INTRODUCTION

Top Abstract Introduction Methods Results Discussion References

Potassium channels constitute a functionally and structurally diverse class of ion channels (for reviews see Salkoff et al., 1992; Hoshi and Zagotta, 1993). The majority of voltage-dependent potassium channels can be classified as either outwardly or inwardly rectifying channels that preferentially conduct outward or inward current, respectively. Cloned outward rectifiers, including the delayed rectifier potassium channels, inactivating Shaker  -family channels (Kamb et al., 1987; Tempel et al., 1987; Pongs et al., 1988), and the Calcium-activated maxi-K channels (Atkinson et al., 1991; Adelman et al., 1992; Butler et al., 1993), share a common putative transmembrane topology consisting of six transmembrane segments, a pore loop between the fifth and sixth transmembrane segments, and an S4 "voltage sensor" region.

Inward rectifier K⁺ channels were first described as "anomalous rectification" currents to emphasize the contrast between these channels and previously described outwardly rectifying currents (Katz, 1949; Hille, 1992). Inwardly rectifying currents have been examined in a variety of preparations, including frog skeletal muscle, and tunicate and starfish eggs (Adrian, 1969; Hagiwara and Takahashi, 1974; Hille and Schwarz, 1978). These channels preferentially conduct inward current while limiting outward current. Recently, the cDNA clones and primary amino acid sequences for several inward rectifiers have been obtained. These include the IRK, ROMK, GIRK, K_ATP, and KAT1 families (Anderson et al., 1992; Sentenac et al., 1992; Dascal et al., 1993; Ho et al., 1993; Kubo et al., 1993a,b; Ashford et al., 1994; Inagaki et al., 1995; Krapivinsky et al., 1995). Unlike the proposed transmembrane topology of outwardly rectifying potassium channels, the "small" inward rectifiers possess two putative transmembrane regions, with a pore region interposed between them. The NH₂ and COOH termini of these channels are relatively long and intracellularly located. This proposed structure has been recently confirmed using x-ray crystallography in the bacterial potassium channel KcsA (Doyle et al., 1998). In the cloned small inward rectifiers, intracellular Mg²⁺ and polyamines (spermine, spermidine, and putrescine) have been demonstrated to be the intracellular blocking particles responsible for inward rectification (Kubo et al., 1993a,b; Ficker et al., 1994; Lopatin et al., 1994; Nichols et al., 1994; Fakler et al., 1995).

Among inwardly rectifying potassium channels, the KAT1 and akt1 channels are unique in both structure and function. KAT1 and the related channel akt1 are inwardly rectifying potassium channels cloned from the plant Arabidopsis thaliana, a member of the mustard family, through a yeast complementation strategy (Anderson et al., 1992). Unlike the small inward rectifiers, KAT1 possesses six putative transmembrane regions, a pore loop between the fifth and sixth transmembrane regions, and an S4 motif. Structurally, KAT1 resembles the Shaker family of potassium channels, and yet functionally it behaves as an inward rectifier (Schachtman et al., 1992). Unlike the small inward rectifiers, KAT1 rectification does not require intracellular cation block (Hoshi, 1995). Inward rectification is not significantly altered upon patch excision, suggesting that polyamine block is also not important in KAT1 rectification (Hoshi, 1995). It is therefore a reasonable conclusion that the gating mechanisms resulting in an inwardly rectifying phenotype in KAT1 are intrinsic to the channel protein itself.

KAT1 appears to have the structural architecture of an outward rectifying channel, yet its functional phenotype is that of an inward rectifier. This suggests that perhaps KAT1 achieves inward rectification through a fast inactivation recovery mechanism, as demonstrated in the eag-family channel herg (Smith et al., 1996; Spector et al., 1996) and in Shaker channels containing mutations that alter activation properties (Miller and Aldrich, 1996). However, NH₂-terminal deletions and permeant ion effects that should affect NH₂-terminal inactivation processes (Demo and Yellen, 1991; Lopez-Barneo et al., 1992) and mutations in residues corresponding to residues critical for C-type inactivation (Hoshi et al., 1991; Heginbotham and MacKinnon, 1992) in Shaker channels have little effect on KAT1 activation (Marten and Hoshi, 1997). Perhaps the KAT1 protein functions similarly to outwardly rectifying channels like Shaker, but is inserted in the membrane in a reversed topology so that the "voltage sensor" is oriented in the electric field in the opposite direction from these other channels. This hypothesis is unlikely, as sequence analysis does not suggest possible signal sequences in the channel protein that differ significantly from those of other channels, and mutations in the NH₂ terminus do not reverse the channel's voltage dependence, as might be expected if there were a crucial signal sequence (Marten and Hoshi, 1997). One can also imagine a channel in which states that are normally closed are conducting states, and vice-versa, resulting in opening at negative voltages. In other words, KAT1 may possess a unique gating mechanism in which the polarity of a critical component of the voltage sensing mechanism is reversed so that hyperpolarization, rather than depolarization, increases open probability. Mutations in both the NH₂- and COOH-terminal domains produce significant effects on the voltage-dependent gating behavior of KAT1, suggesting that these regions of the molecule play an important role in gating (Marten and Hoshi, 1997). On the other hand, the presence of the charged S4 voltage sensor motif implies that KAT1 gating involves the S4 region, as seen in other channels gated by voltage.

