Molecular Vision 2017; 23:416-430 <>
Received 27 December 2016 | Accepted 05 June 2017 | Published 07 July 2017

Origins of the phototransduction delay as inferred from stochastic and deterministic simulation of the amplification cascade

Alexander Yu Rotov,1,2 Luba A. Astakhova,1 Michael L. Firsov,1 Victor I. Govardovskii1

1Institute for Evolutionary Physiology and Biochemistry, Russian Academy of Sciences, St. Petersburg, Russia; 2Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia

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


Purpose: To identify steps of the phototransduction cascade responsible for the delay of the photoresponse.

Methods: Electrical responses of fish (Carassius) cones and Rana ridibunda frog rods and cones were recorded with a suction pipette technique and as an aspartate-isolated mass receptor potential from isolated perfused retinas. Special attention was paid to sufficiently high temporal resolution (1-ms flash, 700 Hz amplification bandpass). Stochastic simulation of the activation steps from photon absorption to the formation of catalytically active phosphodiesterase (PDE) was performed. In addition, a deterministic mathematical model was fit to the experimental responses. The model included a detailed description of the activation steps of the cascade that enabled identification of the role of individual transduction stages in shaping the initial part of the response.

Results: We found that the apparent delay of the photoresponse gets shorter with increasing stimulus intensity and reaches an asymptotic value of approximately 3 ms in cones and greater than or equal to 10 ms in rods. The result seems paradoxical since it is suggested that the delay occurs in the chain of steps from photon absorption to the formation of active transducin (T*) which in cones is, on average, slower than in rods. Stochastic simulation shows that actually the steps from photon absorption to T* may not contribute perceptibly to the delay. Instead, the delay occurs at the stage that couples the cycle of repetitive activation of T by rhodopsin (R*) with the activation of PDE. These steps include formation of T* (= TαGTP) out of TαβγGTP released from the activation cycle and the subsequent interaction of T* with PDE. This poses a problem. The duration of an average cycle of activation of T in rods is approximately 5 ms and is determined by the frequency of collisions between R* and T in the photoreceptor membrane. The frequency is roughly proportional to the surface packing density of T in the membrane. As the packing density of PDE is approximately 12 times lower than that of T, it could be expected that the rate of the T*–PDE interaction were an order of magnitude slower than that of R* and T. As modeling shows, this is the case in rods. However, the delay in cones is approximately 3 ms which could be achieved only at a T*–PDE interaction time of less than or equal to 5 ms. This means that either the frequency of the collisions of T* and PDE, or the efficiency of collisions, or both in cones are approximately ten times higher than in rods. This may be a challenge to the present model of the molecular organization of the photoreceptor membrane.

Conclusions: The delay of the photoresponse is mainly set by the rate of interaction of T* with PDE. In cones, the delay is shorter than in rods and, moreover, shorter than the duration of the cycle of repetitive activation of T by R*. This poses a problem for the present model of diffusion interaction of phototransduction proteins in the photoreceptor membrane.


The photoresponses of vertebrate rods and cones exhibit a delay phase that in amphibians lasts from approximately 3 ms to greater than 10 ms. This is not surprising as the conversion of the photon absorption by rhodopsin into the cell’s electrical response is mediated by a multistep biochemical cascade. The delay is tacitly attributed to the reactions that lead to the generation of the active transducin (T*) and phosphodiesterase (PDE*) [1-3]. Activation of transducin by the active rhodopsin state (R* = Metarhodopsin II) proceeds via at least five steps (Figure 1). At step 1, R* meets an inactive transducin trimer (TαβγGDP) and binds it. Rhodopsin and transducin are thought to freely move in the photoreceptor membrane colliding at the rate of approximately 6,000 s−1, and on average, one of 30 collisions is productive, that is, leads to the formation of the R*–TαβγGDP complex at the rate of approximately 200 s−1 [3]. At step 2, the R*-bound TαβγGDP complex releases GDP thus vacating a binding site for GTP. At step 3, GTP binds to the site. At step 4, the R*–TαβγGTP complex dissociates. After dissociation, free R* can initiate the next cycle of activation. Meanwhile, TαβγGTP dissociates into Tβγ and TαGTP (step 5). TαGTP is the active form of transducin (T*) that at step 6 binds to phosphodiesterase and activates it. Activation of PDE translates into the electrical response of the photoreceptor via cGMP hydrolysis and closure of the cGMP-gated ionic channels (CNGCs) in the plasma membrane of the outer segment (reviewed in [3]).

