§
From the * Department of Psychology, Department of Electrical Engineering, and § Institute of Neurological Sciences, University of
Pennsylvania, Philadelphia, Pennsylvania 19104; and
Institute of Cellular Biophysics, Puschino, Russia 142292
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ABSTRACT |
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The kinetics of the dark-adapted salamander rod photocurrent response to flashes producing from
10 to 105 photoisomerizations () were investigated in normal Ringer's solution, and in a choline solution that
clamps calcium near its resting level. For saturating intensities ranging from ~102 to 104
, the recovery phases of
the responses in choline were nearly invariant in form. Responses in Ringer's were similarly invariant for saturating intensities from ~103 to 104
. In both solutions, recoveries to flashes in these intensity ranges translated on
the time axis a constant amount (
c) per e-fold increment in flash intensity, and exhibited exponentially decaying
"tail phases" with time constant
c. The difference in recovery half-times for responses in choline and Ringer's to
the same saturating flash was 5-7 s. Above ~104
, recoveries in both solutions were systematically slower, and translation invariance broke down. Theoretical analysis of the translation-invariant responses established that
c
must represent the time constant of inactivation of the disc-associated cascade intermediate (R*, G*, or PDE*)
having the longest lifetime, and that the cGMP hydrolysis and cGMP-channel activation reactions are such as to
conserve this time constant. Theoretical analysis also demonstrated that the 5-7-s shift in recovery half-times between responses in Ringer's and in choline is largely (4-6 s) accounted for by the calcium-dependent activation of
guanylyl cyclase, with the residual (1-2 s) likely caused by an effect of calcium on an intermediate with a nondominant time constant. Analytical expressions for the dim-flash response in calcium clamp and Ringer's are derived,
and it is shown that the difference in the responses under the two conditions can be accounted for quantitatively
by cyclase activation. Application of these expressions yields an estimate of the calcium buffering capacity of the
rod at rest of ~20, much lower than previous estimates.
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INTRODUCTION |
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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
).
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METHODS |
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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.
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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 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:
= 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 MatLabTM 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.
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RESULTS |
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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:
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(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:
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(1b) |
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(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
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(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
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(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.
We assume then that E*(t), the number of phosphodiesterase catalytic subunits active in the outer segment at time t in response to a flash given at t = 0 is a linear function ofTheorem 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
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(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):
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(5) |
where RP is the rate of generation of E* per fully active
R*, and CRE = [
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 CRE. 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.
The reactions governing the hydrolysis and synthesis of cGMP in a rod outer segment after an isotropic flash can be written
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(6) |
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(7) |
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(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
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(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)
RPCREe
t/
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
=
/cGdark 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
=
/cGdark that can be derived by combining Eqs. 6 and 9:
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(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/nH).
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 s
1. 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 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
):
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(11) |
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(12) |
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(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 (
fCa F Jdark) and outward pumping by the exchanger (Jex). (Note that Jdark is an inward current, and
therefore a negative quantity, and that while Jex 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 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.
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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 Appendix 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
c of the dominant mechanism of the disc membrane-
associated reactions, providing the inequality
![]() |
(14) |
![]() |
(15) |
and > 0 and 1 >
> 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
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(16) |
where Tcyclase is the predicted shift, µ is given in Eq. 15, and
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(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 and
, 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/
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
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 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
tail over intensity might be obscured by averaging, as
we now explain.
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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
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
(t) =
(t)
dark obtained by Hodgkin and
Nunn (1988
; see Fig. 16) with their IBMX- and lithium-jump methods (see Fig. 8, above). Our estimates of
tail
and theirs of the time constant of decline of
(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 (
100-10,000, depending on the cell),
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
tail deserve attention.
