Abstract
The conformational changes associated with activation gating in Shaker potassium channels are functionally characterized in patchclamp recordings made from Xenopus laevis oocytes expressing Shaker channels with fast inactivation removed. Estimates of the forward and backward rates for transitions are obtained by fitting exponentials to macroscopic ionic and gating current relaxations at voltage extremes, where we assume that transitions are unidirectional. The assignment of different rates is facilitated by using voltage protocols that incorporate prepulses to preload channels into different distributions of states, yielding test currents that reflect different subsets of transitions. These data yield direct estimates of the rate constants and partial charges associated with three forward and three backward transitions, as well as estimates of the partial charges associated with other transitions. The partial charges correspond to an average charge movement of 0.5 e_{0} during each transition in the activation process. This value implies that activation gating involves a large number of transitions to account for the total gating charge displacement of 13 e_{0}. The characterization of the gating transitions here forms the basis for constraining a detailed gating model to be described in a subsequent paper of this series.
Introduction
Voltagegated sodium, calcium, and potassium channels underlie the electrical properties of excitable cells. Insights into the structural changes involved in the voltagedependent opening of these channels first came from functional studies performed in the squid axon (Hodgkin and Huxley, 1952a, 1952b). The steep voltage sensitivity of the observed potassium and sodium conductances suggested that the channel opening process involves the displacement of a large amount of charge across the cell membrane. The long delay in the time courses of activation implied that channel opening involves multiple kinetic steps. The structural origins of these functional properties are now becoming clear. The alpha subunits of voltagegated sodium and calcium channels are each encoded by a single long transcript with four homologous regions (Noda et al., 1984; Tanabe et al., 1987), while voltagegated potassium channels are encoded by a transcript that is approximately onefourth as long (Tempel et al., 1987; Kamb et al., 1988), with a channel complex formed by four subunits (MacKinnon, 1991; Kavanaugh et al., 1992; Li et al., 1994). A requirement for separate activation of each of these units helps to explain the delay in channel activation. Each of the protomers or subunits of these channels contains a putative transmembrane segment S4 that has a number of conserved basic residues and is a candidate for the charged region of the channel that moves in response to voltage, leading to channel activation (Guy and Seetharamulu, 1986; Noda et al., 1986). Mutations in the S4 region yield large effects on the voltage dependence of activation (Stühmer et al., 1989; Liman et al., 1991; Lopez et al., 1991; McCormack et al., 1991; Papazian et al., 1991; Logothetis et al., 1992; summarized in Sigworth, 1994). Also, it has been shown recently that membrane potential changes cause changes in accessibility of S4 residues (Yang and Horn, 1995; Yang et al., 1996; Mannuzzu et al., 1996; Larsson et al., 1996), and that S4 charge changes cause changes in total gating charge movement (Aggarwal and MacKinnon, 1996; Seoh et al., 1996).
Insights into the mechanics of activation gating have also come from extensions of the functional studies that were first performed by Hodgkin and Huxley (1952a, 1952b). In voltageclamp measurements of macroscopic and singlechannel ionic currents, various voltage protocols have been used to emphasize particular steps in the activation process (Cole and Moore, 1960). The voltagedependent transitions among closed states have also been characterized from the time courses of gating currents, which are the direct electrical manifestation of the charge displacements associated with conformational changes (Armstrong and Bezanilla, 1973; Schneider and Chandler, 1973; Keynes and Rojas, 1974). Additionally, the fluctuations in the gating currents can provide information about the size of the charge displacements in single gating transitions (Conti and Stühmer, 1989; Crouzy and Sigworth, 1993; Sigg et al., 1994b). Ultimately, the understanding of voltage gating in ion channels will require combining this detailed functional information with experiments that give more direct structural information.
Examples of the value of detailed functional studies are seen in the recent work on the Shaker potassium channel (Bezanilla et al., 1991; Stühmer et al., 1991; Schoppa et al., 1992; Bezanilla et al., 1994; Hoshi et al., 1994; McCormack et al., 1994; Perozo et al., 1994; Stefani et al., 1994; Zagotta et al., 1994a, 1994b). Shaker channels have been a favorite in the study of activation gating for a variety of reasons. They can be made noninactivating through a NH_{2}terminal truncation (Hoshi et al., 1990) and they express well in Xenopus oocytes, allowing the measurement of gating currents as well as ionic currents. Also, because they are tetramers, the presumed fourfold functional symmetry of Shaker channels can be exploited in developing simpler kinetic models. The major results from the recent studies on these channels may be summarized as follows.
(a) The total gating charge per channel is ∼13 e_{0}. This value was obtained from calibrated measurements of gating currents (Schoppa et al., 1992; Aggarwal and MacKinnon, 1996; Seoh et al., 1996), and corroborated by measurements of limiting voltage sensitivity (Zagotta et al., 1994a; Seoh et al., 1996).
(b) Between the resting and the open state, the channel undergoes a minimum of five kinetic transitions, as estimated from the time course of channel opening in response to a voltage step (Zagotta et al., 1994a).
(c) The time course of the “on” gating current induced by a step depolarization has a rising or plateau phase (Bezanilla et al., 1994), implying that the first kinetic steps in channel activation are slower or less voltage dependent than subsequent steps.
(d) After large depolarizations, the time course of the “off” gating current induced by a voltage step back to the holding potential has a rising phase (Bezanilla et al., 1991) and also decay kinetics that match the time course of channel deactivation (Zagotta et al., 1994a). Thus, the first kinetic steps in channel deactivation are slower than subsequent steps.
(e) At intermediate voltages, the gating currents display a fast component that is followed by a slow exponential component that is correlated with channel opening (Bezanilla et al., 1994). Also, at these voltages, the ionic currents have a relatively short delay, followed by a very slow rise to the peak current (Zagotta et al., 1994a). These phenomena imply that channel opening at these voltages is much slower than the rate that the channel traverses through early closed states.
(f) Shaker's voltage dependence of charge movement (QV) relation is shallow at hyperpolarized voltages but is steeper over the voltage range where channels open (Stefani et al., 1994; Bezanilla et al., 1994).
(g) Components of charge in the QV relation have been shown to be differentially affected by some mutations in the S4 region (Schoppa et al., 1992; Perozo et al., 1994) and also by drug binding (McCormack et al., 1994). The differential effects suggest the existence of different types of voltagedependent conformational changes.
(h) Measurements of gating current fluctuations suggest that elementary charge movements are roughly 2 e_{0} in size (Sigg et al., 1994b; Sigworth, 1994).
(i) The distribution of channel open times is well described by a single exponential function (Hoshi et al., 1994), which is consistent with Shaker channels having a single open state. The single channel data also suggest the presence of closed states that are not in the main activation path (Hoshi et al., 1994).
Several kinetic models have been proposed that take into account subsets of these functional properties for Shaker channels (Schoppa et al., 1992; Tytgat and Hess, 1992; Bezanilla et al., 1994; McCormack et al., 1994; Zagotta et al., 1994b). Interestingly, however, these models have little in common with each other, with different models explaining the same functional data through very different mechanisms. For example, the steep voltage dependence of charge movement and channel opening is explained in the model proposed by Bezanilla et al. (1994) by a transition with a large valence, while the model of Zagotta et al. (1994b) achieves this through smaller charge movements but with a slow channel closing rate. The discrepancies between the models point to the need of further analysis of the activation properties of Shaker channels.
In this paper and the two papers that follow (Schoppa and Sigworth, 1998a, 1998b), we present further functional studies on activation gating in Shaker potassium channels. Our general strategy is to perform a systematic study of the different gating steps in Shaker's activation process. This first paper will focus on measurements of macroscopic ionic and gating currents measured at extreme depolarizations and hyperpolarizations, which yield estimates of forward and backward rates. Some of the described experiments are similar to those that have been reported previously, but new insights into the activation gating process are gained by analyzing the data in new ways, and also by extending the voltage range of the current measurements. The channel studied, which has its NH_{2} terminus truncated to remove fast inactivation, will be referred to as wild type (WT)^{1} to distinguish it from a mutant channel (V2) that will be the focus of the second paper. The third paper will use results from both WT and V2 channels to develop a new kinetic model for Shaker channel activation.
Methods
Channel Expression
The construction of noninactivating WT Shaker 294 cDNAs and in vitro synthesis of cRNA have been described previously (Iverson et al., 1990; Schoppa et al., 1992). Channels were expressed in Xenopus laevis oocytes injected with ∼3 ng cRNA for recordings of macroscopic ionic and gating currents, and 50–100fold less cRNA for single channel recordings. Current measurements were made 4–21 d after injection with the patchclamp technique (Hamill et al., 1981) using an EPC9 patch clamp amplifier (HEKA Electronic, Lambrecht, Germany).
