Molecular Vision 2015; 21:244-263 <>
Received 14 November 2014 | Accepted 05 March 2015 | Published 07 March 2015

Activation and quenching of the phototransduction cascade in retinal cones as inferred from electrophysiology and mathematical modeling

Luba Astakhova, Michael Firsov, Victor Govardovskii

Institute for Evolutionary Physiology and Biochemistry, Russian Academy of Sciences

Correspondence to: Victor Govardovskii, IEPHB RAS, 44 Thorez prospect, 194223 St. Petersburg, Russia; Phone: (7)-(812)-5504989; FAX: (7)-(812)-5504989; email:


Purpose: To experimentally identify and quantify factors responsible for the lower sensitivity of retinal cones compared to rods.

Methods: Electrical responses of frog rods and fish (Carassius) cones to short flashes of light were recorded using the suction pipette technique. A fast solution changer was used to apply a solution that fixed intracellular Ca2+ concentration at the prestimulus level, thereby disabling Ca2+ feedback, to the outer segment (OS). The results were analyzed with a specially designed mathematical model of phototransduction. The model included all basic processes of activation and quenching of the phototransduction cascade but omitted unnecessary mechanistic details of each step.

Results: Judging from the response versus intensity curves, Carassius cones were two to three orders of magnitude less sensitive than frog rods. There was a large scatter in sensitivity among individual cones, with red-sensitive cones being on average approximately two times less sensitive than green-sensitive ones. The scatter was mostly due to different signal amplification, since the kinetic parameters of the responses among cones were far less variable than sensitivity. We argue that the generally accepted definition of the biochemical amplification in phototransduction cannot be used for comparing amplification in rods and cones, since it depends on an irrelevant factor, that is, the cell’s volume. We also show that the routinely used simplified parabolic curve fitting to an initial phase of the response leads to a few-fold underestimate of the amplification. We suggest a new definition of the amplification that only includes molecular parameters of the cascade activation, and show how it can be derived from experimental data. We found that the mathematical model with unrestrained parameters can yield an excellent fit to experimental responses. However, the fits with wildly different sets of parameters can be virtually indistinguishable, and therefore cannot provide meaningful data on underlying mechanisms. Based on results of Ca2+-clamp experiments, we developed an approach to strongly constrain the values of many key parameters that set the time course and sensitivity of the photoresponse (such as the dark turnover rate of cGMP, rates of turnoffs of the photoactivated visual pigment and phosphodiesterase, and kinetics of Ca2+ feedback). We show that applying these constraints to our mathematical model enables accurate determination of the biochemical amplification in phototransduction. It appeared that, contrary to many suggestions, maximum biochemical amplification derived for “best” Carassius cones was as high as in frog rods. On the other hand, all turnoff and recovery reactions in cones proceeded approximately 10 times faster than in rods.

Conclusions: The main cause of the differing sensitivity of rods and cones is cones’ ability to terminate their photoresponse faster.


Vision in vertebrates covers an enormous range of natural illumination levels that spans over 11 orders of magnitude. This unique ability is supported by the presence of two kinds of photoreceptor cells in the retina, namely rods and cones. Rods are photoreceptors of nocturnal vision. They are highly sensitive and can reliably signal arrivals of single photons. High sensitivity, however, makes rods susceptible to saturation; thus, for example, human rod monochromats (persons whose retina lacks functional cones) are light-blinded even at low diurnal levels of illumination [1]. Diurnal photoreceptors, cones, are 100- to 1,000-fold less sensitive than rods. Cones do not saturate and provide useful vision at the maximum intensities available in nature.

The biochemical and physiological mechanisms that ensure high sensitivity of rods are fairly well understood. The rod visual pigment, rhodopsin (R), belongs to the family of G-protein coupled receptors (GPCRs). Upon absorption of light, rhodopsin efficiently interacts with its cognate trimeric GTP-binding protein, transducin (T), and produces its active form, T* (Tα-GTP). Due to the enzymatic nature of the reaction, a single photoactivated rhodopsin (R*) produces hundreds of T*s per second. Each T* activates a catalytic subunit of the effector enzyme cGMP-phosphodiesterase (PDE). This greatly increases the rate of hydrolysis of the cytoplasmic secondary messenger, cGMP. The concentration of cGMP drops, which leads to the closure of the cGMP-gated ionic channels (cyclic nucleotide-gated [CNG] channels) in the plasma membrane of the outer segment (OS), thereby generating the electrical response.

The light-activated cascade is quenched by mechanisms operating at each activation step. The activity of R* is decreased by multiple phosphorylation by rhodopsin kinase and finally blocked by binding of arrestin. The active TαGTP-PDE* is turned off by its intrinsic GTPase activity, which is enhanced by the interaction with the GTPase activating complex, RGS-9/Gβ5. Hydrolyzed cGMP is replenished by continuously running guanylate cyclase (GC), which returns the rod to its prestimulus state. Recovery of the dark state is greatly enhanced by negative feedback regulations that are controlled by the cytoplasmic concentration of Ca2+ ions. Ca2+ enters the OS via the CNG channels and is pumped out by the Na+/Ca2+-K+ exchanger (NCKX). Light closure of the CNG channels reduces the Ca2+ influx, thereby causing a decrease in [Ca2+]in. This decrease accelerates the phosphorylation (hence turnoff) of R* by rhodopsin kinase via the Ca2+-sensing protein recoverin, and increases the production of cGMP by GC via GC-activating proteins (GCAPs). In addition, the decrease of [Ca2+]in increases the affinity of the CNG channels to cGMP so that they stay open at lower cGMP concentration. Ca2+ feedback contributes greatly to shaping photoresponses and plays an important role in light adaptation, the process that prevents the saturation of photoreceptors by steady illumination.

The mechanisms of rod excitation and light adaptation have been studied in unprecedented detail using biochemical, molecular, electrophysiological, and genetic methods, and extensively reviewed (since 2000, see [2-17]). This, however, cannot be said about cones. It is generally believed that the basic principles of cone functioning are similar to those of rods. Nevertheless, the mechanisms by which cones reduce their sensitivity by 100- to 1,000-fold and avoid saturation at maximum possible illuminance levels are poorly understood. Most proteins of the phototransduction cascade exist in a rod- or cone-specific version, and could account for the observed differences between the two types of photoreceptors. Basically, the photoreceptor sensitivity is set by a balance between two factors, the rate of the light-induced activation of the cGMP hydrolysis, and the speed of its turnoff. The latter certainly plays the leading role, as is seen from the fact that the photoresponses in cones are an order of magnitude briefer compared to rods. The role of the former factor, which could be called biochemical amplification, is still controversial.