The first and most salient deviation from the simple
ideal occurs at
10,000, where for most rods
tail becomes systematically much longer, increasing by as
much as twofold over the next 1-log unit range of intensities. This systematic lengthening of
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
< 100, where for a number of rods
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
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
= 10,000. The average
slopes of the lines fitted were 0.15 ± 0.12 s/log10(
) in
the middle intensity range and 1.2 ± 0.2 s/log10(
) 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
tail vs. log
curves occurs at ~
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
tail also have greater slopes.
Theorems 5 and 6 together lead to the conclusion
that c estimated from recovery half-time data and
tail
should be the same for each rod. Fig. 10 D compares
the
's estimated from the tail phase analyses with the
estimates of the dominant time constant
c for all the
rods of this study, as determined in Ringer's (open symbols) and in calcium clamp (closed symbols). The values
of
tail in this figure were obtained from the responses
to the conditioning flashes (
= 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
c and
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
c and the
response tail phases in the same manner. Interestingly,
the rods exhibiting the smallest values of
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 (T0.5),
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
T0.8 in Table IV. The averaged absolute fractional difference, |
T0.5
T0.8 |/
T0.5
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: nH,
the Hill coefficient of the cGMP-activated current, and
BCa,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
nH = 3. Nonetheless, recent experiments on truncated
salamander rods have yielded the estimate nH = 2 (Koutalos et al., 1995a
), and the value nH = 3 must be
called into question. According to Lagnado et al.
(1992, see Eq. 8), BCa in the salamander rod can be
generally expressed as
![]() |
(18) |
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|
|
|
where Cbuff is the concentration of high affinity buffer,
Kbuff 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 Cbuff = 240 µM, Kbuff = 0.7 µM, Ca = Cadark = 0.4 µM, B = 16; these values predict that in the salamander rod BCa,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 BCa,rest = 156 was
problematically high.
Fig. 11 serves to illustrate for rods a and b the problem with BCa,rest arising from the application of Eq. 18,
and shows how we obtained estimates of nH and BCa,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 Cbuff = 100 µM, a
value <1/2 the estimate Cbuff = 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 BCa,rest = 15 and 18, with nH = 2. Also
shown in Fig. 11 are the best fitting theoretical curves
that could be obtained with nH = 3; clearly these curves
fit the data less well than those computed with nH = 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 BCa,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 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
(t) at low flash intensities (see Fig.
10 A). In Table IV, we identify this value as
c.
Finally, with BCa,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 (T0.8) and that predicted by cyclase activation alone (
Tcyclase) 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.
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DISCUSSION |
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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
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/(t) is a time- and intensity-dependent "time constant"
(theorem 4).
Saturating responses in Ringer's over the intensity
range from
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 c and Its Independence of Ca2+i
We have shown that a single time constant, 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,
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
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 (
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
R =
nd
0.5 s) is regulated by the decline in calcium, and if the lifetime of E* is
E =
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
c =
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
= 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
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
c with
R does not apply
to photoresponses measured with Ca2+i clamped near
its resting level.
If we reject the argument for the identification of 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
= 1,000-10,000
where RTI holds reasonably well in both Ringer's and
choline (Fig. 3),
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
as measured by
Hodgkin and Nunn (1988)
, also gets systematically longer for
> 10,000. Mindful that the principal argument in favor of identification of
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
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
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* = Gt -GTP. One such factor implicated in GTPase acceleration is the
-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
= 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
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 Gt
-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, K1/2 = 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
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 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
= 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
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(19) |
while for dim-flash responses in Ringer's, theorem 6 gives
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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 (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 (teff
15 ms) by using the reduced time t
= t
t
eff, where t
eff = teff +
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
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 c and
nd are aliases for
R and
E, though
the identifications remain uncertain, as discussed
above.
c and
c.
