INTRODUCTION

Top Abstract Introduction Methods Results Discussion References

Many G-protein receptor-coupled signal transduction systems comprise a reaction chain linking two or more enzymes; the G-protein cascade of the vertebrate rod is one of the most thoroughly investigated mechanisms of this class. Physiologically realistic models of the rod phototransduction G-protein cascade have been shown to provide quantitative accounts of the activation phases of the photoresponses of rods to flashes over many decades of intensity (Lamb and Pugh, 1992; Pugh and Lamb, 1993; Kraft et al., 1993; Breton et al., 1994; Hood and Birch, 1994; Cideciyan and Jacobson, 1996; Lyubarsky and Pugh, 1996; Smith and Lamb, 1997).

Accounts of the recovery phases of photoresponses have not yet progressed to the same degree as those of activation, despite a wealth of information available about biochemical mechanisms that inactivate or downregulate the different steps of the transduction cascade. Among the reasons for the slower progress in the development of a full account of photoresponse recoveries are the co-occurrence in situ of the various biochemical inactivation mechanisms, the high concentrations of reactants in situ (which cannot be achieved in vitro), and the complexity of the dynamic changes in Ca²⁺_i that accompany light responses and modulate the inactivation biochemistry.

Photoresponse recoveries of intact salamander rods to saturating flashes exhibit a striking kinetic feature that we believe provides a key for unlocking the door to understanding inactivation in situ: over an intensity range that can exceed 100 fold, rod response recoveries to saturating flashes translate on the time axis with a characteristic linear increment (_c) per e-fold increase in flash intensities. Such translatory behavior of photoresponses suggests that recovery is "dominated" by a single biochemical mechanism that inactivates exponentially with the time constant _c (Baylor et al., 1974; Adelson, 1982a, 1982b; Pepperberg et al., 1992).

In a previous investigation (Lyubarsky et al., 1996), we made an unexpected observation: salamander rod photoresponses to saturating flashes measured under conditions that maintain Ca²⁺_i near its resting level were delayed in their recovery by a constant amount of time (typically 5-7 s, depending on the individual rod) relative to those measured in Ringer's, over a substantial range of intensity. Thus, the "dominant time constant" (_c) was statistically the same, whether Ca²⁺_i was clamped to rest, or free to decline to a low level during the period of response saturation. The focus of that previous investigation was on characterizing the method of clamping Ca²⁺_i, and on measuring _c under Ca²⁺_i clamp and with Ca²⁺_i varying freely.

The theoretical goal of this investigation was to provide a rigorous foundation for the concept of a dominant time constant of inactivation, and for interpreting its apparent lack of calcium dependence in the presence of the large effect of declining Ca²⁺_i on overall recovery time. The empirical goals were to examine response recoveries for obedience to the law that we show to define a dominant time constant, and to analyze the contributions of different mechanisms underlying the speed-up of recoveries in Ringer's relative to those in calcium clamp. To achieve these goals, we have done the following. First, we have examined the complete form of the response recoveries in clamped Ca²⁺_i and in Ringer's, determining the extent to which the recoveries to saturating flashes are invariant in shape. Previous experimental protocols have precluded an examination of the complete form of the recoveries in clamped Ca²⁺_i over an adequately wide range of times and intensities. Second, based on the observation that the recoveries are invariant in form for saturating flashes producing up to ~10,000 photoisomerizations, we derive and illustrate several general theoretical results not previously formalized; these mathematical theorems provide a rigorous basis for interpreting results presented here and elsewhere by others. Third, we quantify the contributions of two non-mutually exclusive explanations of the 5-7-s time shift between recoveries to single saturating flashes in clamped Ca²⁺_i and Ringer's (see Fig. 1): (a) calcium-dependent guanylyl cyclase activation, as characterized by Hodgkin and Nunn (1988); (b) calcium-dependent gain-control, as described by Lagnado and Baylor (1994), Murnick and Lamb (1996), Gray-Keller and Detwiler (1996), and Matthews (1996, 1997).