Biochemical amplification is the product of the speed of generation of T* by a single R* and the speed of the cGMP hydrolysis by PDE*. Thus, the amplification in cones can be reduced either by less efficient interaction of cone visual pigments with cone transducin, lower catalytic activity of cone PDE, or both. As for the catalytic properties of cone PDE, they are probably not much different from those of rod PDE. The maximum hydrolysis rate per PDE subunit of the two enzymes is the same, from 2,000 to 2,700 s−1 (frog rods [18], bovine cones [19], chipmunk cone-dominant preparation [20], bovine rod and cone PDE subunits expressed and tested in various combinations [21,22]). Michaelis’ constant of cone PDE may be a bit higher than that of rod PDE (17 to 26 μM in cones [19,22,23] versus 10 to 17 μM in rods [18,23]).

As for the activation of transducin/PDE by a cone visual pigment, the most direct biochemical assays in vitro show that it is apparently two- to fivefold slower than the activation by rhodopsin. This was shown for chicken [24,25], human [26], and mouse green cone pigments [25]. Kawamura’s group [27-29] used the most intact preparations of carp rod and cone photoreceptor membranes. They found that the activation of cone transducin by its cognate cone visual pigment was fivefold slower than the activation of rod transducin by rhodopsin. There is a contradictory result, though. Xenopus violet cone visual pigment expressed in COS-1 cells activated rod transducin as efficiently as rhodopsin did [30]. However, the relevance of the in vitro data to the situation in intact cells is unclear. The rates of production of rod T* by rod R* were substantially lower in these experiments, sometimes by orders of magnitude, than the benchmark value derived from physiological and biochemical data for rods [18]. Obviously, multiple factors other than possible differences between rod and cone visual pigments and transducins affect the results of in vitro measurements.

The efficiency of interaction between R* and T in rods and cones was also studied through physiological methods using a heterologous expression of cone proteins in rods. The results were contradictory and thus inconclusive. For instance, the expression of mouse green cone pigment in mouse rods reduced the amplification approximately fourfold without a major change in the response kinetics [31]. The expression of cone Tα in mouse rods decreased the sensitivity of photoresponse and accelerated its kinetics [32]. On the other hand, expressing the blue-sensitive (S) mouse cone visual pigment in mouse rods had no effect on the single-photon response [33]. Deng et al. [34] expressed cone Tα in mouse rods and rod Tα in mouse cones, and concluded that rod and cone transducins are functionally equivalent. Similarly, no difference between mouse rod photoresponses was observed when testing all four possible patterns of expression of rod and cone visual pigments and rod and cone Tα [35]. To add to the uncertainty, expression of cone PDE in mouse rods increased the sensitivity of the photoresponse [36]. Of special interest is the experiment designed by nature. Amphibian retinas contain two types of blue-sensitive photoreceptors—so-called green rods and blue-sensitive cones that are spectrally identical. In the salamander, both have been shown to express the same short wavelength–sensitive 2 (SWS-2) cone visual pigment, but cones use cone-specific transducin, while green rods use rod transducin [37]. In this study, it appeared that blue cones needed 5 to 28 times more light falling on them than green rods to produce photoresponses of the same amplitude, while the kinetics of the response did not markedly change [37]. Kawakami and Kawamura [38] recently argued that the right way to assess the amplification in the cascade is to compare sensitivities to incident (rather than absorbed) light. We further support their conclusion (see below). Thus, the results in [37] suggest that the amplification in salamander blue cones is substantially lower than in green rods naturally expressing the same visual pigment.

Physiological experiments on photoreceptors heterologously expressing various rod- and cone-specific visual pigments and transducins have their own problems. First, genetic manipulations apparently strictly targeted to the desired protein may also have poorly traceable effects on other components of the phototransduction cascade. Second, and perhaps more importantly, these experiments mostly deal with “unnatural” combinations of visual pigments and transducins that may not be selected by evolution to work together in the same cell. Thus, whatever the results are, they do not answer the main question: What is the reason(s) for different sensitivities of rods and cones naturally expressing their own visual pigments, transducins, and other components of the phototransduction cascade?

To address this question, we compared photoresponses of frog rods and fish Carassius cones recorded at the time resolution sufficient to obtain undistorted activation phases. The results were interpreted within the framework of a specially designed mathematical model of phototransduction. Parameters of the model were constrained by measurements when calcium feedback was disabled by fixing cytoplasmic Ca2+ concentration. These constraints unambiguously allowed the biochemical amplification and characteristic times of the cascade turnoffs to be determined. We found that there was a large scatter among individual cones with respect to their amplification. However, maximum amplification observed in cones was as high as in rods. The approximately three orders of magnitude lower sensitivity of the “worst” cones compared to rods resulted from approximately tenfold lower biochemical amplification and a tenfold acceleration of all turnoff reactions (quenching of photoactivated visual pigment and phosphodiesterase, dark cGMP turnover, and calcium feedback).


Selecting experimental animals

At first glance, it seems ideal to compare amplification in rods and cones from the same species. This, however, may not be the right approach. Different animal species have different visual tasks, and interspecies differences are reflected, for instance, in rod or cone dominance in the retina. This may not only be related to the number and size of photoreceptors, but also to their biochemical and physiologic properties. Thus, one could expect all combinations of “good” and “poor” rods and cones to exist in different retinas. For proper comparison, one needs rods and cones representing the best examples in their class. Another prerequisite is the big size and robustness of the cells that would allow easy experimental manipulations. We tested the frog retina, whose rods are among the largest and most sensitive known. It appeared that it is possible to record from frog cones, but the cells were small and fragile, making Ca2+-clamping experiments unfeasible. A good source of cones could be the retina of fish, for instance, of carp or Carassius. Again, it appeared impossible to carry out Ca2+-clamp experiments on Carassius rods to complement cone data, since the rods were highly fragile and unstable. Thus, we chose frog rods and fish cones for comparison. Working on frog rods and Carassius cones also had an important advantage in relation to any other animal, because a large body of biochemical and physiological data on these and closely related species is available. This facilitated the interpretation of our data in terms of underlying biochemical mechanisms.

Animals and preparations

Experiments were performed on rods and cones of the frog Rana ridibunda (Pelophylax ridibundus) and on cones of the fish Carassius carassius. Frogs were caught from the Volga River (southern Russia) and kept for up to 6 months in tanks with free access to water at 18 to 20 °C on a natural day/night cycle, and fed living cockroaches and dry pet food. Fish were obtained from a local hatchery. They were kept in aerated aquaria on a 12 h:12 h light–dark cycle and fed dry fish food. Animals were treated in accordance with the Guide for the Care and Use of Laboratory Animals (1996, National Academy of Sciences, Washington, DC) and with the rules approved by the local Institutional Animal Care and Use Committee.

