Avian photoreceptor patterns represent a disordered hyperuniform solution to a multiscale packing problem

Optimal spatial sampling of light rigorously requires that identical photoreceptors be arranged in perfectly regular arrays in two dimensions. Examples of such perfect arrays in nature include the compound eyes of insects and the nearly crystalline photoreceptor patterns of some fish and reptiles. Birds are highly visual animals with five different cone photoreceptor subtypes, yet their photoreceptor patterns are not perfectly regular. By analyzing the chicken cone photoreceptor system consisting of five different cell types using a variety of sensitive microstructural descriptors, we find that the disordered photoreceptor patterns are “hyperuniform” (exhibiting vanishing infinite-wavelength density fluctuations), a property that had heretofore been identified in a unique subset of physical systems, but had never been observed in any living organism. Remarkably, the patterns of both the total population and the individual cell types are simultaneously hyperuniform. We term such patterns “multihyperuniform” because multiple distinct subsets of the overall point pattern are themselves hyperuniform. We have devised a unique multiscale cell packing model in two dimensions that suggests that photoreceptor types interact with both short- and long-ranged repulsive forces and that the resultant competition between the types gives rise to the aforementioned singular spatial features characterizing the system, including multihyperuniformity. These findings suggest that a disordered hyperuniform pattern may represent the most uniform sampling arrangement attainable in the avian system, given intrinsic packing constraints within the photoreceptor epithelium. In addition, they show how fundamental physical constraints can change the course of a biological optimization process. Our results suggest that multihyperuniform disordered structures have implications for the design of materials with novel physical properties and therefore may represent a fruitful area for future research.

Figure 1

Spatial arrangements of chicken cone photoreceptors. The leftmost panel shows a flatmount preparation of a post-hatch day 15 chicken retina with colored oil droplets within the inner segments of the five cone photoreceptor types. Size bar = 10 $\mu$m. The additional panels (from left to right) depict the same field of view under illumination with ultraviolet, blue, and green light, respectively. Oil droplet autofluorescence permits subtype classification of the individual photoreceptor cells and determination of their spatial coordinates. These figures derive from Ref. [18].

Figure 2

Experimentally obtained configurations representing the spatial arrangements of chicken cone photoreceptors. Upper panels: The configurations shown from left to right, respectively, correspond to violet, blue, green species. Lower panels: The configurations shown from left to right, respectively, correspond to red, double species, and the overall pattern.

Figure 3

Structure factors $S\left(k\right)$ of the experimentally obtained point configurations representing the spatial arrangements of chicken cone photoreceptors. The experimental data were obtained by averaging 14 independent patterns. The estimated values of $S(k=0)$ by extrapolation for violet, blue, green, red, double, and the overall population in the actual pattern are respectively given by $2.11\times {10}^{-3}$, $6.10\times {10}^{-4}$, $1.06\times {10}^{-3}$, $5.72\times {10}^{-4}$, $1.38\times {10}^{-4}$, and $1.13\times {10}^{-3}$.

Figure 4

The number variance ${\sigma }^2\left(R\right)$ associated with the photoreceptor patterns in chicken retina as well as the associated fitting function of the form ${\sigma }^2\left(R\right)=A{R}^2+BRln\left(R\right)+CR$. We found that the values of the parameter $A$ are several orders of magnitude smaller than the other two parameters, indicating that the associated patterns are effectively hyperuniform. Also shown in each plot is the “surface term” $CR$ for purposes of comparison. The window radius $R$ is normalized with respect to the mean nearest-neighbor distance $d_0$ of the corresponding point configurations.

Figure 5

Pair-correlation functions $g_2\left(r\right)$ of the experimentally obtained point configurations representing the spatial arrangements of chicken cone photoreceptors. The experimental data were obtained by averaging 14 independent patterns. The distance is rescaled by the average nearest-neighbor distance $d_n$ in the system.

Figure 6

Illustration of the hard- and soft-core interactions in a two-species system containing black and red (or light gray in the print version) cells. The left panel shows the exclusion regions (circular disks with two distinct sizes) associated with the two types of cells, which are proportional to the actual sizes of the cells. The black cells have a larger exclusion region than the red cells. The middle panel illustrates the soft-core repulsive interaction (large concentric overlapping circles of the solid black disks) between the black cells. Such a repulsive interaction will drive the black cells to arrange themselves in a perfect triangular lattice in the absence of other species. The right panel illustrates the soft-core repulsive interaction (large concentric overlapping circles of the solid red disks) between the red cells.

Figure 7

Left panel: The bond-orientational order metric $q_6$ of the individual species as a function of the packing fraction $\varphi$ associated with the exclusion regions. Right panel: The translational order metric $T$ of the individual species as a function of the packing fraction $\varphi$ associated with the exclusion regions.

Figure 8

Simulated point configurations representing the spatial arrangements of chicken cone photoreceptors. Upper panels: The configurations shown from left to right, respectively, correspond to violet, blue, and green species. Lower panels: The configurations shown from left to right, respectively, correspond to red, double species, and the overall pattern. The simulated patterns for individual photoreceptor species are virtually indistinguishable from the actual patterns obtained from experimental measurements.

Figure 9

Comparison of the structure factors $S\left(k\right)$ of the experimentally obtained and simulated point configurations representing the spatial arrangements of chicken cone photoreceptors. The simulation data were obtained by averaging 50 independent configurations.

Figure 10

Comparison of the pair-correlation functions $g_2\left(r\right)$ of the experimentally obtained and simulated point configurations representing the spatial arrangements of chicken cone photoreceptors. The simulation data were obtained by averaging 50 independent configurations. The distance is rescaled by the average nearest-neighbor distance $d_n$ in the system.

Figure 11

Left panel: A random-sequential-addition (RSA) packing of hard, identical circular disks in two dimensions with a packing fraction $\varphi =0.54$, which is close to the saturation state. Right panel: An equilibrium system of hard, identical disks at $\varphi =0.54$. The fact that neither of these systems is hyperuniform, as discussed in the text, indicates that hard-core exclusion effects alone are not sufficient to induce hyperuniformity.

Figure 12

Configurations and the associated $S\left(k\right)$ of a three-component system for selected $\varphi$ values. Left panel: At $\varphi =0.35$, the structure factor $S\left(k\right)=0$ for $k<K^{*}$ which indicates the system is stealthy. Right panel: At $\varphi =0.55$, the structure factor $S\left(k\right)\sim k^2$ for small-$k$ values, which indicates that number variance ${\sigma }^2\left(R\right)\sim R$ for large window sizes. The system possesses different degrees of hyperuniformity that can be ascertained from the small-$k$ behavior of $S\left(k\right)$.