View larger version (20K): [in this window] [in a new window]   Fig. 1.   A schematic representation of the rod transduction cascade. Table I identifies the variables and parameters. The notation is that used in previous papers (see for example Pugh and Lamb, 1993; Lyubarsky et al., 1996). The arrows at right point to sites in the cascade at which calcium is known or has been  hypothesized to affect photoresponse recoveries, based on results of biochemical and physiological experiments. These sites are (1) R* inactivation kinetics, via the calcium-binding protein recoverin; (2) R* catalytic gain; (3) guanylyl cyclase activity.

	METHODS

Top Abstract Introduction Methods Results Discussion References

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.

View larger version (45K): [in this window] [in a new window]   Fig. 4.   Averaged photoresponses of three rods (noisy traces) obtained under calcium clamp, fitted with a model (dotted traces) in which the disc-associated reactions are characterized as a linear cascade having two inactivation time constants (Eq. 5). The larger of the two time constants, c, was estimated initially from analysis of the recovery half-times as in Fig. 5 A (i.e., by application of theorem 1), with small variations (~5%) allowed to optimize the fittings. The lesser time constant nd was estimated from the fitting; its value was strongly constrained by the time to peak of the subsaturating responses, though also affected somewhat by the value of dark, as expected from theorem 5. The value of dark was varied between 0.8 and 1.2 to optimize the fittings: the final values were 1.1, 0.8, and 0.8 s1. The fittings were done with the Hill coefficient nH = 2 (left), and also with nH = 3 (right). Holding the value of nH at either 2 or 3 had negligible effect on the estimates of nd (as expected from theorems 2-3), or on the amplification constant, A (Table II). Such invariance of A is expected from previous work (Lamb and Pugh, 1992). Rods a and b are typical in their parameter values. In contrast, rod c was unusual in being about three times more light-sensitive (without having an unusually large value of A); however, the estimate of nd for this rod was about three times greater than the average. The "undershoot" of current after the responses of rod c was modeled by continual activation of cyclase at rate 0.017/nH s1 (Lyubarsky et al., 1996, Eq. 12). The unusual features of the rod suggest that it may have had a higher Ca2+i in Ringer's than the other rods.

View larger version (37K): [in this window] [in a new window]   Fig. 6.   Recovery templates and tail phase data obtained in choline of eight different rods; letter labels correspond to those used in Table II to identify the rods. (top) The noisy traces are the recovery templates of the rods obtained by averaging the responses in choline to saturating flashes up to  = 10,000, as illustrated in Fig. 2. The thicker gray curves lying behind the template traces are the theoretical recovery template forms generated with the model, as in Fig. 4; the parameter values characterizing these theoretical templates are reported in Table II (the value of nH used was 3). The dotted trace is a first-order exponential, fitted to the tail phase of the template data trace, beginning at the point (~0.2) marked with a filled circle; the exponentials were fitted with the simplex fitting algorithm in the MatLabTM software package. The values of the time constants tail for the exponentials are reported in Table II. (bottom) The data traces and fitted exponentials are replotted in semilog coordinates; the traces are truncated at a normalized amplitude of ~0.03-0.04, corresponding to an absolute magnitude of 0.3-0.4 pA (the amplitude of saturated responses under these conditions in choline is ~10-11 pA; see Fig. 2).

View larger version (16K): [in this window] [in a new window]   Fig. 5.   (A) Half-times of recovery for responses of rod a (circles) and rod b (squares) collected in choline (filled symbols) and in Ringer's (open symbols). Regression lines have been fitted to the choline data for flashes up to and including  = 10,000, and extrapolated (dotted lines); regression lines were fitted to the entire set of response half-times obtained in Ringer's. For rod a, the regression slopes (in unit of s per e-fold increase in intensity) are 2.2 and 2.3 for the Ringer's and choline data ( and , respectively); for rod b, the slopes are 2.1 and 2.3 ( and , respectively). The shift T0.5 between the choline and Ringer's recovery data is 7.7 s for the circles, and 7.0 s for the squares. (B) Choline recovery half-time data from eight rods for flashes up to  = 100,000. Linear regression lines as in the left panel were fitted to responses up to and including  = 10,000, and then extrapolated; the plotted points represent the residual deviations from the regression lines. All eight rods exhibit reliable deviations in the intensity range above  = 30,000. The downward triangles represent data of rod c.