To fit the response of each of the rods to
the dimmest flash used to stimulate it, the longer of
the two time constants
R or
E had to be on average
25% shorter than
c; that is, the average ratio
c /
c
was 0.75 ± 0.06 (n = 10). The tail phase analysis and
the data of Hodgkin and Nunn (1988)
presented in
Fig. 10 A are also consistent with the notion that the
longer of the two time constants of the disc-associated reactions is smaller at subsaturating intensities
than in the middle range of intensities, where RTI is
obeyed. Speculation about the apparent shortening of the longer time constant at low intensities seems
premature before the definitive identification of the
biochemical mechanism underlying
c. Nonetheless,
a hypothesis that bears consideration is that longitudinal inhomogeneities in cG, outer segment cGMP, and Ca2+i might play some role. Longitudinal variation in cG is likely present during the responses to
subsaturating flashes such as reported in Figs. 12 and
13, and such variation should produce systematic longitudinal variation in Ca2+i and guanylate cyclase
activity. At present, however, we are uncertain that
such variations would act to produce an apparent shortening of the longer time constant.
nd.
The values of the shorter time constant attributed to the inactivation of the disc-associated reactions
were generally consistent across conditions. Thus, comparing the estimates obtained from the analysis of responses measured in Ringer's and in calcium clamp
(Tables II and IV), we find
nd /
nd = 0.9 ± 0.2 (mean ± SD, n = 9), very close to unity. This ratio omits the
data of rod c, whose nondominant time constant was
unusually long. The very large nondominant time of
rod c (see Fig. 4 C) may have been caused by elevated
Cadark. Such an effect would be consistent with many
observations in the literature (e.g., Torre et al., 1986
;
Baylor and Lagnado, 1994; Sagoo and Lagnado, 1997
),
suggesting that the effective lifetime of R* may increase
with elevated Ca2+i, as mentioned above in the discussion of the biochemical identity of the mechanisms underlying
nd and
c.
nH.
During the activation phase, the Hill coefficient
is absorbed into the amplification constant, A (Lamb
and Pugh, 1992; Pugh and Lamb, 1993
). Since A must
be fixed to fit the rising phase data of any response
family, the value of nH used in the model has no effect
on the quality of the fitting of theory traces to the activation data. The value of nH, however, does affect the fitting of the recovery phases, both in Ringer's and in
choline. We have generally found that the responses in
choline are well fitted with a value of nH of 2.5-3 (Fig.
4; see also Lyubarsky et al., 1996
). However, the responses to low intensity flashes in Ringer's are clearly
better fitted with nH = 2 than 3 (Fig. 11). Given the estimate nH = 2 obtained by Koutalos et al. (1995a)
in experiments on truncated salamander rods, the best estimate for nH in Ringer's responses now has to be taken
as 2.
max, maximum guanylyl cyclase rate.
Estimates of
max
(Table II,
F = 0.1) derived from the combined analysis
of responses in choline and Ringer's (Figs. 8 and 9; Table II) were almost independent of whether nH was
chosen as 2 or 3. This follows from Eq. 10 because,
while
(t) is diminished by the ratio 2/3 (for any value
of
) to fit a particular response when nH is changed
from 3 to 2, the value of (0.1)(1/n H) almost perfectly
compensates. If the dark level of cGMP, cGdark, is taken
to be 2-3 µM, then
max =
max cGdark is predicted to be 20-30 µM s
1, which corresponds to the range of
values obtained by Koutalos et al. (1995a)
in their
study of truncated salamander rods. Another way
to look at
max is to consider the ratio
max/
dark =
max/
dark. Previously published biochemical results
reviewed by Pugh et al. (1997)
and new biochemical
data recently presented by Calvert et al. (1997) show
that
max/
dark in amphibian rods is ~10, when
dark is
taken to be the cyclase activity at ~400 nM Ca2+. Since
dark is ~1 s
1, we again arrive at the expectation
max
10 s
1. Three potential caveats need to be mentioned about our estimates of
max, however. The first,
which also applies to the method of Hodgkin and
Nunn (1988)
, arises because it is likely that at the point
of 10% recovery in Ringer's, Ca2+i may reach 20-30
nM, which should partially inhibit cyclase; correcting
for this effect would lead to a higher estimate of
max.
