From the Department of Cellular and Molecular Physiology, Yale University School of Medicine, New Haven, Connecticut 06520
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ABSTRACT |
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A functional kinetic model is developed to describe the activation gating process of the Shaker potassium channel. The modeling in this paper is constrained by measurements described in the preceding two papers,
including macroscopic ionic and gating currents and single channel ionic currents. These data were obtained
from the normally activating wild-type channel as well as a mutant channel V2, in which the leucine at position
382 has been mutated to a valine. Different classes of models that incorporate Shaker's symmetrical tetrameric
structure are systematically examined. Many simple gating models are clearly inadequate, but a model that can account for all of the qualitative features of the data has the channel open after its four subunits undergo thr
ee transitions in sequence, and two final transitions that reflect the concerted action of the four subunits. In this model,
which we call Scheme 3+2, the channel can also close to several states that are not part of the activation path.
Channel opening involves a large total charge movement (10.8 e0), which is distributed among a large number of
small steps each with rather small charge movements (between 0.6 and 1.05 e0). The final two transitions are different from earlier steps by having slow backward rates. These steps confer a cooperative
mechanism of channel
opening at Shaker's activation voltages. In the context of Scheme 3+2
, significant effects of the V2 mutation are
limited to the backward rates of the final two transitions, implying that L382 plays an important role in the conformational stability of the final two states.
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INTRODUCTION |
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Several functional kinetic models have been proposed
that describe the activation gating process of Shaker potassium channels (Schoppa et al., 1992; Tytgat and
Hess, 1992
; Bezanilla et al., 1994
; McCormack et al.,
1994
; Zagotta et al., 1994
b). However, these models are
fundamentally quite diverse. One of the reasons for the
differences is that no single group has attempted to
model all of the available data that reflect the activation
gating process for Shaker channels; instead, different
groups have modeled different subsets of the data. Another reason is that the activation process is likely to involve a very large number of gating transitions (Zagotta
et al., 1994
a), and data have not been available to constrain all of the transitions of appropriately complicated models.
This paper is the last in a series of three papers in
which we describe our efforts to produce a well-constrained functional gating model for the Shaker potassium channel. The specific channel that we have studied is the Shaker 29-4 channel (Iverson and Rudy, 1990),
which has been truncated at the NH2 terminus to remove rapid inactivation, and which has been expressed
in Xenopus laevis oocytes. Our strategy in the first two
papers (Schoppa and Sigworth, 1998a
, 1998b
) has been
to characterize in detail the electrophysiological properties of the Shaker channel, using a combination of
measurements of macroscopic ionic and gating currents and single channel currents. We have obtained
data from not only the normally activating (wild type,
WT)1 channel, but also from a mutant channel (V2)
having a leucine to valine mutation at position L370 in
the Shaker 29-4 sequence, corresponding to L382 in the
better-known ShB sequence. Data from these channels,
taken together, have yielded starting estimates of rate
constants for several gating transitions.
Our strategy for the modeling here will be first to explore systematically several classes of gating models. All
of these models invoke the tetrameric structure of
Shaker channels (MacKinnon, 1991; Kavanaugh et al.,
1992
; Li et al., 1994
), by having many of their transitions correspond to Shaker's four subunits moving on
e
subunit at a time, and with the subunits acting equivalently. We will show that different lines of data rule out
the most simple models, leading us to our first hypothesis for a gating model, which we call Scheme 2+2
. This
model has the channel open after each of Shaker's four
subunits undergo two transitions in sequence, followed
by two additional concerted conformational changes.
Next, we compare detailed predictions of Scheme 2+2
with data, and find that it is an inadequate model. Finally, we propose a more complicated gating model,
called Scheme 3+2
, as an example of one model that
can account for all of the qualitative features of the
data. In this model, the Shaker channel opens after each of its four subunits undergoes three transitions in sequence, followed by two concerted transitions.
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METHODS |
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Calculations for the Modeling
For the modeling of Shaker's activation gating, we assume continuous-time, discrete-state Markov models, which have performed adequately at describing the gating processes of many other ion channels (for example, see Mc
Manus and Magleby, 1988). For all of the calculations, the numerical techniques described by
Colquhoun and Hawkes (1995)
were used. For a gating scheme
with n states, we constructed an n by n matrix K(0) of rate constants at 0 mV and a matrix Q of partial charges qij that reflect the
voltage dependence of the forward and backward rates of each transition between states i and j. For a given voltage V a matrix K(V ) was constructed with off-diagonal elements kij(V ) calculated from the corresponding elements in K(0) and Q:
(1)
and the diagonals were subsequently computed to cause the rows to sum to zero. The equilibrium and time-dependent state occupancies were derived from the eigenvalues and the eigenvectors of the K(V) matrix, which were found using standard routines (Eispack). Computations were performed within the PowerMod Modula-2 programming environment (Heka Electronic, Lambrecht, Germany) on a Macintosh Centris 650 computer. Please note that the matrix Q is not to be confused with Q, the relative charge movement measured in gating current experiments.
For the fitting of the macroscopic ionic and gating current
time courses, typically two sets of calculations were performed: one to obtain the equilibrium state occupancies at the prepulse voltage, and a second to determine the time-dependent changes in occupancies during the test
pulse. The simulations in Fig. 7, A
and C, were done slightly differently, assuming that all of the
channels reside in the first closed state at the beginning of the
test pulse. This is a reasonable assumption since little charge has
moved in the Q-V relation at the 93-mV prepulse voltage used
in these experiments.
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For ionic current relaxations, the occupancy of the open state
was multiplied by the single channel current amplitude i, and the
number of channels n that we estimated to contribute to the macroscopic current. Estimates of i for the current measurements made in the absence of the external potassium were obtained directly from the ampli
tudes of WT and V2 single-channel currents
measured under the same ionic conditions. For the simulations
of tail currents that were measured with 14 mM K+ in the pipette
(see Fig. 3), no estimate of i was available, but the simulated
curves were scaled to peak at the same value as the measured tail
currents. The value for n was typically fixed to that which best fitted the family of current traces from a given patch, and was kept
constant for all of the traces. However, in experiments in which
currents were measured over 20 min of recording time (e.g.,
in the reactivation measurements in Fig. 10), small variations
(<10%) in n were introduced into the fitting to account for the
gradual run down of current.
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The slow inactivation process in Shaker channels has one rapid
component (with
70 ms; Schoppa and Sigworth, 1998a
) with kinetics that are comparable to activation gating at some voltages. Thus, in the simulations, the current time courses predicted by the activation model were multiplied by a decaying single exponential
reflecting this component. It was implicit in this
approach that slow inactivation occurs during the test pulse independently of activation. Parameter estimates for this transition
were taken from the measured time constant and amplitude of
the fast component of slow inactivation, obtained by fitting a sum
of two exponentials to ionic currents measured during 4-8-s voltage pulses.
For the fitting of the gating-current time courses, the amplitudes derived from the eigenvectors of K(hnbsp;) and the charge
movements were scaled by the number of channels. At the 5-kHz
filtering bandwidth at which the gating currents were usually recorded, we expected that any charge component decaying faster than ~100 µs was likely to be distorted, given that the measured step response of the recor
ding system required a few sample intervals to settle (Schoppa and Sigworth, 1998a). In the simulations, this was accounted for by constraining the rates of the expon
ential relaxations: in each component with a time constant
shorter than 100 µs, the time constant was fixed to 100 µs, and
the amplitude of the component was appropriately adjusted to
maintain the correct amount of charge.