Prior to the experiment, the animals were dark-adapted overnight. They were killed by decapitation and double-pithed under dim red light. Further manipulations were carried out under infrared (IR) TV control. Procedures to prepare the samples for electrophysiological recordings were described in detail earlier [39,40]. Briefly, eyes were enucleated and retinas removed into a Ringer-filled Petri dish. Small retinal pieces were transferred to a drop of the solution on a coverslip and finely chopped with a razor blade. The resulting mixture of tiny retinal pieces and isolated photoreceptors was placed in the perfusion chamber. The photoreceptor current was recorded using the standard suction pipette recording technique [41]. Most of the data were obtained from intact rods and cones protruding from retinal pieces, in the configuration OS in. To conduct measurements at a fixed cytoplasmic Ca2+ concentration, isolated Carassius cones were held in the suction pipette inner segment in, and their OSs were subjected to the Ca2+-clamping solution using a fast solution changer. The changer consisted of an assembly of movable tubings. A two-barrel pipette made of a theta capillary produced two jets of solutions of different composition. The second, single-barrel pipette placed opposite to the theta tubing at the distance of ≈ 0.5 mm sucked the jets in, thus preventing their mixing with the main bath solution that was always perfused with slowly flowing normal Ringer. A computer-controlled stepper motor could move the tubing assembly so that the OS was suddenly immersed into either of the two jets of different composition [39].


The standard Ringer solution used for retinal dissection and as the main solution in the perfusion bath contained for frog the following ingredients (in mM): NaCl 90, KCl 2.5, MgCl2 1.4, glucose 10, CaCl2 1, NaHCO3 5, HEPES 5, bovine serum albumin (BSA) 50 mg/l, and EDTA 0.05, with pH adjusted to 7.6. For fish, the solution contained the following: NaCl 102, KCl 2.6, MgCl2 1, glucose 5, CaCl2 1, NaHCO3 28, HEPES 5, and BSA 50 mg/l, and the pH was adjusted to 7.8–8.0. The composition of each solution was chosen to ensure stable long-lasting recordings, high dark current and maximum sensitivity in corresponding species.

To block Ca2+ feedback onto the cascade, we used the Ca2+-clamping solution based on the recipe in [42] for fish cones. This contained the following: guanidine-Cl 102, KCl 2.6, KHCO3 28, glucose 5, HEPES 5, and EGTA 4, with pH adjusted to 7.8–8.0 by tetramethylammonium hydroxide (TMA-OH). Free Ca2+ concentration in this solution was calculated to be below 1 nM. Ca2+-clamping solution for frog rods was as in [39]. All chemicals were from Sigma-Aldrich (St. Louis, MO).

Light stimulation

The light stimulation system consisted of two independent channels based on high-output light-emitting diodes (LEDs), one with λmax = 519 nm, and the other with λmax = 632 nm. By comparing sensitivities at the two wavelengths, it was possible to discriminate spectral classes of the photoreceptors (red-sensitive versus green-sensitive versus blue-sensitive cones in the goldfish, and red rods versus green rods in the frog). Stimulus intensity was controlled by switchable neutral density (ND) filters and LED current. Standard flash duration was 2 ms. Flash intensity, expressed as the number of isomerizations per flash (R*), was calibrated for each individual frog rod using the statistics of few-photon responses [41]. Cell dimensions were determined from the IR monitor screen, the number of rhodopsin molecules per rod outer segment (ROS) calculated, and intensity was further expressed as the fractional bleach or as the number of photons per square micrometer. The latter was used to estimate the number of activated visual pigment molecules in Carassius cones. Independently, intensity was calibrated using a Burr-Brown OPT-310 integrated optosensor. The two calibrations coincided within 20%.

Data acquisition

Photoresponses were low-pass filtered at 300 Hz (8-pole analog Bessel filter), digitized at 2 ms intervals and stored on the computer hard disk. If necessary, further digital filtering could be applied to the data. Proper reference to the digital filtering is given in the figure legends. Data acquisition, stimulus timing, and stimulus intensity were under LabView hardware and software control (National Instruments, Austin, TX).


To estimate light collecting areas (F) of photoreceptors in the population of animals used in the study, we measured absorbances and OS sizes of cones and rods by microspectrophotometry. The design of the microspectrophotometer (MSP) and the measurement procedures were described in detail earlier [43,44]. Briefly, small pieces of dark-adapted isolated retinas were placed in a drop of appropriate physiological saline on a coverslip and teased apart by needles to obtain solitary photoreceptors or their outer segments. The sample was covered by another coverslip, sealed at the edges with petroleum jelly and placed on the MSP stage. Width of the measuring beam was set to 3 μm for frog rods, and 2 μm for Carassius cones, and its length was about 2/3 of the OS length. Recordings from the OSs were taken at two polarizations of the measuring beam (T, transversal with respect to the OS axis, and L, longitudinal, along the axis). This allowed more accurate estimation of the absorption of the nonpolarized light used for stimulation. Baylor et al.’s [41] formula for the cell light collecting area was used with a small modification:

Equation 1

F = π d 2 4 q ln ( 10 ) f a max S ( λ ) = V O S q n ( 10 ) f a max S ( λ )

Here, d is the ROS diameter (μm), l is the ROS length (μm), and VOS is the volume of a cylindrical OS. For conical OSs, VOS = π·(db2 + db·dt + dt2)·l / 12, where db and dt are OS diameters at the base and tip. In addition, q = 0.67 is the quantum yield of the visual pigment bleaching, and amax is the specific T-density at λmax of the corresponding visual pigment. F is measured in μm2. As estimated from MSP recordings, amax = 0.0151 ± 0.0015 μm−1 for A1-based frog rods, and amax = 0.012 ± 0.0025 μm−1 for A2-based Carassius cones (average ± standard deviation [SD] of 24 rods and 27 cones). The L/T ratio was 0.23 for rods and 0.27 for cones. S(λ) is the relative spectral sensitivity at the wavelength of stimulation as determined from the visual pigment templates [43]. Red- and green-sensitive Carassius cones had λmax = 613 and 539 nm, respectively. Factor f = 0.5·(1+L/T) applies to the situation where OS is illuminated side-on with nonpolarized light, as in our setup.

Definition and experimental determination of biochemical amplification in the phototransduction cascade: an analysis

The generally accepted definition of biochemical amplification in phototransduction has been formulated by Pugh and Lamb [15,45,46]. As shown by the authors, if turnoff processes are neglected, the initial part of a current response r(t) to a short flash of light should follow a squared parabolic time course:

Equation 2

r ( t ) / r max = 1 2 A R * ( t t e f f ) 2

Here, rmax is the maximum response that corresponds to complete closure of the CNG channels, R* is the number of photoactivated visual pigment molecules, teff is the phototransduction delay, and A is the amplification constant (s−2). A is expressed via the parameters of the phototransduction cascade:

Equation 3

A = ν R E k c a t K M n c G 10 15 N A v V c y t o

Here, νRE (s−1) is the rate of production of T*/PDE* by a single R*, kcat is the maximum hydrolytic activity of a PDE* subunit (s−1), KM is the PDE Michaelis constant (here in mol/l), and ncG is the Hill’s coefficient of the CNG channel gating by cGMP. Avogadro’s number, NAv = 6·1023 mol−1, and the cytoplasmic volume of the OS, Vcyto (μm3), convert number of molecules to concentration. Factor 1015 converts liters to μm3. The cytoplasmic volume of the OS (Vcyto) is assumed to be 1/2 VOS [15,46].

