## Abstract

Vertebrate photoreceptors are commonly distinguished based on the shape of their outer segments: those of cones taper, whereas the ones from rods do not. The functional advantages of cone taper, a common occurrence in vertebrate retinas, remain elusive. In this study, we investigate this topic using theoretical analyses aimed at revealing structure–function relationships in photoreceptors. Geometrical optics combined with spectrophotometric and morphological data are used to support the analyses and to test predictions. Three functions are considered for correlations between taper and functionality. The first function proposes that outer segment taper serves to compensate for self-screening of the visual pigment contained within. The second function links outer segment taper to compensation for a signal-to-noise ratio decline along the longitudinal dimension. Both functions are supported by the data: real cones taper more than required for these compensatory roles. The third function relates outer segment taper to the optical properties of the inner compartment whereby the primary determinant is the inner segment’s ability to concentrate light via its ellipsoid. In support of this idea, the rod/cone ratios of primarily diurnal animals are predicted based on a principle of equal light flux gathering between photoreceptors. In addition, ellipsoid concentration factor, a measure of ellipsoid ability to concentrate light onto the outer segment, correlates positively with outer segment taper expressed as a ratio of characteristic lengths, where critical taper is the yardstick. Depending on a light-funneling property and the presence of focusing organelles such as oil droplets, cone outer segments can be reduced in size to various degrees. We conclude that outer segment taper is but one component of a miniaturization process that reduces metabolic costs while improving signal detection. Compromise solutions in the various retinas and retinal regions occur between ellipsoid size and acuity, on the one hand, and faster response time and reduced light sensitivity, on the other.

## INTRODUCTION

Since the early days of vision research, pioneered by the work of Hannover (1840), Müller (1856), and Schultze (1866, 1867), vertebrate photoreceptors have been classified as rods and cones by morphological criteria. Schultze (1866) correlated the visual habits of animals with the relative preponderance of rods and cones in their retinas; this led him to formulate the concept upon which the Duplicity Theory rests. The premise of this theory is that cones are the receptors for photopic (bright light) vision, whereas rods are the receptors for scotopic (dim light) sensing. Schultze (1866, 1867) arrived at the correct conclusion that cones mediate color perception.

Subsequent to Schultze’s time, visual cells have been described with intermediate morphological, physiological, and molecular attributes that tend to blur the distinction between rods and cones (Walls, 1963; Pedler, 1965; Crescitelli, 1972; Kojima et al., 1992; Ma et al., 2001; Collin et al., 2004; Zhang et al., 2006). Nevertheless, the old classification has endured as regards to the vertebrate retina: its photoreceptors are rods and cones wherein cones typically exhibit a tapered outer segment, whereas rods do not. The functional significance of this distinguishing feature, so prevalent in nature, remains largely unexplored.

One of the potential benefits of tapering cone outer segments was introduced by Hodgkin and O’Bryan (1977) with their concept of critical taper. In their study of turtle cone electrical responses, these authors considered two limiting cases of cone geometry: the cylindrical (untapered) form and another, in which the “outer segment tapers in such a way that all molecules have an equal chance of absorbing a quantum” (Hodgkin and O’Bryan, 1977). In the latter case, the outer segment must taper at a specific, critical angle, and light must be funneled by complete internal reflection from the broad to the narrow end of cone outer segments (Hodgkin and O’Bryan, 1977). The significance of critical taper is as follows.

Rods and cones are highly specialized cells with unusual properties. First, the sensory visual pigments that they use are extremely absorbent; i.e., they possess very high extinction coefficients, corresponding to large molecular absorption cross sections (Hárosi and MacNichol, 1974). Second, visual pigment molecules are densely packed in lamellar membranes, which, in turn, are tightly stacked in hundreds of layers within the outer segment (the molecular packing within the membrane and the tightness of lamellar packing are probably as high as functional constraints will allow; see Wen et al., 2009). Consequently, the pigment-laden lamellae in the more proximal layers act as light filters for the more distal layers. This phenomenon is known as self-screening (Brindley, 1970). As a result of self-screening, light quanta arriving in the physiological setting have a greater probability of being absorbed near the base of an outer segment than toward the apex. Thus, in a cylindrical rod, where lamellae are of equal size, signal generation declines steadily in more distal layers with a concomitant decline in efficiency (i.e., photocurrent production per unit volume; Schnapf, 1983). One possible way to improve performance is to trim the volume slices along the length of outer segments in proportion to the fall-off of lamellar absorption caused by self-screening. A conical structure could accomplish this. Tapering is considered critical when the trimming of lamellar cross section along the taper is exactly proportional to the fall-off of absorption rate, resulting in uniform efficiency (Hodgkin and O’Bryan, 1977). This idea, which was neither generalized nor experimentally tested, is the basis for the first potential function, overcoming signal loss caused by self-screening, that we evaluate in this study.

A second function considered is that outer segments taper to enhance the signal-to-noise ratio along their lengths. Accordingly, taper would also be driven by another outer segment function: signal generation. Although the generation of signal and associated noise in photoreceptors are complex phenomena, in part because of the stochastic nature of underlying processes, such as the opening and closing of ionic channels or the binding and release of ligands at receptor sites in the enzymatic cascade of the light response, there is a consensus on the existence of thermal activation of visual pigment molecules and cGMP phosphodiesterases, both components giving rise to noise (Rieke and Baylor, 1996, 2000; Holcman and Korenbrot, 2005). Based on current understanding, the receptor signal consists of a photocurrent generated through a narrow circumferential region of the outer segment membrane in response to the number of quanta absorbed in the adjacent volume containing one or a few lamellae (Baylor, 1987). Noise, on the other hand, is assumed, on the most basic level, to be proportional to the total number of visual pigment molecules or cGMP phosphodiesterases contained in the same volume (Rieke and Baylor 1996, 2000; Sampath and Baylor, 2002; Holcman and Korenbrot, 2005). Either way, the signal-to-noise ratio is expected to diminish along the outer segment length (z direction) in cylindrical cells. With a tapered outer segment, however, consecutive lamellae are progressively reduced in cross section, leading to diminution of noise along the way.

The third and last function that we evaluate, efficient light collection and utilization of biomaterials, is based on the hypothesis that outer segment taper follows the optical properties of the inner segment. Rather than considering outer segment taper to be tied up with strictly outer segment functions, this idea proposes a multifaceted interdependence between inner and outer segments, as suggested by morphology.

Both cones and rods feature three distinct compartments or subcellular organelles: an outer segment (limb), specialized for trapping light; an inner segment (cone ellipsoid), concerned primarily with energy production and homeostatic functions; and a synaptic apparatus that communicates with other neurons (Fein and Szuts, 1982). Cone ellipsoids are usually the most conspicuous of the photoreceptor compartments in practically every retina, with primate foveal cones being a notable exception (Borwein et al., 1980; Packer et al., 1989; Hoang et al., 2002). Cones are always broadest at their ellipsoid and tend to taper toward the outer segment, to which they attach closely (Fein and Szuts, 1982). In some fish retinas, the two cone compartments appear as one confluent unit, so that it is hard to discern through the light microscope where the ellipsoid ends and the base of the outer segment begins. In contrast, rods rarely have any difference in width between the two limbs. Shape and size variation notwithstanding, it is always the inner segment wherefrom light enters the outer segment in the physiological setting. For these reasons, it seems logical to consider the two compartments combined as one optical unit.

