Large-conductance Ca2+-activated K+ channels can be activated by membrane voltage in the absence of Ca2+ binding, indicating that these channels contain an intrinsic voltage sensor. The properties of this voltage sensor and its relationship to channel activation were examined by studying gating charge movement from mSlo Ca2+-activated K+ channels in the virtual absence of Ca2+ (<1 nM). Charge movement was measured in response to voltage steps or sinusoidal voltage commands. The charge–voltage relationship (Q–V) is shallower and shifted to more negative voltages than the voltage-dependent open probability (G–V). Both ON and OFF gating currents evoked by brief (0.5-ms) voltage pulses appear to decay rapidly (τON = 60 μs at +200 mV, τOFF = 16 μs at −80 mV). However, QOFF increases slowly with pulse duration, indicating that a large fraction of ON charge develops with a time course comparable to that of IK activation. The slow onset of this gating charge prevents its detection as a component of IgON, although it represents ∼40% of the total charge moved at +140 mV. The decay of IgOFF is slowed after depolarizations that open mSlo channels. Yet, the majority of open channel charge relaxation is too rapid to be limited by channel closing. These results can be understood in terms of the allosteric voltage-gating scheme developed in the preceding paper (Horrigan, F.T., J. Cui, and R.W. Aldrich. 1999. J. Gen. Physiol. 114:277–304). The model contains five open (O) and five closed (C) states arranged in parallel, and the kinetic and steady-state properties of mSlo gating currents exhibit multiple components associated with C–C, O–O, and C–O transitions.

Large-conductance Ca2+-activated K+ channels (BK channels)1 are sensitive to membrane potential as well as intracellular calcium. Although the voltage dependence of these channels is weak compared with that of many purely voltage-gated K+ (Kv) channels (Barrett et al. 1982; Cui et al. 1997), voltage gating is likely to be of central mechanistic importance to BK channel function. Sequence similarity between the Slo family of BK channels and Kv channels, including the presence of a charged S4 domain, suggests that the basic structure of the BK channel may resemble that of Kv channels, and that Ca2+ acts to regulate the function of this voltage-dependent “core” (Cui et al. 1997; Meera et al. 1996; Wei et al. 1994). Consistent with this hypothesis, BK channels can be maximally activated by voltage in the absence of Ca2+ (Cox et al. 1997; Cui et al. 1997), gating currents are detected under these conditions (Stefani et al. 1997), and mutations in the S4 region alter the voltage dependence of activation (Diaz et al. 1998; Cui, J., and R.W. Aldrich, manuscript in preparation). Thus, BK channel voltage sensitivity reflects the action of an intrinsic voltage sensor, and Ca2+ is not required for channel opening. Ca2+ shifts many of the voltage-dependent parameters of BK channel gating to more negative voltages (Barrett et al. 1982; Cox et al. 1997). These results are consistent with a model in which Ca2+ binding allosterically regulates voltage-dependent channel activation (Cox et al. 1997; Cui et al. 1997). Such a mechanism implies that voltage-dependent gating plays a critical role in the response of BK channels to both Ca2+ and voltage.

In the preceding article (Horrigan et al. 1999), we examined the response of mSlo Ca2+-activated K+ channels to voltage by recording K+ current in the absence of Ca2+. The kinetic and steady-state properties of mSlo IK indicate that the mechanism of voltage gating is complex. In response to a voltage step, IK activates with an exponential time course after a brief, voltage-dependent delay. The exponential relaxation of IK suggests that a rate-limiting step dominates channel activation. However, the delay indicates that rapid voltage-dependent transitions also exist in the activation pathway. The time constant of IK relaxation (τ(IK)) and steady-state open probability (Po) both exhibit complex voltage dependencies that are inconsistent with many conventional sequential gating schemes. A particularly important finding is that τ(IK) and Po become less voltage dependent at very negative voltages. To account for these results, we proposed a voltage-gating scheme based on an allosteric mechanism.

This diagram illustrates the idea that mSlo channels undergo a transition between a closed (C) and open (O) conformation, and that this transition is influenced allosterically by the state of four independent and identical voltage sensors (one for each subunit). We assume each voltage sensor can undergo a transition between a resting (R) and an activated (A) conformation, and the equilibrium constant for the C–O transition (L) increases by a constant factor (D) for each voltage sensor that is activated. Similarly, the equilibrium constant for voltage sensor activation (J) increases D-fold in favor of the activated state, when the channel opens. Therefore, the factor D embodies the allosteric interaction between voltage-sensor activation and channel opening. This mechanism results in a gating scheme (Scheme I) that contains a parallel arrangement of open and closed states.

(Scheme I)

The horizontal transitions (C–C and O–O) reflect the activation or movement of voltage sensors while vertical (C–O) transitions represent channel opening. The closed and open conformations are each represented by five states, with subscripts (0–4) denoting the number of activated voltage sensors.

For this scheme to reproduce IK, it is necessary that voltage-sensor activation is fast and accounts for most of the channel's voltage dependence while C–O transitions are slow and weakly voltage dependent (Horrigan et al. 1999). Closed-state transitions (C–C) must be fast and voltage dependent (zJ = 0.55 e per voltage sensor) to describe the delay in IK activation. C–O transitions must be slow to limit the exponential relaxation of IK. The weak voltage dependence of τ(IK) and Po at negative voltages implies that the charge associated with channel opening is small (zL = 0.4 e). Finally, the equilibrium constant L is small (∼10−6) and the allosteric factor large (D = 17), equivalent to an interaction energy of 2.8 kT, to account for the shape of the Po–V relationship in 0 Ca2+.

Such a model provides mechanistic insight and places constraints on the possible molecular events that link voltage-sensor movement and channel opening (Horrigan et al. 1999). The allosteric relationship between voltage-sensor activation and channel opening requires that the channels can open with any number of voltage sensors activated, including none. Furthermore, it requires that the allosteric transitions from closed to open alter the energetics of voltage-sensor movement such that voltage sensors, present in each subunit, are more easily activated when the channel is open. This effect is analogous to the change in ligand affinity that occurs between the T and R states in an allosteric ligand binding model (Monod et al. 1965).

Although the properties of mSlo IK are consistent with the allosteric model, several aspects of the gating scheme are not tightly constrained by the ionic current data (Horrigan et al. 1999). Transitions among closed and open states (C–C, O–O) do not immediately alter the number of open channels and, therefore, are not observed directly as a change in IK. Instead, these transitions contribute to the delay in IK activation and to the complex voltage dependence of IK kinetics and steady-state activation. However, any voltage-dependent transition must produce a movement of gating charge that can be detected as gating current (Ig). Gating current provides a direct assay of voltage-sensor movement (C–C, O–O transitions) and, therefore, constrains any voltage-dependent gating scheme. The allosteric model makes specific predictions about the kinetic and steady-state properties of gating charge movement and their relationship to IK. Our experiments examine these predictions and provide a critical test of the model.

Our results are consistent with the assumption that mSlo voltage sensors move rapidly and independently while channels are open or closed. Measurements of the charge associated with voltage-sensor movement are in line with previous estimates based on the ionic current data. Our results also support the prediction that channel opening alters the kinetics of voltage-sensor movement. Finally, we show that some complex kinetic and steady-state properties of mSlo charge movement are reproduced by the proposed gating scheme. These include a large slow component of ON charge that is limited by the speed of channel opening, and three components of OFF charge reflecting C–C, O–O, and C–O transitions. The relationships between these components are consistent with the allosteric model and rule out many alternative schemes.

Channel Expression

Experiments were performed with the mbr5 clone of the mouse homologue of the Slo gene (mSlo), kindly provided by Dr. Larry Salkoff (Washington University School of Medicine, St. Louis, MO). The clone was modified to facilitate mutagenesis and was propagated and cRNA transcribed as described previously by Cox et al. 1997. Xenopus oocytes were injected with ∼50 ng of cRNA (50 nl, 1 ng/nl) 3–7 d before recording. mSlo was also subcloned into a mammalian expression vector (SRα; kindly provided by Dr. A.P. Braun, University of Calgary, Calgary, Alberta, Canada) containing the SV40 promoter. HEK 293 cells expressing the large T-antigen of the SV40 virus were cotransfected with mSlo and green fluorescent protein (GFP, as a marker) using LipofectAMINE (GIBCO BRL) 3 d before recording.

Electrophysiology

Currents were recorded using the patch clamp technique in the inside-out configuration (Hamill et al. 1981). Upon excision, patches were transferred to a separate chamber and washed with at least 20 vol of solution. The internal solution contained (in mM) 135 N-methyl-d-glucamine (NMDG)-MeSO3, 6 NMDG-Cl, 20 HEPES. 40 μM (+)-18-crown-6-tetracarboxylic acid (18C6TA) was added to chelate contaminant Ba2+ (Diaz et al. 1996; Neyton 1996; Cox et al. 1997) unless otherwise indicated. In addition, “0 Ca2+” solutions contained 2 mM EGTA, reducing free Ca2+ to an estimated 0.8 nM in the presence of ∼10 μM contaminant Ca2+ (Cox et al. 1997). These solutions are considered Ca2+-free for the purposes of this study since [Ca2+]i < 50 nM does not affect Slo channel activation (Cui et al. 1997; Meera et al. 1996). Solutions containing 60 μM Ca2+ were buffered with 1 mM HEDTA, and free Ca2+ was measured with a Ca2+ electrode (Orion Research, Inc.). The external (pipette) solution contained 125 tetraethylammonium (TEA)-MeSO3, 2 TEA-Cl, 2 MgCl2, 20 HEPES. pH was adjusted to 7.2. Solutions containing 110 mM K+ were as described in the preceding article (Horrigan et al. 1999). Experiments were performed at room temperature (20–22°C).

