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Conceived and designed the experiments: YAK JCC. Performed the experiments: YAK SM JCC. Analyzed the data: YAK JCC. Contributed reagents/materials/analysis tools: YAK JCC. Wrote the paper: JCC.

Current address: The University of Michigan Flint; Flint, Michigan, United States of America

The authors have declared that no competing interests exist.

The avian retina possesses one of the most sophisticated cone photoreceptor systems among vertebrates. Birds have five types of cones including four single cones, which support tetrachromatic color vision and a double cone, which is thought to mediate achromatic motion perception. Despite this richness, very little is known about the spatial organization of avian cones and its adaptive significance. Here we show that the five cone types of the chicken independently tile the retina as highly ordered mosaics with a characteristic spacing between cones of the same type. Measures of topological order indicate that double cones are more highly ordered than single cones, possibly reflecting their posited role in motion detection. Although cones show spacing interactions that are cell type-specific, all cone types use the same density-dependent yardstick to measure intercone distance. We propose a simple developmental model that can account for these observations. We also show that a single parameter, the global regularity index, defines the regularity of all five cone mosaics. Lastly, we demonstrate similar cone distributions in three additional avian species, suggesting that these patterning principles are universal among birds. Since regular photoreceptor spacing is critical for uniform sampling of visual space, the cone mosaics of the avian retina represent an elegant example of the emergence of adaptive global patterning secondary to simple local interactions between individual photoreceptors. Our results indicate that the evolutionary pressures that gave rise to the avian retina's various adaptations for enhanced color discrimination also acted to fine-tune its spatial sampling of color and luminance.

The chicken (

(A) Diagram of the seven photoreceptor cell types of the chicken retina. Oil droplets are colored approximately according to their appearance under brightfield illumination. Rods and the accessory member of double cones lack oil droplets. A hematoxylin and eosin-stained section of an adult chicken retina is shown on the right. The drawing are based on depictions of avian rods and cones by Ramón y Cajal

Prior studies have shown that most non-photoreceptor cell types in the retina tile its surface with varying degrees of regularity

Photoreceptors display the most regular tiling of all neuronal cell types. Many teleost fish and some reptiles have almost perfectly regular ‘crystalline’ arrays of photoreceptors which occur in a variety of patterns

Amongst vertebrate retinas such crystalline regularity of photoreceptors is the exception rather than the rule. The most detailed studies of mammalian photoreceptor spatial distribution to date have been performed on human and ground squirrel retinas

Mounting evidence suggests that the common ancestor of modern reptiles, birds and mammals was a diurnal organism with a highly sophisticated cone visual system comparable to that of present-day birds

During the evolution of placental mammals, many of these specialized adaptations to a strongly diurnal niche were lost

Unlike the case of mammals, a diurnal lifestyle is presumed to have been maintained throughout the evolutionary history of birds from the common amniote ancestor

In order to identify individual cone photoreceptors in the chicken retina, we took advantage of the presence of brightly colored oil droplets in their inner segments (

We examined a total of 28 post-hatch day 15 (P15) chicken retinas including seven mid-peripheral retinal fields from each of four quadrants (

(A) Diagram of a chicken eye cup showing the regions of the mid-peripheral retina (in light blue) from which all fields analyzed in this study were derived. (B) Percentages of cone types from each of four quadrants (n = 7 fields for each quadrant). Data for violet, blue, green and red cones are colored accordingly. Data for double cones are shown in black. Error bars indicate SD.

