\set{final}

\def\Author{Hamer}
\def\author{hamer}
\def\vol{6}
\def\year{2000}
\def\anum{36}
\def\pages{265-286}
\def\txt_title{Analysis of Ca++-dependent gain changes in PDE activation in vertebrate rod phototransduction}
\def\txt_authors{Russell D. Hamer}

\def\rcvd{8 August 2000}
\def\accept{19 December 2000}
\def\publ{31 December 2000}
\def\pdfsize{}
\def\PMID{}


\include{mvstyle.hsm}

\| External links

\| Internal defs

\def\dot{·}


\article{

\title{Analysis of Ca\sup{++}-dependent gain changes in PDE activation
in vertebrate rod phototransduction}

\authors{\mailto{russ@ski.org}{Russell D. Hamer}}

\institutions{Smith-Kettlewell Eye Research Institute, San Francisco,
CA}

\correspondence{Russell D. Hamer, Smith-Kettlewell Eye Research
Institute, 2318 Fillmore Street, San Francisco, CA, 94115; Phone: (415)
345-2056; email: \mailto{russ@ski.org}{russ@ski.org}}

\abstract

\abs_purpose{Recent biochemical and physiological data point to the
existence of one or more Ca\sup{++}-mediated feedback mechanisms
modulating gain at stages early in the vertebrate phototransduction
cascade, i.e., prior to activation of cGMP-phosphodiesterase (PDE). The
present study is a computational analysis that combines quantitative
optimization to key data with a qualitative evaluation of each candidate
model's ability to capture "signature" features of representative rod
responses obtained under a broad range of dark- (DA) and light-adapted
(LA) conditions. The primary data motivating the analyses were the
two-flash data of Murnick \and\ Lamb. These data exhibited strikingly
nonlinear behavior: the period of complete photocurrent saturation
(T\sub{sat}) in response to a Test flash was reduced substantially when
preceded by a less-intense saturating Pre-flash. Depending on the delay
between Pre- and Test flashes, the change in T\sub{sat} (\Delta
T\sub{sat}) could exceed the magnitude of the delay, and could be
reduced by as much as ~50%, corresponding to a large reduction in gain
by a factor of 10-15. The overall goal of the study was to evaluate what
model structure(s) were commensurate with both the Murnick \and\ Lamb
data and the salient qualitative features of rod responses obtained
under a broad range of DA and LA conditions.}

\abs_methods{Three candidate models were quantitatively optimized to the
Murnick \and\ Lamb saturated toad rod flash responses and,
simultaneously, to a set of sub-saturated flash responses. Using the
parameters from these optimizations, each candidate model was then used
to simulate a suite of DA and LA responses.}

\abs_results{The analyses showed that: (1) Within the context of a model
with Ca\sup{++} feedback onto rhodopsin (R*) lifetime (\tau\sub{R}), the
salient features of the Murnick \and\ Lamb data can only be accounted
for if the rate-limiting step is not the Ca\sup{++}-sensitive step in
the early cascade reactions, i.e., if PDE* lifetime, and not
\tau\sub{R}, is rate-limiting. (2) With \tau\sub{R} rate-limiting, the
model cannot account for \Delta T\sub{sat} exceeding the delay. (3) The
Ca\sup{++}-dependent reduction in \tau\sub{R} required to effect the
large gain is incommensurate with the empirical dynamics of dim-flash
responses. (4) Regardless of which reaction is rate-limiting, a model
using solely modulation of R* lifetime puts strong constraints on the
domain of biochemical parameters commensurate with the large gain
changes Murnick \and\ Lamb observed. (5) The analyses show that, in
principle, the Murnick \and\ Lamb data can be accounted for when
\tau\sub{R} is both rate-limiting and Ca\sup{++}-sensitive if, in
addition to the feedback onto \tau\sub{R}, there is an earlier, stronger
Ca\sup{++} feedback that does not affect R* inactivation kinetics (e.g.,
gain at R* activation or transducin (T*) activation). (6)
Ca\sup{++}-modulation of R* activation or T* activation as the sole
early gain mechanism can also account for the Murnick \and\ Lamb data,
but fails to predict the data of Matthews, and can thus be rejected
along with any model of comparable form.}

\abs_conclusions{The results imply that the Murnick \and\ Lamb data per
se are insufficient to rule out rate-limitation by
(Ca\sup{++}-sensitive) R* lifetime; evaluation of a broader set of
responses is required. The analyses illustrate the importance of
evaluating candidate models in relation to sets of data obtained under
the broadest possible range of DA and LA conditions. The analyses are
aided by the presence of reproducible signature, qualitative features in
the data since these tend to constrain the domain of acceptable model
structures and/or parameter sets. Some implications for vertebrate
photoreceptor light-adaptation are discussed.}


\introduction

\p{A recent paper by Murnick \and\ Lamb [1] presented physiological data
with striking nonlinear features. Using a two-flash technique, the
authors found that a saturating Pre-flash applied to toad rods
dramatically reduced the period of complete photocurrent saturation
(T\sub{sat}) elicited by a second, more intense, saturating Test flash.
The interpretation of the data was that the Pre-flash led to a
Ca\sup{++}-dependent reduction in gain early in the phototransduction
cascade. The effective gain reduction was substantial. T\sub{sat} for
the Test flash was reduced by as much as 6-7 s by the Pre-flash,
corresponding to an effective reduction in gain by a factor of 10-15,
depending on the slope of the T\sub{sat} versus ln(I) function. This
interpretation receives support from several lines of recent
experimental evidence that suggest that one or more steps in the
biochemical events leading to activation of cGMP-phosphodiesterase
(PDE*) are regulated dynamically by the level of internal calcium
[1-11].}

\p{The authors suggest that the observed decreases in Test flash
T\sub{sat} could result from Ca\sup{++}-sensitive gain modulation at an
early phototransduction step. They propose that the Ca\sup{++}-sensitive
process could be Ca\sup{++}-modulation of the rate of rhodopsin
(R*)-inactivation (R* phosphorylation). In this model, a decrease in
internal Ca\sup{++} pursuant to a flash of light (and cGMP-gated channel
closure) accelerates the process of phosphorylation of R* by releasing
rhodopsin kinase (RK) from inhibition by the Ca\sup{++}-binding protein,
recoverin (Rec) [4-9].}

\p{Murnick \and\ Lamb [1] propose that their data are consistent with
PDE*-inactivation being the rate-limiting step in photocurrent recovery,
not R* inactivation as proposed by Pepperberg et al. [12-14]. In this
model, R* lifetime would be significantly shorter than PDE* lifetime,
and would decrease further with light-induced decreases in internal
Ca\sup{++}, reducing the effective gain of photocurrent activation
without altering the overall dynamics of photocurrent recovery from
saturation controlled by the slower PDE*-recovery. The critical
observation supporting this interpretation of the data was that the
decrease in T\sub{sat} was found to exceed the delay, \Delta t, between
Pre- and Testflashes (e.g., for \Delta t = 1 s, T\sub{sat} was typically
reduced by 2 s). Murnick \and\ Lamb reason that within a RecRK model
structure, if the light-induced gain reduction is generated at the same
step that forms the rate-limiting reaction, this predicts a decrease in
T\sub{sat} that is, at most, equal to, but never greater than \Delta t.}

\subsection{Goals of the Present Study}

\p{The present article implements and evaluates the model structure (the
RecRK model) suggested by Murnick \and\ Lamb [1] to explain their data.
In addition, two other candidate models are analyzed. The first
alternative model has an early gain mechanism that does not alter the
dynamic of photocurrent recovery, i.e., Ca\sup{++}-modulation of
effective R* catalytic gain without a concomitant modulation of R*
lifetime (R* activation model). This corresponds to an implementation of
a scheme proposed by Lagnado \and\ Baylor [2] to explain their
observations that, during the period of light exposure, experimental
reduction of Ca\sup{++} caused a substantial decrease in transduction
gain, but no apparent change in response kinetics. The second
alternative model combines the two early gain mechanisms, R* lifetime
(RecRK) and R* activation (RecRK-R* activation model).}

\p{The models were evaluated in relation to the Murnick \and\ Lamb data,
as well as in relation to other key data obtained under a broad range of
stimulus conditions. The approach was similar to that used in Hamer
[15], i.e., a combination of quantitative optimization of the models to
one or more sets of data, combined with qualitative evaluation in which
the optimal parameters are used to predict signature qualitative
features of a suite of data sets from other experiments under both
dark-adapted (DA) and light-adapted (LA) conditions.}

\p{The overall goal of the study was to evaluate what model structure(s)
are commensurate with both the Murnick \and\ Lamb data and the broader
suite of representative DA and LA rod data. The analyses confirm that,
within the context of a RecRK model structure in which R* lifetime is
Ca\sup{++}-sensitive, PDE* lifetime indeed must be rate-limiting in
order to account for all the features of the Murnick \and\ Lamb data. In
addition, the analyses show that rate-limitation by R* lifetime in a
RecRK model imposes other fundamental constraints: with \tau\sub{R}
rate-limiting, the change in \tau\sub{R} required to reproduce both the
observed gain changes (10-15x) and the empirical intensity-dependence of
T\sub{sat} [1,12-14] is incommensurate with the dynamics of dim-flash
responses. Moreover, it is shown that, if the large gain change implied
by the Murnick \and\ Lamb data (10-15x) occurs entirely by means of
modulation of R* lifetime, then severe constraints are imposed on some
key biochemical parameters and that these constraints occur regardless
of which is the rate-limiting inactivation reaction, R* or PDE*
lifetime.}

\p{The analyses demonstrate that the Murnick \and\ Lamb data can be
accounted for by a model in which R* lifetime is both rate-limiting and
Ca\sup{++}-sensitive if, in addition to feedback via RecRK, an early,
stronger feedback is present. However, this additional feedback must be
such that it does not significantly alter the recovery kinetics of the
rate-limiting reaction. This result implies that the Murnick \and\ Lamb
data alone are not sufficient to unequivocally identify the
rate-limiting reaction in the early cGMP cascade.}

\p{Finally, the results show that some model structures can be ruled out
despite the fact that they can provide an excellent account of the
Murnick \and\ Lamb data, since they fail to account for robust
qualitative features of data from other experiments. The latter two
results highlight the importance of including as broad a range of data
as possible in model evaluation.}

\methods

\subsection{Analyses}

\p{The analyses were similar to those used in [15]. Candidate models
were quantitatively optimized (Optimization Toolbox, The Mathworks,
Natick, MA) to the Murnick \and\ Lamb [1] saturated toad rod flash
responses and, simultaneously, to a set of sub-saturated flash responses
from Rieke \and\ Baylor [16]. After the optimization, the optimal
parameters were then used to simulate a suite of dark- and light-adapted
empirical rod responses under a range of experimental conditions
(Empirical Response Suites I and II, described below). The adequacy of
each model was then evaluated based on quantitative (optimized) fits to
the data, and on the ability of the model to capture salient qualitative
("signature") features of the broader suite of responses.}

\p{Inclusion of the sub-saturated flash response data in the initial
optimization turned out to be crucial. Without it, the optimization was
not adequately constrained, so that the model could achieve a good fit
to the Murnick \and\ Lamb saturated data with parameter values that
failed to reproduce essential qualitative features of the Empirical
Response Suite, including dim-flash responses and step responses.
Inclusion of the sub-saturating responses constrained the optimization
such that the model was then more likely to be able to reproduce
signature features of the response suite.}

\subsection{Models}

\p{The candidate models each have four core sets of phototransduction
reactions, elements of which have been implemented in a number of
published models [15,17-20].}

\p{\b{(1) Simplified R-, PDE activation scheme (Eqs. 1-3)}. PDE
activation is modeled as two sequential first-order reactions [19-21].
In the scheme depicted by Eqs. 1-3, feedback from RecRK has been
included (q\sub{5}RK* in Eqs. 1,2), plus a slow back-reaction from
inactive R\sub{i} to R* (\tau\sub{b}, \tau\sub{Ri} in Eqs. 1,2; also see
Eqs. A6-A8, \appref{1}). RK* in Eq. 1 represents the amount of activated
rhodopsin kinase at time t. In Eq. 1, it acts to modulate the rate of R*
inactivation. The effect of lowered Ca\sup{++} (in response to light) is
to increase RK*, increasing the rate of R* inactivation. Both the RecRK
feedback reaction and the rhodopsin back-reaction are discussed in
\appref{1}.}

\p{In Eqs. 1 and 2, (1/\tau\sub{b}) and (1/\tau\sub{Ri}) are the
rate-constants for the back-reaction between R and R* and the
disappearance of inactivated rhodopsin. q\sub{5}RK* is a
pseudo-first-order rate constant for R* inactivation. \Phi\ is the
number of photoisomerizations elicited by a brief flash of light.}

\p{In Eq. 3, \nu\sub{rp} is the rate of activation of PDE per rhodopsin
molecule [20,21], and 1/\tau\sub{E} is the rate constant of
PDE*-inactivation (i.e., \tau\sub{E} is the time constant of
PDE*-inactivation).}

\ctr{\gifimage{1}{495}{69}{4}}

\ctr{\gifimage{2}{531}{75}{4}}

\ctr{\gifimage{3}{411}{72}{3}}

\p{\b{(2) Dynamic Ca\sup{++} reactions (Eqs. 4 and 5)}. Equations 4 and
5 describe Ca\sup{++}-influx through the light-sensitive cGMP-gated
membrane cation channels (first term, Eq. 4), efflux via the Na\sup{+}:
Ca\sup{++}, K\sup{+} electrogenic exchanger (with rate \gamma\sub{Ca};
second term, Eq. 4), and dynamic Ca\sup{++}-buffering (c\sub{b}, third
and fourth terms, Eq. 4 and Eq. 5; cf. [17] and [18]). In Eqs. 4 and 5,
c is the concentration of free internal Ca\sup{++} at time t, and
c\sub{b} is the concentration of Ca\sup{++} bound to buffer at time t.
In the first term in Eq. 4, J\sub{d} is the dark circulating current,
and F is the fraction of channels open at time t. The Ca\sup{++}-influx
through the channels is set by the factor
[F\sub{Ca}/(2F\sub{fd}V\sub{cyto})], which gives fraction of current
carried by Ca\sup{++} (F\sub{Ca}), converted to concentration units (by
the Faraday constant, F\sub{fd}, and the cell volume, V\sub{cyto}). The
Ca\sup{++}-buffer binds Ca\sup{++} with an on-rate of k\sub{1}, and an
off-rate of k\sub{2}. The parameter e\sub{T} is the total buffer
concentration.}

\ctr{\gifimage{4}{544}{113}{5}}

\ctr{\gifimage{5}{421}{67}{3}}

\p{\b{(3) Hydrolysis and synthesis of cGMP (Eq. 6)}. PDE* hydrolyzes
cGMP (second and third terms, Eq. 6), while Ca\sup{++}-modulates, in a
cooperative fashion (with Hill coefficient, n\sub{ca}), the synthesis of
cGMP by guanylate cyclase (A\sub{max}; first term, Eq. 6). The
hydrolysis terms in Eq. 6 include a light-activated term
(\beta\sub{sub}E*(t)), and a term to account for basal hydrolysis of
cGMP in the dark (\beta\sub{dark}; [17,18,20,21]). \beta\sub{sub}
converts E* from a unitless number (of molecules) to units of
concentration per unit time. K\sub{ca} is the concentration of
Ca\sup{++} that yields 1/2-maximal synthesis rate of cGMP (when c =
K\sub{Ca}, the first term in Eq. 6 = A\sub{max}/2).}