The A-value can apparently be extracted from a simple parabolic fit to an initial part of the normalized photoresponse. The analysis does not involve any complicated treatment and relies on a simple theory. This is why the amplification constant has gained general acceptance as a tool for characterizing signal amplification in photoreceptors. The definition has its problems, though. The first problem arises from the fact that the amplification constant A depends not only on molecular parameters of the cascade, such as νRE, kcat, KM, and ncG, but also on the OS volume Vcyto (Equation (3)). Obviously, the same biochemical machinery initiated by the absorption of a single photon would produce a greater effect in smaller cells [46]. Thus, the comparison of photoreceptors of different size (e.g., rods versus cones) based on the A-value is meaningless. A higher amplification constant does not necessarily mean that underlying biochemistry works faster, or vice versa.

This problem can be solved by expressing stimulus strength as the intensity of the incident light I(λ) rather than as the number of photoactivated rhodopsins R*. R* in Equation (2) can be calculated as:

Equation 4

R * = I ( λ ) F = 2 V c y t o I ( λ ) q ln ( 10 ) f a max S ( λ )

Here, I(λ) is measured in photons·μm−2, and 2·Vcyto stands for VOS in Equation (1). Inserting this into Equation (2) and taking into account Equation (3), one obtains:

Equation 5

r ( t ) / r max = I ( λ ) q ln ( 10 ) f a max S ( λ ) ν R E k c a t K M n c G 10 15 N A v ( t t e f f ) 2

Thus, Vcyto vanishes from the equation. Let us now define effective light intensity Ie(λ) (photoisomerizations·μm−3) as:

Equation 6

I e ( λ ) = I ( λ ) q ln ( 10 ) f a max S ( λ )

Its meaning is the number of photoactivated visual pigment molecules per unit volume of the OS. To maintain similarity with Equation (2), Equation (5) can be rewritten as:

Equation 7

r ( t ) / r max = I e ( λ ) A m 10 15 N A v ( t t e f f ) 2

Here, the modified amplification constant Am only includes biochemical parameters of the phototransduction cascade and does not depend on the cell’s volume:

Equation 8

A m = ν R E k c a t K M n c G

An additional advantage of this definition is that the effective light intensity does not depend on the cell size. Rather, it is expressed via a directly measurable physical quantity, that is, the surface density of photon flux I(λ). In other words, cells with the same Am value—rods or cones, irrespective of their size—have to exhibit the same rising phase of the response at the same irradiance level.

Recently, Kawakami and Kawamura [38] suggested a slightly different way of eliminating the cell volume from the definition of amplification, by expressing stimulus intensity as fractional bleach. Our modified amplification constant Am is equivalent to their index of the gain G. If Am is expressed in s-2.μM−1, it can be converted to G (fL·s−2) by:

Equation 9

G = A m N A v 10 21

The second, at a first glance technical, problem is how to extract the amplification in either sense from experimental recordings. As clearly stated in [46], the parabolic approximation of the initial phase of the photoresponse is only valid if all recovery processes can be neglected. These processes include the following: R* quenching by phosphorylation, T* quenching by GTP hydrolysis, prestimulus steady cGMP turnover, and active restoration of the cGMP level by Ca2+ feedback via GC. Parameters of these turnoffs are mostly unknown, so it is hard to tell what stretch of the initial phase of the response can be used to determine A. Roughly 10% of the time to the peak of the flash response seems a reasonable guess. This means that in amphibian rods, the proper stretch is about 100 ms, but it may not be longer than 10 to 15 ms in mammalian rods or in the cones of cold-blooded animals. For mammalian cones, the stretch appropriate for fitting does not exceed a few milliseconds. In addition, routinely used 10–20 ms stimuli cannot be considered as flashes for cones; rather, they are steps of light, which invalidates simple analysis based on second-order parabola fit. Third-order parabola fitting of the step responses, as recently introduced [47], partly ameliorates the situation, but the problem of neglected turnoff processes remains.

Unless the recordings are made in voltage-clamp mode, the response waveform is affected by an electrical low-pass filtering by the capacitance and resistance of the cell membrane. For rods, the effect is probably negligible, since the electrical time constant of amphibian rods—which is under 20 ms [15,48,49]—is substantially shorter than the time stretch available for analysis. In cones, however, the high area of the folded membrane of the OS may significantly increase the cell capacitance, hence the time constant. The typical electrical time constant of cones lies between 20 and 80 ms [49-52], thereby spoiling the entire part of the curve appropriate for fitting. An adverse effect may also arise from excessive low-pass analog filtering during recordings, and from digital filtering applied during data analysis.

It seems that only a realistic mathematical model of cone phototransduction that explicitly takes into account all of the turnoff processes and electrical filtering could allow us to quantitatively characterize all the processes that set the speed and sensitivity of the cone photoresponse. The model should be applied to a set of experimental data obtained with short (≤1–2 ms) flashes and at a good time resolution. Thus, our goal was to collect such a dataset and to develop an appropriate mathematical model to analyze it.

The general approach to mathematical modeling

To process and interpret data, we used a mathematical model of phototransduction common for rods and cones [53,54]. In this section, we focus solely on the features of the model that make it suitable for the specific goal of the work, that is, for extracting key parameters of the cone and rod phototransduction cascade with a minimum of ambiguity and arbitrary assumptions. The model includes all basic mechanisms that participate in the cascade activation and quenching, and Ca2+-feedback regulations imposed on it. However, to make the model less redundant and more manageable, we use a phenomenological description of each process, omitting mechanistic details that are important but unnecessary here. For instance, activation of transducin by R* is a multistage process whose detailed description includes five reactions and depends on over 10 (essentially unknown) rate constants [55,56]. For our purpose, it can be convolved into a linear activation with a certain rate νRE (s−1) and possible delay teff [15,45,46]. The task of the modeling is to extract the two parameters’ values from the experimental data. Similarly, the turnoff of light-activated visual pigment depends on its multiple phosphorylation. Yet the resulting time course of the decay of R* catalytic activity derived from a detailed model by Hamer et al. [55] could satisfactorily be approximated with a single exponential. Only the time constant of the exponential is necessary for our model.