Measurements of Macroscopic Ionic and Gating Currents
Macroscopic ionic and gating currents were recorded in insideout membrane patches using conventional oocyte macropatch techniques (Stühmer et al., 1991). Patch recordings used pipettes pulled from Kimax capillary tubes with tip diameters ranging from 3 to 10 μm (0.5–3.0 MΩ resistance). Voltage pulses were applied from a holding potential of −93 mV. Current signals were filtered (unless otherwise indicated) at 10 and 5 kHz (Bessel characteristic) for recordings of ionic and gating currents, respectively. Data were sampled at five to seven times the filtering frequency. For subtraction of linear leak and capacitive currents in recordings of macroscopic ionic currents, alternating positive and negative pulses of 20mV amplitude from a −133mV leak holding potential were applied, and the resulting current was scaled appropriately. For gating currents, a scaled current response induced by only a negative going voltage pulse from −133 to −153 was subtracted. This modification reduced the artifact in leak subtraction from charge movement at voltages positive to −133 mV (Stühmer et al., 1991). To increase the signaltonoise ratio, 10–100 sweeps were averaged before the data were stored. The pulse frequency was set to be high (1–5 Hz) to allow rapid measurement of many sweeps, but we determined that the frequencies used did not induce rundown of the ionic or gating currents, arising from slow inactivation. Displayed gating current traces were sometimes additionally filtered with a Gaussian digital filter to 2.5–3.5 kHz.
Most of the ionic current measurements were made with pipettes filled with 140 mM Nmethyldglucamine (NMDG) aspartate (Asp), 1.8 mM CaCl_{2}, and 10 mM HEPES, and the bath solution contained 139 mM KAsp, 1 mM KCl, 1 mM EGTA, and 10 mM HEPES. Most of the gating current measurements were made with the same solutions, except with 2 μM charybdotoxin (CTx) added to the pipette solution to block ionic currents (Lucchesi et al., 1989). CTx did not appear to alter the properties of the charge movement, as the currents recorded in the presence of CTx were similar to those recorded in the absence of CTx but with NMDG^{+} replacing K^{+} in the bath to remove the ionic current. Membrane potential values were corrected for a liquid junction potential at the interface of the NMDG^{+} pipette and K^{+} bath solutions, which we estimated to be −13 mV (Neher, 1992). All experiments were done at room temperature. The bath chamber was not perfused in these experiments.
Voltage steps to very positive and negative voltages were kept short (≤5 ms at V > +100 mV and V < −120 mV) to reduce contamination of the recorded ionic currents by endogenous oocyte currents. Corresponding gating current recordings made in the presence of CTx showed little background current, suggesting that contamination of our ionic currents by endogenous currents is likely to be negligible.
The large pipette sizes used in recordings of gating currents encouraged the formation of membrane vesicles or partial vesicles when the patch was pulled off the oocyte. Gating current recordings were rejected if the measured on current did not show an instantaneous component of the rising phase: specifically, a component with amplitude at least 50% of the peak on current was expected to rise with the same time course as the measured step response of the recording system (see below). Further, recordings with very large ionic currents (>2.5 nA) were rejected to avoid errors due to series resistance.
To allow the interpretation of the rapid gating events reflected in the recordings of ionic tail currents and reactivation time courses, the step response of the recording system, including the filter, was determined by injecting a step of current into the patch clamp head stage using the test facility of the EPC9 and measuring the current response. At the low gains at which the macroscopic ionic and gating currents were measured (500 MΩ feedback resistor), the response time, defined as the time required for the current to reach 50% of its peak in response to a step current input, was found to be 40 and 25 μs at the 10 and 15 kHz filtering bandwidths used for most ionic current measurements, respectively. Other delays in the stimulus and recording system were expected to be negligible: the stimulus filter risetime was set at 2 μs for measurements of tail currents and reactivation time courses, while the patch membrane charging time constant was expected to be below 1 μs. To account for the total delays, the recorded current traces were offset in time by three sample intervals. In the measured time course of the step response of the recording system, it was also determined that an additional approximately three sample intervals were required for the step response to settle to near its final level. Thus, an additional three sample intervals were always ignored in the fitting of exponentials to the tail current and reactivation time courses.
The time courses of the macroscopic ionic and gating currents were fitted to the sums of exponentials by least squares within the Igor data analysis program (WaveMetrics, Lake Oswego, OR). The derived parameter estimates were consistently independent of the initial guesses supplied.
In the text, errors in all measured quantities are given as the mean ± SEM.
Measurement of SingleChannel Ionic Currents
Singlechannel recordings were made in insideout patches in response to step depolarizations from a −93mV holding potential. Patch pipettes were pulled from 7052 glass (Garner Glass, Claremont, CA) with 1–2μm tip diameters (4–10 MΩ resistance). The recording solutions were identical to those used in the measurements of macroscopic ionic currents. Filtering and sampling frequencies were variable, appropriate for the amplitude of the singlechannel activity at a given voltage. Leak subtraction was performed by subtracting an average of 8–20 of the nearest null traces. To allow the measurement of a large number of single channel sweeps, the pulse frequency was set to be high (1–5 Hz); however, there was no indication of slow inactivation, which would have caused a timedependent increase in the number of null traces. The displayed single channel data are filtered at the frequencies used for the event detection, as described below.
The singlechannel activity apparently arose from Shaker channels since (a) no such single channel activity was observed in patches in which CTx was included in the pipette, and (b) the ensemble averages of WT's singlechannel current traces were kinetically identical to the macroscopic currents in patches from oocytes injected with 50–100fold more cRNA. Infrequently, a patch would display other singlechannel activity, but this activity was easily distinguishable from Shaker's in its voltagedependence, kinetics, and conductance properties.
The analysis of the equilibrium singlechannel closed and open times was performed with the TAC singlechannel analysis program, which is based on THAC (Sigworth, 1983). Data were filtered with a digital Gaussian filter to achieve an appropriate signaltonoise ratio, and event detection was performed using the standard halfamplitude threshold analysis (Colquhoun and Sigworth, 1995). Complicating the analysis of the singlechannel events was the presence of apparent subconductance activity (Hoshi et al., 1994). The event detection for a given trace was stopped at the time point at which the channel first exhibited such behavior. Ignoring these data was unlikely to introduce a significant error in our results since subconductance activity was relatively infrequent: in one patch recording of 613 consecutive traces of single channel activity at +27 mV, the open channel spent only 14% of the total 16.6 s of recorded open time at a subconductance level. Here a subconductance level was defined to be an amplitude level smaller than 75% of the most common amplitude level.
For the construction of closed and opentime histograms, the deadtime (T_{d}) for a given analysis filtering frequency f_{c} was taken to be T_{d} = 0.179/f_{c} and shortevent durations were adjusted appropriately for Gaussian filtering as described (Colquhoun and Sigworth, 1995). Closed and open times were binned logarithmically and the square root of the number of events was plotted (Sigworth and Sine, 1987). The minimumduration bin for each histogram was set to be the deadtime corresponding to the analysis filtering frequency used for the event detection. The event histograms were fitted to a mixture of exponential components with the amplitudes and time constants adjusted using the binned maximum likelihood method (Sigworth and Sine, 1987). The number of exponential components fitted to the histograms was determined by the likelihood ratio test for nested models (Horn and Lange, 1983), which can be applied to the problem of comparing fits with different numbers of exponentials (McManus and Magleby, 1988).
Accounting for Three Artifacts
In the following section, we describe how we accounted for three additional potential artifacts that could affect our interpretation of the data.
Effects of different recording solutions.
While most of the recordings were made with the bath and pipette solutions indicated above, different solutions were sometimes used to facilitate certain types of the measurements. For these instances, it was important to show that changing the recording solution had no effect on activation gating.
First, to allow the measurement of large inward ionic tail currents at very hyperpolarized voltages (see Fig. 7 A), measurements were routinely performed while replacing 14.3 mM of the NMDG^{+} in the pipette with an equimolar amount of K^{+}. High concentrations of external K^{+} have been shown to alter the ionic tail currents for Shaker channels (Stefani et al., 1994), and, indeed, we found that 140 mM K^{+} in the pipette slowed the decay of the ionic tail currents by a factor of two and caused a −15mV shift in the voltage dependence of P_{o}. However, tailcurrent recordings made at −93 mV in four patches with 14.3 mM external K^{+} had decay time constants identical to those measured in five patches with no K^{+} added to the pipette solution. In our analysis, we therefore assume that external K^{+} up to 14.3 mM has no effect on channel gating.
