## Abstract

Mechanisms for the Ca^{2+}-dependent gating of single
large-conductance Ca^{2+}-activated
K^{+} channels from cultured rat skeletal muscle were
developed using two-dimensional analysis of single-channel currents recorded
with the patch clamp technique. To extract and display the essential kinetic
information, the kinetic structure, from the single channel currents, adjacent
open and closed intervals were binned as pairs and plotted as two-dimensional
dwell-time distributions, and the excesses and deficits of the interval pairs
over that expected for independent pairing were plotted as dependency plots. The
basic features of the kinetic structure were generally the same among single
large-conductance Ca^{2+}-activated
K^{+} channels, but channel-specific differences were
readily apparent, suggesting heterogeneities in the gating. Simple gating
schemes drawn from the Monod- Wyman-Changeux (MWC) model for allosteric proteins
could approximate the basic features of the Ca^{2+}
dependence of the kinetic structure. However, consistent differences between the
observed and predicted dependency plots suggested that additional brief lifetime
closed states not included in MWC-type models were involved in the gating.
Adding these additional brief closed states to the MWC-type models, either
beyond the activation pathway (secondary closed states) or within the activation
pathway (intermediate closed states), improved the description of the
Ca^{2+} dependence of the kinetic structure. Secondary
closed states are consistent with the closing of secondary gates or channel
block. Intermediate closed states are consistent with mechanisms in which the
channel activates by passing through a series of intermediate conformations
between the more stable open and closed states. It is the added secondary or
intermediate closed states that give rise to the majority of the brief closings
(flickers) in the gating.

## Introduction

Large-conductance Ca^{2+}-activated K^{+}
channels (maxi-K or BK channels),^{1} which are activated by micromolar
concentrations of intracellular Ca^{2+}
(Ca^{2+}_{i}) and by membrane potential, are present
in a wide variety of tissues (reviewed by Rudy,
1988; Marty, 1989; McManus, 1991; Latorre, 1994). BK channels act to reduce membrane excitability by
allowing K^{+} efflux though their opened pores to drive the
membrane potential more negative. A large number of studies has led to the
development of progressively more detailed models that can describe with increasing
fidelity various features of the gating of BK channels (see McManus and Magleby, 1991; Wu
et al., 1995; Cox et al., 1997,
for recent models, and the references therein for preceding models). The minimal
model of McManus and Magleby (1991), with
three open and five closed states can account for many of the basic features of the
Ca^{2+} dependence of the detailed single-channel gating
kinetics over a 400-fold range of open probability (*P*_{open
}= 0.00137–0.53). The 10-state minimal model of Wu et al. (1995) added two additional closed
states to account for activity at high Ca^{2+}_{i}, and
the 10- and 12-state minimal models of Cox et al.
(1997) can account for the average *P*_{o} (but
not the single-channel kinetics) over a wide range of
Ca^{2+}_{i} and voltage. The alpha subunits of BK
channels assemble as a tetramer (Shen et al.,
1994), and many of the above models that have been examined for the
gating of BK channels can be related to allosteric mechanisms that have been
proposed for tetrameric proteins, where each subunit can bind a single ligand. The
allosteric model of Monod et al. (1965) has
10 states with concerted transitions, whereas the model of Koshland et al. (1966) has five states with sequential
transitions. Both the concerted and sequential models are contained within the
general 35-state model proposed by Eigen
(1968) for tetrameric proteins.

The purpose of our present study is to investigate further the
Ca^{2+} dependence of the detailed single-channel gating
kinetics to examine which of the various allosteric models are consistent with the
gating. As in our previous study (McManus and
Magleby, 1991), simultaneous (global) maximum likelihood fitting of data
obtained at several different Ca^{2+}_{i }was used to
estimate the most likely rate constants and to rank the examined models, but with
the fitting of two dimensional (2-D) rather than 1-D dwell-time distributions, to
take the correlation information between adjacent open and closed interval durations
into account (Fredkin et al., 1985; Magleby and Weiss, 1990*b*).

While full maximum likelihood fitting provides one of the best methods to rank models
(Horn and Lange, 1983; Qin et al., 1997), it gives little detailed
information about where the kinetic schemes may be inadequate, or insight into how
they might be modified to better describe the data. To overcome these difficulties,
the detailed kinetic information contained in the single-channel current record, the
kinetic structure, was extracted and displayed as 2-D dwell-time distributions and
dependency plots (Magleby and Song, 1992).
Comparison of the observed kinetic structure to that predicted by the various
kinetic schemes was then used to assess the ability of the top ranked models to
account for the gating kinetics and to provide insight into how to modify the models
to improve further the description of the data. Our findings identify minimal gating
mechanisms for single BK channels that can account for both the
Ca^{2+} dependence of the kinetics and the correlations
between the durations of adjacent open and closed intervals. These models have
additional brief closed states added to the model of McManus and Magleby (1991), which is a subset of the
Monod-Wyman-Changeux model. These additional states can be located as intermediate
closed states within, or as secondary closed states beyond, the activation pathway.
Most of the flickers (brief closed intervals) are found to arise from these
additional closed states. Our observations suggest that the Monod-Wyman-Changeux and
Koshland-Nemethy-Filmer models are inadequate for BK channels, and that models based
on either extensions of these models or on the more general model proposed by Eigen (1968) or extensions of Eigen's model may
be more appropriate. A preliminary report of some of these findings has appeared
(Rothberg and Magleby, 1998).

## Materials And Methods

### Preparation

Currents flowing through single large-conductance
Ca^{2+}-activated K^{+} channels in
patches of surface membrane excised from primary cultures of rat skeletal muscle
(myotubes) were recorded using the patch clamp technique (Hamill et al., 1981). Pregnant rats were killed by
CO_{2} inhalation, and cultures of myotubes were prepared from the
fetal skeletal muscle as described previously (Barrett et al., 1982; Bello and
Magleby, 1998). All recordings were made using the excised inside-out
configuration in which the intracellular surface of the patch was exposed to the
bathing solution. Kinetic analysis was restricted to patches containing a single
BK channel. Single-channel patches were identified by observing openings to only
a single open channel conductance level during several minutes of recording in
which the open probability was >0.4. Experiments were performed at
room temperature (22–24°C). All recordings were at
+30 mV with the intracellular membrane surface positive.

### Solutions

The solutions bathing both sides of the membrane contained either 150 mM KCl and
5 mM TES (*N*-tris(hydroxymethyl)methyl-2-aminoethane sulfonate)
pH buffer, with the pH of the solutions adjusted to 7.0 (channels B06, B12,
B14), or 144 mM KCl, 2 mM TES, and 1 mM EGTA, with the pH of the solutions
adjusted to either 7.2 (channels M09 and M25) or 7.0 (M24). The solution at the
intracellular side of the membrane also contained added
Ca^{2+} (as CaCl_{2}), to bring the free
Ca^{2+} concentration at the intracellular surface
(Ca^{2+}_{i}) to the indicated levels. No
Ca^{2+} was added to the extracellular (pipette)
solution. Ca^{2+}_{i} in the absence of EGTA was
determined by atomic absorption spectrometry, and in the presence of EGTA was
determined with a Ca^{2+}-sensitive electrode as detailed
previously (McManus and Magleby, 1988).
Solutions were changed though the use of a microchamber (Barrett et al., 1982).

### Recording and Measuring Interval Durations and Identifying Normal Activity

Single-channel currents were recorded on either FM tape (DC-20 kHz) or on a digital data recorder (DC-37 kHz). Data were then low-pass filtered with a four-pole Bessel filter to give a final effective filtering of typically 6–10 kHz (−3 dB) and sampled by computer at a rate of 80–200 kHz. The filtering gave dead times (the duration of an interval that would just reach the 50% current level) for the various channels of (μs): 28.5 B06, 22.9 B12, 17.9 B14, 36.0 M25, 28.6 M09, 24.6 M24. The sampled currents were then analyzed using custom programs written in the laboratory. The methods used to set the level of filtering to exclude false events that could arise from noise, measure interval durations with half-amplitude threshold analysis, test for stability, and identify modes using stability plots, have been described previously, including the precautions taken to prevent artifacts in the analysis (McManus et al., 1987; McManus and Magleby, 1988, 1989; Magleby, 1992). We thank Owen McManus (Merck Sharp and Dohme Research Laboratories, Manukau City, New Zealand) for data from channels M09, M24, and M25, which were originally obtained for the study by McManus and Magleby (1991). The analysis in the present study was restricted to channel activity in the normal mode, which typically involves 96% of the detected intervals (McManus and Magleby, 1988). Activity in modes other than normal, including the low activity mode (Rothberg et al., 1996), was removed before analysis. Since the analysis in this study takes into account the correlation information in the durations of adjacent open and closed intervals, the sites of any removal of intervals due to activity in modes other than normal or artifacts associated with transitions to subconductance levels, were marked to avoid the later juxtaposition of open and closed intervals that were not adjacent in the original record.

### Log Binning and Plotting of 2-D Dwell-Time Distributions

Two types of 2-D dwell-time distributions were generated. The first was 2-D frequency histograms of each pair of successive (adjacent) open and closed intervals. These distributions were used for the maximum likelihood fitting to determine the minimal numbers of components (states) and also for evaluating kinetic schemes to estimate the most likely rate constants and obtain likelihood estimates. The second type was surface plots constructed using interpolated smoothing of the histograms. The 2-D distributions presented in the figures are surface plots.

The first step in generating a 2-D frequency histogram for the dwell-time
distribution was to bin adjacent open and closed intervals. Every open interval
and its following (adjacent) closed interval were binned and every closed
interval and its following (adjacent) open interval were also binned, with the
logs of the open and closed interval durations of each pair locating the bin on
the x and y axes, respectively. Each interval was thus binned twice, but with a
different adjacent interval. Including open–closed and
closed–open interval pairs in each distribution assumes microscopic
reversibility, an assumption that appears consistent with the data (Song and Magleby, 1994). The 2-D frequency
histograms for the 2-D dwell-time distributions were binned at a resolution of
10 per log unit. Further details on log binning of 2-D dwell-time distributions
may be found in Magleby and Weiss
(1990*b*) and Rothberg et al. (1997).

