Kinetics of Recovery of the Dark-adapted Salamander Rod Photoresponse

General Experimental

The experimental methods employed for preparing isolated salamander rods, and for recording and analyzing their electrical responses have been reported (Cobbs and Pugh, 1987; Lyubarsky et al., 1996). For all the experiments whose data are reported here, the circulating currents of rods were recorded by means of suction electrodes into which the rod inner segment was drawn; the outer segment was continually superfused, either with a standard Ringer's solution or by rapid exchange with a test solution.

Calcium Clamping

We made use of recent work showing that Ca²⁺ _i in the outer segments of salamander rods can be maintained near its resting (dark) level by exposing the outer segment to an isotonic choline solution containing very low Ca²⁺ (Matthews, 1995; Lyubarsky et al., 1996). In most of our previous experiments, we employed a “0-Ca²⁺ choline” solution, which, while keeping Ca²⁺ _i near its resting level, allows Ca²⁺ _i to decline slowly in the dark (Lyubarsky et al., 1996; see Figs. 4 and 6); we will report some results and analyses of four rods whose responses were recorded in 0-Ca²⁺ choline. In the present investigation, which reports new data from 19 rods, for calcium clamping we employed exclusively a choline solution containing an estimated 2.3 nM Ca²⁺. This latter concentration of Ca²⁺ _o is in equilibrium with the measured resting concentration in salamander rods, Ca²⁺ _i = 400 nM (Lagnado et al., 1992), and the membrane potential, −67 mV, estimated for the condition in which the outer segment is exposed to a nonpermeant solution while the inner segment is maintained in normal Ringer's (Lyubarsky et al., 1996). While a jump in the dark from Ringer's into choline solution containing 2.3 nM Ca²⁺ _o yields a circulating current whose initial magnitude (∼10 pA) is diminished ∼50% relative to that (∼20 pA) in 0-Ca²⁺ choline, 2.3 nM Ca²⁺ _o serves to maintain a stable circulating current in the dark, allowing the recovery kinetics under calcium clamp to be examined over time intervals up to 40 s or more, as required for examination of the response recovery phase to bright flashes.

Because of intrinsic variability between rods, one would not expect 2.3 nM Ca²⁺ _o (or any particular value) to be in equilibrium for all rods whose outer segments are exposed to choline. In fact, we observe increases or decreases in the circulating current of some rods of up to 20% between 10 s after the jump into choline (when we deliver our first flash) and 45 s (the greatest time at which we deliver a second saturating flash and terminate the exposure to the choline). A 20% increase in circulating current corresponds to a change of <10% in [cGMP], assuming a Hill coefficient of at least 2 for activation of the cGMP channels, and to a change of <5% in Ca²⁺ _i, assuming the cooperativity coefficient for calcium dependence of cyclase activity is also ∼2 (Koutalos et al., 1995a ). A 20% increase in circulating current is also only 0.09 of the average 3.2-fold (220%) increase in circulating current that occurs when the cGMP concentration is strongly elevated before the jump into choline (Lyubarsky et al., 1996).

Stimuli

Stimuli were monochromatic (500 nm, 8 nm full width at half-maximum), circularly polarized light flashes, generated via one of two optical channels: (a) a tungsten/halogen source illuminating a grating monochromator, followed by a shutter; (b) a xenon flashlamp (flash duration, 20 μs) filtered with an interference filter. Intensities are reported in photoisomerizations (symbolized by Φ), obtained by multiplying the physically measured energy density (photons μm⁻²) of the flash at the image plane by an estimated outer segment collecting area of 18 μm². For all new response family data reported here, one of two flash series was used: Φ = 47, 150, 470, 1,500, 4,700, 1.5 × 10⁴, 4.7 × 10⁴ (10-ms flashes); Φ = 23, 94, 300, 940, 3,000, 9,400, 3 × 10⁴, 9.4 × 10⁴ (20-ms flashes); the Φ = 23 flash was not used in all experiments. In general, we avoided flashes of intensity lower than Φ = 47 because of the low amplitude (<2 pA) they evoke in choline (necessitating extra superfusion cycles for reliable data), and because of the focus in this investigation on responses to saturating flashes. Flashes of higher intensities than listed above were generated with the flashlamp channel as needed (for example, to produce strongly saturated responses in choline immediately before the return to Ringer's solution).

Theorems and Model Calculations

The principal theoretical results of this paper are analytical in nature and are cast as “theorems.” Our concept of a theorem is that of a relatively short proposition about well defined variables and quantities, a proposition that can be established by formal reasoning. The theorems are important for providing the context for the presentation of our findings, and thus are given together with the empirical results. However, grasp of the proofs of the theorems is not necessary to understand our conclusions, and so the proofs have been placed in i, where they are available for interested readers. Several of the theorems involve straightforward applications of linear systems theory (e.g., Jaeger, 1966); they have been included, nonetheless, so that readers not familiar with this branch of mathematics may have a self-contained framework for understanding all the theoretical results.

To illustrate certain theoretical results and estimate critical parameters of the rod phototransduction cascade, we employ a computational model developed to characterize responses in clamped-Ca²⁺ _i condition, and written in the MatLab™ programming language (Lyubarsky et al., 1996). The model is generalized here to apply to responses of dark-adapted rods in Ringer's solution, in which Ca²⁺ _i is free to vary. Details of the model calculations will be given as needed in the text, or in ii.

Recovery Translation Invariance

Fig. 2 illustrates the experimental protocol used to obtain response families of rods with Ca²⁺ _i clamped near its resting value. The figure shows three repeated superfusion cycles in which the rod was stimulated first in Ringer's, and then in choline with a test flash producing 3,000 photoisomerizations. To insure that the rod was always in an identical state upon each exposure to choline, a “conditioning flash” of 9,400 photoisomerizations was delivered in Ringer's 40 s before the jump into choline. Unlike the protocol followed in previous calcium-clamping experiments in which a second, saturating flash was delivered at a fixed time after the jump into choline (Fain et al., 1989; Lyubarsky et al., 1996), in the experiments reported here, the timing of the second flash in choline was varied with the intensity of the first flash in such a way as to allow the full recovery to be followed in choline.

Fig. 3 illustrates response families of the rod of Fig. 2 for saturating flashes, obtained in choline (Fig. 3 A) and in Ringer's (Fig. 3 B), and for a second rod (Fig. 3, C and D). The responses are plotted in a nonconventional manner: only the response to the most intense flash is plotted correctly with respect to the time axis; all other responses were translated to coincide at the point of 50% recovery. Here it can be seen that the recovery phases of the responses in Ca²⁺ _i clamp (Fig. 3, A and C) are nearly identical in shape. The responses in Ringer's (Fig. 3, B and D) are also quite similar to one another, though clearly less so than those obtained in choline.