View larger version (21K): [in this window] [in a new window]   Fig. 2.   Protocol used for measuring photoresponses in Ringer's and calcium-clamping choline solution. As illustrated in the inset at right, the inner segment of the rod is held in a suction pipette containing normal Ringer's, while the outer segment is fully exposed to a test solution, which is either Ringer's or isotonic choline containing 2.3 nM Ca2+. At the beginning of each cycle, the rod was exposed to the test flash, in this case producing  = 3,000 photoisomerizations; the rod was then exposed to a standard flash,  = 9,400, and 40 s later the outer segment was jumped into choline. At 10 s after the jump into choline, the test flash was again delivered and, after an appropriate period (which depended on the test flash intensity), given a standard saturating flash and returned to Ringer's. The junction current produced by the jump to choline has been subtracted from the raw records (see Lyubarsky et al., 1996, Fig. 1). The entire cycle was completed three times for flashes spanning the intensity range from  = 94 to 94,000. Over the 2.5-h time period required for the recording, the circulating current declined ~10-15%; the photocurrent traces were normalized before averaging for additional analysis. In addition, the circulating current recovery after the initial test flash in choline increased ~10% over the time course of recording from its magnitude at the time of the first flash, as indicated by the dashed line. Before averaging the photocurrents, a correction for this effect was applied by dividing the overall circulating current (at each time point) by the current course represented by the dashed line. (The line drawing of the rod was made from a videotape record of the experiment, obtained with infrared viewing equipment.)

View larger version (37K): [in this window] [in a new window]   Fig. 9.   (top) Recovery templates (unbroken traces) for responses in Ringer's. Here the template represents the averaged response to a flash producing either  = 4,700 or 9,400 (the number of responses averaged varied between 9 and 28 for different rods, depending on how many times it was possible to repeat the entire response family series). The tail phase of the template was fitted with an exponential, as in Fig. 6, from the point marked with the filled symbol. a-h correspond to labels used in Fig. 6 and in Table II. (bottom) The data and fitted exponentials of the top panel are shown in semilog format.

View larger version (26K): [in this window] [in a new window]   Fig. 8.   (top) Family of saturating responses obtained in Ringer's for rod a (see Table II); the point of 10% circulating current recovery on each trace is marked with a filled symbol. (bottom) Application of the analysis of Fig. 7 to the responses in the top panel: the unbroken traces give the estimates of (t) obtained from the model analysis applied to the responses of the rod to the same flash series in choline (Fig. 3). The dotted curves are the predicted time courses of (t). The curves are only calculated for t such that F(t)  0.1, with the filled symbol marking the value of associated with 10% circulating current recovery. The dotted line labeled max is an estimate of the highest possible rate constant of cGMP hydrolysis, computed as max = PDEtot sub, where PDEtot is the total number of catalytic subunits in the outer segment (Dumke et al., 1994) and sub is the hydrolytic rate constant of a single fully activated catalytic subunit in a well-stirred volume equal to that of the outer segment (Lamb and Pugh, 1992).