Cone ellipsoids tend to taper from the thickest proximal region toward the distal outer limb, and this, most likely, is a ploy to concentrate light (Winston 1970, 1981). And if that is so, the outer segment taper may be dependent on the light-gathering property of the inner segment. This idea is also bolstered by the observation that cones with oil droplets tend to have more tapered outer segments than those without this organelle (Nilsson, 1965; Kolb and Jones, 1982; Röhlich and Szél, 2000; Bailes et al., 2006). In view of the high refractive index values of oil droplets (Ives et al., 1983), there is no doubt about their refractive role (Baylor and Fettiplace, 1975; Young and Martin, 1984). Given some light concentration property, cone ellipsoids could funnel parallel incident light into converging (conical) beams, which, when projected onto smaller lamellar areas, could result in equal photon catch (and signal) maintained at reduced noise. Even in the presence of light losses, increased tapering should be advantageous for the gains to be made in improved signal to noise (by lamellar shrinkage) and in savings in detector material (by volume reduction). A practical solution ought to balance the advantages against the concomitant drawbacks, such as reduced acuity and some light loss by ellipsoid leakage. As such, a standard cone should not exist, but there should be variously tapered structures in nature that represent compromise solutions to different sets of constraints. Although the third function does not lend itself to testing via a single mathematical relationship, its validity can be ascertained by examining structure–function relationships in different species and comparing outer and inner segment taper-related variables that, from the aforementioned reasoning, should be positively correlated.

Our analysis of cone taper focuses on photoreceptor properties that have been routinely selected for during the course of evolution such as improved signal detection and metabolic savings by efficient use of biomaterials (see, for instance, the photoreceptor innovations of anchovies; Novales Flamarique, 2011). Early vertebrates, like extant hagfishes and larval ascidians, evolved ciliary photoreceptors that acted as shadow detectors, presumably conferring some of these animals an advantage in dim light environments (Collin, 2010). Natural selection acting on mutations to these ancestral designs led to a large number of novel photoreceptor features, including changes in outer segment shape (from conical to rodlike and vice versa, the transmutation hypothesis; Walls, 1963), multiple photopigments for color vision (Bowmaker, 2008), phototransduction enzymes with varying response kinetics (Hisatomi and Tokunaga, 2002), and, with a focusing eye, photoreceptor mosaics that improved overall sensitivity and/or visual acuity (van der Meer, 1992). As per other selective traits, the shape and size of photoreceptors are expected to vary, and each form may subserve multiple functions, though perhaps none optimally. Indeed, natural selection may retain a given form because it is either nondeleterious or because it confers some advantage to the individual (Bell, 2009). We therefore surmised that our analysis could reveal various advantages of taper to cone photoreceptor function.

The primary thrust of this study is theoretical. Attention is focused on morphological and biochemical properties of vertebrate photoreceptors. The aim is to gain insight into the principles governing their structure and function. In addition to the analytical approach, experimental results are used for testing theoretical predictions. The empirical data include cellular dimensions, which were derived by light and electron microscopic measurements, in situ visual pigment determinations by microspectrophotometry, in vitro visual pigment data obtained by spectrophotometry, electrophysiological determinations, and comparative anatomy. The cited empirical data are derived from either published articles in the literature or hitherto unpublished work from our laboratories. The three potential functions of cone taper that we evaluate are considered in sequence; for each, the consequences and ramifications are examined.

## MATERIALS AND METHODS

### Animals

The majority of data in this study originated from animals used in published works, either our own or those of others. However, some measurements were taken from studies that have yet to appear in the literature. These measurements originated from goldfish (*Carassius auratus*), common carp (*Cyprinus carpio*), zebrafish (*Danio rerio*), three-spine stickleback (*Gasterosteus acuelatus*), blue gill sunfish (*Lepomis macrochirus*), green sunfish (*Lepomis cyanellus*), rainbow trout (*Oncorhynchus mykiss*), coho salmon (*Oncorhynchus kisutch*), chinook salmon (*Oncorhynchus tschawytscha*), African clawed frog (*Xenopus laevis*), northern leopard frog (*Rana pipiens*), American bullfrog (*Rana catesbeiana*), Canada goose (*Branta canadensis*), green-winged teal (*Anas crecca carolinensis*), red-eared slider turtle (*Trachemys scripta elegans*), and mouse (*Mus musculus*). Animals were obtained from the following locations: zebrafish, local pet shop supplier in Burnaby (British Columbia, Canada); three-spine stickleback, Swan Lake (Victoria, British Columbia, Canada); blue gill sunfish, ponds around the Woods Hole, MA area; rainbow trout, Lower Mainland Trout Hatchery (Abbotsford, British Columbia, Canada); coho salmon and chinook salmon; Capilano River hatchery (North Vancouver, British Columbia, Canada); common carp, green sunfish, African clawed frogs, and northern leopard frogs, Marine Resources Centre of the Marine Biological Laboratory (Woods Hole, MA); and bullfrogs, Aquatic Facility Centre of Simon Fraser University (Burnaby, British Columbia, Canada). The animals were kept in aerated, flow-through water tanks under a 12-h light/dark cycle while experiments were being conducted. Fixed and fresh eyes from red-eared sliders were provided by C. Carr (University of Maryland, College Park, MD) and E. Enos (Marine Resources Centre), respectively. Fixed mouse eyes were obtained from staff at the Animal Care Facility of Simon Fraser University, and fixed eyes from Canada geese and green-winged teals (a species of northern duck) were provided by M. Juhas (Haida Gwaii Archipelago, British Columbia, Canada).

Laboratory animals were killed in a state of light or dark adaptation for histology or microspectrophotometry, respectively, and the retinas were processed as detailed in previous publications (Hárosi, 1987; Cheng et al., 2006; Novales Flamarique, 2011). All experimental procedures were approved by the Animal Care Committee of Simon Fraser University or the Marine Biological Laboratory, which are in compliance with the guidelines set by the Canadian Council for Animal Care and the National Institutes of Health.

### Measurements

We obtained cell dimensions from live retinas (microspectrophotometry experiments) and from fixed, Epon-embedded retinas cut into thin (75 nm) sections and observed using a transmission electron microscope (model 7600; Hitachi; Fig. 1). These measurements were inner segment ellipsoid diameter (d_{i}), outer segment diameter at the base (d_{o}), outer segment diameter at a distance, z, from the base (d_{z}), and outer segment length and tip diameter when possible (Fig. 2). In addition, densities of rods and cones were obtained from thick (1 µm) histological sections.

Besides unpublished data, we obtained similar measurements from studies spanning the last 75 yr of anatomical literature. These included 116 species of fishes, covering the evolutionary spectrum from ancient groups like lampreys, elasmobranchs, and lungfishes to modern teleosts like killifishes, carps, and cichlids; 9 species of amphibians; 33 species of birds; 19 species of reptiles; 31 species of mammals; and 8 species of monotremes and marsupials. Table 1 details the species examined and the works consulted.

Each datum presented in the graphs of this study is the mean from a minimum of 15 cells for species used in live cell recordings (Table 2) and anywhere from three to thousands of cells for species data originating solely from the literature. Some publications failed to report cell numbers for the morphological data presented; in such cases, we took the numbers as averages for the entire retina. In our presentation of figures, we show parallel analyses for species for which measurements from live cells were obtained and those whose measurements originated from histological work, primarily from the published literature.