On the other hand, the description of certain steps in many existing models is oversimplified. For instance, cGMP hydrolysis by PDE and Ca2+ pumping by the exchanger are often treated as linear reactions, neglecting their Michaelis-like kinetics. Turnoff processes are commonly neglected when trying to extract the rate of the activation of the cascade from experimental recordings and so on. The price of these oversimplifications cannot be estimated a priori without more detailed experimental and model analysis. Thus, we constructed a minimum essential model (MEM) based on maximally realistic phenomenological descriptions of each key reaction without involving unnecessary mechanistic details. The complete set of equations comprising the model is given in Appendix 1.

The MEM, however minimal, contains over 30 parameters. Some of their values are (relatively) reliably known from direct measurements (e.g., kcat and KM of the PDE, or the parameters of the control of the GC by Ca2+ and of the CNG channel gating by cGMP; for the sources of the data, see Appendix 1). Some others are mainly interdependent scaling factors that can be relatively freely varied when fitting the model to experimental data. However, many parameters are crucial for the model functioning, but their values are just guessed and can differ by orders of magnitude among the models available in literature. The resulting freedom of fitting makes quantitative conclusions derived from modeling mostly meaningless. Therefore, we conducted a special set of measurements using Ca2+-clamp protocol. It provided a wealth of information on the dark and light-dependent cGMP turnover. This allowed strong experimental constraints to be imposed on many crucial parameters of the model. Corresponding experiments and procedures of analysis are described in detail in Appendix 1. All computations were performed using MathCad 2001i (MathSoft, Cambridge, MA).


Basic features of rod and cone photoresponses

Figure 1 shows a series of photoresponses to flashes of varying intensities recorded from a frog rod (A) and two Carassius cones (B, C). The shapes of the rod and cone responses are basically similar, save the fact that cone responses are an order of magnitude faster and approximately three orders of magnitude less sensitive. However, there is a cone-specific feature of the response that is already incipient in Figure 1B, and fully expressed in Figure 1C. In B, the slope of the front of the response saturates at modest intensities, and further increase of the intensity mostly shortens the delay of the front without making it steeper. An extreme example of this behavior is seen in Figure 1C, where all responses follow the same initial trajectory, peeling away at progressively later times with increasing intensities. This feature was not specific for green-sensitive cones (as shown in Figure 1C). Saturation of the slope of the front was to various extents evident in all spectral types of cones, including a few recorded blue-sensitive cells. Responses of this sort were earlier recorded from cones of fish (Carassius, Danio [57,58]) and salamander [50,52,59]. The response versus intensity function for all types of cones was satisfactorily described by a Michaelis-like relation:

Equation 10

r / r max = I I + I 05

where I0.5 is the flash intensity at which the response amplitude is half-maximum (Figure 2). Rod response versus intensity dependence was steeper and better described by an exponential saturation function:

Equation 10a

r / r max = 1 exp ( I / I 0.63 )

On average, green cones were more light-sensitive than red ones. However, the scatter of sensitivity among individual cells even of the same spectral type was quite large, exceeding an order of magnitude in green cones. Sensitivity did not obviously correlate with the speed of the photoresponse. The scatter of the time to peak (Tpeak) and the response integration time (Ti) was modest, at about 25% (SD/mean). There was no statistically significant difference in these parameters among red and green cones (Table 1).

Mathematical modeling

The series of responses like those in Figure 1B,C clearly show a saturation of the speed of the PDE activation even at modest-intensity stimuli. Thus, the full series cannot be fitted with our model, where the PDE activity is scaled linearly with intensity. Therefore, we only used for fitting the responses to the weakest flashes whose fractional amplitude did not exceed 20%.

Throughout the modeling, many parameters, mostly characterizing the steady state in the dark, were fixed at the values taken from literature (Appendix 1, Table A2). Subject to change during fitting were the parameters that determine the sensitivity and the time course of the photoresponse. These are the rate of transducin/PDE activation by photoactivated visual pigment (νRE), the rates of R* and PDE* quenching (kR and kE), and basal (dark) cGMP turnover rate αdark/cGdark. Crucial factors are also the kinetics and amplification of the Ca2+ feedback loop. With the exception of νRE, all of the parameters were initially assigned the guess values obtained from the Ca2+-clamp experiments (Appendix 1, Table A3). After that, adjusting νRE usually allowed a model flash response approaching the experimental one to be obtained. Further νRE, kR, kE, αdark, τe, and parameters of the Ca2+ buffer were finely tuned for best fit. Tuning was carried out by trial and error, and the quality of the fit was judged from the coefficient of the correlation between the model and the experimental response. Necessary adjustments of kR, kE, αdark, τe, and FB were usually within ± 15% of the initial values, although approximately one-fifth of cones needed larger deviations from the average. The rate of T*/PDE* production νRE, however, varied greatly, in line with a big scatter of sensitivity among individual cones (Figure 2). Figure 3 shows sample fits of the photoresponses of the cone with maximum amplification (A), the cone with minimum amplification (B), and a “typical” rod (C).

Values of the parameters specific for individual cells from Figure 3 are given in Table 2. Average data on the rates of activation and inactivation of the phototransduction cascade in rods and cones are summarized in Table 3. It is obvious that there is a big scatter in the amplification parameters among individual cones. Noticeably, in the most sensitive green cones, the rate of activation of the cGMP hydrolysis (which determines the biochemical amplification) is virtually equal to that in rods. However, both deactivation reactions (R* and PDE* quenching), dark cGMP turnover, and Ca2+ feedback in all cones were about 10 times faster.


Uniqueness of the fits and validity of cascade parameters estimated from the Ca2+ clamp

We found that cone photoresponses could be successfully fitted by our model using values of the parameters partially taken from literature and partially constrained by or derived from our measurements with the Ca2+ clamp. The question arises of whether the fits are unique, and the answer is that they are definitely not. Fits were highly sensitive to change of any single key parameter like νRE, adark, kR, kE, or Kex. However, wildly different parameters of the cascade turnoff might yield equally good fits to the experimental data as soon as compensatory changes to other components are allowed. For instance, the effect of fivefold slowing of the dark turnover rate (αdark/cGdark) and 330-fold reduction of Kex in the model can be compensated by appropriate acceleration of kR and kE and adjustments of the parameters of the Ca2+ buffer (Figure 3B, Table 4). This sensitivity of the model fits to even small changes of a single parameter and the possibility to compensate it by proper changes in others has been observed previously [32,52]. However, the experimental constraints on the set of parameters that we obtained with the Ca2+ clamp allowed us to reduce the ambiguity of the fits and derive robust estimates of biochemical values from the model.

Biochemical amplification: cones versus rods

It would be tempting to calculate the rate of activation of the cascade, νRE, directly from the PDE*(t) derived from the Ca2+-clamp experiments. This would avoid any ambiguity inherent to multiparametric model fits. Unfortunately, this is precluded by the high noise level of the Ca2+-clamp data on cones (see Appendix 1, Figure A3, A5). Reducing the noise by averaging multiple responses is not feasible because the Ca2+-clamp procedure deteriorates the cell. Averaging across many cones makes no sense due to the large scatter of the amplification among individual cells (Table 3).