Measurement of rapid gating current in response to voltage pulses requires accurate subtraction of linear capacitive currents due to the electrode and cell membrane. Electrodes were pulled from thick-walled 1010 glass (World Precision Instruments) and coated with wax (sticky wax; Kerr) to minimize electrode capacitance (∼1 pF). Pipette access resistance (Rs) ranged from 0.7 to 1.5 MV in K-free solutions. Membrane capacitance ranged from 0.25 to 1 pF as determined by the responses to a −10 mV voltage step from −80 mV before and after sealing the electrode tip onto Sylgard (Dow Corning). Data were acquired with an Axopatch 200B amplifier (Axon Instruments, Inc.) in patch mode at a relatively low gain (1–2 mV/pA) to avoid saturation of capacitive transients in response to voltage steps that often exceeded 300 mV. Both the voltage command and current output were filtered at 20 kHz with 8-pole bessel filters (Frequency Devices, Inc.) to limit the speed of fast capacitive transients so that they could be accurately sampled and subtracted. The Axopatch's internal filter was set at 100 kHz. Currents were sampled at 100 kHz with a 16 bit A/D converter (ITC-16; Instrutech). Ig records were typically signal-averaged in response to at least eight voltage pulses, and a P/−4 protocol was used for leak subtraction (Armstrong and Bezanilla 1974) from a holding potential of −80 mV. A Macintosh-based computer system was used in combination with Pulse Control acquisition software (Herrington and Bookman 1995) and Igor Pro for graphing and data analysis (Wavemetrics, Inc.). A Levenberg-Marquardt algorithm was used to perform nonlinear least squares fits.

Simulations

Simulations were calculated at 1-μs intervals using a fifth order Runga-Kutta algorithm with adaptive step size (Press et al. 1992) implemented in Igor Pro (Wavemetrics, Inc.). Voltage commands and simulated currents were convolved with the impulse response of a 20 kHz 8-pole bessel filter to reproduce the experimental condition (see Horrigan et al. 1999).

Admittance Analysis

Admittance (Y) is defined by the expression Y = I/V where V and I represent the amplitude of the sinusoidal voltage command and resultant current at a specific frequency (ƒ). The admittance of a membrane (Ym) is:

\begin{equation*}Y_{{\mathrm{m}}}=G_{{\mathrm{m}}}+j{\mathrm{{\omega}}}C_{{\mathrm{m}}}\end{equation*}
1

where Gm and Cm are membrane conductance and capacitance, respectively (j=−1, ω = 2πƒ). Gm and Cm each represent the sum of a contribution from the lipid bilayer (Gb, Cb) and gating charge movement (Gg(V), Cg (V)) (see results):

\begin{equation*}G_{m}=G_{b}+G_{g} \left \left({\mathit{V}}\right) \right \end{equation*}
\begin{equation*}C_{m}=C_{b}+C_{g} \left \left(V\right) \right \end{equation*}

The total admittance of the patch equivalent circuit is:

\begin{equation*}Y_{{\mathrm{P}}}=j{\mathrm{{\omega}}}C_{{\mathrm{s}}}+\frac{1}{ \left \left[R_{{\mathrm{s}}}+{1}/{Y_{{\mathrm{m}}}}\right] \right }\end{equation*}
2

where Cs is the stray capacitance of the electrode and holder, and Rs is the series resistance. Combining (1) and (2):

\begin{equation*}Y_{{\mathrm{P}}}=\frac{ \left \left[G_{{\mathrm{m}}}+R_{{\mathrm{s}}} \left \left(G_{{\mathrm{m}}}^{2}+{\mathrm{{\omega}}}^{2}C_{{\mathrm{m}}}^{2}\right) \right \right] \right }{T}+j{\mathrm{{\omega}}} \left \left[\frac{C_{{\mathrm{m}}}}{T}+C_{{\mathrm{s}}}\right] \right \end{equation*}
3

where

\begin{equation*}T= \left \left(1+R_{{\mathrm{s}}}G_{{\mathrm{m}}}\right) \right ^{2}+ \left \left({\mathrm{{\omega}}}C_{{\mathrm{m}}}R_{{\mathrm{s}}}\right) \right ^{2}\end{equation*}

Under typical experimental conditions [RS ≅ 106 , Cm ≅ 1 pF, Gm < 1 nS, ω = 5,451 (ƒ = 868 Hz)], T approaches unity, and this expression can be approximated:

\begin{equation*}Y_{{\mathrm{P}}}= \left \left[G_{{\mathrm{g}}} \left \left(V\right) \right +G_{{\mathrm{b}}}\right] \right +j{\mathrm{{\omega}}} \left \left[C_{{\mathrm{g}}} \left \left(V\right) \right +C_{{\mathrm{b}}}+C_{{\mathrm{s}}}\right] \right \end{equation*}
4

Therefore, Cg(V) can be determined directly as the voltage-dependent component of Yp/ω appearing at a phase angle of 90° relative to the command voltage.

For admittance measurements, the membrane was clamped with a sinusoidal voltage command (60 mV peak to peak) generated by the D/A converter of the ITC-16 interface at 18-μs intervals (at least eight samples per cycle of the sinusoid). The voltage command and current signal were both filtered at 20 kHz. Admittance was determined for each cycle of the sinusoid at 0° and 90° after correcting for phase shifts (ΔΦ) due to the filters and amplifier. These were determined at each frequency by measuring the admittance of an electrode in solution (Y = 1/RS), which should appear at an angle of ΔΦ relative to the command voltage. DC current was determined as the mean current over each cycle of the sinusoid.

Gating charge movement was examined in excised macropatches from Xenopus oocytes and HEK 293 cells expressing the pore-forming α subunit of mSlo Ca2+-activated K channels. Several factors combine to make gating currents more difficult to measure for mSlo than for Kv channels such as Shaker. First, mSlo is less voltage dependent than Shaker, and the gating charge is correspondingly smaller. The steady-state GK–V relationships for mSlo in 0 Ca2+ and Shaker can be approximately fit by Boltzmann functions with equivalent charges of 1 e (Cui et al. 1997) and 5.3e (Zagotta et al. 1994b), respectively. A similar difference is observed based on more sophisticated estimates of total gating charge per channel, ranging from 2.6 to 4.4 e for Slo channels (Horrigan et al. 1999; Stefani et al. 1997) and from 12 to 16e for Shaker (Aggarwal and MacKinnon 1996; Schoppa et al. 1992; Zagotta et al. 1994b). In addition to this 5-fold difference in charge, mSlo exhibits a single channel conductance roughly 10-fold greater than that of Shaker. Thus, the ratio of gating charge to ionic current is almost two orders of magnitude smaller for mSlo than for Shaker. Owing to this relationship, measurement of mSlo gating current (Ig) requires stringent conditions for eliminating IK. For example, in this study, Ig was recorded in the presence of isotonic (125 mM) external TEA, to block the channel pore, even though internal and external solutions were nominally K+-free. Experiments performed in the absence of TEA appear to produce similar results (Stefani et al. 1997) but require extensive washing to assure that residual IK is eliminated. Even when TEA was present at a concentration almost 1,000-fold higher than the Ki for mSlo block (Ki = 0.14 mM; Butler et al. 1993), Ig could not be recorded in the presence of normal internal K+ (110 mM) because IK was not abolished. Finally, mSlo channels activate only at very positive voltages in 0 Ca2+ with a half-activation voltage of +190 mV for GK (Cui et al. 1997; Horrigan et al. 1999). The high voltages and large voltage steps needed to activate these channels often proved problematic for leak-free recording of small gating currents.

Two approaches were used to measure mSlo gating charge movement. The first involved clamping the membrane with a sinusoidal voltage command and measuring gating charge as a voltage-dependent component of membrane capacitance using admittance analysis (Fernandez et al. 1982). The second involved conventional measurement of gating currents in response to voltage steps. Although the bulk of the analysis was performed using voltage steps, the admittance analysis is presented briefly first (see Fig. 1 and Fig. 2) to provide an initial characterization of mSlo charge movement and to demonstrate several necessary controls.

Gating Capacitance Measurements and Admittance Analysis

Membrane capacitance (Cm) represents the ability of charge to redistribute across or within the cell membrane in response to a change in voltage. Therefore, Cm includes a nonlinear voltage-dependent contribution from gating charge movement (Cg) as well as a voltage-independent component due to the lipid bilayer. One of the most sensitive methods for measuring capacitance is admittance analysis. The membrane is driven with a sinusoidal voltage command, and the resulting current is analyzed with a phase-sensitive detector to determine Cm as well as other parameters in the membrane equivalent circuit (Gillis 1995; Lindau and Neher 1988). One advantage of this technique is that residual ionic and “leak” currents can be separated from capacitive (gating) currents based on their phase relative to the command voltage. A related advantage is that measurements can be acquired rapidly without the need for leak-subtraction protocols.

