows that this heretofore unknown systematic area variation of ommatidia is present to a similar degree in wild-type and mutants. It occurs, with some eye-to-eye variability, in a direction that does not appear to conform to any of the body axis directions. In the case of wild-type eyes, the mean angle between the dorsoventral body axis and the area gradient axis, ${\theta }_s$, is ≈27° (Fig. 4 b) and exhibits a sample-to-sample standard deviation of ${\sigma }_{{\theta }_s}\approx {14}^{\circ }$. In the mutant eyes, we find ${\theta }_s\approx {36}^{\circ }\pm {16}^{\circ }$ for Ras, ${\theta }_s\approx {36}^{\circ }\pm {12}^{\circ }$ for Sev, and ${\theta }_s\approx {47}^{\circ }\pm {11}^{\circ }$ for Fz.

Figure 4.

(a) Spatial distribution of ommatidium area (color scale) for a wild-type eye, representative of $c_{A,sy}$ values given in Table 2. The long-range area variation is visible. (b) The area gradient axis (red, diagonal line), in comparison with the dorsal-ventral axis (green, vertical line). To see this figure in color, go online.

The systematic area variation is expected to have qualitatively different effects on topology from the statistical area variation. A systematic gradient leads to a variation of area (polydispersity value) $c_{A,sy}$, but an individual ommatidium experiences a much smaller degree of area fluctuation relative to its immediate neighbors, when compared to an ommatidium in a domain with a nominally equal $c_{A,st}$ due to random statistical variation. This can be verified in simulations as follows.

Modified model with systematic area variation

Lattice with varying areas

To validate the effect of the systematic variation, Voronoi constructions are performed as above, but with an altered lattice vertex generation protocol. When a systematic variation is present, the point-to-point lattice distance depends on the coordinate of each vertex. With a linear gradient of area imposed in the x direction of the sample of size $\left[\mathrm{0,2}L\right]\times \left[\mathrm{0,2}L{a}_m\right]$, we can write the area function as $A_s\left(x,y\right)=A_0+A_{1}x$. The mean and standard deviation of the area function are then explicitly

$${\overline{A}}_s=A_0+A_{1}L.$$ (6)

$${\sigma }_{A_s}=c_{A,sy}{\overline{A}}_s=\frac{A_{1}L}{\sqrt{3}}.$$ (7)

The above equations yield expressions for $A_0$ and $A_1$ in terms of ${\overline{A}}_s$ and $c_{A,sy}$, so that

$$A_s\left(x,y\right)={\overline{A}}_s\left[\left(1-\sqrt{3}{c}_{A,sy}\right)+\frac{\sqrt{3}{c}_{A,sy}}{L}x\right].$$ (8)

Conversely, determining the linear area function from a least square fit to experimental data quantifies $c_{A,sy}$.

In the simulations with systematic area variation, we maintain the same average area as before, ${\overline{A}}_s=\sqrt{3}/2$. For a regular periodic lattice, the x coordinate of the ${\left(i+1\right)}^{\text{th}}$ vertex on the x axis is simply i. If we assume that the ${\left(i+1\right)}^{\text{th}}$ cell is located near the original point under the systematic variation, its area is approximated as $A_s\left(i,0\right)$, equivalent to a lattice distance of $s_i=\sqrt{\left(\left(2A_s\left(i,0\right)\right)/\sqrt{3}\right)}$. All lattice distances can thus be expressed in terms of $c_{A,sy}$,

$$s_i=\sqrt{\left(1-\sqrt{3}{c}_{A,sy}\right)+\sqrt{3}{c}_{A,sy}\frac{i}{L}}.$$ (9)

Hence, the location of vertices under the systematic area variation is obtained in terms of $s_i$ as

$${\left(x_0,y_j\right)}_{sy}=\{\begin{array}{cc}\left(0,\frac{\sqrt{3}j}{2}{s}_0\right),& \text{if}j\text{is}\text{even,}\\ \left(\frac{s_0}{2},\frac{\sqrt{3}j}{2}{s}_0\right),& \text{if}j\text{is}\text{odd}.\end{array}$$ (10)

$${\left(x_i,y_j\right)}_{sy}=\{\begin{array}{cc}\left(\frac{s_0}{2}+\stackrel{i-1}{\sum _{k=1}}{s}_k+\frac{s_i}{2},\frac{\sqrt{3}j}{2}{s}_i\right),& \text{if}j\text{is}\text{even,}\\ \left(\frac{s_0}{2}+\stackrel{i}{\sum _{k=1}}{s}_k,\frac{\sqrt{3}j}{2}{s}_i\right),& \text{if}j\text{is}\text{odd}.\end{array}$$ (11)

