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

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 λ_max = 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.

Solutions

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

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 t_R is the time between photon absorption and generation of Meta II = R*, t_G is the time from R* to the appearance of T*, t_E is the delay between the buildup of T* and the buildup of PDE*, t_r is the time for cGMP diffusion, t_B is the time for equilibration of cGMP with fast buffer sites in the rod outer segment (ROS), and t_C 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 (t_R + t_G + t_E 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* (t_G) 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*T_αβγGTP 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 (t_cycle = 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 t_cycle 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/t_cycle), and the final product appears as 1 – exp(-t/t_cycle), 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*–T_αβγGDP 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 t_r 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 T_αβγGTP into T_αGTP and T_βγ, and T_αGTP–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*–T_αβγGDP complex; that is, t_cycle 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, ν_enc is the frequency of the diffusion encounters between single R* and Ts, and c_e is the efficiency of the encounters, that is, the fraction of encounters that lead to formation of R*–T_αβγGDP. ν_enc is approximately 6,000 s⁻¹ [3,4], and ν_RG is approximately 200 s⁻¹; thus, c_e is approximately 1/30. ν_enc 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 c_e 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 ν_RG. 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 ν_RG (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 ν_enc or c_e at the T*–PDE interaction stage is six to 12 times higher than in rods.

Equations

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 ν_RE. 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 ν_RE is given by the output of Equation (7e), and in the absence of turn-off reactions steady ν_RE = 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. t_P is the time constant of R* phosphorylation by rhodopsin kinase, t_Ar is the time constant of binding of phosphorylated rhodopsin by arrestin, and k_E 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.