The probability density functions fitted to the single-channel
open and closed dwell time histograms were calculated using the
methods that have been described previously (Colquhoun and Hawkes, 1981). For the calculations of the open times, a correction was performed for missed closed events, using d
escribed
methods (Crouzy and Sigworth, 1990
).
For the simple characterization of activation time courses, we
sometimes fitted a single exponential function to simulated time
courses in the same manner as was done for the experimental data (Schoppa and Sigworth, 1998a). Briefly, an exponential
function was fitted to the time course, starting at the time at
which the relaxation reached 50% of its final value. The resulting
time constant
a and delay
a parameters have simple interpretations in the case th
at all transitions have negligible reverse rates.
Using Data Obtained from Different Patch Recordings
It has been reported that Shaker channels exhibit variabilities in
gating between different patches (Zagotta et al., 1994b). Indeed,
in our records, WT and V2 channels displayed patch-to-patch variabilities in several gating properties, including the voltage dependences and kinetics of channel opening (Fig. 1, A and B). F
or
both channels, the variabilities in equilibrium Po and in the kinetics of channel opening (as reflected in the activation time constant
a) co
rresponded to a 5-10-mV voltage shift (Fig. 1 C). The
extent of the variabilities was larger than would be expected from
drift in the pipette voltage offset. This offset was corrected at the
beginning of each recording; it was found to change by no more
than 2-3 mV by the end of the experiment, when the offset was
reevaluated. One possible source of the variability in the macroscopic currents of V2 is its modal single channel behavior
(Schoppa and Sigworth, 1998b
). The macroscopic current time
course could reflect the modal behavior if factors exist (e.g., second messengers) that shift the properties of the entire population of channels that contribute to the macroscopic current.
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Since it was impossible to obtain all of the types of data from a
single patch recording, the patch-to-patch variabilities implied
that no single set of parameters could account for all of the data
simultaneously. To account for the variabilities, we allowed the
rate values (at 0 mV) to differ by as much as 20% between
patches, but the charge values were the same for all the simulations. This magnitude of variation accounts for the ~10-mV voltage shifts in Po and
a shown in Fig. 1. We considered this a satisfactory approach since our interest in the modeling was to discriminate between different fundamental mechanisms of channel gating rather than to determi
ne rate constants to high precision. Most gating mechanisms could be quite easily ruled out by simple qualitative criteria or if they produced extremely poor fits of the data (e.g., the fits of Scheme 2+2
to the equilibrium data in
Fig. 11); allowing small variations in rate constants between patches did not obscure our ability to differentiate models. In fact, the number of instances that the rates differed from the values given i
n the appropriate tables is quite small. These are explicitly noted in the legends of Figs. 5, 10, 16, and 18.
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Finding Optimal Parameter Estimates
For previously described gating models, parameter estimates
have typically been found by using a search algorithm that minimizes the error between the fitted curves and the data. In our initial modeling attempts, we employed the simplex search algorithm (Nelder and Mead, 1965; Press et al., 1992
) to optimize fits,
but obtained disappointing results. One problem was a bias in
the fitting toward current traces that were larger in magnitude,
since these yielded the largest error values. A simple weighting
scheme improved things somewhat, but did not solve the problem that fits to individual traces often accounted well for certain
features of the time course but not for others. For example, good
fits of the rising phase and the final value of an ionic time course
would be obtained (since these features account for most of the
data points), but the delay would be poorly represented. The delay, however, was often the more important feature of the current
for constraining models, since it reflects many more rates than
the rest of the current.
Some attempts were made at using an appropriate error weighting function to avoid these problems. In our experience, however, determining the appropriate function was very tedious, and it was more expedient to perform the fitting by simply setting parameters manually and determining the goodness of fit by visual inspection. Our success in deriving a set of parameters with nonautomated methods can be attributed to the availability of good initial estimates for each of the rate constants.
The complexities surrounding the weighting of the errors in the fits, as well as the presence of variabilities between different patch experiments, made it difficult to provide meaningful confidence limits for the different parameter estimates that we give. However, we emphasize that each of the parameter estimates in our model is highly constrained. As we will describe below, we are generally able to identify particular current measurements that isolate each transition, and thus tightly constrain each of the parameters in the model. A good example of how making modest changes in the parameters affects the fits is illustrated in Fig. 10, C and D.
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RESULTS |
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Several tenable classes of activation gating models are
illustrated in Fig. 2. Since Shaker channels have a symmetrical structure composed of four identical subunits
(MacKinnon et al., 1991; Kavanaugh et al., 1992; Li et
al., 1994
), a credible hypothesis for a gating model is
one in which many of the transitions correspond to
Shaker's four subunits moving one subunit at a time,
and with the subunits acting equivalently. The symmetry can also be exploited in the modeling, since it reduces the number of different parameters that have to
be constrained. Having many transitions correspond to
the equivalent movement of single subunits has been a
feature of many of the published gating models for
Shaker (Schoppa et al., 1992
; McCormack et al., 1994
;
Zagotta et al., 1994
b), and, for largely philosophical
reasons, we also favor this formulation (see also below).
Thus, in each of the models in Fig. 2, the channel opens after each of the four subunits undergoes at least
one transition between different states that reflect the
conformation of each of the individual subunits. These
subunit states are designated S0, S1, S2, and S3, and we
will refer to transitions among these states as "subunit
transitions." Some of the models in Fig. 2 have one or
two additional transitions that follow the subunit transitions. These presumably reflect the concerted action of the four subunits. The naming of each of the models
follows the assigned number of subunit transitions and
the number of subsequent concerted transitions. For
example, in Scheme 1+2, the channels open after one
set of four subunit transitions and two concerted transitions.
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In each of the models, the subunit transitions occur
in sequence with each other instead of independently
since sequential subunit movement better accounts for
the long delay in the channel opening time course. Sequential subunit transitions also better account for the
presence of a rising phase in the "on" gating currents,
assuming that different subunit transitions have different rates (Zagotta et al., 1994b).
All of the models in Fig. 2 have a single open state.
Two pieces of evidence in favor of single open state
models have been presented previously by Hoshi et al.
(1994) and Zagotta et al. (1994
a). The first is Shaker's
single-exponential open dwell-time distributions, which
are best explained by a single open state. The second is
the shape of Shaker's voltage dependence of open probability Po. The activation curve becomes increasingly
steep at low Po (see Fig. 11 B), reaching an asymptotic
steepness corresponding with the channel's total charge
movement (Seoh et al., 1996
); this property is inconsistent with the existence of multiple open states with voltage-dependent transitions among them (Sigg and Bez
anilla, 1997
).
Our strategy for the modeling here will be to consider the different models in Fig. 2 systematically, starting with the most simple models and moving to more complicated models, as they are required by the data. The modeling will be divided into four stages, as summarized here.
Stage I. In the first stage, we point out a number of observations that suggest that the correct activation model is more complicated than all but three of the classes of models shown in Fig. 2.
Stage II.