Gating capacitance (Cg) represents the amount of gating charge that moves (ΔQg) in response to a small change in voltage (ΔV) and therefore reflects the slope of the Qg–V relationship (ΔQg/ΔV). Cg is also dependent on the kinetics of charge movement and is therefore sensitive to the frequency (ƒ) of the sinusoidal voltage command. Thus, capacitance measurements provide an assay of gating charge mobility reflecting both voltage-dependent and kinetic properties. When measured in response to a small amplitude, low frequency voltage perturbation Cg approximates the derivative of the steady-state Q–V relationship (Cg(V) = dQss/dV) (Fernandez et al. 1982; Fernandez et al. 1983; Taylor and Bezanilla 1979). Thus, if the Q–V can be described by a Boltzmann function Qss = [1 + exp(−ze(VVh)/kT)]−1 then Cg should exhibit a bell-shaped voltage dependence described by the derivative of a Boltzmann function. This relationship between Cg(V) and Qss(V) is strictly valid only when Cg is measured at a frequency approaching zero (Cg0). However, as discussed below, useful information about mSlo charge movement can be obtained using relatively large amplitude (30 mV) sinusoidal voltage commands at frequencies of hundreds or thousands of Hz.

Fig. 1 A shows the Cg–V relationship for mSlo measured at 868 Hz in 0 Ca2+ (see Materials and Methods) from channels expressed in an excised macropatch. Cg exhibits a bell-shaped voltage dependence and is well fit by the derivative of a Boltzmann function (Fig. 1 A, Fit). In nontransfected cells, the C–V relationship is flat (Fig. 1 A, Control), representing only the uncompensated linear capacitance of the lipid bilayer and electrode (Co = Cb + Cs; see ). These contributions to the record in Fig. 1 A were effectively eliminated by setting the baseline equal to zero at negative voltages (less than −100 mV) where C is voltage independent and presumed equal to Co. The Qg–V relationship was obtained by integrating the Cg–V trace with respect to voltage, and is plotted in Fig. 1 B together with the normalized conductance–voltage (GK–V) relationship for mSlo in 0 Ca2+ (mean ± SEM, n = 23 [Horrigan et al. 1999]). The Qg–V relationship is fit by a Boltzmann function (dashed line) characterized by a half-activation voltage (Vh) of 133 mV, corresponding to the peak voltage of the Cg–V curve, and an equivalent charge (z) of 0.59 e. Fits to Cg–V relationships obtained at 868 Hz from many experiments yielded values of Vh = 127.4 ± 3.4 mV and z = 0.61 ± 0.014 e (mean ± SEM, n = 15) in 0 Ca2+. These parameters are similar to those estimated in the preceding paper to characterize the charge and voltage dependence of mSlo voltage sensors (Vh(J) = 145 mV and zJ = 0.55 e [Horrigan et al., 1999]). The GK–V relationship is steeper and shifted to more positive voltages than the Qg–V, and can be approximated by a Boltzmann function (from the Qg–V fit) raised to the 4th power as shown in Fig. 1 B.

The Cg–V relationship in Fig. 1 A was measured during a 1-s voltage ramp from −160 to +200 mV. A sinusoidal command (ƒ = 868 Hz) was superimposed on the ramp as indicated in Fig. 1 C, and patch admittance (Yp) was measured for each cycle of the sinusoid using a phase-sensitive detector implemented in software (Herrington and Bookman 1995). Patch capacitance (Cp) was determined based on the expression:

\begin{equation*}Y_{{\mathrm{p}}}=G_{{\mathrm{p}}}+j{\mathrm{{\omega}}}C_{{\mathrm{p}}}\end{equation*}
5

where the real and imaginary terms represent the components of Yp appearing at phase angles of 0° and 90°, respectively, relative to the command voltage, and ω = 2πƒ. Cg was then defined as the voltage-dependent component of Cp [Cg(V) = Cp(V) − Co]. The Cg–V relationship was unaffected by the polarity of the voltage ramp (data not shown), indicating that a pseudo steady-state condition was achieved at each voltage. Because the amount of gating charge detected was small (1–30 fC), admittance was typically measured using a relatively large amplitude 30-mV sinusoidal voltage command (60 mV peak to peak) to increase the signal to noise ratio. We were concerned that such a perturbation might alter the shape of the Cg–V relationship relative to that obtained with a small amplitude command. However, reduction of the sinwave amplitude from 30 to 3 mV had no detectable effect on the Cg–V relationship (Fig. 1 D). This result suggests that the Cg–V was not distorted by the size of the sinusoidal command and is consistent with the weak voltage dependence of mSlo channel gating.

Gating Capacitance Represents mSlo Charge Movement

Although a voltage-dependent component of Cm was not detected in uninjected oocytes, it is important to verify that Cg arises from mSlo channels. High levels of heterologous expression of many membrane proteins in Xenopus oocytes have been shown to upregulate expression of endogenous ion channels (Tzounopoulos et al. 1995). It is conceivable that such endogenous channels could contribute to gating charge movement in cells expressing mSlo. However, several lines of evidence argue against such a contribution. First, similar Cg signals were observed using two different expression systems, Xenopus oocytes (Fig. 1 D and 2 C) and HEK 293 cells (Fig. 1 A and 2 A). Furthermore, Cg is Ca2+-sensitive and can be altered by mutating the mSlo channel.

The Ca2+ sensitivity of Cg is examined in Fig. 2 A. Cg–V traces obtained in 0 or 60 μM Ca2+ from the same patch were normalized to peak capacitance and superimposed. The Cg–V relationships are similar in shape but shift to more negative voltages with increasing [Ca2+]i. The GK–V relationship for mSlo also exhibits a negative voltage shift upon application of Ca2+ in this concentration range (Cui et al. 1997).

The Ca2+ sensitivity of Cg suggests that this signal represents mSlo charge movement but does not rule out contributions from endogenous Ca2+-sensitive channels. To eliminate this possibility, we examined the properties of an mSlo mutant. Neutralization of a charged residue in the S4 domain of mSlo (R207Q) shifts the GK–V relationship to more negative voltages and reduces its slope relative to that of the wild-type (Diaz et al. 1998; Horrigan et al. 1999; Cui, J., and R.W. Aldrich, manuscript in preparation). We showed in the preceding paper that these shifts in the G–V can be accounted for by the allosteric voltage-gating scheme if the mutation allows voltage sensors to activate at more negative voltages without altering their charge. Consistent with this hypothesis, the Cg–V relationship for R207Q is approximately the same shape as that for mSlo but is shifted by almost −250 mV (Fig. 2 B). This result also confirms that the Cg signal reflects the gating of mSlo.

The Kinetics of mSlo Gating Charge Movement

To assess the speed of charge movement, we examined the frequency dependence of Cg. In the simplest case, where gating charge movement can be represented by a two-state process, such as voltage-sensor activation from R to A, gating admittance

\begin{equation*}Y_{{\mathrm{g}}}=G_{{\mathrm{g}}} \left \left({\mathrm{{\omega}}}\right) \right +j{\mathrm{{\omega}}}C_{{\mathrm{g}}} \left \left({\mathrm{{\omega}}}\right) \right \end{equation*}
6

can be represented by an equivalent circuit consisting of a capacitor Cg0 in series with a resistor Rg0 where τg = Cg0Rg0 is the time constant of gating charge relaxation at a particular voltage (Fernandez et al. 1982; Taylor and Bezanilla 1979). Where

\begin{equation*}C_{{\mathrm{g}}} \left \left({\mathrm{{\omega}}}\right) \right =C_{{\mathrm{g}}0^{\prime}} \left \left[\frac{1}{1+ \left \left({\mathrm{{\omega}{\tau}}}\right) \right ^{2}}\right] \right \end{equation*}
7
\begin{equation*}G_{{\mathrm{g}}} \left \left({\mathrm{{\tau}}}\right) \right =\frac{C_{{\mathrm{g}}0}}{{\mathrm{{\tau}}}} \left \left[\frac{ \left \left({\mathrm{{\omega}{\tau}}}\right) \right ^{2}}{1+ \left \left({\mathrm{{\omega}{\tau}}}\right) \right ^{2}}\right] \right \end{equation*}
8

When the frequency of the sinusoidal voltage command is low (ω << 1/τ), Yg reduces to a purely capacitive signal (Yg ≅ jωCg0), where Cg0 = dQss/dV. As ω increases, Cg should be attenuated, since the gating charge effectively cannot move fast enough to keep up with the voltage command. Cg(ω) is a Lorenzian function that describes the frequency dependence of Cg. At higher frequencies, the gating current should also change phase with respect to the voltage command such that a component of Yg, described by the function Gg(ω), appears in phase with the membrane voltage.

The frequency dependence of mSlo charge movement is shown in Fig. 2C–F. Consistent with the above predictions, the Cg–V relationship is attenuated as the frequency of the sinusoidal voltage command is increased from 200 to 6,944 Hz (Fig. 2 C). At the same time, a voltage-dependent signal appears in the orthogonal G trace and increases at higher frequencies (Fig. 2 D). The DC current level during the voltage ramp is small and increases in a roughly linear manner with voltage, indicating a constant membrane resistance (Rm) of ∼125 GV (Fig. 2 E). Thus, the G signal in Fig. 2 D represents a component of gating charge movement (Gg) and not a voltage-dependent change in membrane conductance. Gg is almost eliminated at 200 Hz, consistent with the prediction that Yg will reduce to a purely capacitive signal at low frequencies. Fig. 2 F plots the amplitudes of Cg and Gg measured at +120 mV versus frequency for two experiments. Cg and Gg are well fit by and , respectively, with a time constant (τ) of 70 μs. The relative amplitudes of the admittance components are also consistent with a 70-μs time constant, since Cg and Gg were normalized by Cg0 and Cg0/τ, respectively. Thus, a component of mSlo gating charge appears to move much faster than IK activation, which is described by a mean time constant of 1.63 ms at +120 mV (Horrigan et al. 1999). We will demonstrate below that an additional component of gating charge moves with the time course of channel activation but is too slow to be detected with admittance analysis.