\ctr{\gifimage{6}{604}{96}{5}}

\p{\b{(4) Closure of membrane cation channels and generation of
photocurrent (Eq. 7)}. The photocurrent elicited by a light stimulus is
proportional to the number of membrane cation channels opened by the
cooperative action (with Hill coefficient n\sub{cg}) of free cGMP (g).}

\ctr{\gifimage{7}{370}{81}{3}}

\p{To this core set of reactions two forms of additional Ca\sup{++}
feedback have been added, first singly, then in combination.}

\p{RecRK Model: This model introduces Ca\sup{++}-modulation of R*
lifetime via RecRK (Eqs. 8,9). The model assumes a cooperative binding
(with Hill coefficient, w) of free internal Ca\sup{++} to Rec that
happens rapidly in relation to the time scale of the other reactions.
Hence, the interaction between Ca\sup{++} and Rec is treated as if it
was instantaneous and is quantified by the steady-state solution (Eq. 9)
to the differential equations (Eq. A11, \appref{1}). In Eqs. 8,9 below,
Rec* symbolizes Rec bound to w Ca\sup{++} ions (see \appref{1}, Eqs. A9,
A11, A12). K\sub{Rec,Ca} is the Ca\sup{++} concentration at which half
of Rec is bound to Ca\sup{++}.}

\p{Interaction between Rec* and RK (Eq. 8) is treated as a dynamic,
reversible reaction. RK* is activated rhodopsin kinase, i.e., RK that
has been released from inhibition by recoverin. The feedback gain
control is established by an increase in RK* corresponding to a speed-up
of the quenching of activated rhodopsin, i.e. an effective decrease in
the time constant of recovery, \tau\sub{R}. A decrease in Ca\sup{++}
pursuant to a flash of light (and cGMP-gated channel closure) decreases
the amount of bound Rec- Ca\sup{++}, which reduces Rec's inhibition of
RK, thus accelerating the process of phosphorylation of R* and ultimate
capping (quenching) by arrestin [4-9]. Eqs. 8-9 are derived in
\appref{1}, Section B.}

\ctr{\gifimage{8}{616}{68}{4}}

\ctr{\gifimage{9}{424}{95}{4}}

\p{The RecRK model is evaluated for two cases, \tau\sub{E} and
\tau\sub{R*} rate-limiting.}

\p{R* activation Model: This model replaces the RecRK Ca\sup{++}
feedback with a feedback that modulates the effective catalytic gain of
R* without altering its dynamics. This corresponds to an implementation
of a scheme proposed by Lagnado \and\ Baylor [2] to explain their
observations that during the period of light exposure, experimental
reduction of Ca\sup{++} causes a substantial decrease in transduction
gain, but no apparent change in response kinetics. Their results imply
an early gain mechanism that acts as if it reduced the effective light
intensity. The locus of action could be at the "activatability" of
rhodopsin, or in the catalytic activity of R* in activating transducin
and PDE. However, there is currently no known mechanism for this gain
effect, and Lagnado \and\ Baylor's results cannot distinguish between
action at R* activation or transducin activation.}

\p{Hence, in the present implementation of the Lagnado \and\ Baylor
scheme, activation of R* is treated as Ca\sup{++}-sensitive, such that a
reduction in internal Ca\sup{++} reduces the number of R molecules
activated by a flash. The Ca\sup{++}-effect is assumed to be rapid, and
thus the feedback (denominator of the first term in Eq. 10) is written
in the form of a Michaelis-Menten modulation of \Phi, the number of R*
generated by a brief flash. K\sub{r} in Eq. 10 is the K\sub{1/2} for the
Ca\sup{++}-effect on R* activation, with Hill coefficient n\sub{r}.}

\ctr{\gifimage{10}{564}{96}{4}}

\ctr{\gifimage{11}{500}{78}{4}}

\p{Here, 1/\tau\sub{R} is the first-order rate-constant for inactivation
of R*, replacing the pseudo-first order rate-constant, q\sub{5}RK*, of
Eqs. 1 and 2. The rest of the Eqs. in this model are as in Eqs. 3
through 7. Again, two cases are examined (\tau\sub{E}, and \tau\sub{R}
rate-limiting).}

\p{RecRK-R* activation Model: This model contains both of the additional
(non-cyclase) feedback mechanisms: modulation of R* inactivation rate
(q\sub{5}RK*, Eq. 12) and Ca\sup{++}-modulation of R* activation
(denominator of the first term in Eq. 12). Both of these feedback
reactions are embodied in Eq. 12.}

\ctr{\gifimage{12}{612}{96}{5}}

\p{All other Eqs. in this model are the same as in the RecRK model
(i.e., Eqs. 2 through 9).}

\p{Four cases are examined for the RecRK-R* activation model. For each
of the rate-limiting cases (\tau\sub{E} and \tau\sub{R} rate-limiting),
the analysis is carried out twice, first under the assumption that the
RecRK mechanism (modulation of R* lifetime) dominates the early gain
control, and then under the assumption that the R* activation mechanism
(modulation of R* activation gain) dominates the early gain.}

\subsection{Empirical Response Suite I}

\p{After optimization to the Murnick \and\ Lamb saturated two-flash data
and the sub-saturated flash responses, the models described above were
used to "predict" the responses to a suite of DA and LA stimulus
conditions obtained in other studies. These are shown in \figref{1} and
\figref{2}.}

\p{\figref{1}{A} shows sub-saturated flash responses obtained from toad
rods by Rieke \and\ Baylor [16]. Each model was simultaneously optimized
to a subset of these data (responses to the lowest four flash
intensities) and to the Murnick \and\ Lamb saturated data (shown in
\figref{3}).}

\p{\figref{1}{B} shows step responses obtained from newt rods by Forti
et al. [17]. These data have two qualitative, signature features that
have been observed in similar recordings from rods of other species
(e.g., salamander [22]; primate [18]). (1) A "nose" on the leading edge
of the response that recovers slowly to a steady-state level. The
relaxation to a steady-state presumably reflects the action of feedback
gain mechanisms. (2) A pronounced, multi-phasic response at step-offset,
exhibiting a fast recovery phase followed by a slow phase, with some
damped resonant behavior in between. The slow phase may reflect a slow
back-reaction from inactivated to activated rhodopsin [12,17].}

\p{\figref{1}{C} shows a summary of the intensity-dependence of
saturation period (T\sub{sat}) observed by Murnick \and\ Lamb. The
signature feature is the slope of the T\sub{sat} versus ln(I) function.
The thick red dashed line in \figref{1}{C} has a slope of 2.8 s/ln unit,
the slope for the cell presented in Figure 3 of Murnick \and\ Lamb. The
cell presented in their Figure 3 (whose data are analyzed in this paper)
only provided two data points on the T\sub{sat} function (filled red
squares). However, the T\sub{sat} slope for this cell closely matches
data from another rod presented in their Figure 1 (filled red circles),
and is close to the average of the T\sub{sat} slopes of seven cells
presented in their Table 1 (2.7 s/ln unit). Typically, the slope of
T\sub{sat} versus ln(I) observed in amphibian rods is 2-3 s [1,12-14].}

\p{\figref{1}{D} shows the decrease in flash sensitivity as a function
of background light intensity (LA flash sensitivity). The data are from
6 newt rods studied by Torre et al. [23]. A number of studies have shown
that the gain, as measured by the peak amplitude in response to a flash
on a background, decreases according to the Weber-Fechner relation over
several log units in rods [18,23-25], and over a larger range in cones
[26]. The dashed red curve is the Weber-Fechner relation fit to the
Torre et al. data [23]. The intensity that caused the incremental flash
sensitivity to decrease by a factor of two (I\sub{1/2}) was 100
R*s\sup{-1}. The signature feature of note is the relatively large
dynamic range over which flash sensitivity obeys the Weber-Fechner
relation. In this case, the Torre et al. [23] data obeys the
Weber-Fechner relation over a ~4 log unit intensity range.}

\subsection{Empirical Response Suite II}

\p{\figref{2} shows two additional elements of the Empirical Response
Suite. The left panel reproduces the results of a step-flash paradigm
used by Fain et al. (salamander rods; [27]). A saturating flash was
applied after presentation of a 7 s conditioning step of light of
increasing intensity (curves labeled 1, 2, 3). The flash response in the
absence of a conditioning step is shown by the curve labeled DA. The
signature feature to note is that the period of saturation (T\sub{sat})
in response to the flash decreases as the intensity of the conditioning
step increases.}

\p{The right panel in \figref{2}{B} shows the results of a Ca\sup{++}
clamp experiment by Matthews [2]. Tiger salamander rods were exposed to
a 0 Ca\sup{++}/0 Na\sup{+} test solution for brief periods around the
time of presentation of a super-saturating, 20 ms flash. The test
solution minimizes simultaneously the influx and efflux of Ca\sup{++},
thus opposing the light-induced fall in Ca\sup{++} [22,27-29]. Removal
of external Ca\sup{++} minimizes the influx of Ca\sup{++} through the
outer segment cation channels, while removal of external Na\sup{+}
prevents Ca\sup{++}-efflux through the Na\sup{+}: Ca\sup{++},K\sup{+}
exchanger.}

\p{Matthews applied the Ca\sup{++} clamp at four time periods around the
time of flash presentation (t=0): the test solution was applied either
from 1 s before to 1 s after the flash ("Pre \and\ Post" condition), or
from the time of the flash until 1 s after the flash ("Post"), or from 1
s before the flash until the time of the flash ("Pre"), or from 1 s
after the flash until 2 s after the flash ("Late"). The signature
feature is that the Ca\sup{++} clamp significantly prolonged the period
of saturation only when the Ca\sup{++} clamp was applied within a brief
time window around the time of the flash ("Pre \and\ Post" and "Post"),
but not if it was applied too early ("Pre") or too late ("Late").}

\p{The prolongation was interpreted as reflecting a gain increase
(relative to the gain with Ringer's solution) at a Ca\sup{++}-sensitive
step early in the phototransduction cascade. By parametric variation of
the timing of the application of the Ca\sup{++} clamp solution and
return to Ringer's solution, Matthews was able to map out a time course
for the Ca\sup{++}-sensitivity of the gain effect (roughly exponential,
with a time constant of ~0.5 s). Matthews proposed that the observed
time constant of the Ca\sup{++}-effect might correspond to RecRK
modulation of phosphorylation of R*.}

\results

\subsection{RecRK Model: \tau\sub{E} Rate-Limiting}

\subsection{The Murnick \and\ Lamb data can be accounted for with
PDE* lifetime rate-limiting.}

\p{With \tau\sub{E} rate-limiting, the RecRK model provides an excellent
account of the Murnick \and\ Lamb data. The panels in \figref{3}{A} show
the Murnick \and\ Lamb saturated two-flash data (in red) and the model
fits (in blue) for delays of 0, 1, 2, 3 s (left column of panels), and
4, 6, 8, and 10 s (right column of panels). The parameters for the model
are given in \tabref{2}, and a description of the variables and
parameters for all models is given in \tabref{1}.}

\p{In each panel of \figref{3}{A}, the time at which the Test flash was
presented is indicated by an inverted green triangle near the abscissa
(as this triangle moves to the right on the abscissa, the delay between
Pre- and Test flashes increases); the red star near the top of each
panel marks the time at which the response to the Test flash would have
emerged from saturation if the Pre-flash had not affected Test flash
saturation period. Note that the data remain in saturation for
progressively less time as delay increases (i.e., the time between the
saturated data and the red star increases as delay increases).}

\p{The plot of Test flash T\sub{sat} versus delay (derived from
\figref{3}{A}) is shown in \figref{3}{B}. The model responses are shown
as blue filled circles and solid blue curves, and the Murnick \and\ Lamb
data are shown as red filled squares with a red solid curve. The dashed
lines have a slope of -1, and are placed to pass through the first data
point (at delay = 0) for both the model results and the data. The data
fall below this line out to a delay of 6-7 s, illustrating the signature
feature that the change in T\sub{sat} can exceed the delay. The model
results capture this feature, and also fall below the corresponding blue
line of slope -1.}

\p{In addition, after optimization to the Murnick \and\ Lamb data and
the sub-saturating flash responses, the RecRK model (with \tau\sub{E}
rate-limiting), using the same set of optimized parameters, reproduces
all the salient qualitative features of the DA and LA responses in the
Empirical Response Suites shown in \figref{1} and \figref{2}. The fits
to the sub-saturated toad rod flash responses are shown in
\figref{4}{A}. The model is able to provide a reasonable account of the
four sub-saturating responses to which it was optimized (four smallest
responses in \figref{4}{A}) as well as to responses to five higher
intensities (to which the model was not optimized). \figref{4}{B} shows
the model step responses, which exhibit the two qualitative features
seen in the newt rod responses of \figref{1}{B}, namely the "nose" at
step onset and the two-phase response at step offset. The slow phase of
the step-offset response is due to the back-reaction between R and R* in
the model [17,23]. \figref{4}{C} shows the model T\sub{sat} versus ln(I)
function (blue solid curve) along with the empirical T\sub{sat} data
from Murnick \and\ Lamb (dashed red line and red data). The model
T\sub{sat} versus ln(I) function reproduces the same slope (~2.8 s/ln
unit) as the Murnick \and\ Lamb data. This slope is set by the
rate-limiting \tau\sub{E} (\tabref{2}).}

\p{The model also generates a substantial range of Weber's law LA flash
sensitivity (~3 log units; solid blue curve, \figref{4}{D}). For each of
a series of background adaptation levels (I\sub{b}), model LA flash
sensitivity was defined as the amplitude of the response to a flash of
fixed criterion intensity divided by the intensity of the criterion
flash. The criterion was the flash intensity eliciting a DA flash
response amplitude that was 10% of the full range of circulating
current. The dashed red curve is the Weber-Fechner relation from
\figref{1}{D}, shifted horizontally to fit the model output below a
"cutoff" background intensity (cutoff I\sub{b}), above which the model
was judged to deviate from Weber's law. The cutoff I\sub{b} is marked by
the black dashed vertical cursor in \figref{4}{D}, and its definition is
given in the legend to \figref{4}.}

\p{The model LA flash sensitivity obeys Weber's law over a significant
range (cutoff I\sub{b} = 8000 R* s\sup{-1}). At the cutoff I\sub{b}, 14%
of the model DA circulating current (F\sub{ss}) remains, as indicated by
the intersection of the vertical cursor line with the green solid curve.
The latter plots the fraction that the steady-state current is saturated
(i.e., 1 - F\sub{ss}(I\sub{b})), where F\sub{ss} is the steady-state
circulating current defined to be 1.0 in the dark, and zero when all
channels are closed. Also, at the cutoff I\sub{b}, the steady-state
internal Ca\sup{++} level (c\sub{ss} in inset) has dropped by a factor
of 5.1, from a dark value of 0.3 \mu M to 0.059 \mu M.}