We formally model Scheme 2+2, which is
an example of the class of models Scheme 2+2. We
first use the kinetic data at the voltage extremes outlined in the previous two papers to derive starting estimates for the rates in the model, and then model these
same data. We find that Scheme 2+2
accounts for the
kinetic details quite well. Then, as an additional test, we
model the equilibrium voltage dependence of channel
opening and charge movement relations. We find that
Scheme 2+2
accounts poorly for some features of the
equilibrium data. The deviations in the fits suggest that the correct model must have a larger total charge
movement, which can be provided only by adding
more transitions to the scheme.
Stage III.
We consider ways of adding more transitions to Scheme 2+2. We consider two possibilities,
one that adds a single concerted transition with a large
charge (Scheme 2+3
), and another that adds an additional subunit transition (Scheme 3+2
). Predictions of
these two models for V2's Q-V relation indicate that
Scheme 3+2
is a better solution. In this stage, we also
refit all of the kinetic and equilibrium data considered
in Stage II, to obtain a set of parameters for Scheme
3+2
.
Stage IV.
In the last stage of the modeling, we compare the behavior of Scheme 3+2 with other types of
macroscopic current measurements, including kinetic
measurements at intermediate voltages. These experiments act as independent tests for the robustness of
Scheme 3+2
.
Stage I: Evidence Against Simple Models
Several lines of evidence suggest that models that are more simple than Scheme 2+2 in Fig. 2 are inadequate.
Schemes 1+0, 1+1, and 1+2.
Zagotta et al. (1994a)
used the magnitude of the delay in the activation time
course to derive a minimum estimate of the total number of gating transitions. By fitting the current to a sequential model with equal forward rates, these authors
estimated that the Shaker channel undergoes a minimum of five transitions. We performed a similar analysis on Shaker's ionic current time courses, but with currents measured across a broader test voltage rang
e (between
13 and +147 mV) and, also while using a more
negative holding potential (
133 mV). More negative
holding potentials load channels into the earliest
closed states, and thus provide a more reliable estimate
of the total number of transitions. Fits of the currents at
+27 mV to a sequential model with equal forward rates
yielded an average minimum estimate of seven transitions (the values in three patches were six, seven, and
eight). Seven transitions were also required to account
for the ionic currents at +67 and +147 mV (one patch
each). This lower bound of seven transitions rules out all
models in the classes of Schemes 1+0, 1+1, and 1+2,
which have no more than six transitions.
Scheme 2+0.
This model has an activating channel
undergo eight transitions. However, evidence against
Scheme 2+0 is provided by the estimates of the voltage
dependences of different rates (Schoppa and Sigworth, 1998a), which indicate that there are at least three
types of transitions that can be differentiated by the
magnitudes of their associated charges. Scheme 2+0,
however, has only two types of transitions. An assumption in this argument is that all four of a given type of
subunit transition (for example, S1
S2 in Scheme 2+0) have equivalent charge movements. While we
cannot rule out models that invoke "symmetry breaking" in the movement of charge, we prefer models that
do not require this added complexity.
Scheme 3+0.
Evidence against Scheme 3+0 is provided by three observations made in the first paper
(Schoppa and Sigworth, 1998a) that suggest that the final gating transition is qualitatively different from earlier transitions, including the second to last transiti
on.
Schemes 2+1 and 3+1.
Finally, there is one argument
against the models Schemes 2+1 and 3+1, which each
have a single concerted transition. WT's "off" gating
currents after large depolarizations show a rising phase
and a slow decay (Bezanilla et al., 1991; Zagotta et al., 1994a). These features imply that the final two transitions that determine the deactivation kinetics (Schopp
a
and Sigworth, 1998a
) have slower reverse rates than
earlier transitions. Because the magnitudes of these
rates differ from those of earlier transitions, we suggest
that the final two transitions represent qualitatively different transitions. The simplest model producing this
variety of transition types has the last two transitions be
concerted ones. Models with only one concerted transition would require that the movement of one of the
four subunits (S1
S2 in Scheme 2+1, for example) be
much slower than the others. The model of Zagotta et
al. (1994
b) includes this sort of symmetry breaking to
describe the slow reverse rate of the final transition in
their model, which is otherwise like Scheme 2+0. However, in the absence of data that indicate that there is
symmetry breaking in the rates of the final two transitions, we favor a more simple interpretation of these
slow reverse rates.
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Stage II. Modeling Scheme 2+2
Scheme 2+2 has quite a large number of different parameters that must be constrained (16 for the transitions in the activation path, 28 in total). To expedite
the modeling, we consider separately data that reflect
different subsets of transitions, following an approach
similar to that of Vandenburg and Bezanilla (1991)
in
modeling activation gating for the squid sodium channel. In A, we model data that reflect transitions near
the open state. In B, we model data that reflect earlier
transitions, while fixing the parameter estimates of the
transitions near the open state to those obtained in A.
(A) Modeling kinetic data that reflect transitions near the
open state.
We first consider a simplified model (Scheme
0+2)
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(B) Modeling kinetic data that reflect S0 S1 and S1
&n
bsp;S2.
We next model kinetic data that reflect the subunit
transitions S0
S1 and S1
S2 in Scheme 2+2
. We
make use of the complete model while initially constraining the values of the rates of the final transitions
to be the same as those just assigned above. In these
simulations, the forward and backward rates of each
subunit undergoing one of the two subunit transitions
are each scaled by a statistical factor that reflects the
number of available subunits. For example, for the very
first subunit undergoing S0
S1, the forward rate is a1
multiplied by four, and the backward rate is just b1.
Thus, our initial assumption in the modeling is that the
four Shaker subunits gate independently of each other, since this is the simplest model. Subunit-subunit interactions are included later as required by the data.
Derivation of starting estimates of the rates for S0 S1 and
S1
S2
We use as starting estimates for the rates a1
and b1 of the first subunit transition S0
S1 the assigned values
1 and
1 for the very first transition in a
sequential model, as derived in the previous two papers; that is, we make the very first transition be the first
of four S0
S1 transitions. This formulation is the simplest, but it is also favored by an apparent paradox that arises when one
compares the kinetic description of
the first transition with equilibrium data that correspond to the earliest transitions. The
1 and
1 estimates derived from ionic and gating current time
courses give a charge estimate of z1 = 0.9 e0, and a midpoint voltage (V1/2) for the first transition of
53 mV. However,
from the equilibrium q-V relation for WT
(Fig. 6), the first 0.9 e0 of charge that moves in activation occurs at much more negative voltages, below
80
mV. A general way that a transition with a given midpoint voltage can contribute to charge movement at
more negative voltages is if the first transition is one of several like transitions with similar midpoint voltages;
the charge movement at the most negative voltages
would then reflect the sum of the charge associated
with several transitions. Fig. 6 illustrates simulations in
which we assume that the first transition is one of four
S0
S1 transitions, and have fixed the charge and the
midpoint voltage of S0
S1 to that derived from the kinetics. The curve for four transitions (p = 4) accounts
quite well for the magnitude of the charge movement
at voltages below
90 mV. Models with fewer than four
subunits yield charge magnitudes that are too small.