Another way to examine the shape invariance of the recoveries is illustrated in the lower half of each of the four panels (Fig. 3). Here we have taken the average of the traces in each case most closely similar in form (see legend), and then, with smoothing created an empirical template recovery shape; the template was subtracted from each of the individual traces and the residuals were plotted. For the responses in choline, shape invariance is again seen to hold well for flashes that produce up to 15,000–20,000 photoisomerizations. Above 20,000 photoisomerizations, systematic changes in recovery form are observed, most notably for the responses in Ringer's.

Fig. 3 also serves to illustrate another feature of the recoveries: geometric increases in flash intensity give rise to linear increments in recovery time. This feature is revealed by the approximately constant spacing of the rising phases of the translated responses.

The experiment illustrated in Fig. 3 was completed on eight rods, with similar results. (Summary data from all the rods will be reported in Table II, and also in Figs. 6 and 9, below.) We return to consideration of the deviations from shape invariance later. Our immediate goal is explicating the theoretical implications of the shape-invariant recovery behavior.

Theoretical Analysis

We can formulate the observations illustrated in Fig. 3 in terms of the following functional equation:

1a

where F[Φ,t] is the circulating current present at time t after a flash producing Φ photoisomerizations at t = 0. The interval (Φ₀, Φ_max) is the intensity range over which Eq. 1a holds, t ₀ is time at which F begins to show recovery from saturation by the flash Φ₀, s is a positive number and h(s) is an unknown function. F is assumed to obey two boundary conditions:

1b

1c

In words, Eq. 1a states that when the intensity of a saturating flash producing Φ ≥ Φ₀ photoisomerizations is scaled by a factor s ≥ 1, the response recovery at times greater than the fixed time t ₀ is translated on the time axis without change of shape to the right by the amount h(s). Eq. 1b states that for any flash whose intensity lies within the specified range of Φ, at sufficiently long times recovery is complete; Eq. 1c states that even the most intense flash can only drive F to zero. A family {F  [Φ,t]} of photoresponse recoveries satisfying Eq. 1 is said to obey Recovery Translation Invariance (RTI).¹

In i(Lemma 1), we show that Recovery Translation Invariance is sufficient to completely determine the nature of the translation function h(s); specifically, if a family of recovery traces obeys RTI, then the only possible form that h(s) can take is

2

where τ_c is a constant having the units of time. Once it has been established that RTI implies Eq. 2, then it is straightforward to prove the following result:

Theorem 1: Recovery Translation Invariance

A family of circulating current recovery traces {F  [Φ,t]} obeys RTI if and only if

3

where H(x) is a saturation function obeying H(x → Θ) = 0, H(0) = 1, and τ_c is a constant having the units of time.

Put into words, theorem 1 states that obedience of a family of saturating responses to Recovery Translation Invariance is equivalent to the requirement that there exists a transduction intermediate that is produced in an amount proportional to the flash intensity Φ (over the restricted intensity range), and which at appropriately long times decays with the time constant τ_c. Theorem 1 by no means states that the circulating current itself recovers with the time constant τ_c; quite the contrary, a saturating nonlinearity H can (and does) exist between the decaying transduction intermediate and the measured circulating current recovery. (Later, however, we establish conditions under which τ_c can be expected to be directly recoverable as the time constant of the “tail phase” of the recovering circulating current.)

We now note several consequences of theorem 1. First, theorem 1 reveals RTI to be both necessary and sufficient for Eq. 3 to hold. In other words, under the boundary restrictions placed on F, Eq. 3 and RTI are equivalent properties: one cannot exist without the other. This equivalence helps to resolve some confusion in the literature on the conditions under which one can infer the existence of a unique dominant time constant, a point to which we return in the discussion. Second, while theorem 1 appears to place only minimal constraints on the saturation function H, it nonetheless leads to the question of which late steps in the transduction cascade can be demonstrated analytically to preserve a dominant time constant established at an earlier step (Fig. 1) and thus serve jointly as an “H function.” We will address this question directly, and answer it in the section below entitled “The cGMP synthesis and hydrolysis reactions.” Third, the time scale τ_c of the logarithmic function h(Φ/Φ₀) = τ_c ln (Φ/Φ₀) is uniquely determined from the translation of the recovery curves per e-fold change in intensity, as noted by Pepperberg et al. (1992); see also Baylor et al. (1974, Eq. 51 and Fig. 19). In keeping with the terminology used by Pepperberg et al. (1992), we call this scale constant the “dominant time constant of recovery,” and have adopted for it the symbol τ_c, where “c” stands for “critical.” We next examine more fully the conditions under which one might expect the rod phototransduction cascade recovery to be governed by a dominant mechanism. In so doing, we find another characterization of a dominant time constant.

Theorem 2: Dominant Time Constant of a Linear Cascade

Suppose that the impulse-activated activity of an enzymatic effector E*(t) can be represented as a cascade of n reactions, each exhibiting first-order decay, having time constants τ₁ < τ₂ < . . . < τ_n. Then, at sufficiently long times, the reaction with the longest time constant, τ_n, always dominates: that is, given any small number δ, it is always possible to find a time T _δ such that to within error of a term of order δ

4

where e* = E*/Φ, Φ is input strength (flash intensity) and C′ is a constant.

Theorem 2 follows straightforwardly from linear systems theory. Our goal in stating it is to show how to compute T _δ, the time at which “dominance” is established. Based on current knowledge of the reactions of the rod phototransduction cascade, n is not expected to be large; recent models of E*(t) have used n = 3 (Tamura et al., 1991) and n = 2 (Lyubarsky et al., 1996). The model of e*(t) implemented here is that generated by the cascading of two first-order exponentials, one representing R* decay (time constant, τ_R) and one for concurrent G*–E* decay (time constant, τ_E) (Fig. 1):

5

where ν_RP is the rate of generation of E* per fully active R*, and C _RE = [τ_E τ_R/(τ_E − τ_R)] is a constant that renders E*(t) at early times consistent with the activation scheme of Lamb and Pugh (1992). Thus, in this particular case in Eq. 4, C′ = ν_RP C _RE. Use of Eq. 5 as a description of the disc-associated reactions is not without problems, particularly inasmuch as it assumes R* activity to decay with first-order kinetics. In discussion, we address some issues concerning this obvious oversimplification of the biochemical reality of R* inactivation. Nonetheless, in the context of Eq. 5, the value T_δ can be thought of as setting the value of t ₀ in theorem 1. Thus, for the two-stage model of E*(t) kinetics embodied in Eq. 5 and the specific values of the time constants τ_R and τ_E estimated below, we find T _{δ = 0.01} = 2.2 s; that is, 2.2 s after a flash is given, the intermediate R* or E* with the longer lifetime is expected to be strongly dominant, for flashes up to the intensity at which RTI fails.

