Functional significance of the taper of vertebrate cone photoreceptors

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.

Figure 1.

Electron micrographs of vertebrate photoreceptors illustrating diverse outer segment tapers and ellipsoid morphologies. (A) Goldfish single cone (SC) and rod (R) flanking one member of a double cone (DC). The cone ellipsoids are packed with mitochondria (Mi). Cone outer segments (black arrows) taper, whereas those of the rods do not (white arrow; only a portion of the rod outer segment is visible). (B) Single cone, double cone, and rod of coho salmon. In this species, there is a clear gradient in the size of cone mitochondria from smaller, at the level of the myoid, to larger, at the level of the ellipsoid. (C) Double cone from a mummichog killifish showing megamitochondria (M) associated with the ellipsoid of each double cone member. This species also has ellipsosomes, which arise from megamitochondria as the cristae disappear. (D) Rod and single cone from a bullfrog. The rod mitochondria are long and compacted; the single cone exhibits an ellipsosome-like structure (E*) in the ellipsoid. (E) Two single cones among rods in the bullfrog retina; one of the cones contains an oil droplet (*). Note the large difference in size between rods and cones. (F) Single cones and rods from a Canada goose. The single cones show different types of oil droplets. As in the frog, elongated mitochondria pack rod inner segments, and the mean diameter of cone ellipsoids (entrance aperture) is similar to that of rods. (G) Single cones of the red-eared slider turtle showing large oil droplets and pronounced cone taper. (H) Rods of the mouse retina. The cones in this and similar nocturnal species are hard to identify without molecular markers. Bars, 2 µm.

Figure 2.

Drawings of single cones from fresh retinal preparations illustrating the morphological parameters measured as well as the taper angle, τ. The cellular dimensions were obtained from video images recorded via a microscope equipped with a calibrated infrared-sensitive video system. (A) Single cone from blue gill sunfish. (B) Single cone from leopard frog. (C) Cone outer segment (left) from B and an idealized representation of that of the optically equivalent rod (right). The equivalency is based on the assumption that both cells have equal entrance aperture with diameter d_i and that the cone ellipsoid funnels the incident flux to the outer segment without loss. The cellular dimensions (in µm) for these cones were as follows: (A) for the blue gill sunfish, d_i = 8.3, d_o = 5.0, d_z = 2.9, z = 18, and the inner segment length, l_i = 25.2; (B) for the leopard frog, d_i = 7.2, d_o = 2.8, d_z = 1.3, z = 6.3, and l_i = 17.5. The parameter z, in these two cases, equals the outer segment length, and d_z is the diameter at the tip of the outer segment. The asterisk in B depicts an oil droplet.

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.

TABLE 1.