When a field of retinal oil droplets is viewed as a whole, there is little apparent order. However, when cone types are considered individually, they show a highly regular distribution with a relatively uniform distance between neighboring cones (

(A) Digitized image of double cone distribution in a portion of a single field (dorsal-nasal field 7 in ^{st} shell” etc. are explained in the main text. The vertical orange line indicates the average diameter of a double cone oil droplet. (D) Distribution of nearest neighbor distances for each of the five cone types within a single retinal field (dorsal-nasal field 7 in

Progressing farther out from the origin of the autocorrelogram there occur alternating shells of increasing and decreasing cone density which can be better appreciated by graphing the data as a density recovery profile (DRP;

The distances of all the nearest neighbors of a given cone type follow an approximately Gaussian distribution (

One way of assessing topological order in a two-dimensional (2D) distribution of points is to use Voronoi tessellations _{6} = 0.293±0.018 [mean ± SD]), but there was a wide distribution of sizes ranging from 3-sided up to 13-sided. As the degree of order in a Voronoi tiling increases, there is a corresponding increase in P_{6} and a decrease in the width of the polygon distribution. In the limiting case of a perfectly regular tiling, P_{6} = 1 (

(A–C) Voronoi tessellations of a portion of a red cone field (B) and a random (A) and perfect (C) distribution of points of the same density as in (B). (D) Graph showing the average P_{n} distributions for all chicken cone types as well as simulated random and perfect distributions. ‘Epithelia’ indicates the average P_{n} distribution for five different animal and plant epithelia as given in _{n} distribution for the random simulations included a small number of 11-, 12- and 13-sided cells which are not shown. Error bars are SD. (E) Graph showing the topological disorder (μ_{2}) for all five cone types as well as random and perfect distributions. ‘Epithelia’ are as described in (D). Error bars are SD. (F) Graph of P_{6} vs. topological disorder (μ_{2}) for all 140 P15 cone mosaics examined. The solid curve indicates the value of Lemaître's law (equation shown in the graph) in the range, 0.34<P_{6}<0.66.

We found that Voronoi tilings of the four single cones all showed very similar polygon distributions with P_{6} ranging from 0.454±0.019 (mean ± SD) for green cones up to 0.494±0.031 (mean ± SD) for blue cones (_{6} value for epithelia from five different species was 0.460±0.020 (mean ± SD) _{6} = 0.570±0.034 (mean ± SD) (

An alternative measure of topological regularity is the variance of the probability distribution _{n}_{6} serves as a measure of order, μ_{2} is a measure of the spread of the polygon distribution and is therefore a measure of topological disorder. We found that the four single cone types have similar μ_{2} values ranging from 0.634±0.060 (mean ± SD) for blue cones up to 0.734±0.046 (mean ± SD) for green cones (

Next, we studied the relationship between P_{6} and μ_{2} in the chicken cone mosaics. It has been shown that a wide range of 2D cellular mosaics found in nature including examples from metallurgy, geology and ecology as well as mosaics obtained from experimental and computational simulations all obey a quasi-universal topological relation between P_{6} and μ_{2} known as Lemaître's law _{6}<0.66, this law takes the form:_{6} versus μ_{2} (_{6}>∼0.47 showed μ_{2} values which were in close agreement with Lemaître's law (_{6}<∼0.47 tended to have a value for μ_{2} which was less than would be predicted by the law. The near universality of this law is thought to be a consequence of the fact that all mosaics which obey it are statistical ensembles in equilibrium _{6} values therefore suggests that these mosaics may not be in statistical equilibrium. Alternatively, there may be unknown biological constraints which contribute to this deviation.

Given the homotypic spacing observed between cones of the same type, we wished to determine whether similar spacing occurs between cones of different type. We evaluated whether there was any tendency for heterotypic pairs of photoreceptors to repel one another by measuring the effective radius of exclusion (ERE) around individual photoreceptors (

(A) Graph of the effective radius between cones of the same type (homotypic pairs) and different types (heterotypic pairs). Also shown for comparison is the average oil droplet diameter for all cone types. ‘D-D’, ‘Double cone-Double cone’; ‘G-G’, ‘Green cone-Green cone’ etc. Error bars are SD. (B) Graph of the nearest neighbor regularity indices for cones of the same type (homotypic pairs) and different types (heterotypic pairs) (blue bars). Also shown are regularity indices for simulated mosaics as described in the main text (red bars). Abbreviations are as in (A). Error bars are SD.