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Derivation of final estimates for the forward rates a1 and a2. For WT, a1 is the slowest forward rate at voltages near 0 mV, and is therefore constrained by the rise of the ionic current at these voltages (Fig. 7, A and B) and by the decay of the on gating currents (Fig. 7 C). For V2, a1 is also the slowest forward rate at these voltages and is constrained by the decay of the on gating currents. In contrast to WT, the rise of V2's ionic current at these voltages does not directly reflect a1.
For both WT and V2, a2 is the slowest forward rate at very large depolarizations (VSubunit-subunit interactions in S0 S1
While our initial assumption in the modeling is that the different
gating subunits act independently, we find that the addition of a small degree of cooperative interaction between subunits for the first forward subunit transition
S0
S1 causes modest improvements in some of the
fits. In particular, the interaction helps provide a sufficiently broad plateau phase in WT's and V2's on gating
current time course at voltages near 0 mV. While the fact that a1 is smaller than a2 contributes to a plateau
phase in the predicted current (Fig. 7 C), the measured
on current is broader than can be accounted for by
Scheme 2+2
in the absence of these interactions. We
introduced interaction in S0
S1 by multiplying each
of the forward rates by a factor c raised to a power that
reflects the number of subunits that have undergone this transition. This can be depicted in the expanded
form of as:
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An explicit model of Ci states.
Scheme 2+2 accounts for
most of the features of the ionic current time courses,
but is unable to account in detail for the kinetics of the
final approach of WT's and V2's channel opening time
course at voltages
+67 mV (Fig. 7 A). This discrepancy reflects the absence in Scheme 2+2
of trans
itions
to Ci states from closed states in the activation path,
which account for the slow upward creep in the channel opening time course at high voltages (Schoppa and
Sigworth, 1998a
). The discrepancy is larger for V2 because these channels enter Ci states from closed states
more frequently than WT (Schoppa and Sigworth,
1998b
). Nevertheless, good agreement is obtained between the
a values derived from the simulated curren
ts and those derived from fits of WT and V2's measured
currents to the sum of two exponentials (Fig. 7 B). The
value of
a derived in this way reflects the kinetics of the
main activation path.
Derivation of final estimates for the backward rates b1 and
b2.
For estimating the backward rates for the subunit
transitions, we assume for simplicity that the first backward transition S0 S1 is affected by the same degree of
cooperative interaction as the forward transition S0
S1,
as depicted in Scheme I.
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Small adjustments in the parameter estimates for the final
transitions in Scheme 2+2.
Our strategy for constraining the parameters in Scheme 2+2
has been first to
model the transitions near the open state (Scheme 0+2
) and subsequently to add earlier transitions while
fixing the parameters of the final transitions to be the
same as those derived in the context of Scheme 0+2
.
Returning to the data in Figs. 3-5 that reflect the final
transitions, we obtained slightly better fits in making
some minor adjustments in the parameters of the final
transitions (described in Table II). All of the illustrated fits in Figs. 7-10 actually reflect the adjusted parameters for the final transitions.
(C) Modeling WT's and V2's Po-V and Q-V relations with
Scheme 2+2
As shown by the previous simulations, Scheme 2+2 accounts very well for kinetic data obtained at voltage extremes. In the final part of Stage II, we consider WT's
and V2's voltage dependences of channel opening (Po-V)
and charge movement (Q-V).
The simulations of these equilibrium data (Fig. 11)
indicate that, without changing the values for the parameters from those listed in Table II, Scheme 2+2 accounts for many of the features of WT's and V2's Po-V
and Q-V relations. The model predicts WT's characteristic steep voltage dependence of channel opening, as well as the steep component of charge in WT's Q-V relation that has been described by Bezanilla et al.
(1994)
.
Scheme 2+2, however, accounts poorly for some of
the other features of the equilibrium data. For example, the model significantly underestimates the steepness of V2's Q-V relation (Fig. 11 A). It is difficult to interpret this discrepancy, but one e
xplanation is that the
total gating charge in the model is too small. More direct evidence that more charge is required in the
model is obtained from the predictions of the model to
the log-transformed equilibrium data (Fig. 11 B). It is
clearly apparent that Scheme 2+2
underestimates the
steepness of WT's Po curve, which reflects the total
charge. Indeed, Scheme 2+2
contains a total charge
of only 7 e0, much less than the model-independent estimate of 13 e0 that has been obtained from measurements of gating currents (Schoppa et al., 1992
; Aggarwal and MacKinnon, 1996
).
A different test of Scheme 2+2 is provided by evaluating the charge movement at negative voltages (Fig.
11 B). At asymptotically negative voltages, the steepness
of the Q-V relation reflects the amount of charge associated with the first transition in the activation path; in
Scheme 2+2
, this is the first of four subunits undergoing S0
S1. Scheme 2+2
underestimates the steepness of the Q-V relations at negative voltages, which
could imply that our total charge estimate of 0.8 e0 for
S0
S1 is too small. Alternatively, if the charge movement at the lowest voltages where we could measure
currents (between
113 and
83 mV) reflects more than the very first transition, the discrepancy is more
difficult to interpret (see below).
Stage III: Adding Four more Transitions, Yielding
Scheme 3+2
The difference between the charge in Scheme 2+2 (7 e0) and the model-independent estimate of 13 e0 obtained from gating currents suggests that addit
ional
charge must be added to the model. In this stage we
will address ways to increase the total charge.
As a starting point, consider the following two observations from the modeling that has been performed
thus far. The first is that the voltage dependences of
the forward and backward rates in Scheme 2+2 are
very tightly constrained by the kinetics at depolarized
and hyperpolarized voltages; simulations like those shown in Fig. 10 C indicate that increasing the charges
will yield poor fits of the data. Thus, the additional 6 e0
of charge must be introduced by adding more transitions instead of by changing the charges on the transitions in Scheme 2+2
. The second observation is that,
in Scheme 2+2
, the first and final transitions are more tightly constrained than intermediate transitions. Data
can be identified that reflect only the very first transition (gating currents at hyperpolarized voltages) or
only the last transitions (tail currents and reactivation
time courses), but the intermediate transitions are constrained only by kinetic data that reflect the composite
properties of many transitions. Thus, we favor adding new transitions at intermediate positions in the model.
Distinguishing between Scheme 2+3 and Scheme 3+2
.
We are unable to choose among all of the large number of possible ways of adding more intermediate transitions to the model, but here we will consider two of
the most simple solutions. One of these models adds
more charge by adding a single concerted transition
that carries a large charge; this is Scheme 2+3 in Fig. 2,
or Scheme 2+3
when the transitions to states outside of the activation path are added. The large-valenced
transition would be the third to last transition that the
channel undergoes before it opens. The other solution
adds an additional subunit transition (Scheme 3+2
).
|
Scheme 3+2 with a minimal number of unconstrained parameters.
We next assign rates for the new, larger
model Scheme 3+2
, which in its complete form can
be depicted as
|
Assigning a3 and b3
For S2 S3, we choose to fix the
partial charges qa3 and qb3 to be the same as qa2 and qb2,
respectively, and let a3 = ma2 and b3 = mb2. Making the
ratio a3/b3 to be equal to a2/b2 is a simplification that is
not unreasonable. Consider again the shape of V2's Q-V
relation (Fig. 11 A). While Scheme 2+2
underestimates the steepness of V2's Q-V relation, it is able to account for the correct position of the
Q-V relation on
the voltage axis. This would suggest that the equilibrium of S2
S3 in Scheme 3+2
must be such as not to
change the position of V2's Q-V relation on the voltage
axis. Fig. 12 C illustrates simulations in which we have
assigned three different values for the ratio a3(0)/
b3(0). For a ratio that is similar to a2(0)/b2(0), Scheme
3+2
predicts a Q-V relation that maintains the correct
position on the voltage axis, but values for a3(0)/b3(0)
that are larger or smaller by a factor of three yield predictions that are displaced to the right or the left of the
measured Q-V relation.
