INTRODUCTION

Top Abstract Introduction Methods Results Discussion References

Voltage-gated sodium, calcium, and potassium channels underlie the electrical properties of excitable cells. Insights into the structural changes involved in the voltage-dependent 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 voltage-gated 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 voltage-gated potassium channels are encoded by a transcript that is approximately one-fourth 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 voltage-clamp measurements of macroscopic and single-channel ionic currents, various voltage protocols have been used to emphasize particular steps in the activation process (Cole and Moore, 1960). The voltage-dependent 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 NH₂-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 e₀. 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 (Q-V) 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 Q-V 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 voltage-dependent conformational changes.

(h) Measurements of gating current fluctuations suggest that elementary charge movements are roughly 2 e₀ 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 NH₂ 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

Top Abstract Introduction Methods Results Discussion References

Channel Expression

The construction of noninactivating WT Shaker 29-4 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 50-100-fold less cRNA for single channel recordings. Current measurements were made 4-21 d after injection with the patch-clamp technique (Hamill et al., 1981) using an EPC-9 patch clamp amplifier (HEKA Electronic, Lambrecht, Germany).

Measurements of Macroscopic Ionic and Gating Currents

Macroscopic ionic and gating currents were recorded in inside-out 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 M 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 20-mV amplitude from a 133-mV 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 signal-to-noise 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 N-methyl-D-glucamine (NMDG) aspartate (Asp), 1.8 mM CaCl₂, 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 EPC-9 and measuring the current response. At the low gains at which the macroscopic ionic and gating currents were measured (500 M 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 Single-Channel Ionic Currents

Single-channel recordings were made in inside-out patches in response to step depolarizations from a 93-mV holding potential. Patch pipettes were pulled from 7052 glass (Garner Glass, Claremont, CA) with 1-2-µm tip diameters (4-10 M 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 single-channel 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 time-dependent 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 single-channel 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 single-channel current traces were kinetically identical to the macroscopic currents in patches from oocytes injected with 50-100-fold more cRNA. Infrequently, a patch would display other single-channel activity, but this activity was easily distinguishable from Shaker's in its voltage-dependence, kinetics, and conductance properties.

The analysis of the equilibrium single-channel closed and open times was performed with the TAC single-channel analysis program, which is based on THAC (Sigworth, 1983). Data were filtered with a digital Gaussian filter to achieve an appropriate signal-to-noise ratio, and event detection was performed using the standard half-amplitude threshold analysis (Colquhoun and Sigworth, 1995). Complicating the analysis of the single-channel 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 open-time histograms, the deadtime (T_d) for a given analysis filtering frequency f_c was taken to be T_d = 0.179/f_c and short-event 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 minimum-duration 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.

View larger version (20K): [in this window] [in a new window]   Fig. 7.   Estimate of N. (A, top) Inward macroscopic ionic currents elicited by a fixed prepulse to +7 mV followed by test pulses to various hyperpolarizations between 93 and 193 mV. Filtering at 15 kHz. Patch w448. (bottom) A single exponential (dashed curves) poorly accounts for the decay of the tail currents at 153 and 193 mV. (B) An estimate of N was obtained from the faster time constant f in fits of the tail currents at V  153 mV to the sum of two exponentials (Eq. 8). Values reflect five different experiments, and estimates for N(0) and qN were derived from each of the patches (solid lines). These estimates are similar to N (dashed line) derived from fits of Scheme III to the tail current and CN-1 occupancies in Fig. 8. (C) Voltage dependence of the relative amplitude of the f component in fits of the tail currents in the same five patches to Eq. 8. The superimposed dashed curve reflects the amplitude Af(III) of the faster of two tail current relaxation components predicted by Scheme III, assuming values for N and N-1 derived from the fits in Fig. 8. For N-1 = 0, the amplitude is given by Here, f is the faster relaxation eigenvalue that approaches N at very negative voltages.