List of species for which literature data were obtained and sources

Common name	Scientific name	References
Fishes
Narrow-mouthed lamprey	Geotria australis	Collin et al., 2003
South hemisphere lamprey	Mordacia mordax	Collin et al., 1999, 2004; Collin and Pottert, 2000
Sea lamprey	Petromyzon marinus	Dickson and Graves, 1979
River lamprey	Lampetra japonica	Ishikawa et al.,1987, 1989
Southern fiddler ray	Trygonorhina fasciata	Braekevelt, 1992a
Short tail stingray	Dasyatis brevicaudata	Braekevelt, 1994a
Blue spotted mask ray	Dasyatis kuhlii	Kohbara et al., 1987; Theiss et al., 2007
Giant shovelnose ray	Rhinobatus typus	Hart et al., 2004
Eastern shovelnose ray	Aptychotrema rostrata	Hart et al., 2004; Litherland and Collin, 2008
Sting ray	Dasyatis sayi	Hamasaki and Gruber, 1965
Electric ray	Narcine brasilensis	Ali and Anctil, 1974
Freshwater sting ray	Paratrygon motoro	Ali and Anctil, 1974
Red stingray	Dasyatis akaiei	Kohbara et al., 1987
Sepia stingray	Urolophus aurantiacus	Kohbara et al., 1987
Thornback ray	Platyrhina sinensis	Kohbara et al., 1987
Spiny dogfish	Squalus acanthias	Stell, 1972
Mediterranean dogfish	Centroscysmus coelolepis	Bozzano, 2004
Small spotted dogfish	Scyliorhinus canicula	Bozzanao et al., 2001
Black mouth dogfish	Galeus mealstomus	Bozzanao et al., 2001
Longsnout dogfish	Deania eglantina	Kohbara et al., 1987
Nurse shark	Ginglymostoma cirratum	Hamasaki and Gruber, 1965
Lemon shark	Negaprion brevirostris	Gruber et al., 1963
Great white shark	Carcharodon carcharias	Gruber et al., 1975; Gruber and Cohen, 1985
Mako shark	Isurus oxyrinchus	Gruber et al., 1975
Thresher shark	Alopias vulpinus	Gruber et al., 1975
Blue shark	Prionace glauca	Gruber et al., 1975; Kohbara et al., 1987
Banded houndshark	Triakis scyllia	Kohbara et al., 1987
Silky shark	Carcharinus falciformis	Gruber et al., 1975
White tip shark	Carcharinus longimanus	Gruber et al., 1975
Sandbar shark	Carcharinus milberti	Gruber et al., 1975
Brown bamboo shark	Chiloscyllium punctatum	Harahush et al., 2009
White tip reef shark	Triaenodon obesus	Litherland and Collin, 2008
Epaulette shark	Hemiscyllium ocellatum	Litherland and Collin, 2008
Ornate wobbegong	Orectolobus ornatus	Litherland and Collin, 2008
Australian lungfish	Neoceratodus forsteri	Bailes et al., 2006
South American lungfish	Lepidosiren paradoxa	Ali and Anctil, 1973
Coelacanth	Latimeria chalumnae	Locket, 1973
Green sturgeon	Acipenser medirostru	Sillman et al., 2005
Pallid sturgeon	Scaphirhynchus albus	Sillman et al., 2005
Siberian sturgeon	Acipenser baeri	Govardovskii et al., 1992
Amur sturgeon	Acipenser shrenckii	Fang et al., 2004
Shovelnose sturgeon	Scaphirhynchus platorhynchus	Sillman et al., 1999a
Paddlefish	Polyodon spathula	Sillman et al., 1999a; Sillman and Dahlin, 2004
Florida garfish	Lepisosteus platyrhincus	Collin and Collin, 1993
Bowfin	Amia calva	Ali and Anctil, 1976
Goldeye	Hiodon alosoides	Braekevelt, 1982
European eel	Anguilla anguilla	Braekevelt, 1984, 1985, 1988a,b
American eel	Anguila rostrata	Ali and Anctil, 1976
Northern anchovy	Engraulis mordax	Novales Flamarique, 2011
Goldfish	Carassius auratus	Stell and Hárosi, 1976
Carp	Cyprinus carpio	Bunt and Klock, 1980
Japanese dace	Tribolodon hakonensis	Hárosi and Hashimoto, 1983
Roach	Leucisus rutilus	Engström and Rosstorp, 1963; Zaunreiter et al., 1991
Tench	Tinca tinca	Douglas and Wagner, 1982
Zebrafish	Danio rerio	Nawrocki et al., 1985; Forsell et al., 2001; Kim et al.,2005
Cutlips minnow	Exoglossum maxillingua	Collin et al., 1996a
Silverjaw minnow	Ericymba buccata	Moore et al., 1950
Creek chub	Semotilus atromaculatus	Collin et al., 1996b
Gangfish	Loregonus lavateus	Reckel et al., 1999
Common white sucker	Catostomus commersoni	Novales Flamarique and Hárosi, 1997; Novales Flamarique and Hawryshyn, 1998
Razorback sucker	Xyrauchen texanus	Novales Flamarique et al., 2007
Channel catfish	Ictalurus puntatus	Sillman et al., 1993
White catfish	Ictalurus catus	Sillman et al., 1993
Brown bullhead	Ictalurus nebulosus	Welsh and Osborn, 1937; Ali and Anctil, 1976
Coho salmon	Oncorhynchus kisutch	Cheng et al., 2006, 2007
Chum salmon	Oncorhynchus keta	Cheng et al., 2006
Pink salmon	Oncorhynchus gorbuscha	Cheng et al., 2006
Chinook salmon	Oncorhynchus tschawytscha	Cheng et al., 2006
Atlantic salmon	Salmo salar	Novales Flamarique, 2002, 2011; Cheng et al., 2006
Rainbow trout	Oncorhynchus mykiss	Kusmic and Gualtieri, 2000; Cheng and Novales Flamarique, 2007; Cheng et al., 2007, 2009
Smelt	Osmerus eperlanus	Reckel et al., 2003
Salamander fish	Lepidogalaxias salamandroides	Collin and Collin, 1998