Adjustments in the final charge estimates for Scheme
3+2.
Next, we adjust all the parameters in the model
to take into account the addition of S2
S3. As it turns
out, the addition of this single set of rapid transitions
actually makes little difference in most of the kinetic
predictions, so that little change in the parameters is required. The largest difference between Schemes 3+2
and 2+2
is in the estimates of the partial charges of
the backward rates for the subunit transitions. In the
modeling, we first set the charges for S0
S1 and S1
S2 in Scheme 3+2
to be the same as those of the corresponding transitions in Scheme 2+2
(and then m
ade
qa3 and qb3 equal to qa2 = 0.09 e0 and qb2 =
0.33 e0, respectively). Howe
ver, the total gating charge for this
version of Scheme 3+2
would be 8.7 e0, still substantially smaller than the measured total charge of 13 e0.
Indeed, Scheme 3+2
with these parameters underestimates the voltage dependence of WT's Po and underestimates the steepness of V2's Q-V relation
(Fig. 12 C).
To account for the charge discrepancy, we add more total charge by introducing an increase, ~20%, in the
charge associated with each of the three subunit transitions. Arbitrarily, we choose to add most of this charge
to the partial charges of the backward rates. To further
reduce the number of free parameters, we constrained qb1, qb2, and qb3 to have the same values,
0.52 e0. The
final Scheme 3+2
has a total charge of 10.8 e0.
Fits of Scheme 3+2 to kinetics at voltage extremes and to
equilibrium data.
Because the rate constants of the final
transitions are unchanged, the predictions of Scheme
3+2
for the kinetic data that reflect the final two transitions (illustrated in Figs. 3-5) are essentially identical
and are not shown.
|
|
|
|
IV. Additional Tests of Scheme 3+2
In the fourth and final stage of the modeling, we consider the ability of Scheme 3+2 to account for several
additional types of kinetic data. In most of the experiments described here, we make use of test pulses or
prepulses to intermediate voltages (between
73 and
23 mV). The fact that both forward and backward rates are nonnegligible at intermediate voltages complicates the interpretation of many of these data, but
each of the experiments functions as an independent
test of the robustness of Scheme 3+2
.
(A) WT's and V2's macroscopic ionic and gating currents
at intermediate test voltages.
Fig. 13 shows that Scheme
3+2 accounts for WT's ionic currents at intermediate
test voltages quite well, and, in particular, accounts for
WT's characteristic slow and nonsigmoidal channel opening time course at small depolarizations (Zagotta
et al., 1994
a). Scheme 3+2
also accounts for the time
course of the on gating currents at all intermediate test
voltages (Fig. 14). At
33 mV, for example, the model
predicts a plateau phase of WT and V2 currents and also
predicts a slow component in the decay of WT current
(Bezanilla et al., 1994
) that is absent in the case of V2
(Schoppa and Sigworth, 1998b
). The slow component is
associated with WT's channel opening; V2 lacks this
component because V2 fails to open at these voltages.
(B) WT's and V2's channel opening time courses after various prepulses.
Test currents measured after prepulses
that preload channels into different distributions of
states emphasize the kinetics associated with various subsets of transitions. We performed simulations of Scheme
3+2 for an experiment in which we fixed the test pulse voltage and varied the amplitude of the prepulse (Fig.
18). This is an experiment like that performed by Cole
and Moore (1960)
, who showed that changes in the
prepulse altered the delay of the squid potassium current. Fig. 18 A shows that the model accounts quite well
for the kinetics of the rising phase of the Shaker currents,
as well as the delay for each of the different prepulses. The good fits of the current rising phase (Fig. 18, A and
B) implies that the model correctly predicts the kinetics
of the rate-limiting step for different starting conditions.
The good fits of the delay for the different prepulses
(Fig. 18, A and C) indicate that the model correctly predicts the equilibria of the transitions that contribute to
the delay at the test-pulse voltage.
(C) WT and V2 off gating currents after prepulses.
The ef-
fects of prepulses on gating currents are illustrated in
Fig. 19. In these experiments, we used a hyperpolarized
test voltage (93 mV), so that the observed currents reflect the backward kinetics from the set of states that
are occupied during each of the prepulses. Both the
amplitude of the prepulse (Fig. 19 A for WT and V2)
and the duration of the prepulse (Fig. 19 B for WT)
were varied. These data are complementary to the gating current experiments illustrated in Fig. 15, in which
we used a range of test voltages, but prepulses to only
two voltages.
|
![]() |
DISCUSSION |
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![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
In this paper we have developed an activation gating
model for Shaker potassium channels that can describe
a wide range of data obtained from the WT and V2
Shaker channels. In our model (for Scheme 3+2), each
of Shaker's four subunits undergo three transitions in
sequence; the channel opens after two additional transitions that presumably reflect the concerted action of
Shaker's four subunits:
|
Rate constants (at 0 mV) are indicated for both WT and V2 channels; the rates for V2 are delineated by rectangles. The partial charges associated with each of the forward and backward rates are in parentheses.
Scheme 3+2 is a very well-constrained model. The
data we have modeled included measurements made
with voltage protocols that employ prepulses that preload channels into various distributions of states, which
give test current time courses that reflect different subsets of transitions. We have also included data obtained
for the mutant V2 channel, which uniquely constrain
certain features of the model, such as the amount of
charge associated with the final two transitions. Starting
estimates of parameters, obtained by fitting exponentials to selected current time courses, were available for
all of the transitions in Scheme 3+2
except S2
S3.
The similarity in the estimates subsequently obtained
by fitting the entire model to the data supports the idea
that particular types of current measurements can constrain each of the parameters describing individual gating transitions. Scheme 3+2
is also a robust model.
The simulations show that Scheme 3+2
is able to account qualitatively for all of the features of the data
that were modeled, with only minor discrepancies (see
below). Finally, Scheme 3+2
is a relatively unique
model. After the general strategy employed by Zagotta
et al. (1994
b), we systematically considered models of
increasing complexity, in the process ruling out certain classes of models, including all of the schemes shown in
Fig. 2 except Scheme 3+2. Our final model, Scheme
3+2
, is certainly not the only model that can account
for all of the data, but it is better than these other models. With a reliable functional kinetic model in hand,
we can begin to ask detailed questions about the physical basis of gating in Shaker channels.
Properties of Scheme 3+2
In the following, we point out a number of properties
of Scheme 3+2, briefly citing the data that are the
most important for constraining them.