Through rotation of the lattice, we can similarly apply a gradient in an arbitrary direction relative to the lattice symmetry directions. As before, the modeling result is ultimately restricted to an area approximating the shape of the eye epithelium and its number of ommatidia. A Gaussian statistical perturbation of the lattice points can then be added, so that finally the positions of the points for the Voronoi construction are

$$\left(x_i,y_j\right)={\left(x_i,y_j\right)}_{sy}+\left({\xi }_{x_i},{\xi }_{y_j}\right).$$ (12)

Here, the ξ-values are drawn from the normal distribution $N\left(0,\alpha s_i\right)$, i.e., with equal variance relative to the local lattice distance.

Modeling results

In Fig. 5 a, we show the result of a Voronoi construction using systematic area variation only, with no statistical variation present. The value of $c_{A,sy}$ is higher than any observed in the experimental samples ($c_{A,sy}=0.23$, compare to Table 2 for sample data), but it is readily seen that even this high gradient does not induce the formation of any packing defects. The gradual, systematic displacement of lattice points can be accommodated without defects on a lattice with a number of points corresponding to a Drosophila eye. This result is found to be insensitive to the orientation of the gradient relative to lattice symmetry directions (the orientation in Fig. 5 a was chosen as ${\theta }_s={37}^{\circ }$, a typical value for the experimental samples; see Experimental Evidence for Systematic Area Variation, above).

Figure 5.

(a) Result of a Voronoi simulation performed with systematic area variation only ($c_{A,sy}$ ≈ 0.23), and no statistical variation. Even at this high value of $c_{A,sy}$, systematic area variation does not induce any defects. (b) Size-topology correlation for cases where both statistical variation and systematic variation are present; the correlation curve shifts to the right with $c_{A,sy}$. (c) When the statistical area variation is isolated by subtracting the effect of systematic variation, all curves collapse. To see this figure in color, go online.

Modeling with systematic area variation also provides more quantitative insight into the translational correlation function $g_T$ in Fig. 2 d. The dashed line in that figure shows that a lattice with a systematic variation of $c_{A,sy}\approx 0.1$, typical for experimental samples, reproduces the decay and oscillations of the wild-type, Ras, and Sev data. Importantly, it fails to explain the exponential decay of $g_T$ for Fz samples, which is therefore a consequence of statistical area variation.

When applying systematic variation and statistical variation simultaneously in the model, the two effects are independent, so that the total observed $c_A$ simply results from

$$c_A^2=c_{A,sy}^2+c_{A,st}^2,$$ (13)

as the variances of the two effects add up. The approximate independence implied in Eq. 13 is confirmed by the experimental data: when evaluating overall $c_A^2$ and $c_{A,sy}^2$ from the observed area gradient, the difference of these variances is in very good agreement with the variance computed from the local deviations from systematic area, $A-A_s\left(x,y\right)$.

As systematic area variation does not induce any defects for the experimental range $0\le c_{A,sy}\mathrm{\lesssim }0.15$, the resulting $c_n$-$c_A$ correlations (obtained by performing simulations for different polydispersities) display a rescaling of the $c_A$-axis according to the size of $c_{A,sy}$ (Fig. 5 b). More systematically, we can now isolate the effect of statistical variation by subtracting the systematic contribution according to Equation 13, and showing $c_n$ as a function of $c_{A,st}$ only. This operation is applied to modeling and experimental data, and Fig. 5 c shows that the former collapses onto one universal curve.