(A) Graph of photoreceptor density vs. average nearest neighbor distance for all 140 P15 cone mosaics examined (middle curve). The upper and lower curves are graphs of density vs. average nearest neighbor distance for a series of computer-generated perfect and random distributions, respectively. The inset shows data for the three developmental timepoints (i.e., red cones at E18, P0 and P6). It corresponds to the region of the main graph highlighted with a dotted box except that all P15 chicken datapoints shown in the main graph are shown in black to facilitate visualization of the developmental timepoints. (B) Graph of the same datapoints as in (A) but shown as density vs. the inverse-square of the average nearest neighbor distance. The linear correlation coefficients (r) for the best fit line for each of the three datasets are shown.

In order to further explore the possibility of spatial co-regularities between cone mosaics we employed a commonly used measure of geometric order within cellular mosaics known as the regularity index (RI). The RI is equal to the average nearest neighbor distance divided by its standard deviation

Next, we determined the RI for all possible heterotypic (X-Y) pairs of cone by identifying the nearest ‘Y’ neighbor of every ‘X’ cone and then calculating the mean and standard deviation (

In order to control for the effects of steric hindrance and spurious co-regularity due to random spatial registration of mosaics, we carried out computer simulations to assess their effects. We generated random distributions of photoreceptor ‘Y’ that matched the density and mean regularity index of the real ‘Y’ mosaics using a sequential addition, ‘hard disk’ model (see Materials &

In order to obtain further insights into the mechanism of cone spacing, we plotted the average nearest neighbor distance between cones of the same type as a function of photoreceptor density. We found that average nearest neighbor distance decreases as a function of increasing density (

In order to assess at what point in development the orderliness of the cone mosaics first appears, we determined the spatial coordinates of red cones at three earlier developmental timepoints: embryonic day 18 (E18) and post-hatch days 0 and 6 (

In order to quantify the degree of geometric order inherent in the cone photoreceptor mosaics, we next plotted density versus the inverse-square of the average nearest neighbor distance. It can be seen that all of the real datapoints as well as the simulated random and perfect distributions fall on three straight lines with different slopes (

Next, we wished to determine the generality of the relationship between density and average nearest neighbor distance amongst birds. We therefore examined the spatial distribution of a subset of cones from three additional species belonging to three different orders: downy woodpecker (

(A) Graph of photoreceptor density vs. average nearest neighbor distance for three additional species of bird representing three different orders. All P15 chicken datapoints are shown in black for clarity.

In this study we have used the colored oil droplets present in the inner segment of cone photoreceptors to characterize the spatial distribution of the chicken's five functional classes of cone. We found that each type of cone is arrayed as a highly regular mosaic with a characteristic spacing between cones of the same type. All five cone mosaics display a high degree of topological and geometric order but are spatially independent of one another. Remarkably, all cone types use a similar density-dependent yardstick to measure intercone spacing. Based on the relationship between density and nearest neighbor distance, we derived a single parameter that uniquely characterizes the regularity of all the cone mosaics within the retina. The value of this parameter was determined for three additional species of bird, which were found to have cone spatial patterning which was fundamentally similar to that of the chicken. This result suggests that the principles of cone spacing identified in the chicken may be universal among diurnal birds. These results confirm that avian cone photoreceptor have an extremely high degree of spatial organization which is likely the result of evolutionary selection.

In evaluating the spatial distribution of chicken cones, we found that as photoreceptor density decreases, the average nearest neighbor distance between cones of the same type increases. The net result of this scaling is that photoreceptors maintain a relative uniform degree of spatial regularity despite changes in density. A similar scaling relationship between density and average nearest neighbor distance was previously found for a type of ganglion cell in the chicken

The degree of order within two-dimensional cellular mosaics can be characterized by the distribution of values for the area of individual cells within the mosaic (geometric order) or by the distribution of values for _{n} functions (see