The duration of a single cycle of activation is the time between R* initiating the chain (states 1–4) at state 1 and its release from R*TαβγGTP at state 4, thus the time per single activation tt = t1 + t2 + t3 + t4 (Figure 2). Keeping in mind the inherent randomness of all molecular events, the average rate of repetitive production of active T* is νRG = 1/tcycle where the tcycle is the average of the individual tts. The time interval between the photon absorption and the appearance of the first PDE* is equal to the total durations of steps 1 to 6, t1 + t2 + t3 + t4 + t5 + t6, plus a certain delay t0 needed for formation of Meta II from early photointermediates, plus other possible delays between TαGTP–PDE binding and formation of the active PDE*. Thus, the delay time should be td ≥ t0 + t1 + t2 + t3 + t4 + t5 + t6. Here td is an arbitrarily chosen time point at which the minimum deviation of PDE activity from its dark level can be detected. In addition, extra steps may exist between the formation of TαGTP × PDE and the start of the full-speed PDE* production (shown in Figure 2 with the bent line) [1-3].

Pugh and Lamb [1,2] suggested that the time course of the flash-induced PDE activity could approximately be described by a delayed ramp with a certain effective delay teff>td (the red curve in Figure 2). Then the time course of the cGMP hydrolysis (and thus, the electrical response r(t)) is predicted to be a squared parabola starting at teff:

Equation 1

r ( t ) = 1 2 Φ V R E β s u b n c G ( t t e f f ) 2                      t t e f f

Here, Φ is the number of activated rhodopsins, νRE is the steady rate of PDE activation, βsub is the catalytic activity of a single PDE subunit per outer segment cytoplasmic volume, and ncG is the Hill coefficient of the cGMP channel activation. Further, it was argued that the effective delay of the photoresponse is a bit longer than teff of PDE activation, but the difference is supposed to be small ([3], see also the Discussion section). In addition, the coupling between T* and PDE* may be imperfect due to diffusion limitations; therefore, in general νRE < νRG [4]. Later, it was argued that in reality the coupling efficiency is close to unit [3]. Therefore, we assume νRE = νRG = 1/tcycle and use νRE throughout, except the cases when the treatment specifically refers to the chain t1–t4. In addition, in line with [3], we consider two PDE* subunits as separate functional units since they are enzymatically equivalent and can be activated independently [5,6].

The parabolic fitting of the initial part of the photoresponse in the form of Equation (1) is widely used to extract the amplification and the effective delay from the experimental recordings. However, teff is not the real transduction delay td (Figure 2). In addition, the rate of the PDE* production (and thus, amplification) is set by tcycle that comprises part of td and teff. This means that the amplification and delay parameters are interrelated. Thus, it is of significant interest to identify and quantify the steps of the activation chain that shape the delay and amplification. This may shed light on the kinetics of the underlying reactions (t1 to t6) and further our understanding of the molecular mechanics of the cascade activation.

To do this, we collected data on the photoresponse delays in frog (Rana ridibunda) and fish (Carassius) cones and frog rods specifically focusing on high time resolution and low noise. We found that in the cones the delay td is approximately 3 ms and in the rods is greater than or equal to 10 ms. Then we performed stochastic simulation of the reactions that lead to the production of active phosphodiesterase (PDE*). Contrary to expectations, we found that the delay of the averaged photoresponse is shorter than the average duration of the sum of the activation steps. For instance, the time from the appearance of R* to dissociation of the R*–TαβγGTP complex (steps 1 to 4 in Figure 1, Figure 2) obtained from many averaged simulations (multiphoton responses) is shorter than the average tcycle. It depends on the relative durations of the steps. The delay of the appearance of T* is longest when all four steps comprising the cycle are of equal duration (that is, t1 = t2 = t3 = t4 = 1/4 tcycle) but is substantially shorter than the sum of the individual steps (t1 + t2 + t3 +t4). If any of the steps is dominant (slowest), then the delay of the average decreases and may approach zero. The main delay in PDE* production occurs at steps 5 and 6, that is, during the dissociation of the TαβγGTP trimer into Tβγ + TαGTP and TαGTP–PDE binding. We also performed a model simulation of complete photoresponses using a continuous deterministic description of the steps t0–t6; the result was in exact agreement with the results of the stochastic simulation. We determined the restrictions that must be applied to relevant parameters to reproduce experimental responses. It appeared that the average time of TαβγGTP to the TαGTP × PDE* transition (t5 + t6) in cones must be shorter than or equal to the average time of repetitive TαβγGTP production (tcycle). This is quite surprising because the surface packing density of PDE in the photoreceptor membrane in rods and cones is roughly an order of magnitude lower than that of T [3,7]. Thus, it is expected that the frequency of diffusion collisions between T* and PDE would also be an order of magnitude lower than that between R* and T.