In other voltage-dependent ion channels, the role of the S4 region in gating has been substantiated through mutagenesis. Mutations of the charged residues located within the S4 segment have been shown to alter the voltage-dependent gating properties of potassium and sodium channels (Stühmer et al., 1989; Papazian et al., 1991; Logothetis et al., 1992, 1993; Schoppa et al., 1992; Tytgat and Hess, 1992; Aggarwal and MacKinnon, 1994). Cysteine mutagenesis has demonstrated that the S4 region likely moves during the activation of sodium channels (Yang and Horn, 1995) and potassium channels (Larsson et al., 1996). Optical signals from channels with fluorescent labels in the S4 region support the hypothesis that the S4 region moves during activation (Mannuzzu et al., 1996; Cha and Bezanilla, 1997). However, mutations in other regions of the channel protein (Gautam and Tanouye, 1990; Lichtinghagen et al., 1990; MacKinnon, 1991; Schoppa et al., 1992; Papazian et al., 1995) have also been shown to alter the voltage-dependent properties of these channels, demonstrating that other channel regions are also likely to be involved in the gating process. Moreover, the S6 region is believed to be involved in the gate that physically impedes ion flux through the channel pore (Liu et al., 1997).

In this article, the gating properties of the wild-type KAT1 channel will be described through the analysis of single channel patch clamp currents. Statistical analysis of the single channel open and closed durations will provide information about conformational changes that the channel undergoes between closed and open states. Analysis of these data can be used to create a kinetic model that accurately describes the gating behavior of the KAT1 channel, particularly conformational transitions near the open state, over a wide voltage range. This kinetic model can then provide insight into the intrinsic voltage-dependent gating properties of KAT1 that result in inward rectification. This model will serve as a basis for interpreting the effects of two neutralization mutations in the S4 region (R177Q and R176L) on the gating mechanisms of KAT1 that produce inward rectification.

	METHODS

Top Abstract Introduction Methods Results Discussion References

Molecular Biology

All currents were recorded from the KAT1 channel cloned from the plant A. thaliana, which was provided to us by Dr. Richard Gaber (Northwestern University, Evanston, IL). All mutant constructs were made in the wild-type KAT1 clone using the standard PCR-based cassette mutagenesis. DNA sequences for all mutants were confirmed by dideoxy termination sequencing (Sanger et al., 1977). The KAT1 and mutant cDNA clones were propagated in the -YES vector provided by Dr. Gaber in the Escherichia Coli strain DH5-. The cDNA was transcribed in vitro into cRNA using a T7 polymerase (Ambion Inc., Austin, TX). Approximately 40 nl of cRNA per cell was injected into Xenopus oocytes to record macroscopic currents. Single channel patches were obtained by injecting cRNA that was diluted up to 1,000-fold. Recordings were made 2-7 d after cRNA injection.

Electrophysiology

Data from single channel patches were recorded in the inside-out configuration, unless otherwise noted (Hamill et al., 1981). Patch pipettes were composed of borosilicate glass (VWR Micropipettes, West Chester, PA). Their tips were coated with wax (Sticky Wax, Emeryville, CA) and fire polished before use. Data were acquired using an Axopatch 200-A patch clamp amplifier (Axon Instruments, Foster City, CA), and the amplifier output was low-pass filtered through an eight-pole Bessel filter (Frequency Devices, Inc., Haverhill, MA), digitized at a frequency as noted in the figure legends and stored for later analysis. The data were typically filtered at frequencies ranging from 400 to 1,000 Hz, which did not limit the kinetic measurements, as KAT1 channel kinetics are relatively slow. A Digital Equipment Corp. LSI 11/ 73-based minicomputer system (Indec Systems, Sunnyvale, CA) was used for controlling the voltage-clamp protocols and was used for a portion of the data analysis. Experiments were carried out at 20-22°C. The pipette potential was nulled just before seal formation. The voltage error due to junction potentials was estimated to be <5 mV, and no corrections were made for this error. Unless otherwise noted, the holding potential was 40 mV for wild-type KAT1 currents and 0 mV for R177Q and R176L mutant currents.

Data were recorded and stored on the LSI 11/73-based minicomputer system. Voltage pulses were applied every 2-5 s. Linear leak currents and uncompensated capacity currents were subtracted using leak templates made from fits to sweeps with no openings. For the ensemble averages and duration histograms, opening and closing transitions were detected using a 50% amplitude criterion of the single channel amplitude at any given voltage. Open and closed durations were measured from these idealized records. The number of channels in any given patch was determined by observing the maximum number of channel openings at a potential where the probability of the channel being open was high. Unless otherwise noted, single channel patches were used in the data presented in this article.

A confounding factor in the measurement of single channel KAT1 currents is that channels will occasionally exhibit rundown, or a progressive decline in macroscopic current amplitude and open probability over time. It has been noted that rundown is faster in excised patches (Hoshi, 1995). Rundown was indeed occasionally observed in our single channel patches, and the data from those patches were not used.

Duration Fitting and Simulations

Durations of open and closed events were compiled from the idealized single channel data and transferred to a Macintosh-based computer system. Ensemble averages that have been expressed as open probabilities were determined by dividing the current averages by the number of channels in the patch and the unitary current amplitude. The open and closed durations were fitted with sums of exponential probability density functions using the maximum likelihood method. The fits were corrected for the left censor time, or the dead time of the recording system, and for the right censor time, which corresponds to the limited pulse duration (Colquhoun and Hawkes, 1982; Lawless, 1982; Hoshi and Aldrich, 1988). The left censor time was estimated as 0.253/f, where f represents the filter cutoff frequency in Hertz. The first latency distributions were corrected for the filter delay time, which was estimated as 0.506/f. Unless otherwise noted, open- and closed-duration histograms are displayed using a log-binning transformation (Sigworth and Sine, 1987) to optimize the presentation of data and associated fits.