### Methods to evaluate the first biophysical function

Evaluation of this function relied on a comparison of anatomically and physiologically derived measures of two parameters: characteristic length and taper, as described in Appendix 1. Two methods were used in the evaluation. The first involved the calculation of the required characteristic length, a_{h1}^{−1}, from Eq. 6 (Appendix 1) for each cell. The results were then compared with two separate estimates of characteristic length, a_{z}′^{−1} and a_{z}′′^{−1}, obtained with the aid of Eqs. 10 and 16 (Appendix 1). The necessary parameter values for the latter calculations were derived from previous determinations performed on equivalent cells. For example, it was assumed that cones with vitamin A_{2}–type pigments share the properties with those of the yellow perch, goldfish, or Japanese dace and have the following parameter values: ε = 30,000 liter mol^{−1} cm^{−1}, c = 3.5 mM, R = 2.2, and S_{┴} = 0.0125 µm^{−1}. For cones using vitamin A_{1}–type pigments, ε = 42,000 liter mol^{−1} cm^{−1}, c = 3.5 mM, R = 2.5, and S_{┴}= 0.0175 µm^{−1} (Hárosi and MacNichol, 1974; Hárosi, 1975, 1976, 1984, 1985; Hárosi and Hashimoto, 1983). The estimated characteristic lengths were not significantly affected by the particular values assumed for individual parameters. For instance, we could have used Liebman and Granda’s (1971) specific absorbance of 0.0130 µm^{−1} to obtain a_{z}′′^{−1} = 33.4 µm or assumed 3-mM pigment concentration and an infinite dichroic ratio (k = 1.5), as did Hodgkin and O’Bryan (1977), and arrived at a_{z}′^{−1} = 32.2 µm. These values are similar to those reported in Table 2.

As a second method for testing the validity of the first function, we calculated a new diameter (d_{z}′) for each cell that would occur at critical taper by using a_{z}′ and the rest of the geometrical data in Eq. 5 (Appendix 1). The actual taper, τ, obtained with d_{z}, was then recalculated using d_{z}′ in Eq. 17. The resulting τ′ represents taper for each cell that would satisfy invariance of flux density along the outer segment.

### Evaluation of the second biophysical function

Eq. 30 of Appendix 2 is an expression for the signal-to-noise ratio along an outer segment. This equation was scrutinized to assess the validity of the second proposed function.

### Evaluation of the third biophysical function

For the lack of a compact theoretical expression (Appendix 3), this function could not be verified experimentally. Our approach here was to test whether several morphological observations across phyla were consistent with it. For instance, the difference in light-gathering efficiency by tapered versus nontapered photoreceptors (Appendix 3) led us to predictions about the rod/cone ratios of primarily diurnal animals; i.e., animals with both diurnal and crepuscular/nocturnal rhythms, but in which the primary rhythm is diurnal, and in which utilization of both cone and rod pathways occurs during the day (these species are to be contrasted with those that are fully diurnal, e.g., some ground squirrels [West and Dowling, 1975; Anderson and Fisher, 1976], or primarily nocturnal, e.g., the giant African rat; Peichl, 2005). In addition, we postulated that there should be a correlation between outer segment taper and ellipsoid concentration factor (Eq. 33, Appendix 3), the latter being a measure of the ability to concentrate light from inner to outer segment. To test this, we used critical taper (Appendix 1) as a benchmark and expressed actual taper as the ratio a_{z}′^{−1}/a_{h1}^{−1}, indicating how many critical characteristic lengths would equal the characteristic length considered realistic.

## RESULTS

### Cone taper exceeds critical taper

Table 2 presents typical morphometric data obtained from live photoreceptors during microspectrophotometric experiments, in this case from individual photoreceptors of goldfish. Values of derived variables (e.g., taper, characteristic length, and concentration factor) are also presented. Figs. 3–6 show means from analogous datasets for the various species examined.

Comparison of data in columns 8–10 of Table 2 reveals that the characteristic length corresponding to critical taper differs from the characteristic length calculated from known data. Moreover, the difference in most cases goes far beyond the margin of error (∼0.1 µm, from repeated measures). Except for one cone (Table 2, row 1) that came close to having critical taper, the a_{h1}^{−1} values for the rest of the cones are much smaller than expected. A smaller characteristic length, however, means a larger absorption coefficient. We can consider, for instance, the blue-absorbing single cone on line 25 of Table 2. For it to have critical taper, the axial extinction would have to be ∼3.8-fold higher than a realistic value.

As is apparent from columns 6 and 7 of Table 2, the taper of real cones is excessive. For the vast majority of species studied, whether the data originated from live cell measurements (Fig. 3 A) or from fixed, histological material (Fig. 3 B), the mean cone taper (τ) was at least 1.5 times the critical taper (τ′). Only in the case of bats, the giant African rat, and foveal-perifoveal cones of primate retinas was cone taper statistically the same as critical taper (P > 0.05, paired *t* tests). On average, the ratio τ/τ′ ± SD was 4.8 ± 2.4 for fishes, 5.1 ± 1.7 for amphibians, 5.5 ± 2.1 for birds, 5.7 ± 3.8 for reptiles, 4.6 ± 3.2 for mammals, and 4.3 for marsupials (two species). Among the fishes, the mean elasmobranch ratio (7.2 ± 2.7) was about twice that of teleosts (3.8 ± 1.4), and the highest ratios among the teleosts were for lungfishes (5.2) and killifishes (4.8), which, like most amphibians, reptiles, and birds and some fishes and monotremes, have oil droplets (lungfishes) or have ellipsosomes (killifishes) in the ellipsoid region (Fig. 1). Among the reptiles, the ratio for the strictly diurnal garter snake (5.3) was about half that of primarily nocturnal snakes like the boa and ball python (mean of ∼11). Overall, phylogenetic groups with focusing structures in the ellipsoids (e.g., oil droplets, microdroplets, and megamitochondria) had larger τ/τ′ ratios than those lacking them.

Both methods of assessment led to the same general observation: overall, cones are more tapered than they ought to be for critical taper. Therefore, cone outer segment taper overcompensates for the loss of signal caused by self-screening. In contrast, rod outer segments, if tapered at all, have less than critical taper (Table 2), with the potential exception of rods in lampreys and geckos, species in which the two photoreceptor types are hard to classify because they exhibit intermediate morphological and molecular attributes (Collin et al., 2004; Muradov et al., 2008; Zhang et al., 2006).

### Signal-to-noise ratio increases beyond that predicted by critical taper

Examination of Eq. 30 (Appendix 2) reveals the following: (a) when A_{z} is invariant in z (cylindrical outer segment), the signal-to-noise ratio, S/N, diminishes exponentially along z as a result of self-screening; (b) if A_{z} versus z diminishes by tapering, the S/N will undergo proportionate increases; (c) for critical taper, when A_{z} varies in accordance with Eq. 4 (Appendix 1), the exponential terms cancel, and the S/N becomes independent of z. From Eq. 30 we can also deduce that, for excessive taper, i.e., when a cone cross section (A_{z}) diminishes faster than the exponential fall-off of signal, the signal-to-noise ratio may actually increase toward the apex of a cone outer segment. Eq. 30 thus reveals an important tendency: more taper means greater improvement in the signal-to-noise ratio along an outer segment. For this reason, in the absence of other requirements, cones should taper as much as possible.

### Inner segment morphology as a major determinant of outer segment taper

For the lack of a compact theoretical expression (see Appendix 3), the third function linking inner segment morphology to outer segment taper could not be verified experimentally. Nevertheless, several observations were consistent with it, as demonstrated by the following analyses using examples from diverse phylogenetic groups.