View larger version (30K): [in this window] [in a new window]   Fig. 8.   Estimate of N-1. (A) Separation of the reactivation time course into fast and slow components. Selected current traces in Fig. 5 A are shown for th between 100 µs and 1 ms. The time course of the fast component, reflecting channels reopening from CN-1, and the slow component, reflecting channels reopening from all closed states that precede CN-1, were approximated by fits of Eq. 9 (smooth curves). In the fitting, the value for N was fixed to the mean value derived from the reactivation time courses in Fig. 5 D. From the slow component, we derived an estimate of the delay a (shown for th = 1 ms; dashed curve). (B) The occupancies in CN-1 derived from the reactivation time courses, as well as tail currents (C), were fitted to Scheme III (superimposed smooth curves) to obtain estimates of N-1 and N. The occupancy pN-1 in CN-1 for different amplitude Vh and duration th was obtained from fits of Eq. 9 to the reactivation time course as pN1 = Af/ (Iinst + Af + As); that is, taking Af to reflect the amplitude of current due to the return of channels from CN-1 to the open state. This expression for pN-1 is approximate, but is expected to hold in this case. Each of the data points reflects one to four experiments. The parameter estimates obtained were: N(0) = 150 s1, qN = 0.57 e0, and N-1 (0) = 320 s1, qN-1 = 0.30 e0. In the fitting, N was fixed to mean estimate value for N (Fig. 5 D); all channels were assumed to reside in the open state at the beginning of the test pulse.

View larger version (21K): [in this window] [in a new window]   Fig. 5.   Estimate of N from a fast reactivation component. (A, top) WT's macroscopic ionic currents elicited by a triple-pulse stimulus, using a pair of depolarizations to +47 mV separated by a voltage step to a hyperpolarized voltage Vh = 153 mV. The displayed currents correspond to different hyperpolarization durations th between 70 and 1,000 µs. Tail currents during the second pulse were inward since the pipette solution contained 14 mM K+. Data were filtered at 15 kHz. Patch w448. (bottom) The same traces are shown, but expanded and time shifted to align the start of the test pulses. The "upper half" of the reactivating current relaxations for th = 70 µs and 1 ms have been fitted to a single exponential to estimate a (smooth curves). (B) The dependence of the derived values for a on the hyperpolarization duration th, shown for different hyperpolarization amplitudes Vh. Note that for each Vh, the fastest a values appear to approach 100 µs. All data reflect the patch in A. (C) To demonstrate that the fast reactivating component is not a recording artifact, we compare the apparent fast reactivating current in A for th = 100 µs with two different simulated current traces reflecting Eq. 7. The solid and dashed smooth curves, respectively, correspond to x(t) being a single exponential with a fast time constant  = 100 µs (the good fit) or a slower  = 430 µs (the poor fit). The total amplitude (39 pA) associated with x(t) was constrained by fitting the tail current at 153 mV in the same patch to the sum of two exponentials and determining the amount of decrease in current by 100 µs after the beginning of the pulse. The impulse response h(i) was determined by differentiating the step response measured at the 15-kHz bandwidth (see METHODS). (D) The fast reactivation time constant (f) values at different voltages were fitted to an exponential (solid curves) to estimate the voltage dependence of N, yielding estimates for N(0) and qN for each of three patches (N(0) = 7,600, 6,800, and 6,500 s1 and qN = 0.19, 0.16, and 0.18 e0). For each patch, f was obtained from currents measured after a 150-µs hyperpolarization to 153 mV. The reactivation time course for th = 150 µs includes a small slow component (accounting for, on average, 14% of the time course) as well as the dominant fast component; the f values reflect the fast time constant in fits of the reactivation time course to the sum of two exponentials:

View larger version (11K): [in this window] [in a new window]   Fig. 6.   Estimates of 1 and d. (A) Gating currents elicited by voltage steps between 93 and 153 mV. The current recorded at 93 mV (top) reflects the on current measured after a voltage step from 133 mV; the currents recorded at the 153-, 133-, and 113-mV test voltages reflect the off current measured after voltage steps from 93 mV. Currents were fitted to a single exponential (smooth curves) to estimate a decay time constant . Patch w249. (B) To estimate 1, the  values derived from the currents in A were fitted to an exponential (solid curve), yielding 1(0) = 200 s1 and q1 = 0.48 e0. (C) The off gating current measured at 93 mV after a 2-ms depolarization to 33 mV was fitted to a single exponential with  = 0.82 ms. This time constant is similar to the reciprocal of the value of 1 at 93 mV (0.77 ms) estimated in B, consistent with d being similar to 1 at 93 mV.

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 time-dependent 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 ± 1.1-fold longer than tail currents recorded less than 1 min after patch excision. Time-dependent 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 time-dependent 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 time-dependent slowing effect occurred with a time course that was approximated by a single exponential with  = 6.2 ± 1.5 min.

Slow inactivation. A final potential problem for the interpretation of the macroscopic current time courses was slow-inactivation gating (Hoshi et al., 1991; Lopez-Barneo et al., 1993). In recordings made with 4-8-s voltage pulses to between 53 and +67 mV, macroscopic ionic currents decayed with voltage-independent 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 70-ms 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 70-ms component of the slow inactivation process.