Waryfish	Scopelosaurus lepidus	Munk, 1977
Greenland cod	Gadus ogac	Ali and Anctil, 1976
Antarctic cod	Dissostichus mawsoni	Meyer-Rochow and Klyne, 1982
Bald rock cod	Trematomus borchgrevinki	Meyer-Rochow and Klyne, 1982
Emerald rock cod	Trematomus bernacchii	Meyer-Rochow and Klyne, 1982
Common cod	Gadus callarias	Engström, 1961
Burbot	Lota lota	Engström, 1961
Tadpole fish	Raniceps raninus	Engström, 1961
Silver hake	Merluccius bilinearis	Ali and Anctil, 1976
Capelin	Mallotus villosus	Ali and Anctil, 1976
Mummichog killifish	Fundulus heteroclitus	Anctil and Ali, 1976; Novales Flamarique and Hárosi, 2000
Mangrove killifish	Rivulus marmoratus	Ali et al., 1989
Four-eyed fish	Anableps anableps	Borwein and Hollenberg, 1973
Guppy	Lebistes reticulatus	Müller, 1952; Yacob et al., 1977; MacNichol et al., 1978; Kunz et al., 1983
Pipefish	Corythoichthyes paxtoni	Collin and Collin, 1999
Halfbeak	Dermogenys pusillus	Reckel et al., 2002; Reckel and Melzer, 2003
Australian rainbowfish	Melanotaenia maccullochi	Reckel et al., 2002; Reckel and Melzer, 2003
Celebes rainbowfish	Marosatherina ladigesi	Reckel et al., 2002; Reckel and Melzer, 2003
Sea needle	Belone belone	Reckel et al., 2001, 2002; Reckel and Melzer, 2003
Lesser weever	Trachinus vipera	Kunz et al., 1985
European perch	Perca fluviatilis	Ahlbert, 1973; Guma’a, 1982
Yellow perch	Perca flavescens	Ali and Anctil, 1976
Green sunfish	Lepomis cyanellus	Burnside and Ackland, 1984; Dearry and Barlow, 1987
Butterfly fish	Pantodon buchholzi	Braekevelt, 1990a
Black bream	Acanthopagrus butcheri	Shand et al., 1999
Blackstriped cardinalfish	Apogon angustatus	Fishelson et al., 2004
Iridescent cadinalfish	Apogon kallopterus	Fishelson et al., 2004
Yellowstriped cardinalfish	Apogon cyanosoma	Fishelson et al., 2004
Cook’s cardinalfish	Apogon cookie	Fishelson et al., 2004
Rock bass	Ambloplites rupestris	Munz and McFarland, 1977
Large-mouth bass	Micropterus salmoides	García and de Juan, 1999
Striped bass	Morone saxatilis	Paillart et al., 2006
West Australian dhufish	Glaucosoma hebraicum	Shand et al., 2001
Black sea bass	Centropristis striata	Singarajah and Hárosi, 1992
Snake mackerel	Gempylus serpens	Munk, 1985
Walleye	Stizosteidon vitreum vitreum	Zyznar and Ali, 1975; Januschka et al., 1987
Sauger	Stizosteidon canadense	Ali and Anctil, 1977
Nile tilapia	Oreochromis niloticus	Braekevelt et al., 1998
Golden dwarf cichlid	Nannacara anomala	Wagner, 1978; Douglas and Wagner, 1982
Velvet cichlid	Astronotus ocellatus	Braekevelt, 1992b
Burton’s haplochromis	Haplochromis burtoni	Pietzsch-Rohrschneider, 1976
Goldsinny wrasse	Ctenolabrus suillus	Engström, 1963
Corkwing wrasse	Crenilabrus melops	Engström, 1963
Winter flounder	Pseudopleuronectes americanus	Evans and Fernald, 1993
Amphibians
African clawed frog	Xenopus laevis	Kinney and Fisher, 1978a,b; Hollyfield et al., 1984; Röhlich et al., 1989; Röhlich and Szél, 2000
Leopard frog	Rana pipiens	Nilsson, 1965
Bullfrog	Rana catesbeiana	Hisatomi et al., 1998
Edible frog	Rana esculenta	Reichenbach and Fuchs, 1983
Tropical toad	Bufo marinus	Moody and Robertson, 1960; Hárosi, 1975
Tiger salamander	Ambystoma tigrinum	Hárosi, 1975; Mariani, 1986; Braekevelt, 1993a; Sherry et al., 1998; Ma et al., 2001
Axotol	Ambystoma mexicanum	Custer, 1973
Red-back salamander	Plethodon cinereus	Braekevelt, 1992c
Newt	Triturus viridescens	Keefe, 1971
Birds
Chicken	Gallus domesticus	Meyer and May, 1973; Araki et al., 1984; Szél et al., 1986; Oishi et al., 1990
Pigeon	Columba livia	Cohen, 1963; Mariani and Leure-duPree, 1978; Cserháti et al., 1989
Crow	Corvus brachyrhynchos	Braekevelt, 1994b
Great blue heron	Ardea Herodias	Rojas et al., 1999a
Yellow-crowned heron	Nycticorax violaceus	Rojas et al., 1999a
Black-crowned heron	Nycticorax nycticorax	Gondo and Ando, 1995
Cattle egret	Bubulcus ibis	Rojas et al., 1999a
Tricolored egret	Egretta tricolor	Rojas et al., 1999a
American white ibis	Eudocimus ruber	Rojas et al., 1999a
Roseate spoonbill	Ajaia ajaja	Rojas et al., 1999a
Wilson’s plover	Charadrius wilsonia	Rojas et al., 1999b
Short-billed dowitcher	Limnodromus griseus	Rojas et al., 1999b
American woodcock	Scolopax minor	Rojas et al., 1999b
Black-winged stilt	Himantopus himantopus	Rojas et al., 1999b
Willet	Catoptrophorus semipalmatus	Rojas et al., 1999b
Red-tailed hawk	Buteo jamaicensis	Braekevelt, 1993b
Great horned owl	Bubo virginianus	Braekevelt, 1993c
Barred owl	Strix varia	Braekevelt et al., 1996
Mallard duck	Anas platyrhynchos	Braekevelt, 1990b
Australian galah	Eolophus roseicapillus	Braekevelt and Richardson, 1996
Emu	Dromaius novaehollandiae	Braekevelt, 1998