A wide range of post-mitotic animal and plant epithelia show very similar P_{n} functions and hence display similar degrees of topological order _{n} function in such a wide range of epithelia has been posited to arise as a topological consequence of mitosis _{n} functions of the Voronoi tessellations of single cone mosaics in the chicken are similar to those found in epithelia (_{n} functions simply mirror that of the underlying epithelium. If the underlying epithelium had a P_{n} function similar to the one observed in many other epithelia, then a random assignment of cells (i.e., polygons) from the underlying epithelia to each of the individual cone mosaics would, on average, endow the individual mosaics with a similar P_{n} function. However, we know that the individual cone mosaics do not represent random samplings of the underlying epithelium. Furthermore, the fact that double cones have a very different P_{n} function argues against this simple interpretation. These findings suggest that there may be unknown biological reasons for the repeated occurrence of this particular P_{n} function.

The results of the present study constrain the range of possible models that can explain the formation of the chicken's cone mosaics. Any model of mosaic formation must encompass two key aspects of cone photoreceptor patterning. On the one hand, the five cone mosaics are spatially independent and show no evidence of heterotypic repulsion between different cone types. These facts suggest the existence of distinct biochemical mechanisms of spacing unique to each cone type. On the other hand, cone-to-cone spacing, although density-dependent, is independent of cell type, suggesting a mechanism of measuring intercone distance which is shared by all cone types. If cone spacing is established simultaneously or in temporally overlapping waves for the five cone types, it seems necessary to invoke multiple distinct molecular signals mediating homotypic interactions for each of the five types. Such interaction could be mediated either by a diffusible signal or by cell-cell contact. In this scenario, cone-spacing might involve a ‘two-component’ mechanism consisting of a cone type-specific signaling system mediating cell type recognition and a second shared system for measuring the distance between cones.

If spacing occurs in temporally separate waves for the different types of photoreceptors, it is possible to posit models which involve only a single biochemical mechanism for all photoreceptor types (

In this model the individual photoreceptor types establish their spacing in a series of temporally discrete waves. The least abundant photoreceptor type (i.e., violet cones) establishes spacing first, possibly via a lateral inhibition mechanism (far left). Then, the next most abundant photoreceptor type, blue cones, establishes its spacing. This process continues until spacing has been established for all photoreceptor types (the diagram only shows the four single cone types). The addition of subsequent waves of photoreceptors results in a relatively uniform expansion of the epithelium and a concomitant ‘spacing out’ of those photoreceptor types whose spacing was established earlier. Since spacing is established in discrete steps, all photoreceptor types can, in principle, employ the same biochemical mechanism to establish spacing.

A variety of models and theoretical mechanisms have been proposed to explain the development of the nearly crystalline cone photoreceptor mosaics of certain teleost fish species

Another model for the formation of the zebrafish cone mosaic posits that cell-cell signaling between differentiating photoreceptors and adjacent undifferentiated progenitors may be responsible for patterning

The remarkable regularity of the chicken's cone mosaics raises the question of its adaptive significance. Theoretical analyses have suggested that optimal spatial sampling of the visual scene is achieved by perfectly regular, hexagonal arrays of receptors and that any deviation from this pattern results in a decrement in the quality of the reconstructed image

Another potential explanation for cone disorder is that it may be topologically impossible to pack six perfectly hexagonal photoreceptor mosaics (i.e., five cone and one rod mosaic) within a single epithelium. The question then arises whether the photoreceptor mosaics of the chicken retina are as regular as they can be given the ratios of their occurrence and these packing constraints. Under such conditions, any increase in the regularity of one mosaic might necessitate a decrement in the regularity of one or more of the other mosaics. Thus, although the individual cone mosaics are spatially independent, their regularities may depend on packing constraints within the photoreceptor epithelium and therefore be interdependent. Given the ratios and densities of its photoreceptors, it is possible that the chicken's mildly disordered photoreceptor mosaics represent an optimal solution to a 2D packing problem

It has been postulated based on a variety of theoretical considerations that birds use two separate sets of photoreceptors for detection of chromatic and luminance signals, the single cones and double cones, respectively