In the context of theorem 1, and the empirical obedience of rod recoveries to RTI (Fig. 3), the overall significance of Eq. 4 is this: we can tentatively identify the scale constant τ_c of Eq. 3, estimated from recovery half-time data, with the component of the impulse response e*(t) having the longest time constant, τ_n. This identification will provide a satisfactory completion of the meaning of the term “dominant time constant.” However, such identification is premature unless it can be shown that the reactions of the phototransduction cascade subsequent to E* cannot contribute a dominant time constant, and yet are such as to preserve a dominant time constant established at an earlier stage in the cascade.

The cGMP synthesis and hydrolysis reactions.

Our primary goal in this section is to inquire whether the reactions governing cGMP hydrolysis and synthesis are such as to allow a dominant time constant present in e*(t) to be conserved. Our analysis answers this inquiry affirmatively, and also shows that while the hydrolysis/synthesis step of the cascade cannot be the source of the dominant time constant manifest in recoveries from saturating flashes, it nonetheless makes an important contribution to the time to peak of subsaturating responses.

The reactions governing the hydrolysis and synthesis of cGMP in a rod outer segment after an isotropic flash can be written

6

where cG is the concentration of free cGMP, α the rate of cGMP synthesis by guanylyl cyclase, and β the rate constant of hydrolysis. Many investigations have established the applicability and generality of Eq. 6 (reviewed in Pugh and Lamb, 1993).

For a rod in normal Ringer's solution, α is time dependent, due to the decline in Ca²⁺ _i that occurs during the light response and the dependence of guanylyl cyclase activity on Ca²⁺ _i. For the specific condition in which Ca²⁺ _i is held at its resting level (as in Fig. 3, A and C), α ≡ α_dark and we can simplify Eq. 6 to the following:

7

By further restricting attention to the recovery phase of the response when e*(t) is governed by its dominant mechanism, and by application of theorem 2, we can rewrite Eq. 7 as

8

By analysis of Eq. 8, we can establish the following result.

Theorem 3: Conservation of the Dominant Time Constant of Recovery

When α = α_dark, a constant, the family of recovery curves {cG(Φ,t)} generated by solving Eq. 8 for different saturating values of Φ obeys RTI. Thus, there exists a time t ₀ such that for t > t ₀ solutions to Eq. 6 for α = α_dark are isomorphic, and translate on the time axis τ_c units for each e-fold increase in Φ, where τ_c is the largest time constant of the reactions governing the rod transduction cascade up to and including E *.

Before closing this section on the cGMP hydrolysis and synthesis reactions, we emphasize a feature of Eq. 7 important for full appreciation of RTI. While neither Eq. 6 nor 7 is the equation of a linear filter, at sufficiently low response amplitudes, Eq. 7 is in fact linear in Φ. Thus, the behavior of solutions of Eq. 7 is important for understanding the kinetics of photoresponses at low intensities and for understanding the tail phase of recovery from saturating flashes. The behavior is also important for excluding a role of cGMP hydrolysis and synthesis reaction in determining the dominant time constant. Thus, we formalize this behavior as follows.

Theorem 4: Dim-Flash Responses and Tail Phase of Responses in Calcium Clamp: The Filtering Effect of β_dark

At appropriately low response amplitudes (such as those of responses to low intensity flashes), under calcium clamp the cGMP hydrolysis and synthesis reac-tion (Eq. 7) acts as a low pass filter with time constant τ_dark ≈ 1/β_dark; at high intensities the reaction does not contribute a significant time constant to the cascade.

The effect of the Hill equation governing the cGMP-activated current.

The Hill equation governing the relationship between free cGMP, cG(t), and F, the fraction of circulating current present in normal Ringer's at time t, is given by

9

where n _H is the Hill coefficient. Eq. 9 is valid for time scales exceeding a few milliseconds because of the rapid equilibration of cGMP-gated channel currents with the ligand concentration. Inspection of Eq. 9 reveals that its application to a family of theoretical curves {cG(Φ,t)} generated as the solution to Eq. 8 will not alter the relative lateral positions of the members of the family on the time axis; rather, application of Eq. 9 with n _H > 1 serves only to steepen each of the recovery curves in a manner that preserves their relative positions. The same conclusion applies to the modified version of Eq. 9 that governs responses in choline (Lyubarsky et al., 1996; see Eq. 10).

The overall consequence of theorems 2 and 3 is this: the dominant time constant of the reactions governing the time course of activation and inactivation of E* will be conserved through the subsequent reactions of the cascade, and be manifest in the spacing on the time axis of circulating current recovery traces. Illustrating this conclusion, Fig. 4 shows theoretical curves generated with the model, fitted to the photoresponses of the two rods of Fig. 3, and to those of an additional rod whose distinctive pattern of responses provides a basis for useful discussion later. In Fig. 4, we have fitted the response families of rods a–c twice: once (left) with n _H = 2, and again (right) with n _H = 3. As can be seen, other factors constant, the recoveries are somewhat better characterized by theory traces employing the higher value of the Hill coefficient. It is noteworthy that the values of the other parameters (A, τ_c, τ_nd), estimated by fitting the model to response families, were practically independent of whether the Hill coefficient is set to 2 or to 3; thus, for the families shown in Fig. 4, the estimates of the dominant and nondominant time constants giving the best-fitting curves differed in each case by <10%. (The notation adopted in Fig. 4 and Table I for the two time constants τ_R and τ_E of Eq. 5 is noncommittal as to their molecular identity since the constants are formally interchangeable. Thus, τ_c refers to the longer or dominant time constant and τ_nd to the shorter, nondominant time constant. The molecular identity of the mechanisms underlying the time constants will be taken up in discussion.)

The application of the model to the data of Figs. 3 and 4 underscores an important feature: for flashes exceeding ∼20,000 photoisomerizations, the linear E*(t) model (Eq. 5) fails systematically, predicting recoveries that are more rapid than those observed. This failure of RTI is illustrated further in Fig. 5, where we plot recovery half-time data for rods a and b, and the deviations of the recovery half-times from the constancy predicted by RTI for eight rods.