Animals and preparations

Adult Rana ridibunda frogs were caught in the wild in southern Russia, and Carassius fish were obtained from local hatcheries. The frogs were kept for up to 6 months with free access to water at 10 to 15 °C on a natural day/night cycle and fed mealworms. Fish were kept in aerated aquaria at room temperature (near 20 °C) at a 12 h:12 h light-dark cycle and fed dry fish food. Animals were treated in accordance with the European Communities Council Directive (24 November 1986; 86/609/EEC), ARVO Statement for Use of Animals in Research, and the protocol approved by the local Institutional Animal Care and Use Committee. Before the experiment, the animals were dark-adapted overnight. Eyes were enucleated and the retinas extracted under dim red light. All further procedures were conducted at infrared (IR) TV surveillance.

Electrical recordings

Suction pipette recordings were performed from frog rods attached to small pieces of the retina, and from attached or isolated fish cones, sucked in configuration outer segments in. The specific details of the suction setup design and functioning have been previously described [8,9]. The light stimulation system consisted of two independent channels based on high-output light-emitting diodes (LEDs) with λmax = 519 and 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. The shortest flash duration was 1 ms. Responses were low-pass filtered at 700 Hz (eight-pole analog Bessel filter) and recorded at 1-ms digitization intervals. All recordings were further shifted by −1 ms to account for the delay introduced by the filter. Digital filtering was not applied. Flash intensity was calibrated using a Burr-Brown OPT-301 integrated optosensor (Tucson, AZ) and regularly checked during the experiments on individual rods using the Poisson statistics of responses to weak flashes [10]. Rods were video recorded to measure their size, so the stimulus intensity could be specified as fractional bleach per flash (for flashes) or per second (for continuous backgrounds). Data acquisition, stimulus intensity, and timing were controlled by National Instruments (Austin, TX) hardware and LabView software (Austin, TX).

In addition, frog and Carassius cone responses were recorded across the isolated retina placed in an Ussing-type perfusion chamber. The electroretinogram (ERG) b-wave was suppressed and the receptor potential (RP) isolated by adding 12 mM sodium aspartate to the perfusion solution. The illuminating system consisted of three independent light channels, one with a 525 nm LED and two with white LEDs. This allowed the application of background light and independently controlled stimulus color with appropriate filters. Specifically, the responses of the frog red-sensitive cones were recorded with orange stimuli (cut-off at 600 nm) on a 525-nm LED steady background that suppressed the rod response. For the Carassius retina, a 640-nm cut-off filter was used that selectively stimulated red-sensitive cones. Light intensity was calibrated with a Burr-Brown OPT-301 optosensor, and additionally by measuring rhodopsin bleaching in frog rods with microspectrophotometry. In the ERG rig, the shortest flashes were 1 ms, and digitization was at 0.5 ms. Further frequency filtration was as in the suction rig. The ERG rig was controlled via the PCI-DAS-1602 I/O card (Measurement Computing, Norton, MA). The controlling program was custom written in the laboratory using Microsoft Visual Basic 96 and Measurement Computing Universal Library.


Main Ringer solution used to record from frog photoreceptors (further referred to as normal) contained NaCl 90 mM, KCl 2.5 mM, MgCl2 1.4 mM, CaCl2 1.05 mM, NaHCO3 5 mM, HEPES 5 mM, glucose 10 mM, and EDTA 0.05 mM; pH was adjusted to 7.6. Stock aspartate solution (120 mM) was added to the perfusion solution to the final concentration of 12 mM, when needed. All chemicals were from Sigma-Aldrich (St. Louis, MO). Temperature was held at 17 to 19 °C. The perfusion solution for recording from the fish cones contained NaCl 102 mM, KCl 2.6 mM, MgCl2 1 mM, glucose 5 mM, CaCl2 1 mM, NaHCO3 28 mM, HEPES 5 mM, and bovine serum albumin (BSA) 50 mg/l; pH was adjusted to 7.8–8.0.