The experimental data likewise shift upon rescaling, and the $c_{A,st}$ values of wild-type, Sev, and Ras eyes are sufficiently small $\left(c_{A,st}<c_{A,c}\right)$ to explain the absence of defects in these strains. However, the number of defects observed in Fz eyes is significantly higher than what is predicted from the Voronoi simulation. Having tested that the results are insensitive to both the orientation of the systematic area gradient and to the nature of the statistical displacements of centroids (we attempted various non-Gaussian perturbations), we conclude that the model still misses an important feature of eye morphology. In the following section, we argue that this feature is rooted in cell mechanics.

Cone cell exclusion volume

In previous studies, it was shown that the distribution of cone cells inside each ommatidium, as well as the shape of these cells, can be explained from a simple interfacial mechanics model, taking into account a generic cortical contractility and cell-cell adhesion as contributions to the mechanical energy of the system, which is then minimized by specific cell shapes (24, 25). Similar mechanical models have been employed with great success in a wider context of epithelial and epidermal tissues (13, 14, 26, 27, 28, 29). In the case of cells in an ommatidium, modeling revealed that the cells forming the ommatidial “frame” (the secondary and tertiary pigment cells, and the bristle group, compare to Fig. 1 c) are under much greater tension than those cells inside the frame (primary pigment and cone cells). Of these, the cluster of cone cells is under significantly higher tension than the primary pigment cells. The net effect is that the primary cells provide an easily deformable medium between the stiffer elements of the cone cell cluster and the frame. This means that each ommatidium is not a structureless polygon but contains a cone cell core of much lower deformability. It is then reasonable to make reference to Voronoi simulations with exclusion volumes around the seed points, whose statistics has been investigated in various contexts (30, 31). In this limit, we regard the cone cell cluster as an incompressible disk, so that the centroid of a neighboring ommatidium cannot be closer than a minimum distance. Additionally, the rigid frame cells provide a further (much smaller) contribution to the exclusion volume in the ommatidial plane. Displacements of centroid positions are thus almost entirely accommodated by the deformation of the primary cells.

To accurately assess the size and shape of the cone cell clusters and frame cells, we use experimental images of higher magnification (Fig. 6 a). In each image, segmentation of the secondary pigment cells and the cluster of cone cells allows for quantification of a mean frame width $w_f$ (the uniform width at equal secondary pigment cell area) and an equivalent cone cell radius $r_c$ (from a circle of area equal to that of the cluster); compare to Fig. 6 b. For the developmental stage considered here, the cone cell cluster is to good approximation circular. Together, these two exclusion effects of frame and cone cell cluster restrict the minimum distance between two neighboring ommatidia to $2r_c+w_f$. A dimensionless measure of exclusion is then the ratio of this minimum distance to the lattice distance, $v_c=\left(2r_c+w_f\right)/\overline{s}$. The analysis of experimental images results in values of $v_c\approx 0.75\pm 0.03$ with no statistically significant systematic variation between genotypes.

Figure 6.

(a) Image analysis for determining $v_c$. The interfaces between neighboring ommatidia, ommatidial frames, and the cluster of cone cells are identified. (b) Schematic diagram of the distances used in the exclusion volume calculation. (c) $c_n\text{-}{c}_A$ correlation. (Dashed line) Result of a simulation without the exclusion volume. (Black solid line) Simulation of the gradient with $c_{A,sy}$ = 0.1. (Green, dot-dashed line) Including exclusion volume but neglecting systematic area variation leads to similar effects as in plain Voronoi simulations (compare to Fig. 5). (d) $c_n\text{-}{c}_{A,st}$ correlation after subtraction of systematic area variation. All modeling results are consistent and in agreement with experiment. To see this figure in color, go online.

The simulation protocol is now altered as follows: First, the systematic displacement is applied to centroid vertices as above. Every vertex k then has a minimum distance $s_{k,\text{min}}$ given by the position of one of its neighbors. Before a statistical displacement is applied to a vertex k, the connecting vectors ${\overrightarrow{s}}_{\text{km}}$ are determined to all cells m that have already undergone statistical displacement. A statistical displacement $\overrightarrow{\xi }=\left({\xi }_x,{\xi }_y\right)$ of centroid k is accepted if

$$\underset{m}{\text{min}}\left(s_{\text{km}}-\overrightarrow{\xi }\cdot {\overrightarrow{s}}_{\text{km}}/s_{\text{km}}\right)>v_c{s}_{k,\text{min}}.$$ (14)

If the inequality in Eq. 14 is violated, the displacement is rejected and random number generation is repeated until the lattice vertex satisfies the exclusion volume criterion.