Secondly, some of the gating current measurements that we report were made with Cs^{+} replacing K^{+} in the bath. Cesium was originally used in many of our recordings since it is much less permeant in Shaker channels (Heginbotham and MacKinnon, 1993). However, Cs^{+} was observed to cause small changes in the properties of the gating currents at intermediate voltages (near −50 mV): Cs^{+} shifts, by −15 mV, the voltage range where a slow exponential component appears in the on gating current, and also shifts, by −15 mV, the prepulse voltages after which off gating currents begin to decay slowly. Since these phenomena are associated with channel opening (Bezanilla et al., 1991, 1994; Zagotta et al., 1994a), we take these changes to mean that Cs^{+} alters some of Shaker's late gating transitions. Cs^{+}, however, apparently does not alter the early gating transitions. Parallel measurements of gating currents made with bath K^{+} and Cs^{+} showed that gating currents induced by voltage pulses between −73 and −113 mV are unaffected by bath Cs^{+}. Our analysis of gating currents at V ≤ −93 mV thus includes data obtained in Cs^{+} as well as K^{+} (Fig. 6 A). It should be noted that the liquid junction potential in the Cs^{+}/NMDG^{+} solutions (−12 mV) was essentially the same as that for K^{+}/NMDG^{+} solutions.
Nonstationarities in channel behavior.
Sigg et al. (1994a) have reported that the decay of Shaker's ionic tail currents recorded in excised membrane patches slows by a factor of ∼3 during the first several minutes after patch excision. We observed a similar timedependent slowing in WT's tail currents. In eight patches, tail currents at −93 mV recorded at least 6 min after patch excision had decay time constants that were 2.9 ± 1.1fold longer than tail currents recorded less than 1 min after patch excision. Timedependent changes were also observed in the decay of WT's off gating currents after large depolarizations, which also reflect the kinetics of channel deactivation (Bezanilla et al., 1994; Zagotta et al., 1994a). For the analysis of the macroscopic ionic tail currents and off gating currents, we therefore limited the analysis of tail currents to those measured at least 6 min after patch excision, when most of the timedependent effects were apparently over. In seven patch recordings, in which tail currents were measured at different times up to 11 min or more after patch excision, the timedependent slowing effect occurred with a time course that was approximated by a single exponential with τ = 6.2 ± 1.5 min.
No timedependent effects were observed for any other macroscopic current properties. In three patches in which large changes in tail current decay rates occurred, the equilibrium voltage dependence of open probability never shifted by >3 mV and no significant changes in the measured activation time constants (at V ≥ −53 mV) were observed. We observed no nonstationary behavior in WT's single channel activity.
Slow inactivation.
A final potential problem for the interpretation of the macroscopic current time courses was slowinactivation gating (Hoshi et al., 1991; LopezBarneo et al., 1993). In recordings made with 4–8s voltage pulses to between −53 and +67 mV, macroscopic ionic currents decayed with voltageindependent kinetics that were fitted by the sum of two exponentials with time constants (at −13 mV) of 70 ± 13 and 870 ± 76 ms (n = 7); the 70ms component comprised 26 ± 19% of the total amplitude. The slow inactivation time course was much slower than most, but not all, phenomena associated with activation gating. Thus, in the fits of exponentials to the channel opening time courses that are reported, an additional exponential component was always added that reflected the 70ms component of the slow inactivation process.
Results
The goal of this study is to identify kinetic transitions in activation gating and assign rate constants for the WT channel. Experiments were designed to isolate, as much as possible, the rates of individual steps in the activation process. In the first studies that we describe, we make use of data obtained at voltage extremes to study kinetic constants. We consider forward rates, assigning values and voltage dependences to the first forward rate α_{1}, the limiting rate at large positive potentials α_{p}, and the final opening rate α_{N}. An estimate for the voltage dependence q_{αd} of intermediate steps is also obtained. Similarly, the first and the last two backward rates β_{1}, β_{N}, and β_{N1} are determined, along with an estimate of the “average” rate of intermediate steps β_{d}. In the second group of studies, we characterize the transitions to several channelclosed states that are distinct from those traversed in the depolarizationinduced activation process.
General Framework
We assume a discrete, homogenous Markov model for activation gating. Thus, activation is taken to involve transitions between discrete closed and open states separated by large energy barriers. For a sequential gating scheme, this framework can be depicted as (Scheme I)
The forward and backward rates α_{i} and β_{i} are taken to be exponential functions of the membrane potential V, scaled by the partial charges q_{αi} and q_{βi,} 1
(Terms with higher powers of V in the exponent are not included because any charge movement with the expected properties, if present, is very small; see Sigworth, 1994.) The gating charge movement accompanying a transition from state i − 1 to state i is then given from the partial charges as 2
Within this framework, we estimate the forward and backward rate constants α_{i }and β_{i} for various gating transitions. Many of the current measurements were made at voltages where we could presume that either forward or backward rates predominate. Following Zagotta et al. (1994a), we define three voltage ranges. Forward rates are presumed to predominate in the depolarized voltage range, where WT's equilibrium P_{o} and charge movement Q saturate. In WT's P_{o} − V and QV relations in Fig. 1, this appears to occur above −20 mV. (Although small changes in channel P_{o }continue at higher voltages, this property reflects the voltage dependence of a transition to a state that is not in the activation path and which carries only ∼0.3 e_{0} of charge; Zagotta et al., 1994a.) Backward rates are presumed to predominate at all voltages where most channels reside amongst the earliest closed states at equilibrium. These hyperpolarized voltages were taken to be V ≤ −90 mV because only 8% of WT's charge movement occurs negative to −90 mV. A third voltage range, activation voltages, between −90 and −20 mV, is the range in which WT's channel P_{o} and charge movement are undergoing most of their changes.
Given the special condition that all of the backward rates in Scheme I are negligible, information about the forward rate constants in any sequential scheme can be obtained from a simple analysis of the time course of the open probability. At depolarized voltages, where we assume β_{i} → 0 for all i, the mean latency to arrive at the open state O_{N} is given by 3
Now suppose that one of the rates, α_{j}, is much smaller than the others. Then the time course of channel activation can be approximated by the exponential function 4
with time constant τ = α_{j}^{−1} and with a time delay equal to the latency due to the other steps, 5
The approximate time course given by Eq. 4 has a mean latency that is equal to t_{l}; however, the time course is correct only as the limiting rate α_{j} becomes much smaller than the other rates. Useful estimates of the limiting rate and of δ can nevertheless be obtained from fits of Eq. 4 when the limiting rate differs little from the other rates. This is illustrated in Fig. 2, where fits were made to the “upper half” of the time course, starting at the time when P_{o} is equal to half of the final value. In Fig. 2 A, an “n^{4}” scheme where the next slowest rate is only twice that of the slowest one, the limiting rate is estimated with 11% error, while δ is estimated within 1%. Fig. 2 B demonstrates the worst case, in which no rate constant is smaller than the others. Here, the limiting rate is underestimated by a factor of two, while the error in δ is only 20%. Similar deviations between the measurements and theory are obtained from sequential schemes with more transitions. For an “n^{8}” scheme, the deviations in the measured and predicted τ^{−1} and δ are only 13 and 1%, respectively; for eight equivalent rates, the deviations in τ^{−1} and δ are 61 and 13%. These results suggest that fits of Eq. 4 to activation time courses for sequential models can yield reasonable firstpass estimates of the ratelimiting rate constant, though this rate will tend to be underestimated in cases where several rates are comparable in magnitude. Information about the other transitions is also obtained with surprisingly good accuracy from the delay value.
The approximation of Eq. 4 can also be used to obtain a simple characterization of dwell times during activation in branched models. Consider a model like that of Zagotta et al. (1994b) in which four independent subunits each undergo two steps, (Scheme II)
When this scheme is expanded, it is seen that there are many distinct paths (14
in all) leading from closed state C_{0} to the open
The mean latency t_{l} can be computed as a weighted average of the time spent in each path, with the weights being the probabilities of the paths. The time spent in each path is the sum of the dwell times in each state of the path.
Fig. 2 C shows an example of fits to the time course for this scheme in the case where a_{1} = 1 and a_{2} = 4, yielding t_{l} = 2.369 time units. The simple theory predicts τ to be equal to the reciprocal of the slowest rate, and the delay parameter to be given by δ = t_{l} − τ. A singleexponential fit (dotted curve) yields a value for τ^{−1} that differs by 11% from the slowest rate, and a value of δ that differs only 1% from the theoretical value. Fig. 2 D demonstrates the most difficult case, in which a_{1} = a_{2}. As in the corresponding case of a sequential scheme (Fig. 2 B), the error in the time constant is moderate, ∼30%, but the error in δ is small, <1%. Thus, the parameters of a fitted singleexponential function give surprisingly good estimates for the aggregate dwell times in branched models as well as in linear models.