Figure 5.

Open in a new tab

Fig. 6 (top) serves to illustrate the degree to which the two-stage inactivation model accounts for the template recovery shape of each of the rods. The model was first fitted to the response family of each rod (as illustrated in Fig. 4); this generated the theory templates. The fitting also yielded estimates of the nondominant time constant (τ_nd) of each rod; these values are reported in Table II. Fig. 6 also provides evidence for testing a prediction resulting from theorems 2 and 4: providing the dominant time constant τ_c exceeds 1/β_dark, the tail phase of the response recovery from any flash is predicted to decay exponentially with time constant τ_c; this prediction is not dependent on the value of the Hill coefficient, providing that the fitting is begun at a sufficiently low response amplitude.

Theorem 5: Recovery Translation Invariance in Ringer's

If a family {F[Φ,t]} of photoresponse recoveries obtained under conditions that allow α to vary freely obeys RTI, then α(t) itself must obey RTI and recover after a saturating flash in such a manner as to track the recovery of the incremental cGMP hydrolysis rate constant, at long times given by Δβ(t) ≈ Φν_RP C _RE e ^−t/τ_c β_sub.

Figs. 7 and 8 illustrate an application of theorem 5 to our results. Fig. 7 (top) reproduces from the investigation of Hodgkin and Nunn (1988) the response of a rod to a flash they estimated to yield Φ = 40,800, along with the response in Ringer's of the rod of Fig. 2 to the flash producing Φ = 30,000; Fig. 7 (bottom) shows Hodgkin and Nunn's estimates of α′ = α/cG _dark and β, along with estimates of the same two variables obtained in a complementary manner from our data, as we now explain. We first introduce an expression for α′ = α/cG _dark that can be derived by combining Eqs. 6 and 9:

10

In their experiments, Hodgkin and Nunn estimated α′ by measuring the rate of change of the circulating current after rapid exposure of the outer segment to the phosphodiesterase inhibitor IBMX (3-isobutyl-1-methylxanthine); they then estimated β with the steady state approximation of Eq. 10, which neglects the second term; i.e., they used the relation β(t) = α′(t)/F(t)^(1/n _H ⁾. In contrast to Hodgkin and Nunn's approach, we first estimated β by fitting the model to the responses of the rod obtained under Ca²⁺ _i clamp (Fig. 4 A), and then derived α′(t). Thus, from the fitting we obtained β(t) = Φe*(t)β_sub + β_dark (see Eq. 7), and we then computed α′(t) = β(t)F(t)^(1/n _H ⁾, which is plotted along with β(t) in Fig. 7 (bottom).

In Fig. 8, we apply the analysis of Fig. 7 to the complete set of saturating responses obtained in Ringer's of the same rod: unbroken lines are the estimates of β(t) obtained from the fitting of the cascade model to the responses obtained in calcium clamp (Fig. 4 A); gray thickened lines are the estimates of α′(t) obtained with the steady state approximation of Eq. 10, while the dotted trace gives the result of applying the complete equation, including the derivative term. As is seen in Fig. 8, we found generally that the derivative term of Eq. 10 contributed <5% to the estimate of α′(t) at any time after the point of 10% circulating current recovery.

The value of α′(t) at the time of 10% recovery is informative, since the concentration of Ca²⁺ _i should have changed relatively little from the minimal value achieved during the saturated phase of the response; thus α′_{F = 0.1} provides an estimate of α′_max. For the rod of Fig. 8, the average value of α′_{F = 0.1} estimated from the responses to the four highest intensities was 10.2 s⁻¹. In Table II (rightmost column), we report the values of α′_{F = 0.1} obtained in this way for each rod. The average value α′_{F = 0.1} for these rods was 10.4 s⁻¹; this value was the same if the two outlier values (Table II, rods f and g) were eliminated before averaging.

The focal issue of this section is the analysis of the mechanisms that underlie the accelerated recovery kinetics of saturated responses in Ringer's, relative to those measured in calcium clamp. The analysis of Figs. 7 and 8 provides an explanation of this acceleration, inasmuch as it shows that an ∼10-fold increase in cyclase activity during the saturated phase of the responses, along with Eq. 10, suffices to explain the acceleration. However, this analysis provides relatively little insight into the mechanistic details underlying the accelerated recoveries and, moreover, by assuming that none of the early steps in the cascade is affected by the decline in Ca²⁺ _i, begs the question of whether another calcium-dependent process might be involved in the faster recoveries.

To gain deeper insight into the effect of cyclase activation on response recoveries in Ringer's, we adopted and applied three equations that have been used by several investigators to characterize fluxes of Ca²⁺ across the salamander rod outer segment membrane, free Ca²⁺ in the outer segment, and the Ca²⁺-dependent activity of guanylyl cyclase (Lagnado et al., 1992; Miller and Korenbrot, 1994; Koutalos et al., 1995a , 1995b ; see also Tamura et al., 1991; reviewed in Pugh et al., 1997):

11

12

13

In these three equations, Ca represents the concentration of outer segment free calcium (i.e., Ca²⁺ _i); the parameters of the equations are listed in Table III (𝔉 is the Faraday). Eq. 11 describes the dependence of the Na/Ca-K exchange current on Ca²⁺ _i, while Eq. 12 describes the rate of change of Ca²⁺ _i in terms of the balance between inward current through the cGMP-gated channels (−f_Ca F J _dark) and outward pumping by the exchanger (J _ex). (Note that J _dark is an inward current, and therefore a negative quantity, and that while J _ex is also a net-inward charge flow, it corresponds to a decrease in Ca²⁺ _i.) Eq. 13 describes the dependence of the cyclase rate α on Ca²⁺ _i. If these equations provide an adequate characterization of the mechanisms governing Ca²⁺ _i, then, when combined with Eqs. 5, 6, and 9, they should in general yield a quantitative account of the responses in Ringer's and, more specifically, provide an account of the shift in recovery times between responses in calcium clamp and in Ringer's.

We took two approaches to the application of Eqs. 11–13. First, we combined them with Eqs. 5, 6, and 9 and solved the ensemble of six equations numerically; further details, including a description of the initial conditions, are provided in ii. We will return to the numerical analysis below. Second, we expanded each of the six equations into perturbation approximations about the initial (i.e., dark/resting) values of the variables cG and Ca, thereby obtaining an analytic formula for the small signal response, and for the tail-phase response in Ringer's. This analysis yielded the following result.