Mathematical modeling

Stochastic simulation-- To account for the randomness of the activation events, we performed stochastic simulation of the reactions that lead to the production of the active phosphodiesterase (PDE* = TαGTP × PDE,Figure 1). We used the algorithm suggested by Gillespie [11] that is widely used for modeling the variability of single-photon responses, most recently in [12]. A single cycle of the program generated a series of states from 0 to 7 (Figure 1) separated by time intervals t0 to t6 (Figure 2). Each interval varied randomly according to an exponential distribution [4,11-13], with the corresponding average t0 to t6 time constants. Each R* released at state 4 initiated a new loop of activation events, now starting from P1, and the cycle was repeated, for example, 100 times. A large number of sweeps, each consisting of 100 activation loops, was repeated and summed to obtain the response to many photons. Computations were performed with a program written in Microsoft Visual Basic 96.

Deterministic model-- For modeling the entire photoresponse, we used a deterministic model consisting of a set of differential equations. The model was described in [14] and named there the minimum essential model (MEM). However, in the MEM the activation of PDE by R* is considered a single-step reaction with an adjustable delay and rate νRE. To account for the multistep nature of cascade activation, the MEM was supplemented with extra front-end equations (see the Equations section) that provided a detailed description of the Figure 1 steps t0 to t6, similarly to [15]. The model treats the outer segment as a well-stirred volume. We call it the extended MEM (E-MEM). The model was implemented in MathCad (MathSoft, Cambridge, MA).


Figure 3 shows the main experimental finding that prompted this work. It appeared that the delay of the photoresponse in cones is substantially shorter than that in rods, while, at least in this particular case shown in Figure 3, the speed of the activation of the phototransduction cascade in the cone is half of that in the rod. Moreover, although the amplification in the best cones may be as high as in the rods [14], on average, it is two to four times lower [14,16,17]. It apparently contradicts the simplified treatment of the phototransduction delay shown in Figure 2. The rate of steady production of T* and PDE* νRE is inverse to the total duration of steps 1 to 4 (tcycle = t1 + t2 + t3 + t4, Figure 1 and Figure 2). Then one could expect that cone td ≥ rod td (assuming, of course, that the interaction between T* and PDE via steps 5 to 6 occurs at the same rate in the rods and cones).

In each individual single-photon response, the first PDE* appears at t0+ t1 + t2+ t3 + t4 + t5 + t6 (Figure 2). However, the reactions of single molecules are subject to stochastic fluctuations, and each t0….t6 vary randomly among the activation events. To account for this effect, we performed a stochastic simulation of the reactions that lead to the production of active phosphodiesterase (PDE* = TαGTP × PDE; see Methods). As an example, Figure 4A shows two series of responses, each initiated by a single photon. Notice that in the graph the values for TαβγGTP (state 5) correspond to the total number of the molecules produced up to the corresponding time point; the number of TαβγGTPs existing at this moment is lower due to its further processing via states 6 and 7. The figure shows that the delays in the production of the first TαβγGTP and PDE* are different in the two program sweeps, as expected from the random process.

Summing multiple sweeps (which is equivalent to the response to many photons) reveals that there is actually no pure delay in the activation of the cascade (Figure 4B). This is because there always is a probability that, due to the randomness of the process, the first active T* or PDE* appears almost immediately after a photon absorption. Remarkably, the time of the R to R* transition t0, which is 1 ms in rods [18] and 0.5 ms in cones [19], does not contribute perceptibly to the delay (see the initial part of the T* curve in Figure 4B). In addition, although in the example simulation the average duration of the cycle of activation tcycle = t1 + t2 + t3 + t4 = 5 ms is equal to the duration of the final steps 5 and 6 (t5 + t6), the main delay of PDE* production occurs at the two latter steps (cf. the curves for T* and PDE* in Figure 4B).

More detailed analysis of the activation cycle t1 to t4 shows that the delay depends not on the entire duration of the cycle (t1 + t2 + t3 + t4) but on the presence of a single dominant (slowest) reaction in it. We fixed the total duration of the cycle at 5 ms, close to that determined in direct biochemical measurements [20], and varied the contribution of the individual steps to the total duration. If a single step is rate-limiting, then the delay of the (average) TαβγGTP production is much shorter than the average t1 + t2 + t3 + t4 and may approach zero (Figure 5, red symbols). However, if all four intermediate steps are of equal duration, then the delay of about 1 ms is the longest but still substantially shorter than t1 + t2 + t3 + t4 (Figure 5, black symbols). The results of the stochastic simulation are in agreement with the prediction of the deterministic continuous model based on Equations (1e) to (5e) with t5 = ∞ (smooth lines through the simulated points).