\p{Also shown is the LA flash sensitivity of the model under two types
of simulated Ca\sup{++} clamp conditions: (1) LA flash sensitivity with
Ca\sup{++} clamped at its dark value (Ca\sub{dark}\sup{++} clamp; blue
dash-dot curve). Ca\sup{++} -feedback is fully disabled over the entire
dynamic range, with only static saturation contributing to flash
desensitization. Ca\sup{++} was fixed at its dark value in the model,
and I\sub{b} was adjusted to achieve the same steady-state current as in
the unclamped case, ensuring that the steady-state currents were at the
same level in relation to static saturation (i.e., cGMP-gated channel).
Differences in flash sensitivity then can be ascribed to the differing
states of Ca\sup{++} in the unclamped and clamped cases.}

\p{The Ca\sub{dark}\sup{++} clamp analysis equates steady-state current
levels (F\sub{ss}), but does not equate internal Ca\sup{++} levels at
the time of presentation of the flash. This is achieved in a second
analysis: (2) LA flash sensitivity with Ca\sup{++} clamped at the new
steady-state level reached in response to each I\sub{b}
(Ca\sub{ss}\sup{++} clamp; blue dashed curve). This approach equates the
F\sub{ss} (and hence equates the effect of channel saturation), and
equates Ca\sup{++} at the time of the flash. Thus, in comparing the
unclamped and the Ca\sub{ss}\sup{++}-clamped flash sensitivity, the LA
flash response in each case is affected equally by saturation and by the
steady-state level of Ca\sup{++}-mediated gain. The only additional
factor shaping the LA flash response in the unclamped case is the
dynamic Ca\sup{++}-mediated gain change evoked by the flash.}

\p{Note that at high I\sub{b} (I\sub{b}\gt\ cutoff I\sub{b}), the
unclamped model flash sensitivity falls more steeply than a Weber's law
slope of -1, and eventually follows a steep function that parallels the
high - I\sub{b} behavior of both Ca\sup{++} clamped curves. In fact, all
3 curves asymptote to a slope of -(n\sub{cg} + 1), which is predicted by
the instantaneous compressive saturation of the cGMP-gated channels
[30].}

\p{The analyses in \figref{4}{D} (which are recapitulated in all
subsequent similar figures) aid in seeing the magnitude of the
Ca\sup{++}-mediated adaptational gain control and the intensity range
over which it is exerted. They also help dissect the model sensitivity
losses due to dynamic gain mechanisms from those due to static (channel)
saturation of the model. For example, the Ca\sub{dark}\sup{++} clamp
analysis (blue dash-dot curve) shows vividly how much the two Ca\sup{++}
feedback mechanisms protect the rod from rapidly losing sensitivity (as
I\sub{b} increases) due to the static saturation of the cGMP-gated
channel. Moreover, both Ca\sup{++} clamp analyses show that, although
the dynamic Ca\sup{++}-mediated gain mechanisms lead to some absolute
loss of sensitivity (especially at low I\sub{b}), the rate of decrease
of sensitivity with I\sub{b} is greatly diminished by the gain
mechanisms, permitting Weberian desensitization over a large dynamic
range. Finally, it can be seen from the analyses that a significant
amount of cGMP-gated membrane channels remain open even at the cutoff
I\sub{b} where the model begins to fall off more sharply than Weber's
law (F\sub{ss} at the cutoff I\sub{b} is 0.14, and is close to this
value in all subsequent similar figures).}

\p{With the same parameters, the RecRK model with \tau\sub{E}
rate-limiting also reproduces the signature features of the Fain et al.
[27] data; the model saturated flash response emerges out of saturation
faster as the intensity of a conditioning step is increased
(\figref{5}{A}). Under simulated Ca\sup{++} clamp, the model also
reproduces the qualitative behavior of the Matthews [3] data
(\figref{5}{B}); the period of saturation (blue solid curve) is
prolonged (blue dashed curve) by the application of the Ca\sup{++} clamp
near the time of the flash ("Pre \and\ Post", "Post"), but not if the
flash occurs too early ("Pre") or too late ("Late").}

\subsection{RecRK Model: \tau\sub{R} Rate-Limiting}

\subsection{The rate-limiting step cannot also be the Ca\sup{++}-sensitive
step in a RecRK model.}

\p{The analyses show that R* lifetime cannot be rate-limiting in a model
in which RecRK is the only early gain mechanism. In simulating the data
with the RecRK model, the salient features of the Murnick \and\ Lamb
data can only be accounted for if PDE* lifetime, and not R* lifetime is
rate-limiting [1]. One critical failure of the model is that, with
\tau\sub{R} rate-limiting, model Testflash T\sub{sat} cannot exceed the
magnitude of the delay between Pre- and Test flashes. This is
illustrated in \figref{6}, which shows Test flash T\sub{sat} versus
delay for the model (blue) and the Murnick \and\ Lamb data (red). The
model results are always above the dashed blue line with slope -1,
depicting the fact that the change in model T\sub{sat} never equals or
exceeds the magnitude of the delay. This failure rules out a model
structure in which R* lifetime is rate-limiting and in which early
Ca\sup{++} feedback occurs solely at R* lifetime (via RecRK; [1]).}

\subsection{Rate-limitation by R* lifetime in a RecRK model is incommensurate
with the kinetics of the dim-flash response.}

\p{Other fundamental constraints mitigate against R* lifetime being
rate-limiting in a model with RecRK as the only early gain mechanism. In
order to achieve the gain change observed by Murnick \and\ Lamb
(10-15x), the time constant of R* inactivation must decrease by 10-15
times from its dark level to its minimum (\tau\sub{Rmin}) as Ca\sup{++}
approaches its physiological minimum (c\sub{min}, Eq. 4). Empirical data
from measurement of the intensity dependence of T\sub{sat} (T\sub{sat}
versus ln(I); [1,12-14,20]) place bounds on the value of the
rate-limiting recovery time constant, i.e., ~2-3 s. This means that when
Ca\sup{++} approaches c\sub{min}, \tau\sub{R} can fall to no less than
2-3 s to remain the rate-limiting reaction with a time constant
commensurate with empirical measures of T\sub{sat}. Thus, the dark
values of \tau\sub{R} (\tau\sub{Rdark}) must be greater than or equal to
20-30 s, forcing the dim-flash flash response to be anomalously
prolonged. This problem is illustrated in \figref{7}. The top panel
shows the relationship between \tau\sub{R} and the level of Ca\sup{++}
as it falls from a dark value of 0.3 \mu M to a minimum of 0.02 \mu M.
Over this range of Ca\sup{++}, \tau\sub{R} falls from 20 s to ~2 s. The
bottom panel of \figref{7} shows the resulting dim-flash response (blue
curve) in comparison to an empirical response (red). The model response
was the result of optimizing the RecRK model to the Murnick \and\ Lamb
data and to the sub-saturating flash responses with the RecRK parameters
set to generate the profile in the top panel of \figref{7}. The
resulting model dim-flash response can capture much of the early
response of the cell due to the influence of Ca\sup{++} feedback on
recovery, but it has a very prolonged recovery "shelf" reflecting the 20
s (DA) rhodopsin inactivation time constant. This prolonged response
profile is not seen in normal, healthy rod responses.}

\subsection{RecRK Model: General Considerations}

\subsection{The need for a large front-end Ca\sup{++}-mediated gain severely
constrains RecRK parameters.}

\p{If the full gain modulation underlying the Murnick \and\ Lamb data
(~10-15x) is assumed to be generated via modulation of \tau\sub{R}, this
puts strong constraints on some of the biochemical parameters of the
RecRK reactions. This is true regardless of which is the rate-limiting
reaction, R* or PDE* lifetime.}

\p{The model can help to illustrate this problem. \figref{8} shows how
the maximum effective "front-end" gain change depends on some of the key
biochemical parameters governing the interaction between Rec and
Ca\sup{++}, and between Rec* and RK (here Rec* symbolizes Rec bound to
wCa\sup{++} ions, the species that inhibits RK).}

\p{The ordinate in \figref{8} is the maximum, steady-state change in
effective R* Gain, defined as the calculated steady-state change in R*
lifetime when Ca\sup{++} falls from its dark level (0.3 \mu M in this
example; [31]) to its physiological minimum (0.02 \mu M in this example
[32-34]; \Delta Gain = \tau\sub{R,dark}/\tau\sub{R,Ca_min}). It can be
seen from Eq. 1 that any process that increases RK* effectively
increases the rate of inactivation of R* (decreases R* lifetime,
\tau\sub{R}). From Eq. 8 it can be seen that the steady-state level of
RK* will be set by the ratio of q\sub{4} to q\sub{3}, the dissociation
constant (K\sub{D} = q\sub{4}/q\sub{3}) for the interaction between RK
and Rec* (see \appref{1}, Eqs. A13 and A14). Hence K\sub{D} is plotted
on the abscissa of \figref{8}.}

\p{The level of RK* is also determined by the parameters governing the
interaction of Rec and Ca\sup{++}. For a given Ca\sup{++} level, the
amount by which RK* can change depends on the affinity of Rec for
Ca\sup{++} and the resultant amount of Rec* available. Hence, the
different curves in \figref{8} show the Gain versus K\sub{D} for
different values of K\sub{Rec,Ca}. The latter parameter is the
K\sub{1/2} for the interaction between Ca\sup{++} and Rec, i.e., the
Ca\sup{++} concentration at which 1/2 of the Rec is bound to Ca\sup{++}
(the state in which it inhibits RK, thus slowing R* inactivation). The
curves shown correspond to candidate K\sub{Rec,Ca} values ranging from
0.2 to 1.2 \mu M (from top to bottom, in 0.1 \mu M increments). The red
curve corresponds to a biochemical estimate for this parameter
(K\sub{Rec,Ca}=0.9 \mu M) obtained by Klenchin et al. [8]. The
horizontal dashed line occurs at a gain change of 10.}

\p{The curves in \figref{8} illustrate that K\sub{D} must be very small
if other RecRK and Ca\sup{++} parameters are set to values estimated in
the literature. For a value of K\sub{Rec,Ca} = 0.9 \mu M (the in vivo
estimate by Klenchin et al. [8]), K\sub{D} must be less than ~0.25 \mu M
in order to achieve a gain change of 10x or more (the gain change
implied by the large decreases in T\sub{sat} in the Murnick \and\ Lamb
data). This value for K\sub{D} is ~14 times less than the value
estimated by Klenchin et al. (~3.4 \mu M; [8]) in an in vitro study with
bovine tissue. K\sub{D} will be even more severely constrained (to even
smaller values) if it is assumed that the minimum physiological value
for Ca\sup{++} is greater than 0.02 \mu M.}

\p{Note that each of the curves is quite steep in the region of high
gain values (10x or more). This means that, not only is K\sub{D}
restricted to small values, but the overall range of (small) K\sub{D}
values that are permissible in order to achieve the empirically observed
gain changes is highly restricted.}

\p{If we knew a priori that the RecRK feedback only generated gain
changes on the order of 2-4x, a significantly broader range of values
for K\sub{D} would be permissible. However, in this case, to achieve the
an overall gain change of 10x or more, there would need to be an
additional feedback mechanism(s) contributing to the overall large gain
change, such as a gain affecting R* catalytic activity [2]. Two models
that include this kind of gain mechanism are examined in the following
sections.}

\subsection{R* activation Model: General Considerations}

\subsection{An R* activation model accounts for the Murnick \and\ Lamb data
regardless of what is the rate-limiting reaction}

\p{\b{\tau\sub{E} Rate-Limiting:} A model containing
Ca\sup{++}-modulation of R* activation as the sole early gain mechanism
(Eqs. 3-7, 10,11) can account for the Murnick \and\ Lamb data regardless
of whether \tau\sub{R} or \tau\sub{E} is rate-limiting. \figref{9} shows
the results for \tau\sub{E} rate-limiting. The left panels
(\figref{9}{A}) show the excellent fit of the model to the Murnick \and\
Lamb saturated two-flash data. The right panel (\figref{9}{B}) shows the
corresponding model Testflash T\sub{sat} versus delay along with the
Murnick \and\ Lamb data. As was seen in comparable results from the
RecRK model (\figref{3}{B}), the model is able to capture the critical
feature of the data, i.e., that the decrease model Test flash T\sub{sat}
in response to the Pre-flash can exceed the delay (blue curve falls
below dashed blue line with slope = -1).}

\p{\figref{10}{A} shows the model fit to the sub-saturating flash
responses, while \figref{10}{B-D} shows the model step responses,
T\sub{sat} versus ln(I), and LA flash sensitivity, respectively. Clearly
the model was able to achieve a reasonable fit to the Murnick \and\ Lamb
data, while providing a good account of the sub-saturating flash
responses. In addition, the model captures the salient features of the
other DA and LA responses, including a relatively large range of LA
flash sensitivity adhering to Weber's law (\figref{10}{B-D}).}

\p{\b{\tau\sub{R} Rate-Limiting:} When \tau\sub{R} is rate-limiting, the
R* activation model can achieve a good fit to the Murnick \and\ Lamb
data (\figref{11}), as well as to the sub-saturated flash responses
(\figref{12}{A}).}

\p{These results alone demonstrate that the Murnick \and\ Lamb results
can be accounted for by a model in which the rate-limiting step early in
the cascade is also Ca\sup{++}-sensitive, as long as Ca\sup{++} does not
act on the recovery time constant of the target reaction.}

\p{With the same parameters, the R* activation model (with \tau\sub{R}
rate-limiting) also reproduces the salient features of amphibian rod
step responses (\figref{12}{B}), the empirical T\sub{sat} versus ln(I)
function (\figref{12}{C}), as well as a large dynamic range of Weber's
law LA flash sensitivity (\figref{12}{D}). In addition, it reproduces
the qualitative behavior observed by Fain et al. [27] in their
step-flash paradigm, regardless of whether \tau\sub{E} (top panel of
\figref{13}{A}) or \tau\sub{R} (bottom panel \figref{13}{A}) is
rate-limiting.}

\subsection{A pure R* activation model can be ruled out}

\p{Despite its ability to account for the Murnick \and\ Lamb data and
other representative response profiles, the R* activation model can,
nevertheless, be ruled out because it fails to predict the Matthews [3]
Ca\sup{++} clamp data. This is shown in the right panels of
\figref{13}{B}. The top pair and bottom pair of panels are for
\tau\sub{E} and \tau\sub{R} rate-limiting cases, respectively. Note that
in both cases, the model does not predict a significant extension of
saturation period under Ca\sup{++} clamp conditions. This failure
implies that the R* activation model, and any similar model (with an
early Ca\sup{++} feedback up to and including PDE activation) that does
not alter the recovery kinetics of the target reaction, can be ruled
out.}

\subsection{RecRK-R* activation Model: Dominance by Gain at
R* activation}

\p{The RecRK-R* activation model combines both early feedbacks, i.e., at
R* activation and at R* lifetime (Eq. 12). As was seen with the R*
activation model, the analyses show that the Murnick \and\ Lamb data can
be accounted for when \tau\sub{R} is both rate-limiting and
Ca\sup{++}-sensitive if, in addition to the RecRK feedback, there is an
earlier, stronger Ca\sup{++} feedback (i.e., Ca\sup{++} feedback at R*
activation) that does not affect R* inactivation kinetics. This is shown
in \figref{14}, \figref{15}, and \figref{16}.}