Japanese quail	Coturnix coturnix japonica	Konishi, 1965; Oishi et al., 1990; Rojas et al., 2007
Eastern tree sparrow	Passer montanus	Gondo and Ando, 1995
House swallow	Hirundo rustica	Gondo and Ando, 1995
Great tit	Parus major	Engström, 1958
American robin	Turdus migratorius	McNeil et al., 2005
Hermit thrush	Catharus guttatus	McNeil et al., 2005
Mourning dove	Zenaida macroura	McNeil et al., 2005
Common grackle	Quiscalus quiscula	McNeil et al., 2005
Oil bird	Steatoinis caripensis	Martin et al., 2004; Rojas et al., 2004
Common pauraque	Nyctidromus albicellis	Rojas et al., 2004
Ring billed gull	Larus delawarensis	Emond et al., 2006
Gray gull	Larus modestus	Emond et al., 2006
Reptiles
Red-eared slider	Pseudemys scripta elegans	Baylor and Fettiplace, 1975; Leeper, 1978; Kolb and Jones, 1982, 1987; Ohtsuka and Kawamata, 1990
Snapping turtle	Chelydra serpentina	Baylor and Fettiplace; 1975; Leeper; 1978
Reeve’s turtle	Geoclemys reevesii	Ohtsuka, 1985; Ohtsuka and Kawamata; 1990
Garter snake	Thamnophis sirtalis	Wong; 1989; Sillman et al., 1997
Ball python	Python regius	Sillman et al., 1999b
Common boa	Boa constrictor imperator	Sillman et al., 2001
Mississippi alligator	Alligator mississippiensis	Kalberer and Pedler, 1963; Sillman et al., 1991
Caiman	Caiman crocodilus	Govardovskii et al., 1988
Tokay gecko	Gekko gecko	Pedler and Tilly, 1964; Crescitelli, 1972
Blue-tailed day gecko	Phesulma inunguis	Pedler and Tansley, 1963; Pedler and Tilly, 1964
Coastal banded gecko	Coleonyx variegates	Dunn, 1966
Mediterranean gecko	Hemydactilus turcicus	Pedler and Tilly, 1964; Loew et al., 1996
House gecko	Hemydactilus garnotii	Loew et al., 1996
Scinc gecko	Teratoscincus scincus	Govardovskii et al., 1984; Loew et al., 1996
Chameleon	Chamaleo chamaleo	Armengol et al., 1981
Western fence lizard	Sceloporpus occidentalis	Young, 1977; Bernstein et al., 1984
Ornate dragon lizard	Ctenophorus ornatus	Barbour et al., 2002
Tuatara	Sphenodon punctatu	Meyer-Rochow et al., 2005
Bouton’s skink	Cryptoblepharus boutonii	Röll, 2001
Mammals
Stumptail macaque	Macaca arctoides	Hoang et al., 2002
Rhesus monkey	Macaca mulatta	Cohen, 1961; Leach, 1963; Dowling, 1965; Young, 1971; Borwein et al., 1980
Crab-eating macaque	Macaca fascicularis	Borwein et al., 1980
Pigtail macaque	Macaca nemestrina	Packer et al., 1989
Owl monkey	Aotes trivirgatus	Jones, 1965; Murray et al., 1973; Ogden, 1975
Vervet monkey	Cercopithecus aethiops	Braekevelt, 1987
Human	Homo sapiens	Missotten, 1966; Dieterich and Rohen, 1970; Steinberg et al., 1977; Curcio et al., 1990; Hoang et al., 2002
Dog	Canis domesticus	Shively et al., 1970; Hebel, 1971
Wolf	Canis lupus	Peichl et al., 2001
Cat	Felis domesticus	Steinberg et al., 1973; Braekevelt, 1990c
Ferret	Mustela putorius furo	Braekevelt, 1983a
Mink	Mustela vison	Dubin and Turner, 1977; Braekevelt, 1990d
Spotted hyena	Crocuta crocuta	Calderone et al., 2003
Rabbit	Oryctologus cuniculus	Szél et al., 1988
Cow	Bos taurus	Szél et al., 1988
Pig	Sus scrofa domesticus	Szél et al., 1988; Hendrickson and Hicks, 2002
Domestic sheep	Ovis aries	Braekevelt, 1983b
Mouflon	Ovis musimon	Peichl, 2005
Long-finned pilot whale	Globicephala melaena	Peichl et al., 2001
13-lined squirrel	Spermophilus tridecemlineatus	West and Dowling, 1975; Anderson and Fisher, 1976
Mexican ground squirrel	Spermophilus mexicanus	West and Dowling, 1975; Anderson and Fisher, 1976
California ground squirrel	Spermophilus beecheyi	Anderson and Fisher, 1976
Eastern gray squirrel	Sciurus carolinensis	Cohen, 1964; West and Dowling, 1975; Anderson and Fisher, 1976
Western gray squirrel	Sciurus griseus	Anderson and Fisher, 1976
Prairie dog	Cynomys ludovicianus	West and Dowling, 1975
Mouse	Mus musculus	Carter-Dawson and LaVail, 1979
Tree shrew	Tupaia belangeri	Kühne, 1983; Foelix et al.,1987; Müller and Peichl, 1989; Petry and Hárosi, 1990; Petry et al., 1993; Knabe et al., 1997
African giant rat	Cricetomys gambianus	Peichl, 2005
Madagascar rousette	Rousettus madagascariensis	Müller et al., 2007
Seba’s short-tailed bat	Carollia perspicillata	Müller et al., 2009
Greater horseshoe bat	Rhinolophus ferrumequinum	Kim et al., 2008
Monotremes and marsupials
Echidna	Tachyglossus aculeatus	Young and Pettigrew, 1991
Fat-tailed dunnart	Sminthopsis crassicaudata	Arrese et al., 2002, 2003
Honey possum	Tarsipes rostratus	Arrese et al., 2002, 2003
Mouse opossum	Thylamis elegans	Palacios et al., 2010
Tammar wallaby	Macropus eugenii	Hemmi and Grünert, 1999
Quokka	Setonix brachyurus	Arrese et al., 2005
Quenda	Isoodon obesulus	Arrese et al., 2005
Agouti	Dasyprocta aguti	Rocha et al., 2009