Estimates of Forward Rates
Estimates of α_{1}.
The forward rate constant α_{1} for the first transition was evaluated from the time courses of ionic currents and gating currents at depolarizing voltages, as illustrated in Fig. 3. Fits of the singleexponential function (Eq. 4) to the ionic current from the 50% amplitude level to its final value yielded the “activation time constant” τ_{a} and the activation delay δ_{a} (Fig. 3 A). From the exponential decay of gating currents at depolarized voltages, the time constant τ_{on} was determined (Fig. 3 B).
It has been shown previously that Shaker's on gating current displays a rising phase (Bezanilla et al., 1994), which implies that the forward rate of the first transition is slower than subsequent transitions or that the first transition has less associated charge movement. The additional observation that τ_{a} and τ_{on }have similar values at voltages near 0 mV (Fig. 3 C) suggests that the first step is the slowest at these voltages, being rate limiting for both channel opening and the movement of the charge.
The relationship between τ_{a} and τ_{on }is shown for a fivestate sequential gating scheme in Fig. 3 D. In these simulations, we set the rates of all but one of the transitions to be equivalent and fast, and varied the position of the slow step. Only the case in which the slow step was the first step does the gating current decay with a single exponential with a time constant τ_{on} that is nearly identical to the activation time constant τ_{a}. We also show the relationship between τ_{a} and τ_{on }for the branched model in Fig. 3 E (bottom). For the case in which we made the rate a_{1} of the first “subunit transition” slow and rate limiting, the τ_{a} and τ_{on }values are nearly equivalent; however, for the case in which a_{2} is slow, the τ_{a} and τ_{on }values differ by a factor of 2. The results of these simulations suggest that it is reasonable to take the similarity between τ_{a} and τ_{on} for Shaker to mean that the first gating step is indeed the slowest. Activation at depolarized voltages then mirrors channel deactivation at hyperpolarized voltages, where the similar off gating current and tail current decay time courses imply that the first steps in channel closing are the slowest at these voltages (Bezanilla et al., 1991; Zagotta et al., 1994a).
An estimate of the voltage dependence of the first forward rate α_{1} can be obtained from the τ_{a} or the τ_{on }values at voltages up to +67 mV (Fig. 3 C). The reciprocal of the τ_{a} values derived from ionic currents measured between −13 and +67 mV in seven different patches yielded α_{1}(0) = 1,200 ± 90 s^{−1} and q_{α1} = 0.36 ± 0.02 e_{0}. As derived from τ_{a}, the value of α_{1} might be underestimated if, as in the case illustrated in Fig. 2 B, other transitions have very similar rates. However, the close correspondence of τ_{a} and τ_{on} argues against the presence of this error.
Estimate of α_{p}.
If one or more transitions have forward rates with smaller voltage dependences than α_{1}, one of these should be rate limiting at sufficiently high voltages, and should be reflected in the voltage dependence of τ_{a} at very large positive voltages. Thus, we obtained current recordings at voltages up to +147 mV (Fig. 4 A). The analysis of the channel opening time course at high depolarized voltages is complicated by the fact that the rising phase of the current is usually not well fitted by a single exponential. After a rapid rise, a slow “creep” up to the final value is observed at these voltages. In a more complete discussion of this phenomenon below, we will show that this slow component reflects an alternate activation path that a small fraction of the channels enter before opening. Here, to estimate the kinetics of the main activation path, the currents were fitted to the sum of two exponentials, 6
τ_{a} was taken to be the time constant of the faster decay, whose amplitude A_{f} accounted for ∼90% of the total. The logtransformed values of τ_{a} up to +147 mV from two patches are plotted in Fig. 4 B. These plots do not show the linear dependence on voltage expected if a single transition is rate limiting at all of the voltages; instead, the dependence of τ_{a} on voltage becomes increasingly shallow at high voltages. Zagotta et al. (1994a) made recordings of activation time courses only at voltages up to +50 mV, so these authors did not observe this deviation from simple behavior.
Estimates of the forward rate α_{p }of the ratelimiting step at high depolarized voltages were obtained as the reciprocals of τ_{a} values derived from the ionic current time courses measured between +87 and +147 mV, yielding α_{p}(0) = 2,100 ± 100 s^{−1} (n = 6) and q_{αp }= 0.17 ± 0.02 e_{0}. It is not clear where in the sequence of gating steps this ratelimiting step occurs. The time course of gating currents at these high voltages could provide some information about this, but we did not record gating currents in this voltage range.
Estimate of α_{N}.
Macroscopic ionic currents were measured with one additional voltage protocol to attempt to estimate the forward rate α_{N} of the very last step in the activation path (Fig. 5). A sequence of three voltage pulses was applied (Fig. 5 A): an initial depolarization to +47 mV to open most of the channels, a second pulse to a hyperpolarized voltage V_{h} (−153 mV in this case) of duration t_{h }to close a fraction of the channels, and a third pulse to +47 mV to reopen the channels that closed during the second pulse. In principle, for small enough t_{h}, channels should not have time to close to states beyond the first closed state, and the kinetics of channel reactivation should reflect the forward rate of the last transition in the activation path. This time course will be distinctly faster if the forward rate of the last transition is faster than that of the earlier gating steps. Zagotta et al. (1994a) measured currents using a similar voltage protocol with t_{h} as small as 1 ms, but we employed secondpulse hyperpolarizations that were much shorter, as short as 70 μs. In our experiments, briefer hyperpolarizing pulses make the reactivating currents much faster (Fig. 5, A and B). Compared with the 400μs activation time constant τ_{a }taken from the reactivation time course after a relatively long t_{h} = 1 ms hyperpolarization, single exponential fits of the reactivation time courses for shorter t_{h }yielded τ_{a }values of 300, 170, and 110 μs for t_{h }= 300, 150, and 70 μs, respectively.
Several lines of evidence suggest that these fast relaxations reflect genuine current changes due to gating and are not artifacts of the recording system. First, the singleexponential fits just described were performed while ignoring the first 60 μs of the pulse, which at the 15kHz bandwidth is longer than the settling time of the recording system (see methods). Second, the hyperpolarization to −153 mV produces a fast “instantaneous” current change reflecting the large change in the driving force between −153 and +47 mV. In the reactivation time course after a 1ms hyperpolarization in Fig. 5 A, this current change is clearly complete within 40 μs and distinguishable from the slower kinetics of the current changes during the reactivating pulse. As a final check, we simulated reactivating current time courses I(t) by convolving theoretical functions x(t) reflecting current changes due to gating, with the impulse response of the recording system h(i): 7
The impulse response h(i) was obtained by differentiating the step response at the 15kHz bandwidth at which these currents were measured (see methods) and was evaluated for n = 15 sample intervals. An additional constraint in this analysis was provided by the fact that an amplitude for x(t) can be predicted given the measured tail current time course during the preceding hyperpolarization. When it is assumed that x(t) is a single exponential with a time constant of 430 μs, which is the τ_{a} value taken from the activation time course during the first voltage pulse, I(t) poorly reproduces the data; but an exponential x(t) with a time constant of 100 μs accounts for the current time course very well (Fig. 5 C).
The 100μs time constant at +47 mV is much faster than the reciprocal of the values that we calculate for both α_{1} and α_{p }at +47 mV (430 and 400 μs, respectively) and thus reflects a distinct forward rate. It is difficult to prove that it is the very final step that is reflected by this time constant. More careful measurements using shorter hyperpolarizations might have revealed an even faster reactivation component associated with a later step. Nevertheless, consistent with the fast component reflecting the final step is that the effect of shortening t_{h} on the measured reactivation time constant appears to saturate at a value near 100 μs by the shortest −153mV hyperpolarizations used (Fig. 5 B). The same saturating relationship between t_{h} and τ_{a} is observed when V_{h} is varied between −93 and −193 mV, which may be expected to produce varying occupancies in the closed states near the open state.
We define the activation time constant τ_{a} observed for the shortest hyperpolarization (t_{h }= 70 μs) to be the fast reactivation time constant τ_{f} reflecting the final forward rate α_{N}. The reciprocal of τ_{f} at +47 mV gives α_{N}(+47) = 9,100 s^{−1}. The voltage dependence of α_{N} was evaluated by varying the voltage of the test depolarization after a short duration hyperpolarization. From the voltage dependence of τ_{f} in this and two other patches (Fig. 5 D), we estimate α_{N}(0) = 7,000 ± 300 s^{−1} and q_{αN }= 0.18 ± 0.01 e_{0}.
