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.

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.

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.

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.

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.

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.

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.

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.