In the case of the northern anchovy, only retinal areas with regular cones were evaluated (Novales Flamarique, 2011).

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.

TABLE 2.

Measured parameters and computed variables from fresh preparations of goldfish retina used in microspectrophotometry experiments

Cell type	di	do	dz	z	τ	τ′	ah1−1	az′−1	az′′−1	az′−1/ah1−1	Fc
	μm	μm	μm	μm	°	°	μm	μm	μm
DC1-R	8.9	6.1	4.6	18.3	2.3	2.26	32.4	33.8	34.7	1.04	2.12
DC1-G	8.4	5.3	3.0	18.3	3.6	1.97	16.1	33.8	34.7	2.10	2.51
DC2-R	9.6	6.9	4.6	19.8	3.3	2.53	24.4	33.8	34.7	1.38	1.93
DC2-G	9.1	6.1	3.8	12.2	5.4	2.36	12.9	33.8	34.7	2.62	2.22
DC3-R	9.4	6.9	4.6	13.7	4.8	2.64	16.9	33.8	34.7	2.00	1.85
DC3-G	8.8	6.1	4.6	10.7	4.0	2.39	19.0	33.8	34.7	1.78	2.08
DC4-R	9.2	4.6	3.0	10.7	4.3	1.80	12.5	33.8	34.7	2.70	4.00
DC4-G	9.8	6.1	4.6	7.6	5.6	2.44	13.5	33.8	34.7	2.51	2.58
DC5-R	9.1	5.4	2.9	19.6	3.6	2.00	15.8	33.8	34.7	2.14	2.84
DC5-G	8.4	6.9	4.1	21.4	3.7	2.50	20.5	33.8	34.7	1.64	1.48
DC6-R	9.2	7.6	4.7	19.4	4.3	2.79	20.2	33.8	34.7	1.67	1.46
DC6-G	8.1	6.8	4.5	18.3	3.6	2.52	22.1	33.8	34.7	1.52	1.41
DC7-R	6.5	3.8	2.4	12.9	3.1	1.46	14.0	33.8	34.7	2.41	2.92
DC7-G	6.4	3.3	2.1	12.3	2.8	1.28	13.6	33.8	34.7	2.48	3.76
DC8-R	8.3	5.5	3.7	13.3	3.9	2.11	16.8	33.8	34.7	2.01	2.28
DC8-G	7.8	5.2	3.6	12.8	3.6	2.01	17.4	33.8	34.7	1.94	2.25
DC9-R	8.7	5.8	3.5	18.2	3.6	2.15	18.0	33.8	34.7	1.88	2.25
DC9-G	8.4	6.1	4.2	19.4	2.8	2.24	26.0	33.8	34.7	1.30	1.89
DC10-R	10.2	7.6	4.5	18.4	4.8	2.82	17.6	33.8	34.7	1.93	1.80
DC10-G	9.6	7.2	4.6	17.4	4.3	2.69	19.4	33.8	34.7	1.74	1.77
SC1-R	9.1	6.1	3.8	15.2	4.3	2.31	16.1	33.8	34.7	2.10	2.22
SC2-R	8.8	6.1	3.8	16.7	3.9	2.29	17.6	33.8	34.7	1.92	2.08
SC3-R	6.2	4.5	3.0	12.6	3.4	1.74	15.5	33.8	34.7	2.17	1.89
SC4-R	8.2	4.1	2.8	12.8	2.9	1.58	16.8	33.8	34.7	2.01	4.00
SC5-B	9.3	6.1	3.8	8.4	7.8	2.43	8.87	33.8	34.7	3.81	2.32
SC6-B	9.4	6.1	4.6	7.6	5.6	2.44	13.5	33.8	34.7	2.51	2.37
SC7-B	9.2	6.5	4.8	12.2	4.0	2.52	20.1	33.8	34.7	1.68	2.00
SC8-B	9.3	6.2	4.1	14.8	4.1	2.36	17.9	33.8	34.7	1.89	2.25
SC9-B	8.8	5.8	3.9	13.6	4.0	2.22	17.1	33.8	34.7	1.97	2.30
SC10-B	8.7	6.1	3.6	10.1	7.1	2.40	9.58	33.8	34.7	3.53	2.03
Rod1	2.2	2.2	2.1	42.0	0.1	0.69	451	33.8	34.7	0.07	1.00
Rod2	2.0	2.0	2.0	39.1	0.0	0.64	∞	33.8	34.7	0.00	1.00
Rod3	2.1	2.1	2.0	38.4	0.1	0.67	389	33.8	34.7	0.09	1.00
Rod4	2.3	2.3	2.3	44.0	0.0	0.71	∞	33.8	34.7	0.00	1.00
Rod5	1.9	1.8	1.8	36.6	0.0	0.59	∞	33.8	34.7	0.00	1.11
Rod6	1.9	1.9	1.9	39.7	0.0	0.60	∞	33.8	34.7	0.00	1.00
Rod7	2.1	2.0	2.0	40.3	0.0	0.63	∞	33.8	34.7	0.00	1.10
Rod8	2.2	2.1	2.1	41.3	0.0	0.67	∞	33.8	34.7	0.00	1.10
Rod9	2.0	2.0	2.0	43.5	0.0	0.62	∞	33.8	34.7	0.00	1.00
Rod10	1.9	1.7	1.7	35.5	0.0	0.57	∞	33.8	34.7	0.00	1.25
Rod11	2.2	2.2	2.2	38.0	0.0	0.71	∞	33.8	34.7	0.00	1.00
Rod12	2.1	2.0	2.0	37.7	0.0	0.65	∞	33.8	34.7	0.00	1.10
Rod13	2.2	2.2	2.2	41.8	0.0	0.70	∞	33.8	34.7	0.00	1.00

