Correspondence to: Karl L. Magleby, Dept. of Physiology and Biophysics (R-430), University of Miami School of Medicine, Miami, FL 33101-6430., kmagleby{at}miami.edu (E-mail), Fax: 305-243-6898; (fax)
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Abstract |
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The Ca2+-dependent gating mechanism of large-conductance calcium-activated K+ (BK) channels from cultured rat skeletal muscle was examined from low (4 µM) to high (1,024 µM) intracellular concentrations of calcium (Ca2+i) using single-channel recording. Open probability (Po) increased with increasing Ca2+i (K0.5 11.2 ± 0.3 µM at +30 mV, Hill coefficient of 3.5 ± 0.3), reaching a maximum of ~0.97 for Ca2+i ~ 100 µM. Increasing Ca2+i further to 1,024 µM had little additional effect on either Po or the single-channel kinetics. The channels gated among at least three to four open and four to five closed states at high levels of Ca2+i (>100 µM), compared with three to four open and five to seven closed states at lower Ca2+i. The ability of kinetic schemes to account for the single-channel kinetics was examined with simultaneous maximum likelihood fitting of two-dimensional (2-D) dwell-time distributions obtained from low to high Ca2+i. Kinetic schemes drawn from the 10-state Monod-Wyman-Changeux model could not describe the dwell-time distributions from low to high Ca2+i. Kinetic schemes drawn from Eigen's general model for a ligand-activated tetrameric protein could approximate the dwell-time distributions but not the dependency (correlations) between adjacent intervals at high Ca2+i. However, models drawn from a general 50 state two-tiered scheme, in which there were 25 closed states on the upper tier and 25 open states on the lower tier, could approximate both the dwell-time distributions and the dependency from low to high Ca2+i. In the two-tiered model, the BK channel can open directly from each closed state, and a minimum of five open and five closed states are available for gating at any given Ca2+i. A model that assumed that the apparent Ca2+-binding steps can reach a maximum rate at high Ca2+i could also approximate the gating from low to high Ca2+i. The considered models can serve as working hypotheses for the gating of BK channels.
Key Words: BK channel, KCa channel, Monod-Wyman-Changeux, Eigen, Markov
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Introduction |
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Large-conductance Ca2+-activated K+ (BK)1 channels, which are activated by micromolar concentrations of intracellular Ca2+ (Ca2+i) and by depolarization (
Information about the gating mechanism of BK channels has accumulated from kinetic studies on native and heterologously expressed channels. BK channels are homotetramers, formed from four alpha subunits (
Models drawn from the Monod-Wyman-Changeux (MWC) model (
Despite these successes, MWC-type models appear too simple to account for certain details of BK channel gating.
We now test further whether the above extensions of the MWC model are consistent with the underlying gating mechanism. Our approach is to examine the activity of single BK channels from low (4 µM) to very high (~1 mM) intracellular concentrations of Ca2+ (Ca2+i). By driving the channel toward its fully liganded states, we estimated the numbers of open and closed states involved in the gating at high Ca2+i, and through analysis of two-dimensional dwell-time distributions and dependency plots, we determined the minimum number of independent transition pathways connecting the fully liganded open and closed states.
We found that at high Ca2+i, the BK channel gates among three to four open and four to five closed states, with two or more effective transition pathways (gateway states) connecting the open and closed states. Neither the general model of
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Methods |
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Preparation
Currents flowing through single BK channels in patches of surface membrane excised from primary cultures of rat skeletal muscle (myotubes) were recorded using the patch clamp technique (
Solutions
The solutions bathing both sides of the membrane contained 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. Contaminant Ca2+i was determined by atomic absorption spectrometry. The solution at the intracellular side of the membrane also contained added Ca2+ (as CaCl2), to bring the Ca2+ concentration at the intracellular surface (Ca2+i) to the indicated levels. (The solutions did not contain Ca2+ buffers.) No Ca2+ was added to the extracellular (pipette) solution. Solutions were changed through the use of a microchamber (
Recording and Measuring Interval Durations and Identifying Normal Activity
Single-channel currents were recorded on a digital data recorder (DC-37 kHz; Instrutech Corp.), low-pass filtered with a four-pole Bessel filter to give a final effective filtering of 610 kHz (-3 dB), and sampled by computer at a rate of 125200 kHz. The effective filtering is expressed in terms of dead time, which is the duration of an underlying interval before filtering that would just reach 50% of the single-channel current amplitude with filtering (
Restriction of Analysis to Activity in the Normal Mode
The analysis in the present study was restricted to channel activity in the normal mode, which typically involves ~96% of the detected intervals (
Log Binning and Plotting of One- and Two-Dimensional Dwell-Time Distributions
Both one- (1-D) and two-dimensional (2-D) dwell-time distributions were analyzed. The 1-D distributions of open and closed interval durations were log-binned as described previously (
With filtering, detected intervals with durations less than about twice the dead time are narrowed (
Dependency Plots
Dependency plots were constructed from the 2-D dwell-time distributions as detailed in
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(1) |
where NObs(tO,tC) is the experimentally observed number of interval pairs in bin (tO,tC), and NInd(tO,tC) is the calculated number of interval pairs in bin (tO,tC) if adjacent open and closed intervals pair independently (at random). The method of calculating expected frequencies for observations that are independent (contingency tables) is a common statistical procedure (compare
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(2) |
where P(tO) is the probability of an open interval falling in the row of bins with a mean open duration of tO, and P(tC) is the probability of a closed interval falling in the column of bins with a mean closed duration of tC. P(tO) is given by the number of open intervals in row tO divided by the total number of open intervals in all rows, and P(tC) is given by the number of closed intervals in column tC 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. Each open and closed interval forms two pairs: one with the preceding interval and one with the following interval. Hence, the number of interval pairs in a 2-D dwell-time distribution is equal to the number of open plus closed intervals minus one.