(A) The sum of the charges for each of the transitions in Scheme 3+2 yields a total gating charge q = 10.8 e0. This value for q is require
d to account for WT's
voltage dependence of Po at low Po (Fig. 17 B). This estimate of q is somewhat smaller than the model-independent estimate of q
13 e0 that has been obtained from measurements of the absolute charge movement
per channel (Schoppa et al., 1992
; Aggarwal and
MacKinnon, 1996
; Seoh et al., 1996
) and from the voltage dependence of channel open probability (Zagotta
et al., 1994
a; Seoh et al., 1996
). Evidence from our modeling also suggests that the total charge is likely to
be larger than 10.8 e0. Scheme 3+2
slightly underestimates the steepness of WT's voltage dependence of Po
at low Po (Fig. 11 B), which could be accounted for by
adding more charge to the model.
(B) A count of the number of transitions in Scheme
3+2 indicates that a channel that begins in the first
closed state undergoes 14 transitions to the open state,
and each of the transitions is associated with the movement of a small amount of charge (0.6-1 e0). Small partial charges are required to account for the shallow voltage dependences of kinetic features at extrem
e voltages (Figs. 13-16). Given the small partial charges, a
large number of transitions is required to achieve a sufficiently large total gating charge.
(C) An activating channel in Scheme 3+2 undergoes
two fundamentally different types of transitions before it
opens. The first 12 transitions in Scheme 3+2
correspond to transitions occurring in each of Shaker's four
subunits, while the last two transitions are distinct and are
taken to reflect the concerted movement of the four subunits. The presence of the subunit transitions with equivalent rates allows the required large number of transitions while greatly reducing the number of parameters
that need to be constrained in the modeling; the last two
transitions have kinetic properties that suggest that they
are fundamentally different from the subunit transitions.
Further tests of whether the different transitions reflect
subunit transitions or concerted transitions could use different types of mutant channels. Preliminary data from
tandem concatemeric constructs of WT and A2 channels
(having the L382A mutation) are consistent with the final steps being concerted ones (Lin et al., 1994
).
(D) The rates of the second and third subunit transitions (the intermediate transitions) are several times faster than the rates for the first subunit transition, except at very large depolarizations (V > +80 mV). Faster forward rates for intermediate transitions help account for the rising phase in the on gating current time course (Fig. 14). The most direct evidence for fast intermediate backward rates comes from V2's off gating currents, which, after intermediate depolarizations, display a fast component at some test voltages (Fig. 15 B).
(E) A small cooperative interaction between subunits
(corresponding to c = 1.3 in Scheme I) has been included in S0 S1 to help account for the rising phase
in the on gating currents (Fig. 7). This amount of interaction reflects a small deviation from independence,
especially when compared with the large interactions (c = 6) in the model of Tytgat and Hess (1992)
proposed for the mammalian Shaker (rKv1.1) channel. It
should be kept in mind that we started from the assumption that subunit transitions occur independently.
In the detailed modeling, we found no compelling reason to include more than a small interaction; indeed,
the inclusion of larger interactions of the form depicted in Scheme I yielded poor fits of gating current
time courses (Fig. 7 D).
(F) The final two transitions are associated with a charge movement of 1.8 e0. This value is constrained by data from the V2 channel: its Po-V relation (Fig. 5 B) has a steepness that corresponds to ~2 e0 and its q-V relation displays a distinct component 2 e0 in magnitude at depolarized voltages.
(G) The final transitions confer functional cooperativity. In Scheme 3+2, the final two transitions have
backward rates (
N-1 and
N) that are much slower than
those of the preceding transitions. The specific values
for the rates are constrained by tail currents at very hyperpolarized voltages and by reactivation time courses (Fig. 3). Small backward rates for the last two transitions also account for the presence of
a rising phase
and the slow decay of off gating currents after large,
long depolarizations (Fig. 19).
The most interesting implication of slow final backward rates occurs at intermediate voltages. Small backward rates give the last two transitions (and especially the
final transition) values for their equilibrium constants at
these voltages that are substantially larger than for earlier transitions. At WT's midpoint activation voltage
(48 mV), the equilibrium for CN-1
ON is 7.8, which
compares to values of 1, 1.6, 1.6, and 2.8 for S0
S1, S1
S2, S2
S3, and CN-2
CN-1, respectively. The final
two favorable transitions drive channel opening to occur with a steep voltage dependence (Fig. 17 A), in effect inducing cooperativity in the subunit charge movements.
The same process results in WT's slow and nonsigmoidal
kinetics of activation at intermediate voltages (Fig. 13).
(H) Scheme 3+2 also includes transitions to three
states (Cf1, Cf2, and CiN) that are not in the activation path.
These transitions account for additional closed-time components in Shaker's single channel behavior at large depolarizations (Fig. 4) and account for the shape of WT and
V2's Po-V relation at depolarized voltages (Fig. 5).
Discrepancies in the Predictions of Scheme 3+2
There are several discrepancies in the fits of Scheme
3+2, which suggest strategies for future improvements
in the modeling.
The most important discrepancy occurs in the fits of
the kinetic data at hyperpolarized voltages, in particular the reactivation time courses as shown in Fig. 16.
For the most negative hyperpolarizing prepulses, the
model predicts a delay in the subsequent reactivation time course that is too long. This discrepancy indicates
that the backward rates of the early and intermediate
transitions are too large at very negative voltages; that
is, backward rates are too voltage dependent. This
arises from a compromise in the modeling. As a way to
constrain the ambiguously defined third subunit transition S2 S3, we assigned S2
S3 to have the same
charges as S1
S2, and then introduced a small increase in the charges for each of the subunit transitions. A better model might make
the charge for S2
S3 larger than that for S1
&nb
sp;S2. Alternatively, there could be even more transitions than in Scheme 3+2
.
A second discrepancy is in the fits of the Q-V relation at
hyperpolarized voltages. The simulations in Fig. 17 B indicate that Scheme 3+2 is able to account for these data
quite well, but these good fits reflect the assigned small
increase in the charge for S0
S1, which, as just described, has detrimental effects on the fits of the kinetics.
One solution that could simultaneously account for the
Q-V relation and kinetics would be to increase the magnitude of cooperative interactions in this first subunit transition. The time course of the on gating currents restricts the amount of interaction that can be co
nferred
on the forward rate constant, but there is more freedom
for interactions that affect the reverse rate constant.
Two further discrepancies concern our characterization of the transitions to states that are not in the activation path. Scheme 3+2 fails to predict two discernible
rapid components that are present in the closed dwell-time distributions at depolarized voltages (Fig. 4).
These components might be better accounted for if the
rates f1 and f2 were more disparate than the values assigned. The basic version of Scheme 3+2
, which lacks
transitions to Ci states from closed states, also fails to account for a slow component in the activation time
course at large depolarized voltages (the simulations of
activation time courses of this version of Scheme 3+2
superimpose with the simulations of Scheme 2+2
in Fig. 7 A). In this paper, we have demonstrated one
plausible way of incorporating these additional transitions into the scheme (Figs. 8 and 13); in the modified
model, an additional state CiN-1 can be entered from
the last closed state. However, a proper characterization of the transitions into Ci states will require additional experimental constraints.
Generally, to obtain a better kinetic model that can
account for these discrepancies in a well-constrained
way, it would be useful to model other types of data.
These would include measurements of gating current
fluctuations (Crouzy and Sigworth, 1993; Sigg et al.,
1994
), and the analysis of rapid gating events, for example by the use of hidden Markov techniques (Venkataramanan, L., R. Kuc, and F.J. Sigworth, manuscript
submitted for publication). Also, in the same way that
we have used V2 here, current measurements from
other mutant Shaker channels (Lopez et al., 1991
; McCormack et al., 1991
; Papazian et al., 1991
; Perozo et
al., 1994
) may be used to help constrain the properties
of particular transitions in the model.