Correction for Missed Events

Rate constant values in the three-state burst model described in this article were estimated with corrections for events too short to be resolved (Blatz and Magleby, 1986). Given the model considered and the dead time of the recording system, the four effective rate constants were calculated by estimating the fraction of all events in each data set both longer and shorter than the dead time of the recording system. The rate constant values were then optimized separately for the open and closed-time parameters.

Solutions

All currents were recorded in symmetrical 140-mM K⁺ solutions. The standard extracellular solution contained (mM): 140 KCl, 6 MgCl₂, 5 HEPES, pH 7.2. The intracellular solution contained (mM): 140 KCl, 11 EGTA, 2 MgCl₂, 1 CaCl₂, 10 HEPES, pH 7.2.

	RESULTS

Top Abstract Introduction Methods Results Discussion References

Single Channel Currents

Although macroscopic patch data can provide a great deal of insight into the gating behavior of KAT1 channels, single channel analysis allows the observation of real-time conformational changes of individual KAT1 protein molecules that can constrain a potential kinetic model. Fig. 1 A displays representative KAT1 single channel currents over a voltage range from 110 through 190 mV. After pulse initiation, there is a short delay before the channel initially opens. This delay, or first latency to opening, becomes faster with increasingly negative voltages. Once open, the channel flickers, or bursts, between open and closed states before occasionally entering a longer-lived closed state. From this longer-lived closed state, the channel may begin bursting again. With increasingly negative voltages, KAT1 channels are more likely to continue bursting through the entire pulse duration. The combined effect of shorter latencies to first opening and longer burst durations is an increasing overall open probability with increasing hyperpolarization.

View larger version (22K): [in this window] [in a new window]   Fig. 1.   (A) Representative single channel currents from KAT1 channels recorded in the inside-out patch configuration in response to hyperpolarizing voltage pulses. The voltages are indicated in the figure. The data at 190, 180, 160, 140, 120, and 110 mV were filtered at 0.9, 0.6, 0.6, 0.6, 0.6, and 0.7 kHz, respectively, and sampled at 1.54 kHz. The voltage pulses were delivered every 2-6 s. The prepulse, tail, and holding potentials were 40 mV. (B) The single open channel i(V) for KAT1 channels over the voltage range from 80 mV to 190 mV. Current amplitudes were obtained as follows: at each given voltage, all sampled data points from a representative sweep were compiled into histograms plotting the number of data points as a function of current amplitude. These current amplitude histograms were then fitted by eye to determine peak amplitudes, which represent closed and open state current values. The single channel current amplitude was then calculated as the difference between the open and closed state current levels at each given potential. The i(V) curve has been fitted to a linear function, which yields a unitary conductance of 7.5 pS.

A current-voltage relationship [i(V)]¹ constructed from single channel currents yields a unitary conductance of ~7.5 pS in symmetrical 140-mM K⁺ solutions (Fig. 1 B). The i(V) relationship is linear over the range of activating voltages examined for KAT1 single channels, confirming that, in this voltage range, gating mechanisms, not permeation properties, determine inward rectification. At depolarized potentials (greater than +20 mV), outward current is not observed in KAT1 channels (data not shown), which is in agreement with the observation of open-channel rectification of outward currents in macroscopic tail currents (Zei, 1998). KAT1 currents at voltages more hyperpolarized than 200 mV were difficult to obtain, a result of both the extreme voltages and the long pulse durations needed.

The overall behavior of single KAT1 channels at a given potential is demonstrated by the ensemble averages expressed as open probabilities in Fig. 2 A over a range of voltages. The overall time course of activation is very similar to currents recorded in macroscopic patches (Hoshi, 1995; Zei, 1998), in the whole-cell voltage clamp configuration (Schachtman et al., 1992; Véry et al., 1995; Marten and Hoshi, 1997), and in the cut-open oocyte system (our unpublished results), confirming that the single channel records accurately represent the gating behavior of wild-type KAT1. The channels activate slowly after a small but noticeable delay, with a time scale on the order of several hundred milliseconds, in contrast with activation in Shaker channels, which occurs on the order of 1-10 ms. This delay suggests the existence of multiple closed states along the activation pathway. The time course of activation becomes faster with increasingly negative voltages ranging from 110 to 190 mV, and there is no appreciable decay or inactivation in the ensemble average currents. The steady state probability that the KAT1 channel is open during the pulse increases with increasingly negative voltages, as shown in Fig. 2 B. The open probability increases steeply between 120 and 180 mV, saturating between 190 and 200 mV. The maximum open probability is ~0.7, reflecting a significant contribution of closed events to the steady state channel behavior even at the most hyperpolarized potentials. The open probability voltage dependence is fitted by a Boltzmann function:

(1)

View larger version (20K): [in this window] [in a new window]   Fig. 2.   (A) Ensemble averages of single-channel patch currents expressed as open probabilities at several voltages. Open probability is calculated by dividing the ensemble average current by the number of channels in the patch (single channel patches were used in most of the ensemble averages shown) and the measured unitary current amplitude. The voltages are indicated in the panel. Recording conditions were as described in Fig. 1. (B) Voltage dependence of the steady state open probability. Peak probabilities recorded from the ensemble averages are plotted against voltage. Data from several patches are shown. The smooth curve represents a fit to a Boltzmann function expressed by the following equation: where Pomax is the maximum open probability (0.7), V is membrane voltage, V1/2 is the voltage where the Boltzmann distribution is equal to 0.5 (143 mV), z is the equivalent charge movement associated with the Boltzmann distribution (1.36 electronic charges), F is Faraday's constant, R is the universal gas constant, and T is the absolute temperature. The dotted line represents the Boltzmann fit to the macroscopic G(V) relationship (Zei, 1998). The macroscopic G(V) fit has been scaled to the maximum single channel open probability, Pomax.