DC, double cone; SC, single cone. R, G, and B indicate the presence of visual pigments with maximum wavelength of absorption in the red (LWS opsin), green (RH2 opsin), or blue (SWS2 opsin) regions of the spectrum.

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 A₂–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 A₁–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.

Figure 3.

Ratio of observed taper to critical taper (τ/τ′) in relation to observed taper (τ). (A) Species from which live cell measurements were obtained. (B) Species for which measurements originated from the literature. Recall that taper is defined as the angle between the axis of the cone and the inclination of the contour line, which, upon precession, describes the conical surface. Critical taper is the taper required to exactly compensate for light flux diminution by absorption (self-screening) so that flux density remains invariant throughout the outer segment. In general, taper was highest for species with focusing organelles in the ellipsoids.

Figure 6.

Regressions of the ratio of realistic to critical characteristic length (a_z′⁻¹/a_h1⁻¹) as a function of concentration factor (F_C). Recall that the characteristic length is the distance along the outer segment at which the light flux, Φ, falls to 0.368 of the incident light flux at the base, Φ_o. The concentration factor is the square of the ratio between ellipsoid diameter at its largest cross section and outer segment base diameter, i.e., F_C = (d_i/d_o)², and represents coupling of light flux without loss from ellipsoid to outer segment. (A) Data for species from which live cell measurements were obtained. (B) Data for species for which measurements originated from the literature.

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 µm³. The volume of a cylinder with equal base diameter is 399 µm³. 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.

Figure 4.

Regressions of expected rod/cone ratios as a function of those observed for primarily diurnal (though birythmic) species. (A) Data for species from which live cell measurements were obtained. (B) Data for species for which measurements originated from the literature.

Figure 5.

Regression of expected rod/cone ratios as a function of those observed for fully diurnal and primarily nocturnal species. Data originated from the literature.

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 µm³, V_r = 39.6 µm³, F_G = 1.97, and V_R = 11.4. The optically equivalent cylinder, as indicated in Fig. 2 C, has a volume of ∼231 µm³. 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 µm³. 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 µm² 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 [SCE₁]). 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 SCE₁. 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 SCE₁ (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.

TABLE 3.

List of symbols and definitions pertaining to the first biophysical function

Symbol	Definition
Φ	Total light flux (photons s−1)
Φo, Φz	Value of Φ at z = 0 and at distance z from base
A	Cross-sectional area of outer segment, assumed to be equal with the surface area of a transverse membrane, referred to as a lamella (Eckmiller, 1987). One rod disk, also called flattened saccule, has the surface of two lamellae
Ao, Az	Value of A at the base and at distance z from it
A┴, A║	Outer segment absorbance for transversely and axially polarized light
Jo	Incident flux density (photons s−1 μm−2); Jo = Φo/Ao
Jz	Transmitted flux density through level z from base; Jz = Φz/Az
R	Dichroic ratio: absorbance ratio at two orthogonal polarizations; R = A┴/A║
S┴	Specific absorbance at transverse polarization; S┴ = A┴/d
d	Outer segment diameter (mean)
do, dz	Outer segment diameter at base and at a distance z from the base
c	Concentration of visual pigment in appropriate units
α	Molecular extinction coefficient (µm2 per molecule; Eqs. 7 and 9)
ε	Molar extinction coefficient (liter mole−1 cm−1; Eqs. 8 and 9)
λmax	Wavelength of peak extinction
k	Anisotropy factor: ratio of anisotropic to isotropic absorbance
a	Absorption coefficient (usually in cm−1; Eq. 3)
a−1	Characteristic length (reciprocal of a) expressed in μm (signifies the distance at which Φz reduces to 36.8% of Φo)
ah1	Absorption coefficient required for invariance of light flux along the outer segment, as by Eq. 6
ah1−1	Characteristic length, based on invariance of light flux along an outer segment (μm)
az′	Absorption coefficient obtained by Eq. 10
az′′	Absorption coefficient obtained by Eq. 16
az′−1, az′′−1	Characteristic lengths (μm) obtained from az′ and az′′
τ	Taper, as defined by Eq. 17
Θ	Angle of incidence with respect to cell axis
Θc	Critical angle of Θ, defined as Θc = 90° − ϕc
ϕc	Critical angle with respect to normal of boundary (Eq. 19)
ϕi, ϕo	Angles of incidence and refraction to normal of boundary
n1, n2	Refractive indices of two optical media forming a boundary
z	Axial distance within outer segments (μm), with z = 0 at the base