View larger version (17K): [in this window] [in a new window]   Fig. 7.   (top) This shows two photoresponses: the trace with open symbols attached is copied without alteration from Fig. 15 of Hodgkin and Nunn (1988); they obtained it as the response of a salamander rod to a flash estimated to yield  = 40,800. The second trace, with the filled symbol attached is from rod a (Table II) of this paper to a flash estimated to yield  = 30,000. The filled symbol indicates the point of 10% circulating current recovery. (bottom) Estimates of  and = /cGdark. The unbroken curves through the open symbols reproduce the estimates of the time course of these variables obtained by Hodgkin and Nunn (1988), based on the application of the IBMX jump method during the response of the rod of the top panel to the  = 40,800 flash at the points marked with the open circles. The unbroken trace labeled "" gives an estimate of (t) for the response of rod a to the  = 30,000 flash in the top panel; this estimate was obtained from the curve fitting analysis of Fig. 4. The dotted trace is the time course of (t) predicted from the relation (t) = (t)F(t)(1/nH), as described in the text. The purpose of reproducing the Hodgkin and Nunn (1988) data is to show how similar the estimates of and obtained here are to theirs. Note that we have reproduced the original figure scales of the Hodgkin and Nunn (1988) figure to the right of both panels.

View larger version (33K): [in this window] [in a new window]   Fig. 3.   Experimental results examining Recovery Translation Invariance for two rods. A and B show photoresponses collected from the rod of Fig. 2 (rod a); C and D show photoresponses collected from a second (rod b). In the upper part of each panel, the responses are shown translated on the time axis to coincide with the point of 50% recovery, which is indicated by a dotted vertical line; in the lower part of each panel a template recovery shape has been subtracted from each trace; the template was made by averaging the three responses in the midrange of intensities (940- 9,400) that have the most closely identical shapes. For rod a, the template curve was essentially identical to the responses to the flash  = 9,400; for rod b, the template curve was most closely similar to the responses to the flash  = 4,700. The responses of rod a are the averages of three individual responses to each intensity; those of the rod b to two flashes. The flashes delivered to rod a were 20 ms in duration; those to rod b were 10 ms. Full obedience to RTI (Eq. 1) requires not only that the recovery shapes be identical, but also that the spacing between the activation phases in the upper part of each panel be uniform.

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 ), obtained by multiplying the physically measured energy density (photons µm²) 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:  = 47, 150, 470, 1,500, 4,700, 1.5 × 10⁴, 4.7 × 10⁴ (10-ms flashes);  = 23, 94, 300, 940, 3,000, 9,400, 3 × 10⁴, 9.4 × 10⁴ (20-ms flashes); the  = 23 flash was not used in all experiments. In general, we avoided flashes of intensity lower than  = 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 Appendix 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^TM 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 Appendix ii.

	RESULTS

Top Abstract Introduction Methods Results Discussion References

The general framework and notation adopted for the variables and parameters describing the reactions of the rod G-protein cascade have been presented previously (Lamb and Pugh, 1992; Pugh and Lamb, 1993; Lyubarsky et al., 1996), and thus are summarized in an abbreviated manner in Table I and in Fig. 1.

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[,t] is the circulating current present at time t after a flash producing  photoisomerizations at t = 0. The interval (₀, _max) is the intensity range over which Eq. 1a holds, t₀ is time at which F begins to show recovery from saturation by the flash ₀, 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   ₀ photoisomerizations is scaled by a factor s  1, the response recovery at times greater than the fixed time t₀ 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 , 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  [,t]} of photoresponse recoveries satisfying Eq. 1 is said to obey Recovery Translation Invariance (RTI).¹

In Appendix 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 _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  [,t]} obeys RTI if and only if

(3)

where H(x) is a saturation function obeying H(x  ) = 0, H(0) = 1, and _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  (over the restricted intensity range), and which at appropriately long times decays with the time constant _c. Theorem 1 by no means states that the circulating current itself recovers with the time constant _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 _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 _c of the logarithmic function h(/₀) = _c ln (/₀) 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 _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.

Phosphodiesterase activity modeled as a linear system. The fact that rod photoresponse recoveries to saturating flashes obey RTI (Fig. 3) lends support to the hypothesis that during such recoveries the underlying process is being "dominated" by the first-order inactivation of a single molecular species. Based on general considerations about the established reactions of the transduction cascade (and specific considerations taken up below in presentation of the cGMP synthesis/hydrolysis reactions), it is reasonable to look to the reactions that occur at the disc surface for the identity of this molecular species. For mathematical purposes, we thus represent the disc-associated reactions of the transduction cascade as a linear system. Further support for this representation will be mentioned in the DISCUSSION.