Estimating the Most Likely Rate Constants for Kinetic Schemes
The most likely rate constants for the examined kinetic schemes were estimated from the simultaneous fitting of the 2-D dwell-time distributions obtained at six different Ca2+i using the iterative maximum likelihood fitting procedure described in
The number of underlying 2-D exponential components that sum to form a 2-D dwell-time distribution is given by the product of the numbers of open and closed states (
For a given model and rate constants, the equilibrium occupancies of the states could be calculated as described in
Evaluating Kinetic Schemes
Normalized likelihood ratios (NLR1000) have been used to indicate how well any given kinetic scheme describes the 2-D dwell-time distributions when compared with a theoretical best description of the data. Normalization corrects for the differences in numbers of interval pairs among experiments so that comparisons can be made among channels. The normalized likelihood ratio per 1,000 interval pairs is defined as
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(3) |
where lnS is the natural logarithm of the maximum likelihood estimate for the observed 2-D dwell-time distributions given the kinetic scheme, lnT is the natural logarithm of the likelihood of 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 (
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, models were ranked using an information criteria approach (
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(4) |
then model g is ranked above model f, where mg and mf are the maximum likelihood estimates for models g and f, and kg and kf are the number of free parameters for each scheme. We also compared the Akaike rankings to those of the
Estimating the Theoretical Best Descriptions of the 2-D Dwell-Time Distributions
To evaluate models, it was useful to have an estimate of the theoretical best descriptions of the dwell-time distributions for comparison to the distributions predicted by the various examined gating mechanisms. For gating consistent with a discrete state Markov process (the rate constants remain constant in time for constant experimental conditions), as appears to be the case for BK channels (
Using Simulation to Generate Predicted Single-Channel Data
To make comparisons between the observed distributions and those predicted by the kinetic models, simulated single-channel current records were generated with filtering equivalent to that used in the analysis of the experimental current records and with noise similar to that in the experimental current record. The simulated single-channel currents were then analyzed in the same way that the experimental currents were in order to obtain the predicted 2-D distributions, dependency plots, and numbers of exponential components observed in the predicted dwell-time distributions. The method used to simulate single-channel currents with filtering and noise is detailed in
Online Supplemental Material
Figure S1 presents the equilibrium occupancy of the states in Figure 17 at low (5.5 µM) and high (1,024 µM) Ca2+i. At low Ca2+i, the channel spent most of its time (91.7%) in closed states C8C11, considerably less time (7.5%) in open states O1O6, and even less time (0.8%) in the intermediate closed states C12C17. At high Ca2+i, the channel spent essentially all (99.4%) of its time in the fully liganded open and closed states, with most (96.6%) of the time spent in open states O1 and O4, considerably less time (2.8%) in closed states C7, C12, and C15. A very small amount of time (0.5%) was spent in open states O2 and O5, with negligible time (0.1%) spent in all of the remaining open and closed states. Thus, at high Ca2+i, Figure 17 gates as though there were one effective transition pathway between open and closed states (C15O4), inconsistent with the observed significant dependencies at high Ca2+i. Available at http://www.jgp.org/cgi/content/full/114/1/93/DC1
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Figure S2 presents the estimated rate constants for the three channels examined in detail for Figure 29 Figure 15 Figure 16 Figure 17, Figure 22, and Figure 23. Available at http://www.jgp.org/cgi/content/full/114/1/93/DC1
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Results |
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The findings presented in this paper are based on a complete single-channel analysis of data obtained from three different BK channels, each studied at six different Ca2+i, from low to high (41,024 µM). In addition to these three channels, two additional channels were analyzed at three Ca2+i, including high Ca2+i, and an additional four channels were also examined, with findings consistent with the channels analyzed in more detail. All of the data, analysis, and figures presented in this paper are restricted to data collected during normal (mode) activity, which typically includes ~96% of the intervals (
Saturation of Po and the Mean Open and Closed Interval Durations at High Ca2+i
Currents flowing through a single BK channel in an inside-out patch of membrane excised from a cultured rat skeletal muscle cell are shown in Figure 1 A. These traces are representative of BK channel gating during normal activity over a range of Ca2+i, and illustrate a small fraction of the ~25 min of continuous recording from this channel. The current traces are excerpts of data obtained with 5.5, 12.3, 132, and 1,024 µM Ca2+i that gave mean open probabilities (Po) during normal activity of 0.061, 0.50, 0.97, and 0.97, respectively. Thus, while the 2.2-fold increase in Ca2+i from 5.5 to 12.3 µM gave an 8.2-fold increase in Po, the 7.8-fold increase in Ca2+i from 132 to 1,024 µM resulted in no increase in Po, indicating that the channel had reached a maximum level of activation. The saturation in Po at high Ca2+i is readily apparent in the current traces in Figure 1.
The effect of Ca2+ on Po for five different single BK channels during normal activity is illustrated in Figure 2 A. For these channels, increasing Ca2+i in the 420 µM range increased Po from ~0.02 to 0.85, while further increases in Ca2+i to either 132 or 1,024 µM led to a maximum Po ranging from 0.93 to 0.98. Maximum Pos within this range have been observed previously (
Fitting Po versus Ca2+i with the Hill equation for data from these five channels gave a K0.5 (the Ca2+i for a Po of 0.5) of 11.1 ± 0.7 µM, with a Hill coefficient of 3.5 ± 0.6, consistent with at least four Ca2+-binding steps contributing to maximal channel activation. The mean of the maximal fitted Po was 0.95 ± 0.03, and the fitted line indicated that a Ca2+ of ~40 µM was sufficient to drive the channel to within 1% of the maximum Po. Hill coefficients ranging from 24 are a common feature of BK channels (see INTRODUCTION), and higher slopes have been reported (
The Po did not reach 1.0 during normal activity at high Ca2+i because of frequent sojourns to brief closed states that generated flickers, and also because of much less frequent sojourns to longer closed intervals with durations of typically 110 ms. Examples of the frequent flickers and of the less frequent longer closed intervals during normal activity at high Ca2+i (>100 µM) are presented in Figure 1 B for data obtained at 132 and 1,024 µM Ca2+i. Analysis of the current records obtained at 132 and 1,024 µM Ca2+i indicated that only 2.1 ± 1.0% of the closed intervals had durations >1 ms, and even fewer, 0.026 ± 0.023%, had durations >10 ms. (The estimates obtained at both 132 and 1,024 µM Ca2+i from each of five channels were combined, as the percentages of longer closed intervals at the two different high Ca2+i were not significantly different.)
The increase in Po with increasing Ca2+i was associated with increases in the mean open interval durations and decreases in the mean closed interval durations (Figure 2B and Figure C), consistent with previous reports for BK channels from skeletal muscle (
The increase in mean open interval duration (Figure 2 B, thick line; Hill coefficient of 1.02) was much less Ca2+ sensitive than the decrease in mean closed interval duration (Figure 2 C, thick line; Hill coefficient of 3.48), suggesting that the major effect of Ca2+i is to decrease the durations of the closed intervals. The decreased filtering for channel B04 (Figure 2 B, ) would contribute to the briefer observed open times for this channel, as more of the flickers would be captured.
Just as Po saturated at high Ca2+i, the mean durations of the open and closed intervals also saturated in high Ca2+i at ~3.6 and ~0.15 ms, respectively (Figure 2B and Figure C, thick line). Thus, any viable mechanism for the gating of the BK channel must account for a saturation in mean open times, mean closed times, and in Po at high Ca2+i.
The Gating Kinetics Remain Unchanged at High Ca2+i
To further characterize the effect of high Ca2+i on gating, the open and closed interval durations during normal activity were measured and plotted as 1-D dwell-time distributions in Figure 3. The thick lines in the open and closed distributions in Figure 3 are the fits with mixtures (sums) of three open and five closed exponential components, respectively. Increasing Ca2+i from 5.5 to 132 µM shifted the open intervals to longer durations (note rightward shift of the major peak for the open times) and the closed intervals to briefer durations (note leftward shift and decreasing amplitude of the peak describing the longer closed times.). A further increase in Ca2+i from 132 to 1,024 µM had little additional affect on either the open or closed distributions, as seen in Figure 3D and Figure H, where the fits to the distributions at 132 µM Ca2+i are plotted as thin lines on the fits to the distributions at 1,024 µM Ca2+i. The thin lines essentially superimpose the thick lines, indicating essentially unchanged gating kinetics at high Ca2+i. Figure 3 H, inset, where the ordinate is plotted on a log scale to present the tails of the distributions at high gain, shows that the thin line also superimposes the thick line (within the range of the data) at the longer intervals where the frequency of occurrence of intervals is low. Results consistent with those in Figure 3 were observed for four additional channels.