Conclusions from Capacitance Measurements

Admittance analysis reveals several important properties of mSlo charge movement. Comparison of the Qg–V and GK–V relationships (Fig. 1 B) suggests that charge movement can occur at voltages where most channels are closed. The frequency dependence of Cg shows that charge relaxes with a time constant that is much faster than that of IK activation (Fig. 2 F). Together, these results suggest that admittance analysis detects charge movement associated with rapid closed-state transitions that precede channel opening. In terms of the allosteric voltage-gating scheme, such transitions result from voltage-sensor movement. That the Qg–V relationship can be fit by a Boltzmann function is consistent with the movement of each voltage sensor being described by a two-state model with a single transition between a resting (R) and an activated state (A). The simple voltage dependence of Qg also supports the notion that the voltage sensors, in different subunits of the mSlo homotetramer, behave identically and act independently. The approximate 4th power relationship between Qg–V and G–V is consistent with the assumption that channel opening is linked to the activation of four voltage sensors. However, as discussed below, this relationship may be affected by the different ionic conditions under which IK and Ig are measured.

Advantages and Limitations of Admittance Analysis

Our results show that admittance analysis provides a sensitive method for detecting and characterizing some aspects of mSlo gating charge movement. By using a large amplitude sinusoidal voltage command in combination with a voltage ramp, we were able to acquire the Cg–V relationship rapidly, and to determine Qg(V) at submillivolt intervals. The speed of mSlo charge movement is advantageous for admittance analysis because it allows measurements to be performed at hundreds or thousands of Hz where the signal to noise ratio is high (Gillis 1995; Lindau and Neher 1988; Lollike et al. 1995). By the same token, as discussed below, this technique is not well suited to detecting slow components of charge movement and may present difficulties in dissecting complex kinetic behavior.

The charge movement detected with capacitance measurements is much faster than IK activation. However, any scheme that assumes the C–O conformational change is voltage dependent or that channel opening affects the ability of voltage sensors to move requires that a component of gating charge will relax with the kinetics of IK activation. The frequency dependence of Cg (Fig. 2 F) can be adequately fit by a single Lorenzian function between 200 and 7,000 Hz and therefore provides no evidence for a slow component of gating charge, which should appear as an additional Lorenzian component at low frequencies. However, the frequency range of our measurements may limit our ability to detect such components. For example, charge that moves with a time constant of 2 ms would produce a Cg component that is attenuated by ∼85% at frequencies >200 Hz.

Admittance analysis is also not an ideal method for dissecting a model as complex as the one we have proposed for mSlo. The allosteric scheme predicts that multiple kinetic components of charge movement will result from C–C, C–O, and O–O transitions. Admittance analysis detects charge movement associated with perturbations about an equilibrium distribution of channel states, and will therefore contain contributions from all of these sources. Slow transitions associated with channel opening should contribute little to Cg at the frequencies used in our experiments. However, fast transitions among closed or open states (C–C, O–O) should be detected. At voltages less than +100 mV, most channels are closed in 0 Ca2+, and Cg will mainly reflect C–C transitions. However, at more positive voltages, Cg should represent a combination of open- and closed-state charge movement. For this reason, gating currents measured in response to step depolarizations provide a better method for isolating the various transitions predicted by the model.

A Fast Component of Gating Current

Fig. 3 A shows Ig evoked in response to a 0.5-ms pulse to +160 mV from a holding potential of −80 mV in 0 Ca2+. The ON current decays rapidly with a time course that is well fit by an exponential function (dashed line) with a time constant of 59 μs, similar to that determined with admittance analysis at +120 mV (70 μs). The OFF current measured at −80 mV decays more quickly, with a time constant of 17 μs. A family of Ig evoked at different voltages (0 to +160 mV) in response to 1-ms pulses is shown in Fig. 3 B. The QON–V and QOFF–V relationships obtained by integrating IgON and IgOFF are plotted in Fig. 3 C (open symbols) together with the Qg–V relationship obtained from capacitance measurements at 868 Hz in the same patch (solid line). At all voltages, QON and QOFF are equivalent, as expected for gating charge. The gating current and capacitance measurements superimpose from 0 to +120 mV but diverge at +160 mV.

Similar results were obtained with brief voltage pulses and capacitance measurements because both methods mainly detect fast charge movement. Fig. 3 D compares the time course of Ig evoked at +160 mV to the initial activation of IK measured at the same voltage from a different experiment. Ig decays, to a large extent, before IK begins to increase. After 1 ms, IK increases to 31% of its steady-state amplitude, representing only 7% of maximum Po. Thus, the channel does not achieve a steady state during a 1-ms pulse, and Ig should reflect little if any slow charge movement that might be associated with channel opening. An important difference between the gating current and capacitance measurements is that the initial decay of IgON represents charge moved when most channels are closed, while Cg is measured after Po has reached a steady state and therefore reflects the behavior of both open and closed channels. Thus, Ig measurements allow better isolation of closed-state transitions owing to the large kinetic difference between IgON and IK.

Fast Ig: Isolation and Voltage Dependence

According to the allosteric model, the initial decay of Ig represents activation of voltage sensors from a resting (R) to an activated state (A) while channels are closed (i.e., C–C transitions). The exponential decay of IgON is consistent with such a two-state model. Moreover, in Fig. 3 D, IK achieves an exponential time course (dashed line) at a time (arrows) when the gating current has almost completely decayed. This correlation between Ig and the delay in IK activation is consistent with Ig reflecting closed-state transitions in the activation pathway. However, QON measured during a 1-ms pulse is not only an assay of closed-state charge movement, as some channels do open during this time (Fig. 3 D). QON measurements can also be contaminated by outward leak currents that often are observed at voltages greater than +200 mV. To better characterize closed-state transitions, the fast component of ON charge was isolated by fitting an exponential function to the decay of Ig during the first 200 μs of the voltage pulse when most channels are closed. The area under the fit (Qfast), as indicated by the shaded region in Fig. 3 D, was used as an estimate of closed-state charge movement (QC). The Qfast–V relation in Fig. 3 C (filled symbols) is similar to the Qg–V relation. When data were acquired over a larger voltage range (Fig. 3 E), Qg and Qfast diverge at more positive voltages (Fig. 3 F). The Qfast–V relationship in Fig. 3 F is well fit by a Boltzmann function (z = 0.57 e, Vh = 136 mV). The difference between Qg and Qfast is expected, as it occurs at voltages (>100 mV) where channels begin to open, and Qg therefore cannot be equivalent to QC.

Fig. 4 A1 plots the normalized Qfast–V relationships for many experiments. The data were initially fit with Boltzmann functions where all parameters were allowed to vary, yielding a mean equivalent charge <z> = 0.59 ± 0.03 e (mean ± SEM, n = 10). The Q–Vs were then refit with z = <z> and normalized as shown in Fig. 4 A1. Although the individual plots are reasonably fit using identical values of z, they are scattered in their position along the voltage axis, similar to the mSlo GK–V relationships (Horrigan et al. 1999). To compare the shapes of the Q–Vs, the individual records were aligned as shown in Fig. 4 A2 (open symbols) by shifting them along the voltage axis by ΔV = 〈Vh〉 −Vh where Vh is the half-activation voltage of an individual Q–V and 〈Vh〉 is the mean (155 ± 6.5 mV, n = 10) determined from Fig. 4 A1. These voltage-shifted plots were then used to determine the average Q–V (Fig. 4 A2, filled symbols). A Boltzmann function with z = 0.59 e and Vh = 155 is superimposed on the data (solid line).

Voltage Dependence of Fast Ig Kinetics

To further characterize the properties of closed-channel charge movement, we examined the voltage dependence of fast Ig kinetics. Time constants of fast Ig relaxation (τgFast) were determined from exponential fits to ON and OFF currents for the experiments in Fig. 4 A1 and are plotted in Fig. 4 B1. OFF currents, measured at voltages less than +40 mV, were evoked after very brief pulses (0.05–0.25 ms) to +160 or +200 mV and therefore should represent the relaxation of closed channels. τgFast exhibits a bell-shaped voltage dependence, consistent with a two-state model of voltage-sensor activation where forward and backward rate constants are voltage dependent. τgFast–V relationships from three experiments that covered a large voltage range are compared in Fig. 4 C1. The individual plots are similar in shape but shifted relative to each other along both axes. The amplitude differences resemble those described previously for the delay in IK activation (Horrigan et al. 1999) and may reflect temperature variation between experiments conducted at room temperature. To better compare the shape of the τgFast–Vs, the plots were first shifted along the voltage axis based on the Q–V shifts determined in Fig. 4 A. The data were then normalized to the mean τgFast determined over an interval around the peak of the τgFast–V (59.0 ± 2.2 μs, n = 10, from +100 to +180 mV). The resulting records, corresponding to Fig. 4, B1 and C1, are plotted in Fig. 4, B2 and C2, respectively, and exhibit improved alignment of the τgFast–V relationships.