	RESULTS

Top Abstract Introduction Methods Results Discussion References

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 ₁, the limiting rate at large positive potentials _p, and the final opening rate _N. An estimate for the voltage dependence q_d of intermediate steps is also obtained. Similarly, the first and the last two backward rates ₁, _N, and _N-1 are determined, along with an estimate of the "average" rate of intermediate steps _d. In the second group of studies, we characterize the transitions to several channel-closed states that are distinct from those traversed in the depolarization-induced 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

View larger version (5K): [in this window] [in a new window]   Scheme I.

The forward and backward rates _i and _i are taken to be exponential functions of the membrane potential V, scaled by the partial charges q_i and q_i,

(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 _i and _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 Q-V 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 e₀ 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.

View larger version (15K): [in this window] [in a new window]   Fig. 1.   Voltage dependence of relative channel open probability Po and charge movement Q. Relative Po estimates () at V  +67 mV were derived from measurements of ionic currents made using a double pulse protocol, in which currents were measured at a fixed amplitude voltage pulse (at 13 or +7 mV) that followed different test pulses. The values at +107 and +147 mV were obtained by measuring the magnitude of the current relaxation elicited by voltage jumps from +67 mV (in experiments similar to those illustrated in Fig. 12 A). Each plotted value reflects one to eight experiments. Relative Q estimates () were obtained as described previously (Schoppa et al., 1992), by numerically integrating the on gating current at each voltage. Each value reflects three to five experiments. The vertical dashed lines at 20 and 90 mV mark the boundaries of the defined hyperpolarized (V  90 mV), activation (90 mV < V < 20 mV), and depolarized (V  20 mV) voltage ranges.

View larger version (18K): [in this window] [in a new window]   Fig. 12.   Characterization of the transitions to Cf1 and Cf2. (A, top) Macroscopic ionic currents elicited by voltage steps between pairs of depolarized voltages. The smaller current (at both the prepulse and test voltages) reflects a voltage step from +7 to +127 mV, and the larger current reflects a voltage step from +47 to +147 mV. Data were filtered at 15 kHz. Patch w447. (bottom) The same traces are expanded to show just the current relaxation during the test pulse; zero time is the start of the test pulse. The current relaxations are fitted to a single exponential to estimate the time constant r (solid curves). At +127 and +147 mV, the time constants are r = 230 and 180 µs, respectively, and the amplitudes of the fitted exponentials are 32 and 21 pA. These amplitudes can be compared with the size of the rest of the test current (278 and 323 pA, respectively) to yield estimates of the change in Po induced by these voltage steps (10 and 6%). (B) The rate f1 from Cf1 to the open state was estimated by fitting an exponential (solid curve) to the r values () taken from the current relaxations measured in the experiment in A and from similar measurements made in one other patch. (C) Estimates of absolute Po at voltages between +47 and +147 mV were derived from the mean measured channel closed and open dwell times in the equilibrium single channel activity. The data points reflect measurements made in four patches. Absolute Po apparently saturates near 0.9.

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 _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, _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  = _j¹ 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 _j becomes much smaller than the other rates. Useful estimates of the limiting rate and of  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 "n⁴" scheme where the next slowest rate is only twice that of the slowest one, the limiting rate is estimated with 11% error, while  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  is only 20%. Similar deviations between the measurements and theory are obtained from sequential schemes with more transitions. For an "n⁸" scheme, the deviations in the measured and predicted ¹ and  are only 13 and 1%, respectively; for eight equivalent rates, the deviations in ¹ and  are 61 and 13%. These results suggest that fits of Eq. 4 to activation time courses for sequential models can yield reasonable first-pass estimates of the rate-limiting 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.

View larger version (17K): [in this window] [in a new window]   Fig. 2.   Fits of a single exponential function and delay (Eq. 4) to the time course of open probability Po predicted by different models. (A) Fits to the time course of Po (solid curve) from the n4 scheme Note that the order of rate constants in this scheme does not influence the time course. A fit of Eq. 4, taking only points where Po  0.5, is shown as the dotted curve; it yields 1 = 0.89 and  = 1.09. The values expected from the approximate theory, 1 = 1 and  = 1.083, yield the dashed curve. (B) Fits to the time course from the same scheme but with all four rate constants equal to 1. The fit (dotted curve, fitted for Po  0.5) yielded 1 = 0.53 and  = 2.45; the expected values are 1 = 1 and  = 3 (dashed curve). (C) Fits to the time course of Po from Scheme II, in which each of four independent subunits undergoes two transitions, with the forward rate constants a1 and a2 equal to 1 and 4, respectively. The fit (dotted curve, fitted for Po  0.5) yielded 1 = 0.89 and  = 1.38. The expected values are 1 = 1 and  = 1.37 (dashed curve); these were obtained from the mean latency to opening tl that was computed by use of a recursive subroutine as the expectation value over all possible paths in Scheme II of the sum of dwell times in states in each path. (D) Fits to the time course from Scheme II with a1 and a2 both equal to 1. The fit (dotted curve) yielded 1 = 0.71 and  = 2.38; the expected values are 1 = 1 and  = 2.55 (dashed curve).