Estimate of q_{αd}.
The long delay in Shaker's channel opening time course (Zagotta et al., 1994a) is expected to reflect the forward rates of a large number of transitions. While we were unable to determine directly the forward rates of most of the transitions that come between the very first and last transitions, information about the partial charges q_{αd} associated with the forward rates of these intermediate transitions is available from the voltage dependence of the delay δ_{a} in the channel opening time course (Fig. 4 B). The δ_{a }value represents the sum of contributions from multiple steps; its voltage dependence is not well fitted by a single exponential across the wide voltage range, which is consistent with transitions with forward rates with different voltage dependences predominating at different voltages. Nevertheless, to approximate the partial charges of a large number of these transitions here, the δ_{a} values between −13 and +147 mV were fitted to a single exponential (not shown), which yielded a charge estimate of 0.25 e_{0} for q_{αd}.
Estimates of Backward Rates
Estimates of the voltage dependences of the backward rates of three transitions were obtained from the kinetics of macroscopic ionic and gating currents at hyperpolarized voltages.
Estimate of β_{1}.
The small fraction (<8%) of the total charge movement that occurs at voltages negative to −90 mV (Fig. 1) is consistent with the idea that at these voltages channels reside among the earliest closed states at equilibrium. If it is assumed that at −93 mV, most channels reside in either the first or second closed states, the off gating currents measured with voltage pulses from −93 mV to more hyperpolarized voltages will decay with a time course given by the backward rate β_{1} of the first transition. Consistent with the expected twostate behavior, the measured currents (Fig. 6 A) rise instantaneously and have a single exponential decay that becomes faster at more hyperpolarized voltages. The time constants of the single exponentials fitted to these gating currents (Fig. 6 B) in three patches yielded an estimate for a backward rate of the earliest gating step β_{1}(0) = 190 ± 60 s^{−1} and q_{β1} = −0.53 ± 0.03 e_{0}. Numerous reports of Shaker's gating currents exist in the literature (Bezanilla et al., 1991, 1994; McCormack et al., 1994; Zagotta et al., 1994a), but none of these include gating currents measured with voltage steps between different hyperpolarized voltages, which are required to estimate β_{1}.
The estimates of the backward rate β_{1} obtained here and the forward rate α_{1} obtained earlier can be compared with the QV relation. The charge of the first step z_{1} = q_{α1 }− q_{β1} should match the steepness of the exponential QV relation at extreme negative voltages. Measurements of Q obtained between −133 and −83 mV were used, where Q/Q_{max }≤ 0.13. Fitting the average data to an exponential function of voltage yielded an estimate of the first transition valence of z_{Q }= 1.1 e_{0} (not shown). This z_{Q} estimate is similar to the estimate z_{1} = 0.9 e_{0} obtained from the voltage dependences of the forward and backward rates.
Estimate of β_{d}(−93).
It has been previously shown that Shaker's off gating currents after large depolarizations have a rising phase and decay kinetics dominated by the slow kinetics of channel closing from the open state (Bezanilla et al., 1991; Zagotta et al., 1994a). However, the off current kinetics after short depolarizations that fail to open channels are much more rapid, presumably reflecting the relatively rapid backward kinetics of the transitions that come before the last transition (Zagotta et al., 1994a). We use this rapid off current time course here to obtain an approximate estimate β_{d} for the backward rate of intermediate transitions. Fig. 6 C illustrates the off current measured after a short 2ms depolarization to −33 mV. From the time integral of the gating current at this voltage, we estimate that 80% of the total gating charge movement occurs by the end of the 2ms depolarization, indicating that most channels reside in relatively late activation states; however, from the channel opening time course, we estimate that only 15% of the channels are open. Exponential fits of the off currents after the 2ms depolarization to −33 mV in two patches yielded time constants of 0.82 and 0.62 ms.
The decay of this off current reflects the contribution of the backward rates of many transitions, including the transitions between intermediate states that are preferentially occupied at the end of the prepulse depolarization, and also β_{1} of the first transition. The decay time constant of the current is in fact indistinguishable from the reciprocal of β_{1} at −93 mV (0.77 ms), suggesting that the rate of the intermediate steps β_{d} is similar to β_{1} at −93 mV. This analysis gives a rough estimate of β_{d}(−93 mV) = 1,300 s^{−1}.
Estimate of β_{N}.
Zagotta et al. (1994a) made estimates of the channel closing rate β_{N} from macroscopic ionic tail currents measured between −60 and −140 mV. For our estimates of β_{N}, we chose to use tail currents measured at even more negative voltages. While the position of the QV relation on the voltage axis would suggest that most of the forward rates are negligible in the hyperpolarizing voltage range (≤−93 mV), it has been shown above that the forward rate of the last transition α_{N} is considerably faster than the forward rates of the preceding transitions, and could have values at voltages near −93 mV that are comparable to backward rates. Indeed, according to our fits, α_{N}(−93) = 3,600 s ^{−} ^{1} is greater than β_{1}(−93) = 1,300 s ^{−} ^{1}. Considering a partial scheme consisting of the last three states of Scheme I, (Scheme III)
the channel deactivation time course at these voltages might involve many reopenings from state C_{N1} before closing further to state C_{N2}. The large forward rate of the last transition implies that estimates of β_{N} must be obtained from currents measured at voltages more negative than −93 mV.
Inward tail currents were measured between −153 and −203 mV using 14.3 mM external K^{+} (Fig. 7 A). One criterion to determine that the tail current time course entirely reflects the channel closing rate β_{N} is that it should be well fitted by a single exponential, but even at the most negative voltages, the tail currents were poorly fitted by a single exponential. Assuming that the complicated channel deactivation time course reflects channel reopenings from the last closed state in the activation path, as reflected in Scheme III, estimates of β_{N} could nevertheless be obtained by fitting the tail currents to the sum of two exponentials: 8
with the faster time constant τ_{f} reflecting β_{N} at sufficiently negative voltages.
In practice, only tail currents at V ≤ −153 mV were considered in this analysis. At these voltages, τ_{f} displays the singleexponential dependence on voltage expected for β_{N} (Fig. 7 B). Additionally, the amplitude A_{f }(Fig. 7 C) of the fitted fast exponential increases monotonically below −153 mV, expected for a tail current time course that reflects channel reopenings from the last closed state in the activation path but reflecting increasingly fewer reopenings at more negative voltages. The reciprocal of the τ_{f} values at different voltages below −153 mV from five separate patches yielded estimates of β_{N}(0) = 290 ± 90 s^{−1} and q_{βN} = −0.50 ± 0.04 e_{0}.
Estimate of β_{N1}.
If WT's channel deactivation time course is determined at most voltages by the channel closing rate and by reopenings from the last closed state C_{N1}, this time course provides one way to estimate the backward rate β_{N1 }from the pentultimate state. A second measure of β_{N1} is provided by the reactivation time courses (Fig. 5 A). The amplitude of the fast reactivation component should reflect the occupancy in C_{N1} during the preceding hyperpolarization. The voltage dependences of the rates of the transitions out of C_{N1}, including β_{N1}, will determine the way in which this amplitude varies with the duration t_{h} and amplitude V_{h} of the preceding hyperpolarization.
Fig. 8 A illustrates the strategy we used to estimate the size of the fast component in the reactivation time course. In this analysis, we used an observation made in a different experiment, that changing the amplitude of a prepulse has a negligible effect on the kinetics of the final approach of the measured ionic current, estimated by τ_{a}. (These data are not shown here, but see Fig. 18 B in Schoppa and Sigworth, 1998b). This result suggests that a reasonable approximation to the fast and slow components in the reactivation time course is the function 9
Varying the exponent r in the third term (while allowing it to take fractional values) yields an opening time course with varying amounts of delay, but with a nearly invariant final approach. The fits of Eq. 9 to the reactivation time courses in Fig. 8 A have been separated into the respective fast and slow components. Increasing t_{h} from 100 to 200 μs yields a larger fast current, but the magnitude of this current decreases with even longer t_{h}, reflecting channels entering closed states that precede C_{N1 }during the hyperpolarization. For the slow component, increasing t_{h} produces a monotonic increase in amplitude, and also a longer delay, reflecting channels entering into relatively early closed states.