The surface plots for display of the 2-D dwell-time distributions were constructed from the 2-D frequency histograms in a series of steps. The first step was to smooth the histograms using a 2-D moving bin average with three bins per side, with the number of events in each bin weighted as the inverse of the distance from the central bin. Thus, the numbers of events in the four corner bins in the three-by-three moving array were multiplied by 0.707 before being added to the events in the other bins of the moving array. The total was then divided by 7.828 (4 × 0.707 to weight the corner bins plus 5 × 1 to weight the center and noncorner bins) to obtain the weighted average for the bin in the position of the center bin in the new smoothed distribution. The process was then repeated for all bins in the unsmoothed distribution to obtain the values for the new smoothed distribution. The Sigworth and Sine (1987) transform, which plots the square-root of the numbers of observations per log bin, where the bin widths are constant on a log scale, was then applied to the smoothed distribution.

Once the data were transformed, the 2-D surface plots for display were generated with the program Surfer (Golden Software, Golden, Colorado). The interpolation for the gridding with Surfer was performed using the inverse distance to a power method with smoothing, where the power was 2.0 and the smoothing factor was 0.1. Applying these smoothing procedures to distributions generated from different numbers of simulated intervals indicated that the smoothing procedures reduced features that might be expected to arise from stochastic variation, while having little effect on the basic features. The smoothing procedures were used only for visual display. The fitting was performed on the 2-D frequency histograms without averaging or smoothing. To simplify the writing, the text will refer to the fitting of 2-D dwell-time distributions presented in the figures, when, in reality, it was the 2-D frequency histograms that were fitted.

With filtering, detected intervals with durations less than approximately twice the dead time are narrowed (McManus et al., 1987; Colquhoun and Sigworth, 1995). For the fitting of kinetic models using 2-D frequency histograms, the measured durations of these intervals were corrected to their estimated true durations before binning and fitting, using the numerical method in Colquhoun and Sigworth (1995). For the surface plots presented in the figures, the measured durations were not corrected for narrowing before binning and plotting.

### Dependency plots

Dependency plots were constructed from the 2-D dwell-time distributions as
detailed in Magleby and Song (1992).
Briefly, the dependency for each bin of open-closed interval pairs with mean
durations *t*_{O} and *t*_{C} is
1

where
*N*_{Obs}(*t*_{O},*t*_{C})
is the experimentally observed number of interval pairs in bin
(*t*_{O},*t*_{C}), and
*N*_{Ind}(*t*_{O},*t*_{C})
is the calculated number of interval pairs in bin
(*t*_{O},*t*_{C}) if adjacent
open and closed intervals pair independently (at random). The method of
calculating expected frequencies for criteria that are independent (contingency
tables) is a common statistical procedure (see Mendenhall et al., 1981). The expected number of interval pairs in
bin (*t*_{O},*t*_{C}) for
independent pairing is
2

where *P*(*t*_{O}) is the probability of an
open interval falling in the row of bins with a mean open duration of
*t*_{O}, and
*P*(*t*_{C}) is the probability of a
closed interval falling in the column of bins with a mean closed duration of
*t*_{C}.
*P*(*t*_{O}) is given by the number of
open intervals in row *t*_{O} divided by the total number
of open intervals in all rows, and
*P*(*t*_{C}) is given by the number of
closed intervals in the column in *t*_{C} divided by the
total number of closed intervals in all columns. Since open and closed intervals
are paired, the total number of open intervals is equal to the total number of
closed intervals, which is equal to the total number of interval pairs in the
2-D dwell-time distribution.

### Estimating the Most Likely Rate Constants for Kinetic Schemes

The most likely rate constants for the examined kinetic schemes were determined
from the simultaneous fitting of the 2-D frequency histograms (dwell-time
distributions) obtained at three different
Ca^{2+}_{i} using an iterative maximum
likelihood fitting procedure similar to the one detailed in McManus and Magleby (1991), except that 2-D
dwell-time distributions replaced the 1-D dwell-time distributions, and the
correction method of Crouzy and Sigworth
(1990) for missed events due to filtering replaced our previous
correction method. The steps in the fitting were: (*a*) for the
given kinetic scheme and starting rate constants, an equivalent uncoupled
kinetic scheme (Kienker, 1989) with
additional kinetic states to account for missed events was calculated based on
the dead time and Ca^{2+}_{i}; (*b*)
the time constants and volumes of the 2-D components underlying the predicted
2-D dwell-time distributions for the given kinetic scheme and filtering were
calculated from the equivalent kinetic scheme using 2-D Q-matrix methods (Fredkin et al., 1985; Colquhoun and Hawkes, 1995*b*);
(*c*) the likelihood that the interval pairs in the observed
2-D dwell-time distribution were drawn from the predicted distribution was then
calculated using the predicted underlying 2-D components, as detailed in Rothberg et al. (1997);
(*d*) steps *a–c* were repeated for
the 2-D distribution obtained at each different
Ca^{2+}_{i}, and the global log likelihood for
the simultaneously fitted 2-D dwell-time distributions was then the sum of the
log likelihoods for the individual distributions; and (*e*) the
rate constants were then changed using a maximization routine. Steps
*a–e* were repeated until the rate constants for
the given scheme and dead time were found that maximized the likelihood.

Precautions were taken during the fitting to diminish the chance that the rate constants for a given fitted scheme represented a local maximum on the likelihood surface. For example, schemes were typically refit using different initial rate constants, and the size of the step change for each rate constant was varied and periodically reset during the maximization routine to increase the possibility of jumping over local maxima. In spite of these precautions, we cannot exclude that more likely fits might be found in some cases.

### Estimating the Significance of the Dependencies

The significance of the dependencies was obtained by comparing the numbers of
interval pairs in the various bins of the observed 2-D dwell-time distribution
with the number expected if adjacent open and closed intervals paired
independently. The comparison was made using a moving paired *t*
test for nine bins at a time in corresponding three-by-three arrays from the
observed and expected distributions. After each comparison, both arrays were
moved one bin, until the entire surface of the 2-D distribution was covered. The
calculated *P* value was determined from a *t*
table with eight degrees of freedom, and then converted to the log of the
*P* value times the sign of the dependency. This dependency
significance was then plotted at the centers of the moving three-by-three arrays
to generate 2-D dependency significance plots. With this transform, dependency
significance values >1.3 or <−1.3 would
indicate *P* values <0.05. Heavy lines at
±1.3 were included on the dependency significance plots to indicate
when the dependencies were significant for *P* <
0.05.

### Estimating the Theoretical Best Description of the 2-D Dwell-Time Distributions

To evaluate models, it is useful to have an estimate of the theoretical best description of the dwell-time distributions. This theoretical best description can then be compared with the best description generated by any given kinetic scheme in order to evaluate how well the kinetic scheme describes the data. If the assumption is made that the gating of the BK channel is consistent with a discrete state Markov process, such that the rate constants do not change with time (McManus and Magleby, 1989; Petracchi et al., 1991), then two different methods can be used to obtain an estimate of the theoretical best description of the 2-D dwell-time distributions that would be obtained if the discrete state Makov gating mechanism were known.

In the first method, the 2-D dwell-time distributions were fitted with sums of 2-D exponential components with all free parameters, except for the volume of one component, as the volumes of the components must sum to 1.0 (Rothberg et al., 1997). The number of components was increased until there was no longer a significant increase in likelihood. The maximum likelihood for this fit would then approximate that of the theoretical best description for a discrete state Markov model fit to the exact same data.

In the second method, an uncoupled kinetic scheme equivalent to the unknown gating mechanism was used to estimate the theoretical best fit to the data. This approach is based on an extension of the observation of Kienker (1989), who found that any given kinetic scheme can be transformed into an equivalent uncoupled kinetic scheme. Since the form of the uncoupled scheme depends only on the number of open and closed states, then it follows that the uncoupled scheme for a channel can be determined without knowing the gating mechanism, provided that the numbers of open and closed states are known. Although the gating mechanism of the uncoupled scheme is different from the unknown gating mechanism, the uncoupled scheme with appropriate rate constants should give descriptions of the single-channel data that are identical to those that would be obtained from the (unknown) underlying kinetic scheme. Hence, fitting a 2-D dwell-time distribution with an uncoupled scheme should give the same theoretical best description of the distribution as the unknown kinetic scheme, assuming a discrete state Markov model and provided that both schemes have the same number of states. To estimate the theoretical best likelihood for the simultaneous fitting of 2-D dwell-time distributions obtained under different experimental conditions, each distribution was fitted separately with an uncoupled scheme, and then the log likelihoods for the separate distributions were summed together.

Estimating the theoretical best likelihood by fitting with uncoupled schemes has
an advantage over fitting with sums of 2-D components in that uncoupled schemes
can be used to simulate single-channel data with filtering and noise, provided
that none of the rate constants in the fitted uncoupled schemes are negative,
which appears to be the case so far. While the uncoupled schemes can give an
estimate of the theoretical best likelihood, they do not have predictive value
beyond the specific experimental conditions for the data they are fitted to, as
there are no Ca^{2+}- or voltage-dependent rate constants in
the uncoupled schemes.

### Ranking the Kinetic Schemes

Normalized likelihood ratios (NLR) have been used to indicate how well any given kinetic scheme describes the 2-D dwell-time distributions when compared with the theoretical best description of the data. Normalization accounts for the differences in numbers of interval pairs among experiments, so that comparisons can be made between channels. The normalized likelihood ratio per 1,000 interval pairs is defined as 3

where ln *S* is the natural logarithm of the maximum likelihood
estimate for the observed 2-D dwell-time distributions given the kinetic scheme,
ln *T* is the natural logarithm of the maximum-likelihood
estimate for the theoretical best description of the observed distributions, and
*N* is the total number of fitted interval pairs (events) in
the observed dwell-time distributions (McManus
and Magleby, 1991; Weiss and Magleby,
1992).