The data in Fig. 4 B2 were fit with a function τgFast = 1/(α+β), representing the predicted τgFast–V relationship for a two-state process where the forward (α) and backward (β) rate constants are exponential functions of voltage [α=α0ez α e kt ,β=β0ez β e kt ]. Fits were constrained such that the equilibrium constant J = α/β equals one at the half-activation voltage of the Qfast–V (Vh(J) = 155 mV). The solid line in Fig. 4 B2 represents the best fit and is characterized by a total equivalent charge of zJ = 0.51e (zα = +0.30 e, zβ = −0.21 e).

Estimates of the charge associated with voltage-sensor activation (zJ) based on fits to the Qfast–V and τgFast–V relationships (0.59 and 0.51 e, respectively) apparently differ. However, both relationships can be reasonably fit using the average of these two estimates (0.55 e) (Fig. 4, A2, B2, and C2; dashed lines). This value of zJ was also used in the preceding article to reproduce the ionic current data using the allosteric voltage-gating scheme (Horrigan et al. 1999). One difference is that the value of Vh(J) used to fit the Qfast–V (155 mV) is 10 mV greater than that previously used to fit IK. In addition, the values of zα and zβ used to fit the τgFast–V relationship (zα = +0.33 e, zβ = −0.22 e) indicate that the R–A transition in the allosteric model is not symmetrically voltage dependent as previously assumed.

A Slow Component of ON Charge Movement

Although the ON currents in Fig. 3 appear to decay with a single-exponential time course, there is a significant slow component of charge movement. Fig. 5 A plots a family of Ig evoked at +140 mV in response to voltage pulses of different duration (see also Fig. 6 A). The peak amplitude of IgOFF increases rapidly with pulse duration, paralleling the rapid decay of IgON, and then remains relatively constant for pulses longer than 0.5 ms. The total gating charge moved during the pulse (Qp) was determined by integrating IgOFF and is plotted versus pulse duration in Fig. 5 B. Qp increases with a time course that can be fit by a double-exponential function (solid line) with a fast phase (QpFast) corresponding to the rapid decay of IgON, and an additional phase that is roughly 100-fold slower. The slow component (QpSlow) relaxes with a time constant (τgSlow) of 4.22 ms and represents a significant fraction of the total gating charge movement at +140 mV (43%) but is too slow to be observed as a component of IgON. This point is illustrated in Fig. 5 C, which compares IgON evoked at +140 mV to Qp′(t) (dashed line). Qp′(t) is the time derivative of the double-exponential fit to Qp(t) and should represent the time course of IgON (Qp′(t) = dQON/dt = IgON). These two relationships superimpose, demonstrating that observed IgON kinetics are consistent with the presence of a large slow component of ON charge movement.

The predicted amplitude of the slow component of IgON, determined from Qp′(t), is small (2.1 pA) because it decays slowly. For similar reasons, the slow component of ON charge could not be reliably measured from IgON. Small sustained outward currents on the order of a few pA were often observed at high voltages, presumably representing residual ionic or leak current. For example, the current trace in Fig. 5 C decays to a steady-state level of 2.2 pA at the end of the pulse. While such small currents have little effect on measurement of Qfast they can contaminate estimates of slow charge determined by integrating IgON over a 20-ms pulse. The slow component of Qp from Fig. 5 B is only 8 fC, equivalent to a 0.4 pA current for 20 ms. Measurements of OFF charge (Qp) provide a more reliable estimate of slow charge movement because leak is constant at the holding potential.

The voltage dependence of Qp(t) is examined in Fig. 6. Families of Ig evoked at different voltages in response to pulses of 0.06–20 ms duration are shown in Fig. 6 A. At each voltage, Qp was plotted versus pulse duration (Fig. 6 B) and fit with a double-exponential function as in Fig. 5 B. The plots represent data from three experiments and were normalized to the total fast charge movement QTfast estimated from the amplitude of a Boltzmann fit to the Qfast–V relationship for each experiment. The indicated voltages were corrected for shifts in the Qfast–V relationship as determined in Fig. 4 A.

A slow component of Qp is observed in Fig. 6 B for V ≥ +100 mV. The time constant of QpSlowgSlow) is comparable to that for IK activation (τ(IK)) measured from +140 to +240 mV (Fig. 6 C). The similar magnitude and voltage dependence of τgSlow and τ(IK) suggest that slow charge movement is limited by channel opening. These kinetics also show that gating charge and open probability equilibrate on a similar time scale. Therefore, QOFF determined with a 1-ms voltage pulse, as in Fig. 3 C, can underestimate steady-state QOFF (Qss), determined with a 20-ms pulse, by as much as 40%. Despite this difference, the Qss–V and Qfast–V relationships are similar in shape. Fig. 6 D compares normalized Qss–Vs from four experiments to the normalized Qfast–V and GK–V relationships. Qss–V almost superimposes with Qfast–V, and the steady-state data were fit with Boltzmann functions with an equivalent charge z = 0.65 ± 0.03 e (mean ± SEM, n = 4), indicating a slightly steeper voltage dependence than Qfast.

The Relationship between Slow Charge Movement and Channel Activation

The predominantly exponential time course of mSlo IK suggests that the kinetics of voltage-dependent activation are dominated by a rate-limiting transition (Horrigan et al. 1999). The similar kinetics of QpSlow and IK relaxation implies that slow gating charge movement also reflects this rate-limiting step. It is important to distinguish between two possible sources of slow charge movement. First, the rate-limiting step may represent a voltage-dependent conformational change and therefore contribute directly to QpSlow. Second, the rate-limiting step may contribute indirectly to QpSlow by limiting the speed of other voltage-dependent transitions in the activation pathway. The data suggest that both of these mechanisms contribute to slow charge movement in mSlo.

We have previously concluded that the transition from a closed to open conformation represents the rate-limiting step in mSlo activation and is weakly voltage dependent (Horrigan et al. 1999). Hence, the rate-limiting step should contribute directly to slow charge movement. However, the charge associated with the C–O transition (zL = 0.4 e) was estimated to represent only 15% of the total charge per channel. In contrast, slow charge movement in mSlo can represent >40% of the total ON charge (Fig. 6 B). These results are inconsistent with the idea that QpSlow merely represents the charge moved during the C–O transition, but they can be understood in terms of the allosteric voltage-gating scheme (Scheme I).

The allosteric model predicts that the majority of charge movement can be attributed to voltage-sensor activation. Fast Ig is evoked in response to a voltage step as sensors initially equilibrate between resting (R) and activated (A) conformations while the channel is closed. Qfast is determined by the voltage-dependent equilibrium constant (J) that characterizes the R–A transition. In addition, a slow component of charge movement should be produced as channels open, representing the C–O transition. However, voltage-sensor movement can also contribute to QpSlow. When a channel opens, the equilibrium constant for voltage-sensor activation increases by the allosteric factor D, causing sensors to reequilibrate between R and A and produce additional charge movement. This charge movement will be slow because the voltage-sensor reequilibration is limited by the speed of channel opening.

The amplitude of QpSlow should depend upon the number of channels that open as well as the fraction of voltage sensors that are initially activated before channels open. For example, at very positive voltages (approximately +300 mV), the model predicts that voltage sensors can be completely activated with channels closed. In this case, channel opening cannot cause additional voltage sensors to be activated so QpSlow will represent only the charge associated with the C–O transition (zL). At less positive voltages, however, QpSlow will represent a combination of channel opening and voltage-sensor reequilibration and may therefore be larger than zL. We will demonstrate later that the magnitude and voltage dependence of QpSlow are consistent with the allosteric gating scheme (Scheme I). The notion that the C–O transition limits slow charge movement is also important in understanding the properties of IgOFF as discussed below.

Three Components of OFF Gating Charge Movement

The large slow component of Qp(t) observed at V ≥ +140 mV in Fig. 6 B indicates that QOFF increases with pulse duration. In contrast, the peak amplitude of IgOFF remains roughly constant or decreases with pulse duration at the same voltages (Fig. 6 A). That IgOFF can decrease or remain constant while its integral (QOFF) increases implies that the kinetics of OFF current change with pulse duration. This change is obvious in Fig. 7 A, which compares OFF currents evoked at −100 mV after pulses to +140 mV of different duration (0.06–20 ms). Two components of IgOFF are evident from these records. After brief pulses (0.06 or 0.11 ms), OFF current decays with a rapid exponential time course, but an additional slower component appears as pulse duration is increased. The decay of IgOFF at all pulse durations can be well fit by double-exponential functions with time constants of 15.5 and 59 μs (Fig. 7 B). Both components decay within 300 μs and therefore appear to be fast relative to the time course of channel closing. Potassium tail currents decay with a time constant of 172 ± 15 μs at −80 mV (Horrigan et al. 1999) and therefore require approximately 5τ(IK) = 900 μs to decay completely. However, a slower component of OFF charge movement can be detected by plotting the integral of IgOFF (QOFF(t); Fig. 7 C). QOFF(t) measured after a brief (0.06 ms) voltage pulse achieves a steady state within 300 μs (Fig. 7 C, arrow), consistent with the rapid decay of IgOFF. In contrast, QOFF(t) measured after a 20-ms pulse requires >1 ms to reach a steady state, indicating a slow component of charge relaxation. This component of QOFF is not evident in the corresponding IgOFF trace because it is slow and represents <20% of the total OFF charge.