Table 1 presents the time constants and areas of the three open and five closed exponential components fitted to the 1-D distributions in Figure 3. For increases in Ca2+i up to 132 µM, the time constant of the briefest open component remained relatively unchanged, the time constants of the two longer open components increased with increasing Ca2+i, and the area from the briefest open component shifted into the longer open components. In contrast, the time constants of the longer closed components became briefer and their areas shifted from the longer to the briefer closed components. Increasing the Ca2+i from 132 to 1,024 µM then had little additional effect on the open and closed components, when compared with the large effects observed for lower concentrations of Ca2+i.
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Table 2 presents the mean ± SD of the time constants and areas of the exponential components for data obtained from five channels at two levels of high Ca2+i. For four of the channels, the data were obtained at 132 and 1,024 µM Ca2+i, and for the fifth channel the data were obtained at 100 and 1,000 µM Ca2+i. The data from the fifth channel was pooled with the data from the other four since there was no apparent difference in the findings. Increasing the Ca2+i from 132 to 1,024 µM had no significant effect on either the time constants or areas of any of the open or closed components (Table 2, P > 0.05; paired t test). The measured increase in the time constant of the slowest closed component in Table 1 for channel B06 would be consistent with stochastic variation, as there were <20 intervals contributing to this component at each of the two levels of high Ca2+i.
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The apparent lack of effect of an eightfold increase in Ca2+i on the dwell-time distributions at high Ca2+i suggest that the rates for the transitions between states that dominate the gating at high Ca2+ are either Ca2+ independent or saturate at high Ca2+. These possibilities will be considered later.
Lack of Discrete Ca Block with High Ca2+i
It is well established that Ba2+ produces discrete block of BK channels (
As a more critical test for discrete block, we tabulated the frequency of occurrence of closed intervals >1 and >10 ms at the two different Ca2+i. For the five examined BK channels, the frequency of closed intervals with durations >1 ms was 7.4 ± 3.6 s-1 for 132 µM Ca2+i and 5.6 ± 2.5 s-1 for 1,024 µM Ca2+i, values that were not significantly different (P = 0.098, paired t test). The frequency of closed intervals with durations >10 ms was 0.12 ± 0.11 s-1 with 132 µM Ca2+i and 0.076 ± 0.086 s-1 with 1,024 µM Ca2+i, values that were also not significantly different (P = 0.44, paired t test).
Thus, the lack of effect of Ca2+ on the closed dwell-time distributions from brief to long times, and also on all closed intervals >1 and >10 ms, suggests that discrete Ca2+ block did not contribute to the closed intervals. As expected, increasing Ca2+i to 1,024 µM did decrease the conductance of the channel by ~10%, presumably due to a screening (fast blocking) effect (
At Least Three to Four Open and Four to Five Closed States Are Entered during Gating in High Ca2+i
Assuming that the gating is consistent with a discrete-state Markov model (
In general, the 1-D and 2-D methods gave similar estimates of the numbers of exponential components, but there were some differences. 2-D fitting can have an increased ability to detect components over 1-D fitting when the same numbers of intervals are analyzed (
At the lower concentrations of Ca2+i, the distributions were typically described by three to four open and four to seven closed components for fitting with either 1- or 2-D distributions (Figure 4), consistent with previous observations (
Theoretical Models for the Gating of Tetrameric Ligand-gated Channels
Functional BK channels can be formed by four alpha subunits (
Although relatively simple in concept, with four subunits and at least one Ca2+-binding site per subunit, theoretical models for the gating of a ligand-activated homotetramer, such as the BK channel, can be highly complex, with 3555 potential states (Eigen, 1969;
For the gating of the channel, the states in the top row of Figure 26 are assumed to represent closed states of the channel, and the states in the bottom row are assumed to represent open states. The conductance of the states in the middle three rows is less clear, but may be open, closed, or partially conducting (
If it is further assumed that each collection of isoforms has the same properties, and that conformational changes are concerted so that all four subunits undergo conformational changes simultaneously or that the lifetimes of the middle three rows of states in Figure 26 and Figure 27 are very brief, then Figure 26 and Figure 27 reduce to the 10-state Monod-Wyman-Changeux model for allosteric proteins (
Simple Models Fail to Predict the Single-Channel Kinetics at High Ca2+i
It has been shown previously for low to intermediate levels of Ca2+i that the Ca2+ dependence of the single-channel kinetics of BK channels in rat skeletal muscle can be approximated by the gating mechanisms described by Figure 29 Figure 15 Figure 16 (
Figure 29 is drawn from the MWC model. Figure 16 is an expansion of Figure 29 because it includes closed states beyond the activation pathway. Figure 15 can be viewed as a condensed version of Figure 26 and Figure 27 if it is assumed that the intermediate states in Figure 26 and Figure 27 are too brief to be detected or that only a subset can be detected. It is the additional brief closed states C9, C10, and C11 that generate most of the flickers (brief closings) in Figure 15 and Figure 16, whereas flickers for Figure 29 are generated mainly by sojourns between states O2 and C5. Flickers are highly characteristic of single-channel currents and can be seen in Figure 1.
Before examining whether Figure 29 Figure 15 Figure 16 could account for the gating from low through high Ca2+i, we first examined whether they could describe the 1-D dwell-time distributions from low to intermediate levels of Ca2+i, as reported previously (
We next examined whether Figure 29 Figure 15 Figure 16 could predict the distributions from low to high Ca2+i. Rate constants for each scheme were estimated by the simultaneous fitting of 2-D dwell-time distributions obtained at six different Ca2+i of 5.5, 8.3, 12.3, 20.3, 132, and 1,024 µM. These rate constants were then used to predict the observed distributions over the full range of Ca2+i. As shown in Figure 5, Figure 29 could not simultaneously describe the dwell-time distributions from low to high Ca2+i. Figure 15 gave a better description than Figure 29, but still could not describe the distributions. The predictions of Figure 16 were very slightly better than those of Figure 15, and are not shown. Table 3 shows the schemes rankings: Figure 16 > Figure 15 > Figure 29 (Akaike rankings, Equation 4).
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The inability of these schemes to account for the single-channel kinetics from low to high Ca2+i indicates that the models described by Figure 29 Figure 15 Figure 16 are too simple. Such a finding is, perhaps, not surprising since Figure 29 with eight states and Figure 15 and Figure 16 with 11 states include only a small subset of the minimal 55 potential states for the gating of a ligand-activated homotetrameric channel based on theoretical considerations (Figure 26). Nevertheless, it is possible that the channel does not gate as described by the theoretical 55-state model or, if it does, that only a fraction of the potential 55 states actually contribute to the gating. Consequently, we examined the differences between the observed and predicted responses for these schemes to determine how the minimal schemes might be expanded to be more consistent with the gating of the channel.