The second arises because of the gain effect characterized by Murnick and Lamb (1996)
, and by Matthews
(1997)
, which will act to diminish the magnitude of
in Ringer's relative to that estimated in choline by the
fitting analysis. Correcting for this effect would lead to
a diminution of our estimate of
max (Table IV); we estimate that this correction would not exceed 30%. A third caveat arises because of a possible decrease in the
K1/2 of the cGMP channels for cGMP during the saturated phase of the light response when Ca2+i is very
low. This shift in the K1/2 of the channels, effected by
calmodulin binding (Hsu and Molday, 1993
; Koutalos
and Yau, 1996
), if present at the point of 10% circulating current recovery, would also cause
max to be overestimated.
BCa,rest.
Perhaps the greatest surprise of the modeling
of the responses to low intensity flashes in Ringer's is
the estimate of the calcium buffering capacity at or
near rest, BCa,rest = 17.5 ± 7.2 (Table IV). The analysis
of Fig. 11 shows that BCa,rest must be far lower than that
predicted by the investigation of Lagnado et al. (1992),
though in fact our estimate corresponds to that, 17, which they obtained as the low affinity buffer capacity
"at high calcium." Theoretical curves such as those in
Figs. 11-13 are very sensitive to BCa,rest, which by retarding the change in Ca2+i increases the peak amplitude of
subsaturating responses, and also causes the "noselike"
behavior of the response immediately after the peak.
While BCa,rest is thus likely to be <20, both modeling efforts and previous work with calcium dyes indicates that
BCa is surely much higher when Ca2+i declines below its
resting level. Estimates of BCa at all levels of Ca2+i will be
crucial for the development of a complete account of response families in Ringer's.
![]() |
FOOTNOTES |
---|
Address correspondence to E.N. Pugh, Jr., Department of Psychology, University of Pennsylvania, 3815 Walnut Street, Philadelphia, PA 19104-6196. Fax: 215-573-3892; E-mail: pugh{at}psych.upenn.edu
Received for publication 23 July 1997 and accepted in revised form 29 October 1997.
1 Abbreviations used in this paper: PDE, phosphodiesterase; RTI, Recovery Translation Invariance. ![]() |
APPENDIX I |
---|
Proofs of Theorems
We begin with a result that makes it straightforward to prove theorem 1.
Lemma.
If a family of recovery functions {F[, t]}
obeys RTI (i.e., Eq. 1), then h(s) =
ln(s), where ln()
is the natural logarithm and
is a time unit.
Proof.
Suppose that RTI (Eq. 1) holds. Then, for any
0,
= s
0 for some s
1, and so we can write
![]() |
(A1.1) |
![]() |
(A1.2) |
![]() |
(A1.3) |
![]() |
(A1.4) |
![]() |
(A1.5) |
Theorem 1: Recovery Translation Invariance.
A family of
photoresponse recovery traces {F [, t]} obeys RTI if
and only if F [
,t] = H[
e
t/
],
0
max, t
t0
where H(x) is a saturation function obeying H(x
) = 0, H(0) = 1, and
a constant having the units of time.
Proof. Sufficiency of RTI. Suppose RTI (Eq. 1) holds. Then, we have
![]() |
(A1.6) |
![]() |
(A1.7) |
E*(t) Modeled as a Cascade of First-Order Exponentials
E*(t), the number of active catalytic subunits of phosphodiesterase in the outer segment at time t generated
by a brief flash at t = 0 has been modeled as a cascade
of reactions having first-order exponential inactivations. To present theorem 2 in a generalized form, we
now consider a system formed of a cascade of n reactions having first-order exponential decays; in linear
systems terminology (Jaeger, 1966), we consider a system that cascades n low pass filters. Each stage of such a
system has impulse response
![]() |
(A2.1) |
where subscript i refers to the ith filter (or ith stage), i
is the time constant of the ith filter, and Ci is the peak value of response of this stage to a Dirac delta function
impulse input. We now assume that the n time constants
are all different (i.e., nonrepeating), and without loss
of generality that they satisfy the following inequalities:
![]() |
(A2.2) |
Taking ai
i
1, the impulse response can be expressed (Jaeger, 1966
) as
![]() |
![]() |
(A2.3) |
where, C C1C2C3 . . . Cn, and * indicates the convolution operator. For this system, which we may call an n-d-LP
system (where "d" stands for "different"), the following
theorem holds:
Theorem 2: dominant time constant of a linear cascade.