\p{\b{\tau\sub{R} is rate-limiting:} \figref{14} shows results for the
case where \tau\sub{R} is rate-limiting. Since the results for
\tau\sub{E} rate-limiting were virtually the same, and since
rate-limitation by \tau\sub{R} is the more challenging of the two cases
(i.e., when \tau\sub{R} was rate-limiting, the RecRK model failed to
capture the Murnick \and\ Lamb data; \figref{6}), only the results for
\tau\sub{R} rate-limiting case will be shown.}

\p{The model provides a good overall fit to the Murnick \and\ Lamb data
(\figref{14}{A}). Moreover, the corresponding plot of Testflash
T\sub{sat} versus delay (\figref{14}{B}) shows that, in contrast to the
RecRK model (\figref{6}), with \tau\sub{R} rate-limiting, the model can
generate a decrease in Testflash T\sub{sat} that exceeds the magnitude
of the delay (model results fall below the blue dashed line with slope
-1).}

\p{In addition, the model fits the sub-saturated flash responses
(\figref{15}{A}), and accounts for the signature qualitative features of
step responses (\figref{15}{B}) and T\sub{sat} versus ln(I)
(\figref{15}{C}). The model also generates a relatively large dynamic
range of LA flash sensitivity adhering to the Weber-Fechner relation
(\figref{15}{D}). Finally, the model captures the qualitative features
of both the step-flash data of Fain et al. [27] (\figref{16}{A}), and
the Ca\sup{++} clamp data of Matthews [3] (\figref{16}{B}).}

\subsection{RecRK-R* activation Model: Dominance by Gain at R* lifetime}

\p{In contrast to the case where R* activation gain is dominant, if
Ca\sup{++}-modulation of R* lifetime is the dominant of the two early
gain mechanisms in the combined model, the large decreases in Testflash
T\sub{sat} observed by Murnick \and\ Lamb cannot be reproduced (as was
seen in the analyses of the RecRK model; \figref{6}). This is shown in
\figref{17} (Testflash T\sub{sat} versus delay). Unlike the data (red),
the model never falls below the dashed blue line with slope -1 because
the change in Testflash T\sub{sat} never exceeds the delay (compare with
\figref{3}{B} and \figref{14}{B}). Thus, if both R* activation and R*
lifetime (RecRK) gains are present, the R* lifetime gain cannot
dominate.}

\discussion

\p{The overall goal of the present study was to evaluate model
structure(s) that might be commensurate with both the highly nonlinear
data of Murnick \and\ Lamb and a suite of representative DA and LA rod
data. A standard model of phototransduction, with only one Ca\sup{++}
feedback locus at a late stage in the cascade (i.e., cGMP-synthesis),
cannot generate the primary observations of Murnick \and\ Lamb's
two-flash data [1] or the Fain et al. step-flash data [27]; no
combination of parameters will permit such a model to predict a
reduction in Testflash saturation period as a result of prior
stimulation, nor can such a model account for the Matthews [3]
Ca\sup{++} clamp data.}

\p{In order to account for the striking experimental observations of
Murnick \and\ Lamb [1], as well as those of Fain et al. [27] and
Matthews [3], at least one additional feedback mechanism is required.
Three candidate models were implemented and evaluated: feedback at R*
lifetime (RecRK model), feedback at R* activation with no change in R*
inactivation kinetics (R* activation model), and a combined model with
both early feedbacks (RecRK-R* activation model).}

\subsection{RecRK Model}

\subsection{\tau\sub{R} cannot be rate-limiting if it is the only early,
Ca\sup{++}-sensitive gain mechanism}

\p{Analysis of the RecRK model confirmed Murnick \and\ Lamb's [1]
prediction that when R* lifetime is Ca\sup{++}-sensitive, PDE* lifetime
must be rate-limiting in order to account for all the features of their
data (\figref{4}, \figref{5}, and \figref{6}). Ca\sup{++}-regulation of
a rate-limiting \tau\sub{R} fails to capture at least one crucial
feature of the Murnick \and\ Lamb data: the large change in T\sub{sat}
that can exceed the magnitude of the delay between Pre- and Test
flashes. This failure rules out a model in which Ca\sup{++}-mediated
modulation of a rate-limiting \tau\sub{R} is the single early gain
mechanism.}

\p{With PDE* lifetime rate-limiting (i.e., the dominant early recovery
reaction), the effective time constant of the non-dominant recovery
reaction (i.e., \tau\sub{R}) would be constrained to be less than ~1 s
in the dark, and would have to decrease by 10-15x as Ca\sup{++} level
dropped to its physiological minimum. Estimates of the non-dominant
recovery time constant with Ca\sup{++} clamped at its dark level have
been reported to be 0.4-0.5 s for amphibian rods [19,20]. This means
that \tau\sub{R} would have to fall to quite small values (40 to 50 ms)
during photocurrent saturation in order to effect the requisite gain
change.}

\p{The analyses also showed that rate-limitation by R* lifetime in a
RecRK model imposes other serious constraints. With \tau\sub{R}
rate-limiting, the change in \tau\sub{R} required to reproduce both the
observed gain changes (10-15x) and the empirical intensity-dependence of
T\sub{sat} (2-3 s/ln unit [1,12,14,22]) is incommensurate with the
dynamics of dim-flash responses (\figref{7}). In order to be compatible
with both empirical T\sub{sat} data and the Murnick \and\ Lamb data,
\tau\sub{R} would have to transition between 20-30 s in the dark to 2-3
s in complete photocurrent saturation (when Ca\sup{++} approaches
c\sub{min}), thus yielding an inordinately prolonged dim-flash response
(\figref{7}).}

\p{Finally, the analysis of the RecRK model showed that it is difficult
to account for the full gain change (~10-15x) implied by the Murnick
\and\ Lamb data by modulation of R* lifetime using parameters in the
range of current empirical estimates (\figref{8}). This is true
regardless of which is the rate-limiting recovery reaction (PDE*- or R*
lifetime).}

\subsection{R* activation Model}

\subsection{In principle, \tau\sub{R} can be rate-limiting and Ca\sup{++}-sensitive
if the gain occurs at R* activation.}

\p{The limitations of the RecRK model led us to examine an alternative
scheme suggested by Lagnado \and\ Baylor [2] (R* activation model;
described in Methods, Models). The results showed that when the
Ca\sup{++}-dependent gain change occurs at the locus of effective R*
catalytic gain, all the key features of the Murnick \and\ Lamb data can
be captured regardless of whether \tau\sub{E} or \tau\sub{R} is the
rate-limiting recovery time constant (\figref{8}, \figref{9},
\figref{10}, and \figref{11}). Moreover, the model provides a reasonable
account of several of the responses included in the Empirical Response
Suites. These results demonstrate that, in principle, the Murnick \and\
Lamb results can be explained by a model in which Ca\sup{++} feedback
modulates effective R* catalytic gain as long as the gain change does
not alter R* inactivation kinetics. This same conclusion applies to any
reaction prior to PDE activation, as long as the gain change does not
alter the inactivation kinetics of the target reaction.}

\subsection{The R* activation model can be rejected}

\p{Despite the success at capturing the Murnick \and\ Lamb data as well
as many of the salient features represented in the Empirical Response
Suites (\figref{9},\figref{10},\figref{11},\figref{12}, and
\figref{13}{A}), the R* activation model is unable to simulate one key
data set, i.e. the extension of saturation period observed by Matthews
[3] when changes in internal Ca\sup{++} are minimized for a brief period
around the time of the saturating flash (\figref{13}{B}). This failure
rules out this specific model, and a whole class of models in which the
feedback action of a Ca\sup{++}-decrease is to decrease the gain of any
stage up to and including PDE* activation without changing the effective
dynamics of the target reaction's inactivation.}

\p{Pugh et al. [35] also provide evidence that an R* activation model
can be rejected. If Ca\sup{++} feedback significantly decreased R*
catalytic gain, Pugh et al. reason that it would cause a decrease in the
slope of the early rising phase of LA flash responses. However, they
present evidence that the early rising phase of the photocurrent of LA
flash responses follows an invariant trajectory independent of
background intensity. However, several other studies have found evidence
that the early rising phase of the LA flash response is not invariant
with intensity [10,11,36,37].}

\p{If, indeed, the rising phase is invariant with background intensity,
this argues against the existence of any decrease in R* catalytic gain,
and thus argues against two of the three models analyzed in the present
study (either the pure R* activation model or the combined RecRK-R*
activation model). However, as discussed in the section above, the
remaining model (RecRK) has serious limitations in accounting for all
the data using available estimates for key biochemical parameters. This
suggests the need to implement other feedback mechanisms. Ca\sup{++}
feedback onto the affinity or Hill coefficient of the channel for cGMP
[38-41] have not been implemented in the present study, but these
mechanisms cannot generate the Murnick \and\ Lamb results. Hence, other,
as yet unknown mechanisms that can affect early transduction gain may be
needed.}

\subsection{Combined RecRK-R* activation Model}

\subsection{A Ca\sup{++}-sensitive \tau\sub{R} can be rate-limiting if gain at
R* activation is dominant}

\p{When both early Ca\sup{++} feedbacks are included (RecRK-R*
activation model), the model can account for the Murnick \and\ Lamb data
as well as many of the features of the empirical suite of responses
regardless of whether \tau\sub{E} or \tau\sub{R} is rate-limiting, as
long as Ca\sup{++}-modulation of R* activation is the dominant of the
two gain mechanisms (\figref{14}, \figref{15}, and \figref{16}). If the
gain at \tau\sub{R} is dominant, as was seen for the RecRK model
(\figref{6}), the large decreases in Testflash T\sub{sat} observed by
Murnick \and\ Lamb cannot be reproduced, thus eliminating this as a
viable scheme (\figref{17}).}

\subsection{General Considerations}

\subsection{Is PDE* lifetime or R* lifetime rate-limiting?}

\p{Overall, the models are able to capture a broader range of DA and LA
features with parameters in the range of empirical estimates when
PDE*-inactivation is the rate-limiting reaction. However, the analyses
of the R* activation and RecRK-R* activation models showed that, in
principle, the Murnick \and\ Lamb data alone are not sufficient to
unequivocally identify the rate-limiting reaction in the early cGMP
cascade. The present analyses demonstrated that the Murnick \and\ Lamb
data can be accounted for by a model in which R* lifetime is both
rate-limiting and Ca\sup{++}-sensitive if, in addition to feedback via
RecRK, an early, stronger feedback is present. The stronger feedback
must be such that it does not significantly alter the recovery kinetics
of the rate-limiting reaction. Hence, the question of whether
\tau\sub{R} [12-14] or \tau\sub{E} [1,3,19,20,42] is the rate-limiting
recovery time constant still remains open.}

\subsection{Role of some key biochemical parameters}

\p{In a recent article, Hamer [15] showed that two models (one based on
the Nikonov et al. [20] model which used an instantaneous
Ca\sup{++}-buffer, and one with a dynamic Ca\sup{++}-buffer) failed to
be able to capture a suite of rod responses when some key biochemical
parameters were held near their current best empirical estimates. One
key failure resulted when n\sub{Ca} (the Hill coefficient for Ca\sup{++}
feedback onto guanylate cyclase) was held to ~2.2 [43-46]. In this case,
the models were able to generate LA flash sensitivity adhering to
Weber's law over only a small intensity range. Only when n\sub{Ca} was
large (i.e., approaching 4) could the models generate a large dynamic
range of Weberian LA similar to that observed by Forti et al. [17].}

\p{This result (and other failures using "modern" parameters values)
implied that the models analyzed in Hamer [15] lacked one or more
important mechanisms. In particular, the failure to reproduce empirical
LA behavior using "modern" cyclase Ca\sup{++} feedback parameters
suggested the possibility that with additional Ca\sup{++} feedback, a
model might be able to reproduce the empirical LA behavior with
n\sub{Ca} held at or near its modern empirical estimate (~2.2).}

\p{The present work has analyzed models with additional feedback
mechanisms in place. One finding (not described in the Results) was that
with Ca\sup{++} feedback at R* lifetime in place (RecRK model), the
model was able to fit the Murnick \and\ Lamb [1] data (\figref{3}), the
sub-saturated flash responses (\figref{4}{A}), and all the other key
qualitative features of the Empirical Response Suites (\figref{4}{B,C}
and \figref{5}), including generation of a large range of LA flash
sensitivity adhering to Weber's law (\figref{4}{D}) using a value of
n\sub{Ca} = 2.5 instead of ~4 [15] (see \tabref{2}). Moreover, these
results were achieved with 10 other key biochemical parameters set to
within \pom\ ~60% (0.2 log units) of their modern empirical estimates
(\tabref{2}): Rec\sub{tot} = 35 \mu M (empirical estimate = 34 \mu M
[8]); RK\sub{tot} = 7 \mu M (~1/5 Rec\sub{tot} [8]); K\sub{Rec,Ca} =
0.53 \mu M (0.87 \mu M [8]); w = 2.5 (empirical estimate = 2 [8]);
\beta\sub{dark} = 0.5 s\sup{-1} (empirical estimate = 0.5 - 1 s\sup{-1}
[16,20,47,48], reviewed in [15]); K\sub{Ca} = 0.14 \mu M (empirical
estimate = 0.1 to 0.2 \mu M [43-45,49,50], reviewed in [15] and [51]);
n\sub{cg} = 2.59 (empirical estimate = 1.6-3 [24,52-56]); f\sub{Ca} =
0.3 (empirical estimate = 0.10 - 0.25 [17,22,28,57,58] Korenbrot,
personal communication); c\sub{dark} = 0.3 \mu M (0.2 - 0.7 \mu M
[31,34,59-62]); c\sub{min} = 0.02 \mu M (empirical estimate = ~0 - 0.05
\mu M [33,34,63-65]).}

\subsection{Some implications for vertebrate photoreceptor light adaptation}

\p{Weber's law and the value of n\sub{Ca}: This paper has evaluated
models having one or two Ca\sup{++} feedback mechanisms in addition to
the more well-studied feedback onto cGMP-synthesis (via
Ca\sup{++}-mediated stimulation of guanylate cyclase). Each model was
able to generate a relatively large range of LA flash sensitivity
adhering to Weber's law (see \figref{4}{D}, \figref{10}{D},
\figref{12}{D}, and \figref{15}{D}). These results are in contrast to
the results of an earlier analysis which found that the Nikonov et al.
model [20] was not able to generate a large range of Weberian LA and,
with the same set of parameters, capture the features of other DA and LA
responses [15]. Moreover, with the Hill coefficient for the cooperative
interaction between Ca\sup{++} and cyclase held to its modern empirical
estimate ~2 (n\sub{Ca} = ~2; [43-45]), the Nikonov et al. model [20]
generates only a modest range of Weber's law LA flash sensitivity.}

\p{Addition of a dynamic Ca\sup{++}-buffer to the Nikonov et al. model
[20] conferred the ability to capture a number of signature empirical
features and generate a significantly larger range of Weber's law LA
flash sensitivity [15]. However, the model had only the one Ca\sup{++}
-feedback (onto cyclase), and the extended range of LA behavior could
only be obtained when the Hill coefficient for cooperative action of
Ca\sup{++} onto cyclase activity (n\sub{Ca}) was high (~4).}