The components of QOFF(t) relaxation were further analyzed by plotting the quantity (QOFF(t) − QOFFss) where QOFFss is the steady-state value of QOFF(t) measured 3 ms after the voltage pulse (Fig. 7 D). The relaxation of (QOFF(t)− QOFFss) after a brief pulse (0.06 ms) can be fit by a single-exponential function as indicated by a linear relationship on this semilog plot (τF = 15.5 μs). The relaxation of (QOFF(t) − QOFFss) after a prolonged pulse (average of 10–20-ms records) is more complicated and was best fit by three exponential components (τF = 15.5 μs, τM = 59 μs, τS = 448 μs), indicated by dashed lines in Fig. 7 D, where τF was constrained to that used to fit the 0.06-ms record. On average, time constants of 15.7 ± 1.3, 64.7 ± 10.6, and 580 ± 50 μs were measured at −80 mV (mean ± SEM, n = 6).

The time course of development of the OFF charge components were examined by fitting (QOFF(t)− QOFFss) with triple-exponential functions for all pulse durations (Fig. 7 E). The time constants (termed Fast, Medium, and Slow) were determined from the 0.06- and 10–20-ms traces as in Fig. 7 D, and component amplitudes were varied to fit the other records. The QOFF component amplitudes (QOFFfast, QOFFmed, and QOFFslow) are plotted versus pulse duration in Fig. 7 F. The Fast component develops rapidly and then slowly decreases in amplitude as pulse duration is increased. At the same time, a parallel increase in the Medium and Slow components is observed. The slow relaxations in the development of all three components were fit by exponential functions (solid lines) with a time constant of 4.2 ms. This time constant is identical to that used to fit QpSlow (Fig. 5 B) and is therefore assumed to represent the time course of channel opening. As discussed below, the results in Fig. 7 F suggest that the Fast component of OFF charge movement represents the relaxation of closed channels, while the Medium and Slow components represent the relaxation of open channels.

Factors Influencing OFF Component Characterization

Accurate separation of QOFF components depends on several factors, including the estimation of their time constants. τF is most easily determined because the fast component is large and can be examined in isolation using brief voltage pulses. The Slow component can also be effectively isolated because τS is almost 10-fold larger than τM. However, the small amplitude of the Slow component and its sensitivity to baseline drift make τS more difficult to determine than τF. The relaxation of QOFF(t) to a steady state in Fig. 7 C indicates that IgOFF decays to the baseline level after ∼1 ms. A small offset or drift in baseline current can prevent QOFF(t) from achieving such a steady state and affects determination of τS and QOFFslow. To minimize such artifacts, the Ig baseline was typically set equal to the mean current measured during an interval 4–5 ms after the end of the pulse. Despite this precaution, drift in QOFFss was observed in some experiments (data not shown) and contributes to variability in the estimate of τS.

The medium time constant (τM) was also difficult to determine because it is only fourfold slower than τF and cannot be studied under conditions where the Fast and Slow components are absent. Thus, estimates of τM from triple exponential fits to QOFF relaxation were sensitive to the estimates of τF and τS. Error bars in Fig. 7 F indicate the effect of ±10% changes in τM on the estimated amplitudes of the different OFF components (with τF and τS held constant). Such variation still allows reasonable fits to QOFF(t) (data not shown); however, an increase in τM results in a decrease in the measured QOFFmed and a complimentary increase in QOFFfast. Larger changes in τM produce inadequate fits to QOFF(t), and the time course of the Medium component development becomes biphasic as the separation of Fast and Medium components is compromised.

Measurements of QOFFmed can be affected by baseline drift or variation in τS. Therefore, the development of Fast and Medium components were also studied by fitting IgOFF with double-exponential functions (Fig. 7 B), a procedure that is less sensitive to the slow component. Fig. 7 G plots the amplitude of the IgOFF components versus pulse duration, indicating a time course of Fast and Medium component development similar to that determined from QOFF (Fig. 7 F). Exponential fits in Fig. 7F and Fig. G, used identical values of τgSlow. However, in experiments where baseline drift was a problem, fits to IgOFF produced more consistent results and were used to determine τgSlow.

mSlo Charge Movement and Allosteric Voltage Gating

The presence of three components of OFF gating charge movement, their kinetics, and development with pulse duration can be understood in terms of the allosteric voltage-gating scheme (Scheme I). As indicated in Fig. 7 H, the allosteric model predicts that OFF charge relaxation will be characterized by Fast, Medium, and Slow components that reflect C–C, O–O, and O–C transitions, respectively. When mSlo channels are closed, OFF currents should represent the relaxation of voltage sensors from an activated to a resting state, corresponding to C–C transitions in the gating scheme. Since brief voltage pulses allow few channels to open, the fast relaxation of IgOFF after such a pulse (τF) mainly reflects the kinetics of this closed-state relaxation pathway. As pulse duration is increased, channels open and their deactivation after the pulse reflects a more complex relaxation pathway involving O–O and O–C transitions. The model predicts that voltage sensors can move even when channels are open. Therefore, the OFF current should exhibit a component that reflects relaxation of voltage sensors from an activated to a resting state, corresponding to O–O transitions in the gating scheme. If these open-state transitions account for the Medium QOFF component, to account for the difference between τM and τF, we must assume that voltage-sensor relaxation is slower when the channel is open than when it is closed. This is a reasonable assumption because the allosteric mechanism requires that channel opening increase the equilibrium constant for voltage-sensor movement D-fold, stabilizing the activated state (A) relative to the resting state (R). Finally, the model predicts that there will be a slow component of OFF charge movement associated with the transition of open channels back to the closed state. Therefore, the slow component should have the same time course as channel deactivation. We will argue later that differences in the observed time course of IK deactivation and slow charge movement (τS) may reflect effects of ionic conditions on channel gating.

If the Fast component of OFF charge movement represents the relaxation of closed channels while Medium and Slow components represent the relaxation of open channels, the effect of pulse duration on the relative amplitude of these components can be understood in terms of the kinetics of channel activation. QOFFfast increases initially because voltage sensors can be activated rapidly during brief pulses while channels are closed. As pulse duration increases, the number of closed channels is reduced and QOFFfast decreases with the time course of channel activation. At the same time, both Medium and Slow components increase, reflecting an increase in the number of open channels.

Charge Movement Measurements Are Not Contaminated by Ionic Currents

An important conclusion from the above analysis is that the slow components of ON and OFF charge movement are limited by channel opening and closing. Since the kinetics of these components are similar to those of IK, it is critical to establish that they do not represent contamination of Ig by residual ionic currents. The slow component of ON charge movement was detected as an increase in QOFF measured after pulses of different duration, whereas the slow OFF charge was seen as a component of QOFF relaxation. Thus, the presence of an inward potassium tail current could potentially contribute to both measurements. This possibility appears unlikely because gating current records that give rise to large slow components of ON charge movement (e.g., Fig. 5 and Fig. 6) do not exhibit appreciable sustained (ionic) current during the voltage pulse. In addition, the slow increase in Qp with pulse duration involves simultaneous changes in the amplitudes of all three components of QOFF relaxation. The Fast component decreases while the Medium and Slow components increase (Fig. 7 F). Although the Slow component relaxes with kinetics similar to that of ionic tail currents, it accounts for only a small fraction of QpSlow. Finally, as discussed below, the relative amplitudes and voltage dependence of the different QOFF components are consistent with previous estimates of the charge and equilibrium properties of C–C, O–O, and C–O transitions in the allosteric scheme (Scheme I).

Testing the Allosteric Voltage-gating Scheme

The relationship between Po and QOFFfast.

The allosteric model predicts that the fast component of OFF charge movement should be eliminated after voltage pulses that open all channels. One way to increase Po is by stepping to more positive voltages. Fig. 8 A plots the time course of QOFF component development at +240 mV. The decay of QOFFfast is more complete than at +140 mV (Fig. 7 F), consistent with a voltage-dependent increase in Po. It is likely that the fast component was not eliminated because, in the absence of Ca2+, mSlo channels are maximally activated only at very positive voltages (greater than +300 mV) (Horrigan et al. 1999). However, in the presence of 60 μM Ca2+, channels can be fully activated at +160 mV. Fig. 8 B compares the relaxation of QOFFQOFFss after a 0.1- or 20-ms pulse under these conditions. The 0.1-ms trace decays rapidly and is fit by a triple exponential function (τF = 23.8 μs, τM = 150 μs, τS = 822 μs), with the Fast component representing the majority of OFF charge (91%). However, the 20-ms record is well fit by a double-exponential function using only τM and τS. This confirms that the Fast component can be eliminated and that the relaxation of open channels back to the closed state contributes only to the Medium and Slow components of QOFF.

Voltage dependence of QOFF component amplitudes.

To further test the allosteric model, we examined the effect of repolarization voltage on the relative amplitudes of QOFF components. Ig was evoked in response to pulses of different duration to +160 mV (0.1–20 ms). After each pulse, the membrane was repolarized to either −80 or 0 mV, and OFF currents were analyzed as in Fig. 7. The amplitudes of the three QOFF components are plotted versus pulse duration in Fig. 8C and Fig. D, for −80 and 0 mV, respectively. The component amplitudes were normalized to Qfast measured in response to a pulse from −80 to +160 mV because a 20% increase in this quantity was observed during the course of the experiment. In the absence of Ca2+, steady-state open probability at 0 mV is expected to be small (<10−4) (Horrigan et al. 1999). Therefore, both repolarization voltages should be sufficiently negative to close most channels. The Qss–V relationship (Fig. 6 D) indicates that there is also little change in the steady-state charge distribution between −80 and 0 mV, so total QOFF is similar at −80 or 0 mV. Fig. 8C and Fig. D, shows that the time course of total OFF charge development (Qp(t), open symbols) is also unaffected by repolarization voltage. This is expected, since Qp(t) represents the time course of ON charge movement and should depend only on the voltage during the pulse (+160 mV). Similarly, the development time course of the three QOFF components and the amplitude of the fast component are unaffected by repolarization voltage. However, a change in the relative amplitudes of the Medium and Slow components is observed. QOFFslow increased 2.4-fold at 0 mV while QOFFmed decreased, such that total QOFF remained constant. This complementary change in QOFFmed and QOFFslow supports the idea that both represent charge movement and are not contaminated by ionic currents.