Estimates of β_{N1} were then obtained by simultaneously fitting the threestate Scheme III to the time course of occupancy in C_{N1 }(Fig. 8 B), estimates of which were derived from the amplitude of the fast reactivation component, as well as to tail currents (Fig. 8 C). We restricted our analysis to voltages lower than −90 mV, where we assume that the tail current and occupancy time courses are functions of just α_{N}, β_{N}, and β_{N1}; that is, we take the reverse rate β_{N2} to be much greater than the forward rate α_{N1}, so that only reverse transitions occur out of C_{N2}. In the fitting, α_{N} was fixed to the estimate obtained in Fig. 6, but β_{N} was allowed to vary along with β_{N1}. Good fits of the tail current time courses and the derived occupancy estimates are obtained with the parameter estimates for β_{N} and β_{N1} indicated in the legend. The smaller estimate for the charge q_{βN1} compared with q_{βN} accounts for the relatively higher occupancies in C_{N1} at the most hyperpolarized voltages (Fig. 8 B). We note that the derived estimates for β_{N1}(0) and q_{βN1} may be too large if the assumption that β_{N2 }is much greater than α_{N1} at V ≤ −93 mV is not valid.
The fits of the data in Fig. 8 to Scheme III are expected to provide a more reliable estimate of β_{N} than that derived from fitting the tail currents to the sum of two exponentials in Fig. 7. However, the estimate for β_{N} derived from the fits of Scheme III is nearly identical to that derived directly from the tail currents (Fig. 7 B). Additionally, the fast tail current component predicted by Scheme III has approximately the same amplitude as the faster component in the twoexponential fits of the tail currents (Fig. 7 C). These observations confirm our direct estimates of β_{N}.
Estimate of q_{βd}.
A measure of the partial charge q_{βd }associated with the backward rates of intermediate transitions was obtained from measurements of macroscopic ionic currents, using a strategy that is analogous to that used above to estimate q_{αd}. In that case, the voltage dependence of the delay in channel activation gave a rough estimate of the voltage dependence of the underlying rate constants of intermediate transitions. Here we measure the activation delay as initial state occupancies are varied using repolarizing pulses of duration t_{h} and amplitude V_{h} that follow an initial depolarization that loads the channels into the open state. Let the initial occupancy of state i be p_{i}, and let α _{i} be the forward rate from state i to state i + 1. Then Eq. 5 can be generalized to give the delay with arbitrary occupancies, 10
where again we assume the unidirectional sequential Scheme I and that rate α _{j} is rate limiting. The dependence of δ_{a} on t_{h} will reflect the time dependence of the occupancies during the hyperpolarizing pulse and therefore depend on the backward rates of the transitions that contribute to the delay. The dependence on V_{h} of this accumulation will be a function of the partial charges that determine these backward rates.
The analysis of the accumulation of the delay requires defining three parameters. The first is the delay δ_{a} derived by fitting a single exponential to the slow component in the reactivation time course (Fig. 8 A), which has been approximated by the second term in Eq. 9. The second parameter is τ_{acc}(V_{h}), which is the time constant of a single exponential fit to the dependence of δ_{a }on t_{h} at a given V_{h} (Fig. 9). The third parameter is the estimate of the charge q_{acc} that reflects the voltage dependence of the accumulation of the delay, obtained by fitting the τ_{acc} values to an exponential function of V_{h} (Fig. 9 B).
Changing the hyperpolarization voltage V_{h} from −93 to −193 mV (Fig. 9 A) causes the delay to accumulate more quickly, which is expected for backward rates with associated negative partial charges. Notably, however, the time course of the accumulation of the delay is only about twice as fast at −193 as at −93 mV, consistent with backward rates having only small partial charges. Evaluating reactivation time courses for various V_{h} and t_{h} in four patches, the τ_{acc}(V_{h}) values across the voltage range showed a voltage dependence corresponding to a charge of −0.24 e_{0} (Fig. 9 B), which we take as an estimate of q_{βd}. Additionally, we can use this estimate of q_{βd }to derive an estimate of the rate β_{d}(0) at 0 mV, given the estimate of β_{d}(−93) = 1,300 s^{−1} obtained above from gating currents. This gives β_{d}(0) = 540 s^{−1}.
Transitions to States that Are Not in the Activation Path
Up to this point, we have characterized the transitions that the Shaker channel undergoes in the depolarizationinduced activation process. However, Hoshi et al. (1994) have shown from Shaker's single channel data that there are at least two additional closed states, called C_{f} and C_{i}, into which the channel enters only after it has opened. The state C_{f} corresponds to a predominant, fast 200–300μs component in their single channel closed dwelltime histograms at depolarized voltages, and the other closed state C_{i }accounts for a voltageindependent, 1–2ms component in the closed time histograms at depolarized voltages. In the following, we use a combination of single channel and macroscopic current measurements to further characterize the transitions to these additional states.
We begin by outlining the general properties observed in our single channel measurements at depolarized voltages. As was observed by Hoshi et al. (1994), we find that the channel opens quickly after the beginning of a voltage pulse (Fig. 10 A) and remains open for most of the trace, occasionally closing briefly into shortlived closed states. Infrequently, the channel displays longer closures (see the third trace at +67 mV). The closed time histograms constructed from our single channel activity are always fitted by a mixture of three exponentials (Fig. 10 B), including one largeamplitude component with a very fast (τ_{1} = 40–100 μs) time constant, a second component with an intermediate amplitude and duration (τ_{2} = 200–500 μs), and a third component with a 1–3ms time constant (τ_{3}) and a very small amplitude (comprising ∼1% of the closures). Our fit of two exponentials to closures with durations of <500 μs differs from the single exponential fitted to these closures by Hoshi et al. (1994). It is difficult to interpret the voltage dependences of the two faster closed time components (Fig. 10 D), since the rapid closures are near the limit of time resolution and, additionally, the two rapid closedtime components are not well resolved from each other. The third, slowest component has a voltageindependent duration and corresponds to the C_{i} closures observed by Hoshi et al. (1994).
As shown previously (Hoshi et al., 1994), the open times (Fig. 10 C) are well fitted by a single exponential with time constants near 3 ms, consistent with the presence of a single open state. The measured open times display little voltage dependence (Fig. 10 E). No effort was made to correct the open times for missed fast closures because the missed event correction would rely on the amplitude and time constant estimates for the fastest closures, which are quite unreliable.
Characterization of the Transitions to C_{i }
The first new property that we assign to C_{i }is that it is a state that can be entered from not only the open state but also from closed states in the activation path. This property for C_{i }is required by the fact that C_{i} contributes to occasionally observed long first latencies to channel opening at depolarized voltages.
An instance in which the channel enters into a longlived closed state before it opens is evident in the second trace at +67 mV in Fig. 10 A. Entries into longlived closed states before first opening are also apparent in the shape of the cumulative first latency histogram (Fig. 11 A), which displays a gradual rise superimposed on the dominant faster component. The same slow component was also usually observed in the time course of macroscopic ionic currents at high depolarized voltages (V ≥ +67 mV; Fig. 4 A). While a singleexponential fit may account for the time course for even a complex gating scheme depending on the way that the fitting is performed, the observed deviation from a singleexponential time course in the final approach of the current in our fits is good evidence that individual channels are traversing different ratelimiting steps to opening. The observed two components in our time courses apparently are not, however, due to differences in the initial conditions for different channels, because the size of the slow component in the macroscopic current does not change with a very hyperpolarizing prepulse to −143 mV (in three patches; data not shown); this prepulse is expected to preload virtually all channels into the very first closed state. The slow component is also probably not due to modal gating, as it was found in one single channel patch recording that there was no apparent pulse–pulse correlation in the appearance of long first latency. Instead, we take the presence of two components in the final approach of the ionic current to imply that there are two pathways to channel opening; that is, a fraction of channels pass through closed states in a slow path that is distinct from the main activation path.
In fits of two singlechannel first latency histograms at V ≥ +67 mV to the sum of two exponentials (Eq. 6), the fitted slow exponential had time constant τ_{s} values of 1.6 and 1.8 ms (Fig. 11, A and B), similar to the 1.9 ± 0.5ms mean duration of the τ_{3 }component in the closed dwelltime histograms at the same voltages (n = 3; Fig. 10 D). Fits of 21 macroscopic ionic current time courses to Eq. 6 yielded τ_{s} = 1.5 ± 2 ms, also similar to τ_{3}. (The time course of the slow component in the macroscopic ionic current can be compared with τ_{3} because backward rates at high voltages are negligible, making the ionic current have essentially the same time course as the cumulative first latency histogram; Hoshi et al., 1994.) We take the similarity in the latency time constant and τ_{3} to imply that the long latencies to opening reflect sojourns in the same family of states as the C_{i} closures that follow the first opening, except these sojourns in C_{i} are made from closed states in the activation path. The experiments of Hoshi et al. (1994) were not able to discriminate transitions into C_{i }before opening because their single channel measurements were not made at sufficiently high voltages; at lower voltages, the kinetics of the transitions through this secondary activation path are not substantially slower than the kinetics of the main activation path.
