INTRODUCTION

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Large-conductance Ca²⁺-activated K⁺ channels (maxi-K or BK channels),¹ which are activated by micromolar concentrations of intracellular Ca²⁺ (Ca²⁺_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 Ca²⁺ 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 Ca²⁺_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 Ca²⁺_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 Ca²⁺ 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 Ca²⁺_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, 1990b).

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 Ca²⁺ 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

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Preparation

Currents flowing through single large-conductance Ca²⁺-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 CO₂ 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 Ca²⁺ (as CaCl₂), to bring the free Ca²⁺ concentration at the intracellular surface (Ca²⁺_i) to the indicated levels. No Ca²⁺ was added to the extracellular (pipette) solution. Ca²⁺_i in the absence of EGTA was determined by atomic absorption spectrometry, and in the presence of EGTA was determined with a Ca²⁺-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 (1990b) 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 Ca²⁺_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 Ca²⁺_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, 1995b); (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 Ca²⁺_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 Ca²⁺- 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 NLR₁₀₀₀ 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 (1990a).

	RESULTS

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Currents flowing through a single large-conductance Ca²⁺-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, 1995a), 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 Ca²⁺-dependent gating mechanisms.

View larger version (28K): [in this window] [in a new window]   Fig. 1.   Current records and stability plots from a single BK channel. (A and B) Current records at low and high time resolution recorded from an inside-out patch of membrane excised from cultured rat skeletal muscle. (C and D) Stability plots of the mean open and closed interval durations, averaged 100 at a time, after excluding any artifacts and shifts to modes other than normal. The effective low-pass filtering was 7.8 kHz; channel B12; membrane potential was +30 mV (intracellular membrane surface positive) in this and the following figures.

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 Ca²⁺_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).

View larger version (55K): [in this window] [in a new window]   Fig. 2.   2-D dwell-time distributions for six different BK channels. Adjacent open and closed intervals were binned as pairs, with the logs of the open and closed interval durations locating the bins on the x and y axis, respectively. The z axis plots the square root of the number of interval pairs in each bin. The Ca2+i, open probabilities, numbers of plotted interval pairs in each distribution, and effective level of low-pass filtering were: channel B06: 12.3 µM, 0.50, 28,560, 6.3 kHz; B12: 12.3 µM, 0.54, 46,652, 7.8 kHz; B14: 12.3 µM, 0.61, 21,904, 10 kHz; M25: 21.6 µM, 0.52, 118,006, 5 kHz; M09: 7.46 µM, 0.58, 194,000, 6.3 kHz; M24: 23.1 µM, 0.52, 158,238, 7.3 kHz.

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 Ca²⁺ 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).

View larger version (49K): [in this window] [in a new window]   Fig. 3.   Dependency plots for six different BK channels. The fractional excess or deficit of interval pairs between the 2-D dwell-time distributions in Fig. 2 and the 2-D dwell-time distributions that would be expected for independent pairing of open and closed intervals is plotted as dependency (Eqs. 1 and 2). The heavy solid lines indicate a dependency of zero. Dependencies of +0.5 or 0.5 would indicate a 50% excess or deficit, respectively, of observed interval pairs over that expected for independent pairing of open and closed intervals.

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 Ca²⁺_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 Ca²⁺_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.

View larger version (3K): [in this window] [in a new window]   Scheme I.

View larger version (34K): [in this window] [in a new window]   Fig. 4.   Estimating the contribution of stochastic variation, filtering, and noise to the dependency plots. The 2-D dwell-time distribution for channel B06 in Fig. 2 was fitted with Scheme I to obtain the most likely rate constants. These rate constants were then used with Scheme I to simulate (with filtering and noise) the same number of detected interval pairs as for the plot for channel B06 in Fig. 2. The simulated single-channel current was then analyzed to obtain the predicted dwell-time distribution (A), the predicted dependency plot (B), and the predicted dependency significance (C). As expected for Scheme I, which should have zero dependency, none of the dependencies were significant. The variation in dependency thus reflects contributions from stochastic variation, filtering, and noise.

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, 1990b).

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.

View larger version (54K): [in this window] [in a new window]   View larger version (26K): [in this window] [in a new window]   Fig. 5.   Significance of the basic features of the dependency plots. (A) The significance of the dependencies in the dependency plots in Fig. 3 is plotted as dependency significance, which indicates the log of the P values times the sign of the dependency. Values of dependency significance greater than the heavy solid line at +1.3 and less than the heavy solid lines at 1.3 indicate that the dependency values are significant (P < 0.05). Absolute values of dependency significance >2, 3, and 4 would indicate P values < 0.01, < 0.001, and < 0.0001, respectively. (B) Backside views of the dependency significance plots in A for the indicated channels. There is a significant deficit of long closed intervals adjacent to long open intervals.

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.

View larger version (18K): [in this window] [in a new window]   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).