Fortunately, it appears that the uncertainty of the values of the shape-forming parameters (αdark/cGMPdark, kR, kE, and Ca2+ buffering, constrained by the Ca2+ clamp) had only a weak effect on the estimate of νRE, which determines the speed of the activation of the cascade (hence, amplification). As long as the set of the shape parameters provided a good fit of the model to the experimental response, variation of νRE did not exceed a few percent. However, membrane filtering had a stronger effect on the estimate of amplification. Amplification and τe were to a certain extent interchangeable, so an approximate 10% change of amplification can be compensated by an adjustment of τe and possibly other parameters without markedly reducing the quality of the fit. However, further reduction of νRE below the optimum value distinctly deteriorated fit statistics. Thus, we believe that the uncertainty of our values of amplification is within about 15%.

We found that the biochemical amplification, as defined in this work, can vary by about an order of magnitude among individual cones. In addition, the red-sensitive cones are on average less sensitive than green-sensitive cells (Figure 1, Table 1), and this is mostly due to their lower amplification (Table 3). We made no attempt to identify the morphological types of the cones we recorded from (large or small single cones, members of double cones, etc.). However, differences in properties of individual cone types may be the cause of the observed variability. This idea is in line with large variability of cones’ absolute sensitivity found earlier on the striped bass retina [60]. The variability was clearly related to the morphological and spectral type of the cell, red-sensitive “fast” twins being 40 times less sensitive than green-sensitive singles.

Variation in sensitivity and amplification among frog red rods was far less prominent (Figure 2; Table 3). The “best” cells yielded Am = 12.1·104 s-2.μM−1, which corresponded to νRE = 220 s−1, kcat = 2200 s−1, and KM = 10 μM, that is, close to the benchmark values established for amphibian rods in a specially designed biochemical and electrophysiological study [18]. Remarkably, the cones with maximum amplification yielded virtually the same result, as follows: Am = 11.8·104 s-2.μM−1 which stemmed from νRE = 430 s−1, kcat = 2,200 s−1, and KM = 20 μM (Table 2, cone 1). It should be noted that the amplification is an integrated parameter that depends on νRE, the rate of activation of PDE by R*, and on PDE* catalytic properties kcat and KM (Equation (8)). The three properties are completely interchangeable, and only Am could unambiguously be determined from modeling. The exact values of cone kcat and KM are not quite certain [19,21,23], so for instance, if one assumes cone kcat = 2,200 s−1 and KM = 10 μM, like in rods, the resulting νRE = 215 s−1 also becomes identical to that in rods.

Our Am values can be compared with equivalent G-values obtained in [38] on carp rods and red-sensitive cones, as well as bullfrog rods. The authors found that the gain in frog and carp rods is approximately four times higher than in red-sensitive carp cones. This is similar to the 4.2:1 ratio found by us for Ams of frog rod and average Carassius red cone (Table 3). In addition, amplification in best Carassius green-sensitive cones was four times higher than the Carassius average red cone, which makes them similar to bullfrog and carp rods. However, if converted to G, our absolute values of amplification are about two times higher than those found by Kawakami and Kawamura [38] in the carp and bullfrog. This apparent discrepancy can be explained by the fact that the authors determined amplification by routine parabolic fitting, which neglects turnoff processes. As we further show, neglecting turnoffs leads to an approximately 1.6-fold underestimation of amplification. Taking into account the scatter of experimental data, our results are in fair agreement with the results in [38].

Equal amplification values of the “best” green-sensitive cones and frog rods (Table 3) suggest that the R*-T-PDE interaction in cones may proceed as fast as in rods. A wide scatter of the amplification among individual cones can plausibly be explained by varying levels of expression of transducin, among other possibilities. It is also obvious that biochemical samples, in which many cones are pooled, should on average exhibit a lower rate of PDE activation than rods or the “best” cones. This may explain the biochemical results from Kawamura’s group [10,27-29] that consistently found two- to fivefold lower rate of T*/PDE* production in carp cone preparations compared to rods.

Reactions of the cascade turnoff in cones

The reactions that control the restoration of the dark state work in a concerted way. Therefore, the three main parameters, αdark/cGMPdark, kR, and kE, are also interchangeable to a great extent and cannot unambiguously be found solely from model fitting without additional experimental constraints. Fortunately, Ca2+-clamp data provided the constraints; it appeared that the three rates are close to each other and are an order of magnitude faster than in rods (Table 3).

The dark cGMP turnover rate

The rate of the turnover of the cytoplasmic cGMP pool is an important factor that contributes to the control of the speed of the photoresponse and to light adaptation [61]. The dark turnover rate Tr (s−1) can be found as:

Equation 11

T r = a d a r k c G d a r k

Here, the rate of cGMP synthesis (dark activity of GC) αdark is expressed in μM·s−1 and the concentration of cGMP is expressed in μM. Since the cone responses are approximately 10-fold faster than rod responses (Figure 1), it could be expected that the activity of GC in cones is correspondingly higher. This is indeed the case, as shown by biochemical measurements on carp cones [62].

Equation (11) is a bit deceptive, though, since it gives the impression that the dark turnover rate is proportional to the activity of GC, and conceals the fact that αdark affects cGdark as well. From the equation of cGMP turnover:

Equation 12

d c G ( t ) d t = α ( t ) β ( t ) c G ( t ) c G ( t ) + K M

one can find:

Equation 13

T r = β d a r k c G d a r k + K M

At cGdark << KM, that is, at low αdark, this yields:

Equation 14

T r = β d a r k K M

This means that the turnover rate is set by the catalytic activity of phosphodiesterase rather than GC, which is lacking from Equation (14). Excess activation of synthesis (hence an increase of cGdark) actually decreases Tr (Equation (13)). This suggests that the primary cause of the accelerated cGMP turnover in cones is high dark PDE activity.

Our estimate of Tr in Carassius cones based on Ca2+-clamp measurements and supported by model fitting is about 10 s−1; values below 5 s−1 can decidedly be rejected (see Appendix 1, Figure A5A). This is in agreement with more direct measurements with the isobutylmethylxanthine (IBMX)-jump on salamander cones. In these cells, whose flash response is two to three times slower than in Carassius cones, Tr ≈ 2 to 3 s−1 [63-65]. As for the increased activity of GC in cones found in [62], it may be a compensatory effect caused by the need to keep a proper dark cGMP concentration.

Quenching the photoactivated cascade

The flash-evoked wave of the PDE* activity is shaped by quenching of R* and active Tα-GTP/PDE complexes at the rates of kR and kE, respectively. At a fixed [Ca2+]in, each reaction can be approximated by an exponential, which finally yields a two-exponential PDE*(t) curve:

Equation 15

P D E * ( t ) = A ( exp ( t / τ 2 ) exp ( t / τ 2 ) )

Here, A is a scaling factor that is proportional to stimulus intensity and amplification, and τ1 and τ2 are the time constants of the two turnoff reactions. If kR > kE, τ1=1/kE and τ2=1/kR; otherwise, τ1 and τ2 are swapped. We determined kR and kE in Ca2+-clamp experiments (Appendix 1, Figure A5B). Similarly to the dark cGMP turnover, they appeared approximately 10 times faster than in rods (Table 2, Table 3).