Teleosts, like the goldfish, have typically large, plump inner segments and shorter outer segments compared with the rods (Fig. 1). The average goldfish cone in this study had an entrance aperture (assumed to be equal to the broadest region of the ellipsoid) of 8.7-µm diameter, wherefrom light would be funneled to the base of the outer segment with a mean diameter of 5.9 µm (Table 2). The concentration factor, F_{C}, by Eq. 33 (Appendix 3) is 2.2. The outer segment volume calculated by Eq. 34 yields 277 µm^{3}. The volume of a cylinder with equal base diameter is 399 µm^{3}. Their ratio gives 1.4 for the geometry factor, F_{G}, by Eq. 35. Additionally, the volume reduction ratio, V_{R}, is obtained at 3.1 by Eq. 36. Thus, the cone outer segment uses ∼32% of the volume of that of the hypothetical optically equivalent rod. The rod cells measured in these goldfish preparations were quite uniform, with an average outer segment diameter of 2.1 µm and lengths in the range of 36–44 µm. For a 40-µm-long rod outer segment, we can calculate the axial absorptance by Eqs. 3 and 22 using a_{z}′ (Appendix 1). Accordingly, such a rod would absorb ∼69% of the light incident at its base. By a similar calculation, the 15-µm-long cone outer segment would absorb only 35% of the flux it receives. Therefore, the rod is the better light detector of the two. However, this rod collects a mere fraction of what a single cone can gather from the retinal illumination. In fact, it would take (8.7/2.1)^{2} = 17.2 rods to intercept as much incident flux as the average cone considered here. The rod/cone ratio in goldfish was reported to be 15:1 (Stell and Hárosi, 1976). The reasonably satisfactory agreement between predicted and observed ratios suggests that, in this animal, interception of equal light flux, a property linked to inner segment aperture, appears to be the criterion driving rod and cone densities.

When this analysis was applied to the other, predominantly diurnal, species studied, the correlations between observed and expected rod/cone ratios were surprisingly good (Fig. 4, A and B). On average, the rod/cone ratio expected and observed ± SD for the various phylogenetic groups were as follows: 6.9 ± 10 and 7.6 ± 9.2 (fishes), 1.4 ± 1.6 and 1.5 ± 0.58 (amphibians), 2.0 ± 2.1 and 1.4 ± 2.0 (birds), 5.0 ± 3.4 and 6.8 ± 7.4 (reptiles), 12 ± 7.4 and 13 ± 10 (mammals), and 12 and 16 (marsupials, two species). Among the fishes, the rod/cone ratios of teleosts were better predicted than those of elasmobranchs; the mean predicted and observed ratios for these two groups were 6.2 ± 9.0 and 6.4 ± 9.1 (teleosts) and 8.5 ± 12 and 10 ± 9.1 (elasmobranchs). Overall, the highest rod/cone ratios occurred for the walleye (a teleost with “remarkably large cones”; Januschka et al., 1987) and the mink. The best predictions occurred within the teleosts and mammals, especially for primates (predicted and expected mean rod/cone ratios were 16 ± 7.3 and 17 ± 8.8). The worst predictions (i.e., ratio of expected to predicted, or vice versa, >2) occurred within the birds (owls) and reptiles (snakes) as well as for elasmobranchs and one teleost, the snake mackerel. Some of these animals (e.g., owls, several species of elasmobranchs, and large snakes) were included in the analysis because of reported crepuscular (cone driven) activity, although they may be primarily nocturnal and perhaps not very appropriate for inclusion in these regressions. Indeed, the rod/cone ratios of primarily nocturnal or fully diurnal animals are not very well predicted based on our principle of equal flux sharing between photoreceptor types (Fig. 5). Primarily nocturnal or dark habitat–dwelling species, which include deep ocean sharks and eels, several mammals (e.g., the mouse, African giant rat, bats, and the spotted hyena), the oil bird, and marsupials (opossums and the tammar wallaby), have large to very large rod/cone ratios (some exceeding 100:1). At the other end of the spectrum, strictly diurnal species like some ground squirrels, the prairie dog, and the tree shrew have very small rod/cone rations (≤0.1). None of these species, located at either extreme of the rod/cone ratio range, were included in the analysis leading to Fig. 4 B.

In contrast to the comparable size of photoreceptors in teleosts and mammals, the retinas of amphibians have colossal rods and minute cones (Fig. 1). The dichotomy in outer segment size is seldom as pronounced as in frog species, with the possible exception of the salamanders (Crescitelli, 1972). However, even in these animals, the important feature determining rod and cone densities appears to be the equal sampling of the plane, where, presumably, the image of the outside world is formed within the eye.

As an example, the mean cone ellipsoid diameter (entrance aperture) of the African clawed frog was found to be 7 µm, whereas the mean diameter of the outer segment base was 2.9 µm. Repeating the procedure used previously, we can calculate, by using Eqs. 33–36 (Appendix 3), the following: F_{C} = 5.8, V_{c} = 20.1 µm^{3}, V_{r} = 39.6 µm^{3}, F_{G} = 1.97, and V_{R} = 11.4. The optically equivalent cylinder, as indicated in Fig. 2 C, has a volume of ∼231 µm^{3}. Thus, the cone outer segment volume is only ∼9% of that of the cylinder.

The preparations where frog single cones were located also had rhodopsin-containing red rods with an outer segment diameter equal to the widest region of the ellipsoid, with a mean diameter of 7 µm. This finding may be interpreted in terms of the cone ellipsoid’s light-gathering property and suggests that the entrance aperture of single cones and red rods are equal in frogs. Further support for the idea of equal flux collection by these cells in the frog retina can be found in the electron microscopic studies of Nilsson (1964, 1965). He showed, among other things, that red rods and single cones have about equal areas in cross section at the inner segment level (Nilsson, 1964). These photoreceptors, therefore, may indeed gather nearly equal fluxes from a uniform retinal illumination.

The actual volume of a red rod outer segment (at a length of 66 µm) is ∼2,500 µm^{3}. Using known parameters and Eqs. 3 and 22 (Appendix 1), we can calculate that these rods may catch 94% of axially incident light, whereas the average cone catches only ∼23% of it. Thus, a typical red rod may be a fourfold better absorber of axial light than a single cone (at the λ_{max} of their respective visual pigments). However, the cone outer segment volume, V_{c}, as calculated previously, is nearly 100-fold smaller than that of the rod. The rod/cone ratio expected, 1, is similar to that assessed from published micrographs (means of 1–1.5; Kinney and Fisher, 1978a; Hollyfield et al., 1984; Röhlich and Szél, 2000). The fact that cones absorb a smaller portion of the incident light is not a handicap under daylight conditions, when high absolute sensitivity is not required. For the same reason, the loss of light by leakage from cone ellipsoids can be tolerated when light is plentiful. Therefore, frog rods and cones can serve the retina equally well despite their large disparity in size. By making cones with large entrance aperture and an ability to concentrate light, outer segment size reduction becomes possible at some cost in detector efficiency and sacrifice in acuity.