Functional Effects of the V2 Mutation
On the basis of the assumption that the V2 mutation
changes activation energies but not charge movements
in the various transitions (Schoppa and Sigworth, 1998b),
we have incorporated data from the V2 channel to better constrain features of the model. The model accounts well for V2 channel behavior and provides insight into the functional effects of the mutation.
A comparison of WT and V2 parameters for Scheme
3+2 shows that V2 has virtually no effect on the rates
of the subunit transitions. Similar forward rates account for V2's similar channel opening time courses
(at V
+67 mV) and on gating currents (Figs. 13 and 14). Similar reverse rates account for V2's similar off
gating currents at hyperpolarized voltages after small
or intermediate amplitude depolarizations (Fig. 15).
Similar equilibria for the subunit transitions account
for the fact that V2 causes little voltage shift in most of
the charge in the Q-V relation (Fig. 17 A) and little voltage shift in the relationship between prepulse voltage
and the delay in channel opening (Fig. 18 C). For the
subunit transitions, V2's forward rates are 15% smaller
than WT's, which accounts for V2's slightly slower
channel opening time course (reflected in the
a and
a
values in Fig. 13 C).
In contrast to its small effects on the subunit transitions, V2 has substantial effects on the rates of the final
concerted transitions. In V2 channels, the rates N-1,
N, and
N, are changed 2-3-fold, while the backward
rate
N-1 of the pentultimate transition is increased 56-fold. The large increase in
N-1 acco
unts for nearly all
of the qualitative differences between WT's and V2's behavior, which we will outline here.
As outlined in the previous paper (Schoppa and Sigworth, 1998b), two of V2's main kinetic effects are to accelerate the tail current and off gating current decay
(Figs. 3 and 19), and to slow the channel opening time
course (Fig. 7 B). An increase in
N-1 accelerates the deactivation time course by decreasing the channel reopening rate. The increas
e in
N-1 slows V2's channel
opening time course by a similar mechanism: a large
N-1 decreases the rate at which an opening channel in
the last closed state precedes into the open state. One
other kinetic effect of V2 is to change the channel
opening time course at the activation voltages from nonsigmoidal to sigmoidal. This can be understood
from the fact that the V2 mutation disrupts the functionally cooperative mechanism of channel opening
that is present for the normally activating Shaker channel, as described below.
The shift in the voltage dependence of V2's equilibrium Po and a component of charge in its Q-V relation
(Fig. 17 A) both directly result from the 150-fold reduction in the equilibrium of the second to last transition.
The reduction in the voltage sensitivity of charge movement (Fig. 13 A) can be explained by considering the equilibrium constants of V2's transitions at 48 mV
(WT's activation midpoint voltage). The equilibrium
constants for the transitions S0
S1, S1
S
2, S2 S3,
CN-2
CN-1, and CN-1
ON are 0.9,
1.4, 1.4, 0.03, and
1.6, respectively. The very small equilibrium of the second to last transition blocks the cooperative effect that
steepens WT's Q-V relation, causing V2's charge movement to occur across a broader voltage range. Similarly,
the reduction in the voltage sensitivity in Po can be explained by considering the equilibrium constants at
V2's activation midpoint voltage (+15 mV), which are
10, 6, 6, 0.4, and 9. At this voltage, the early transitions
are already favorable for V2; they contribute little to
the voltage dependence of Po since they do not move in
a functionally cooperative fashion as they do for WT.
The ability of Scheme 3+2 to account for all V2's
data with effects that are essentially limited to the final
two transitions has one important mechanistic implication. It supports the idea that the last two transitions
are in some way distinct from the earlier transitions
(Schoppa and Sigworth, 1998a
) and arise from different structural changes.
Comparison of Scheme 3+2 to Other Activation Gating
Models for Shaker Channels
We next compare Scheme 3+2 to other published
models that have been proposed for Shaker or Shaker-like channels. Most of the published models fall into
the classes of models depicted in Fig. 2. These include
(a) models like Scheme 1+0 (Koren et al., 1990
; Zagotta et al., 1990; Tytgat and Hess, 1992
); (b) models
like
Scheme 1+1 (Schoppa et al., 1992
; McCormack et al.,
1994
); and (c) a model like Scheme 2+0 (Zagotta et al.,
1994
b). One additional model not depicted in Fig. 2
has the opening channel undergo seven sequential
transitions (Bezanilla et al., 1994
). Most of the models that have been proposed can be ruled out by the criteria that were outlined in Stage I of the modeling in this
paper. With the exception of the model of Bezanilla et
al. (1994)
, all of the proposed models have either an
inadequate number of transitions (fewer than seven)
or an inadequate number of types of transitions that
are distinguished by their partial charges (fewer than
three).
The most fundamental difference between Scheme
3+2 and the other published models is its smaller partial charges. The voltage sensitivities of the kinetic data
at voltage extremes constrain the charge movements
accompanying transitions in Scheme 3+2
to the range
0.6-1.05 e0. The charge values assigned to transitions in
the other models are generally larger, ranging from 1.2 to 2.3 e0. Simulations such as those illustrated in Fig. 10
C show that such models with larger charges cannot account for kinetic data at voltage extremes. Also, because
of its smaller individual transition charges, Scheme 3+2
is required to have many more gating transitions than
other models to account for the steep voltage sensitivity
of gating at intermediate voltages. Stripped of the various closed states not in the activation pathway, the resulting Scheme 3+2 has a total of 37 states. However,
because of the high degree of symmetry and some simplifying assumptions about relationships between rate
constants, the rate constants and partial charges in the
scheme are determined by only 16 free parameters.
In the next two sections, we compare Scheme 3+2
to two of the most detailed models that have been proposed for Shaker activation gating. These models were
shown to account for activation, deactivation, and gating current time courses at many voltages.
Comparison of Scheme 3+2 with the ZHA Model
The first model to be compared with Scheme 3+2 is
one proposed by Zagotta et al. (1994
b):
|
Strictly speaking, this model belongs to the Scheme
2+0 class, where the configuration of all four subunits
being in the active (A)-state corresponds to the channel-open state. It has one important modification: for a
deactivating channel, the rate at which the first subunit
moves from the A to the R2 state is slowed by a factor
= 10 compared with the rates
of the other subunits. This property for the final transition ma
kes
Model ZHA behave functionally more like a model of
the class of Scheme 2+1, with subunit independence
and a forward-biased final concerted step, than a
Scheme 2+0 model with subunit independence. The
values for the rates at 0 mV (in s
1) and partial charges
(in parentheses, units of e0) are indicated. For the activation pathway (i.e., excluding the closed state B), this
model has nine free parameters.
General architecture.
The development of Model ZHA
was based on the same presuppositions as ours, invoking a fourfold kinetic symmetry to explain the existence of a large number of transitions having similar rates and voltage dependences, while minimizing the
number of free parameters. It differs from Scheme
3+2 in that it has only two subunit transitions and
lacks the two final concerted transitions.
Subunit transitions.