\p{The present results show that, with additional Ca\sup{++} feedback, a
large range of Weber's law LA flash sensitivity can be generated with a
relatively low cyclase Hill coefficient. For example, the combined
RecRK-R* activation model (\figref{14}, \figref{15}, and \figref{16})
captures the Murnick \and\ Lamb data as well as the array of signature
features of the suite of DA and LA empirical responses with a value of
2.38 for n\sub{Ca}, close to the modern estimates [43-45].}

\subsection{Effects of steady increases in Ca\sup{++} feedback on flash
sensitivity}

\p{Pugh et al. [35] point out that a steady increase in Ca\sup{++}
feedback (due to a steady I\sub{b}) extends the photoreceptor's
operating range by "protecting" the cell from sensitivity losses due to
the instantaneous saturation imposed by the cGMP-gated channels. In
addition, they claim that steady increases in Ca\sup{++} feedback will
either increase or decrease LA flash sensitivity depending on the
specific locus of the feedback.}

\p{For example, Pugh et al. [35] claim that a steady increase in cyclase
activity will \b{increase} flash sensitivity. The analyses of flash
sensitivity presented in Hamer [15] argue against this being a general
rule. In that paper, in which \b{the models had only one feedback at
guanylate cyclase}, the Ca\sub{dark}\sup{++}-clamped sensitivity was
higher than the unclamped sensitivity at low to moderate I\sub{b}, and
lower than the unclamped sensitivity at high I\sub{b}. This means that
the effect of steady increases in cyclase activity was to \b{decrease}
flash sensitivity over low to moderate I\sub{b}, and to increase flash
sensitivity at high I\sub{b}. In fact, the relative increase in
sensitivity conferred by the steady increase in cyclase activity could
be ascribed, in large part, to the extended operating range. Without
Ca\sup{++} feedback, the Ca\sub{dark}\sup{++}-clamped sensitivity fell
below the unclamped sensitivity due to static saturation of the
channels, i.e., when the steady circulating current in the clamped
condition began to approach saturation (\lt~20% circulating current
remaining).}

\p{The same pattern can be seen in the present analyses (\figref{4}{D},
\figref{10}{D}, \figref{12}{D}, and \figref{15}{D}). However, in the
present models, Ca\sup{++} acts at one or more of two additional loci
(RecRK or R* activation or both). Pugh et al. [35] point out that steady
increases in feedback at these loci will have the effect of
\b{decreasing} flash sensitivity. Hence, the lower sensitivity of the
unclamped function at low/moderate I\sub{b} could be due to the effect
of Ca\sup{++} feedback at RecRK and/or R* activation. Nevertheless, the
net combined effect on sensitivity of all the active Ca\sup{++}
feedbacks in the present models was to decrease sensitivity for all
I\sub{b} causing less than or equal to ~15% reduction in (unclamped)
circulating current, or ~75% reduction in current in the
Ca\sub{dark}\sup{++}-clamped condition.}

\p{In order to check the relative contribution of RecRK and/or R*
activation gain (versus feedback gain at cyclase), the models were run
with the RecRK and/or R* activation feedbacks disabled, (using the same
parameters as in \tabref{2}, \tabref{3}, and \tabref{4}), leaving only
the feedback at cyclase functional. The result was a small increase in
flash sensitivity at low to moderate I\sub{b} (as expected). Under these
conditions, the effect of a steady increase in cyclase activity was (as
in Hamer [15]) to \b{decrease} flash sensitivity for all I\sub{b}
causing less than or equal to ~70% reduction in current under
Ca\sub{dark}\sup{++} clamp. This result implies that the relatively
lower sensitivity under \b{unclamped} conditions at low/moderate
I\sub{b} was not due simply to a desensitization imposed by decreases in
\tau\sub{R} and/or R* activation gain. The desensitization must be due
to the feedback at cyclase. A more direct demonstration of this was
achieved by re-running the models while disabling Ca\sup{++} feedback
onto cyclase. This resulted in a virtual elimination of the difference
between the unclamped and Ca\sub{dark}\sup{++}-clamped sensitivities at
low/moderate I\sub{b}.}

\p{The above analyses show that steady-state increases in cyclase
activity do not necessarily lead to an increase in LA flash sensitivity
as claimed by Pugh et al. [35]. Light adapted flash sensitivity is more
complex, with sensitivity depending on the net balance between dynamic
sensitivity regulation mechanisms and static saturation
influences,including pigment bleaching. It is also important to note
that the balance between these is not necessarily a monotonic function
of background intensity.}

\subsection{Effects of transient increases in Ca\sup{++} feedback on flash
sensitivity}

\p{Pugh et al. [35] claim that when Ca\sup{++} is clamped at its new
steady-state level set by each background, the resulting flash
sensitivity function has approximately the same shape
(intensity-dependence) as the unclamped function. The analyses in the
present article and in Hamer [15] show that, strictly speaking, this is
not the case. The shape of the Ca\sub{ss}\sup{++} clamp (blue dashed
curves in \figref{4}{D}, \figref{10}{D}, \figref{12}{D}, and
\figref{15}{D}) is not the same as either the unclamped model responses
(solid blue curves) or the theoretical Weber-Fechner relation (red
dashed curves in same Figures); it falls off more quickly with
intensity. The only difference between the Ca\sub{ss}\sup{++}-clamped
and unclamped conditions is presence or absence of transient increases
in Ca\sup{++} feedback due to the flash. Hence, the model results serve
to illustrate that the effect of these relatively small transient
increases in Ca\sup{++} feedback can be important for light adaptation,
in that they tend to make the flash sensitivity fall off more slowly
with background intensity.}

\subsection{Importance of modeling a broad range of data}

\p{The results of Murnick \and\ Lamb are reminiscent of some earlier
data by Fain et al. [27] reproduced in the present article (\figref{2}).
Although the reduction in saturation period observed by Fain et al. [27]
may be explained by the same mechanism that reproduces the Murnick \and\
Lamb results, the Fain et al. data did not have features that could be
used to identify the rate-limiting and Ca\sup{++}-sensitive reactions
within the context of a model of the form of the RecRK model.}

\p{The limitations of the Fain et al.data, as well as the results of the
present analyses illustrate the importance of evaluating candidate
models in relation to sets of data obtained under the broadest possible
range of DA and LA conditions. A model may be able to generate an
excellent quantitative fit to one or more sets of data (e.g.,
\figref{9}, \figref{10}, and \figref{12}) and yet fail to capture even
qualitative features of other representative responses (e.g., failure of
the R* activation model to reproduce the Matthews data; \figref{13}{B}).
To the extent that the critical data are truly representative of the
cell's response repertoire, such failures can rule out the candidate
model.}

\p{Hence, the data to be simulated must be carefully chosen, with some
attention to reliability. Moreover, analyses are aided by the presence
of reproducible signature, qualitative features in the data since these
tend to constrain the domain of acceptable model structures and/or
parameter sets.}

\acknowledgements

\p{This work was supported by NEI Grant EY11513 and Smith-Kettlewell Eye
Research Institute Grant 0100-10-52.}

\p{I would like to thank Juan I. Korenbrot, Ph.D. and Norberto M.
Grzywacz, Ph.D. for their helpful comments about the physiological data
and analyses. In addition, I am grateful for the skillful assistance of
my programmer, Spero C. Nicholas, M.S. in carrying out the analyses, and
technical assistance of Robert D. Flint, III, B.S.E.}

\references

\p{1. Murnick JG, Lamb TD. Kinetics of desensitization induced by
saturating flashes in toad and salamander rods. J Physiol 1996;
495:1-13. \pubmed{8866347}}

\p{2. Lagnado L, Baylor DA. Calcium controls light-triggered formation
of catalytically active rhodopsin. Nature 1994; 367:273-7.
\pubmed{8121492}}

\p{3. Matthews HR. Actions of Ca2+ on an early stage in
phototransduction revealed by the dynamic fall in Ca2+ concentration
during the bright flash response. J Gen Physiol 1997; 109:141-6.
\pubmed{9041444}}

\p{4. Kawamura S, Murakami M. Calcium-dependent regulation of cyclic GMP
phosphodiesterase by a protein from frog retinal rods. Nature 1991;
349:420-3. \pubmed{1846944}}

\p{5. Kawamura S. Rhodopsin phosphorylation as a mechanism of cyclic GMP
phosphodiesterase regulation by S-modulin. Nature 1993; 362:855-7.
\pubmed{8386803}}

\p{6. Kawamura S, Hisatomi O, Kayada S, Tokunaga F, Kuo CH. Recoverin
has S-modulin activity in frog rods. J Biol Chem 1993; 268:14579-82.
\pubmed{8392055}}

\p{7. Chen CK, Inglese J, Lefkowitz RJ, Hurley JB. Ca(2+)-dependent
interaction of recoverin with rhodopsin kinase. J Biol Chem 1995;
270:18060-6. \pubmed{7629115}}

\p{8. Klenchin VA, Calvert PD, Bownds MD. Inhibition of rhodopsin kinase
by recoverin. Further evidence for a negative feedback system in
phototransduction. J Biol Chem 1995; 270:16147-52. \pubmed{7608179}}

\p{9. Calvert PD, Klenchin VA, Bownds MD. Rhodopsin kinase inhibition by
recoverin. Function of recoverin myristoylation. J Biol Chem 1995;
270:24127-9. \pubmed{7592614}}

\p{10. Jones GJ. Light adaptation and the rising phase of the flash
photocurrent of salamander retinal rods. J Physiol 1995; 487:441-51.
\pubmed{8558475}}

\p{11. Gray-Keller MP, Detwiler PB. Ca2+ dependence of dark- and
light-adapted flash responses in rod photoreceptors. Neuron 1996;
17:323-31. \pubmed{8780655}}

\p{12. Pepperberg DR, Cornwall MC, Kahlert M, Hofmann KP, Jin J, Jones
GJ, Ripps H. Light-dependent delay in the falling phase of the retinal
rod photoresponse. Vis Neurosci 1992; 8:9-18. \pubmed{1739680}}

\p{13. Pepperberg DR, Jin J, Jones GJ. Modulation of transduction gain
in light adaptation of retinal rods. Vis Neurosci 1994; 11:53-62.
\pubmed{8011583}}

\p{14. Pepperberg DR, Birch DG, Hofmann KP, Hood DC. Recovery kinetics
of human rod phototransduction inferred from the two-branched alpha-wave
saturation function. J Opt Soc Am A 1996; 13:586-600. \pubmed{8627416}}

\p{15. Hamer RD. Computational analysis of vertebrate phototransduction:
combined quantitative \and\ qualitative modeling of dark- and
light-adapted responses in amphibian rods. Vis Neurosci 2000;
17:679-699.}

\p{16. Rieke F, Baylor DA. Single photon detection by rod cells of the
retina. Reviews of Modern Physics 1998; 70:1027-36.}

\p{17. Forti S, Menini A, Rispoli G, Torre V. Kinetics of
phototransduction in retinal rods of the newt Triturus cristatus. J
Physiol 1989; 419:265-95. \pubmed{2621632}}

\p{18. Tamura T, Nakatani K, Yau KW. Calcium feedback and sensitivity
regulation in primate rods. J Gen Physiol 1991; 98:95-130.
\pubmed{1719127}}

\p{19. Lyubarsky A, Nikonov S, Pugh EN Jr. The kinetics of inactivation
of the rod phototransduction cascade with constant Ca2+i. J Gen Physiol
1996; 107:19-34. \pubmed{8741728}}

\p{20. Nikonov S, Engheta N, Pugh EN Jr. Kinetics of recovery of the
dark-adapted salamander rod photoresponse. J Gen Physiol 1998; 111:7-37.
\pubmed{9417132}}

\p{21. Lamb TD, Pugh EN Jr. A quantitative account of the activation
steps involved in phototransduction in amphibian photoreceptors. J
Physiol 1992; 449:719-58. \pubmed{1326052}}

\p{22. Nakatani K, Yau KW. Calcium and light adaptation in retinal rods
and cones. Nature 1988; 334:69-71. \pubmed{3386743}}

\p{23. Torre V, Forti S, Menini A, Campani M. Model of phototransduction
in retinal rods. In: The Brain: Cold Spring Harbor Symposia on
Quantitative Biology, Volume LV. Cold Spring Harbor (NY): Cold Spring
Harbor Laboratory Press; 1991. p. 563-73.}

\p{24. Koutalos Y, Nakatani K, Yau KW. The cGMP-phosphodiesterase and
its contribution to sensitivity regulation in retinal rods. J Gen
Physiol 1995; 106:891-921. \pubmed{8648297}}

\p{25. Koutalos Y, Nakatani K, Tamura T, Yau KW. Characterization of
guanylate cyclase activity in single retinal rod outer segments. J Gen
Physiol 1995; 106:863-90. \pubmed{8648296}}

\p{26. Burkhardt DA. Light adaptation and photopigment bleaching in cone
photoreceptors in situ in the retina of the turtle. J Neurosci 1994;
14:1091-105. \pubmed{8120614}}

\p{27. Fain GL, Lamb TD, Matthews HR, Murphy RL. Cytoplasmic calcium as
the messenger for light adaptation in salamander rods. J Physiol 1989;
416:215-43. \pubmed{2607449}}

\p{28. Yau KW, Nakatani K. Light-induced reduction of cytoplasmic free
calcium in retinal rod outer segment. Nature 1985; 313:579-82.
\pubmed{2578628}}

\p{29. Matthews HR, Murphy RL, Fain GL, Lamb TD. Photoreceptor light
adaptation is mediated by cytoplasmic calcium concentration. Nature
1988; 334:67-9. \pubmed{2455234}}

\p{30. Matthews HR, Fain GL, Murphy RL, Lamb TD. Light adaptation in
cone photoreceptors of the salamander: a role for cytoplasmic calcium. J
Physiol 1990; 420:447-69. \pubmed{2109062}}

\p{31. McCarthy ST, Younger JP, Owen WG. Dynamic, spatially nonuniform
calcium regulation in frog rods exposed to light. J Neurophysiol 1996;
76:1991-2004. \pubmed{8890309}}

\p{32. Sampath AP, Matthews HR, Cornwall MC, Bandarchi J, Fain GL.
Light-dependent changes in outer segment free-Ca2+ concentration in
salamander cone photoreceptors. J Gen Physiol 1999; 113:267-77.
\pubmed{9925824}}

\p{33. Gray-Keller MP, Detwiler PB. The calcium feedback signal in the
phototransduction cascade of vertebrate rods. Neuron 1994; 13:849-61.
\pubmed{7524559}}

\p{34. McCarthy ST, Younger JP, Owen WG. Free calcium concentrations in
bullfrog rods determined in the presence of multiple forms of Fura-2.
Biophys J 1994; 67:2076-89. \pubmed{7858145}}

\p{35. Pugh EN Jr, Nikonov S, Lamb TD. Molecular mechanisms of
vertebrate photoreceptor light adaptation. Curr Opin Neurobiol 1999;
9:410-8. \pubmed{10448166}}

\p{36. Detwiler PB, Gray-Keller MP. The mechanisms of vertebrate light
adaptation: speeded recovery versus slowed activation. Curr Opin
Neurobiol 1996; 6:440-4. \pubmed{8794098}}