Theorem 6: Tail Phase of Saturating Responses in Ringer's: Apparent Gain-control Effect of Cyclase Activation

The tail phase of the photoresponse in Ringer's will decay as a first-order exponential with the time constant τ_c of the dominant mechanism of the disc membrane– associated reactions, providing the inequality

14

is satisfied, where

15

and γ > 0 and 1 > η > 0. Moreover, if Eq. 14 is satisfied, the effect of cyclase activation alone on the position on the time axis of the late phase of recovery from a saturating photoresponse in Ringer's relative to that in calcium clamp can be expressed as

16

where ΔT _cyclase is the predicted shift, μ is given in Eq. 15, and

17

Eqs. 14–17 in theorem 6 yield quantitative constraints for the theory of recovery, which we now explore. The constraint embodied in Eq. 14 appears generally satisfied, since the terms γ and η, and, therefore, μ are positive, and the values of the parameters involved yield an estimate for μ of 2.7 s⁻¹ (Table III); thus, μ is more than fourfold larger than (1/τ_c), which is ∼0.5 s⁻¹ (Table I). But does the prediction of theorem 6 hold that the tail phase of the responses in Ringer's should decay as a first-order exponential with time constant τ_c? And how does the prediction of Eq. 16 compare with the observed shift in recoveries between saturating responses in calcium clamp and Ringer's?

Figs. 9 and 10 address the first question. In Fig. 9, we show an averaged response of each rod in Ringer's, along with a decaying exponential fitted to the tail phase to estimate τ_tail. This analysis was similar to that used to analyze the calcium-clamp response tail phases (Fig. 6), except that we did not average responses obtained at different flash intensities. For most cells, we analyzed only the response to the conditioning flash, which was repeated many times over the course of an experiment (Table II, column 11). We adopted this procedure because of concern that systematic variation in τ_tail over intensity might be obscured by averaging, as we now explain.

Since for some rods of this and of our previous investigation we had five or more responses to flashes of different intensities in Ringer's (obtained over the time course of a 2–3-h experiment), we were able to estimate τ_tail reliably from responses to these different flash intensities; these estimates are illustrated in Fig. 10 A (open symbols). Fig. 10 A (shaded circles) represents data from a rod of our previous investigation. Also plotted along with our data in Fig. 10 A as symbols with embedded crosses are estimates of the time constant of decline of Δβ(t) = β(t) − β_dark obtained by Hodgkin and Nunn (1988; see Fig. 16) with their IBMX- and lithium-jump methods (see Fig. 8, above). Our estimates of τ_tail and theirs of the time constant of decline of Δβ(t) are in good agreement, as theorems 5 and 6 leads us to expect. Visual inspection of Fig. 10 A reveals that, in the middle range of intensities (Φ ≅ 100–10,000, depending on the cell), τ_tail is approximately constant, as expected from analysis of the recovery half-times (Fig. 5, open symbols; Lyubarsky et al., 1996). However, three systematic deviations from the simple ideal of an intensity-independent value of τ_tail deserve attention.

The first and most salient deviation from the simple ideal occurs at Φ ≅ 10,000, where for most rods τ_tail becomes systematically much longer, increasing by as much as twofold over the next 1-log unit range of intensities. This systematic lengthening of τ_tail occurs at approximately the same intensities at which RTI fails for calcium-clamp responses (Fig. 5). The second deviation from the simple ideal occurs at intensities Φ < 100, where for a number of rods τ_tail becomes systematically shorter. We will consider these latter deviations in more detail below. The third deviation from ideality occurs exactly in the middle range of intensities, and is characterized by a gradual increase of τ_tail. To put the deviations of the first and third kind into relative perspective, in Fig. 5 B we have fitted straight lines to points in the middle and upper range, picking (somewhat arbitrarily) a “break point” near Φ = 10,000. The average slopes of the lines fitted were 0.15 ± 0.12 s/log₁₀(Φ) in the middle intensity range and 1.2 ± 0.2 s/log₁₀(Φ) in the upper intensity range. In an effort to obviate the arbitrariness of first choosing a breakpoint to determine the slopes, we also derived local slopes from the data in Fig. 5 A, numerically estimating the derivative at each point; these running slopes are plotted in Fig. 5 C. The analyses in Fig. 5, B and C support the conclusion that a highly reliable increase in slope in the τ_tail vs. logΦ curves occurs at ∼Φ ≅ 10,000 for all rods, and that in the middle range the slope is relatively shallow or negligible. Also interesting is that the rods having the larger absolute values of τ_tail also have greater slopes.

Theorems 5 and 6 together lead to the conclusion that τ_c estimated from recovery half-time data and τ_tail should be the same for each rod. Fig. 10 D compares the τ's estimated from the tail phase analyses with the estimates of the dominant time constant τ_c for all the rods of this study, as determined in Ringer's (open symbols) and in calcium clamp (closed symbols). The values of τ_tail in this figure were obtained from the responses to the conditioning flashes (Φ = 4,700 or 9,400), which were repeated many times (Table II, column 4). Based on the relatively shallow slopes in Fig. 10 B, these estimates should be appropriate for examining the prediction that τ_c and τ_tail should be the same for each rod. To the data from the eight principal rods of this study, we have added to the figure points (gray symbols) obtained from responses in Ringer's of 15 additional rods involved in related experiments. The symbols in Fig. 10 D fall near the line of slope 1 through the origin, suggesting that the mechanism(s) underlying variation across rods (and, implicitly, over animals) affects τ_c and the response tail phases in the same manner. Interestingly, the rods exhibiting the smallest values of τ_c were obtained from animals obtained in the early Spring.