The statistics of Voronoi simulation with the exclusion volume determined by $v_c=0.75$ explains the topological statistics of all experimental samples, including Fz mutants. A comparison of Fig. 6, c and d, shows that the effect of systematic area variation is approximately independent of that of exclusion volume, so that the rescaling to $c_{A,st}$ in Fig. 6 d is appropriate. The exclusion volume has almost no effect for small statistical variation so that the domain remains defect-free up to $c_{A,st}\approx c_{A,c}$. As the perturbation, and thus the polydispersity, increases, packing defects are generated. For a fixed magnitude α of the displacement perturbation, the area variation in the domain becomes smaller due to the exclusion volume while the number of defects is not greatly influenced. Hence, a higher degree of perturbation is required to generate a given $c_A$ and the corresponding $c_n$ is larger; see Fig. 6, c and d. This leads to a steeper slope of the $c_n$-$c_A$ correlation, confirming that exclusion volume is an important factor in assessing size-topology correlation in the Drosophila eye, which cannot be fully understood by modeling ommatidia as structureless polygons. All experimental data are in agreement with this model; we emphasize that the model parameters (both $c_{A,sy}$ and $v_c$) were obtained directly from the analysis of experimental images and are not freely adjustable.

Order measures in the model

Having developed a model that reproduces the topological disorder of the samples using the characteristics of area variation, we now ask whether the conventional statistical physics measures of order discussed in Translational and Orientational Order agree with the modeling. To provide a closer one-to-one comparison, we evaluate modeling runs for individual samples, i.e., those that reproduce closely the systematic area variation $c_{A,sy}$, the statistical area variation $c_{A,st}$, and the topological variation $c_n$.

The pair correlation function $g_2\left(r\right)$ is indeed well represented in the model, as demonstrated by Fig. 7, a and b, for typical specimens of wild-type and Fz mutants, respectively. The conclusion of significantly more rapid loss of correlation in Fz specimens as compared with wild-types is borne out. The corresponding translational correlation functions $g_T\left(r\right)$ in Fig. 7, c and d, respectively, are in agreement with this conclusion: the different scales of decay are captured in the modeling solutions. The orientational correlation functions $g_6\left(r\right)$ are in good agreement for wild-types (Fig. 7 e), where no significant decay occurs over the size of the sample, but the agreement is not as good for the Fz mutants. While the majority of the decay, which occurs over the first 2–3 ommatidial distances, is captured well, the experiments tend to show further loss of correlation at larger r not seen in the model. However, this discrepancy has no direct effect on the agreement of translational order, or on the agreement of the size-topology measures $c_{A,sy}$, $c_{A,st}$, and $c_n$. This shows that the critical aspect for explaining topological disorder in the Drosophila eye is the perturbation of translational order, not an exact representation of orientational disorder.

Figure 7.

Comparison of order measures from experimental data (symbols) with simulation results (solid lines). (a and b) The features of the pair correlation functions $g_2\left(r\right)$ from Fig. 2, a and b are captured very well in the model. (c and d) The decay of the translational correlation functions $g_T\left(r\right)$ from experimental wild-type (c) and Fz (d) samples is well represented by the modeling results. (e) The orientational correlation function $g_6\left(r\right)$ of wild-type samples hardly decays in experiment or theory. (f) The initial decay of $g_6\left(r\right)$ in modeled Fz mutant samples agrees with experiment, but larger distances show stronger correlation than the experiments. Scalar order parameter values T and $g_6$ are reported in (a), (b), (e), and (f). To see this figure in color, go online.

As convenient scalar measures of translational and orientational order, we furthermore compare the order metrics (19)

$$T\equiv \frac{1}{{\eta }_c}{\int }_0^{{\eta }_c}\left|g_2\left(r\right)-1\right|dr,$$ (15)

where ${\eta }_c$ is a length cutoff, and

$$q_6\equiv \left|\frac{1}{N}\sum _k{\text{Ψ}}_6\left({\mathbf{r}}_k\right)\right|,$$ (16)

an average comprising all nearest-neighbor pairs of ommatidia (compare to Eq. 3).