\p{37. Rispoli G. Calcium regulation of phototransduction in vertebrate
rod outer segments. J Photochem Photobiol B 1998; 44:1-20.
\pubmed{9745724}}

\p{38. Rebrik TI, Korenbrot JI. In intact cone photoreceptors, a
Ca2+-dependent, diffusible factor modulates the cGMP-gated ion channels
differently than in rods. J Gen Physiol 1998; 112:537-48.
\pubmed{9806963}}

\p{39. Hackos DH, Korenbrot JI. Calcium modulation of ligand affinity in
the cyclic GMP-gated ion channels of cone photoreceptors. J Gen Physiol
1997; 110:515-28. \pubmed{9348324}}

\p{40. Hsu YT, Molday RS. Modulation of the cGMP-gated channel of rod
photoreceptor cells by calmodulin. Nature 1993; 361:76-9.
\pubmed{7678445}}

\p{41. Hsu YT, Molday RS. Interaction of calmodulin with the cyclic
GMP-gated channel of rod photoreceptor cells. Modulation of activity,
affinity purification, and localization. J Biol Chem 1994; 269:29765-70.
\pubmed{7525588}}

\p{42. Sagoo MS, Lagnado L. G-protein deactivation is rate-limiting for
shut-off of the phototransduction cascade. Nature 1997; 389:392-5.
\pubmed{9311782}}

\p{43. Gorczyca WA, Gray-Keller MP, Detwiler PB, Palczewski K.
Purification and physiological evaluation of a guanylate cyclase
activating protein from retinal rods. Proc Natl Acad Sci U S A 1994;
91:4014-8. \pubmed{7909609}}

\p{44. Dizhoor AM, Lowe DG, Olshevskaya EV, Laura RP, Hurley JB. The
human photoreceptor membrane guanylyl cyclase, RetGC, is present in
outer segments and is regulated by calcium and a soluble activator.
Neuron 1994; 12:1345-52. \pubmed{7912093}}

\p{45. Calvert PD, Ho TW, LeFebvre YM, Arshavsky VY. Onset of feedback
reactions underlying vertebrate rod photoreceptor light adaptation. J
Gen Physiol 1998; 111:39-51. \pubmed{9417133}}

\p{47. Hodgkin AL, Nunn BJ. Control of light-sensitive current in
salamander rods. J Physiol 1988; 403:439-71. \pubmed{2473195}}

\p{48. Cornwall MC, Matthews HR, Crouch RK, Fain GL. Bleached pigment
activates transduction in salamander cones. J Gen Physiol 1995;
106:543-57. \pubmed{8786347}}

\p{49. Koch KW, Stryer L. Highly cooperative feedback control of retinal
rod guanylate cyclase by calcium ions. Nature 1988; 334:64-6.
\pubmed{2455233}}

\p{50. Miller JL, Korenbrot JI. Differences in calcium homeostasis
between retinal rod and cone photoreceptors revealed by the effects of
voltage on the cGMP-gated conductance in intact cells. J Gen Physiol
1994; 104:909-40. \pubmed{7876828}}

\p{46. Dizhoor AM, Olshevskaya EV, Henzel WJ, Wong SC, Stults JT,
Ankoudinova I, Hurley JB. Cloning, sequencing, and expression of a
24-kDa Ca(2+)-binding protein activating photoreceptor guanylyl cyclase.
J Biol Chem 1995; 270:25200-6. \pubmed{7559656}}

\p{51. Pugh EN Jr, Duda T, Sitaramayya A, Sharma RK. Photoreceptor
guanylate cyclases: a review. Biosci Rep 1997; 17:429-73.
\pubmed{9419388}}

\p{52. Fesenko EE, Kolesnikov SS, Lyubarsky AL. Induction by cyclic GMP
of cationic conductance in plasma membrane of retinal rod outer segment.
Nature 1985; 313:310-3. \pubmed{2578616}}

\p{53. Haynes LW, Kay AR, Yau KW. Single cyclic GMP-activated channel
activity in excised patches of rod outer segment membrane. Nature 1986;
321:66-70. \pubmed{2422558}}

\p{54. Zimmerman AL, Baylor DA. Cyclic GMP-sensitive conductance of
retinal rods consists of aqueous pores. Nature 1986; 321:70-2.
\pubmed{2422559}}

\p{55. Watanabe S, Matthews G. Dose-response relation of cyclic
GMP-activated channels in the retinal rod photoreceptor. Neurosci Res
Suppl 1989; 10:S1-7. \pubmed{2480557}}

\p{56. Yau KW, Baylor DA. Cyclic GMP-activated conductance of retinal
photoreceptor cells. Annu Rev Neurosci 1989; 12:289-327.
\pubmed{2467600}}

\p{57. Lagnado L, Cervetto L, McNaughton PA. Calcium homeostasis in the
outer segments of retinal rods from the tiger salamander. J Physiol
1992; 455:111-42. \pubmed{1282928}}

\p{58. Menini A, Rispoli G, Torre V. The ionic selectivity of the
light-sensitive current in isolated rods of the tiger salamander. J
Physiol 1988; 402:279-300. \pubmed{2466983}}

\p{59. Ratto GM, Payne R, Owen WG, Tsien RY. The concentration of
cytosolic free calcium in vertebrate rod outer segments measured with
fura-2. J Neurosci 1988; 8:3240-6. \pubmed{2459322}}

\p{60. Korenbrot JI. Ca2+ flux in retinal rod and cone outer segments:
differences in Ca2+ selectivity of the cGMP-gated ion channels and Ca2+
clearance rates. Cell Calcium 1995; 18:285-300. \pubmed{8556768}}

\p{61. Miller JL, Korenbrot JI. In retinal cones, membrane
depolarization in darkness activates the cGMP-dependent conductance. A
model of Ca homeostasis and the regulation of guanylate cyclase. J Gen
Physiol 1993; 101:933-60. \pubmed{8101210}}

\p{62. Detwiler PB, Gray-Keller MP. Some unresolved issues in the
physiology and biochemistry of phototransduction. Curr Opin Neurobiol
1992; 2:433-8. \pubmed{1525539}}

\p{63. Sampath AP, Matthews HR, Cornwall MC, Fain GL. Bleached pigment
produces a maintained decrease in outer segment Ca2+ in salamander rods.
J Gen Physiol 1998; 111:53-64. \pubmed{9417134}}

\p{64. Younger JP, McCarthy ST, Owen WG. Weak adapting backgrounds
reduce the resting level of cytosolic free calcium in rod
photoreceptors. Invest Ophthalmol Vis Sci 1991; 32:907.}

\p{65. Younger JP, McCarthy ST, Owen WG. Light-dependent control of
calcium in intact rods of the bullfrog Rana catesbeiana. J Neurophysiol
1996; 75:354-66. \pubmed{8822563}}

\p{66. Cobbs WH, Pugh EN Jr. Kinetics and components of the flash
photocurrent of isolated retinal rods of the larval salamander,
Ambystoma tigrinum. J Physiol 1987; 394:529-72. \pubmed{2832596}}

\p{67. Baylor DA, Lamb TD, Yau KW. Responses of retinal rods to single
photons. J Physiol 1979; 288:613-34. \pubmed{112243}}

\p{68. Hamer RD, Tyler CW. Does the saturation period of vertebrate
photocurrent flash response reflect only rhodopsin lifetimes? Invest
Ophthalmol Vis Sci 1996; 37:S811.}

\p{69. Hamer RD. Computational modeling of the full flash response
dynamics of dark-adapted tiger salamander rod. Invest Ophthalmol Vis Sci
1998; 39:S955.}

\p{70. Hamer RD. How "signature" features in dark- \and\ light-adapted
photocurrent responses constrain models of vertebrate phototransduction.
Invest Ophthalmol Vis Sci 1999; 40:S208.}

\p{71. Cornwall MC, Fain GL. Bleached pigment activates transduction in
isolated rods of the salamander retina. J Physiol 1994; 480:261-79.
\pubmed{7532713}}

\p{72. Fain GL, Matthews HR, Cornwall MC. Dark adaptation in vertebrate
photoreceptors. Trends Neurosci 1996; 19:502-7. \pubmed{8931277}}

\p{73. Matthews HR, Cornwall MC, Fain GL. Persistent activation of
transducin by bleached rhodopsin in salamander rods. J Gen Physiol 1996;
108:557-63. \pubmed{8972393}}

\p{74. Bownds MD, Arshavsky VY. What are the mechanisms of photoreceptor
adaptation? Behav Brain Sci 1995; 18:415-424.}

\endreferences

\note_correction

}

\correction{

\p{28 February 2001:}

\p{In the paragraph immediately preceeding Equation 3:}

\block{In Eq. 3, n\sub{rp} is the rate of activation of...}

\p{"n" was changed to "\nu":}

\block{In Eq. 3, \nu\sub{rp} is the rate of activation of...}

\p{In the paragraph immediately preceeding Equation A3 of \appref{1}:}

\block{An ordinary differential equation for PDE; activation may be
derived from a simplified enzymatic [19,20]...}

\p{the word "reaction" was added:}

\block{An ordinary differential equation for PDE; activation may be
derived from a simplified enzymatic reaction [19,20]...}

\p{In the paragraph immediately preceeding Equation A5 of \appref{1}:}

\block{...may be approximated as a pseudo-first-order rate constant,
n\sub{rp}...}

\p{"n" was changed to "\nu":}

\block{...may be approximated as a pseudo-first-order rate constant,
\nu\sub{rp}...}

\p{In the paragraph immediately following Equation A11 of \appref{1}:}

\block{where Rectot = Rec* + Rec...}

\p{"tot" was made a subscript:}

\block{where Rec\sub{tot} = Rec* + Rec...}

\p{In the last paragraph of \appref{1}:}

\block{The quantity (q5RK*) in Eq. A23 corresponds to the R* decay time
constant, (1/tR) in Eqs. A2, A7-8.}

\p{"5" was made a subscript and "1/tR" was corrected to
"1/\tau\sub{R}":}

\block{The quantity (q\sub{5}RK*) in Eq. A23 corresponds to the R* decay
time constant, (1/\tau\sub{R}) in Eqs. A2, A7-8.}

\p{In the third paragraph of the caption for \figref{2}:}

\block{The signature qualitative feature to note is that in the "Pre
\and\ Post" and "Post" conditions in (but not in the "Pre" and "Late"
conditions)...}

\p{the word "in" was removed:}

\block{The signature qualitative feature to note is that in the "Pre
\and\ Post" and "Post" conditions (but not in the "Pre" and "Late"
conditions)...}

\p{In the third paragraph of the caption for \figref{4}:}

\block{including the highest I\sub{b} where the model slope still exceeded -1.}

\p{the equality was added to the condition:}

\block{including the highest I\sub{b} where the model slope was still \ge-1.}

\beginfigures

\p{In \figref{4}{D}, the solid blue curve largely obscured the solid red
curve. Also, the curve for the Ca\sub{ss}\sup{++} clamp is identified in
the caption as a "blue dashed curve," but appeared as a solid blue
curve. The image:}

\box{\gifimage{4z}{537}{717}{12}}

\p{was changed to:}

\box{\gifimage{4}{406}{446}{26}}

\p{In \figref{10}{D}, the solid blue curve largely obscured the solid
red curve. The image:}

\box{\gifimage{10z}{537}{717}{19}}

\p{was changed to:}

\box{\gifimage{10}{425}{443}{25}}

\p{In \figref{12}{D}, the solid blue curve largely obscured the solid
red curve. The image:}

\box{\gifimage{12z}{746}{540}{19}}

\p{was changed to:}

\box{\gifimage{12}{415}{435}{25}}

\p{In \figref{15}{D}, the solid blue curve largely obscured the solid
red curve. The image:}

\box{\gifimage{15z}{820}{594}{21}}

\p{was changed to:}

\box{\gifimage{15}{410}{436}{24}}

\begintables

\p{The time dependence of F, Rec*, and RK* should not have been shown in
the definition of the variables in \tabref{1}. The table:}

\box{\gifimage{1z}{571}{864}{38}}

\p{was changed to:}

\box{\gifimage{1}{571}{864}{38}}

\p{In the title of \tabref{4}:}

\block{Parameters for Analyses of Combined RecRK-R* activation Model
\tau\sub{R} Rate Limiting}

\p{a colon was added:}

\block{Parameters for Analyses of Combined RecRK-R* activation Model:
\tau\sub{R} Rate Limiting}

\beginbody

}

\appfile{1}{
\apptitle{1}{Equation Derivations}

\subsection{A. Derivation of PDE activation, including an R*\lt--R
Back-Reaction Without Ca\sup{++}-Feedback Onto R* lifetime}

\p{Rhodopsin activation and depletion are treated as first-order process
[17,19-21],}

\ctr{\gifimage{13}{282}{51}{3}}

\p{with a corresponding differential equation,}

\ctr{\gifimage{14}{382}{74}{3}}

\p{Here \Phi represents the number of photoisomerizations elicited by a
brief flash of light.}

\p{An ordinary differential equation for PDE; activation may be derived
from a simplified enzymatic reaction [19,20]:}

\ctr{\gifimage{15}{385}{48}{3}}

\p{The quantity E represents the number of molecules of a complex of
transducin (T) and PDE, [T\dot PDE] = E, which is activated
enzymatically in a single step by R* with rate K.}

\p{The corresponding differential equation describing the rate of change
of activated E* is given by Eq. A4}

\ctr{\gifimage{16}{450}{72}{4}}

\p{This scheme assumes that the depletion of E* is a first-order process
with rate constant 1/\tau\sub{E}. If it is also assumed that the above
reaction is not stoichiometrically limited by the amount of E available,
then the coefficient governing the accretion of E*, KE, may be
approximated as a pseudo-first-order rate constant, \nu\sub{rp},
yielding}

\ctr{\gifimage{17}{422}{74}{3}}

\p{The scheme depicted in Eqs. A1 through A5 is not, strictly speaking,
correct. The actual chemistry involves a number of additional steps,
including enzymatic activation of the G-protein, or transducin (T). The
full PDE activation process may consist of 6-8 "micro-steps" [21,66].
Nevertheless, over the past few years, it has been proposed that the PDE
activation process is well approximated by summarizing all the reactions
(the internal dynamics of which are, for the most part, unknown) as a
two-stage process.}

\subsection{Introduction of R*\lt--R\sub{i} back-reaction}

\p{It has been observed in several species that the response to a
prolonged step of light has multi-phasic dynamics at step-offset (early
fast recovery, followed by a much slower recovery with some occasional
resonant behavior in the transition; toad: [67]; salamander: [22,28];
monkey: [18]). One mechanism that can reproduce this behavior is the
inclusion of a back-reaction from R\sub{i} to R* [17]. Recently, Hamer
\and\ Tyler [68] and Hamer [15,69,70] incorporated this back-reaction in
computational phototransduction models and were able to capture a number
of salient features of amphibian rod responses, including the
acceleration of the T\sub{sat} versus ln(I) function at high intensities
[12] and the multi-phasic step-offset response. The differential
equations for accretion and deletion of R* and R\sub{i} may be derived
from the chemical scheme in Eqs. A6a through A6b (equivalent to scheme
by Forti et al. [17]). It is worth noting that this reaction scheme is
functionally equivalent to other schemes in which additional forms of
inactivated rhodopsin (apoprotein) may continue to activate PDE at a
slow rate [71-73].}