Upon review of the preceding examples, the data suggest a positive relationship between outer segment taper and ellipsoid concentration factor, F_{C}. The latter variable is a measure of light concentration from the ellipsoid into the outer segment. We have plotted the ratio a_{z}′^{−1}/a_{h1}^{−1}, indicating how many critical characteristic lengths would equal the characteristic length considered realistic, against F_{C} for the various species examined (Fig. 6, A and B). The correlations are positive, though not particularly strong, especially when all the species studied are considered (Fig. 6 B). The lack of a strong correlation for the ensemble of species may in part reflect the lower accuracy of data obtained from histological material (especially from published figures) as opposed to live cell measurements. In addition, and perhaps more importantly, it may be an indication of other factors, such as the presence of oil droplets or light losses caused by ellipsoid leakage, affecting the extent of outer segment taper, none of which were considered in the calculations (for an instance of light loss estimation, see Baylor and Fettiplace, 1975). Nevertheless, the correlations, albeit approximate ones, between the cone ellipsoid concentration factor and outer segment taper expressed as a multiple of critical taper support the third function.

## DISCUSSION

### Emerging view on structure–function relations in vertebrate photoreceptors

Our analysis supports the three postulated functions for cone taper: (1) compensation for light loss resulting from self-screening, (2) increased signal-to-noise ratio along the length of the outer segment, and (3) improved light capture and material savings by shape continuity between inner and outer segment. Cone taper, as determined in this work, was found quite variable and, for the most part, beyond that predicted for critical taper (Table 2 and Fig. 3). Consequently, cones appear to achieve full compensation for the light loss caused by self-screening while improving signal-to-noise ratio along their outer segments. Support for the third function points to a dominant role of the inner segment in shaping the outer segment, leading to a process of miniaturization and, as a result, metabolic savings in biomaterials. The trade-offs in such structural modifications are in absolute sensitivity and visual acuity, with spatial resolution being inversely related to receptor cross section (Snyder and Miller, 1977; Neave, 1984; van der Meer, 1992; Haug et al., 2010). Large cone ellipsoids combined with excessively tapered outer segments could achieve miniaturization, but only in the presence of light funneling. Therefore, an ellipsoid light concentration property was suggested by logic and implied by the third function.

In every retinal region, a trade-off is expected between acuity and receptor size. The typical example is the retina of some diurnal teleosts where the highest visual acuity, found in the mid to upper frontal field, is subserved by smaller, densely packed cones in the centro- and ventro-temporal areas of the retina (Beaudet et al., 1997; Novales Flamarique, 2005, 2011; Cheng and Novales Flamarique, 2007). Other more extreme examples are the specialized foveas of diurnal lizards (Röll, 2001; Barbour et al., 2002) and primates (Borwein, 1981). In primates, high visual acuity is the product of tightly packed, long, rodlike foveal cone outer segments, each having a diameter of ∼1 µm; these are connected by cone fibers to the rest of their compartments that are laterally displaced and squeezed out to the slopes of the foveal pit (Borwein, 1981; Packer et al., 1989). Foveal cones have no ellipsoids, and, therefore, they must collect light without the aid of any other structure. However, parafoveal cones, and other cones in general, are built with joint outer and inner compartments, where the latter is the thicker of the two to intercept a larger area of retinal illumination at the cost of reduced regional acuity (Packer et al., 1989; Hoang et al., 2002). As flux is funneled from the entrance aperture to a smaller exit aperture at the distal end of ellipsoids, structural modifications become feasible in outer segment tapering and size. Cone miniaturization has several advantages, including improved signal-to-noise ratio, faster visual pigment regeneration (i.e., recovery from bleaching), and material savings as a result of reduced volume. The drawbacks are reduction in absolute sensitivity vis-à-vis rods and some light losses from detection as a result of ellipsoid leakage.

Rods, on the other hand, are specialized for high absolute sensitivity and not for rapid response (Burns and Lamb, 2003). The eyes of deep-sea fishes provide exquisite examples for such sensitive detector structures (Locket, 1977). The cylindrical form appears well suited for on-demand tailoring of outer segment length, which may reach hundreds of micrometers in fish species living at great depths. The use of multiple banks and assemblages of rods, in addition to long ones, also appears to be aimed at catching scarce photons in the darkness of the deep ocean (Locket, 1977; Munk, 1977; Collin et al., 1998). Rods do not jeopardize detection efficiency by the use of light-concentrating schemes at the ellipsoid level; they have wider acceptance angles for oblique rays than cones do. Other mechanisms, however, may operate to concentrate light onto the outer segment. For instance, some nocturnal mammals, like the mouse, show an inverted chromatin nuclear pattern that, together with a quasi-columnar organization of rod nuclei, serves to channel light from the outer nuclear layer toward the outer segments (Solovei et al., 2009). Signal pooling from multiple rods (at a sacrifice of spatial resolution) and coincidence detection should further improve scotopic performance as well as signal-to-noise ratio (Peichl, 2005).

### Improved light funneling by oil droplets

Oil droplets are optical devices used by most cones in avian, amphibian, and reptilian eyes. Oil droplets are highly refractive spherical globules (Ives et al., 1983); they may be clear or colored (Kolb and Jones, 1987; Hart et al., 2006). Because of their location at the distal end of inner segments anterior to the base of outer segments, they may serve as light filters as well as focusing devices (Young and Martin, 1984; Vorobyev, 2003). A spherical body with high refractive index immersed in a lower index medium will behave as a positive lens with a short focal length. Thus, an oil droplet (or a concentrated group of microdroplets, as in diurnal snakes; Wong, 1989) is expected to further concentrate the light funneled by the ellipsoid. As such, increased outer segment tapering and size reduction become feasible. And indeed, observational evidence bears this out: the most extreme cases of outer segment volume reduction are found in oil droplet–containing cones (Fig. 1). Oil droplet focusing might be advantageous not only in allowing further reductions in detector cross section, but also in quickening recovery from blinding exposures, as when experiencing glare.

For an elaboration of the last point, note the following: whereas scotopic sensitivity in primate and teleost vision may take tens of minutes to regain dark-adapted levels after a bright “bleaching” exposure, photopic sensitivity returns to former levels in a few minutes (Rushton, 1965; Thomas and Lamb, 1999; Kenkre et al., 2005; and unpublished data for teleosts). This indicates that cones recover their sensitivity in vivo faster than rods do. Experiments in vitro also reveal the same tendency in chemical regeneration of visual pigment: in the presence of copious amounts of exogenous 11-cis retinal, whereas rods regenerate slowly, cones recover rapidly and repeatedly after several bleaching exposures, and, in each case, they regain most of their visual pigment in a few seconds (Hárosi, 1984). Consider now the oil droplet–equipped cones in a fishing bird’s eye. Given the short focal lengths of spherical lenses, an oil droplet will focus light in a specific region of an outer segment lamella, which would receive intense illumination. Let’s assume, for the sake of the argument, that only 10% of a lamella is illuminated. When the bird skims over a body of water, reflected sunlight could be a blinding experience. In case of a human observer, extensive bleaching in retinal receptors would occur, resulting in temporary blindness. However, in cones with oil droplet focusing, only a fraction of the visual pigment complement would get bleached, and, in the assumed lamella, 90% of the visual pigment would remain unexposed. Because lateral and rotational diffusion drives visual pigment molecules rapidly in the receptor membranes (Wang et al., 2008), the bleached molecules would quickly be exchanged with unbleached ones from the adjacent membrane area. Therefore, most of the cone’s sensitivity could recover in milliseconds, at least three orders of magnitude faster than a recovery based on normal biochemical regeneration (Kenkre et al., 2005).