A value of 1.0 for the NLR_{1000} indicates that the given kinetic scheme
describes the observed distributions as well as theoretically possible for a
discrete state Markov model. A value of 0.05 would indicate that the probability
that the observed data were generated by the given kinetic scheme is only 5% per
1,000 interval pairs of the probability that the observed data were derived from
the theoretical best description of the distributions.

The NLR gives a measure of how well different kinetic schemes describe the
distributions, but it cannot be used to directly rank schemes, since no penalty
is applied for the numbers of free parameters. To overcome this difficulty, the
Schwartz criterion has been used to apply penalties and rank models (Schwarz, 1978; Ball and Sansom, 1989). The Schwarz criterion
(*SC*) was calculated from
4

where *L* is the log-likelihood value, *F* is the
number of free parameters, and *N* is the number of interval
pairs. The scheme with the smallest *SC* is the top ranked
scheme.

### Simulation

Experimental single-channel data is distorted by the combined effects of noise
and low-pass filtering. Thus, to make valid comparisons between the observed
distributions and the distributions predicted by the kinetic models, simulated
single-channel current records were generated with filtering equivalent to that
used to analyze the single-channel current and with noise similar to that in the
single-channel current. The simulated single-channel currents were then analyzed
in the same manner used to analyze the experimental currents. The method used to
simulate single-channel currents with true filtering and noise is detailed in
Magleby and Weiss
(1990*a*).

## Results

Currents flowing through a single large-conductance
Ca^{2+}-activated channel in an inside-out patch of membrane
excised from a cultured rat skeletal muscle cell are shown in Fig. 1, *A* and *B,* at
two different time bases. The complexity of the underlying gating process is
reflected in the wide range of the durations of the open and closed intervals and
the apparent grouping of the intervals into bursts. Because of the stochastic nature
of single-channel gating (Colquhoun and Hawkes,
1995*a*), extracting the essential kinetic information
about the underlying gating process requires large amounts of stable single-channel
data to overcome the stochastic variation. Fig. 1,*C* and *D,* presents stability plots of
the mean open and closed interval durations during activity in the normal mode,
which includes ∼96% of the intervals (McManus and Magleby, 1988). The stability plots shown in Fig. 1 are based on 57.6 s of stable data after
artifacts and transitions to modes other than normal were removed. These stability
plots indicate that the analyzed data are reasonably stable, and are representative
of the data analyzed in this study to investigate
Ca^{2+}-dependent gating mechanisms.

### 2-D Dwell-Time Distributions

For channels that gate between two conductance levels, open and closed, 2-D
dwell-time distributions contain essential kinetic information from the
single-channel current record, including correlation information that gives
information about transition pathways among states (Fredkin et al., 1985; Rothberg et al., 1997). Fig. 2
shows 2-D dwell-time distributions for six single BK channels, each from a
different inside-out patch of surface membrane. The membrane potential in each
case was +30 mV and the Ca^{2+}_{i} was
selected to give a *P*_{o} near 0.5. The 2-D dwell-time
distributions plot how frequently open intervals of a specified duration occur
next (adjacent) to closed intervals of a specified duration. The log of the
durations of each adjacent open and closed interval locate the bin on the x and
y axes, respectively, and the z axis plots the square root of the number of
observations per bin (see materials and methods).

The 2-D dwell-time distributions in Fig. 2 can be described by the sums of 2-D exponential components, where the number of 2-D components is given by the product of the numbers of open and closed states (Fredkin et al., 1985; Rothberg et al., 1997). Since BK channels typically enter a minimum of three to four open and five to seven closed states during normal activity (McManus and Magleby, 1988), there would be from 15–28 possible 2-D components underlying each 2-D dwell-time distribution. The square-root transformation (Sigworth and Sine, 1987) used for the 2-D dwell-time distributions would generate peaks at the time constants of the 2-D exponential components (Rothberg et al., 1997). However, only the components with the largest volumes or whose time constants are well separated from the other components would generate visually detectable peaks.

To facilitate reference to the various peaks and regions of the 2-D plots, the 2-D distributions in Fig. 2 and in the subsequent figures are divided into six general regions indicated by the numbers 1–6 in the figures and referred to as #1–#6 in the text. For example, #1, #2, and #3 indicate the regions of brief openings adjacent to brief closings, intermediate closings, and long closings, respectively, while #4, #5, and #6 indicate the regions of long openings adjacent to brief closings, intermediate closings, and long closings, respectively.

The highest peak (#4) in the 2-D dwell-time distributions in Fig. 2 is located in the same general position for
each channel and indicates that the most frequent interval pairs for all the
examined channels consisted of long (∼2-ms) openings adjacent to
brief (∼0.05-ms) closings. These dominant interval pairs are readily
apparent in the single-channel record in Fig. 1
*B* as longer open intervals adjacent to the brief closed
intervals (flickers). Other peaks and inflections are also apparent in the
plots, indicating that additional components can be detected visually. For
example, each plot contains a visible peak indicating a component of long
(∼2-ms) openings adjacent to long (∼10-ms) closings
(#6), and a component of brief openings (∼0.1 ms) also adjacent to
long closings (10 ms) (#3).

The 2-D dwell-time distributions in Fig. 2 indicate the relative frequency of occurrence of the various classes of adjacent open- and closed-interval durations that must be accounted for by kinetic gating mechanisms. The channels for Fig. 2 were selected to be representative of the more than 12 channels examined in this manner. Channels M25, M09, and M24 are channels 1, 2, and 5, respectively, in McManus and Magleby (1991), and were included to allow comparison of the 2-D analysis in this present study with the 1-D methods used previously.

### Kinetic Similarities and Heterogeneities for BK Channels from the Same Preparation

While there are a number of basic similarities in the 2-D dwell-time
distributions from the six different BK channels in Fig. 2, there are also a number of differences. For example,
channels M25 and M09 have a prominent middle ridge (#5), indicating a component
of long openings (∼2 ms) adjacent to intermediate duration closed
intervals (∼0.5 ms). This component is less apparent or appears to
be missing for the other four channels. Although there were some differences in
the level of filtering and *P*_{o} among the different
channels (see Fig. 2,
*legend*), it is unlikely that this would account for the
differences in the 2-D dwell-time distributions as there was no evident
relationship between the observed differences and the small differences
*P*_{o}, or the levels of filtering for the various
channels.

The obvious kinetic differences among the channels in Fig. 2 are consistent with previous studies showing differences
in Ca^{2+} sensitivity and/or gating among different native
BK channels from the same tissue (McManus and
Magleby, 1991; Wu et al.,
1996). Since the six different channels in Fig. 2 were obtained from native tissue, it is possible that the
differences in kinetics might reflect channels with different splice variations
(Atkinson et al., 1991; Adelman et al., 1992; Butler et al., 1993; Lagrutta et al., 1994). Alternatively, other factors may be involved
(see discussion) since expressed cloned channels without the potential
for alternative splicing can also display kinetic differences among channels
(Silberberg et al., 1996). Kinetic
heterogeneity has been observed for other types of channels as well (e.g., Auerbach and Lingle, 1986; Patlak et al., 1986).

### Displaying the Correlations between Adjacent Open and Closed Intervals with Dependency Plots

Although information on the correlation between adjacent open and closed intervals is contained within the 2-D dwell-time distributions, this information is not readily apparent from visual inspection. Dependency plots provide a means to extract this correlation information in a form that can give insight into the connections among open and closed states involved in the gating (Magleby and Song, 1992).

Dependency plots for the six channels shown in Fig. 2 are presented in Fig. 3. The plots present the fractional differences between the observed number of adjacent open and closed intervals of indicated durations and the hypothetical number that would be observed if all the open and closed intervals paired independently. Dependencies of +0.5 or −0.5 would indicate a 50% excess or deficit, respectively, of interval pairs over that expected for random pairing (see Eq. 1). Positive dependencies suggest that the open and closed states underlying the interval pairs in excess are effectively connected, and negative dependencies suggest that the open and closed states underlying the interval pairs in deficit are not effectively connected (Magleby and Song, 1992).

The dependency plots in Fig. 3 show some common kinetic features for the six different BK channels: a deficit of brief open intervals adjacent to brief closed intervals (#1), an excess of brief open intervals adjacent to long closed intervals (#3), an excess of long open intervals adjacent to brief closed intervals (#4), and a deficit of long open intervals adjacent to long closed intervals (#6). It will be shown in the next section that these specific dependencies are significant. Thus, the basic features of the dependency plots in Fig. 3 suggest that kinetic models for the gating should contain dominant transition pathways between the open and closed states underlying the brief open intervals and the long closed intervals (#3), and between the open and closed states underlying the long open intervals and brief closed intervals (#4). These dominant transition pathways would generate the positive dependencies. In addition, there should not be dominant transition pathways between the open and closed states underlying the brief open and closed intervals (#1) and between the open and closed states underlying the long open intervals and long closed intervals (#6). Whether a transition pathway is dominant or not depends on the relative probability of whether that pathway is taken among the possible pathways from any given state.

Interestingly, the dependency plot of channel M09 showed features that were not
observed in any of the other BK channels: most notably, a smaller excess of
brief open intervals adjacent to longer closed intervals (#3). This atypical
kinetic structure of channel M09 is consistent with differences in the gating
mechanism for this channel when compared with four other channels, determined in
a previous study (channel 2 vs. channels 1, 3, 4, and 5 in McManus and Magleby, 1991). That channel M09 is atypical is
readily apparent from the dependency plots in Fig. 3 obtained at a single Ca^{2+}_{i},
indicating the power of dependency plots. The previous determination that
channel M09 was atypical required hundreds of hours of analysis of 1-D
dwell-time distributions obtained at three or more
Ca^{2+}_{i} for each channel.

### Determining the Significant Features of Dependency Plots

While the basic features of the dependency plots were consistent among most
channels, there were also channel-specific features in the plots. Therefore, it
was of interest to determine which features were part of the kinetic structure
and which might have arisen from factors such as stochastic variation, noise in
the single-channel records, and distortions produced by low-pass filtering. The
significance of the dependencies were estimated by two different approaches:
using simulation and applying a paired *t* test.