represented by the solid line in Fig. 2 B, where P_o is open probability, P_o^max is maximal open probability observed, V_1/2 is the voltage at which the open probability is 0.5 of P_o^max, z is the apparent gating valence, and F, R, and T have their usual meanings. The midpoint of activation (V_1/2) is extremely negative, estimated at 143 mV, and the apparent charge is relatively small, at 1.4 e. In Fig. 2 B, the Boltzmann fit obtained from the macroscopic conductance-voltage [G(V)] data (Zei, 1998) is superimposed and scaled to the maximum open probability derived from the Boltzmann fit to the ensemble average G(V). This comparison demonstrates good agreement between the steady state activation properties observed in both macroscopic and single channel patches.

The voltage dependence of KAT1 activation derived from macroscopic currents is far weaker than that seen in other voltage-gated ion channels. For instance, the apparent gating valence of the Shaker channel as measured by limiting slope, Boltzmann fits to tail currents, gating current, and toxin labeling measurements all yield a gating valence of ~12-16 elementary charges per channel (Schoppa et al., 1992; Aggarwal and MacKinnon, 1994; Zagotta et al., 1994a). The apparent gating valence of the Ca²⁺-activated K⁺ channel (slo, BK, or maxi-K channel) using limiting slope and Boltzmann fit techniques is between 1.1 and 1.8 e (Cox et al., 1997; Cui et al., 1997). However, the slo channel is gated by both voltage and calcium ions.

Transitions Before First Opening

As shown in the representative single channel currents in Fig. 1 A, the time to first opening, or first latency, becomes shorter with increasingly negative potentials. During these first latencies, the channel is presumed to traverse closed states along the activation pathway. In Fig. 3 A (top), cumulative distributions of latencies to first opening are displayed for voltages from 110 through 190 mV. Fig. 3 A (bottom) depicts the same cumulative first latency distributions scaled to the same steady state probability to facilitate comparison of their time courses. Similar to the activation of macroscopic ionic currents, there is only a small amount of sigmoidicity in the first latency time courses across all voltages examined, indicating only a few discernible closed states in the transitions before first opening. The first latency time course becomes faster with increasingly negative potentials, consistent with the voltage dependence of the macroscopic activation time course.

View larger version (26K): [in this window] [in a new window]   Fig. 3.   (A) Cumulative distributions of first latencies recorded at several voltages. The distributions show the probabilities that the channel first opened by the times indicated. The openings were elicited in response to 1,000-ms pulses from a prepulse potential of 40 mV. (Bottom) The cumulative first latency distributions have been scaled to a probability of 1 to compare their time courses. (B) Median first latencies measured from the distributions in A and several additional patches are plotted as a function of pulse potential. Several different patches are represented among the data points. (C) The cumulative first latency distributions are superimposed on the ensemble averages from the same patch at each voltage indicated. The thin, relatively noisy lines represent the ensemble averages, and the thick, smoother lines represent the first latency distributions. The curves are scaled so that the steady state levels coincide.

The voltage dependence of first latencies can be quantified by plotting the median first latency as a function of voltage, as in Fig. 3 B. The median first latency is highly voltage dependent, which implies that much of the voltage dependence of activation can be accounted for by the transitions before first opening. Furthermore, if the transition rate constants leaving the open state do not significantly contribute to the activation kinetics of KAT1, then the time courses of first latencies and overall activation will be similar. Superimposed cumulative first latency distributions and ensemble averages are shown in Fig. 3 C at several voltages. The first latency distributions are normalized to the ensemble averages to facilitate comparison of their time courses. The observed similarity between the first latency and ensemble average time courses over the entire voltage range of activation indicates that indeed much of the voltage dependence of activation results from transitions before the channel first opens.

As seen in Fig. 3 A, the first latency distributions saturate at a probability of one only at the most negative voltages, reflecting the fact that, at most voltages, there are several sweeps during which the KAT1 channel fails to open. In fact, the fraction of sweeps that fail to elicit at least one channel opening is larger at more positive potentials and decreases with increasingly negative voltages. There are several mechanistic possibilities, individually or in combination, that can account for these blank sweeps. First, the channel may enter an inactivated state from a closed state located along the activation pathway. Second, the channel may occasionally enter an inactivated state located after the open state and outside the activation pathway during the previous pulse, from which it does not recover. Third, the first latencies may occasionally be longer than the pulse durations used, particularly at depolarized voltages.

In the first two possibilities, an inactivated state with a relatively slow rate of recovery and, therefore, a long dwell time would result in blank sweeps that tend to cluster together. The tendency for blank sweeps to cluster can be tested with runs analysis (Horn et al., 1984). In runs analysis, a run is defined as a contiguous series of like events. In a system with two types of events, the probability of having R runs in an experiment can be calculated. Given a data set with at least ~40 events, the distribution of R can be approximated by an asymptotic distribution, from which a standard variable, Z, can be defined with a mean of zero and variance of one (Horn et al., 1984):