\ctr{\gifimage{18a}{356}{53}{3}}

\ctr{\gifimage{18b}{293}{51}{3}}

\p{These may be written in differential equation form as}

\ctr{\gifimage{19}{447}{72}{4}}

\ctr{\gifimage{20}{500}{78}{4}}

\subsection{B. Derivation of
Ca\sup{++}-Feedback Onto R* lifetime Via Rec/RK in the Presence of
an R*\lt--R Back-Reaction}

\p{The RecRK model adds Ca-sensitivity to the inactivation of R*. The
first reaction is the cooperative binding of Ca\sup{++} by Rec:}

\ctr{\gifimage{21}{505}{63}{4}}

\p{The complex [Rec\dot wCa\sup{++}] will be symbolized by Rec*.}

\p{The second reaction is a reversible disinhibition of RK to its active
form (RK*) by interaction with Rec*:}

\ctr{\gifimage{22}{472}{65}{4}}

\p{where the complex [Rec*\dot RK] is the inhibited form of RK. The
first reaction (Eq. A9) leads to a differential equation for Rec*:}

\ctr{\gifimage{23}{621}{97}{5}}

\p{where Rec\sub{tot} = Rec* + Rec, the total amount of Rec in the rod.
It was assumed that this reaction reached equilibrium rapidly, such that
the formation of Rec* follows the instantaneous level of internal
Ca\sup{++}. Setting Eq. A11 equal to zero and solving for Rec*, we get
the steady-state equation}

\ctr{\gifimage{24}{465}{94}{4}}

\p{where K\sub{Rec,Ca} = (q\sub{2}/q\sub{1})\sup{1/w}, the Ca\sup{++}
concentration at which half of Rec is bound to Ca\sup{++}.}

\p{The second reaction (Eq. A10), the disinhibition of RK, can be
written in differential equation form as}

\ctr{\gifimage{25}{649}{68}{5}}

\p{RK\sub{tot} is the total amount of RK in the cell, and (RK\sub{tot} -
RK*) is the amount of inactive RK.}

\p{Eq. A13 has the steady-state solution,}

\ctr{\gifimage{26}{421}{95}{4}}

\p{where KD = q4/q3, the concentration of Rec* corresponding to
1/2-activation of RK.}

\p{For simplicity, it was assumed that RK* leads to R* phosphorylation
(-inactivation) in two steps. Step 1 is the interaction between R* and
the disinhibited (activated) RK (=RK*) to form a complex, [R*\dot RK*]}

\ctr{\gifimage{27}{437}{46}{3}}

\p{Step 2 is incorporation of a number of phosphates (P) onto the
[R*\dot RK*] [74], yielding a new complex [nP\dot R*] and releasing free
RK*. The complex [nP\dot R*] is treated as inactivated rhodopsin
(R\sub{i})}

\ctr{\gifimage{28}{562}{64}{4}}

\p{Only Step 2 is assumed to be reversible, and q\sub{5} is assumed to
be rate-limiting in the two forward reactions.}

\p{Capping by arrestin, the final mechanism of R* inactivation, is not
explicitly modeled [74], but is assumed to be simultaneous with, and
stoichiometrically equivalent to the phosphorylation process.}

\p{Under these assumptions, the following differential equations
result:}

\ctr{\gifimage{29}{568}{100}{5}}

\p{where q'\sub{6} = q\sub{6}\sup{Pn}, and P\sup{n} is a constant.}

\p{Under the assumption that reaction A15 is rate-limiting,
phosphorylation will proceed at a rate governed by the instantaneous
level of the complex [RK*\dot R*], and a steady-state solution to Eq.
A17 may be used:}

\ctr{\gifimage{30}{545}{77}{4}}

\p{The differential Eqs. for R* and R\sub{i} dynamics may now be written
as:}

\ctr{\gifimage{31}{428}{70}{3}}

\ctr{\gifimage{32}{544}{70}{4}}

\p{Substituting Eq A18 into Eq A20, we get}

\ctr{\gifimage{33}{609}{101}{4}}

\p{In Eq. A19, the Ca\sup{++}-sensitivity is expressed in the modulation
of R* lifetime by RK*.}

\p{In addition to decay of R* to R\sub{i} (with rate-constant
1/\tau\sub{R}; Eq. A6a), depletion of R\sub{i} and the back-reaction to
R* from R\sub{i} are assumed to occur by separate pathways.}

\ctr{\gifimage{34}{355}{54}{3}}

\p{The back-reaction to R* proceeds with rate 1/\tau\sub{b}, and
R\sub{i} is depleted with rate 1/\tau\sub{Ri}. The resulting
differential equations for R* and R\sub{i} with Ca\sup{++}-modulation by
RK are analogous to Eqs. A7-8:}

\ctr{\gifimage{35}{527}{73}{4}}

\ctr{\gifimage{36}{567}{78}{4}}

\p{The quantity (q\sub{5}RK*) in Eq. A23 corresponds to the R* decay
time constant, (1/\tau\sub{R}) in Eqs. A2, A7-8. However, in Eq. A23,
the rate of decay of R* is a time-dependent function of Ca\sup{++}. In
the presence of light, Ca\sup{++} decreases, leading to a decrease in
the steady-state amount of Rec* (Eq. A12). A decrease in Rec* leads, in
turn, to an increase in the amount of RK* (Eq. A14), which translates to
an effective increase in the rate-constant governing the depletion of R*
(Eq. A23).}

}

\beginfigures

\figfile{1}{
\figtitle{1}{Empirical Response Suite I}

\p{\panel{A}: Sub-saturated flash responses obtained from toad rods
obtained by Rieke \and\ Baylor [16]. Each model was simultaneously
optimized to a subset of these data and to the Murnick \and\ Lamb
saturated data (shown in \figref{3}). The sub-saturating responses used
for optimization were the responses to the lowest four intensities:
0.10, 0.35, 1.29, and 4.35 photons/m\sup{2}/10 ms flash, or 2.26, 7.91,
29.15, and 98.3 R*/10 ms flash assuming an effective collecting area of
22.6 m\sup{2}.}

\p{\panel{B}: Step responses obtained from newt rods by Forti et al.
[17]. These data have two qualitative, signature features that have been
observed in step responses from rods of other species (e.g., salamander
[22]; primate [18]). (1) A "nose" on the leading edge of the response
that recovers slowly to a steady-state level. (2) A pronounced,
multi-phasic response at step-offset, exhibiting a fast recovery phase
followed by a slow phase, with some damped resonant behavior in
between.}

\p{\panel{C}: A summary of the intensity-dependence of saturation period
(T\sub{sat}) observed by Murnick \and\ Lamb. The signature feature is
the slope of the T\sub{sat} versus ln(I) function. The thick red dashed
line has a slope of 2.8 s/ln unit, the slope for the cell presented in
Figure 3 of Murnick \and\ Lamb. The cell presented in their Figure 3
(whose data are analyzed in this paper) only provided two data points on
the T\sub{sat} function (filled red squares). However, the T\sub{sat}
slope for this cell closely matches data from another rod presented in
their Figure 1 (filled red circles), and is close to the average of the
T\sub{sat} slopes of seven cells presented in their Table 1 (2.7 s/ln
unit).}

\p{\panel{D}: The decrease in flash sensitivity as a function of
background light intensity (LA flash sensitivity). The data (filled red
circles) are from 6 newt rods studied by Torre et al. [23]. The dashed
red curve is the Weber-Fechner relation fit to the Torre et al. data:
i.e., R/R\sub{dark} = (1 + I/I\sub{1/2})\sup{-1}. The intensity that
caused the incremental flash sensitivity to decrease by a factor of 2
(I\sub{1/2}) was 100R*s\sup{-1}. The signature feature of note is the
relatively large dynamic range over which flash sensitivity obeys the
Weber-Fechner relation. In this case, the Torre et al. [23] data obeys
the Weber-Fechner relation over a ~4 log unit intensity range.}

\ctr{\gifimage{1}{537}{717}{18}}

}

\figfile{2}{
\figtitle{2}{Empirical Response Suite II}

\p{\panel{A}: Results of the Fain et al. [27] step-flash paradigm. A
saturating flash was applied after presentation of a 7 s conditioning
step of light of increasing intensity (curves labeled 1, 2, and 3). The
flash response in the absence of a conditioning step is shown by the
curve labeled DA. The signature feature to note is that the period of
saturation in response to the flash decreases as the intensity of the
conditioning step increases.}

\p{\panel{B}: Results of a "Ca\sup{++} clamp" experiment by Matthews
[2]. Tiger salamander rods were exposed to a 0 Ca\sup{++}/0 Na\sup{+}
test solution for brief periods around the time of presentation of a
super-saturating, 20 ms flash. Removal of external Ca\sup{++} minimizes
the influx of Ca\sup{++} through the outer segment cation channels,
while removal of external Na\sup{+} prevents Ca\sup{++}-efflux through
the Na\sup{+}: Ca\sup{++}, K\sup{+} exchanger [22,27-29].}

\p{The Ca\sup{++} clamp was applied at one of four time periods: from 1
s before, to 1 s after the flash ("Pre \and\ Post" condition); from the
time of the flash until 1 s after the flash ("Post" condition); from 1 s
before the flash until the time of the flash ("Pre" condition), or from
1 s after the flash until 2 s after the flash ("Late" condition; green
trace above abscissa in each panel). The signature qualitative feature
to note is that in the "Pre \and\ Post" and "Post" conditions (but not
in the "Pre" and "Late" conditions), the period of photocurrent
saturation is significantly prolonged under Ca\sup{++} clamp (right-most
curves in each case).}

\ctr{\gifimage{2}{537}{717}{7}}

}

\figfile{3}{
\figtitle{3}{RecRK model accounts for Murnick \and\ Lamb data
with \tau\sub{E} rate-limiting}

\p{\panel{A}: Murnick \and\ Lamb saturated two-flash data (red) with the
model fits (blue) for delays of 0, 1, 2, 3 s (left column of panels),
and 4, 6, 8, and 10 s (right column of panels), where delay equals time
between saturating Pre-flash and a more intense saturating Test flash.
The parameters for the model are given in \tabref{2}.}

\p{In each panel, the time at which the Test flash was presented is
indicated by an inverted green triangle near the abscissa. The red star
near the top of each panel marks the time at which the response to the
Test flash would have emerged from saturation if the Pre-flash had had
no effect. Note that the data remain in saturation for progressively
less time as delay increases (the time between the saturated data and
the red star increases as delay increases). The model fits the Murnick
\and\ Lamb data quite well.}

\p{\panel{B}: Plot of Test flash T\sub{sat} versus delay for both the
data and model (derived from \panel{A}). T\sub{sat} was defined as the
time between application of the Test flash and the time at which the
response first recovered from saturation, i.e., fell to less than 90%
saturation. The model responses are shown as blue filled circles and the
solid blue curve, and the Murnick \and\ Lamb data are shown as red
filled squares with a red solid curve. The dashed lines have a slope of
-1, and are placed to pass through the first data point (at delay = 0)
for both the model results and the data. The data fall below this line
out to a delay of 6-7 s, illustrating the signature feature that the
change in T\sub{sat} can exceed the delay. The model results capture
this feature, and also fall below the corresponding blue line of slope
-1.}

\ctr{\gifimage{3}{537}{717}{8}}

}

\figfile{4}{
\figtitle{4}{RecRK model accounts for Empirical Response Suite
I with \tau\sub{E} rate-limiting}

\p{\panel{A}: The RecRK model (with \tau\sub{E} rate-limiting) was
optimized to both the Murnick \and\ Lamb saturated two-flash data and to
four sub-saturating flash responses from toad rods (four smallest
responses). The model is able to provide a reasonable account of the 4
sub-saturating responses as well as responses to 5 higher intensities
(to which the model was not optimized). \panel{B}: Model step responses
reproduce the "nose" at step onset and the two-phase response at step
offset. \panel{C}: The model T\sub{sat} versus ln(I) function (blue
solid curve) reproduces the same slope as the Murnick \and\ Lamb data
(~2.8 s/ln unit; red data and dashed line as in \figref{1}{B}).}

\p{Finally, the model generates a substantial range of Weber's law LA
flash sensitivity (~3 log units; solid blue curve, panel \panel{D}). For
each of a series of background adaptation levels (I\sub{b}), model LA
flash sensitivity was defined as the amplitude of the response to a
flash of fixed criterion intensity divided by the intensity of the
criterion flash. The criterion was the flash intensity eliciting a DA
flash response amplitude that was 10% of the full range of circulating
current. The dashed red curve is the Weber-Fechner relation from
\figref{1}{D}, shifted horizontally to fit the model output below a
"cutoff" background intensity (cutoff I\sub{b}), marked by a dotted
vertical cursor, above which the model was judged to deviate from
Weber's law.}

\p{The cutoff I\sub{b} was defined as the highest I\sub{b} at which the
value of the model deviated from the best-fit Weber-Fechner curve by a
criterion amount (0.05 log units). The Weber-Fechner curve was fit
(least-squares) to the model over all I\sub{b} values up to and
including the highest I\sub{b} where the model slope was still \ge-1.
The model slope was estimated by fitting a line to a moving window of 3
adjacent model points (sampling every 2\sup{0.25} R*/s, or 0.075 log
units).}

\p{Note that the model LA flash sensitivity obeys Weber's law over a
significant range (cutoff I\sub{b} = 8000 R* s\sup{-1}). At the cutoff
I\sub{b}, 14% of the model DA circulating current (F\sub{ss}) remains,
as indicated by the intersection of the vertical cursor line with the
green solid curve. The latter plots the fraction that the steady-state
current is saturated (i.e., 1 - F\sub{ss}(I\sub{b})), where F\sub{ss} is
the steady-state circulating current defined to be 1.0 in the dark, and
zero when all channels are closed. Also, at the cutoff I\sub{b}, the
steady-state internal Ca\sup{++} level (c\sub{ss} in inset) has dropped
by a factor of 5.1, from a dark value of 0.3 \mu M to 0.059 \mu M.}

\p{Also shown is the LA flash sensitivity of the model under two types
of simulated Ca\sup{++} clamp conditions: (1) LA flash sensitivity with
Ca\sup{++} clamped at its dark value (Ca\sup{++}\sub{dark} clamp; blue
dash-dot curve). Ca\sup{++} feedback is fully disabled over the entire
dynamic range, with only static saturation contributing to flash
desensitization. Ca\sup{++} was fixed at its dark value in the model,
and I\sub{b} was adjusted to achieve the same steady-state current as in
the unclamped case, ensuring that the steady-state currents were at the
same level in relation to static saturation (i.e., cGMP-gated channel).
Differences in flash sensitivity then can be ascribed to the differing
states of Ca\sup{++} in the unclamped and clamped cases.}