Suppose that the impulse response e *(t) = E *(t)/ of
an enzymatic effector E* can be represented as an n-d-LP
linear cascade. Then at sufficiently long times the stage
with the longest time constant,
n always dominates.
Specifically, given any small number
where 0 <
<< 1, it is always possible to find time T
such that for t > T
,
the impulse response of the system is given approximately as
![]() |
(A2.4) |
![]() |
![]() |
Proof.
Since 1 <
2 <
3 < . . .<
n, it is clear that
a1 > a2 > a3 >. . . > an. In Eq. A2.3, there are n exponential functions with different time constants. If we
consider any two consecutive terms having time constants
i and
i + 1, we can find the time beyond which
the magnitude of the term with exp(
ait) is always less
than any given fraction
i + 1 of the magnitude of the term
with exp(-ai + 1t). If this time is denoted by Ti + 1, then
for t > Ti + 1, we have
![]() |
(A2.5) |
![]() |
(A2.6) |
![]() |
(A2.7) |
![]() |
(A2.8) |
![]() |
(A2.9) |
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. That is,
there exists a time t0 such that for t > t0 solutions of Eq. 7 conserve the dominant time constant
c of a set of linear reactions governing the rod transduction cascade
up to and including E *.
Proof.
An intuitive proof comes from consideration
of the differential Eq. 8: the time-dependent coefficient
of cG in the right-hand side includes the term e
t/
c,
which obeys RTI; i.e., solving Eq. 8 for a flash of intensity s
(s > 1) and initial time t0 is equivalent to solving
Eq. 8 for a flash of intensity
, after shifting the initial
condition to the right by the amount
c ln(s). Unfortunately, with this approach no single initial condition
applies to the whole family of response recoveries. A
more satisfactory proof requires care in dealing with
the initial condition.
![]() |
(A3.1) |
![]() |
(A3.2) |
![]() |
(A3.3) |
![]() |
(A3.4) |
![]() |
(A3.5) |
![]() |
(A3.6) |
![]() |
(A3.7) |
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
dim flashes), under calcium clamp the cGMP hydrolysis and synthesis reaction, 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.
Proof. Into Eq. 7,
![]() |
(7) |
![]() |
(A4.1) |
![]() |
(A4.2) |
![]() |
(A4.3) |
Theorem 5: Recovery Translation Invariance in Ringer's.
If a family {F[, t]} of photoresponse recovery curves
obtained under conditions that allow
to vary freely
obeys RTI, then
(t) itself must obey RTI and recover
from a flash in such a manner as to track the recovery
of the cGMP hydrolysis rate constant
(t).
Proof.
If the response family {F[, t]} obeys RTI, then
because Eq. 9 is such as to preserve RTI, the corresponding family of recovery curves {cG(
, t)} of cGMP
also obeys RTI. Thus, we focus attention on the general
solution to Eq. 9 over the intensity range and time period t > t0 when RTI is obeyed. Letting
= s
0, we begin with the analogue of Eq. A3.4:
![]() |
(A5.1) |
![]() |
(A5.2) |
Theorem 6: dim-flash responses and 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 c of the dominant mechanism of the disc membrane-associated reactions, providing the inequality
µ > 1/
c is satisfied, where µ is given by Eq. 15. Moreover, the effect of cyclase activation per se on the recovery in Ringer's from a saturating flash at long times, relative to its position in calcium clamp, is to shift the
curve to shorter time, by a time factor given by Eq. 16.