The effect of voltage on the relative amplitude of Slow and Medium components of QOFF can be understood in terms of the allosteric gating scheme (Scheme I). According to the model, the Medium component represents open state (O–O) transitions while the Slow component is limited by channel closing (O–C). Therefore, QOFFmed reflects the voltage-dependent reequilibration of channels among open states. If the membrane is repolarized to a sufficiently negative voltage, QOFFmed will be maximal because open channels will rapidly occupy the leftmost open state (O0) before closing. Under these conditions QOFFslow will be small, representing only the charge moved during the transition from O0 to C0 (zL). However, if the membrane is repolarized to a less negative voltage, the open-channel equilibrium may favor occupancy of intermediate open states (Oi) rather than O0, and QOFFmed will be reduced. At the same time, QOFFslow will increase to reflect relaxation from Oi to the resting closed state (C0).

To examine the quantitative predictions of the allosteric scheme (Scheme I), it is convenient to compare the charge distributions predicted for Closed and Open channels (Fig. 8 E, QC(V) and QO(V)). QC can be expressed in terms of the voltage-sensor equilibrium constant J(V) and charge zJ.

\begin{equation*}Q_{{\mathrm{C}}} \left \left(V\right) \right =4z_{{\mathrm{J}}} \left \left[\frac{J \left \left(V\right) \right }{1+J \left \left(V\right) \right }\right] \right \end{equation*}
9

Therefore, QC(V) has the same shape as the Qfast–V relation, with a maximum amplitude of 2.2 e (4 zJ) when zJ = 0.55 e. QO(V) is determined by the open-channel voltage-sensor equilibrium constant (DJ), the voltage-sensor charge zJ, and the charge for the C–O transition (zL = 0.4 e):

\begin{equation*}Q_{{\mathrm{O}}} \left \left(V\right) \right =z_{{\mathrm{L}}}+4z_{{\mathrm{J}}} \left \left[\frac{DJ \left \left(V\right) \right }{1+DJ \left \left(V\right) \right }\right] \right \end{equation*}
10

When D is assigned a value of 17, as in the preceding paper, the half-activation voltage for QO(V) is shifted by −130 mV relative to that of QC(V), indicating that voltage sensors are easier to activate when channels are open (ΔΔG2.83 kT). The relative amplitudes of QOFFmed and QOFFslow predicted by the model are indicated by arrows in Fig. 8 E at repolarization voltages of −80 and 0 mV. If voltage sensors equilibrate before channels close, the Medium OFF component evoked from an open channel can be expressed in terms of QO:

\begin{equation*}Q_{{\mathrm{OFFmed}}}=Q_{{\mathrm{O}}} \left \left(V_{{\mathrm{P}}}\right) \right -Q_{{\mathrm{O}}} \left \left(V_{{\mathrm{R}}}\right) \right \end{equation*}
11

where VP is the pulse voltage and VR is the repolarization voltage. The Slow OFF component is determined by the difference of QO and QC.

\begin{equation*}Q_{{\mathrm{OFFslow}}}=Q_{{\mathrm{O}}} \left \left(V_{{\mathrm{R}}}\right) \right -Q_{{\mathrm{C}}} \left \left(V_{{\mathrm{R}}}\right) \right \end{equation*}
12

As illustrated in Fig. 8 E, the model predicts that QOFFslow will increase 1.93-fold when OFF charge is measured at 0 mV rather than −80 mV, similar to the 2.38-fold change observed in Fig. 8C and Fig. D.

Simulations of the allosteric model.

The results discussed thus far are qualitatively consistent with the behavior of the allosteric gating scheme (Scheme I). Simulations based on the model as shown in Fig. 9, Fig. 10, and Fig. 11 also reproduce the major features of the data. However, the parameters that were ultimately used to fit Ig differ from those used to describe ionic currents (Horrigan et al. 1999). Some of these differences are small and may simply reflect a greater accuracy in characterizing fast voltage-sensor movement with gating currents. Other differences, relating to the slow charge movement, suggest that ionic conditions alter mSlo channel gating.

In the preceding article, parameters for the allosteric scheme (Scheme I) were estimated based on fits to the GK–V relationship, and the voltage dependence of both IK relaxation kinetics (τ(IK)−V) and the delay in IK activation (Δt(IK)−V). The charge assigned to voltage-sensor movement (zJ = 0.55 e) was identical to that used here to fit the Qfast–V (Fig. 4 A2) and τgFast–V (Fig. 4 B2) relationships. However, the half-activation voltage of the Qfast–V (Vh(J) = 155 mV) determined from gating current measurements is 10 mV more positive than previously estimated. Although this discrepancy is small, it is useful to consider several factors that could potentially contribute to such a difference. First, patch to patch variability is observed for both ionic and gating current data in the position of relationships such as the GK–V and Qfast–V along the voltage axis (Fig. 4; see also Horrigan et al. 1999; Stefani et al. 1997). We have attempted to minimize the effects of such shifts by averaging results from many experiments. Nonetheless, such variation could contribute to differences between ionic and gating current data. In addition, the estimate of Vh(J) based on IK recordings is less direct and therefore likely to be less accurate than that determined from gating currents. The previous estimate of Vh(J) was based, in part, on the ability of the allosteric scheme (Scheme I) to fit the Δt(IK)–V relationship. Parameters were assigned to the model with the simplifying assumption that the rate constants for voltage-sensor movement (α, β) are symmetrically voltage dependent (zα = −zβ). Under this condition, with Vh(J) = 145 mV, the model reproduces the observation that the maximum delay is observed at approximately +153 mV (Vmaxt)). However, the τgFast–V relationship (Fig. 4 B2) indicates that zα (0.33 e) is greater than −zβ (0.22 e). Under this condition the predicted relationship between Q–V and Δt(IK)–V changes such that Vh(J) > Vmaxt). Thus, Vh(J) is not merely determined by the Δt(IK)–V relationship but is also influenced by zα and zβ. Finally, experimental conditions were different for IK and Ig measurements and might contribute to a real difference in channel gating. For example, Δt(IK) was measured at a lower temperature (5°C) than Ig (20–22°C). In Shaker K+ channels, decreased temperature has been shown to shift the Q–V relationship to more negative voltages (Rodriguez et al. 1998), consistent with the difference in Vh(J) estimated for mSlo from IK and Ig data. In addition, IK was recorded in symmetrical 110 mM K+ while Ig was recorded with NMDG and TEA replacing internal and external K+, respectively. Permeant and blocking ions are known to alter the gating of many K channels (Armstrong 1971; Chen et al. 1997; Fedida et al. 1999; Matteson and Swenson 1986; Sala and Matteson 1991; Swenson and Armstrong 1981; Wang et al. 1999; Yeh and Armstrong 1978), including BK channels (Demo and Yellen 1992; Miller et al. 1987; Neyton and Pelleschi 1991) (see discussion).

Initial Ig simulations (Fig. 9) were generated using parameters determined from a combination of ionic and gating current measurements. The parameters describing the R–A transition for closed channels (zJ = 0.55, Vh(J) = 155 mV, zα = 0.33, zβ = −0.22) were determined from Qfast–V and τgFast–V relationships as described above. The R–A equilibrium for open channels was assumed to differ from that for closed channels by the allosteric factor D = 17, estimated in the preceding article. The rate constants for this transition were assumed to be symmetrically affected by channel opening such that the forward rate is increased f-fold f=D and the backward rate is decreased by the same factor. Rate constants for the C–O transitions were identical to those used to fit the IK data at 20°C (Horrigan et al. 1999). Finally, simulated Ig was scaled to experimental records by estimating the number of channels (N) based on the expression N = QTfast/4zJ.

Fig. 9 A plots a family of IgON evoked at different voltages and compares them to simulations of the allosteric scheme (solid lines). The model reproduces the fast decay and relative amplitudes of these ON currents. The amplitudes of fast gating currents are sensitive to filtering; therefore, the voltage command used in the simulation and the resulting current were filtered at 20 kHz to reproduce experimental conditions (see Materials and Methods). Fig. 9 B plots a family of gating currents evoked at +140 mV in response to pulses of different duration (from Fig. 5 A). The model (solid lines) reproduces the time course and relative amplitudes of ON and OFF currents in response to brief pulses.

The time constants of Fast and Medium charge movement (τF and τM), predicted by the model, are plotted in Fig. 9 C (solid lines). The τF–V relationship is defined (τF = [α + β]-1) by the parameters assigned to the R–A transition when the channel is closed, and is identical to the fit of the τgFast data in Fig. 4, B2 and C2 (dashed lines). τgFast measured from simulated currents (Fig. 9 C, solid symbols) superimposes on τF, confirming that exponential fits to fast Ig can be used to estimate the properties of closed-channel voltage-sensor movement. Similarly, the Qfast–V relationship, determined from these fits, superimposes on the QC–V relationship (Fig. 9 D) defined by the model ().