Comparing the theoretical predictions with delays measured experimentally meets a problem, though. As seen in Figure 4, there is no pure delay in the multiphoton PDE*(t) response. The delay td marked in Figure 2 is arbitrary and depends on the ability to detect the minimum deviation of PDE* from its prestimulus level. Further, PDE activity translates into the electrical response via a sequence of reactions that greatly modify the shape of the response and may introduce extra delays. Estimating the delay of the experimental responses may also suffer from arbitrariness. The delay depends on the ability to detect the minimum deviation from the prestimulus dark current level in the averaged responses. The apparent delay depends on the steepness of the curve and thus decreases with increasing stimulus intensity (Figure 6). Fortunately, at a certain intensity the steepness of the response front saturates, and the delay reaches the minimum [21,22]. Determining this minimum delay value depends on the experimental noise. To avoid ambiguity, we define the start of the response as the first post-stimulus point that lies at or beyond the 0.0025 fractional response, provided that it is also above two standard deviations of the prestimulus record stretch.

The average values of delays in rods and cones in the experimental recordings are summarized in Table 1. To determine what parameters of individual reactions in the activation chain are compatible with the observed photoresponse delay, a complete phototransduction model was necessary. A stochastic simulation can easily be run to the stage of production of PDE*. However, simulating further processes this way is impractical. It has been shown that the average of many stochastic simulations is equivalent to the solution of a set of deterministic differential equations each describing the corresponding reaction [11,13]. Thus, to quantitatively analyze the factors that shape the phototransduction delay we used the MEM of phototransduction [14] updated to the E-MEM (see the Methods and Equations sections). For fitting, we started from the values of the parameters of the rod and cone cascade determined in our previous work [14]. Further, the fit was refined by trial and error, and the quality of the fit was judged by computing the coefficient of correlation between the experimental and model responses. A sample fit to a frog rod and cone photoresponse is shown in Figure 7A,B. The fits in Figure 7 yield r = 0.9997 for cones and r = 0.9987 for rods. The parameters of the fits in Figure 7 are given in Table 2.


Definition and experimental determination of the phototransduction delay

The fact that the (apparent) delay of the flash response in the fish cones is shorter than in the rods was observed recently in [17]. Actually, the photoresponse delays were measured for decades, and a reference list would include more than 30 directly relevant papers. However, it is hard to compare most of the old results with recent data because of different experimental procedures and different definition(s) of the delay. The main shortcomings of the classical works (in the context of the present problem) were insufficiently wide recording bandpass and/or too long stimuli. Only a few papers are free of this flaw. Cobbs and Pugh [21], applying very intense brief flashes, found an irreducible delay of approximately 7 ms in salamander rods. However, such bright flashes (up to 10% rhodopsin bleach) should evoke a noticeable early receptor potential/current (ERP/ERC) that would contaminate the initial part of the response. Thus, the 7-ms delay in rods may well be an underestimate. The contamination by the ERC is clearly seen in recordings from salamander cones and rods in [22]; therefore, the moment of the beginning of the cGMP-gated photoresponse cannot be determined with certainty. The value of the delay reported by the authors (approximately 8 ms) is derived from a certain fitting to the rising phase of the response and is a clear overestimate. Yet Hestrin and Korenbrot’s general conclusion [22] was that the values for the delay are nearly identical in rods and cones. Delays reported in [14] are not experimental values but are the fitting parameters of a model in which the activation of PDE was considered a linear function starting abruptly with an adjustable delay after the stimulating flash. Their relation to the experimental delays (as e.g., in Figure 3 and Figure 7) and to the scheme shown in Figure 2 is unclear. Facing this uncertainty, we had to make a series of experiments especially dedicated to reliably estimating photoresponse delays in rods and cones.