The first approach used simulation to estimate the magnitude of the expected
variations in the dependency that would arise from stochastic variation,
filtering, and noise. A 2-D dwell-time distribution and associated dependency
plot were simulated for Scheme I, a
gating mechanism that would give theoretical dependencies of zero (Magleby and Song, 1992). Scheme I was first fitted to the 2-D dwell-time
distribution for channel B06 to estimate the most likely rate constants. These
rate constants were then used with Scheme I to simulate a single-channel current record with noise and
filtering like that in the experimental data and with the same number of
intervals as for channel B06. The simulated current record was then analyzed to
obtain the 2-D dwell-time distribution and dependency plot shown in Fig. 4. The deviations of the dependency plot from
zero in Fig. 4
*B* then give an estimate of the variations that would be
expected due to the combined effects of noise, filtering, and stochastic
variation, since the expected theoretical dependencies for Scheme I would be zero.
(Scheme I)

From Fig. 4 and similar simulations of this type, the magnitudes of the variations of the dependency from zero were found to depend on the location in the plot. There was little deviation from zero for the dependency of long open intervals adjacent to brief closed intervals (#4) because of the large numbers of interval pairs that contributed to the calculation of dependency for this location, as seen in the 2-D dwell-time distribution. Elsewhere in the plot, the dependencies that would be expected to arise from stochastic variation typically fell within ±0.2 from zero.

A comparison of the predicted dependency plot in Fig. 4
*B* to the observed dependency plots in Fig. 3 indicates that Scheme I is inconsistent with the gating of BK channels. Nevertheless, the
2-D dwell-time distribution predicted by Scheme I in Fig. 4
*A* appears similar (but not identical) to the observed
dwell-time distribution in Fig. 3 for
channel B06. Hence, the ability of a model to approximate the 2-D dwell-time
distribution of the data does not necessarily establish that even the basic
features of the proposed gating mechanism are correct. 1-D distributions can be
even less sensitive for model discrimination (Magleby and Weiss, 1990*b*).

The second approach to estimate the significance of the dependencies involved a
direct calculation of significance. Figs. 4
*C* and 5 plot the
statistical significance of the dependencies. The significance was estimated by
comparing the numbers of intervals in the observed 2-D dwell-time distribution
with the number expected if adjacent open and closed intervals paired
independently of one another. The comparison was made using a paired
*t* test (details in materials and methods). The
distributions of dependency significance in Fig. 5 plot the significance of the dependencies in Fig. 3 as the logarithm of the estimated
*P* value, which is then multiplied by the sign of the
dependency to indicate whether the paired intervals are in excess or deficit.
The heavy lines on the plots at −1.3 and 1.3 indicate a significance
level of *P* = 0.05. Absolute values of dependency
significance >1.3, 2, 3, and 4 would indicate *P*
< 0.05, < 0.01, < 0.001, and <
0.0001, respectively.

The dependency significance plots in Fig. 5
*A* are for the same six channels and orientation as in Fig.
3 (front view). Fig. 5
*B* presents the backside views of the dependency significance
plots for two of the six channels to show the significant deficit of long
openings adjacent to long closings (#6). Similar significant deficits were seen
for the other four channels. It is important not to confuse the significance of
the dependency with the magnitude of the dependency. Fig. 3 shows the magnitude of the dependencies. Fig. 5 shows whether the indicated magnitudes are
significant. Fig. 4
*C* provides an independent measure of the applied significance
test, showing that none of the dependencies arising from stochastic variation
were significant, as would be expected for Scheme I.

### Kinetic Structure of BK Channels

The 2-D dwell-time distributions and dependency plots in Figs. 2 and 3 and the significance of the dependencies in Fig. 5 are representative of dependency plots obtained from more than 12 channels. These plots present the essential kinetic information contained in the single-channel current records, indicating the kinetic structure of the BK channels. It is this information that must be accounted for by proposed gating mechanisms.

### Idealized Dependency Plots from Single-Channel Data

It would be useful if there were a means to eliminate the variation in dependency plots arising from the analysis of limited amounts of single-channel data. We have developed an approximate means to do this, by fitting the data with an uncoupled (generic) kinetic scheme. Since the uncoupled scheme allows direct transitions from each open state to each closed state (Kienker, 1989), the correlations between the adjacent open and closed intervals can be described by such a scheme. The uncoupled kinetic scheme for any discrete state Markov model with four open and six closed states is given by Scheme II. The scheme is uncoupled because there are no direct transition pathways from one open state to another or from one closed state to another. (Scheme II)

Although the actual kinetic scheme for channel B06 is not known, if the data are described by four open and six closed exponential components, which is the case for the distribution in Fig. 3, then Scheme II with most likely rate constants should give the same description of the experimental data as the unknown kinetic scheme (see materials and methods), and should thus be capable of describing the kinetic structure. Scheme II can then be used to generate simulated single-channel currents with different levels of noise and filtering, to examine the effects of these variables on the kinetic structure.

Scheme II was fitted to the 2-D
dwell-time distribution for channel B06 to obtain the most likely rate
constants. Scheme II was then used to
simulate single-channel currents with filtering and noise similar to that in the
experimental data, and also without filtering and noise. 1,000,000 detected
intervals were simulated in each case to reduce stochastic variation to
negligible levels. The simulated single-channel currents were then analyzed to
generate the idealized 2-D dwell-time distributions and dependency plots
presented in Fig. 6. These idealized
distributions give an estimate of what the experimental distributions would look
like without stochastic variation (Fig. 6
*A*), and without filtering, noise, or stochastic variation
(*B*). A comparison of the idealized distributions in Fig.
6
*A* to those for channel B06 in Figs. 2 and 3 suggest that
the minor variations in the experimental dependency plots most likely arise from
stochastic variation due to the analysis of limited amounts of data. A
comparison of the distributions in Fig. 6
*A* with filtering and noise to those in Fig. 6
*B* without filtering and noise indicates that filtering and
noise do not change the basic features of the kinetic structure, except for
those features involving adjacent intervals in which one or both intervals have
durations less than two dead times, where the filtering attenuates the durations
(McManus et al., 1987; Colquhoun and Sigworth, 1995).

### Calcium Dependence of the Kinetic Structure

McManus and Magleby (1991) have previously
detailed the Ca^{2+} dependence of the 1-D dwell-time
distributions of open and closed interval durations for BK channels from rat
skeletal muscle. The Ca^{2+} dependence of the 1-D
distributions for the seven additional BK channels analyzed in this manner for
the present study (data not shown) were consistent with those from the BK
channels analyzed previously. Increasing
[Ca^{2+}]_{i} increased
*P*_{o} by increasing the mean open interval duration
and decreasing the mean closed interval duration. The
Ca^{2+}-dependent shifts in mean interval durations
arose mainly from decreases in the time constants and areas of the longer closed
components and increases in the time constants and areas of the longer open
components. In contrast to the changes in the time constants of the longer open
and closed components, the time constants of the shortest open and closed
components appeared relatively independent of
Ca^{2+}_{i}.

To gain further insight into the Ca^{2+} dependence of the
gating, the Ca^{2+} dependence of the kinetic structure was
examined. Results are presented in Fig. 7
for a representative channel (B06) at three different
Ca^{2+}_{i} of 5.5, 8.3, and 12.3
μM, which resulted in open probabilities of 0.061, 0.202, and 0.504.
Examples of single-channel current records at each level of activity are shown
in Fig. 7
*A* and the kinetic structures are shown in Fig. 7
*B*. The greater variation in the dependency at 5.5 and 8.3
μM Ca^{2+}_{i} is most likely due to the
fewer intervals obtained for analysis at these distributions due to the lower
levels of activity. The idealized kinetic structures obtained after removing the
effects of stochastic variation (as discussed for Fig. 6
*A*) are shown in Fig. 7
*C*. These idealized plots show the dominant features of the
kinetic structure.

The 2-D dwell-time distributions show a shift in the time constants of the longer
closed intervals towards shorter durations with increasing
Ca^{2+}_{i}, and also a shift in the frequency
of occurrence of the longer closed intervals towards briefer closed intervals,
and of briefer open intervals towards longer open intervals. The characteristic
saddle-like appearance of the dependency plots, which indicates an inverse
relationship between the durations of adjacent open and closed intervals (McManus et al., 1985), was maintained at
each level of activity, suggesting that the basic underlying gating mechanism
remained unchanged over the examined range of
*P*_{o}.

### A Simple Gating Mechanism Approximates the Basic Features of the
Ca^{2+} Dependence of the Kinetic Structure

The basic features of the Ca^{2+} dependence of the 1-D
dwell-time distributions from BK channels from cultured rat skeletal muscle can
be described by Scheme III, which
contains three open and five closed states (McManus and Magleby, 1991). To test whether this scheme might also
account for the basic features of the kinetic structure, 2-D distributions
obtained at three different Ca^{2+}_{i} from each
channel were simultaneously fitted to Scheme III to estimate the most likely rate constants for each channel.
These rate constants were then used with Scheme III to simulate a single-channel current record that was then
analyzed to obtain the predicted kinetic structure. The current record was
simulated with filtering and noise like that in the experimental data, and
10^{6} simulated intervals were analyzed for each distribution.
(Scheme III)

Fig. 8 shows the 2-D distributions and
dependency plots predicted by Scheme III for channel B06. Comparison of these predicted kinetic
structures to the observed and idealized kinetic structures in Fig. 7, *B* and *C*,
shows that Scheme III approximates the
basic features of the Ca^{2+} dependence of the kinetics.
However, there are some clear differences between the observed and predicted
distributions. Scheme III predicted a
greater deficit of brief open intervals adjacent to brief closed intervals for
5.5 μM Ca^{2+}_{i} than was observed in
the experimental data (#1). That is, Scheme III generated an insufficient number of brief open intervals
adjacent to brief closed intervals. This underprediction by Scheme III prompted a search for adjacent brief
open and closed intervals in the single-channel current record. Fig. 9 shows examples of such pairings during
normal gating with 5.5 μM Ca^{2+}_{i}.
Approximately 20– 30% of adjacent brief open and closed intervals
were found at the beginnings and endings of bursts and 70– 80% were
found within bursts.