For high Ca2+i, Figure 29 Figure 15 Figure 16 predicted fewer brief openings than were observed in the single-channel data (Figure 5). Brief openings in Figure 15 typically arose from sojourns such as -C11-O3-C11-, and in Figure 16 from sojourns such as -C11-O3-C11- or -C6-O3-C6-, as O3 was the open state with the briefest lifetime in both schemes. For these schemes, the channel would be unlikely to reach O3 or the associated closed states at high Ca2+i, as the high Ca2+i would drive the gating towards the fully liganded states. Thus, to generate more brief open intervals at high Ca2+i, there needs to be a means for the gating to reach directly one or more brief open states from the fully liganded closed states. In addition, the underprediction of the long closed intervals at high Ca2+i suggests that the channel may also gate among additional closed states at high Ca2+i not included in Figure 29 Figure 15 Figure 16. Further evidence that the channel gates among additional states at high Ca2+i arises from the observation in Figure 4 that the channels typically entered at least three to four open and four to five closed states at high Ca2+i, while analysis of the dwell-time distributions predicted by Figure 29 Figure 15 Figure 16 at high Ca2+i indicated that the distributions were described by only one open and two to three closed components.
Figure 17 Can Describe the Open and Closed 1-D Dwell-Time Distributions from Low to High Ca2+i
We next examined how Figure 15 might be expanded to provide the required access to more open and closed states at high Ca2+i. Figure 15 is contained within the general Figure 26 and Figure 27 if it is assumed that the three rows of intermediate states in the general schemes are closed states that can be collapsed into one row of brief lifetime intermediate closed states. However, since it is possible that the last two rows of states in Figure 26 and Figure 27 are open states and that the remaining intermediate closed states do not collapse into one row of closed states, we investigated whether simplified schemes drawn from Figure 27, with an assumption of three rows of closed states and two rows of open states, could account for the data.
Figure 17 presents a gating mechanism of this type, where open states O4-O5-O6 (with brief lifetimes) and closed states C12-C13-C14 are the additional rows of open and closed states when compared with Figure 15. With high Ca2+i, the channel could now make sojourns from closed states to brief open states, such as -C15-O4-C15-, giving brief open intervals, and also make sojourns from closed states to longer open (and compound open) states, such as -C15-O4-O1-O4-C15-, giving longer open intervals. The extra row of closed states should also allow the generation of longer closed intervals at high Ca2+i by allowing more sojourns among closed states between openings.
The most likely rate constants for Figure 17 were determined from the simultaneous fitting of 2-D dwell-time distributions at six different Ca2+i (5.5, 8.3, 12.3, 20.3, 132, and 1,024 µM). The thick lines in Figure 5 show that the additional states allowed Figure 17 to describe the 1-D open and closed dwell-time distributions from low to high Ca2+i. In this scheme, the mean lifetimes at 5.5 µM Ca2+i of states O4, O5, and O6, of 0.11, 0.09, and 0.03 ms, tend to be brief compared with the lifetimes of the final row of open states O1, O2, and O3 of 0.50, 0.18, and 0.06 ms. The improved ability of Figure 17 to describe the single-channel gating when compared with Figure 29 Figure 15 Figure 16 (compare thick line to dotted and thin lines in Figure 5) is also reflected in the greatly improved likelihood ratios in Table 3.
We also examined whether a scheme like Figure 17, but with one less row of intermediate closed states, could account for the data. For the three channels examined, the likelihood estimates were two to four orders of magnitude less than for Figure 17, and the reduced scheme ranked below Figure 17 for all three channels (not shown).
Using Kinetic Structure to Evaluate the Gating Mechanism
Figure 17 accounts for the 1-D open and closed dwell-time distributions from low to very high Ca2+i (Figure 5). Such a description would be sufficient to predict Po as a function of Ca2+i over a wide range of activity, but the 1-D distributions do not take into account the correlation information between adjacent intervals, which can give insight into the connections (transition pathways) among the various states (
The kinetic structure for the same channel featured in the previous figures (channel B06) is shown in Figure 6 at four different Ca2+i. The 2-D dwell-time distributions (Figure 6, left) are plotted on loglog coordinates with the logs of the durations of adjacent open and closed intervals locating the position of the bin on the x and y axis, respectively. The z axis plots the square root of the numbers of intervals in each bin. These 2-D dwell-time distributions thus extend the
The dependency plots in Figure 6 (right) present the fractional excess or deficit of interval pairs of specified durations over that expected if the intervals paired at random. Dependencies of +0.5 or -0.5 would indicate a 50% excess or 50% deficit of interval pairs over the number expected if open and closed intervals paired independently. The thick lines indicate a dependency of zero. Because the dependency plots present magnified representations of excesses and deficits in the numbers of observed interval pairs relative to the numbers expected for independent pairing, they must be interpreted with some caution, as the estimates of dependency can be unreliable where the numbers of observed interval pairs per bin in the 2-D dwell-time distributions are small. Consequently, references to dependency will only be made when the referenced dependencies are known to be significantly different from zero. Such dependencies will be referred to by numbers on the dependency plots. Examples of which areas of the dependency plots are significantly different from zero are presented in
The dependency plots in Figure 6 (right) indicate that over the wide range of examined Ca2+i there was a deficit of brief open intervals adjacent to brief closed intervals (position 1), an excess of brief open intervals adjacent to both intermediate and long closed intervals (positions 2 and 3), and a deficit of long open intervals adjacent to long closed intervals (position 6; not clearly visible in the presented orientation, but visible when the plots were rotated). At the lower Ca2+i of 5.5 and 12.3 µM, there was also an excess of longer open intervals adjacent to brief closed intervals (position 4). These specific excesses and deficits of interval pairs give rise to the characteristic saddle shape of the dependency plots for BK channels (
While the kinetic structure at intermediate levels of Ca2+i for six different BK channels has been presented previously (
Dependency significance plots were made for the data obtained at high Ca2+i to estimate which dependencies were significantly different from zero. A paired t test was used to compare the number of interval pairs in each bin of the observed 2-D dwell-time distribution with the number expected if adjacent open and closed intervals paired independently, by using a moving 3 x 3 bin array as detailed in
From Figure 8 it can be seen that the dependencies at the numbered positions 1, 2, 4, and 5 were significantly different from zero at both 132 and 1,024 µM Ca2+i. Importantly, there was a significant excess of long open intervals adjacent to brief closed intervals (position 4), even though the fractional excess of these interval pairs was small in the dependency plots (Figure 7E and Figure F). The reason for this apparent discrepancy is that most of the interval pairs fall at position 4 in high Ca2+i, as can be seen from the 2-D dwell-time distributions (Figure 7, AD), so that even an appreciable excess of interval pairs at position 4 would still appear small when plotted as dependency, which plots the fractional excess of intervals (Equation 1). In six of six dependency significance plots that were examined at high Ca2+i, the dependencies at positions 1, 2, and 5 were significant, and the dependency at position 4 was significant in five of six plots. The one plot where significance was not observed at position 4 had fewer numbers of analyzed intervals.
The observations that the numbers of detected kinetic states remained relatively unchanged from low to high Ca2+i (Figure 4) and that the general shapes of the dependency plots also remained relatively unchanged from low to high Ca2+i (Figure 6 and Figure 7) raise the possibility that the basic gating mechanism remains relatively unchanged from low to high Ca2+i.