As we show, within the accepted scheme of phototransduction there is no distinct delay between a flash and (averaged) cGMP-gated photoresponse. However, the multistep sequence of reactions that couple photon absorption with the CNGC closure results in a steep initial phase of the response. It may follow a high-order polynomial function of time making an impression of a true delay [1,21] (Figure 3 and Figure 5). The shortest measureable latency is then determined by the ability to detect a minimum deviation in the cell’s current from the dark level. To avoid ambiguity, we define the start of the response as the first post-stimulus point that lies at or beyond the 0.0025 fractional response, provided that it is also higher than two standard deviations of the prestimulus record stretch. In addition to providing a fixed reference point, the criterion also takes into account the experimental noise. Recordings that did not meet the criterion were excluded from further processing.

Thus defined, the delay still clearly depends on the intensity of the stimulus (Figure 6). However, due to limitations within the biochemical cascade and electrical filtering by the cell membrane, the delay at high intensities saturates and reaches a certain minimum value which is approximately 3 ms in Carassius and frog cones and approximately 10 ms in frog rods (Table 1). There is no statistically significant difference between the delays obtained with single-cell suction recordings and trans-retinal recordings of the mass receptor potential.

Rate-limiting and delay-setting steps in the phototransduction cascade

Pugh and Lamb [3] suggested that the total transduction delay, from photon absorption to the initiation of the electrical response, can be represented as

Equation 2

t d = t R + t G + t E + t r + t B + t C

where tR is the time between photon absorption and generation of Meta II = R*, tG is the time from R* to the appearance of T*, tE is the delay between the buildup of T* and the buildup of PDE*, tr is the time for cGMP diffusion, tB is the time for equilibration of cGMP with fast buffer sites in the rod outer segment (ROS), and tC is the delay of the CNGC gating by cGMP.

As we argue, the simple treatment of the (biochemical) delay as the time between photon absorption and the first T*–PDE* appearance (tR + tG + tE in Equation (2)) is valid only for individual single-photon responses (Figure 2, Figure 4). For the averaged responses to many photons, the duration of the repetitive cycle of activation of T* by R* (tG) has no relation to the delay whatsoever. Due to the stochastic nature of the events, there always is a probability that the first R*, via a fast chain of random intermediate reactions, would virtually immediately produce T*. The exact time course of the average R*TαβγGTP production depends on the presence of a rate-limiting step in the chain (from steps 1 to 4, Figure 1). Provided that the total duration of the activation cycle (tcycle = t1 + t2 + t3 + t4) is kept constant, equal durations of all steps (t1 = t2 = t3 = t4) result in the most threshold-like curve with the longest apparent delay (Figure 5, curve 3). If one of the steps is rate-limiting (say, t1 >> t2 + t3 + t4), then the delay almost vanishes becoming much shorter than the average time of Meta II generation (Figure 5, curve 1).

This idea can easily be understood. Since the total duration tcycle is kept constant, prolonging one of the steps reduces the durations of the three others, eventually making them negligible. Then the entire chain t1 to t4 reduces to a single step. The rate-limiting product decays along a single exponential exp(–t/tcycle), and the final product appears as 1 – exp(-t/tcycle), with zero delay. The conclusion does not depend on which particular step in the cycle is rate-limiting. This is because steps 1 to 4 comprise a chain of first-order reactions whose output does not depend on the order of the steps. The reasoning is illustrated with a deterministic continuous model of the activation cycle (steps 1 to 4, Equations (2e) to (5e) which is equivalent to the averaged stochastic computations (Figure 5).

The idea that there is a rate-limiting stage in the activation cycle and that this stage is the (average) interval between productive R*–TαβγGDP encounters t1 (Figure 1 and Figure 2) is supported by experimental data. It was found that the rate of the activation of the cascade increases almost twofold in hemizygous mouse rods with half the normal rhodopsin content, presumably due to reduced molecular crowding in the membrane and the correspondingly increased frequency of R*–T collisions [23]. In addition, the frequency of the R*–T collisions can be decreased tenfold by prolonged light adaptation when 90% of the transducin α- and β-subunits translocate to the inner segment. This causes a corresponding reduction in amplification [24]. Thus, it seems that the R* to TGDP binding limits the rate of T–PDE activation. In this case, the entire cycle of T activation by R* does not introduce a noticeable delay in the photoresponse (Figure 4 and Figure 5).