We return now to the second question posed above: can cyclase activation alone account for the shift between the response recoveries in calcium clamp and those in Ringer's? One issue that needs to be addressed first concerns the amplitude of the response recovery at which the shift should be measured: in Table II, we reported the shift at the point of 50% recovery (ΔT _0.5), but the theoretical prediction of Eq. 16 is valid only for response tail phases. To address this issue, we remeasured the shifts between the responses to saturating flashes in choline and in Ringer's at the points of 80% recovery; these values are reported as ΔT _0.8 in Table IV. The averaged absolute fractional difference, |ΔT _0.5 − T _0.8 |/ΔT _0.5 was 4%, with the maximum fractional difference being 10%. A second issue that must be addressed in order to apply Eq. 16 is the specific values of the various parameters in Eqs. 14–17. For the most part, the needed parameters have already been estimated for each rod or were estimated in previous investigations by others; these are given in Table III. Two particular parameters, however, stand out as requiring special attention: n _H, the Hill coefficient of the cGMP-activated current, and B_Ca,rest, the calcium-buffering capacity of the rod near rest. Based on the quality of the fittings of the theoretical curves in Fig. 4, and on estimates of the Hill coefficient of cGMP-activated currents of excised patches of outer segment membrane, we might prefer the value n _H = 3. Nonetheless, recent experiments on truncated salamander rods have yielded the estimate n _H = 2 (Koutalos et al., 1995a ), and the value n _H = 3 must be called into question. According to Lagnado et al. (1992, see Eq. 8), B_Ca in the salamander rod can be generally expressed as

18

where C _buff is the concentration of high affinity buffer, K _buff is the dissociation constant of the high affinity buffer, Ca the calcium concentration, and B the buffer capacity of the rod “at high Ca²⁺ _i.” Lagnado et al. (1992) provide the estimates C _buff = 240 μM, K _buff = 0.7 μM, Ca = Ca _dark = 0.4 μM, B = 16; these values predict that in the salamander rod B_Ca,rest should be 156. Calculations with the analytical model of the dim flash response resulting from theorem 6 (see discussion, Eq. 20) led us to suspect that the value B_Ca,rest = 156 was problematically high.

Fig. 11 serves to illustrate for rods a and b the problem with B_Ca,rest arising from the application of Eq. 18, and shows how we obtained estimates of n _H and B_Ca,rest. In brief, we numerically solved Eqs. 5, 6, 9, and 11–13 describing the transduction cascade in Ringer's, fitting the solution curve to the response of each rod to the least intense flash used to stimulate the rod, and using the optimized fittings to estimate the parameters. The theory predictions (Fig. 11, dashed lines) are seen to fail seriously if Eq. 18 is applied with C _buff = 100 μM, a value <1/2 the estimate C _buff = 240 μM reported by Lagnado et al. (1992). In contrast, the theoretical calculations with all other parameters unchanged appear to give an excellent account of the responses on the assumption that B_Ca,rest = 15 and 18, with n _H = 2. Also shown in Fig. 11 are the best fitting theoretical curves that could be obtained with n _H = 3; clearly these curves fit the data less well than those computed with n _H = 2.

Fig. 12 shows the application of the theoretical analysis to the lowest intensity flash responses obtained in Ringer's from all the remaining rods of this investigation (Fig. 12, c–h), and from two rods from the previous investigation (Fig. 12, i and j). In Table IV, the resulting parameter estimates are given. The average estimated value of B_Ca,rest is 17.5 ± 7.2. To fit the responses well, the value of the “dominant” or larger time constant of Eq. 5 for every rod had to be set to a value systematically lower than the estimate τ_c obtained from the translation and tail-phase analysis of saturating responses. A similar observation was reported by Hodgkin and Nunn (1988) as lower estimates of the time constant of decay of Δβ(t) at low flash intensities (see Fig. 10 A). In Table IV, we identify this value as τ′_c.

Finally, with B_Ca,rest (and all other relevant parameters) now estimated, we can examine the prediction of theorem 6, Eq. 16. Thus, in Table IV we report the predicted shift between the recoveries of saturating responses in calcium clamp and in Ringer's, predicted on the hypothesis that cyclase activation alone underlies the shifts. The average residual difference between the observed shift (ΔT _0.8) and that predicted by cyclase activation alone (ΔT _cyclase) is 1.5 ± 0.9 s; the residuals range from −0.5 to 2.8 s.

As noted at the beginning of this section, recent evidence has supported the existence of a calcium-sensitive mechanism that affects the gain of an early activation step. Because it appears that such an effect can provide a reasonable account of the residual shift not accounted for by cyclase activation, it is useful to conclude by formalizing the manner in which calcium, acting on a nondominant mechanism, will affect the recoveries to saturating responses in Ringer's.

Theorem 7: Gain Control Via a Nondominant Mechanism

If calcium feedback acts to diminish the gain or shorten the lifetime of a nondominant component of the cascade up to and including E*, then such an effect will be manifested in the recoveries of saturating photoresponses in Ringer's only as a shifting of the family of recoveries to shorter times, with no change in the spacing on the time axis of the members of the family.

Linearity of the Phosphodiesterase Response Revealed by Recovery Translation Invariance

An earlier investigation concluded that during the rising phase of the salamander rod photoresponse the number of active phosphodiesterase catalytic subunits, E*(t), is linear in intensity up to Φ ≅ 10,000–20,000 photoisomerizations per rod, equivalent to 1 photoisomerization for each 7–16 μm² of disc membrane (Lamb and Pugh, 1992). The phenomenon of Recovery Translation Invariance (Figs. 3 and 4) now leads via theorem 1 to the conclusion that for responses obtained in clamped Ca²⁺ _i, linearity holds for the entire time course of E*(t), both activation and inactivation, for flashes up to approximately the same intensity. The likely explanation of this linearity is that the reactions governing the activation and inactivation of R*, G*, and E* (Fig. 1) for such intensities occur in completely nonoverlapping domains on the disc membranes, and involve no significant competition for cascade reactants.

Two essential nonlinearities intervene between E*(t) and the suppression of circulating current in clamped Ca²⁺ _i, the reactions governing cGMP hydrolysis/synthesis (Eq. 6) and the Hill relation (Eq. 9). Theorem 3 establishes that these nonlinearities are such as to conserve a dominant time constant in the disc-associated inactivation reactions; i.e., that these nonlinearities can serve as an appropriate “H” saturation function in theorem 3. It bears emphasis in this context that the nonlinearity represented by Eqs. 6 and 7 cannot be considered an “instantaneous saturating nonlinearity” of the sort often used in modeling photoresponses; quite the contrary, these latter equations act as filters in which 1/β(t) is a time- and intensity-dependent “time constant” (theorem 4).

Saturating responses in Ringer's over the intensity range from Φ ≅ 1,000–10,000 also obey Recovery Translation Invariance approximately, and thus we conclude that for such saturating responses the entire time course of E*(t) of rods in Ringer's is also to a good approximation a linear function of flash intensity, despite the changes in Ca²⁺ _i that necessarily occur. Theorems 5 and 7 show that it is reasonable to expect such linear behavior, provided declining Ca²⁺ _i acts on the lifetime or gain of a nondominant disc-associated intermediate, as previously proposed by Murnick and Lamb (1996) and Matthews (1996, 1997).