Proof. The framework of the theorem is provided by Eqs. 11-13, along with Eqs. 6 and 9; moreover, E*(t) is assumed to be a linear cascade, so that theorem 2 is in force. The first step in proving the theorem is the expansion of Eqs. 11-13 into perturbation approximations. To do this, we introduce the four perturbation variables
![]() |
![]() |
(A6.1) |
![]() |
(A6.2) |
![]() |
(A6.3) |
![]() |
(A6.4) |
![]() |
(A6.5) |
![]() |
![]() |
(A6.6) |
![]() |
(A6.7) |
![]() |
(A6.8) |
![]() |
(A6.9) |
![]() |
(A6.10) |
![]() |
![]() |
![]() |
(A6.11) |
![]() |
(A6.12) |
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 manifest in the recoveries of saturating photoresponses in Ringer's only as a shifting of the family of recoveries, with no change in the spacing on the time axis of the members of the family.
Proof.
Theorem 2 shows that in the absence of calcium feedback at adequately long times and for sufficiently intense flashes, e*(t) satisfies Eq. A2.4. The factors C i represent the "gains" of each of the steps involved, while i = 1/ai are the time constants of the
stages. Even with calcium feedback operative in Ringer's, the dominant time constant remains unchanged, so
that during the time period and for the intensities for
which RTI is obeyed, e*(t) = C
e
t/
c. Here the exact expression Eq. A2.4 becomes important, because the constant
![]() |
![]() |
(A6.13) |
![]() |
APPENDIX II |
---|
Considerations for Numerical Solutions
To solve the differential equations governing the cascade
under calcium clamp and in Ringer's, in addition to selection of the parameters, assumptions must be made
about initial conditions. In calcium clamp, the only initial condition (from Eq. 6) is that dark =
darkcGdark; this
condition must also be met for the solutions governing
the responses in Ringer's.
For responses in Ringer's, initial conditions dictated by
Eqs. 11-13 must also be met and these conditions must be
mutually consistent. We took the following approach.
(a) We fixed cGdark = 2 µM; (b) Since dark was varied
(between 0.8 and 1.2 s
1) to optimize the fittings, we set
dark =
dark cGdark = 2
dark (µM s
1); (c) The dark exchange current was calculated from Jex,dark = fCa Jdark (Eq. 12, dCa/dt = 0), and then the initial calcium concentration was computed from Eq. 11 as Cadark =
Kex/(1
),
where
= Jex,dark/Jex,sat. With the parameters listed in
Table III, this yielded Cadark = 385 nM, very near the estimates in the literature (reviewed in Pugh et al., 1997
).
(d) Finally, the maximum cyclase activity was calculated
from Eq. 13 as
![]() |
While the value of max is not required as an initial condition, it implicitly enters into the perturbation analysis
of the dim-flash response in Ringer's (Eq. A6.3).
The same initialization procedure was used for computing both numerical and analytical solutions (Eqs. 19
and 20). Numerical solutions to the coupled differential
Eqs. 6 and 12, combined with Eqs. 11 and 13 were computed with the fourth- and fifth-order Runge-Kutta routine
ode45 of the MatLabTM software package. Once the solution cG(t) was obtained, Eq. 9 was used to compute the
fraction of cGMP current present for responses in
Ringer's, while Eq. 10 of Lyubarsky et al. (1996) was used
for responses in choline. The normalized current response
was convolved with a first-order filter representing the
membrane time constant, 20-30 ms. For adequately low
intensity flashes (
< 5) and typical parameters, the numerical solutions agreed exactly with the analytical solutions, Eqs. 19 and 20; (compare Fig. 13, left and right).
We are grateful to A. Lyubarsky and L. Zhang for helpful criticisms.
Supported by National Institutes of Health grant EY-02660.
![]() |
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