The next natural candidate for the delay step in rods looks at the diffusion of cGMP from the outer membrane to the sites of hydrolysis on the disk surface. This would possibly explain a shorter delay in cones because the CNGC and PDE in cones reside in the immediate vicinity within the plasma membrane. The diffusion delay tr consists of two phases. It starts from the radial diffusion in interdisc spaces. Then, during the generation of a single-photon response, the decline in the cGMP concentration at the outer membrane should spread along the outer segment. In rods, the excitation may spread along tens of discs, which, in addition, form a baffling system for the diffusion. This might markedly contribute to the delay. However, the longitudinal diffusion is significant only at weak stimuli (a few photons per rod). At the intensities used in the present experiments (tens of photons per each disc), the reduction of cGMP in the outer segment is virtually uniform, and the longitudinal diffusion can be neglected [3].

The radial diffusion in the interdisc spaces of rods is usually considered to be very fast [3,15,25-27]. However, the only direct experimental evidence for this is the result in [25] where the temporal resolution, a few hundred milliseconds, was not sufficiently high to make any conclusion on the less than or equal to 10 ms time scale. If the radial diffusion were the source of the delay, it is expected that the delay would be longer the bigger the diameter of the ROS. Thus, it could be predicted that in frog rods of approximately 7-μm diameter the delay would be greater than five times longer than in fish rods whose diameter is less than or equal to 3 μm ((7/3)2 = 5.4). This is not the case, though. In both species, the delay is virtually the same, approximately 10 ms (fish [17], frog, present paper). An estimate of the radial diffusion time can be obtained from the crude model shown in Figure 8. In darkness, the concentration of cGMP in the interdisc spaces and in the gap l between the disk rim and the plasma membrane is uniform. A bright flash leads to a virtually instantaneous drop in the GMP concentration in the cytoplasm between the discs to a new steady low level. The time course of the decrease in the near-channel concentration is determined by the diffusion equilibration of the gap l with the light-produced sink on the disc surface. The characteristic time of equilibration is t = l2/D ([28], Equation 2.67). Assuming the diffusion constant for cGMP in the cytoplasm D = 40 μm2s−1 [12,26,27,29] and l = 0.03 μm, one obtains that the concentration of cGMP at the plasma membrane drops to 0.5 of the dark level in 8.5 s–6 which is negligible compared to the experimental delay.

The next possible candidates are cGMP buffering and the release of cGMP from the ionic channels of the plasma membrane. Both processes are supposed to be fast [3]. It is found that the interaction of cGMP with channels takes milliseconds, possibly approximately 1 ms [30]. However fast, this may comprise a substantial part of the delay in cones. In rods, it can be neglected. This means that the main component of the transduction delay resides in the amplification cascade itself.

Stochastic modeling shows that an appropriate candidate for the delay-setting stage(s) is the dissociation of TαβγGTP into TαGTP and Tβγ, and TαGTP–PDE binding (Figure 4). To test this idea, we used a mathematical model of the photoresponse that included a detailed description of the activation chain between the photon absorption and formation of PDE* (E-MEM, see the Equations section). The model was fit to a cone response (Figure 7B), and then the time constants t1 … t4 and t5… t6 were varied to test their effect on the phototransduction delay. Changing the time of the limiting step (t1) between approximately 2.5 and 10 ms, as expected, resulted in inversely proportional changes in amplification (and thus, the response amplitude). Correspondingly, the delay as defined above decreased, although by only approximately 1 ms (inset in Figure 9A). However, the responses become virtually congruent after normalizing to their peaks and show no noticeable effect on the delay (Figure 9A). In contrast, similar variations in the time of the T*–PDE interaction (t6) had virtually no effect on amplification but substantially changed the delay (Figure 9B).

Thus, contrary to intuition, the average duration of the cycle of T*–PDE* production does not significantly contribute to the photoresponse delay. Instead, the delay occurs at the stage of transmitting the signal from the activation cycle via T* to PDE.

PDE activation: A paradox

The short transduction delay in cones poses an important problem. We argue that the limiting step in T* production is the formation of the R*–TαβγGDP complex; that is, tcycle is approximately t1. The rate of the process can be expressed as

Equation 3

v R G = 1 / t c y c l e 1 / t 1 = c e v e n c .