The aforementioned mechanism could improve the foraging performance of birds like the black phoebe, a sit-and-wait predator whose visual searching increases significantly under bright light conditions, likely as a result of the negative effects of glare (Gall and Fernández-Juricic, 2009). Other bird species like herons and osprey routinely hunt in shady habitat or at crepuscular periods (unpublished data), reducing their exposure to glare and other blinding factors such as the light flickers produced by waves near the water surface. In fact, such flicker may be used by fish for camouflage, as multiple species have developed body markings that resemble the light patterns (McFarland and Loew, 1983). Fast-moving predators, like some insects and birds, have critical fusion frequencies that surpass the predominant flicker occurring in surface waters (in the range of 1–5 Hz), improving visual contrast of underwater targets (McFarland and Loew, 1983). Given a mean diffusion coefficient of 0.4 µm^{2} s^{−1} for activated opsins (Wang et al., 2008), between ∼3 and 13% of the (bleached) visual pigment molecules in a 1-µm-diameter lamellae would get replaced by intact ones between flickers, contributing to fast recovery from bleaching.

### On light collection mechanisms

The idea that cone inner segments concentrate light has been around for a long time. For instance, O’Brien (1946) proposed it in his theory explaining the Stiles-Crawford effect (O’Brien, 1951; Johnson and Tansley, 1956; Enoch, 1963). Concerning vertebrate photoreceptors, turtle cone ellipsoids have been compared with ideal light collectors (Baylor and Fettiplace, 1975; Winston, 1981). The name refers to nonimaging optical devices that concentrate light by internal reflection onto the smallest possible exit aperture (Winston, 1970). In the case of an invertebrate eye, Levi-Setti et al. (1975) suggested that the crystalline cone in an ommatidium of *Limulus* is an ideal light collector. Subsequent work by Land (1979), however, showed that the crystalline cones can form images and that they pass light through a refractive index variation scheme. Similar refractive index gradients have not been uncovered in vertebrate photoreceptors, and the mechanism of importance remains to be sorted out. Internal reflection, refraction, and even diffraction may play some role. Cone ellipsoids are not truly homogeneous, as they contain abundant mitochondria that might function akin to a Fresnel zone plate. The one conclusion that is certain at present is that an oil droplet must have a major contribution to the refractive power of the system. Another open question is whether investigations of waveguide modal patterns in vertebrate visual cells would facilitate the understanding of structure–function relationships between photoreceptor compartments (Snyder and Menzel, 1975).

### Oblique incidence and the Stiles-Crawford effect

Stiles and Crawford (1933) discovered that the visual sensitivity of the human eye depends on the direction from which light enters the pupil (known as the Stiles-Crawford effect of the first kind [SCE_{1}]). O’Brien (1946) was the first to advance a theory to explain it, and there have been others (e.g., Snyder and Pask, 1973; reviewed by Enoch and Bedell, 1981). The third hypothesis is consistent with the SCE_{1}. Accordingly, cone ellipsoids are imperfect concentrators; they lose light by leakage. The lower refractive indices of cone inner and outer segments provide reduced critical angles as compared with those of rods (Appendix 1). But rods, with more uniform and denser distal compartments, have larger critical angles and thus can tolerate a wider range of off-axis rays. The combination of these properties may be sufficient to account, at least qualitatively, for the larger photopic and smaller scotopic SCE_{1} (Enoch and Bedell, 1981). Further support for the third function comes from a finding by Westheimer (1967) that foveal cones of the human eye have a reduced (i.e., rodlike) Stiles-Crawford effect when compared with parafoveal cones. Also supportive is the small directional sensitivity displayed by the human achromat, an abnormal condition in which vision depends entirely on rod function (Nordby and Sharpe, 1988).

### Summary

We have provided evidence to support the notion that cone outer segment taper follows from the shape of the ellipsoid, a structure that serves to concentrate light onto the outer segment. The advantages of taper and cone miniaturization include compensation for light loss caused by self-screening, metabolic savings in structural components, higher signal-to-noise ratio, and accelerated regeneration of visual pigment. The trade-offs are in absolute sensitivity and visual acuity. As suggested by the different sizes of photoreceptors and their relative densities across phylogenetic groups, species have evolved visual cells that presumably optimize these trade-offs for life in particular environments. As such, there is no model cone or rod but a range of sizes and shapes dictated by the ecological constraints guiding the evolution of each species.

## APPENDIX 1

### Derivation of equations to test the first biophysical function: Cone outer segments taper to compensate for light flux diminution by absorption (self-screening) so that flux density remains invariant or increases with axial distance along the outer segment

The list of symbols and definitions used in the testing of this function are shown in Table 3. The following sections present geometric and spectrophotometric definitions of key variables (e.g., absorption coefficient and taper) used to evaluate whether a cone taper is critical, i.e., whether it compensates exactly for light flux diminution by self-screening (Hodgkin and O’Bryan, 1977). As a consequence, taper that is equal to or exceeds the critical value indicates compensation for self-screening.

### Geometric derivation of the absorption coefficient

Perfect compensation for light absorption caused by self-screening imposes two requirements. The first is invariance of flux density with respect to z, so that J_{o} = J_{z}; in other words, the light flux impinging on the base (Φ_{o}) divided by the cross-sectional area (A_{o}) is equal to the transmitted flux (Φ_{z}) divided by the corresponding cross section (A_{z}) along the entire length of the outer segment. Accordingly,

The second requirement is that visual pigment absorption be the only reason for the diminution of flux (i.e., light leakage is absent). This means that the incident light rays interact with the cell boundary at angles (Θ) below the critical angle (Θ_{c}) so that total internal reflection prevents light from escaping (see the section Critical angle estimations for data confirming this). Thus,

The exponential law of absorption,_{z}) in terms of the incident flux (Φ_{o}) multiplied by an exponential function of variable (z) and a characteristic constant of the medium, called the absorption coefficient (a). The latter is inversely proportional to the attenuation, or characteristic, length, at which Φ falls to exp(−1) = 0.368 of Φ_{o}. Upon combining Eqs. 1 and 3, we obtain

The circular cross section of a cone with diameter d is A = d^{2}π/4, and thus, Eq. 4 may be written as

wherefrom the absorption coefficient for testing the first function is expressible as

The surprising outcome gleaned from Eq. 6 is that the absorption coefficient, a purely spectroscopic quantity, may be determined from measurements of distance involving the base diameter, d_{o}, a second diameter, d_{z}, and their separation along the z coordinate.

### Spectrophotometric determinations of the absorption coefficient

Two methods have been established for the determination of the axial absorption coefficient of a vertebrate photoreceptor outer segment in situ (Hárosi and MacNichol, 1974; Hárosi, 1975, 1982). The first one requires knowledge of the visual pigment’s molar extinction coefficient, ε, its equivalent (random) molar concentration, c, and the dichroic ratio, R, or some equivalent measure of anisotropy. The second method makes use of the transverse specific absorbance, S_{┴}, determined in a side-on oriented outer segment. The two methods are interdependent by the dichroic ratio, R, as shown in the section Anisotropic absorbance by visual cells.

### Isotropic absorbance by solutions

The light-absorbing property of a homogeneous, isotropic medium is expressed in terms of either the natural or the decadic logarithm of incident to transmitted light fluxes. Thus, absorbance (optical density) may be written as

in which c is the concentration, z is the path length, and α is the molecular or chromophoric (Napierian) and ε is the molar (decadic) extinction coefficient of the light-absorbing substance. Substituting Eq. 3 in Eqs. 7 and 8, the absorption coefficient for a random ensemble of absorbing molecules can be related to α and ε as

Note that a, α, and ε are applicable to isotropic pigments and that they are also wavelength dependent. Their values at λ_{max} are often used as single-valued parameters (e.g., Warrant and Nilsson, 1998).