Figure 4 supports the validity of the turnoff parameters estimated this way. Panel A shows a model fit of a cone flash response in normal Ringer. Fitting was initiated from crude average estimates of αdark/cGMPdark, kR, and kE, found as explained in Appendix 1, Figures A3, A5, A6, Table A3 and refined by small adjustments of the three parameters, νRE, and Ca2+-buffering. The wave of PDE activity was then derived from the model (smooth red line in panel B). The light-induced PDE activity was also found from the response of the same cell in Ca2+-clamp condition using the model-derived αdark/cGMPdark value (gray noisy curve in B). The black line in B represents a two-exponential least square fit to the gray (Ca2+-clamp) curve. It is seen that the βl(t) time courses obtained experimentally from the Ca2+-clamped response and from model fitting are in fair agreement.

Both reactions of the quenching of the photoactivated cascade, at R* and at the T* stage, proceed in cones roughly 10 times faster than in rods (Table 3). It has been shown that the active conformation of a visual pigment, meta II, decays in cones almost 100 times faster than in rods [66-68]. Still, the characteristic time of decay of meta II of Carassius red- and green-sensitive cones is about 5 s, which is 50-fold slower than necessary for timely turnoff of transducin activation. Obviously, the fast quenching of cone R* is ensured by phosphorylation via cone-specific rhodopsin-kinase (GRK7), whose level of expression and specific activity are substantially higher than those of GRK1 in rods [69-73].

As for the rate of T*/PDE* turnoff, it is controlled in rods by the level of expression of the GTPase activating complex RGS9/Gβ5 [74-78]. It seems that this is true of cones as well, since the level of RGS9/Gβ5 in cones is much higher than that in rods [20,28,79].

The two rates of quenching, kR and kE, are close to each other (Table 2) and are basically interchangeable in the model. Therefore, neither of the two can unambiguously be assigned to R* or PDE* quenching. Recently, it was found [42,80] that the so-called dominant time constant of the cascade turnoff in salamander cones is Ca-dependent. This points to Ca2+-recoverin-dependent R* quenching as the slower process. However, in rods, PDE* quenching may also be adaptation- (supposedly Ca2+-) dependent [59,81,82]. Besides, our analysis suggests that kR and kE are almost identical, which calls into question the notion of the dominant turnoff process, at least in Carassius cones.

Calcium turnover

Ca2+ feedback is decisive in shaping the dark-adapted photoresponse and supporting light adaptation [59,83-87]. Thus, the speed of cytoplasmic Ca2+ changes is one of the crucial factors for proper model (and cone) functioning. In our model, the time course of [Ca2+]in is set by a balance between Ca2+ influx via the CNG channels, its extrusion by the Ca, K/Na exchanger, and interaction with two sort of intracellular buffers [53,88]. One of the buffers equilibrates with [Ca2+]in quickly and can be characterized by its buffering power FB [89]. The second buffer, of a larger capacity, operates slowly and is mostly responsible for shaping the tail of the flash response (Appendix 1 equations (A11), (A12)). Correspondingly, after instant closure of the CNG channels, the intracellular free Ca2+ concentration ([Ca2+]in) declines along an approximately two-exponential curve. As derived from the modeling, the fast phase has an average time constant of 51 ms and an amplitude of about 70% of the total [Ca2+]i decline. The slow phase has a time constant of 1.6 s and an amplitude of about 30% of the total [Ca2+]i decline (Figure 5A, Table 3). Ca2+-sensitive dye measurements also show a two-exponential [Ca2+]in decline of similar parameters, at least in regard to the major fast phase. Its time constant is about 140 ms in salamander cones [90], about 160 ms in visible light-sensitive cones of zebrafish [91], and 255 ms in zebrafish ultraviolet (UV)-sensitive cones [92]. Taking into account that the Carassius cone responses are two to three times faster than in salamander and zebrafish, our model-derived values for the fast phase are in good agreement with the results of direct measurements. As for the slow phase, it is hard to tell how the model data for low-intensity responses may be related to the slow phase of [Ca2+]in decline seen in fluorescent measurements mostly performed with high-intensity stimuli [90-92]. The slow phase in bright light conditions may result from Ca2+ release from a different sort of high-capacity binding site.

In rods, modeled Ca2+ decline also exhibited two-exponential kinetics. The fast phase had an average time constant of 408 ms and an amplitude of about 66% of the total [Ca2+]i decline (Table 3). The slow phase had a time constant of 9.6 s and an amplitude of about 34% of the total [Ca2+]i decline. Thus, the major component of the Ca2+-feedback signal in cones was approximately eight times faster than in rods.

There is a technical problem with determining Ca2+ buffering parameters from modeling. The rate of the cytoplasmic Ca2+ turnover is set by the ratio of ionic influx through the CNG channels to the OS cytoplasmic volume Vcyto and to FB. In most models, including ours, Vcyto is fixed at a certain average value. However, modeled cells may differ widely in respect to the dark current, and hence in respect to the Ca2+ influx, which is a fixed fraction of the current. Therefore, the FB value should be scaled proportionally to the dark current if the kinetics of [Ca2+]in changes is to be preserved. Figure 5B shows that indeed there is an approximately linear relationship between cone experimental jdark and FB derived from the model. This suggests that the Ca-buffering properties are approximately constant among the cells. The cross in Figure 5B marks the “average” cone with an average dark current of 17.6 pA and average FB = 33 ± 3 (standard error of the mean [SEM]). If it also corresponds to the average Vcyto = 0.06 pl assumed in the model, the amount of Ca2+ bound to it in darkness is 16.5 ± 1.5 μmol per liter of the cytoplasmic volume.

As for the slow buffer, its binding capacity Bmax shows no clear correlation with jdark (not shown). The average value of Bmax in the model is approximately 280 μM.