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,

View larger version (4K): [in this window] [in a new window]   Scheme II.

When this scheme is expanded, it is seen that there are many distinct paths (14 in all) leading from closed state C₀ to the open

View larger version (16K): [in this window]

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 a₁ = 1 and a₂ = 4, yielding t_l = 2.369 time units. The simple theory predicts  to be equal to the reciprocal of the slowest rate, and the delay parameter to be given by = t_l  . A single-exponential fit (dotted curve) yields a value for ¹ that differs by 11% from the slowest rate, and a value of  that differs only 1% from the theoretical value. Fig. 2 D demonstrates the most difficult case, in which a₁ = a₂. 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  is small, <1%. Thus, the parameters of a fitted single-exponential 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 ₁. The forward rate constant 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 single-exponential function (Eq. 4) to the ionic current from the 50% amplitude level to its final value yielded the "activation time constant" _a and the activation delay _a (Fig. 3 A). From the exponential decay of gating currents at depolarized voltages, the time constant _on was determined (Fig. 3 B).

View larger version (29K): [in this window] [in a new window]   Fig. 3.   Estimate of 1. (A) Macroscopic ionic currents at 13 and +37 mV were fitted to a single exponential (smooth curves) to estimate an activation time constant a. Extrapolating the fitted exponential to zero current yielded an estimate of the activation delay a. Patch w312. (B) The decay of WT's on gating currents at the same voltages was fitted to a single exponential (smooth curves) to estimate the decay time constant on. Patch w212. For the fitting, a baseline was first calculated from the mean current measured at the end of the voltage pulse (horizontal lines); the fitting began at the time point at which the current had decayed by 20% from the peak value. (C) The values of a () and on () derived from the fitting have nearly equal values at voltages between 13 and +67 mV. Each data point reflects the average from two to eight experiments. Superimposed curve reflects the average voltage dependence of 1, obtained by fitting the a values between 13 and +67 mV in seven different patches. (D) Exponential fits of the activation time courses and on gating currents for a five-state sequential scheme, as depicted in the legend for Fig. 2 A. For these simulations, the rates for all but one of the transitions was set to be 4; the slowest transition had a rate constant equal to 1. The position of the slowest transition was varied to yield the different curves. The activation time courses (top) were identical for each of the conditions; these were fitted to an exponential (dotted curve), yielding an activation time constant a = 1.04. The gating current time courses for each of the conditions (bottom), however, differed. For C1  C2, C2  C3 = 1, the decay of the currents (dashed curves) was not well described by a single exponential. For C3  C4 = 1, the decay of the current (solid curve) was well described by a single exponential (dotted curve), but the fitted time constant on = 0.69 was faster than a. For C0  C1 = 1, the decay time constant on = 1.09 was similar to a. (E) Exponential fits of the activation time courses and on gating currents for Scheme II, with the forward rate constants a1 and a2 equal to 1 and 4, respectively, or with a1 = 4 and a2 = 1. The two cases yielded identical activation time courses (top), which were fitted to an exponential with a = 1.12 (dotted curve). The two cases, however, yielded different gating currents (bottom). The decay of the current for the case of a1 = 1 and a2 = 4 (solid curve) was well described by a single exponential with a time constant on = 1.05, similar to a. The case of a1 = 4 and a2 = 1, however, yielded a more complex gating current decay time course, which, when approximated by a single exponential (dotted curve), yielded a decay time constant on = 0.51 that was much faster than a. In all simulations, each transition carried an equivalent charge movement.

1

1

Estimate of _p. If one or more transitions have forward rates with smaller voltage dependences than ₁, one of these should be rate limiting at sufficiently high voltages, and should be reflected in the voltage dependence of _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,

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View larger version (28K): [in this window] [in a new window]   Fig. 4.   Estimate of p. (A) The upper half of the macroscopic ionic current time course at +87 and +147 mV was fitted to one or two exponentials (Eq. 6), respectively, to estimate