(2)

where R is the number of runs in the experiment, n is the total number of events, and P is the probability of observing a sweep with at least one opening. Negative Z values indicate a tendency for sweeps to alternate between one or the other type of event, while positive Z values indicate the tendency for like events to cluster in long runs. A Z value greater than +1.6 indicates statistically significant clustering (Horn et al., 1984). Fig. 4 A plots the number of openings per sweep contiguously for two experiments at 110 and 140 mV. It might seem qualitatively apparent that there are several clusters of blank sweeps in these experiments. However, in Fig. 4 B, the Z values at all voltages with a significant number of blank sweeps do not demonstrate statistically significant clustering, implying that a long-lived inactivated state is unlikely to account for the observation of blank sweeps in KAT1. Furthermore, variation of the interpulse interval between 1 and 6 s did not have a significant effect on the observed frequency of blank sweeps (data not shown). These observations suggest that, if the channel enters an inactivated state either along the activation pathway or after the open state, then the recovery rate from that state is fast enough so that the probability of observing blank sweeps is not significantly affected by the interpulse duration, and clustering of blank sweeps is not observed. The possibility that first latencies longer than the pulse duration contribute to the observation of blank sweeps must also be considered. The potential contribution of these various mechanisms to the occurrence of blank sweeps will be discussed further when specific models of KAT1 gating are considered.

View larger version (14K): [in this window] [in a new window]   Fig. 4.   (A) The number of openings per sweep over an entire experiment are plotted contiguously at 110 and 140 mV. The x-axis indicates consecutive sweep number. The data represent two different patches. (B) The test parameter (Z) for runs analysis (see text) is plotted at several voltages. Only those experiments with a significant number of blank sweeps are shown. The dotted line represents a Z value of 1.6. Z values greater than or equal to 1.6 indicate statistically significant clustering of blank sweeps.

Transitions After First Opening: Open States

Opening and closing events were assumed to behave as time-homogeneous Markov processes, so the distribution of open and closed durations can therefore be fitted with a sum of exponential functions. The number of exponential functions that best fits each distribution indicates the minimum number of discernible states at each particular current amplitude. Fig. 5 A shows the distributions of open durations at voltages between 110 and 190 mV fitted with single exponential functions. Fits to sums of greater than one exponential did not yield significantly improved fits to the open time distributions using the maximum likelihood method. A good fit of the open time distribution with a single exponential function indicates that there is only one discernible open state. The time constants derived from fits to open time distributions are plotted as a function of voltage in Fig. 5 B. Open times become shorter with increasing hyperpolarization, even after the data are corrected for missed events. Since the overall rate of leaving the open state becomes faster with increasing hyperpolarization, at least one of the individual leaving rates should reside outside the activation pathway towards first opening. Otherwise, if all transitions away from the open state were to lie within the activation pathway, as shown below (Scheme I), then the voltage dependence of the sole transition rate constant leaving the open state would be in the opposite direction to that of overall activation, which is mechanistically unlikely, although certainly possible. On the other hand, if at least one of the closed states lies outside of the activation pathway (Scheme II), then the transition from the open to the closed state outside the activation pathway (rightward arrow) can become faster with hyperpolarization, which is congruent with the overall voltage dependence of activation, and yet still accounts for the voltage dependence of the mean open times. More complex models that incorporate opposite voltage dependences of individual rate constants and overall activation are certainly physically possible. However, the simpler scheme (Scheme I) was adequate to explain the features of gating (see below). A voltage-dependent open time distribution differs from results seen in Shaker channels, where the transitions between the open and closed states outside the activation pathway (C_f and C_i) are essentially voltage independent (Hoshi et al., 1994).

View larger version (14K): [in this window] [in a new window]   (scheme I)

View larger version (14K): [in this window] [in a new window]   (scheme II)

View larger version (26K): [in this window] [in a new window]   Fig. 5.   (A) Frequency histograms of open durations measured at several voltages from KAT1 channels. The data were fitted with single exponential functions, represented by the solid curves. The histograms are plotted using the log-binning transformation of Sigworth and Sine (1987) as described in METHODS. (B) Time constants derived from the exponential fits to the open time distributions are plotted as a function of voltage. The data points represent several patches. The exponential fits have been corrected for the left and right censor times, as described in METHODS.

Transitions After First Opening: Closed States

The distribution of closed times that occur after the channel first opens is shown in Fig. 6 A at voltages ranging from 110 to 190 mV. The data are well fit by a sum of two exponential functions using the maximum likelihood method, and greater than two exponentials did not significantly improve the fits. This implies that there are at least two discernible closed states in close topological proximity to the open state. As seen in the representative traces in Fig. 1 A, the two kinetically distinct closed states can be qualitatively discerned as long and brief populations of closed durations. This type of behavior can be described as bursting behavior, as the channel flickers rapidly between closed and open states in "bursts," with occasional sojourns into a longer-lived closed state, an interburst interval. In Fig. 6 B, the long and brief component time constants from the two-exponential fits to closed-time distributions are plotted separately as functions of voltage. The long time constant becomes shorter with more negative voltages, while the brief component is essentially voltage independent. The relative amplitudes of the long and brief components are also shown as functions of voltage (Fig. 6 B). Over the entire voltage range examined, the contribution of brief closed events is much greater than that of long closed events.

View larger version (29K): [in this window] [in a new window]   Fig. 6.   (A) Frequency histograms of closed-time durations measured at several voltages from KAT1 channels. The data were fitted with a sum of two exponential functions and displayed as described in METHODS. The solid curves represent the overall fits, and the dashed curves represent the individual exponential components. (B) The time constants and relative amplitudes for the brief and long components of the fits to closed time distributions are plotted as functions of voltage.