The time course and magnitude of light-induced Ca2+ changes depend on the properties of the Na+/Ca2+-K+ exchanger. The exchanger is characterized by its maximum activity jexsat and Michaelis’ constant Kex. As soon as Kex, dark Ca2+ concentration and fCa, the fraction of channel current carried by Ca2+, are fixed—as in our model—jexsat should be chosen to counterbalance Ca2+ influx at a given jdark:

Equation 16

j e x s a t = f C a j d a r k ( 1 + K e x / [ C a ] d a r k ) / 2

If Kex >> [Ca2+]dark, the exchanger operates in a linear mode. This means that after a sudden closure of a fraction of the CNG channels, [Ca2+]i decreases exponentially to a new steady level that is proportional to the fraction of the current left. Notably, the magnitude of the [Ca2+]i decline does not depend on fCa. On the other hand, if Kex << [Ca2+]dark, the exchanger operates mostly at its maximum speed. Then, in the light, [Ca2+]i decreases at a constant rate until it reaches a level close to or below Kex << [Ca2+]dark, where the speed of the extrusion becomes dependent on [Ca2+]in. Only then can a new balance between inward and outward fluxes be achieved. This behavior does not depend either on the fraction of the channels closed or on fCa, so even a small steady decrease of jdark would result in a drastic reduction of [Ca2+]in.

The linear or nearly linear mode of operation of the exchanger ([Ca2+]dark/Kex <1) is assumed in most available models, including ours. This is strongly supported by the results in [84] and [93]. Here, the authors showed that loading rods or cones with a saturating Ca2+ concentration results in an exchanger current that is several fold larger than the current seen in dark-adapted, non-loaded cells. In line with this are approximately exponential (rather than linear) light-induced [Ca2+]in declines, measured experimentally by fluorescent dyes [90-92]. During steady background illumination, the Ca2+ concentration falls in proportion to the dark current blocked [92], which also points to an approximately linear mode of the exchanger operation.

A strongly nonlinear regime is postulated by Korenbrot [51], who assumes [Ca2+]dark / Kex ratio from 16 to 80. This assumption contradicts the experimental data on Ca2+ kinetics (above). In addition, as shown in Appendix 1 (Equation (A24)), Michaelis-like nonlinearity at the stages of cGMP hydrolysis and Ca2+ extrusion affects the loop gain of the Ca2+ feedback. Our measurements of the gain set the limit of (1 + cGMPdark / KM)·(1 + [Ca2+]dark / Kex) ≤ 1.7. This excludes a situation where Kex < [Ca2+]dark.

The speed of the Ca2+ feedback is crucial for properly shaping the photoresponse. The feedback signal should match the time courses of other turnoff processes like R* and PDE* quenching. Feedback that is too rapid (where the FB value is too low) makes the response slower rather than faster, and reduces sensitivity (dot-dash curve in Figure 5C). Feedback that is too slow (where the FB value is too high) makes the response oscillatory (Figure 5C, solid red curve). This suggests that Ca2+ changes in cones should be roughly tenfold faster than in rods, proportionally to the speed of their photoresponse. As our modeling shows, this is indeed the case.

Interestingly, drastically different parameters of Ca2+ turnover in rods and cones may stem from a single simple factor, the OS volume. The turnover rate is set by the ratio of the Ca2+ influx to the volume of the OS cytoplasm. Thus, a smaller size of cone OS at approximately the same dark current inevitably makes the feedback in cones faster than in rods. The difference in the OS membrane topology, and hence the big difference in surface-to-volume ratio between cone and rod OSs, is as such irrelevant to setting the speed of the Ca2+ feedback. This has already been noticed [48,84]. A higher area of the OS membrane in cones may be secondary to the task of maintaining proper dark current through a smaller OS.

The price of oversimplification

Whatever the definition of the amplification is, its value can be found from a parabolic fitting of the initial part of the experimental flash response (Equation (2)) if all complicating factors (turnoff reactions, membrane filtering, and technical problems) can be neglected. The question is, what is the price of the oversimplification in real life?

Figure 6A shows an initial phase of a cone response (noisy curve) with the best-fitting model curve superimposed on it (smooth line 1). Dot-dashed line 2 is a parabolic fit (Equation (2)) to the 0–40 ms stretch of the experimental response; it accurately fits the model response as well. However, the definition of A (in either sense) deals with the responses normalized to the light-suppressible current jcGdark. Thus, it should exclude the exchanger current which is 1/2·fCa jcGdark. If fCa = 0.2, as we assumed, then an 11% correction has to be applied. Further, the model allows disabling of the calcium feedback and recovery reactions (dark cGMP turnover, R* and PDE* turnoffs). The model then generates the response shown by the curve 3. Its steepness is 66% higher than the steepness of parabolic fit 2. If the filtering by the cell’s electrical time constant (22 ms in this cell) is also eliminated, the resulting model response follows dot-dashed curve 4. Line 4 is the response that would be generated by a cell that meets prerequisite for the applicability of Pugh and Lamb [15,45,46] analysis (no turnoffs, no feedback, no low-pass filtering, normalized to the dark cGMP-gated current). Fitting it with the parabola would yield the “real” amplification value, which is exactly equal to νRE ·(kcat/KM) ·ncG derived from the model. It appears four times higher than derived from the parabolic fitting to the raw response (curve 1). Thus, the routine parabolic fitting results in a fourfold underestimate of the biochemical amplification in Carassius cones. The error must be even higher in mammalian cones, whose photoresponses are substantially briefer than those of fish cones [94-98]. In amphibian rods, the error introduced by routine analysis is approximately 1.6-fold (Figure 6B).

Why are cones less sensitive than rods?

What is the contribution of individual factors to lower sensitivity of cones compared to rods? We suggested a new definition of biochemical amplification in phototransduction that is independent of the cell’s size and only includes molecular parameters of the cascade. These are the rate of PDE activation by R* (νRE), the catalytic activity of PDE* (kcat/KM), and cooperativity of channel gating ncG (Equation (8); see also [38]). Only their product Am could be determined from physiological experiments, and it appeared that in the most sensitive cones, the amplification can be as high as in rods (Table 3). This suggests that there is nothing inherently different between the processes of activation of the cone and rod phototransduction. Hence, the main factor that reduces sensitivity of cones is their faster turnoff. This can be illustrated by a simple example.

The red-sensitive cone 2 shown in Figure 3B is 2,260 times less sensitive than the rod in the same figure, panel C. (Sensitivities are compared using effective light intensities, to eliminate differences in light capture between rhodopsin rods and porphyropsin cones.) Its amplification, though, is just 14 times lower. The remaining 160-fold sensitivity reduction is due to an approximately 10-fold acceleration of all turnoff reactions, as well as electrical filtering (Table 2). The leading role of the speed of turnoff reactions in setting cones’ sensitivity has already been suggested based on biochemical data [10,27,28,38]. Keeping amplification high and accelerating the response termination appears to be a wise evolutionary strategy. It exchanges sensitivity for the speed of the reaction in the most efficient way.

Appendix 1.


The authors declare no conflict of interest. This work was supported by grant 1B-05 from the Biologic Branch of the Russian Academy of Sciences and grant 13–04–00701a from the Russian Foundation for Basic Research to VG. Preliminary results were presented at the 12th International Conference on Vision (Visionarium-XII, Tvärminne, Finland, 2013). The authors are thankful to Ian Dick for his help with English text, and to anonymous Molecular Vision reviewers whose comments improved the manuscript.


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