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 ₁ < ₂ < . . . < _n. Then, at sufficiently long times, the reaction with the longest time constant, _n, always dominates: that is, given any small number , it is always possible to find a time T such that to within error of a term of order  

(4)

where e* = E*/,  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, 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, _R) and one for concurrent G*-E* decay (time constant, _E) (Fig. 1):

(5)

where _RP is the rate of generation of E* per fully active R*, and C_RE = [_{E R}/(_E  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 = _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 can be thought of as setting the value of t₀ 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 _R and _E estimated below, we find T _= 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 _c of Eq. 3, estimated from recovery half-time data, with the component of the impulse response e*(t) having the longest time constant, _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.

(6)

(7)

(8)

Theorem 3: Conservation of the Dominant Time Constant of Recovery

When  = _dark, a constant, the family of recovery curves {cG(,t)} generated by solving Eq. 8 for different saturating values of  obeys RTI. Thus, there exists a time t₀ such that for t > t₀ solutions to Eq. 6 for  = _dark are isomorphic, and translate on the time axis _c units for each e-fold increase in , where _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 . 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 _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 _dark 1/_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)

Responses in Ringer's: Ca²⁺_i free to vary An important goal of characterizing response recoveries in normal Ringer's solution is the determination of the way in which the decline in Ca²⁺_i that accompanies the light response affects the various cascade steps. At least two distinct sites of action of the decline of Ca²⁺_i have been described in previous physiological experiments (see Fig. 1): an increase of , the rate of cGMP synthesis (Hodgkin and Nunn, 1988; Kawamura and Murakami, 1989; Koutalos et al., 1995a); an apparent change in gain or amplification of an early transduction stage (Lagnado and Baylor, 1994; Pepperberg et al., 1994; Jones, 1995; Koutalos et al., 1995b; Matthews, 1996; Murnick and Lamb, 1996; Gray-Keller and Detwiler, 1996). Our goal in this section is to provide evidence and analysis that will help dissect the relative contributions of these two actions of Ca²⁺_i to the speeding up of the recoveries to saturating flashes in Ringer's, relative to the same flashes in calcium clamp.

Theorem 5: Recovery Translation Invariance in Ringer's

If a family {F[,t]} of photoresponse recoveries obtained under conditions that allow  to vary freely obeys RTI, then (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 (t)  RPCREet/c 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  = 40,800, along with the response in Ringer's of the rod of Fig. 2 to the flash producing  = 30,000; Fig. 7 (bottom) shows Hodgkin and Nunn's estimates of = /cG_dark and , 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 = /cG_dark that can be derived by combining Eqs. 6 and 9:

(10)

In their experiments, Hodgkin and Nunn estimated 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  with the steady state approximation of Eq. 10, which neglects the second term; i.e., they used the relation (t) = (t)/F(t)^(1/n_H⁾. In contrast to Hodgkin and Nunn's approach, we first estimated  by fitting the model to the responses of the rod obtained under Ca²⁺_i clamp (Fig. 4 A), and then derived (t). Thus, from the fitting we obtained (t) = e*(t)_sub + _dark (see Eq. 7), and we then computed (t) = (t)F(t)^(1/n _H⁾, which is plotted along with (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 (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 (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 (t) at any time after the point of 10% circulating current recovery.

The value of (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 _{F = 0.1} provides an estimate of _max. For the rod of Fig. 8, the average value of _{F = 0.1} estimated from the responses to the four highest intensities was 10.2 s¹. In Table II (rightmost column), we report the values of _{F = 0.1} obtained in this way for each rod. The average value _{F = 0.1} for these rods was 10.4 s¹; 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 ( 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  on Ca²⁺_i. If these equations provide an adequate characterization of the mechanisms governing Ca