The gating behavior of KAT1 single channel currents can be analyzed in greater detail using burst analysis (Colquhoun and Hawkes, 1995). As shown in the representative traces of Fig. 1 A, bursts appear to become longer, the number of bursts per sweep appears to increase, and the time between each burst appears to decrease with increasing hyperpolarization. These qualitative observations will be confirmed by quantitative burst analysis. In burst analysis, observed closed events are assigned to one of two kinetically distinguishable closed states using a burst criterion. Closed events longer than the burst criterion are considered closures into an interburst closed state, while closed events shorter than the burst criterion are designated as closures into a within-burst closed state. A burst is then defined as any series of open and closed events for which the closed events are all shorter than the burst criterion. However, it must be noted that any method used to classify closed events as either within-burst or interburst closed events will necessarily misclassify a small fraction of long within-burst closed events as interburst events and a small fraction of short interburst closed events as within-burst events.

The selection of an appropriate burst criterion will minimize false classification of closed events into the inappropriate closed-time distribution (Jackson et al., 1983), and several methods have been described to determine an appropriate burst criterion for a given data set (Magleby and Pallotta, 1983). The method used in this article is to examine the closed-time distributions and select a burst criterion that produces reasonable exponential distributions for both the within-burst closed times and the interburst closed times. A burst criterion of 20 ms was chosen for KAT1, approximately four to five times faster than the brief component time constant of the closed-time distributions. Variation of the burst criterion in this range will not alter the final conclusions derived from the burst analysis, but will alter the specific values of rate constants in the kinetic model by only a modest amount (~20-25% with variation of the burst criteria from 10 to 50 ms).

In Fig. 7 A, the distribution of interburst closed times at voltages ranging from 110 to 190 mV are shown. An important limitation of burst analysis is that the distribution of interburst closed times is a valid measure of transitions outside the burst event only if true single channel patches are used, as is the case in Fig. 7. A second limitation to the utility of the interburst closed-time data is that the time constants measured are sensitive to the pulse duration used, as some interburst durations span the pulse duration after the initial burst event and are cut short (the right censor time). This right censor time error was corrected in the exponential fits to the duration distributions (Lawless, 1982; Hoshi and Aldrich, 1988). Fig. 7 B plots the time constants from fits to interburst closed-duration distributions as a function of voltage. The interburst closed durations become shorter with increasing hyperpolarization, implying that the channel will spend less time outside of the bursting states with increasing hyperpolarization. The voltage dependence of interburst closed times suggests that the interburst closed-to-open-state transition will proceed in the same direction as overall channel activation, as this transition becomes faster with hyperpolarization. This is most parsimoniously accomplished with a model that places the interburst closed state along the activation pathway, leading directly to the open state. Since it was argued earlier that at least one of the burst closed states is likely to lie outside of the activation pathway, the within-burst closed state will therefore reside outside of the activation pathway in the final kinetic model.

View larger version (24K): [in this window] [in a new window]   Fig. 7.   (A) Frequency histograms of interburst closed durations measured at several voltages from KAT1 channels. The data have been fitted with single exponential functions, represented by the solid lines. (B) The interburst duration distribution time constants derived from the single exponential fits are plotted as a function of voltage.

Fig. 8 A shows the distribution of within-burst closed times at voltages ranging from 110 to 190 mV. Each distribution has been fitted by a single exponential function, and the time constants from these fits are plotted as a function of voltage in Fig. 8 B. There is very little voltage dependence to the within-burst closed times, and because this transition lacks significant voltage dependence, conclusions about the location of the within-burst closed state along the activation pathway cannot be inferred. However, as stated earlier, the within-burst closed state is likely to reside after the open state in the final kinetic model.

View larger version (24K): [in this window] [in a new window]   Fig. 8.   (A) Frequency histograms of within-burst closed durations measured at several voltages from KAT1 channels. A burst criterion of 20 ms was used. The solid lines represent single exponential fits to the data. (B) Time constants derived from the exponential fits to the within-burst closed time duration histograms are plotted as a function of voltage.

A Kinetic Model of KAT1 Gating: Burst Analysis

A kinetic scheme that incorporates bursting behavior will contain at least three kinetically distinguishable states: two closed (the interburst and within-burst closed states) and one open state. With two closed states and one open state, there are exactly three possible configurations for arranging these states with respect to each other. The first is a triangular arrangement, where each of the three states can be entered from any of the two other states directly. This configuration was not entertained because (a) there is no experimental evidence in the voltage dependence of the duration distributions that favors a triangular configuration over a linear configuration, and (b) such a configuration introduces an additional free parameter into the model. The second configuration is a "CCO" model, as shown in Scheme III, where the two closed states are adjacent and only the shorter-lived closed state is in direct communication with the open state (the bursting states are outlined by the box).

View larger version (8K): [in this window] [in a new window]   Scheme III.

The third configuration is a "COC" model, as shown in Scheme IV, where the open state resides in between the two closed states, allowing direct communication between the open state and both closed states.

View larger version (8K): [in this window] [in a new window]   Scheme IV.

Note that the numbering system used for the states and transitions in Schemes III and IV are labeled to be consistent with the final kinetic scheme determined later. At any given voltage, it is not possible to discriminate experimentally the CCO and COC models, as both can be fitted equally well to any single channel bursting data set. However, several factors favor the COC model for KAT1 channels. First, given a single open state, as proposed for KAT1, the distribution of open times will be a single exponential with a mean duration of 1/(the sum of all rates leaving the open state). Given the observation that (a) KAT1 activation kinetics become faster with increasing hyperpolarization, and (b) the sum of leaving rates from the open states also becomes faster with increasing hyperpolarization (Fig. 5 B), at least one of the transitions from the open state will likely lie outside the activation pathway, as suggested earlier. Therefore, the minimal kinetic scheme that accounts for both the voltage dependences of these transitions as well as the channel's bursting behavior is most likely the COC scheme. A counter-argument might be that the data can be explained by a CCO or COC bursting model where both burst closed states reside outside the activation pathway, as shown in Scheme V, A and B, respectively.