In addition, Scheme III predicted a
greater excess of brief open intervals adjacent to intermediate duration closed
intervals than were observed, which was apparent at all three levels of
Ca^{2+}_{i} in both the dependency plots (#2)
and the 2-D dwell-time distributions. Finally, Scheme SIII underpredicted the observed excess of long open
intervals adjacent to the brief closed intervals (#4).

Similar results were found for four additional channels examined in detail.
Scheme III captured the basic features
of the Ca^{2+} dependence of the kinetic structure while
giving the same types of over- and underpredictions, often with greater
differences than those detailed for channel B06 above.

### The Dependency Plots Suggest how Scheme III Might Be Modified to Better Describe the Kinetic Structure

To gain possible insight into why Scheme III did not account for all the features of the kinetic structure,
the mean lifetimes of the kinetic states in Scheme SIII were calculated from the most likely rate constants.
The results are presented in Scheme III(8.3) for channel B06 with 8.3
μM Ca^{2+}. The solid line encloses the major
gating pathways, and the mean lifetimes of the various states are indicated (in
milliseconds). For kinetic schemes with compound open and closed states, it can
be difficult, if not impossible, to designate which states contribute to the
observed components of interval durations (Colquhoun and Hawkes, 1981, 1995). Nevertheless, for certain schemes and rate constants, such as
Scheme III, such assignments can be
tentatively made for the purposes of investigating why the scheme did not
account for the complete features of the kinetic structure. The assignments were
made by changing the lifetimes of the states one at a time, in small amounts, to
determine which components were affected in the calculated distributions. The
details of this approach will be presented elsewhere.
(Scheme III(8.3))

Scheme III predicted too few brief open
intervals adjacent to brief closed intervals in low
Ca^{2+}_{i} (#1 in Figs. 7 and 8). For Scheme
III, brief openings, which are
mainly from O_{3}, would occur infrequently adjacent to brief closings,
which are mainly from C_{5}, since transitions from O_{3} and
C_{5} must pass through either the intermediate closed state,
C_{6}, or the intermediate open state, O_{2}. Transitions
through either of these intermediate states would extend the mean duration of
the open or closed intervals so that the intervals would no longer be brief. To
compensate for the inability of Scheme III to generate a sufficient number of brief open intervals
adjacent to brief closed intervals, a transition pathway to an additional brief
duration closed state could be added to O_{3}, as in Scheme IIIA. This
would increase the number of brief openings adjacent to brief closings.
(Scheme IIIA)

Scheme III also generated an excess of
brief openings adjacent to intermediate-duration closings (#2 in Figs. 7 and 8). This excess is likely to arise from direct transition between
O_{3} and C_{6}. While the lifetime of C_{6} is
∼0.4 ms, transitions such as
O_{3}-C_{6}-C_{5}-C_{6}-O_{3} would
double the mean duration of the closed interval to ∼1 ms.
Transitions such as O_{3}-C_{6}-O_{3}-C_{6}
could also increase the observed mean duration of the closed interval associated
with state C_{6}, since some of the dwell-times in O_{3} would
be too brief to detect due to the filtering, making the apparent closed interval
longer due to missed events (Blatz and Magleby,
1986; Colquhoun and Sigworth,
1995). The brief duration closed state C_{11} connected to
O_{3} in Scheme IIIA would increase the number of brief openings
adjacent to brief closings. C_{11} would also act to decrease the excess
of brief openings adjacent to intermediate closings (mainly from C_{6})
by diverting some of the transitions from O_{3} away from
C_{6}.

Finally, Scheme III generated an
insufficient number of brief closed intervals adjacent to long open intervals
(#4 in Figs. 7 and 8). In Scheme III,
brief closings adjacent to long openings would arise mainly from transitions
such as O_{1}-O_{2}-C_{5}-O_{2}-O_{1}.
Adding the brief duration closed states C_{9} and C_{10} to
O_{1} and O_{2}, as in Scheme IIIA, would increase the
number of brief closings adjacent to long openings, as was observed in the
experimental data.

Thus, a comparison of the differences between the observed dependency plots and
those predicted by Scheme III suggest
that there may be additional brief closed states directly connected to the open
states, as proposed in Scheme IIIA. Transitions to these brief closed states
would add flickers to the single-channel current record. Since these additional
closed states are not on the activation pathway, they will be referred to as
secondary closed states, giving rise to secondary flickers. The primary flickers
would involve transitions to C_{5} on the activation pathway. The
addition of the secondary closed states would represent a minimal extension of
Scheme III, so that the modified Scheme
IIIA should still capture the basic features of the gating, while reducing some
of the differences between the observed and predicted dependency plots. Further
reason for investigating closed states beyond the activation pathway comes from
the observation of Wu et al. (1995) that
such states were required to account for activity with high
Ca^{2+}_{i} for the gating for BK channels in
hair cells. Such secondary closed states could arise from either a secondary
gate or channel block, as will be considered in the discussion.

### Scheme IIIA, which has Secondary Closed States, Improves the Description of the Kinetic Structure

To examine whether Scheme IIIA improved the descriptions of the kinetic structure, the 2-D dwell-time distributions and dependency plots predicted by Scheme IIIA were plotted in Fig. 10 for channel B06. During the fitting, the mean lifetimes of the three secondary closed states were made identical by constraining the rate constants for the transitions from the three secondary closed states to the open states to have the same value. Thus, the secondary closed states in Scheme IIIA might be expected to add one additional closed component.

For all five examined channels, Scheme IIIA gave a better description of the
Ca^{2+}-dependent kinetics than Scheme III, as indicated in Table I by both the normalized likelihood ratio
values, NLR_{1000}, and the rankings, R, by the Schwarz criterion, which
applies a penalty for additional free parameters (see Eqs. 3 and 4 in materials and methods).

As would be expected from the improved NLR values and rankings, the plots of the
kinetic structure predicted by Scheme IIIA more closely approximated the
experimental data, as can be seen by comparing the kinetic structures predicted
by Schemes III and IIIA in Figs. 8 and
10, respectively, to the observed and
idealized kinetic structures in Fig. 7,
*B* and *C*. Scheme IIIA predicted the
observed excess of long open intervals adjacent to brief closed intervals (#4),
which was underpredicted by Scheme III.
Scheme IIIA also predicted the magnitude of the deficit of brief open intervals
adjacent to brief closed intervals better than Scheme SIII, which overpredicted the deficit (#1). Scheme IIIA
also predicted the magnitude of the excess of brief open intervals adjacent to
intermediate closed intervals (#2) better than Scheme SIII, although the excess was still overpredicted by
Scheme IIIA. Thus, although Scheme IIIA gave a greatly improved description of
the kinetic structure over Scheme III
(the NLR_{1000} was increased an average of ∼1,000-fold for
the five examined channels), Scheme IIIA still did not capture all of the
features.

A measure of the less than perfect description of the gating kinetics by Scheme
IIIA is given by the NLR_{1000}. The values of the NLR_{1000}
for Scheme IIIA ranged from 8.9 × 10^{−7} for
channel B14 to 0.0019 for channel M25 (Table I). A value of 1.0 for the NLR_{1000} indicates that the
given kinetic scheme describes the observed distributions as well as
theoretically possible for a discrete state Markov model. A value of 0.0019
indicates that the probability that the observed 2-D dwell-time distributions
were generated by the given kinetic scheme is only 0.19% per 1,000 interval
pairs of the probability that the observed data were derived from the
theoretical best description of the distributions. This suggests that Scheme
IIIA is still too simple.

Interestingly, an NLR_{1000} as small as 10^{−7} can
give, by visual inspection, reasonable descriptions of the 2-D dwell-time
distributions. The reason for this can be seen by calculating the
NLR_{1} for a single interval pair from the NLR_{1000} with:
5

An NLR_{1000} of 10^{−7} translates to an
NLR_{1} of 0.984 per interval pair, indicating an average difference
in likelihood of 1.6% per interval pair between the predicted and theoretical
best likelihood estimates, whereas an NLR_{1000} of 0.0019 translates to
an NLR_{1} of 0.994 per interval pair. Thus, small differences between
the observed and predicted distributions can reduce the NLR_{1000} far
below 1.0. (Note that NLR_{1} gives average likelihood differences and
not average errors, which tend to be greater.)

The small differences in likelihood per interval pair between the predicted and
theoretical best likelihood estimates suggests that Scheme IIIA should generate
single-channel current records like those from the experimental data. That this
is the case is shown in Fig. 11, which
presents simulated single-channel currents predicted by Scheme IIIA for channel
B06 for comparison to the experimental records in Figs. 7
*A* and 9. The simulated
records include noise and filtering equivalent to that of the experimental data.
Due to stochastic variation, the simulated and experimental current records
could never look identical, even if the true gating mechanism were known, but
the simulated data do capture the kinetic features of the experimental records,
including the calcium dependence (Fig. 11
*A*) and the occasional observation of adjacent brief open and
closed intervals (Fig. 11
*B*).

The rate constants for Schemes III and IIIA are indicated in Fig. 12, *A* and
*B,* for five examined channels. In general, the values of
the specific rate constants were similar for the five channels, although there
were notable exceptions. For example, for Schemes III and IIIA, rates
*k*_{1−4} and
*k*_{4−1} had values of ∼100
s^{−1} and 2,000 s^{−1},
respectively, for some of the channels, and values near zero for the other
channels.