Figure 17 Describes the Kinetic Structure at Low but not High Ca2+i
To determine whether Figure 17 could account for the kinetic structure, the most likely rate constants for Figure 17, determined from the simultaneous fitting of 2-D dwell-time distributions at six different Ca2+i (5.5, 8.3, 12.3, 20.3, 132, and 1,024 µM), were used to simulate single-channel data for Figure 17 with noise and filtering equivalent to that of the experimental data. The simulated current records were then analyzed in the same manner as the experimental data to determine the predicted kinetic structure shown in Figure 9.
Figure 17 captured the basic features of the kinetic structure at 5.5 and 12.3 µM Ca2+i (compare the predicted 2-D dwell-time distributions and dependency plots in Figure 9A, Figure B, Figure E and Figure F, to the observed plots in Figure 6, A, B, E, and F). Figure 17 also captured the basic features of the 2-D dwell-time distributions at the high Ca2+i of 132 and 1,024 µM (compare Figure 9C and Figure D with Figure 6C and Figure D). However, Figure 17 predicted that little or no dependence would be observed at high Ca2+i (Figure 9G and Figure H), in contrast to the significant dependencies observed in the experimental data (Figure 6G and Figure H, and Figure 7 and Figure 8). Thus, Figure 17 predicted that open and closed intervals would pair randomly at high Ca2+i, in contrast to the dependent pairing observed in the data. These observations indicate that Figure 17 is too simple to capture the features of the gating at high Ca2+i.
The Gating Is Effectively Restricted to the Fully Liganded States of Figure 17 at High Ca2+i
To explore why Figure 17 did not predict the dependency at high Ca2+i, we calculated the equilibrium occupancies of the various open and closed states for this scheme. At low to intermediate Ca2+i (5.512.3 µM), the channel readily entered all the states in Figure 17, with occupancy biased towards the closed states C8C11 at the low Ca2+i. At high Ca2+i (132 and 1,025 µM), the channel spent 99.6% of its time in the fully liganded states, with 96.6% in open states 1 and 4 and 2.8% in closed states 7, 12, and 15. (The calculated equilibrium occupancies for Figure 17 can be found in online supplemental Figure S1 [http://www.jgp.org/cgi/content/full/114/1/93/DC1].)
The reason that Figure 17 predicted a lack of dependence between adjacent open and closed intervals at high Ca2+i is now readily apparent. Because the channel spent 99.4% of its time in the fully liganded column of states at high Ca2+i, the channel would essentially gate as Figure 18. Figure 18 has a single effective transition pathway between the open and closed states, given by C15-O4. A single effective transition pathway gives a single gateway state, which would lead to independent pairing of open and closed intervals and lack of significant dependencies (
If Figure 17 does effectively gate in the fully liganded column of states at high Ca2+i, as indicated by Figure 18, then a maximum of two open and three closed components would be detected in the dwell-time distributions predicted by Figure 17 at high Ca2+i. This was found to be the case. Fitting exponentials to dwell-time distributions simulated with Figure 17 at high Ca2+i gave two open and three closed components, compared with the typically three to four open and four to five closed states detected in the experimental data. Figure 17 also predicted a lack of dependence and too few components at high Ca2+i for the two other channels analyzed in detail.
Figure 26, Figure 26, and Figure 17 Can Be Rejected
The above findings indicate that Figure 17 can thus be rejected as a model for gating, as it cannot describe the dependencies at high Ca2+i. By analogy, Figure 26 and Figure 27 and all schemes based on subsets of states drawn from these schemes, such as Figure 27 Figure 28 Figure 29 Figure 15 Figure 16, can also be rejected as such schemes would also not describe the dependencies at high Ca2+i, provided that the forward rate constants for binding of Ca2+ are sufficiently rapid at high Ca2+ to effectively keep the gating in the fully liganded column of states.
The Kinetic Structure at High Ca2+i Suggests a Two-Tiered Gating Mechanism
The above results suggest that models for gating at high Ca2+i must: (a) allow effective transitions among at least three to four open and four to five closed states at high Ca2+i to generate the required numbers of exponential components in the dwell-time distributions (Figure 4), (b) have two or more independent transition pathways between open and closed states (two or more gateway states) at high Ca2+i to generate a dependent relationship between the durations of adjacent intervals (Figure 6 Figure 7 Figure 8), and (c) gate among the three to four open and four to five closed states at high Ca2+i in a manner that is essentially independent of Ca2+i for Ca2+i > 100 µM to account for the observed lack of effect of Ca2+ on the gating at high Ca2+i (Figure 1 Figure 2 Figure 3 and Table 1 and Table 2).
Since Figure 26 and Figure 27 are theoretical schemes based on a ligand-activated homotetramer, it might be useful if these schemes could be modified to account for gating at high Ca2+i. As pointed out by
Figure 20 Can Approximate the Kinetic Structure From Low to High Ca2+i
A difficulty with Figure 19, as was also the case for Figure 26 and Figure 27, is that Figure 19 has so many rate constants that it would be difficult if not impossible to determine unique rate constants for this scheme, even by the simultaneous fitting of 2-D dwell-time distributions over a range of Ca2+i (see METHODS). Consequently, we examined whether a reduced version of Figure 19 might be sufficient to approximate the kinetic structure. The number of rows of closed states in the upper tier was reduced from five to three, and the number of rows of open states in the lower tier was reduced from five to two to obtain the reduced Figure 20. Some of the open and closed states with zero and one bound Ca2+ were also omitted in Figure 20 to reduce further the number of rate constants because, over the range of Ca2+i examined in this paper, it might be expected that these states would contribute little to the gating (
Figure 20 is like Figure 17, except that Figure 20 has a total of six independent transition pathways between the open and closed states, compared with three in Figure 17. Two of the independent transition pathways connect fully liganded open and closed states, which would then allow at least two functional gateway states for gating with high Ca2+i. Two or more gateway states would be needed to generate the dependencies observed in high Ca2+i.
Figure 20 was tested by determining the most likely rate constants from the simultaneous fitting of 2-D dwell-time distributions obtained at six different Ca2+i ranging from 5.5 to 1,024 µM. The most likely rate constants were then used with Figure 20 to obtain the predicted kinetic structure in Figure 10. The predicted kinetic structure (compare to the observed in Figure 6) shows that Figure 20 captured the major features from low to high Ca2+i, including the general shapes of the dependency plots at high Ca2+i. Figure 20 predicted the excess of brief open intervals adjacent to the longer closed intervals (position 2) and the deficits of brief open intervals adjacent to brief closed intervals (position 1) that were not predicted by Figure 17 at high Ca2+i (compare Figure 10G and Figure H, with Figure 9 and Figure 6G and Figure H). For intervals with durations less than ~0.05 ms, Figure 20 predicted too great of a deficit at position 1. This could reflect an inadequacy of the model or it could reflect the fact that dwell times <0.05 ms were not fitted, so that the predictions of the model were not constrained below this time.
As would be expected from the reasonable descriptions of the kinetic structure, Figure 20 also described the Ca2+ dependence of the 1-D dwell-time distributions. The distributions predicted by Figure 20 essentially superimposed the thick lines in Figure 5. Figure 20 also described the Ca2+ dependence of Po and of the mean open and closed interval durations (Figure 2, thin, dotted, and dashed lines).