Here, νenc is the frequency of the diffusion encounters between single R* and Ts, and ce is the efficiency of the encounters, that is, the fraction of encounters that lead to formation of R*–TαβγGDP. νenc is approximately 6,000 s−1 [3,4], and νRG is approximately 200 s−1; thus, ce is approximately 1/30. νenc is proportional to the surface packing density of T in the photoreceptor membrane, approximately 2,500 μm−2. The surface packing density of PDE in amphibian rods is approximately 200 catalytic subunits μm−2, that is, more than an order the magnitude lower than that of T [3]. Thus, if ce of the T*–PDE collisions were the same as that of R* and T, then the rate of PDE activation by T* (1/t6,Figure 2) should be 12 times lower than νRG. The E-MEM shows that the delay in the rods (10 to 15 ms, Figure 3 and Figure 8) can be reproduced by 1/t6 approximately 20 s−1 which is ten times slower than the rate of the R*–T interaction (1/t1 is approximately 200 s−1, Table 2). This result is in coarse agreement with the idea that the overall efficiencies of the R*–T and T*–PDE interaction are approximately the same, at least in rods.

However, the transduction delay in cones is approximately three times shorter than that in rods (Table 1). Modeling shows that this delay is compatible only with 1/t6 greater than or equal to 200 s−1 which approximately equal to νRG (Table 2, compared with 1/t6 = 20 s−1 in rods). It is believed that the PDE:T surface packing density ratio in cones is approximately the same as in rods (approximately 1:12 [3,7]) although in mammalian cones it could be two times higher (1:6 [31]). Then the question arises, how is it possible that in cones either νenc or ce at the T*–PDE interaction stage is six to 12 times higher than in rods.


Deterministic description of the activation of the cascade and a complete model of the photoresponse-- The average of many stochastic simulations is equivalent to the solution of a set of deterministic differential equations each describing the corresponding reaction [11]. Thus, to quantitatively analyze the factors that shape the phototransduction delay, we used an updated version of our MEM of phototransduction [14]. In the MEM, activation of PDE by R* is considered a single-step reaction with the (average) rate constant νRE. The phototransduction delay in the MEM is implemented as an adjustable pure delay between the real and model stimuli. To account for the multistep activation of PDE responsible for the delay, we constructed an extended MEM (E-MEM) by including in the MEM 11 extra differential equations describing steps 0 to 6 in the scheme Figure 1 (similar but not identical to those used in [15]). Now the rate of PDE activation νRE is given by the output of Equation (7e), and in the absence of turn-off reactions steady νRE = 1/(t1 + t2 + t3 + t4). Hamer et al. [15] considered R*–TGDP binding and GDP release from the R* TGDP complex as reversible (their Equations A3a and A3b) while we treat them as irreversible transitions (our Equations (2e) and (3e)). This has no visible effect on the results.

Equation 1e

d R * d t = R t 0 R * t 1 + R * T α β γ G T P t 4 R * t p

Equation 2e

d ( R * T α β γ G D P ) d t = R * t 1 R * T α β γ G D P t 2

Equation 3e

d ( R * T α β γ ) d t = R * T α β γ G D P t 2 R * T α β γ t 3

Equation 4e

d ( R * T α β γ G T P ) d t = R * T α β γ t 3 R * T α β γ G T P t 4

Equation 5e

d T α β γ G T P d t = R * T α β γ G T P t 4 T α β γ G T P t 5 + R p T α β γ G T P t 4

Equation 6e

T α G T P d t = T α β γ G T P t 5 T α G T P t 6

Equation 7e

d P D E * d t = T α G T P t 6 P D E * t E

Equation 8e

d R p d t = R * t p + R p T α β γ G T P t 4 R p t A r a A r R p t 1

Equation 9e

d R p T α β γ G D P d t = a A r R p t 1 R p T α β γ G D P t 2

Equation 10e

d R p T α β γ d t = R p T α β γ G D P t 2 R p T α β γ t 3

Equation 11e

d R p T α β γ G T P d t = R p T α β γ t 3 R p T α β γ G T P t 4

Here the parameters t0 to t6 correspond to those in Figure 1. tP is the time constant of R* phosphorylation by rhodopsin kinase, tAr is the time constant of binding of phosphorylated rhodopsin by arrestin, and kE is the time constant of the PDE* turn-off by the GAP complex, as in [14]. The pure delay between the real and model stimuli was set to 0, and proper fit to the initial phase of the response was achieved by varying t0 to t6.


The authors declare no conflict of interest. This work was supported by grant # 15–04–03726 from the Russian Foundation for Basic Research to VG. Preliminary results were presented at the 15th International Conference on Vision (Visionarium-XV, Tvärminne, Finland, 2016).


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