Generality of τ_c and Its Independence of Ca²⁺ _i

We have shown that a single time constant, τ_c, governs two major features of the photoresponse recovery to saturating flashes, the spacing of traces obeying Recovery Translation Invariance (Figs. 3–5) and the tail phase kinetics (Figs. 6 and 9). Moreover, to a very good approximation, τ_c is the same whether the responses are measured under calcium clamp or in Ringer's, in which Ca²⁺ _i is free to vary (Figs. 5 A and 10 D). These observations further strengthen the conclusion that the biochemical mechanism underlying τ_c is not sensitive to calcium (Lyubarsky et al., 1996). Because the recovery half-time to a saturating flash given in Ringer's is typically 5–7 s shorter than the recovery half-time to the same flash given in calcium clamping choline, it may seem surprising that the time constants of the tail phases of the recoveries in both solutions are equal. Theorem 6 defines a quantitative condition (Eq. 14) under which such equality will occur, and this condition is met by the parameters of the salamander rod (Table III).

Partitioning the Overall Recovery Speed-Up Produced by Changing Ca²⁺ _i

As just noted, recoveries to saturating flashes in Ringer's are typically sped up 5–7 s relative to those to the same flash obtained with Ca²⁺ _i maintained near its resting value (Fig. 5 A). Our results and analysis lead to the likely conclusion that ∼4–6 s of the total shift is due solely to the activation of cyclase by the decline in Ca²⁺ _i (Table IV). The residual shift not accounted for by cyclase activation is 1.5 s ± 0.9 s. This latter value is greater, though not significantly different from that (0.8 ± 0.2 s) obtained by Matthews (1997). Matthews (1997) rapidly jumped salamander rod outer segments into calcium-clamping solution before a saturating flash (Φ ≈ 11,000), and then restored them to Ringer's 1.7 s after the flash, while the photoresponse was still in saturation. Matthews (1997) measured the shift of the recovery of rods exposed to calcium-clamping (choline) solution, relative to the recovery of the control response the same flash delivered in Ringer's alone. Since the saturated phase of the response in choline continued after the return jump into Ringer's for an additional 5–6 s before circulating current recovery commenced, cyclase activity should have been equalized and maximal for the rod at the time recovery from saturation commenced, in both the control (Ringer's) and experimental (choline) conditions (see Figs. 7 and 8). Thus, the shift in recovery time courses was unlikely due to differential cyclase activation, and a calcium-sensitive step of transduction was uncovered. Matthews (1997) then went on to show that this calcium sensitivity decayed with a time constant of 0.5 s. The closeness of the time constant obtained, 0.5 s, to the value of the nondominant time constant, 0.48 ± 0.27 s, estimated from the analysis of responses in calcium clamp (Fig. 4; Table II) and in Ringer's (Fig. 12; Table IV), supports the hypothesis that the calcium sensitivity of one and the same nondominant mechanism underlies the residual shift between responses in Ringer's and choline not accounted for by cyclase activation.

It seems highly likely that the calcium-sensitive mechanism described by Matthews (1997) is the same as that characterized by Murnick and Lamb (1996). Calculations based on a simple model of the Murnick/Lamb effect (along the lines they discuss) show that if the gain of R* (with τ_R = τ_nd ≅ 0.5 s) is regulated by the decline in calcium, and if the lifetime of E* is τ_E = τ_c = 2 s, then one would expect this gain effect alone to produce a shift of 0.8 s in our experiments. In contrast, if E* were nondominant and its gain/lifetime were the target of the Murnick/Lamb effect, then the shift predicted is 2.7 s, nearly twice as large as the mean shift in our experiments not accounted for by cyclase activation. In other words, based on the time course and magnitude of the effect of Murnick and Lamb (1996), the decline in Ca²⁺ _i that accompanies a single saturating response of a dark-adapted salamander rod in Ringer's can be predicted to feed back on the activity of R* in such a way as to produce a leftward shift of ∼1 s of the recovery, relative to what the recovery would be were this effect not present. It seems then, that the combination of the 4–6-s shift effect due to cyclase activation (Table IV) and an ∼1-s shift due to the gain–control effect characterized by Murnick and Lamb (1996) and Matthews (1997) can provide a full account of the total 6.4 ± 1.2-s shift between recoveries of saturating responses in calcium-clamp and Ringer's.

Breakdown in E* Linearity and Its Significance for Identifying the Mechanism of the Dominant Time Constant

Pepperberg et al. (1992) argued that the mechanism responsible for the dominant time constant was R* inactivation; i.e., that τ_c = τ_R. The principal evidence they cited in favor of this identification was that the 10% point of the recovery phase of the salamander rod response in Ringer's translated on the time axis by approximately the same magnitude per geometric increment in flash intensities up to Φ = 10⁶ or more; in contrast to R*, the disc-associated cascade intermediates G* and E* would be expected to saturate at lower intensities. Their argument needs to be reevaluated in the light of our results and analysis, and other recent results. In Figs. 3–6, we have presented evidence that Recovery Translation Invariance fails for responses in calcium clamp for Φ ≥ 20,000. Our theoretical analysis shows that in the absence of RTI no unequivocal conclusion can be drawn about the existence of a unique dominant time constant. Thus, the principal argument in favor of the identification of τ_c with τ_R does not apply to photoresponses measured with Ca²⁺ _i clamped near its resting level.

If we reject the argument for the identification of τ_c as the lifetime of R* as being valid for responses obtained in choline, there remains little reason to accept the argument as valid for responses in Ringer's, particularly since, in the intensity regime Φ = 1,000–10,000 where RTI holds reasonably well in both Ringer's and choline (Fig. 3), τ_c has the same value (Fig. 10, A and B; Lyubarsky et al., 1996; see Fig. 9). Further reason to reject the argument of Pepperberg et al. (1992) is also provided in Fig. 10, where it is shown that the time constant of the tail phases of responses in Ringer's, as well as the time constant of decay of Δβ as measured by Hodgkin and Nunn (1988), also gets systematically longer for Φ > 10,000. Mindful that the principal argument in favor of identification of τ_c with the lifetime of R* activity is now in doubt, we now evaluate other evidence pertinent to the identification of the mechanisms underlying the dominant and nondominant time constants.