View larger version (44K): [in this window] [in a new window]   Fig. 6.   Idealized 2-D dwell-time distributions and dependency plots for channel B06. The 2-D dwell-time distribution in Fig. 2 for channel B06 was fitted with the uncoupled Scheme II. The estimated most likely rate constants were then used with Scheme II to simulate a single-channel current with 106 detected intervals that were then analyzed to obtain the 2-D dwell-time distribution and dependency plots. For A, the current was simulated with filtering and noise similar to that in the experimental record, and for B the current was simulated without filtering and noise. The effects of stochastic variation are removed by the large numbers of analyzed intervals. The differences between A and B indicate the effects of filtering and noise in A.

Calcium Dependence of the Kinetic Structure

McManus and Magleby (1991) have previously detailed the Ca²⁺ dependence of the 1-D dwell-time distributions of open and closed interval durations for BK channels from rat skeletal muscle. The Ca²⁺ 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 [Ca²⁺]_i increased P_o by increasing the mean open interval duration and decreasing the mean closed interval duration. The Ca²⁺-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 Ca²⁺_i.

To gain further insight into the Ca²⁺ dependence of the gating, the Ca²⁺ dependence of the kinetic structure was examined. Results are presented in Fig. 7 for a representative channel (B06) at three different Ca²⁺_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 Ca²⁺_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.

View larger version (44K): [in this window] [in a new window]   View larger version (55K): [in this window] [in a new window]   View larger version (57K): [in this window] [in a new window]   Fig. 7.   Ca2+ dependence of the kinetic structure. (A) Single-channel current records at the indicated Ca2+i. (B) 2-D dwell-time distributions and dependency plots. The numbers of pairs of intervals in the 2-D distributions for the indicated Ca2+i were: 5.5 µM, 10,168; 8.3 µM, 11,848; 12.3 µM, 28,560. (C) Idealized kinetic structures at the indicated Ca2+i, generated as described for Fig. 6 A. Channel B06.

The 2-D dwell-time distributions show a shift in the time constants of the longer closed intervals towards shorter durations with increasing Ca²⁺_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 Ca²⁺ Dependence of the Kinetic Structure

The basic features of the Ca²⁺ 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 Ca²⁺_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 10⁶ simulated intervals were analyzed for each distribution.

View larger version (8K): [in this window] [in a new window]   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 Ca²⁺ 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 Ca²⁺_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 Ca²⁺_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.

View larger version (57K): [in this window] [in a new window]   Fig. 8.   Kinetic structure predicted by Scheme III for channel B06. The three 2-D dwell-time distributions in Fig. 7 B were simultaneously fitted with Scheme III to estimate the most likely rate constants. Scheme III with these rate constants and using noise and filtering similar to that in the experimental record was then used to generate the predicted kinetic structure at each Ca2+i. Scheme III captures the basic features of the kinetic structure, but has a number of deficiencies. Compare to Fig. 7, B and C. Channel B06.

View larger version (15K): [in this window] [in a new window]   Fig. 9.   Brief open and closed intervals can occur adjacent to each other. Typical examples of single-channel current records at 5.5 µM Ca2+i. *Adjacent brief open and closed intervals both within bursts and at the beginning and end of bursts. Channel B06.

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 Ca²⁺_i in both the dependency plots (#2) and the 2-D dwell-time distributions. Finally, Scheme III 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 Ca²⁺ 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 III were calculated from the most likely rate constants. The results are presented in Scheme III(8.3) for channel B06 with 8.3 µM Ca²⁺. 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.

(III(8.3))

Scheme III predicted too few brief open intervals adjacent to brief closed intervals in low Ca²⁺_i (#1 in Figs. 7 and 8). For Scheme III, brief openings, which are mainly from O₃, would occur infrequently adjacent to brief closings, which are mainly from C₅, since transitions from O₃ and C₅ must pass through either the intermediate closed state, C₆, or the intermediate open state, O₂. 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 O₃, as in Scheme IIIA. This would increase the number of brief openings adjacent to brief closings.

View larger version (10K): [in this window] [in a new window]   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 O₃ and C₆. While the lifetime of C₆ is ~0.4 ms, transitions such as O₃-C₆-C₅-C₆-O₃ would double the mean duration of the closed interval to ~1 ms. Transitions such as O₃-C₆-O₃-C₆ could also increase the observed mean duration of the closed interval associated with state C₆, since some of the dwell-times in O₃ 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 C₁₁ connected to O₃ in Scheme IIIA would increase the number of brief openings adjacent to brief closings. C₁₁ would also act to decrease the excess of brief openings adjacent to intermediate closings (mainly from C₆) by diverting some of the transitions from O₃ away from C₆.

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 O₁-O₂-C₅-O₂-O₁. Adding the brief duration closed states C₉ and C₁₀ to O₁ and O₂, 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 C₅ 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 Ca²⁺_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.

View larger version (56K): [in this window] [in a new window]   Fig. 10.   Kinetic structure predicted by Scheme IIIA for channel B06. The kinetic structures predicted by Scheme IIIA at the indicated Ca2+i are plotted for comparison to the observed kinetic structures in Fig. 7, B and C.

For all five examined channels, Scheme IIIA gave a better description of the Ca²⁺-dependent kinetics than Scheme III, as indicated in Table I by both the normalized likelihood ratio values, NLR₁₀₀₀, 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