View larger version (11K): [in this window] [in a new window]   Scheme V.

These models are less favorable because they require the introduction of additional states and would also require the voltage dependence of rate constants leaving the interburst closed state to be opposite the voltage dependence of overall activation. Kinetic models as depicted in Scheme V will have more than the minimal number of free parameters necessary to describe the behavior of KAT1 channels. Therefore, a within-burst closed state residing after the open state is the simplest kinetic model to satisfy a COC configuration.

If the interburst closed state were indeed to lie within the activation pathway, then the distribution of interburst closed times must be shorter than the distribution of first latencies at any given voltage, as sojourns into the interburst closed state will include a subset of all possible states along the activation pathway. In Fig. 9 A, the interburst durations are displayed as tail distributions and superimposed with the first latency tail distributions at 140, 160, and 190 mV. For a population of sweeps at a given voltage, the interburst durations are indeed shorter than the first latencies, consistent with a model that has the interburst closed state located within the activation pathway. The consistently longer distributions for first latencies compared with the interburst closed times also implies that there are multiple closed states along the activation pathway. Moreover, if the interburst closed state were to reside within the activation pathway, then the overall voltage dependence of the interburst closed durations must always be less than or equal to that of the voltage dependence of first latencies. Fig. 9 B superimposes the mean interburst closed durations with the mean first latencies as functions of voltage, confirming this prediction.

View larger version (14K): [in this window] [in a new window]   Fig. 9.   (A) Comparison of interburst closed durations (thin lines) and first latency (thick lines) distributions plotted as tail distributions at 140, 160, and 190 mV. The distributions represent the probabilities that the interburst and first latency durations are longer than the times indicated on the axis. (B) Mean interburst closed durations and mean first latencies are plotted together as functions of voltage.

Therefore, the KAT1 kinetic scheme will at this point have the following features: (a) bursting behavior that is described by a COC model, (b) a within-burst closed state that lies outside the activation pathway and whose entering rate from the open state becomes faster with increasing hyperpolarization, (c) an interburst state that lies within the activation pathway with a transition into the open state that becomes faster with hyperpolarization, (d) a relatively voltage-independent transition from the within-burst closed state to the open state, and (e) a transition from the open state to the interburst closed state that becomes slower with hyperpolarization. Scheme VI satisfies these criteria.

View larger version (7K): [in this window] [in a new window]   Scheme VI.

Using the COC model (Scheme IV) and the general Markov property that the mean dwell time in any particular kinetic state is equal to the inverse of the sum of all rate constants leaving that state, we can now use the single channel data to assign specific rate constants to transitions in this bursting scheme. Using Scheme IV, the mean open time (Fig. 5) can be described by a single exponential function with a time constant equal to the inverse of the sum of the rates leaving the open state:

(3)

The within-burst closed-time distribution is described by a single exponential function with a time constant equal to:

(4)

The interburst closed-time distribution can be described by an exponential function with a time constant as follows, assuming that any other transitions out of this state are negligible:

(5)

Note that Eq. 5 is an approximation if Scheme VI is entertained, as it assumes that all long closed events after first opening only involve transitions into the interburst state without additional transitions into adjacent closed states along the activation pathway. This is a reasonable assumption at the hyperpolarized potentials studied, where the interburst closed-time distributions are well fitted with single exponentials and the backward rates away from the bursting states are expected to be relatively slow.

Furthermore, according to Scheme IV, an open channel will continue to burst with a probability of

and it will terminate a burst with a probability of

The number of openings per burst (n) will be described by a geometric distribution:

(6)

where

is the probability of terminating a burst. The distribution described by Eq. 6 gives a mean number of openings per burst equal to

In Fig. 10 A, the distributions of the number of openings per burst are shown for voltages ranging from 110 to 190 mV. The distributions of the number of openings per burst are fitted with geometric distributions, and the probability of terminating a burst (q) derived from these fits are plotted as a function of voltage in Fig. 10 B. Eqs. 3-6, along with measurements of open time, closed time, and burst durations over a wide voltage range can then be used to determine every rate constant of Scheme IV by solving for each individual rate constant as a function of the various measured dwell time distributions. In Table I, the equations for determining each rate constant in the COC bursting scheme are shown.

View larger version (26K): [in this window] [in a new window]   Fig. 10.   (A) Frequency histograms of the number of openings per burst at several voltages, using a burst criterion of 20 ms. The data are fitted by a geometric distribution (Eq. 6), represented by the dots. The probability of terminating a burst (q) and the total number of openings are shown for each voltage. (B) The probability of terminating a burst (q), derived from the geometric fits, is plotted as a function of voltage.

The interburst closed-duration distributions were not as well determined as the other distributions because of the relatively small number of events in each distribution. Therefore, the associated rate constant (k₂₃) was allowed to relax in fitting the ensemble averages and cumulative first latency distributions (see below). However, the final values determined for this rate constant are comparable with the original values determined by burst analysis. The calculated burst rate constants are plotted as functions of voltage in Fig. 11 A. As expected, the forward rate constants k₂₃ and k₃₄ become faster with increasing hyperpolarization, while the rate constant leaving the burst (k₃₂) becomes slower with increasing hyperpolarization. The rate constant leaving the within-burst closed state (k₄₃) is essentially voltage independent. Interestingly, the two individual rate constants leaving the open state demonstrate relatively strong voltage dependence, but because their voltage dependences are in opposite directions, the open time distributions are much less voltage dependent. Moreover, all four rate constants appear to follow a single exponential voltage dependence.