### Anisotropic absorbance by visual cells

Rod and cone outer segments in side view exhibit intrinsic linear dichroism, revealing their anisotropic nature. Because the absorption vectors of the visual pigment molecules lie nearly parallel with the lamellar (x-y) planes, axially traveling light (z direction) is well absorbed; this makes photoreceptors more effective light catchers in this direction than an isotropic solution of the same pigment would be at the same concentration and path length.

By a generalization of Eq. 9, the first formula to express the end-on (axial) absorption coefficient along the z direction is

in which k is the anisotropy factor accounting for the gain by the ordered distribution; k may be as large as 1.5 (perfect two-dimensional random array) and as small as 1.0 (three-dimensional random array). By polarized absorbance measurements of side-on oriented receptors, the dichroic ratio, defined as R = A_{┴}/A_{║}, can be determined, and k can be expressed as a function of R, as follows.

If a cylindrical vessel is homogeneously filled with N absorbing molecules, each possessing a transition moment M, the total extinction (E = NM) may be written as E = E_{x} + E_{y} + E_{z}. The fractions expressed in the three spatial coordinates are equal for a three-dimensional random array so that E_{x}/E = E_{y}/E = E_{z}/E = 1/3. The measurable fraction of extinction in the z direction is half of the two orthogonal components:

In a similar model, a cylindrical photoreceptor may be regarded, in its simplest form, as an imperfect two-dimensional random array for which E_{x} = E_{y} and E_{x}/E_{z} = R. With these, the z fraction is expressible as

Viewed from the z direction (i.e., end-on), the measurable fraction is

The anisotropy factor (k) in the z direction is obtainable from the above as the ratio of end-on extinctions (the two-dimensional versus the three-dimensional fractions) and is given by (see Hárosi, 1975, 1982)

Thus, ε, c, and R yield the axial absorption coefficient of a photoreceptor with the aid of Eqs. 10 and 14.

The second formula for the calculation of the axial absorption coefficient makes use of the microspectrophotometrically determined transverse specific absorbance, S_{┴} = A_{┴}/d, in which A_{┴} is the peak absorbance (of the α band) for transversely polarized light of the outer segment in side-on orientation and d is the mean cell diameter, presumed to be equal with transverse path length. The following relationship,

has been found useful in relating in vitro spectrophotometric data to those obtained by in situ microspectrophotometry (Hárosi, 1975). The substitution of Eqs. 14 and 15 into Eq. 10 leads to the desired equation:

Therefore, the axial absorption coefficient is also attainable from the transverse specific absorbance by Eq. 16. Anisotropy is implicit in S_{┴} since Eq. 15 may also be written as S_{┴} = k(εc). The latter relationship shows the interdependence of Eqs. 10 and 16, which, therefore, lead to dependent estimates of the axial absorption coefficient, even though the two are based on different determinations.

### A definition of taper

As a geometrical characterization of a cone taper, τ is defined here as the angle between the axis and the inclination of the contour line, which, upon precession, describes the conical surface:

The axial distance, z, is the separation between the base of a right cone, with diameter d_{o}, and a parallel slice, with diameter d_{z}. For a cone cell, τ describes an average inclination of the contour line between two slices (corresponding to d_{o} and d_{z}), and it cannot account for a point-by-point variation that Eq. 5 describes.

### Critical angle estimations

The refractive indices for rod and cone inner and outer segments have been measured only for a few animal species in a handful of studies. The obtained values are as follows: the refractive index for rod ellipsoid and outer segment, 1.40 and 1.41, respectively; for cone ellipsoid and outer segment, 1.39 and 1.385, respectively; and for the extracellular matrix (mucopolysaccharide), 1.34 (see Borwein, 1981 for primary citations).

At a boundary between two transparent media, one denser with refractive index n_{1} (inside), the other rarer with refractive index n_{2} (outside), light rays will refract according to Snell’s law:

in which ϕ_{i} and ϕ_{o} are the corresponding angles of incidence and refraction with respect to the normal. Beyond a certain angle of incidence in the denser medium, no refraction occurs, and there is total internal reflection. The critical angle (for which sin ϕ_{o} = 1) can be expressed from Eq. 18 as

From this, Θ_{c} of Eq. 2 can now be obtained as

Substitution of the numerical values cited above in Eqs. 19 and 20 leads to the following critical angles in Θ_{c}: 15.4° and 14.6° for cone ellipsoid and outer segment and 16.8° and 18.1° for rod ellipsoid and outer segment, respectively. In these calculations, we assumed that the relevant optical parameters used in critical angle calculations are those that have been determined (Borwein, 1981), notwithstanding the fact that photoreceptors and their surroundings are neither homogeneous nor isotropic. What constitutes the surrounding medium is especially questionable. This, sometimes referred to as interstitial matrix, is commonly equated with mucopolysaccharide, as done here. However, microvilli of the pigmented epithelial cells and the calycal processes that come in contact with at least parts of inner and outer segments may play a role in setting the refractive properties of these cells.

Outer segment taper, as defined by Eq. 17 and determined from our observations and published photomicrographs (see Results), indicate angles <14°. These values are within the requirement of Eq. 2, and thus, they justify the original assumption concerning the funneling of axial rays by total internal reflection.

## APPENDIX 2

### Derivation of equations to test the second biophysical function: Cone outer segments taper to improve signal-to-noise ratio along their lengths

The list of symbols and definitions used to evaluate this function are presented in Table 4. We make the following five assumptions. (1) Incident light consists of an axial flux of parallel, uniform, and steady illumination at the base of outer segments with negligible reflection and scattering losses. (2) Visual pigment molecules are packed uniformly in transverse membranes of all rods and cones with constant surface density: J_{n} = n/A_{t} = n_{o}/A_{o} = n_{z}/A_{z}. (3) Activation of visual pigment molecules requires not only light absorption but also chromophore isomerization, commonly referred to as bleaching. To take this into account, the quantum efficiency of bleaching, γ, must be factored in (Dartnall et al., 1936). (4) Signal is constituted by a photocurrent generated through the cell envelope at each lamella, independent of other lamellae. Although there clearly is an observable spreading of excitation, for the sake of simplicity, all messengers for signal are assumed to originate in one lamella. (5) Noise is generated like signal in lamellae under the conditions set forth in the preceding assumption. Whether thermal isomerization of the chromophore or subsequent biochemical steps are the cause (Rieke and Baylor 1996, 2000; Sampath and Baylor, 2002; Holcman and Korenbrot, 2005), it is assumed here that noise is proportional to the number of visual pigment molecules contained within each layer.

The objective of this appendix is to relate the generation of signal and noise to outer segment taper. Given an absorbing medium, an incident flux will either be absorbed or transmitted, provided that assumption 1 holds. In general terms,

When normalized to the incident flux, Eq. 21 may be rearranged as

The left-hand side in Eq. 22 is usually referred to as absorptance, A_{b} = Φ_{a}/Φ_{o}, whereas the fraction on the right-hand side is called transmittance, T = Φ_{z}/Φ_{o}. Optical density (or absorbance), as defined previously in Eq. 7, is the logarithm of T^{−1}. After substitution of Eq. 3 in Eq. 22 and rearrangement, the rate of photon absorption by a monolayer of thickness δ, with incident flux Φ_{o}, is

Because visual pigments are hydrophobic chromoproteins, they must be membrane bound. For this reason, the shortest meaningful axial path length of the visual pigment in a vertebrate photoreceptor is a single layer. Although the exact numerical value of δ for the present is immaterial, it is assumed to be 15 nm. This is half the repeat distance of the 30 nm obtained for rod disks by electron microscopic and x-ray crystallographic determinations (for references, see Fein and Szuts, 1982). The value of δ = 1.5 × 10^{−6} cm permits estimation of the magnitude of the exponent in Eq. 23. Based on Eqs. 10 and 14, the absorption coefficient for a “typical” rhodopsin-containing cone is expected to be near 420 cm^{−1} (a^{−1} = 24 µm), which would make the value of the exponent ∼6.3 × 10^{−4}.