The channeling of spectral and spatial signals through the same set of photoreceptors may also help explain the absence of colored oil droplets in primate retinas. Although oil droplets improve color discrimination, they reduce photoreceptor sensitivity

All animals studies were conducted in accordance with the Guide for the Care and Use of Laboratory Animals and the Animal Welfare Act and were approved by the Washington University in St. Louis Institutional Animal Care and Use Committee. Post-hatch chickens (

A single individual of each of three additional species (

All computational analyses and calculations were performed using custom Matlab scripts and Microsoft Excel. Voronoi tessellations of photoreceptor distributions were created with a custom script using a Matlab function called ‘Voronoi’. In order to avoid edge effects, only those Voronoi cells whose vertices all lie within the field were included in subsequent analyses. P_{n} distributions were calculated from the number of vertices of the individual Voronoi cells of a given photoreceptor distribution.

Nearest neighbor analysis, spatial autocorrelograms, density recovery profiles and effective radii were all calculated as described previously

In order to create simulated distributions of points with a defined nearest neighbor regularity index for purposes of the cross-correlation analysis (

As a control for the cross-correlation analysis, the regularity indices for heterotypic pairs ‘X-Y’ were calculated by using the coordinates of the real ‘X’ cells and comparing them to the simulated ‘Y’ mosaics. An additional feature of the simulated ‘Y’ mosaics was that they were created on a field already containing the real ‘X’ cells. Thus, newly placed ‘Y’ cells not only had to be at least one hard disk diameter from every previously placed ‘Y’ cell, but they also had to not overlap any ‘X’ cells. For the purpose of this simulation, cell diameter was assumed to be equal to oil droplet diameter (see

Spatial distributions, autocorrelograms and density recovery profiles for all five cone types. (A–O) This figure depicts data in the same format as in

(2.21 MB TIF)

Cone photoreceptor mosaics with P_{6}>∼0.47 obey Lemaître's law. (A and B) These two graphs depict the same data as in _{6}<∼0.47 in (A) and those with P_{6}> = ∼0.47 in (B). The best fit power curve for both datasets are shown as dotted lines, and the equations are given in the box. The R-squared value for the goodness of fit to these curves is also shown. The solid line in both figures represents Lemaître's law. The value of the coefficient ([2π]^{−1}) is shown numerically for comparison with the equation of the fit curve. The cone mosaics with P_{6}>∼0.47 fit a curve which is almost directly superimposed on that representing Lemaître's law. In contrast, the cone mosaics with P_{6}<∼0.47 show a relatively poor agreement with Lemaître's law.

(5.75 MB TIF)

Determining the global regularity indices for all four bird species. (A–F) Graphs of photoreceptor density vs. the inverse-square of the average nearest neighbor distance for the following datasets: computer-generated random mosaics (A), chicken cone mosaics (B), computer-generated perfect mosaics (C),

(5.15 MB TIF)

Data and coordinates for cone mosaics from all four species. This file contains a total of 35 worksheets. ‘Summary’ includes a variety of data about all the P15 chicken mosaics (NND, nearest neighbor distance). Worksheets labeled ‘DN1’ (‘Dorsal-Nasal field #1’), ‘DT1’ (‘Dorsal-Temporal field #1’), ‘VN1’ (‘Ventral-Nasal field #1’), ‘NT1’ (‘Ventral-Temporal field #1’) etc. contain the raw coordinates for all P15 chicken fields examined in the present study. Worksheets labeled ‘E18’, ‘P0’ and ‘P6’ contain the raw coordinates for the chicken mosaics examined at the indicated developmental stages. Worksheets labeled ‘P. pubescens’, ‘P. domesticus’ and ‘C. livia’ contain the raw coordinates for the three additional species examined.

(6.86 MB XLS)

Thanks to C. Diaconu who was involved in the very earliest phases of this project. Also thanks to C. Montana, K. Lawrence, V. Kefalov, D. Kerschensteiner, P. Lukasiewicz, S. Johnson, C. Micchelli and R. Kopan for valuable advice and input on this project.