View larger version (23K): [in this window] [in a new window]   Fig. 11.   (A) The rate constants k23, k34, k32, and k43 from Scheme IV, calculated using Eqs. 3-6, are plotted as functions of voltage. The voltage dependences of the four rate constants are plotted on logarithmic scales. (B) Comparison of the mean burst durations measured directly using a burst criterion of 20 ms () and calculated using Eq. 7 and the rate constants derived from burst analysis (). The COC bursting scheme is shown at right for reference.

As stated earlier, the probability of terminating a burst (q) decreases significantly with more negative voltages, suggesting that the channel is more apt to continue bursting at hyperpolarized potentials. Moreover, this voltage dependence of q is consistent with the interburst closed state residing within the activation pathway and the within-burst closed state residing outside of the activation pathway. These data also imply that the increased steady state open probability with more negative voltages is in large part a result of longer bursting periods. This can be confirmed by examining the distribution of burst durations. The mean burst duration can be measured directly and calculated from the rate constants determined in Scheme IV, as follows. The product of the mean number of openings per burst and the mean open time will give the total time a channel spends in the open state per burst:

Likewise, the total time spent per burst in the within-burst closed state is the product of the mean within-burst closed time and the mean number of openings per burst, less one:

The mean burst duration is then the sum of the mean total open and closed times per burst:

(7)

Fig. 11 B superimposes the measured mean burst durations and the mean burst durations calculated by applying the rate constants in Fig. 11 A to Eq. 7 as functions of voltage, which serves as an internal check for correctness of fit and theory. Fig. 11 B demonstrates close agreement between the measured and calculated burst durations, confirming that Scheme IV describes the bursting behavior of KAT1 reasonably well.

A Kinetic Model of KAT1 Gating

The nucleus of a kinetic model for KAT1 gating can be described by Scheme IV, with rate constants derived from burst analysis (Fig. 11 A). The next step in creating a kinetic model is to account for the behavior of the KAT1 channel before first opening. This model will have the following properties: (a) a COC bursting scheme as described in the previous section, (b) any additional closed states in the activation pathway will be located to the left of the interburst closed state, as shown earlier in Scheme VI, and (c) the voltage dependence of each rate constant is assumed to be in the same direction as the overall voltage dependence of the KAT1 channel, where increasing hyperpolarization produces faster rates in the general direction of activation (left-to-right in Scheme VI) and slower rates in the reverse direction (right-to-left in Scheme VI).

The number of closed states in the activation pathway and the values and voltage dependences of their rate constants must account for the first latency and ensemble average data. Macroscopic currents, ensemble averages, and first latency distributions all display a small but measurable sigmoidicity in the activation time course that must be incorporated in the model. Fits of the data to several candidate models suggest that a minimum of three closed states, including the interburst closed state, are located along the activation pathway. Furthermore, the transition from the interburst closed state to the open state is not a free parameter at this point, as its values in the voltage range of KAT1 activation have been determined by the burst data. Fewer than three closed states in the activation pathway failed to produce adequate fits to the ensemble average and first latency data (data not shown). The minimal configuration necessary to account for these data is shown in Scheme VII.

View larger version (7K): [in this window] [in a new window]   Scheme VII.

Inactivated States

As discussed previously, the first latency distributions (Fig. 3 A) do not always reach a steady state probability of one, reflecting either an inactivated state entered directly from one of the closed states, right-censoring of the data, or incomplete recovery from closed or inactivated state(s) entered after the channel opens. To produce good qualitative fits to the steady state open probabilities derived from ensemble averages and the distributions of first latencies in the positive voltage range, a transition to an inactivated state directly entered from state C₀ was introduced. The contribution of blank sweeps is significant only at depolarized potentials. The minimal contribution of the inactivated state was determined to be the contribution required to account for steady state cumulative first latencies less than one in the fits not accounted for by first latencies longer than the right censor time. This small correction can then be incorporated into the model as an inactivated or accessory closed state that is entered directly from the closed state(s) along the activation pathway. In our analysis, only the minimal contribution of the inactivated state has been determined.

Although inactivation could, in principle, occur from a number of closed states, state C₀ in Scheme VII was chosen as the state connected directly with the inactivated state for several reasons. First, among the closed states along the activation pathway, locating the inactivated state at state C₀ is least likely to affect the burst kinetics previously described, which do not suggest the existence of an inactivated state near the burst states. Second, it is unlikely that the inactivated state can be entered either from the open state (O₃) or the closed state located outside the activation pathway (C₄). Given the fraction of blank sweeps observed at depolarized voltages and given the fit of Scheme VII to the activation kinetics and burst data, the rate constant required for entering an inactivated state located in direct connection with either states O₃ or C₄ would result in an observable time-dependent inactivation process in the ensemble averages. Furthermore, this time-dependent inactivation process would only be apparent at depolarized potentials where the entry rate into the inactivated state is significant. A time-dependent inactivation process is not observed in KAT1 ensemble averages (Fig. 2 A), which makes such a kinetic scheme unlikely; however, a model in which the inactivated state can be accessed from more than one state along the activation pathway cannot be excluded.

The Final Kinetic Model

Given the rate constants derived for Scheme VII, the observation of blank sweeps in KAT1 currents at depolarized voltages is best accounted for by the combined contribution of first latencies longer than the pulse potential and by an inactivated state that can be directly entered by