Biochemical Identities of the Intermediates Underlying the Dominant and Nondominant Constants

Support for identifying the simultaneous decay of the G*/E* complex (Fig. 1) as the mechanism underlying τ_c includes (a) that the decay of G*/E* activity measured under appropriate in vitro conditions has a lifetime approximately equal to the value of τ_c (Arshavsky and Bownds, 1992; Arshavksy et al., 1994; He et al., 1997), and (b) that the decay of G*/E* activity has been shown not to be calcium sensitive (Arshavsky et al., 1991). Another argument in favor of this identification can be made based on our data, and the finding that GTPase-activating factors or proteins, available in the rod outer segment in limited supply, are required for rapid hydrolysis of the terminal phosphate of G* = G_t -GTP. One such factor implicated in GTPase acceleration is the γ-subunit of the PDE (Arshavsky and Bownds, 1992; Arshavsky et al., 1994). Evidence for a second factor was presented by Angleson and Wensel (1994). He et al. (1997) have now identified this latter factor as a novel protein, RGS9, a member of the RGS family of GTPase-activating proteins (GAPs), have established its localization in rod outer segments, and have demonstrated that it can accelerate the G* GTPase rate constant to 1 s⁻¹ at room temperature. Calculations based on the estimated rate of activation of G* per R* in situ (Lamb and Pugh, 1992) suggest that exhaustion of either or both of these GAP factors should occur by Φ = 20,000 photoisomerizations/rod; a relative slow down of recovery kinetics should occur at greater intensities (Figs. 5 and 10). The identification of G*/E* decay as the mechanism of τ_c also finds support in the recent work of Sagoo and Lagnado (1997) on truncated, dialyzed salamander rod outer segments, who make the case that the slowest step in circulating current recovery in their preparation is G_tα-GTP terminal phosphate hydrolysis.

Several additional lines of argumentation suggest that the mechanism underlying the nondominant time constant is the decay of R* enzymatic activity. First, in vitro experiments have shown that R* activity is sensitive to calcium via an effect of the calcium-binding protein recoverin on rhodopsin kinase (Kawamura, 1993; Klenchin et al., 1995; Chen et al., 1995), though it remains moot whether the relatively high calcium sensitivity of this mechanism, K _1/2 = 1.5–3 μM, would produce much effect when Ca²⁺ _i declines from its resting level near 400 nM (Erickson et al., 1996). Second, independently of whether or not the calcium sensitivity of activation gain involves recoverin, there is substantial physiological evidence of an early activation intermediate that is calcium sensitive (Lagnado and Baylor, 1994; Pepperberg et al., 1994; Jones, 1995; Koutalos et al., 1995a ; Matthews, 1996; Murnick and Lamb, 1996; Gray-Keller and Detwiler, 1996; Matthews, 1997; Sagoo and Lagnado, 1997). The simplest reconciliation of this body of evidence with the insensitivity of τ_c to calcium is, as argued by Murnick and Lamb (1996) and Matthews (1997), that the lifetime and/or gain of the mechanism underlying the nondominant time constant is calcium sensitive. Theorem 7 embodies this conclusion.

In sum, it is natural to identify the primary decay of R* activity as the mechanism underlying the nondominant time constant, and the R* lifetime and/or catalytic gain as calcium sensitive, and G*/E* decay as the mechanism of τ_c. Nonetheless, we caution that these identifications remain tentative until a definitive experiment is performed in which the dominant time constant is shortened in situ by a biochemical manipulation highly specific for R* or G*/E* decay.

In the context of discussion of the biochemical identities of the mechanisms underlying the dominant and nondominant time constants, the question naturally arises, Why should the decay of R* activity be describable in terms of a single time constant, as assumed in Eq. 5? The simplest answer is this: while a first-order R* decay model certainly oversimplifies the well established biochemistry of R* inactivation by phosphorylation and arrestin binding, the conditions of the present investigation are not such as would be expected to reveal evidence for such “biochemical fine structure.” Specifically, the responses reported here were to flashes that produced at least 10 photoisomerizations, and typically a number of such responses were averaged. Only at the single-photon response level might the detailed (and possibly stochastic) character of R* decay manifest its structure, as it clearly has in the responses of rods of mice with mutations affecting R* inactivation biochemistry (Chen et al., 1995; Xu et al., 1997). On the other hand, a “slowdown” in R* decay kinetics could be responsible for the systematic failure of the two-time constant linear model of E*(t) above Φ = 20,000 (Figs. 4 and 5). For example, there could be an accumulation at such intensities of a relatively long lived but intrinsically low activity decay product of R*.

Parameters Governing Photoresponse Recoveries in Calcium Clamp and Ringer's

To gain perspective on the factors that govern the time course of recovery of the photoresponse, it is useful to compare analytical expressions for the dim-flash responses in calcium clamp and Ringer's. For calcium-clamp responses, theorem 4 (see Eq. A4.3) yields

19

while for dim-flash responses in Ringer's, theorem 6 gives

20

The parameters of Eqs. 19 and 20 have been identified in Tables I and III, respectively; g(s) is a second- order polynomial that arises in obtaining the perturbation solution of theorem 6 (see Eq. A6.8). Fig. 13 shows application of expressions closely relating Eqs. 19 and 20 to dim-flash responses of rods i and j.

Each of the expressions (Eqs. 19 and 20) has three terms; a fourth first-order term due to the filtering of the membrane time constant (τ_mem ≈ 20 ms) has been left out for simplicity. For the dim-flash responses to which Eqs. 19 and 20 apply, the effect of the membrane time constant can be incorporated along with an absolute transduction delay (t_eff ≈ 15 ms) by using the reduced time t′ = t − t′_eff, where t′_eff = t_eff + τ_mem ≅ 35 ms (Lamb and Pugh, 1992; Pugh and Lamb, 1993). While the terms corresponding to R* and E* decay are similar in Eqs. 19 and 20, the first-order decay term due to β_dark, the dark rate of cyclic GMP hydrolysis in calcium clamp in Eq. 19 is supplanted for responses in Ringer's by a second-order oscillating term, as previously noted by Hodgkin (1988). This oscillating term arises because of the negative feedback coupling between cGMP and Ca²⁺ _i through guanylyl cyclase. This latter term embodies the calcium-dependent cyclase activation and appears solely responsible for the threefold change in sensitivity and time-to-peak of the dim-flash response in calcium clamp and Ringer's, and likewise responsible for most of the 5–7-s shift in the recoveries of saturated responses (theorem 6).

In closing, we now consider briefly specific parameters governing the time course of inactivation whose values deserve note. In what follows, it should be kept in mind that τ_c and τ_nd are aliases for τ_R and τ_E, though the identifications remain uncertain, as discussed above.