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,

$${\Phi }_{\text{o}}{/\text{A}}_{\text{o}}={\Phi }_{\text{z}}{/\text{A}}_{\text{z}}.$$ (1)

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,

$$\Theta \le {\Theta }_{\text{c}}.$$ (2)

The exponential law of absorption,

$${\Phi }_{\text{z}}={\Phi }_{\text{o}}\text{exp}\left(-\text{az}\right),$$ (3)

sets forth the variation of flux in an absorbing medium along the z direction (Φ_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

$${\Phi }_{\text{z}}/{\Phi }_{\text{o}}={\text{A}}_{\text{z}}{/\text{A}}_{\text{o}}=\text{exp}\left(-\text{az}\right).$$ (4)

The circular cross section of a cone with diameter d is A = d²π/4, and thus, Eq. 4 may be written as

$${\text{d}}_{\text{z}}={\text{d}}_{\text{o}}\text{exp}\left(-\text{az}/2\right),$$ (5)

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

$${\text{a}}_{\text{h}1}=\left(2/\text{z}\right)\text{ln}\left({\text{d}}_{\text{o}}{/\text{d}}_{\text{z}}\right).$$ (6)

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

$${\text{D}}_1=\text{ln}({\Phi }_{\text{o}}/{\Phi }_{\text{z}})=\alpha \text{cz}$$ (7)

$${\text{or D}}_2=\text{log}({\Phi }_{\text{o}}/{\Phi }_{\text{z}})=\epsilon \text{cz},$$ (8)

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

$$\text{a}=\alpha \text{c}=2.303\epsilon \text{c}.$$ (9)

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

$${\text{a}}_{\text{z}}\prime =\text{k}\alpha \text{c}=2.303\text{k}\epsilon \text{c},$$ (10)

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:

$$0.5\left({\text{E}}_{\text{x}}+{\text{E}}_{\text{y}}\right)/\text{E}=1/3.$$ (11)

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

$${\text{E}}_{\text{z}}/\text{E}={\text{E}}_{\text{z}}/\left({2\text{E}}_{\text{x}}+{\text{E}}_{\text{z}}\right)=1/\left(1+2\text{R}\right).$$ (12)

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

$$\begin{array}{l}0.5\left({\text{E}}_{\text{x}}+{\text{E}}_{\text{y}}\right)/\text{E}=0.5\left(\text{E}-{\text{E}}_{\text{z}}\right)/\text{E}=\\ 0.5\left[1-1/\left(1+2\text{R}\right)\right]=\text{R}/\left(1+2\text{R}\right).\end{array}$$ (13)

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)

$$\text{k}=3\text{R}/\left(1+2\text{R}\right).$$ (14)

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,

$$(\epsilon \text{c})={\text{S}}_{\perp }\left(1+2\text{R}\right)/3\text{R},$$ (15)

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:

$${\text{a}}_{\text{z}}″=2.30{3\text{S}}_{\perp }\text{.}$$ (16)

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:

$$\tau ={\text{tan}}^{–1}\left[\right({\text{d}}_{\text{o}}-{\text{d}}_{\text{z}})/2\text{z}].$$ (17)

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 n₁ (inside), the other rarer with refractive index n₂ (outside), light rays will refract according to Snell’s law:

$${\text{n}}_1\sin {\varphi }_{\text{i}}={\text{n}}_2\sin {\varphi }_{\text{o}},$$ (18)

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

$${\varphi }_{\text{c}}={\text{sin}}^{–\text{1}}\left({\text{n}}_2{/\text{n}}_1\right).$$ (19)

From this, Θ_c of Eq. 2 can now be obtained as

$${\Theta }_{\text{c}}=90\mathrm{°}-{\varphi }_{\text{c}}.$$ (20)

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.

TABLE 4.