\p{The Ca\sup{++}\sub{dark} clamp analysis equates steady-state current
levels (F\sub{ss}), but does not equate internal Ca\sup{++} levels at
the time of presentation of the flash. This is achieved in a second
analysis: (2) LA flash sensitivity with Ca\sup{++} clamped at the new
steady-state level reached in response to each I\sub{b}
(Ca\sub{ss}\sup{++} clamp; blue dashed curve). This approach equates the
F\sub{ss} (and hence equates the effect of channel saturation), and
equates Ca\sup{++} at the time of the flash. Thus, in comparing the
unclamped and the Ca\sub{ss}\sup{++}-clamped flash sensitivity, the LA
flash response in each case is affected equally by saturation and by the
steady-state level of Ca\sup{++}-mediated gain. The only additional
factor shaping the LA flash response in the unclamped case is the
dynamic Ca\sup{++}-mediated gain change evoked by the flash.}

\p{Note that at high I\sub{b} (I\sub{b}\gt\ cutoff I\sub{b}), the
unclamped model flash sensitivity falls more steeply than a Weber's law
slope of -1, and eventually follows a steep function that parallels the
high - I\sub{b} behavior of both Ca\sup{++} clamped curves. In fact, all
3 curves asymptote to a slope of -(n\sub{cg} + 1), which is predicted by
the instantaneous compressive saturation of the cGMP-gated channels
[30].}

\ctr{\gifimage{4}{406}{446}{26}}

}

\figfile{5}{
\figtitle{5}{With \tau\sub{E} rate-limiting, the RecRK model accounts
for the qualitative features of the Fain et al. [27] step-flash data
and the Matthews [3] Ca\sup{++} clamp data}

\p{With the same parameters as in \figref{3} and \figref{4}, the RecRK
model with \tau\sub{E} rate-limiting also reproduces the signature
features of the Fain et al. [27] data; the model saturated flash
response emerges out of saturation faster as the intensity of a
conditioning step is increased (\figref{5}{A}). Under simulated
Ca\sup{++} clamp, the model reproduces the qualitative behavior of the
Matthews [3] data (\figref{5}{B}); the period of saturation (blue solid
curve) is prolonged (blue dashed curve) by the application of the
Ca\sup{++} clamp near the time of the flash ("Pre \and\ Post", "Post"),
but not if the flash occurs too early ("Pre") or too late ("Late").
Compare with \figref{2}.}

\ctr{\gifimage{5}{537}{717}{8}}

}

\figfile{6}{
\figtitle{6}{RecRK model does not account for Murnick \and\
Lamb data with \tau\sub{R} rate-limiting}

\p{With \tau\sub{R} rate-limiting in the RecRK model, the reduction in
model Testflash T\sub{sat} (blue curve) cannot exceed the magnitude of
the delay between Pre- and Test flashes; i.e., the model results are
always above the dashed blue line with slope -1. This contrasts with the
Murnick \and\ Lamb data (red curve versus red dashed line). This failure
rules out a model structure in which R* lifetime is rate-limiting and in
which early Ca\sup{++} feedback occurs solely at R* lifetime (via RecRK;
[1]).}

\ctr{\gifimage{6}{537}{717}{6}}

}

\figfile{7}{
\figtitle{7}{Consequences for the dim-flash response when \tau\sub{R} is
rate-limiting in a RecRK model.}

\p{Within the context of a RecRK model structure, rate-limitation by R*
lifetime is incommensurate with the kinetics of the dim-flash response.}

\p{Top panel: The relationship between \tau\sub{R} and the level of
Ca\sup{++} as it falls from a dark value of 0.3 \mu M to a minimum of
0.02 \mu M. In this example, parameters in the RecRK model were set such
that as Ca\sup{++} fell from its dark value to its theoretical
physiological minimum, the model \tau\sub{R} decreased from 20 s to 2
s.}

\p{Bottom panel: Resulting dim-flash response from model in top panel
(blue curve) in comparison to an empirical dim-flash response (red;
curve 2 from Rieke \and\ Baylor [16]; see \figref{1}{A}). The model
response was the result of optimizing the RecRK model to the Murnick
\and\ Lamb data and to the sub-saturating flash responses with the RecRK
parameters set to generate the profile in the top panel. Clearly, with
\tau\sub{R} being Ca\sup{++}-sensitive, and the requirement that the
change in \tau\sub{R} effect a large (10x) gain change, and that, at its
minimum, \tau\sub{R} remain rate-limiting with a time constant of ~2 s,
the model dim-flash response recovers much too slowly in comparison with
empirical rod responses.}

\ctr{\gifimage{7}{537}{717}{6}}

}

\figfile{8}{
\figtitle{8}{The need for a large front-end Ca\sup{++}-mediated
gain severely constrains RecRK parameters}

\p{The relationship between R* lifetime gain change and the dissociation
constant for the interaction between Ca\sup{++}-Rec complex and RK
(i.e., K\sub{D} = q\sub{4}/q\sub{3} in Eq. 8, A10). The gain change
(ordinate) is defined as the ratio of \tau\sub{Rdark} (theoretical value
of \tau\sub{R} when Ca\sup{++} = c\sub{dark} = 0.3 \mu M) to
\tau\sub{Rmin} (\tau\sub{R} when Ca\sup{++} = c\sub{min} = 0.02 \mu M).
The curves shown correspond to a family of K\sub{Rec,Ca} values ranging
from 0.2 to 1.2 \mu M (from top to bottom, in 0.1 \mu M increments). The
red curve is for K\sub{Rec,Ca} = 0.9 \mu M, a value obtained by Klenchin
et al. [8]. The horizontal dashed line marks a gain of 10. See text for
details.}

\p{Note that K\sub{D} must be very small if other RecRK- and Ca\sup{++}
parameters are set to values estimated in the literature. In this
example, Rec\sub{tot} = 34 \mu M [8] and the Hill coefficient, w = 2
[8]. For K\sub{Rec,Ca} = ~0.9 \mu M (red curve) [8], K\sub{D} must be
less than ~0.25 \mu M in order to get a gain change of 10x. This value
for K\sub{D} is ~14 times less than the value estimated by Klenchin et
al. (~3.4 \mu M; [8]). Analyses show that K\sub{D} will be even more
severely constrained (to even smaller values) if it is assumed that the
minimum physiological value for Ca\sup{++} is greater than 0.02 \mu M,
as estimated in some studies [33,63-65].}

\p{Not only is K\sub{D} restricted to small values, but, due to the
steepness of the functions in the region of high gain values (10x or
more), the overall range of permissible values is highly restricted.
Hence, for a given K\sub{Rec,Ca} value, only a very restricted range of
(small) values for K\sub{D} are permissible in order to achieve the
empirically observed gain changes.}

\ctr{\gifimage{8}{537}{717}{11}}

}

\figfile{9}{
\figtitle{9}{An R* activation model with \tau\sub{E} rate-limiting
accounts for the Murnick \and\ Lamb data}

\p{\panel{A}: With \tau\sub{E} rate-limiting, the R* activation model
provides a good fit to the Murnick \and\ Lamb saturated two-flash data.
\panel{B}: The corresponding model Testflash T\sub{sat} versus delay
along with the Murnick \and\ Lamb data. As was seen in comparable
results from the RecRK model (\figref{3}{B}), the model is able to
capture the critical feature of the data, i.e., that the decrease model
Test flash T\sub{sat} in response to the Pre-flash can exceed the delay
(blue curve falls below dashed blue line with slope = -1). The format
and symbology for this figure is the same as for \figref{3}. Parameters
for this model are given in \tabref{3}.}

\ctr{\gifimage{9}{537}{717}{12}}

}

\figfile{10}{
\figtitle{10}{An R* activation model with \tau\sub{E}
rate-limiting accounts for the Empirical Response Suite I}

\p{\panel{A}: With the same parameters yielding a fit to the Murnick
\and\ Lamb saturated data, the model provides a reasonable fit to the
sub-saturating flash responses, as well as for the signature features of
step responses (\panel{B}), T\sub{sat} versus ln(I) (\panel{C}).
Moreover, it generates a relatively large range of LA flash sensitivity
adhering to Weber's law (\panel{D}); cutoff I\sub{b} = 12,100 R*
s\sup{-1}. Format and symbology as in \figref{4}.}

\ctr{\gifimage{10}{425}{443}{25}}

}

\figfile{11}{
\figtitle{11}{An R* activation model also accounts for the
Murnick \and\ Lamb data when \tau\sub{R} is rate-limiting}

\p{\panel{A}: With \tau\sub{R} rate-limiting, the R* activation model
provides a good fit to the Murnick \and\ Lamb saturated two-flash data.
This is also reflected in (\panel{B}), which shows the corresponding
model Testflash T\sub{sat} versus delay along with the analogous Murnick
\and\ Lamb data. As for the case with \tau\sub{E} rate-limiting
(\figref{9}), as well as the comparable results for the RecRK model
(\figref{3}{B}), the model is able to capture the critical feature of
the data, i.e., that the decrease model Test flash T\sub{sat} in
response to the Pre-flash can exceed the delay (blue curve falls below
dashed blue line with slope = -1). Format and symbology as in
\figref{5}. Parameters are given in \tabref{3}.}

\ctr{\gifimage{11}{537}{717}{12}}

}

\figfile{12}{
\figtitle{12}{An R* activation model with \tau\sub{R} rate-limiting
accounts for the Empirical Response Suite I}

\p{\panel{A}: The model provides a reasonable fit to the sub-saturating
flash responses. \panel{B}-\panel{D}: With the same parameters, the
model also accounts for the signature features of step responses
(\panel{B}), T\sub{sat} versus ln(I) (\panel{C}), and generates a
relatively large range of LA flash sensitivity adhering to Weber's law
(\panel{D}).}

\ctr{\gifimage{12}{415}{435}{25}}

}

\figfile{13}{
\figtitle{13}{An R* activation model can be ruled out}

\p{\panel{A}: The R* activation model readily reproduces the qualitative
features of the Fain et al. [27] step-flash data regardless of whether
\tau\sub{E} (top panel) or \tau\sub{R} (bottom panel) is rate-limiting.
\panel{B}: However, for neither of the rate-limiting cases (\tau\sub{E}:
top two panels; \tau\sub{R}: bottom two panels) can the model capture
the key feature of the Matthews [3] data, i.e., when a Ca\sup{++} clamp
is applied for a brief period around the time of the flash, the model
does not generate any significant extension of saturation period
(compare with data, \figref{2}{B}). This failure implies that the R*
activation model, and any similar model (with an early Ca\sup{++}
feedback up to and including PDE activation) that does not alter the
recovery kinetics of the target reaction, can be ruled out.}

\ctr{\gifimage{13}{537}{717}{17}}

}

\figfile{14}{
\figtitle{14}{The combined model with \tau\sub{R} rate-limiting can
account for the Murnick \and\ Lamb data}

\p{With \tau\sub{R} rate-limiting, the combined (R*-activation-RecRK)
model can provide a good account of the Murnick \and\ Lamb data when the
dominant of the two front-end gains is at R* activation (compare with
\figref{3}, \figref{9}, and \figref{11}).}

\ctr{\gifimage{14}{537}{717}{12}}

}

\figfile{15}{
\figtitle{15}{The combined model with \tau\sub{R} rate-limiting can
account for the Empirical Response Suite I}

\p{With \tau\sub{R} rate-limiting and R* activation the dominant of the
two front-end gains, using the same parameters as in \figref{14}, the
combined (R*-activation-RecRK) model provides a good account of the
sub-saturated flash responses (\panel{A}), reproduces the signature
qualitative features of the step responses (\panel{B}), T\sub{sat}
versus ln(I) (\panel{C}), and generates a relatively large range of
Weber's law LA (\panel{D}; cutoff I\sub{b} = 5000 R* s\sup{-1}). Format
and symbology as in \figref{4}.}

\ctr{\gifimage{15}{410}{436}{24}}

}

\figfile{16}{
\figtitle{16}{A combined model with \tau\sub{R} rate-limiting accounts
for Empirical Response Suite II}

\p{With \tau\sub{R} rate-limiting and R* activation the dominant of the
two front-end gains, using the same parameters as in \figref{14}, the
combined (R*-activation-RecRK) model reproduces the signature
qualitative features of the step-flash paradigm of Fain et al. [27]
(\panel{A}). As in the Fain et al. data (red), the period of saturation
of the model to a saturating flash (blue curves) decreases progressively
as the intensity of a prior conditioning step of light in increased.
(\panel{B}) With the same parameters, the model also reproduces
Matthews' extension of saturation period when a Ca\sup{++} clamp is
applied near the time of the flash ("Pre \and\ Post" and "Post"
conditions; dashed blue curve), but not when it is applied too early
("Pre") or too late ("Late") (compare with \figref{2}{B}).}

\ctr{\gifimage{16}{537}{717}{12}}

}

\figfile{17}{
\figtitle{17}{A combined model with \tau\sub{R} rate-limiting cannot
account for the Murnick \and\ Lamb data if R* lifetime gain dominates}

\p{Analogous to what was seen for the RecRK model (\figref{6}), when
\tau\sub{R} is rate-limiting \b{and} the stronger of the two front-end
gains modulates \tau\sub{R}, the combined (R*-activation-RecRK) model
cannot account for the Murnick \and\ Lamb [1] data. It cannot generate
large enough changes in Test flash T\sub{sat}, and the model T\sub{sat}
versus delay (blue curve) never falls below the slope of -1. This rules
out such a model. Thus, if both R* activation and R* lifetime gains are
present, the R* lifetime gain cannot dominate.}

\ctr{\gifimage{17}{537}{717}{7}}

}

\begintables

\tabfile{1}{
\tabtitle{1}{Description of Model Variables \and\ Parameters}

\box{\gifimage{1}{571}{864}{118}}

\p{The symbol "#" means that the quantity is a unitless number.}

}

\tabfile{2}{
\tabtitle{2}{Parameters for Analysis of RecRK Model}

\box{\gifimage{2}{540}{727}{64}}

\p{Parameters are grouped according to functional elements of the model.
All descriptions of parameters are given in \tabref{1}. For each of the
two rate-limiting cases analyzed (\tau\sub{E}, \tau\sub{R}), the columns
of numbers show the optimized parameter value, and the lower and upper
bounds used in the optimization. The latter are shown only for
parameters that were actually optimized. The errors listed at the bottom
of the Table are RMS errors for the fits to the Murnick \and\ Lamb [2]
saturated two-flash data (ERR\sub{M\and L}; data from their Figure 3)
and to the sub-saturated flash responses from Rieke \and\ Baylor [16]
(ERR\sub{flash}; responses to the 4 lowest flash intensities shown in
their Figure 4). The combined error (ERR\sub{combo}) was calculated as
the square root of the product of the two other errors.}

}

\tabfile{3}{
\tabtitle{3}{Parameters for Analyses of R* activation Model}

\box{\gifimage{3}{666}{690}{62}}

\p{Table format as in \tabref{2}. Only parameters optimized have lower
and upper bounds specified.}

}

\tabfile{4}{
\tabtitle{4}{Parameters for Analyses of Combined RecRK-R* activation Model:
\tau\sub{R} Rate Limiting}

\box{\gifimage{4}{595}{766}{67}}

\p{Table format as in \tabref{2} and \tabref{3}. The optimized parameter
values and upper and lower bounds are shown for the \tau\sub{R}
rate-limiting case only, under two conditions: dominance by gain at R*
activation (\Delta R* gain dominant), and dominance by gain at
R*lifetime (\Delta\tau\sub{R} dominant). Only parameters optimized have
lower and upper bounds specified.}

}