In view of the probable magnitude of δ and assumption 2, and also considering pigment anisotropy according to Eq. 10 and expressing the concentration as

we can derive the following formula from Eq. 23:

in which the exponential term in Eq. 23 was represented by the first two terms of its series expansion (the second and higher order terms being negligibly small). Thus, the rate of absorption in a pigment layer is proportional to the incident light flux times the products of anisotropy factor (k), pigment type (α), and surface density of the pigment (J_{n} = n_{z}/A_{z}). By extending Eq. 25 to describe a multilayered system with self-screening, the rate of absorption by one layer at depth z is

A second method of obtaining the rate of quantum absorption by a single layer of visual pigment is based on the interpretation of the chromophoric absorption coefficient (α) as the absorption cross section of one molecule in a random array (Dartnall, 1972). The probability of a quantum at λ_{max} to be caught by a single layer of visual pigment, with anisotropy factor k, is expressible as a ratio of the sum of all the molecular cross sections to that of the available total area:

When we scale up from one photon to an incident photon flux of Φ_{o}, Eq. 27 will reproduce the preceding relationship described by Eq. 25.

The rate of signal generation in a lamella at z is expected to be proportional to the rate of photon absorption in that layer (assumptions 3 and 4). Thus, the signal should be formally similar to Eq. 26:

with C_{S} being a conversion factor between photon absorption and corresponding photocurrent.

The noise produced at layer z may be given (assumption 5) as

in which C_{N} is again a conversion factor. In taking the ratio between Eqs. 28 and 29, n_{z} will cancel out, and the signal-to-noise ratio becomes (with C′ as a new constant)

## APPENDIX 3

### Derivation of equations to test the third biophysical function: Cone outer segments taper in accordance with the optical properties of their inner segments, facilitating light capture and reducing use of biomaterials

The list of symbols and definitions used to evaluate this function are presented in Table 5. The following sections introduce definitions of key variables (e.g., concentration factor) for its evaluation.

Assuming a perfect two-dimensional random array of absorbers, the angle of incidence with respect to the optic axis (Θ) will reduce the absorption probability of unpolarized light by a factor (Winston, 1981):

Evaluated for Θ = 15°, Eq. 31 yields ƒ = 0.97. Thus, oblique incidence for angles up to 15° causes only a small drop (3%) in absorption efficiency. In the following treatment, we assume, therefore, that oblique incidence does not significantly affect the in situ absorption efficiency of the visual pigment molecules embedded in the transverse lamellae.

### Flux concentration in a tapered outer segment

In the absence of absorption, the flux density of a converging (conical) beam increases in the direction of convergence. The conceptually simplest case occurs when the incident cone of light matches exactly the outer segment taper. This, however, leads to the same analysis covered previously, where an axially parallel beam was assumed to be incident on the base of a tapered outer segment, in which total internal reflection prevailed. In terms of flux densities, Eq. 26 may also be written as

to indicate a rate of absorption density increase by the factor A_{o}/A_{z}. For obtaining the signal generation in a lamella at z, however, the total absorbed flux is needed, not the flux density. Multiplying Eq. 32 by A_{z}, though, takes us back to Eq. 26. Therefore, the preceding analysis leading to Eq. 30 is also valid for this case.

Besides the matching case, the convergence of the beam incident upon the outer segment may also be lesser or greater than that of the structure. The slightly convergent case is the simplest one, and it may be handled as the axially parallel beam was above (see previous paragraph), necessitating no new analysis. The greater beam convergence, however, warrants further considerations. Some aspects of the latter case are discussed in the manuscript in connection with the effect of oil droplet focusing.

### Flux concentration in a cone ellipsoid

The third function presupposes the existence of a mechanism whereby cone ellipsoids funnel light from a broader, proximal portion toward a narrower, distal end and that this property makes a significant impact on the structure and function of outer segments. Consider a flux Φ_{i} incident at the entrance aperture (largest cross section) of an ellipsoid to produce a flux density J_{i}. If this flux is coupled without loss to a smaller exit aperture, where the flux density is J_{o}, the following relationships hold (provided the cross sections are circular with respective areas and diameters of A_{i}, d_{i} and A_{o}, d_{o}):

The significance of F_{C}, named here concentration factor, is that it shows the proportion by which the base of a photoreceptor can be reduced in area while still capable of detecting all of the incident flux Φ_{i}. Although the issue of light losses by leakage remains to be considered, this property reveals the feasibility of detector miniaturization.

### Volume reduction of a cone outer segment

Whereas the outer limb of rods approaches the cylindrical form in nearly all instances, cone outer segments usually appear truncated, not pointed. Therefore, the frustum of a cone is a more realistic representation of a cone outer segment. The volume of the frustum of a cone is defined with a diameter of base d_{o}, of tip d_{t}, and altitude h as

Compared with the volume of a cylindrical rod, V_{r} = (π/4)hd_{o}^{2}, their ratio is defined as

in which F_{G} is named the geometry factor. Experience shows that the value of F_{G} is variable and tends to fall between 1.5 and 3, the latter being the largest for a right cone (d_{t} = 0). With these two factors combined, the volume reduction ratio is defined as

V_{R} is an indicator of proportion between the outer limb volumes of a cone and an equivalent rod, when both have equal inner segment entrance aperture and incident flux. Discounting light losses, these two cells could produce the excitation of an equal number of visual pigment molecules to equal illumination. Clearly, the cone is the more efficient receptor of the two because it uses only a fraction of the rod’s detector apparatus. This means reduced amounts in lipid membrane, visual pigment, and all the other components of the enzymatic cascade required for generating signals in terms of photocurrent modulation.

## Acknowledgments

Dr. Ferenc I. Hárosi passed away in November 2008. During his career, he was the leading innovator in the field of microspectrophotometry, starting with the making of the first computerized dichroic microspectrophotometer in the 1970s. His insights into photoreceptor physiology and visual pigment properties opened new fields of investigation that are actively pursued by many a prominent scientist today. He was my friend and mentor and an inspiration to vision scientists of all ages. Dr. Hárosi is, and will always be, profoundly missed; thank you for everything, friend.

We are grateful to Drs. Gregor J. Jones, Edward F. MacNichol Jr., and Ete Z. Szuts for critical reading and helpful comments on earlier versions of the manuscript and to Dr. Juan Korenbrot for stimulating discussions on the subject. We also thank Dr. Catherine Carr, Ed Enos, and Mike Juhas for specimens or tissue used in the study and Lisa Grebinsky for logistical help.

This work was funded by the Natural Sciences and Engineering Research Council of Canada Discovery Grant 238886 and a Grass-Marine Biological Laboratory Sabbatical Fellowship in Neurosciences to I. Novales Flamarique.

Edward N. Pugh Jr. served as editor.

- Submitted: 20 July 2011
- Accepted: 21 December 2011

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