Additional list of symbols and definitions pertaining to the second biophysical function

Symbol	Definition
Φa, Φaz	Absorbed flux in one layer at the base and at depth z
Jn	Surface density of pigment (molecules μm−2); Jn = n/At
δ	In situ path length in visual cells corresponding to a single pigment-laden lipid bilayer (lamella); δ = 15 nm is taken as its equivalent thickness
At	Total lamellar surface area of an outer segment
c	Concentration of visual pigment; c = n/Atδ = Jn/δ
n	Total number of visual pigment molecules contained in an outer segment
no, nz	Number of visual pigment molecules in a lamella at the base and at z
pz	Probability of absorption of a quantum by a single layer at z
γ	Quantum efficiency of bleaching; number of photons absorbed per number of photoactivations that result in bleaching
Sz	Signal from light activation (ionic current) at level z
Nz	Noise produced in the dark at level z (ionic current)

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,

$${\Phi }_{\text{o}}={\Phi }_{\text{a}}+{\Phi }_{\text{z}}.$$ (21)

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

$${\Phi }_{\text{a}}/{\Phi }_{\text{o}}=1-{\Phi }_{\text{z}}/{\Phi }_{\text{o}}.$$ (22)

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

$${\Phi }_{\text{a}}={\Phi }_{\text{o}}[1-\text{exp}(-\text{a}\delta \left)\right].$$ (23)

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

$$\text{c}={\text{n}/\text{A}}_{\text{t}}\delta ={\text{J}}_{\text{n}}/\delta ={\text{n}}_{\text{z}}{/\text{A}}_{\text{z}}\delta ,$$ (24)

we can derive the following formula from Eq. 23:

$${\Phi }_{\text{a}}={\Phi }_{\text{o}}[1-(1-\text{k}\alpha \text{c}\delta \left)\right]={\Phi }_{\text{o}}\text{k}\alpha {\text{n}}_{\text{z}}{/\text{A}}_{\text{z}},$$ (25)

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

$${\Phi }_{\text{az}}=\text{k}\alpha \left({\text{n}}_{\text{z}}{/\text{A}}_{\text{z}}\right){\Phi }_{\text{o}}\text{exp}\left(-\text{az}\right).$$ (26)

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:

$${\text{p}}_{\text{z}}=\text{k}\alpha {\text{n}}_{\text{z}}{/\text{A}}_{\text{z}}.$$ (27)

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:

$${\text{S}}_{\text{z}}={\text{C}}_{\text{S}}\text{k}\alpha \gamma \left({\text{n}}_{\text{z}}{/\text{A}}_{\text{z}}\right){\Phi }_{\text{o}}\text{exp}\left(-\text{az}\right),$$ (28)

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

$${\text{N}}_{\text{z}}={\text{C}}_{\text{N}}{\text{n}}_{\text{z}},$$ (29)

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)

$${\text{S}}_{\text{z}}{/\text{N}}_{\text{z}}={\text{C}}^{\prime }\text{k}\alpha \gamma \left({\Phi }_{\text{o}}{/\text{A}}_{\text{z}}\right)\left[\text{exp}\left(-\text{az}\right)\right].$$ (30)

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.

TABLE 5.

Additional list of symbols and definitions pertaining to the third biophysical function

Symbol	Definition
ƒ	Absorption efficiency factor due to oblique incidence (Eq. 31)
Ai	Collection area (circular); largest cross section of inner segment
di	Diameter of Ai
Φi	Total flux at entrance aperture of an inner segment
Ji	Flux density at entrance aperture of an inner segment
Jaz	Rate of flux density absorption
FC	Concentration factor (Eq. 33)
FG	Geometry factor (Eq. 35)
dt	Diameter at the tip of a frustum
h	Altitude or height of a cone or cylinder
Vr	Volume of a rod outer segment
Vc	Volume of a cone outer segment
Vreq	Volume of an optically equivalent rod outer segment
VR	Volume reduction ratio as defined by Eq. 36

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):

$$ƒ=0.5(1+{\text{cos}}^2\Theta ).$$ (31)

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

$${\text{J}}_{\text{az}}={\text{J}}_{\text{o}}\left({\text{A}}_{\text{o}}{/\text{A}}_{\text{z}}\right)\text{k}\alpha \left({\text{n}}_{\text{z}}{/\text{A}}_{\text{z}}\right)\text{exp}\left(-\text{az}\right)$$ (32)

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):

$${\text{J}}_{\text{o}}{/\text{J}}_{\text{i}}={\text{A}}_{\text{i}}{/\text{A}}_{\text{o}}={\left({\text{d}}_{\text{i}}{/\text{d}}_{\text{o}}\right)}^2={\text{F}}_{\text{C}}.$$ (33)

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

$${\text{V}}_{\text{c}}=(\text{π}/4)(\text{h}/3)[{\text{d}}_{\text{o}}^2+{\text{d}}_{\text{o}}{\text{d}}_{\text{t}}+{\text{d}}_{\text{t}}^2].$$ (34)

Compared with the volume of a cylindrical rod, V_r = (π/4)hd_o², their ratio is defined as

$${\text{V}}_{\text{r}}{/\text{V}}_{\text{c}}={\text{F}}_{\text{G}},$$ (35)

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

$${\text{V}}_{\text{R}}={\text{V}}_{\text{req}}{/\text{V}}_{\text{c}}={\text{F}}_{\text{G}}{\text{F}}_{\text{C}}.$$ (36)

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.