Assessing the Various Gating Mechanisms
Figure 20 could also describe the kinetic structure obtained from the two other channels examined over a wide range of Ca2+i. The rankings of the various kinetic schemes for the three channels together with the NLR1000 are presented in Table 3. The NLR1000, which gives a measure of how well the schemes describe the data (see below and METHODS), indicated that Figure 20 was more likely than the other examined schemes for all three channels (larger values of NLR1000 are more likely).
While the NLR1000 can indicate which schemes are most likely, it does not apply any penalties for additional free parameters. Consequently, the schemes were ranked by the Akaike criteria, which applies a penalty for additional free parameters (Equation 4). The general rankings were: Figure 20 > Figure 17 > Figure 29 Figure 15 Figure 16 (Table 3). The Akaike test ranks schemes, but does not give the significance of the rankings. The likelihood ratio test can be used to estimate the significance of rankings for nested models (
The apparent discrepancy between visual observations and the significance of some of the rankings may reflect the necessarily conservative nature of statistical tests. Alternatively, the discrepancy may reflect that Figure 20 is still too simple, so that obvious improvements in some aspects of the gating, such as in the dependency plots, are countered by minimal improvements or even small detrimental changes in other aspects of the gating. Thus, Figure 20 may have to be expanded into Figure 19 to obtain sufficiently improved descriptions of the data to outweigh the heavy penalty imposed by the ranking tests. Consistent with this possibility, Figure 20 predicts only two open and three closed exponential components at high Ca2+i. It will be shown in a later section that the gating of the fully liganded channel is described better by models that more closely approximate Figure 19.
The NLR1000 values in Table 3 give a numerical measure of how well the various schemes describe the experimental data. A NLR of 1.0 indicates that a kinetic scheme describes the 2-D dwell-time distributions as well as the theoretical best description for a discrete state Markov model (see METHODS). The thick lines in Figure 3 show the theoretical best description of the 1-D distributions. For channel B06, the values of NLR1000 (normalized to 1,000 interval pairs) ranged from 3.15 x 10-33 for Figure 29 to 3.25 x 10-4 for Figure 20. These values give likelihood ratios per interval pair of 0.93 [(3.15 x 10-33)0.001] for Figure 29 and 0.99 [(3.25 x 10-4)0.001] for Figure 20. Such values suggest an average likelihood difference per interval pair between the predicted and theoretical best fits of 7% for Figure 29 and only 1% for Figure 20. The 7% difference per interval pair is readily seen (Figure 5, dotted line), while the small 1% difference is still visually apparent as less than perfect descriptions of the data (Figure 5, thick line, and compare Figure 10 to Figure 6).
Figure 20 Predicts too Few Components at High Ca2+i
In spite of its relative success, Figure 20 is still too simple. Analysis of simulated data indicated that Figure 20 predicted only two significant open and four significant closed components at high Ca2+i, compared with the three to four open and five closed components in the experimental data. This underprediction is not surprising, since for Figure 20 the high Ca2+i would effectively drive the gating towards the two open and three closed fully liganded states.
Models for the Gating of the Fully Liganded Channel
Since Figure 20 predicts too few components at high Ca2+i, we explored what types of gating mechanisms might be consistent with the gating at high Ca2+i. In theory, Figure 19 could be examined directly, but the data would be insufficient to constrain the large numbers of rate constants for the 50-state model. Consequently, we explored the reduced models given by Figure 21 Figure 22 Figure 23, which are all composed of fully liganded states.
Figure 21 is drawn from the 10-state model describing the fully liganded states in Figure 19 (the rightmost column of states) with two open states excluded because they had no significant effect on the likelihood estimates. Figure 22 expands Figure 21 so that transitions to the open states pass through intermediate states, and Figure 23 has closed states beyond the activation pathway. Figure 22 and Figure 23 were examined because BK channels gate with large numbers of brief closings (flickers) at high Ca2+i, just as they do at lower Ca2+i (Figure 1, Figure 3, and Figure 6). Intermediate and/or secondary states are associated with the generation of flickers at lower Ca2+i (
Figure 21 Figure 22 Figure 23 were fitted to the 2-D dwell-time distributions obtained at a single high Ca2+i of 1,024 µM for each of three channels. All three of these schemes gave reasonable descriptions of the kinetic structure at high Ca2+i (not shown, but slightly better visually when compared with the experimental data than that shown in Figure 10D and Figure H), and all three schemes gave detected numbers of open and closed states within the range observed in the experimental data at high Ca2+i. Figure 21 gave three open and four closed components, and Figure 22 and Figure 23 each gave three open and five closed components. For purposes of comparison, a simpler Figure 24 with only two open and three closed states, which describes the fully liganded states in Figure 20, was also examined.
Both the NLR1000 and Akaike criteria ranked the schemes in the order: Figure 22 ~ Figure 23 > Figure 21 >> Figure 24 (Table 4). The likelihood ratio test was applied to the nested Figure 23, Figure 21, and Figure 24, and gave highly significant (P < 0.001) rankings of: Figure 23 > Figure 21 > Figure 24.
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The values of the NLR1000 for Figure 21, Figure 22, and Figure 23 (Table 4) indicated that the fits given by these schemes were considerably better than for the simpler Figure 24, and approached the theoretical best descriptions of the single data sets for discrete state Markov models. The values of the NLR1000 for these schemes ranged from 0.061 to 0.959, giving likelihood ratios per interval pair ranging from 0.9972 (0.0610.001) to 0.9999 (0.8990.001), suggesting little difference in likelihood per interval pair between the observed and theoretical best descriptions of the data.
These findings indicate that the gating of BK channels at high Ca2+i can be approximated by models based on the fully liganded states in Figure 19. Figure 21 was drawn from the fully liganded states in Figure 19. Adding three additional brief states as either intermediate states (Figure 22) or secondary states (Figure 23) to generate additional flickers significantly improved the description of the data (Table 4). Whether the intermediate or secondary states are needed, or whether these additional states simply provide a means to compensate for the fact that fitting only the fully liganded states excludes potential contributions to the gating from transitions back to the states with fewer than four bound Ca2+ is not yet clear. What is clear, however, is that at least three open and five to eight closed states, as described by Figure 21 Figure 22 Figure 23, are required to describe the gating at high Ca2+i equivalent to the theoretical best description. These schemes lack Ca2+-dependent rate constants and apply only for Ca2+i > ~100 µM, where the gating kinetics are little affected by Ca2+i.