In Fig. 7, a, b, e, and f, the T and $q_6$ values for the experimental samples and modeling runs are reported (the cutoff length ${\eta }_c=8$ was chosen to exclude insignificant fluctuations at larger scales). The agreement of T between model and experiment is good, while there are again discrepancies in the orientational-order measure $q_6$, indicating that the experimental structure shows a smaller degree of local orientational order than the model system.

Discussion

In this work, we have investigated the statistics of packing defects of ommatidia in the Drosophila eye, which in wild-type is a perfect honeycomb lattice. Mutations that change the number of cells, internal arrangement, and orientation of ommatidia can disrupt this order to a greater or lesser extent. The frequency of packing defects is explained by their necessary generation as a function of local area variation of ommatidia. Specifically, it is a statistically uncorrelated local variation that generates topological disorder, and it is this type of variation that is significantly enhanced in the Fz mutant, which exhibits the most severe disruption of the honeycomb pattern. The number of defects can be quantitatively explained by taking into account that the ommatidia contain cells of differing mechanical properties, so that deformation of the polygonal ommatidia is only possible if frame and cone cells are approximately undeformed. Our findings underscore the need for robust internal development of every ommatidium in an eye for the overall packing pattern of ommatidia to be virtually error-free.

Conversely, mutants in non-PCP-related pathways (Ras and Sev) show, like wild-type specimens, sufficient uniformity of ommatidial area to avoid any topological defects in the pattern. It is noteworthy that these specimens show statistical area variations just marginally below the critical value that would be required to generate defects (compare to Fig. 6 d). This can be interpreted as a result of evolutional optimization: the expenditure of biological resources needed to ensure uniformity of ommatidial patterns does not exceed what is necessary to yield defect-free patterns.

As the Voronoi patterns faithfully reproduce neighbor relations observed in the eye epithelium, this powerful modeling tool quantifies the correlation between polydispersity of areas and number of defects, and confirms that all experimental samples carry a number of defects explained within this framework. Generally, we have shown that analysis of any two-dimensional cellular structure (biological or inanimate) in terms of the statistics of topological disorder requires careful analysis of potential long-range spatial correlations in the domain, to ensure that local statistical variations are not biased. In particular, the onset of disorder in the form of defects can appear delayed if part of the area variation in the pattern is systematic. Conversely, this makes this set of modeling tools a sensitive probe for long-range variability in tissues. To our knowledge, a novel discovery from this analysis is that such a long-range systematic area variation in ommatidia exists across the entire eye. This variation is in itself not sufficient to cause packing defects in the retina. We are not presently aware of any biological mechanism that would lead to such an area gradient in the direction observed (relative to the body axes), but it provides, to our knowledge, a new and interesting challenge to discover how it is formed and what its purpose might be.

Within the scope of our study, the PCP mutant Fz shows the largest effects on ommatidial area variation and hence, packing disorder. A direct analysis of order measures shows that the disorder is associated with rapid decay of translational order, while the long-range orientation of the ommatidial lattice remains partially intact in Fz mutant eyes. The ommatidial structure in Fz mutants can thus be thought of as a hexatic phase, where the translational order is strongly perturbed, but identifiable directions of lattice orientation in relation to the body axes prevail.

This article has hinted at the role of cellular and tissue mechanics in determining the regularity of ommatidial patterns, with the acknowledgment of stiffer exclusion volumes. This is a starting point for future work combining the statistical analysis outlined here with mechanical models of the Drosophila retina (24, 25); not only will this give insight into the physical processes leading to the varying degrees of disorder in the specimens, but parameter changes over time may reflect the dynamical changes occurring during morphogenesis. For this purpose, analogous data from earlier stages of pupal development will show whether the increased disorder in PCP mutants is initiated early on. Quantification of topological disorder is valuable in this respect because it is a robust measure that relies only on neighbor relations between ommatidial centroids. The modeling tools introduced here are a promising addition to such quantitative studies of morphogenetic development.

Author Contributions

S.K. analyzed data, performed modeling, and wrote the article; J.J.C. acquired experimental data; B.Y. analyzed data and developed image analysis tools; and R.W.C. and S.H. designed research and wrote the article.

Editor: Stanislav Shvartsman.