The slow component accounts for on average ∼10% of the total current relaxation (Fig. 11 C), implying that an opening channel has a 10% chance of entering one of a family of C_{i} states from closed states in the activation path before opening. For example, in the following scheme, (Scheme IV)
the opening channel might move from the resting state C_{0 }into C_{N1}, and then into C_{iN1} and C_{iN}, before reaching the open state O_{N}. From which particular set of closed states the channel can enter the C_{i} states remains unclear. Scheme IV does not include transitions between C_{iN} and C_{f}, because we prefer a simple model for these states.
If it is assumed that most of the C_{i }closures observed at equilibrium at depolarized voltages correspond to sojourns in the state C_{iN}, which is entered directly from the open state, we can use the closed and open dwelltime histograms to assign estimates for the rates c and d of this transition. The values of the time constants fitted to the C_{i} closures at different voltages give an estimate for the rate d from C_{iN }to the open state (d(0) = 300 s^{−1} and q_{d} = 0.07 e_{0}; Fig. 11 D). The open times and amplitude of the C_{i} closed time component (Fig. 10 F) give estimates for the rate c from the open state into C_{iN} (c(0) = 5 s^{−1} and q_{c} = 0.09 e_{0}). This estimate for c is an upper limit, if some of the C_{i} closures observed under equilibrium conditions reflect sojourns in states entered from closed states in the activation path.
Evidence for C_{f1} and C_{f2}
We next evaluate the transitions that correspond to the two rapid components of the closedtime histograms in Fig. 10 B. Since Hoshi et al. (1994) have shown that many of the rapid closures at depolarized voltages are to states that are not in the activation path, our starting hypothesis will be that the two rapid components correspond to two different such states: C_{f1} and C_{f2}. The alternate possibility is that one of the components corresponds to closures in the activation path (e.g., to the last closed state C_{N1}). We can address this second possibility by comparing the measured channel open times and the reciprocal of our estimated channel closing rate β_{N} (Fig. 10 E). At the most depolarized voltage where currents were measured (+147 mV), the measured channel open time (3 ms) is 60 times shorter than the expected value of 1/β_{N} = 180 ms at that voltage. (Were we to correct the open time for missed brief closures, the discrepancy with 1/β_{N} would be even larger.) This large difference implies that virtually all of the measured closures at this voltage reflect closures to states that are not in the activation path. Since the two rapid components of the closed time histograms at +147 mV comprise 73 and 25% of the total measured closures, both very much larger than the ∼2% component expected for closures to C_{N1}, we take both components to reflect closures to states C_{f1} and C_{f2} not in the activation path.
While there are a number of different ways to model these two states, we suggest that C_{f1} and C_{f2 }are states into which the open channel closes directly: (Scheme V)
We cannot easily rule out alternative models (e.g., having channels in O_{N} close first into C_{f2} and then into C_{f1}), but we choose Scheme V because of its simplicity.
Since the rapid closures in the single channel recordings are not well resolved, the parameter values derived from exponential fits to the histograms do not yield quantitatively reliable estimates of the rates associated with the transitions to C_{f1} and C_{f2}. We nevertheless obtain rough estimates of these rates by using various strategies.
Estimating Rates for the Transition to C_{f1}
If the transition to either C_{f1} or C_{f2} carries any amount of charge, the rates for this transition can be estimated from the relaxation of macroscopic ionic currents that are elicited by a double pulse protocol. To achieve test currents that predominately reflect the transition to C_{f1} or C_{f2}, prepulses are required that are large enough that virtually no channels reside in states in the activation path at the end of the prepulse. A comparison of the measured channel open times and the reciprocal of the closing rate β_{N} in Fig. 10 E indicates that at voltages ≥+7 mV, open times are approximately the same or shorter than 1/β_{N}; this would imply that much of the test current relaxation after prepulses above +7 mV reflects transitions to states outside the activation path.
Fig. 12 A illustrates an experiment in which we apply voltage steps from +7 to +127 mV, and also from +47 to +147 mV. These voltage jumps elicited small current relaxations with activation time constants of τ_{r }= 230 and 180 μs. The time course of the current relaxation at +147 mV (τ_{r }= 180 μs) is threefold slower than the fast reactivation time constant that corresponds to the last transition in the activation path (Fig. 5 D); this indicates that the observed current must reflect a transition that is distinct from the last activation transition. The observed relaxation also cannot reflect the transition to C_{iN}. Given the estimates for c and d assigned above, the transition to C_{iN} would contribute to a change in open probability ΔP_{o} << 1% during these voltage jumps, but the amplitudes of the relaxations correspond to ΔP_{o} ≥ 6% (see legend to Fig. 12). Thus, these relaxations almost certainly reflect transitions to C_{f1 }or C_{f2}, and we assign them to C_{f1}. Since these currents are recorded at depolarized voltages, we assume that the current time course mostly reflects the rate f_{1} from C_{f1} to the open state, yielding estimates of f_{1}(0) = 1,600 s^{−1} and q_{f1} = 0.2 e_{0} (Fig. 12 B).
The value of the rate constant e_{1} for the entry into C_{f1 }can be determined using the amplitude of the current relaxation in the voltage jump experiment in Fig. 12 A. The 10% change in relative P_{o} induced by the voltage jump from +7 to +127 mV, together with the value for f_{1} derived above, gives e_{1 }= 190 s^{−1} at +7 mV (assuming that the transition to C_{f1} is saturated at +127 mV). We assume that this transition is voltage independent.
The current relaxations in Fig. 12 A verify one feature of data presented earlier in this paper. In the voltage dependence of channel opening for Shaker (Fig. 1), P_{o} rises steeply to ∼80% of its maximal value, but displays a slow upward drift at higher voltages. The presence of the gradual rise, however, has been reported to be dependent on the method by which P_{o }is measured (Zagotta et al., 1994a). The 12% change in relative P_{o} that we estimate to occur between +7 and +67 mV using the tailcurrent method for estimating relative P_{o} (described in the legend of Fig. 1) is, notably, similar to the 10% change in relative P_{o} induced by the voltage jump from +7 to +127 mV. In our modeling paper (Schoppa and Sigworth, 1998b), we will use the shape of the P_{o}V relation at depolarized voltages to help constrain the total charge associated with the transition to C_{f1}.
Estimating Rates of the Transition to C_{f2}
The relaxations of the macroscopic currents in Fig. 12 A that correspond to C_{f1} → O have similar timeconstant values as the intermediate duration τ_{2} component of the singlechannel closed dwelltime histograms at similar voltages (Fig. 10 D); at V > +100 mV, τ_{2} is between 200 and 300 μs. If the transition from C_{f1} corresponds roughly to the intermediate duration closures in the dwelltime histograms, the transition from C_{f2} must correspond to the very rapid 40–100 μs closures. These closures are poorly resolved, but the short apparent time constant implies that the rate f_{2} from C_{f2} to the open state is very rapid (near 10^{4} s^{−1}). Information about this transition can also be obtained from estimates of absolute P_{o}, which can be derived from the mean open and closed times in the single channel measurements (Fig. 12 C). From measurements obtained at voltages as large as +147 mV, the derived absolute P_{o} values apparently saturate near 0.9. The fact that P_{o} at very high voltages saturates at a value less than unity implies that the transition to C_{f2} is nearly voltage independent (giving q_{e2 }+ q_{f2 }≈ 0.0). A saturating value of P_{o }= 0.9 implies a ratio of the two rates e_{2}/f_{2} near 0.1, giving e_{2 }≈ 10^{3} s^{−1}.
Discussion
In this and the following two papers (Schoppa and Sigworth, 1998a, 1998b), we characterize activation gating for the normally activating Shaker channel (WT) and a mutant version of the channel to arrive at a relatively wellconstrained activation gating model. In this first paper, we have undertaken a detailed analysis of the kinetics of WT's macroscopic ionic and gating currents and single channel currents in an effort to obtain estimates of several rate constants. Previous efforts at characterizing Shaker's activation gating process exploited experimental strategies that were similar to ours in order to isolate particular activation gating transitions (Hoshi et al., 1994; Zagotta et al., 1994a; Bezanilla et al., 1994a). However, we have gained new insights by analyzing the current time courses in novel ways and by extending the voltage range of the current measurements.