Figure 15 with an Assumption of Saturation in the Ca2+-dependent Rate Constants Could Approximate the Gating from Low to High Ca2+i
We also explored an alternative explanation to account for the lack of effect of Ca2+i on the gating at high Ca2+i. In all of the above considered schemes, the binding rate of Ca2+i was assumed to be a first order reaction, increasing linearly with Ca2+i. Thus, the effective rate constants for binding are given by the product of Ca2+i times the rate constants expressed per micromole per second. An upper limit for the rate constant for such a diffusion-controlled process is ~109 M-1 s-1 (
Alternatively, if the binding, which is represented by a one-step process in the kinetic schemes, is actually a two-step process that involves binding followed by a conformational change, then the apparent binding rate would saturate if the second step becomes rate limiting at high Ca2+i (
Since the physical details involving Ca2+ binding and action are not known, we explored these two saturation models by using an approach that was independent of a detailed physical model. As both processes would have the effect of reducing the effective concentration of Ca2+i at high Ca2+i, we examined whether the kinetic structure from low to high Ca2+i could be described by letting the effective Ca2+i at high Ca2+i be less than the actual Ca2+i, to mimic apparent saturation of the binding step. Since it is not known what the value of the effective Ca2+i would be at high Ca2+i, this value was estimated by iterative fitting. The 2-D dwell-time distributions obtained at six different Ca2+i (5.5, 8.3, 12.3, 20.3, 132, and 1,024 µM) were simultaneously fitted to estimate the most likely rate constants for Figure 15, and also the most likely effective concentrations of Ca2+i for the data obtained at 132 and 1,024 µM Ca2+i. When fitting, the Ca2+i used for the data obtained at the four lower Ca2+i was fixed to the experimental values.
Figure 15 with effective Ca2+i of 56.9 and 60.0 µM for the data obtained at 132 and 1,024 µM Ca2+i, respectively, and the actual Ca2+i for the other four data sets obtained at lower Ca2+i could approximate the basic features of the kinetic structure from low to high Ca2+i. The predicted kinetic structure at 1,024 µM Ca2+i is shown in Figure 11 and was visually indistinguishable from the predicted kinetic structure at 132 µM Ca2+i. Comparison of the predicted kinetic structure in Figure 11 to that in Figure 6C, Figure D, Figure G, and Figure H, showed that Figure 15 with an assumption of saturation could approximate the data at high Ca2+i. Figure 15 with saturation also described the data at lower Ca2+i, with the predicted structure similar to that in Figure 10 (not shown). For the other two channels studied in a similar manner, the effective Ca2+is at 132 and 1,024 µM Ca2+i were 67.8 and 114 µM (channel B12) and 35.2 and 39.4 µM (channel B14). The likelihoods indicated that Figure 15 with an effective saturation in the binding rate (Scheme V-sat) described the kinetic structure from low to high Ca2+i slightly less well than Figure 17 and Figure 20 (Table 3). The observation that the effective Ca2+i was greater for all three channels for the data collected at 1,024 µM than for the data collected at 132 µM suggests that increasing Ca2+i from 132 to 1,024 µM may have some additional effects on the gating, but any effects would be small since the exponential components describing the dwell-time distributions at 132 and 1,024 µM Ca2+i were not significantly different.
The results in this section show that a relatively simple gating mechanism (Figure 15) with the added assumption of apparent saturation in the Ca2+ binding steps at high Ca2+i can approximate the gating from low to high Ca2+i. It will be discussed later that saturating models may be less appropriate than two-tiered models.
Estimated Rate Constants for the Examined Kinetic Schemes
Figure 12 presents the estimated rate constants for Figure 20 and Figure 21 for the three channels examined in detail. Online supplemental Figure S2 (http://www. jgp.org/cgi/full/114/1/93/DC1) presents estimated rate constants for these same three channels for most of the other examined schemes. The rate constants for the examined schemes typically ranged from ~0 to 45,000/s, indicating a large range in the height of the energy barriers between the various states. Rate constants were limited so as not to exceed 45,000/s, as letting them go higher gave little improvement in the fits. Estimated rate constants for the simpler models (Schemes IVVI and XIXIII) were relatively consistent from channel to channel. For the most complex gating mechanism examined (Figure 20), there could be considerable variability in the estimates, depending on the specific rate constants. In those cases where there was considerable variability in estimates of the rate constants among channels, the variability was typically associated with poorly defined rate constants, as these rate constants could be fixed to various values with little effect on the likelihood values after refitting. The rate constants for Figure 20 will be used in the DISCUSSION to describe how the channel gates at low and high Ca2+.
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Discussion |
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This study used detailed single-channel analysis to examine the Ca2+-dependent gating of native BK channels in cultured rat skeletal muscle. We have extended previous studies by examining the effects of high Ca2+i to obtain critical information about mechanism when the gating is driven towards the fully liganded states. Maximum likelihood fitting together with comparisons of the observed and predicted 2-D dwell-time distributions and dependency plots (the kinetic structure) were used to evaluate gating mechanisms.
Properties of Gating in High Ca2+i
For low to intermediate levels of Ca2+i, the gating was highly Ca2+i dependent, with a Hill coefficient of ~3.5, within the range of 24 typically observed for BK channels (see INTRODUCTION). In contrast, the gating was Ca2+ independent for high levels of Ca2+i (>100 µM). Increasing Ca2+i 810-fold to 1,024 µM had little effect on Po (~0.97), the mean open and closed times, the 1- and 2-D dwell-time distributions, and the dependency plots (Figure 1 Figure 2 Figure 3 Figure 4, Figure 6 Figure 7 Figure 8, and Table 1 and Table 2). Estimates of the numbers of significant exponential components in the dwell-time distributions indicated that the channel entered at least three to four open and four to five closed states during normal activity at high Ca2+i, and that the estimated numbers of states did not change when Ca2+i was increased from 100 to 1,024 µM (Figure 4). Significant dependencies (correlations) between the durations of adjacent open and closed intervals in the dependency plots (Figure 6 Figure 7 Figure 8) indicated that transitions between the open and closed states occurred over at least two independent transition pathways (two or more gateway states) at high Ca2+i.
Thus, as a first approximation, gating at high Ca2+i involves Ca2+-independent transitions among at least three to four open and four to five closed states, with two or more independent transition pathways among the open and closed states. Models with these characteristics (Schemes XIXIII) gave excellent descriptions of the kinetic structure for data limited to high Ca2+i (Table 4).
The MWC Model Is Inconsistent With Gating from Low through High Ca2+i
The MWC model (
Models with a Single Effective Gateway State at High Ca2+i Can Be Rejected
Since models based on the MWC model were inconsistent with the data at high Ca2+i, we examined more complex models based on
The lack of dependence for Figure 17 at high Ca2+i results because the high Ca2+i drives the gating towards the fully liganded states, as summarized by Figure 18, where there is a single effective transition pathway between the open and closed states. The observation that Figure 17 predicted no dependence at high Ca2+i suggests by analogy that the more complex 2555-state models (Figure 26 and Figure 27), and all schemes in which there would be a single effective transition pathway between open and closed states at high Ca2+i, can be rejected.