For the forward rates, the largest
difference is in the partial charges of the intermediate
transitions; the charges qa2 and qa3 = 0.09 e0 in Scheme
3+2 are one-third the size of q
= 0.3 e0 in ZHA. In
Scheme 3+2
, small values for qa2 and qa3 are required
to account for the very shallow voltage sensitivity of
channel opening at very large depolarizations, up to
+147 mV. Zagotta et al. (1994
b) modeled activation
time courses only up to +50 mV.
Transitions near the open state.
Scheme 3+2 and ZHA
are similar in that they both assign reverse rates for the
final transition(s) that are slower than those in preceding transitions. Slow reverse rates account for the slow channel deactivation. ZHA accounts for the slow deactivation by making the explicit channel closing rate very
slow (the rate at which the first subunit undergoes R2
A). Scheme 3+2
makes channel deactivation slow by
the combined action of the final two transitions, due to
a moderately slow explicit channel closing rate
N, and a very high reopening frequency from CN-1. Evidence
that favors our two-transition explanation for the slow
channel deactivation includes the non-single-exponential tail current decay time course (Schoppa and
Sigworth, 1998a
), and also channel reactivation time courses, which constrain
N to be large and
b) model both reactivation
and tail current time courses, but not those measured
at the very extreme voltages (down to
203 mV), which
are the most important in constraining these transitions.
Transitions to states that are not in the activation path.
Both Scheme 3+2 and Model ZHA include at least
one transition to a closed state outside the activation
path. The transition to B in ZHA corresponds to Cf1 in
our model. We find that transitions to a second closed
state Cf2 accounts for a brief component in the closed
dwell-time histograms at depolarized voltages, and also
for the fact that Po saturates at a value of 0.9 rather than 1(Fig. 5 A). After Zagotta et al., we invoke a class of intermediate closed states Ci to account for another
closed-time component.
Comparison of Scheme 3+2 to Model BPSS
The second model that we consider in some detail is
the model proposed by Bezanilla et al. (1994):
|
In this model, the channel opens after undergoing seven sequential transitions, with the indicated forward and backward rates and partial charges. The rate constants and partial charges represent a total of 20 free parameters. In the discussion of the model, the authors speculate that the first four transitions could reflect transitions occurring in each of Shaker's four subunits, while the last three would be concerted transitions. We will assume this description here.
General architecture.
Scheme 3+2 has three subunit
transitions, but Model BPSS has only one such transition. The larger number of subunit transitions in
Scheme 3+2
arose as a consequence of the smaller
transition charges, which, again, are constrained by the
kinetics at extreme voltages. Model BPSS includes a
number of concerted transitions, like Scheme 3+2
,
but their properties are quite different.
Subunit transitions.
The two models have similar forward rates for the very first transition (C0 C1 in
Model BPSS and the first subunit undergoing S0
S1
in Scheme 3+2
), but otherwise there are few similarities in the parameter estimates for the presumed subunit transitions. The most striking difference is in the
estimates for the partial charges for the forward rates:
except for the very first subunit transition, BPSS incorporates much larger partial charge values (1.0 e0 in
Model BPSS vs. qa1 = 0.47 e0 and qa2, qa3 = 0.08 e0 in
Scheme 3+2
). As discussed above, small partial
charges for forward rates are required to account for
the macroscopic ionic current time courses at depolarized voltages. Bezanilla et al. (1994)
do not model the
ionic current time courses, but, in our simulations,
Model BPSS predicts time courses that deviate markedly from what is observed. For example, at +147 mV,
BPSS predicts an activation time course with a very
short delay (
a = 40 µs) compared with the observed
value
a = 250 µs.
Transitions near the open state.
BPSS includes a concerted transition in the second to last position in the activation path that has a large valence. As evidence for a
large charge movement, Bezanilla et al. (1994) cite the steep component in Shaker's Q-V curve, and the large
gating current fluctuations that are observed at intermediate voltages. In contrast, we favor a model that does
not incorporate a transition with a large charge movement, on two grounds. The first is that a large valence
transition is not required to account for the two properties that Bezanilla et al. (1994)
cite. We and Zagotta et
al. (1994
b) have shown that models lacking a large-
valence transition can account for the steep component
of charge in the Q-V relation (Fig. 17 A), if there exists
functional cooperativity in the sequence of transitions.
Further, it turns out that the large "Stage II" gating-current variance observed by Sigg et al. (1994)
at small depolarizations can be explained by rapidly reversible
transitions carrying small charge movements. A direct
comparison of gating-current fluctuations with the predictions of Scheme 3+2
, however, awaits a later study.
Transitions to states that are outside of the activation
path.
Model BPSS includes no transitions to states outside of the activation path. In Scheme 3+2, these transitions account for Shaker's single channel data at depolarized voltages, whic
h were not modeled by Bezanilla
et al. (1994)
.
Structural Correlates to Scheme 3+2
While our data provide no direct structural information, they provide grounds for speculation about the
structural changes underlying the various transitions.
Given the recent evidence implicating the S4 transmembrane region as the major voltage sensor in activation gating (Yang and Horn, 1995; Yang et al., 1996
; Mannuzzu et al., 1996
; Larsson et al., 1996
), a reasonable
hypothesis is that the subunit transitions in Scheme
3+2
correspond to the movement of S4 in each of
Shaker's four subunits. The three subunit transitions in
Scheme 3+2
involve charge movements of 0.99, 0.6, and 0.6 e0, respectively; the main charge movement in
activation gating can therefore be pictured as occurring in small steps, involving the transport of one or
less than one elementary charge at a time. In view of
the triplet repeat motif in S4, one imagines that this region undergoes displacements of three residues at a
time to produce pseudoequivalent structures while
causing the individual charge movements. These displacements could arise as a stepwise secondary structure change, for example.
The final transitions in Scheme 3+2 couple the
movement of the main voltage sensors to channel
opening. The assigned slow backward rates for the final
two transitions imply that the last two conformational
states CN-1 and ON are uniquely stable. From the effects
of agents such as D2O, hyperosmotic solutions, and hydrostatic pressure, previous experimenters have speculated that the final transitions in activation gating correspond to large structural changes in the chann
el protein that are associated with an increase in the number
of bound water molecules (Conti et al., 1984
; Alicata et
al., 1990
; Schauf and Bullock, 1979
; Schauf and Chuman, 1986
; Zim
merberg et al., 1990
; but see Starkus et
al., 1995
). One stabilizing factor for the last two states could, then, be water-protein interactions, if these are
favorable.
The functional effects of the V2 mutation reported
here imply that the mutated leucine (L382 in the ShB sequence) is involved in stabilizing the final two states, especially the last closed state CN-1. In a previous study
from our laboratory (McCormack et al., 1993), this residue was mutated to a series of different amino acids. The
rank order of the effects of the substitutions suggested
that the final states are stabilized by specific hydrophobic interactions with the side chain of this residue.
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FOOTNOTES |
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Address correspondence to Fred J. Sigworth, Department of Cellular and Molecular Physiology, Yale University School of Medicine, 333 Cedar Street, New Haven, CT 06520. Fax: 203-785-4951; E-mail: fred.sigworth{at}yale.edu
Received for publication 3 June 1997 and accepted in revised form 24 November 1997.
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Abbreviation used in this paper |
---|
WT, wild type.
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