Since the kinetic structure predicted by Scheme IIIA did not approach the
theoretical best description of the Ca^{2+}-dependent
gating, a more complex version of Scheme IIIA was considered. Scheme IIIA, with
three open and eight closed states, would generate three open and six closed
components in the dwell-time distributions because the three secondary closed
states have the same lifetimes and would be detected as a single component. Yet,
estimates of the minimal number of components for BK channels suggest up to four
open and seven to eight closed components (McManus and Magleby, 1988, 1991), suggesting that there may be more detectable states than in
Scheme IIIA. Thus, it is possible that the secondary closed states have
different lifetimes, since each secondary closed state has a different number of
Ca^{2+} bound, which could alter its stability.

To test this possibility, the lifetimes of the secondary closed states were no
longer constrained to be identical during the fitting. The results are indicated
under Scheme IIIA′ in Table I, where Scheme IIIA′ improved the NLR for all five channels
and ranked higher than Scheme IIIA for four of the five channels. In general,
for Scheme IIIA′, the lifetimes of the three secondary closed states
differed approximately twofold, but were still brief. For example, the lifetimes
of C_{9}, C_{10}, and C_{11} for Scheme
IIIA′ for channel B06 were 0.036, 0.038, and 0.066 ms,
respectively.

We also examined whether it was necessary to have three secondary closed states
in Scheme IIIA′ or whether two would be sufficient. Deleting the
leftmost secondary closed state in Scheme IIIA′ resulted in a lower
ranking for four of the five channels (not shown), suggesting that at least
three, rather than two, additional closed states may be required. Modifying
Scheme IIIA′ to have two secondary closed states gives a model
similar to the model of Wu et al. (1995),
except that in the model of Wu et al.
(1995) the transitions to the secondary closed states are
Ca^{2+} dependent. We examined whether the model of
Wu et al. (1995) with
Ca^{2+}- dependent transitions to the secondary closed
states would improve the descriptions of the kinetic structure. For all five
examined channels, this model ranked below Scheme IIIA (not shown). This
observation does not necessarily exclude the model of Wu et al. (1995), as they studied BK channels in cochlea
hair cells, which may well have different gating properties than BK channels in
cultured rat skeletal muscle, and it is also unclear whether they excluded
activity in modes other than normal, as we have done.

While Scheme IIIA′ improved the descriptions of the kinetic structure when compared with Scheme IIIA, the increases were small, so that the predicted kinetic structure was still inadequate, appearing similar to that in Fig. 10. Thus, Scheme IIIA′ was still too simple to account entirely for the observed kinetic structure.

### Scheme IIIB with Intermediate Closed States Gave Similar Descriptions of the Kinetic Structure as Scheme IIIA′

The above sections explored the effects of adding needed additional brief closed
states as secondary states to form Scheme IIIA, in which the secondary closed
states would be consistent with channel block or the closing of secondary gates.
However, such additional brief closed states might also arise by other
mechanisms. Theoretical considerations of gating for a channel with four
subunits in which each subunit can bind an agonist and also undergo a
conformational change suggests that there may be large numbers of intermediate
closed states within the activation pathway (Eigen, 1968; Fersht, 1977;
McManus and Magleby, 1991; Cox et al., 1997). It would be difficult,
if not impossible, to define the rate constants for such a large number of
additional states, but an examination of whether such models might account for
gating was made by exploring whether the addition of a few intermediate closed
states would improve the description of the kinetic structure. Scheme IIIB
extends Scheme III into a minimal model
with the three intermediate closed states C_{9}, C_{10}, and
C_{11}. The horizontal transition pathways between the three
intermediate states have been excluded to reduce the numbers of parameters in
the fitting.
(Scheme IIIB)

Scheme IIIB, which was fitted without placing constraints on the lifetimes of the
intermediate closed states, typically gave likelihood values slightly less than
Scheme IIIA′*.* Since Scheme IIIB had the same number
of free parameters as Scheme IIIA′, Scheme IIIB typically ranked, on
average, just below Scheme IIIA′ (Table I), and predicted a similar kinetic structure (not shown).
Thus, Scheme IIIA′ with secondary closed states and Scheme IIIB with
intermediate closed states gave similar descriptions of the data. The rate
constants for Scheme IIIB are indicated in Fig. 12
*C* for five examined channels, where it can be seen that the
lifetimes of the intermediate states are brief. Similar descriptions of the data
were also found (when examined) for the more complex schemes to be presented
below when the secondary closed states were replaced with the same number of
intermediate closed states. Hoshi et al.
(1994) have previously indicated that intermediate closed states can
be difficult to distinguish from secondary closed states for steady state data.
Thus, in the following sections, only the models with the secondary closed
states will be presented in detail, with the implication that the intermediate
state models would give similar results.

### More Complex Models with Additional States

Four additional classes of models were considered, both with and without secondary closed states. These models explored the effect of adding additional open states in different positions to Scheme III to give four open components, as can often be observed in the experimental data.

Schemes IV and V allow channel opening with one or no Ca^{2+}
bound. Experimental data suggest that such openings may occur (Meera et al., 1996; Cui et al., 1997). Scheme V is the Monod, Wyman, and Changeux (MWC; Monod et al., 1965) model for allosteric proteins. Many of
the examined schemes for the BK channel are subsets of this model (McManus and Magleby, 1991; Cox et al., 1997). Scheme VI adds an open and closed state by
assuming that the binding of the second Ca^{2+} can lead to
closed states with different properties, depending on whether the second
Ca^{2+} binds to a subunit adjacent to or diagonal from
the subunit with the first bound Ca^{2+}. The open states
reached from these two different closed states could have different properties,
leading to a fourth open component. Scheme VII adds an open and closed state by assuming that the additional
open and closed states are reached by a Ca^{2+}-independent
conformational change after the binding of four Ca^{2+}.
Finally, the effects of adding a secondary closed state to each of the open
states for each scheme was examined to obtain Schemes IVA–VIIA.
(Scheme IV)
(Scheme IVA)

Based on the average of the rankings for the five channels, Scheme VIIA was the top ranked scheme, followed by Scheme VII, and then Schemes VIA, IIIA′, IIIB, IVA, IIIA, IV, V (MWC), and III. Schemes with secondary closed states ranked above the same scheme without secondary closed states, even though there was a penalty for the additional free parameters associated with the secondary closed states. Replacing the secondary closed states with intermediate closed states (when examined) gave similar descriptions of the data. Thus, these observations are consistent with models in which additional brief closed states, as either secondary or intermediate closed states, contribute to the gating. (Scheme V) (Scheme VA) (Scheme VI) (Scheme VIA) (Scheme VII) Scheme VIIA -->

The predicted kinetic structures for the schemes ranked above Scheme IIIA in
Table I appeared similar to that for
Scheme IIIA shown in Fig. 10, so have not
been presented. As with Scheme IIIA, the predicted kinetic structures for the
higher ranked schemes captured most of the features of the kinetic structure.
The observation that Schemes VII and VIIA were the top ranked schemes suggests
that the gating may include at least one open and one closed state on the
activation pathway that are reached after a
Ca^{2+}-independent conformational change. The
NLR_{1000} for Scheme VIIA (Eq. 3) of 0.0067 for channel B06 (Table I) becomes 0.995 for a single interval pair (Eq. 5), suggesting an average likelihood
difference per interval pair of only 0.005 between the description of the gating
by Scheme VIIA and the theoretical best description for a discrete state Markov
model. For all schemes in Table I except
Schemes IIIA′ and IIIB, the lifetimes of the additional brief closed
states were constrained to be the same for any particular channel. Allowing the
lifetimes of the additional brief closed states to vary independently of one
another (as was done for Scheme IIIA′) typically improved the
descriptions of the kinetic structure and the rankings over the same models in
which the brief lifetimes were constrained.

Interestingly, Scheme VIA, the second ranked scheme for four of the five channels, was the top ranked scheme for the atypical channel M09. The predicted kinetic structure for channel M09 for Scheme VIA is shown in Fig. 13, and captures most of the features for this channel (compare with Figs. 2 and 3). The other schemes gave noticeably worse descriptions for channel M09 (not shown). Thus, channel M09 both looks different in kinetic structure and may have some differences in gating mechanism other than differences in rate constants.

In general, for the various examined schemes with secondary closed states, ∼30–40% of the brief closings with durations <110 μs (flickers) arose from transitions to the primary closed states and 60–70% arose from transitions to the secondary closed states. Thus, in terms of the models with secondary closed states, the majority of flickers arose from the secondary closed states. In models with intermediate closed states, essentially all the flickers arose from the intermediate closed states.

### Additional Secondary Closed States Can Improve the Rankings

It has been suggested for GABA_{A} channels (Rogers et al., 1994) and for *Shaker*
K^{+} channels (Schoppa
and Sigworth, 1998) that two closed states may be connected directly
to the open states beyond the activation pathway. Consequently, we expanded some
of the schemes to include two secondary closed states connected to each open
state to determine if this improved the description of the kinetic structure for
BK channels. For the fitting, all first secondary closed states connected to
each open state were constrained to have identical lifetimes and all second
secondary closed states connected to each open state were also constrained to
have identical lifetimes. In general, such schemes typically ranked above
schemes with only a single secondary closed state connected to each open state,
even though there were additional penalties for the additional states. Such
schemes also gave some minor visual improvements in the kinetic structure. For
example, for channel B06, Scheme VIIA had an NLR_{1000} of 0.0067 with
one secondary closed state per open state, compared with 0.0091 after adding an
additional secondary closed state per open state. Such schemes have not been
presented in detail because the improvement in the kinetic structure was slight
for the additional complexity. For experimental data collected over a wider
range of experimental conditions, it may well be necessary to include such
additional secondary closed states to account for the kinetic structure.

## Discussion

This study employed comparisons of the observed and predicted kinetic structures together with the quantitative ranking of models by simultaneous maximum likelihood fitting of 2-D dwell-time distributions to assess and develop gating mechanisms for BK channels. These approaches extracted correlation information from the single-channel current records, both visually and through likelihood, to place restrictions on the transition pathways and rates for the various kinetic states. In theory, all possible models could be tried blindly and ranked by likelihood to obtain the most likely schemes, but we have found it more efficient to use the kinetic structures to guide the development of the models and assess the ability of the models to describe the single-channel kinetics.