Estimates of Forward and Backward Rates for Transitions in the Activation Path
We have assumed that activation gating is described by a Markov process with transitions between discrete states separated by large energy barriers. Fits of exponentials to selected ionic and gating current time courses at extreme voltages yielded estimates of the rate constants depicted here. (Scheme VI)
For the forward rates, we have obtained estimates of the first forward rate α_{1}, the limiting rate at large positive potentials α_{p}, the final opening rate α_{N}, and an estimate of the partial charge q_{αd} that determines the voltage dependence of forward rates of intermediate steps. For backward rates, we have obtained estimates of the first and last two backward rates (β_{1}, β_{N1}, and β_{N}), and an estimate of the backward rates of intermediate steps β_{d}. The values obtained are given in Table I. We have also extended on the work of Hoshi et al. (1994) by characterizing the transitions to several closed states that are not normally traversed before the channel opens. Hoshi et al. (1994) showed that there were at least two such states C_{i} and C_{f}; from an analysis of single channel data, we have shown that there are at least four such states. These results are summarized by Scheme V above and the list of rate estimates in Table II.
It should be emphasized that the parameters q_{αd}, q_{βd}, and β_{d}(0) that we ascribe to intermediate transitions should be taken as quite rough approximations; these are shown in Scheme VI with parentheses. The charge parameters q_{αd} and q_{βd} were derived from the delay in the channel opening time course, which reflects the composite properties of many transitions. These parameters are also somewhat difficult to interpret because transitions with the fastest forward rates contribute little to the delay. Thus, these charge parameters are not likely to reflect the partial charges associated with the most rapid transitions. However, the number of rates with partial charges similar to q_{αd} and q_{βd} is likely to be quite large. Zagotta et al. (1994b) estimated that a minimum of five transitions are required to account for the delay in the ionic current time course; our own analysis gives a minimum estimate of seven transitions (Schoppa et al., 1998b), implying that at least six transitions contribute significantly to the delay.
Assumptions Used in the Derivation of Rate Constants
The accuracy of these rate estimates has relied on a number of assumptions, besides the assumption of a Markov process. The first of these is that our current measurements were made at sufficiently positive and negative voltages to yield unidirectional transitions, so that our current measurements reflect only a single microscopic rate constant. All but one of the forward rates were estimated from currents measured at V ≥ +67 mV, which is 80 mV higher than where the QV relation saturates, and all backward rates were estimated from currents measured at V ≤ −93 mV, where <8% of the charge movement occurs. While we cannot be certain that transitions are unidirectional at these voltages, our current measurements were made at more extreme voltages than those that have been published previously. Thus, for example, in the estimate of the channel closing rate β_{N}, our estimates are likely to be more reliable than those that have been reported previously.
The forward rate α_{1} of the first transition was estimated from current measurements made at voltages between −13 and +67 mV. The reason for concern about voltage ranges for this estimate is the following. Whereas the equilibrium P_{o}V and QV relations appear to saturate near 0 mV (Fig. 1), previous models for Shaker's gating (Bezanilla et al., 1994; Zagotta et al., 1994b) have included at least one transition that has a relatively large backward rate at this voltage; the channel nevertheless opens due to the driving action of a later forwardbiased transition. Thus, for example, nonnegligible backward rates at voltages near 0 mV will yield an estimate for α_{1} from the final approach of the ionic current that is smaller than its actual value.
A second assumption that we have made for simplicity is that activation gating follows a linear sequence of transitions as in Scheme SVI. Many of the models that have been used to describe the activation gating process for Shaker channels, however, are branched models that reflect the tetrameric structure of the channels (McCormick et al., 1994; Zagotta et al., 1994b); we also will propose branched models in a subsequent paper. The estimates for rate constants given here are likely to be valid for such branched models. Many of our rate estimates rely on fits of exponentials to macroscopic ionic current time courses yielding a time constant τ_{a} and a delay parameter δ_{a}. As we have shown in the simulations of branched models in Fig. 2, C and D, the rates of ratelimiting steps are estimated without large errors by this approach.
In this paper, we have considered evidence that activation gating for Shaker deviates from a sequential model in that there are three closed states (C_{iN}, C_{f1}, and C_{f2}) that the channel enters after the channel opens, and at least one additional closed state C_{i} that the channel can enter from closed states in the activation path. The transitions to the three states C_{iN}, C_{f1}, and C_{f2} are problematic to our analysis because these transitions could affect the macroscopic tail current and reactivation time courses that yielded our estimates of α_{N}, β_{N}, and β_{N1}. For example, channels undergoing slightly voltagedependent transitions from the open state to C_{iN} or C_{f1} could contribute to additional decay components in the tail current (Zagotta et al., 1994a). However, these components are expected to be small due to the transitions' small voltage dependences; also, at hyperpolarized voltages (−193 mV), the rates of the transitions from the open state to C_{iN }and C_{f1} (c = 3 s^{−1} and e_{1 }= 10^{3} s^{−1}) are much smaller than the rate of channel closing into the activation path (β_{N}(−193) = 11,400 s^{−1}). The additional states C_{i }entered from closed states in the activation path make a small contribution to the channel activation time course at high voltages, and thus could affect our estimate of the high voltage ratelimiting step α_{p}; however, we explicitly account for these transitions by fitting the ionic currents to the sum of two exponentials.
The best evidence that the simplified analysis of the Shaker's gating process in this paper is reasonable comes from comparing the firstpass estimates of various rates obtained here (in Tables I and II) with the results of the modeling in the third paper in this series (Schoppa and Sigworth, 1998b). There, we consider a number of different branched models that are similar to Scheme II. We derive starting rate estimates for the transitions in the models from each of the rate estimates obtained here; the rate values that then yield the best fits of the data are quite similar to these starting estimates, being at most a factor of 2–3 different.
Activation Gating Involves Many Transitions with Small Valences
The characterization of the different activation gating transitions performed in this paper leads to a few general results about the activation gating process. The first is a rough estimate of the average amount of charge associated with each of the transitions. We use here the estimated partial charges (q_{αd }and q_{βd}) for the forward and backward rates, which were derived from the delay in the channel opening time courses. The derived q_{αd} and q_{βd} estimates (0.25 and −0.24 e_{0}) yield a single transition charge estimate (q_{αd} − q_{βd}) of 0.5 e_{0}. This estimate of the average amount of charge associated with each transition, in turn, leads to an estimate of the number of gating transitions n. If an activating Shaker channel moves a total charge of 13 e_{0} (Schoppa et al., 1992; Aggarwal and MacKinnon, 1996; Seoh et al., 1996), our estimate of 0.5 e_{0 }for the transition charge implies that an opening channel undergoes n ≈ 26 gating transitions.
The value of 0.5 e_{0} for the transition charge may be too small since, as described above, the delay does not provide a good measure of the transitions with the most rapid rates. Indeed, fluctuation analysis of gating currents at depolarized voltages (Sigg et al., 1994b; 1996) suggests that there are some transitions that have valences near 2. These considerations also imply that the estimate of 26 transitions may be somewhat too large.
Activation Gating Involves at least Three Different Types of Gating Transitions
A second insight that we obtain here is an estimate of the number of different types of gating transitions. This is derived from the partial charges that define the voltage dependences of the different rates. The list in Table I includes q_{α1} = 0.36 e_{0}, q_{αN} = 0.18 e_{0}, q_{β1} = −0.53 e_{0}, q_{βN1 }= −0.30 e_{0}, and q_{βN} = −0.57 e_{0}. From these estimates, two types of transitions can be distinguished by having different partial charges for their forward rates. These correspond to the very first and very last transitions, with charges q_{α1} and q_{αN}. A third transition has a partial charge for its backward rate (q_{βN1 }= −0.30 e_{0}) that is different from that for the backward rate of either the first or last transition (q_{β1} = −0.53 e_{0} and q_{βN} = −0.57 e_{0}). These considerations will be important in the discrimination between different models in the third paper in this series (Schoppa and Sigworth, 1998b).
Acknowledgments
We thank L. Lin for oocyte preparation; E. Moczydlowski and R. MacKinnon for CTx; and R.K. Ayer, W.K. Chandler, K. McCormack, Y. Yang, and W.N. Zagotta for helpful discussions.
This study was supported by National Institutes of Health grant NS21501 to F.J. Sigworth.
Footnotes

Dr. Schoppa's present address is Vollum Institute, Oregon Health Sciences University L474, Portland, OR 972013098.

1 Abbreviations used in this paper: CTx, charybdotoxin; NMDG, Nmethyl dglucamine; WT, wild type.
 Submitted: 3 June 1997
 Accepted: 24 November 1997