A General Two-Tiered Model Can Account for the Gating from Low to High Ca2+i
Figure 26 Figure 27 Figure 28 Figure 29 Figure 15 Figure 16 Figure 17 can be classified as one tiered because the open and closed states can be contained within a single plane. Since the examined one-tiered models were inadequate to describe the data at high Ca2+i, we considered what type of model would be required. If all the states in Figure 26 and Figure 27 are closed states, then there would be a sufficient number of fully liganded closed states at high Ca2+ to generate the four to five observed closed exponential components observed at high Ca2+. If each of the closed states in Figure 26 and Figure 27 can make a direct transition to an open state, then this would give enough fully liganded open states to generate the three to four observed open components at high Ca2+. If the gating is effectively confined to the fully liganded states at high Ca2+i, then the gating at high Ca2+i would be described by Figure 25, where all the subunits are bound with Ca2+. Each subunit in this scheme can exist in two conformational states, and a concerted conformational change of all subunits is required for opening. Figure 25 has five independent transition pathways between the fully liganded closed and open states (upper and lower tiers, respectively) that would allow dependence to be generated between open and closed intervals at high Ca2+i. Figure 21, consistent with Figure 25, gave excellent descriptions of the gating in high Ca2+i (Table 4)
Extending Figure 25 to include states with zero to three bound Ca2+ would give the general two-tiered model described by Figure 19, with 25 closed states on the upper tier and 25 open states on the lower tier. Figure 20, a reduced version of Figure 19, could describe the kinetic structure of the channel from low to high Ca2+i (Figure 10). Figure 20 also generated single-channel current records that closely mimicked, except for stochastic variation, the experimental current records, as can be seen by comparing the simulated (predicted) records in Figure 13 to the experimental records in Figure 1. The Ca2+-independent gating kinetics at high Ca2+i together with the few longer closed intervals are present in the simulated records.
Although the more complex Figure 19 was not tested directly, this general scheme should give an even better description of the gating than Figure 20, as Figure 20 is contained within Figure 19. Hence, Figure 19 can serve as a working hypothesis for the Ca2+-dependent gating of BK channels. Horrigan and Aldrich (personal communication), based on analysis of macroscopic ionic and gating currents from mSlo, have also found evidence for two-tiered gating mechanisms.
Gating in Figure 19 Occurs Among Five Subschemes that Differ in the Number of Bound Ca2+
In the context of Figure 19, it can be seen that, for any fixed number (04) of Ca2+ bound to the channel, the channel could gate among at least five open and five closed states. Thus, Figure 19 can be viewed as being comprised of five subschemes (one of which is shown in Figure 25). The subunit conformations of the analogous states in each of the subschemes are the same, as are the transition pathways among the various open and closed states. Hence, both the conformations of the subunits and the connections among states are identical for each subscheme, independent of the number of bound Ca. Ca2+i acts by driving the gating from the subscheme comprised of the states with zero bound Ca2+ towards the subscheme comprised of the states with four bound Ca2+. The binding of Ca2+ stabilizes the open states. The dynamics of this Ca2+-dependent shift will be presented in a later section.
Additional support for Figure 19 comes from the observation that BK channels gate at very low Ca2+i (
Why the Dependency Plots Appear Similar from Low to High Ca2+i
The saddle shape of the dependency plots (Figure 6 and Figure 7) reflects the inverse relationship between the durations of adjacent open and closed intervals. This inverse relationship suggests that more stable (longer duration) open states are effectively connected to less stable (briefer duration) closed states (
Gating Dynamics at Low and High Ca2+i
Since Figure 20 (drawn from Figure 19) could account for the kinetic structure from low to high Ca2+i, Figure 20 was examined to gain insight into how Ca2+ activates the channel. Figure 14 presents the equilibrium occupancy, mean lifetime, and frequency of entry for each state for gating at low Ca2+i (5.5 µM, Po = 0.061) and high Ca2+i (1,024 µM, Po = 0.97) for Figure 20. Notice that bars of considerable height in these plots can have small values due to the logarithmic ordinates.
At low Ca2+i (Figure 14, AC), the channel readily gates among all of the states in Figure 20, as indicated by the frequencies of entry into each state (C), but spends most of its time in the closed states with zero (82.9%) and one (8.2%) bound Ca2+i (A). At high Ca2+i, the gating of the channel is effectively confined to the fully liganded states with four bound Ca2+, where it spends 99.2% of its time. Little time (0.76%) is spent in the states with three bound Ca2+, and negligible time (<0.04%) is spent in the states with 0, 1, or 2 bound Ca2+i (Figure 14, DF). The high Po in high Ca2+ reflects that the channel spends 96.6% of its time in the open states with four bound Ca2+, where Ca2+ stabilizes the open states. The flickers (brief closings) in the single-channel current record arise mainly from transitions to the brief closed states C12 C16 at lower Ca2+i and C12 and C15 at high Ca2+i.
The Considered Schemes Are Only Approximations of the Gating Mechanism
As emphasized by
However, even the general 50-state Figure 19 is a reduced model compared with what the actual gating mechanism is likely to be. Figure 19 excludes the isoforms of the various open and closed states. Including all the isoforms of each state (see Figure 26) would expand Figure 19 from 50 to 110 states.
BK channels may have additional closed states beyond the activation pathway (secondary states).
The skeletal muscle BK channel often passes through a brief lifetime subconductance state upon opening and closing (
Changing Ca2+i more than two orders of magnitude, as was done in our experiments, would be expected to alter surface charge (
Although Figure 20 could describe many features of the data, this does not exclude the possibility that other rather different mechanisms might also account for the data. A model with fewer states than Figure 20, but with apparent saturation in Ca2+ binding rate could also give reasonable descriptions of the Ca2+ dependence of the single-channel kinetics from low to high Ca2+i (Figure 11 and Table 3). Nevertheless, we prefer the general two-tiered approach based on Figure 19 to the more ad hoc saturation models, as two-tiered models provide a means to account for the complexity of the gating in zero Ca2+i and are consistent with a tetrameric protein.
BK channels can gate in a number of different modes, with 96% of the intervals occurring during activity in the normal mode (
Voltage Dependence of Gating
The voltage dependence of BK channels is an intrinsic property of the channel and does not appear to arise through voltage-dependent increases in Ca2+ binding (
Recent preliminary observations on large multistate models using analysis of macroscopic ionic and gating currents (
Conclusion
This study develops gating mechanisms that can describe the Ca2+ dependence of the kinetic structure of BK channels from low to kinetically saturating levels of Ca2+i. These models are drawn from a general 50-state two-tiered model in which each closed state in the upper tier can make a direct transition to an open state in the lower tier. Our previous models that describe the Ca2+-dependent gating over more limited conditions are contained within the general 50-state model. Thus, the 50-state model serves to unify previous studies, and can provide a framework for further studies on mechanism through single-channel analysis of gating at very low Ca2+i and of the voltage dependence of the gating.
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Footnotes |
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The online version of this article contains supplemental material.
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Acknowledgements |
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This work was supported in part by grants from the National Institutes of Health (NS007044 and AR32805) and the Muscular Dystrophy Association.
Note added in Proof. Forthcoming kinetic studies lend additional support to Figure 19. Gating currents and macroscopic ionic currents recorded from mSlo BK channels in the absence of Ca2+ are consistent with a model that is described by the unliganded states in Figure 19. (Horrigan, F.T., J. Cui, and R.W. Aldrich. 1999. Allosteric voltage-gating of potassium channels I: mSlo ionic currents in the absence of Ca2+. J. Gen. Physiol. In press. Horrigan, F.T., and R.W. Aldrich. 1999. Allosteric voltage-gating of potassium channels II: mSlo channel gating charge movement in the absence of Ca2+. J. Gen. Physiol. In press.)
Submitted: March 5, 1999; Revised: May 6, 1999; Accepted: May 7, 1999.
1used in this paper: 1-D, 1-dimensional; BK channel, large-conductance calcium-activated K+ channel; MWC, Monod-Wyman-Changeux; NLR1000, normalized likelihood ratio
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