### Similarities and Differences in the Kinetic Structure of BK Channels

Similarities and differences in the gating of different BK channels from the same preparation (cultured rat skeletal muscle) were readily apparent in the plots of kinetic structure. While the basic features of the kinetic structure were the same for the different individual BK channels, there were also channel-specific differences (Figs. 2 and 3). The reason for the channel-specific differences in kinetic structure was not established, but alternative splicing of BK channels can give rise to a large family of functionally diverse channels (Atkinson et al., 1991; Adelman et al., 1992; Butler et al., 1993; Lagrutta et al., 1994; Pallanck and Ganetzky, 1994; Tseng-Crank et al., 1994), so it is possible that the individual channels examined in our study may have had different physical structures.

Other factors might also give rise to the differences in kinetic structure. The
association of the modulatory beta subunit with the pore forming alpha subunit
of BK channels can change the Ca^{2+} sensitivity and gating
kinetics (McManus et al., 1995; Dworetzky et al., 1996; Meera et al., 1996; Nimigean and Magleby, 1998). The extent, if any, to which
differential expression of beta (or other possible) subunits might contribute to
the differences in kinetic structure is not known for the examined channels. It
is also possible that individual channels may have been in different redox
states or levels of phosphorylation, which can alter activity (Reinhart et al., 1991; Bielefeldt and Jackson, 1994; DiChara and
Reinhart, 1997; Thuringer and Findlay,
1997). Differences in Ca^{2+} sensitivity and/or
kinetics among BK channels from the same preparation have been described
previously (McManus and Magleby, 1991;
Sansom and Stockand, 1994; Wu et al., 1996). Some of the differences
in kinetic structure might also arise from small differences in the experimental
conditions for some of the channels, but, as indicated in the results,
this is unlikely to be a major contributing factor.

### Simple Models Can Approximate many of the Features of the Kinetic Structure of BK Channels

Most of the models that have been examined to describe the gating of BK channels
can be related to the MWC model for binding of ligands to a tetrameric protein
(McManus and Magleby, 1991; Cox et al., 1997). This model has five
closed and five open states, as indicated in Scheme SV. McManus and Magleby
(1991) found that Scheme III, a simplified version of Scheme V with two fewer open states, was the simplest model that could
describe the basic features of the Ca^{2+} activation of BK
channels (determined with 1-D dwell-time distributions) over an
∼400-fold range of *P*_{o}, from very low
(0.00137) to intermediate (0.53) levels.

As a more critical test of Scheme III,
we examined to what extent it could account for the Ca^{2+}
dependence of the detailed kinetic structure. While Scheme SIII captured the basic features, it underpredicted the
number of brief closed intervals adjacent to longer open intervals and
overpredicted the number of brief open intervals adjacent to intermediate (1 ms)
closed intervals (Fig. 8), suggesting that
Scheme III was too simple. Based on
differences between the observed and predicted kinetic structures, increasingly
more complex models with additional states were then examined. Scheme VIIA was
the top-ranked scheme in Table I for four
of the five examined channels and the second ranked scheme for the atypical
channel M09, which had a markedly different kinetic structure from the other
channels. Scheme VIA was the top ranked scheme for channel M09.

These observations suggest that the basic underlying gating mechanism is the same for at least four of the five BK channels examined in detail. On this basis, the differences in kinetic structure among these channels would then arise from some differences in the rate constants (Fig. 12). The atypical channel M09 may well have some differences in gating mechanism. Another possibility is that the actual gating mechanism is more complex than any of the examined schemes (see below). If this more complex mechanism were known, then perhaps it would be top ranked for all channels.

### Secondary and Intermediate Closed States

A feature that consistently improved the descriptions of the kinetic structure was the incorporation of additional brief closed states connecting directly to the open states. These additional closed states could be outside the activation pathway as secondary closed states, as in the A versions of the various examined schemes, or within the activation pathways as intermediate closed states, as in Scheme IIIB. It was difficult to distinguish between these two types of models. The secondary and intermediate closed states were of brief duration, with typical mean dwell-times of 30–150 μs. The models with the secondary or intermediate closed states could account for additional features of the kinetic structure that were missing with the simpler models (compare Fig. 10 to 7 and 8). For models with secondary closed states, calculations from the most likely rate constants indicated that ∼30–40% of the brief closings with durations <110 μs (flickers) arose from transitions to closed states within the activation pathway and 60–70% arose from transitions to the secondary closed states. For models with intermediate closed states, essentially all of the flickers involved the intermediate closed states.

Consistent with our findings that there are additional closed states, Cox et al. (1997) have found that the basic
MWC model can account for the effects of voltage and Ca^{2+}
on *P*_{o} for mSlo BK channels, but cannot account for
the detailed single-channel kinetics. They suggest that additional states are
needed. Wu et al. (1995) have added
closed states beyond the activation pathway to account for activity in high
Ca^{2+} for BK channels from hair cells in the cochlea.
The two added closed states in the model of Wu
et al. (1995) are Ca^{2+} dependent and had
intermediate duration lifetimes, compared with the
Ca^{2+}-independent briefer lifetimes of the added closed
states in our study. Moss et al. (1998)
have found evidence for additional brief closed states in dslo BK channels
expressed in *Xenopus* oocytes.

Closed states beyond the activation pathway (secondary closed states) have been
proposed for a number of channels including acetylcholine receptors in BC3H-1
cells (Sine and Steinbach, 1987),
GABA_{A} receptor channels (Rogers
et al., 1994), *Shaker* potassium channels (Hoshi et al., 1994; Schoppa and Sigworth, 1998) and NMDA
(*N*-methyl-d-aspartate) receptors (Kleckner and Pallotta, 1995). It would be
of interest to determine whether models with intermediate, rather than
secondary, closed states might also be consistent with the gating of one or more
of these channels.

### Possible Mechanisms for Intermediate Closed States in BK Channels

The possibility of intermediate closed states follows directly from proposed mechanisms for conformational changes in tetrameric allosteric proteins, in which each subunit can both bind a ligand and undergo one or more conformational changes (Eigen, 1968; Fersht, 1977; McManus and Magleby, 1991; Cox et al., 1997). Transitions from the unliganded channel with all the subunits in a basal conformation to the fully liganded and activated conformations could pass through intermediate states in which the four subunits are in various combinations of conformation with and without bound ligand. Models for tetrameric proteins without intermediate states, such as the MWC model, have been found to be inadequate for hemoglobin, as Ackers et al. (1992) have presented evidence for intermediate states in this tetramer. Thus, unless the intermediate states are too brief or infrequent to detect, it might be expected that some of the intervals during the gating would arise from intermediate states.

If the properties of one or more of the four alpha subunits forming the pore of the channel differ due to alternative splicing (Atkinson et al., 1991; Adelman et al., 1992; Butler et al., 1994), associations with accessory subunits (McManus et al., 1995; Dworetzky et al., 1996; Meera et al., 1996), and/or various levels of phosphorylation and redox (e.g., Reinhart et al., 1991; Bielefeldt and Jackson, 1994; DiChiara and Reinhart, 1997; Thuringer and Findlay, 1997), then the number of potential intermediate states could be many times greater than predicted on the basis of four subunits.

It has been suggested for drk1 and also mutated *Shaker* channels
that intermediate states can give rise to subconductance levels (Chapman et al., 1997; Zheng and Sigworth,
1997). Hence, if the brief closings (flickers) for BK channels arise from
transitions to intermediate states, then it might be expected that brief
closings would be to subconductance levels. Consistent with this possibility,
the findings of Ferguson et al. (1993)
suggest that brief closings (flickers) for BK channels are, on average, partial
closings to ∼90–95% of the fully closed conductance
level.

If the intermediate states are partially conducting, then, since transitions to and from the open states in Scheme IIIB must pass through the intermediate states, it would be expected that the transitions between the fully open and fully closed current levels in the single-channel records would pass through brief lifetime subconductance levels. This hypothesis is consistent with the observations of Ferguson et al. (1993) that openings from longer duration closed intervals first passed through, on average, a brief (50 μs) subconductance level at the 90–95% closed level before opening fully, and that closings to the longer duration closed intervals first closed, on average, to the 90–95% closed level before closing completely over the next 50 μs.

### Possible Mechanisms for Secondary Closed States in BK Channels

While there is a theoretical underpinning for intermediate states within the activation pathway (see above), the basis for potential secondary closed states, those beyond the activation pathway, is less clear, but some possibilities include either blocking ions or secondary gates that obstruct the channel for brief periods of time when the primary gate is open. In either case, if secondary states give rise to the brief closings (flickers), then transitions to the secondary closed states, on average, would need to be associated with subconductance levels to be consistent with the observations of Ferguson et al. (1993) discussed above. (Furthermore, to account for the subconductance levels that can be observed, on average, for transitions between the fully open and fully closed current levels in the single-channel records, some states in the activation pathway would still need to be partially conducting.)

Although some blocking ions such as Ba^{2+} decrease the
conductance to the zero current level without measurable delay (Ferguson et al., 1993), blocking ions can
also give partial conductances (Schild and
Moczydlowski, 1994; Premkumar and
Auerbach, 1996; Shneggenburger and
Ascher, 1997). Thus, blocks could serve as a mechanism for secondary
closed states with either full or partial closures. While
Na^{+} produces a fast flickery block in BK channels
(Yellen, 1984), our experimental
solutions contained negligible Na^{+}, so
Na^{+} block can be excluded for secondary closed
states.

It is also unlikely that the large numbers of brief closed intervals in our study
arose from fast Ca^{2+} blocks since brief closed intervals
do not become more frequent with increasing
Ca^{2+}_{i} (Rothberg et al., 1996), and millimolar concentrations of
Ca^{2+}_{i} are required to reduce the
conductance by apparent screening effects (Ferguson, 1991). Intracellular Ba^{2+} applied
to BK channels can produce long lasting blocks with durations of seconds (Vergara and Latorre, 1983; Rothberg et al., 1996) as well as fast
flickery blocks (Sohma et al., 1996;
Bello and Magleby, 1998). While an
occasional flicker in our records may arise from a Ba^{2+}
block, it is unlikely that an appreciable number do, as the expected
submicromolar concentrations of contaminant Ba^{2+} in our
solutions would be three to four orders of magnitude less than the concentration
required for appreciable fast Ba^{2+} blocks, and the mean
durations of slow Ba^{2+} blocks are five orders of
magnitude longer than the flickers (Rothberg et
al., 1996). Thus, the flickers in our study do not appear to arise
from ion block of the channel by Ba^{2+},
Ca^{2+}, or Na^{+}.

Therefore, the flickers might arise from the closing of a secondary gate
intrinsic to the channel or the closing of the primary gate to a secondary
conformation. Mienville and Clay (1996)
have proposed a secondary gate that acts independently of the main gate to
describe the flickering in BK channels in embryonic rat telencephalon. They
found that decreasing intracellular K^{+} increased the
flicker rate, leading to a decrease in mean open time, suggesting that the open
state may be destabilized when the pore is unoccupied by the permeating ion
(Swenson and Armstrong, 1981; Demo and Yellen, 1992). Any secondary gates
associated with the secondary closed states would have properties that differ
from the much slower inactivation gate observed by Solaro and Lingle (1992) for BK channels in rat adrenal
chromaffin cells.

### Minimal Working Hypothesis for the Gating in the Normal Mode

Of the examined models, Schemes IIIA*,* IIIA′, and
IIIB can be considered as minimal models that capture the major features of the
Ca^{2+} dependence of the gating, as revealed by plots
of the kinetic structure. Fig. 14
summarizes the features of the gating for Scheme IIIA with the rate constants in
Fig. 12
*B*. Fig. 14
*A* plots the rate constants for binding of the first through
fourth Ca^{2+} to the channel. The binding of the second and
third Ca^{2+} to the closed states was highly cooperative,
with the second Ca^{2+} binding rate constant, on average,
9× faster than the first, and the third Ca^{2+}
binding rate constant, on average, ∼100× faster than the
first. The fourth Ca^{2+} binding rate constant was not as
fast as the third, but was still ∼80× faster than the
first. For a channel with four identical subunits with one
Ca^{2+}-binding site per subunit, the
Ca^{2+}-binding rate constants per binding site would be
1/4, 1/3, 1/2, and 1/1 of the indicated rate constants in Fig. 14
*A*, increasing the forward binding rate for the successive
bindings of the second, third, and fourth Ca^{2+} even
further. The cooperativity in the binding of successive
Ca^{2+} to the closed states for Schemes
IIIA′, IIIB, IV, IVA, V, VA, VII, and VIIA was similar to that shown
in Fig. 14
*A* for Scheme IIIA. Cooperativity requires that the subunits
interact to facilitate the binding of additional Ca^{2+}. If
the channel opened after binding three Ca^{2+}, the rate
constant for binding the fourth Ca^{2+} to the open state
was, depending on the model, typically similar to or several times faster than
the rate constant for the binding of the fourth Ca^{2+} to
the closed state.

Fig. 14, *B* and
*C,* plots the mean lifetimes of the open and closed states
for Scheme IIIA. In general, increasing the number of bound
Ca^{2+} increased the stability (lifetimes) of the open
states and decreased the lifetimes of the longer closed states. This was also
generally the case for the other schemes, including Scheme IIIB. It is this
general inverse relationship between the durations of the longer closed states
(including compound closed states) and the open states (including compound open
states) that gives rise to the characteristic saddle shape of the dependency
plots.

In Scheme IIIA, the main functional gating pathway was typically
C_{8}-C_{7}-C_{6}-C_{5}-O_{2}-O_{1},
with brief transitions to the secondary closed states C_{9} and
C_{10} interrupting the openings, but having little effect on
*P*_{o}. For Scheme VIIA, the main gating pathway was
typically
C_{10}-C_{9}-C_{8}-C_{7}-O_{3}-O_{2}-O_{1},
with brief transitions to the secondary closed states C_{11},
C_{12}, and C_{13} interrupting the openings. The
equilibrium occupancies of the various states as a function of
Ca^{2+}_{i} are plotted in Fig. 14, *D* and
*E*. Ca^{2+}_{i} drives the
channel towards the closed states C_{4} and C_{9} and the open
state O_{1}.

The Ca^{2+}-binding rates, state lifetimes, and equilibrium
occupancies for Scheme IIIB were similar to those shown in Fig. 14 for IIIA, except that the equilibrium
occupancies of the intermediate closed states C_{9}, C_{10}, and
C_{11} were typically about twice those for the secondary closed
states C_{9}, C_{10}, and C_{11} for Scheme IIIA. For
Scheme IIIB, the main functional gating pathway was typically
C_{8}-C_{7}-C_{6}-C_{5}-C_{10}-O_{2}-O_{1},
with brief transitions to the intermediate closed states C_{9},
C_{10}, and C_{11} interrupting the openings, but having
little effect on *P*_{o}.

Interestingly, while some of the models we considered are similar to those
considered by Cox et al. (1997) for the
gating of mslo, there is little agreement between the two studies for the
estimated parameters in the considered models. This lack of agreement is not
surprising. Cox et al. (1997) fit
macroscopic currents over wide ranges of Ca^{2+}_{i}
and voltage, but did not include single-channel kinetic data, and consequently
could not account for the single-channel kinetics. We fit single-channel data
over a lesser range of Ca^{2+}_{i} at a single
voltage (+30 mV), and consequently could account for single channel
kinetics, but not voltage. The models from both studies summarize the important
observations for the examined experimental conditions, while placing some
restrictions on mechanism. These models can serve as minimal models for future
studies.

### The Top Ranked Models Are Still Too Simple to Account for Normal Activity

BK channels can open at nominally zero Ca^{2+}_{i}
(Rothberg and Magleby, 1996; Cox et al., 1997; Meera et al., 1997). Of
those considered, only Schemes V and VA would open in the absence of
Ca^{2+}_{i}. Thus, to account for the gating at
very low Ca^{2+}_{i}, the more complex Scheme VA, or
a version of this scheme in which the secondary closed states are replaced with
intermediate closed states, could serve as a minimal model. Open states could
also be added to the left-most closed states in the other schemes to allow
opening in the absence of Ca^{2+}_{i}.

Alternatively, while the horizontal steps in our considered schemes are listed as
Ca^{2+}-binding steps, each step may actually represent
a Ca^{2+}-facilitated conformational change. If the
conformational changes can occur spontaneously, then this could contribute to
some of the gating at very low Ca^{2+}_{i}. It is
also possible that the various cations in the bathing solution might facilitate
spontaneous conformational changes, but at a much lower efficacy than
Ca^{2+}. If this is the case, then a spontaneous
transition rate for the forward directed horizontal transitions could be added
to the models.

While the top ranked models gave excellent descriptions of the kinetic structure
over the examined range of Ca^{2+}_{i}, they still
did not describe the data as well as theoretically possible for a discrete state
Markov model. Although the differences between the predicted and observed
kinetics were small, such differences are important, as they may give additional
insight into the mechanism. Small amounts of drift in the single channel data,
minor errors in the estimated Ca^{2+} concentrations, and
errors associated with the assumptions used to correct for missed events (Crouzy and Sigworth, 1990) could contribute
some error. The assumptions of discrete state models with rate constants that do
not change in time may also contribute some error. Although the gating of BK
channels appears consistent with these simple assumptions (McManus and Magleby, 1989; Petracchi et al., 1991), conformational changes in proteins would be
expected to be more complex (see discussion in McManus and Magleby, 1989).

Alternatively, the predictions of the models may be less than complete because
the examined models are still too simple, having too few kinetic states. As
mentioned above, theoretical considerations predict far more complex models for
BK channels than the minimal models examined in this study (McManus and Magleby, 1991; Cox et
al.*,* 1997). If each subunit has two
Ca^{2+} binding sites (Golowasch et al., 1986; Schreiber
and Salkoff, 1997), then the complexity of the models would increase
even further.

The trouble with more complicated models is that it becomes increasingly more difficult to define the parameters as more states are added. Consequently, even though the more complex models may give significantly better descriptions of the data, the number of parameters that are not well defined typically increases with complexity, and the rankings of more complex models can become inconsistent among channels with so many free parameters, so that it may not be clear which of the more complex models should be used.

For example, we found that models with two secondary closed states connected to
each open state typically improved the description of the kinetic structures and
also ranked higher than models with only one secondary closed state connected to
each open state. Based on this observation, it would also be expected that
models with additional intermediate closed states, or with both intermediate and
secondary closed states, would rank higher than models with only secondary or
intermediate closed states alone. However, rankings of the more complex models
that were examined were no longer as consistent among the different channels as
for the simpler models. Another complicating factor is that more complex
schemes, even if correct, may be ranked lower by the Schwarz (1978) criteria if the additional states do not
significantly improve the fit. Thus, the detailed testing of more complex models
seems best delayed until there are more data to constrain the models, such as
detailed single-channel currents obtained over wider ranges of
Ca^{2+}_{i} than used in these experiments and
also over a range of voltages as well as following step changes in voltage. Such
data should provide further constraints on the parameters.

In conclusion, our study describes and presents minimal models to account for the kinetic structure of BK channels. These models have additional brief closed states, either in the activation pathway as intermediate states or beyond the activation pathway as secondary states. It is these additional brief closed states that give rise to the majority of the brief closed intervals (flickers) in the gating.

## Acknowledgments

This work was supported in part by grants from the National Institutes of Health (AR-32805, NS-30584, NS-007044) and the Muscular Dystrophy Association.

## Footnotes

- Abbreviations used in this paper:
- 2-D
- two dimensional
- BK
- large-conductance Ca
^{2+}-activated K^{+}channels - MWC
- Monod-Wyman-Changeux
- NLR
- normalized likelihood ratios

- Submitted: 20 February 1998
- Accepted: 13 April 1998