The τM–V relationship predicted by the model is the same shape as the τF –V but is shifted to more negative voltages owing to the allosteric interaction between channel opening and voltage-sensor movement (τM = (αf + βf/D)-1). Measurements of τM from several experiments (Fig. 9 C, open symbols) are similar to those predicted by the model, consistent with the assumption that the forward and backward rate constants for voltage-sensor activation are symmetrically affected by channel opening (i.e., f=D=4.13). A better fit to the data is obtained if f is increased to 4.8 (dashed line) but, given the limited number and voltage range of τM measurements, we continue to assume f = 4.13 in the following simulations. The similar voltage dependence of the τM and τF data supports the conclusion that both Fast and Medium components of OFF charge represent voltage-sensor movement.

In addition to reproducing Ig in response to brief pulses, the model exhibits a slowing of IgOFF with increased pulse duration (Fig. 9 B). However, this effect is more prominent in the data, suggesting that the model underestimates the amount of slow charge movement. To examine the time course and magnitude of slow charge predicted by the model, IgON was simulated in response to 20-ms pulses to different voltages and then integrated to obtain QON(t) (Fig. 9 E, solid lines). The time course of QON is biphasic and the fast component matches the data (Qp(t); Fig. 9 E, symbols), but the slow component is too small, especially at lower voltages. One possible explanation for this underestimate of QpSlow is that the model underestimates the number of channels that open at different voltages. In other words, the shape of the Po–V relationship and/or its position along the voltage axis may not be accurately reproduced. Since the GK–V relationship was well fit by the allosteric scheme in the preceding paper, this situation could occur if channel opening is enhanced under the ionic conditions where Ig is measured. To test this possibility, we further analyzed the voltage dependence and kinetics of the different charge movement components.

Estimating Po from charge movement.

The allosteric model predicts that slow changes in both ON and OFF charge movement components are related to channel opening and closing. Therefore, the amplitudes of these components are related to open probability as well the charge distribution for open (QO) and closed (QC) channels. For example, the fast component of OFF charge depends on QC and the number of closed channels at the end of a voltage pulse (1 − Po):

\begin{equation*}Q_{{\mathrm{OFFfast}}} \left \left(V_{{\mathrm{P}}}\right) \right = \left \left[1-P_{{\mathrm{o}}} \left \left(V_{{\mathrm{P}}}\right) \right \right] \right \left \left[Q_{{\mathrm{C}}} \left \left(V_{{\mathrm{P}}}\right) \right -Q_{{\mathrm{C}}} \left \left(HP\right) \right \right] \right \end{equation*}
13

For a particular pulse voltage (VP) and holding potential (HP), the second term in this expression can be determined by measuring the fast component of ON charge:

\begin{equation*}Q_{{\mathrm{ONfast}}} \left \left(V_{{\mathrm{P}}}\right) \right = \left \left[Q_{{\mathrm{C}}} \left \left(V_{{\mathrm{P}}}\right) \right -Q_{{\mathrm{C}}} \left \left(HP\right) \right \right] \right \end{equation*}
14

Therefore, Po can be estimated by comparing fast components of ON and OFF charge:

\begin{equation*}P_{{\mathrm{o}}}=1- \left \left[{ \left \left[Q_{{\mathrm{OFFfast}}} \left \left(V_{{\mathrm{P}}}\right) \right \right] \right }/{ \left \left[Q_{{\mathrm{ONfast}}} \left \left(V_{{\mathrm{P}}}\right) \right \right] \right }\right] \right \end{equation*}
15

Fig. 10 A plots the steady-state Po–V relationship estimated in this way for three experiments where QOFFfast(VP) was measured after a 20-ms pulse and QONfast(VP) was determined from an exponential fit to IgON (i.e., Qfast). Although measurements are scattered, reflecting, in part, the previously noted difficulties in separating QOFF components, the data generally follow the shape of the Po–V relationship predicted by the original model parameters (Fig. 10 A, Case A) but are shifted to more negative voltages. Two additional Po–V relationships (Cases B and C) are superimposed on the data and will be used throughout this analysis. Case B indicates the prediction of the allosteric scheme (Scheme I) when the equilibrium constant L is increased 12-fold (equivalent to ΔΔG = 2.5 kT) while leaving the other parameters unchanged. The Po–V relationship indicated by Case C is roughly the same shape as Case A but is shifted along the voltage axis. Case C was not generated by a gating scheme but can be used in combination with the QC and QO relationships defined by the original model to make predictions about the voltage dependence of different charge movement components. As discussed below, various aspects of the data are consistent with these altered Po–V relationships.

The predicted amplitude of the slow component of OFF charge is directly proportional to Po:

\begin{equation*}Q_{{\mathrm{OFFslow}}} \left \left(V_{{\mathrm{P}}}\right) \right =P_{{\mathrm{o}}} \left \left(V_{{\mathrm{P}}}\right) \right \left \left[Q_{{\mathrm{O}}} \left \left(HP\right) \right -Q_{{\mathrm{C}}} \left \left(HP\right) \right \right] \right \end{equation*}
16

Fig. 10 B plots normalized QOFFslow versus voltage for the same experiments as in Fig. 10 A. Again, the data follow the general shape of the Po–V relationship predicted by Case A but appear shifted to more negative voltages. The model relationships were generated from the above expression for QOFFslow where Po was specified by Case A, B, or C in Fig. 10 A and QO and QC were determined from the parameters assigned to the original model as illustrated in Fig. 8 E. The data and model traces were normalized to the total fast charge movement QTfast for each experiment. According to the model QTfast = 4zJ, therefore, the maximum amplitude of the normalized data should be [QO(HP) − QC(HP)]/4zJ. That the data fall within the amplitude range predicted by the model is therefore consistent with the relative amplitudes of QO(HP), QC(HP), and zJ specified in the model.

The Medium component of OFF charge is larger and therefore easier to measure than QOFFslow but its voltage dependence is determined by QO(V) as well as Po(V):

\begin{equation*}Q_{{\mathrm{OFFmed}}} \left \left(V_{{\mathrm{P}}}\right) \right =P_{{\mathrm{o}}} \left \left(V_{{\mathrm{P}}}\right) \right \left \left[Q_{{\mathrm{O}}} \left \left(V_{{\mathrm{P}}}\right) \right -Q_{{\mathrm{O}}} \left \left(HP\right) \right \right] \right \end{equation*}
17

Fig. 10 C compares the normalized QOFFmed–V relationships to the model predictions. Again, the data plots are similar in shape and magnitude to the prediction of Case A but are shifted to more negative voltages. Both data and model predictions were normalized to QTfast as in Fig. 10 B such that the maximum amplitude should be [QO(VP) − QO(HP)]/4zJ. Therefore, the magnitude of QOFFmed is consistent with QO(V) and zJ specified in the model.

The amplitude of the data and model predictions in Fig. 10A, Fig. B, and Fig. C, as noted above, are influenced by several factors in addition to Po. These include model parameters (QO(V), QC(V), zJ, zL) as well as our ability to separate QOFF components and determine QTfast. To better examine the voltage dependence of the data, IgOFFmed–V relationships from several experiments were normalized together with the model predictions to a maximum amplitude of one (Fig. 10 D). IgOFFmed is proportional to QOFFmed, so the model relationships represent normalized versions of those used in Fig. 10 C. IgOFFmed was normalized based on a Boltzmann fit to the IgOFFmed–V relationship for each experiment (z = 0.98 e). When scaled in this way, the data from different experiments superimpose. Case C represents a Boltzmann fit to these normalized data (z = 0.98 e, Vh = 121 mV). The Po–V relationship for Case C (Fig. 10 A) was derived from this fit and the expression for QOFFmed ().

The slow component of ON charge movement (QpSlow) should exhibit a complex voltage dependence that is determined by Po(V), QC(V), and QO(V):

\begin{equation*}Q_{{\mathrm{pSlow}}} \left \left(V_{{\mathrm{P}}}\right) \right =P_{{\mathrm{o}}} \left \left(V_{{\mathrm{P}}}\right) \right \left \left[Q_{{\mathrm{O}}} \left \left(V_{{\mathrm{P}}}\right) \right -Q_{{\mathrm{C}}} \left \left(V_{{\mathrm{P}}}\right) \right \right] \right \end{equation*}
18

The QpSlow–V relationships plotted in Fig. 10 E were normalized by QTfast and exhibit amplitudes that are larger than predicted by Case A, but are similar to those specified by Cases B and C. The model predicts that QpSlow will have a bell-shaped voltage dependence and that QpSlow/QTfast approaches a limiting value of zL/4zJ at positive voltages (Fig. 10 E, dashed line). Our measurements do not extend to high enough voltages to test these predictions. However, the data fall close to the relationships determined by Cases B and C over the voltage range tested, and appear to trend downward at the highest voltages. Importantly, the comparison of QpSlow–V relationships for Cases A, B, and C demonstrate that the amount of slow charge movement is highly sensitive to Po and that QpSlow can be considerably larger than the charge associated with the C–O transition.

Finally, we examined the ability of the model to reproduce slow charge movement kinetics. Fig. 10 F plots τgSlow over a large voltage range. At positive voltages, τgSlow was measured from the time course of development of the Medium component of IgOFF as in Fig. 7 G. At negative voltages, τgSlow was determined from the relaxation of Slow QOFF(t) (τS). The dashed line in Fig. 10 F represents a fit of the allosteric scheme (Scheme I) to the time constants of IK relaxation (τ(IK)), measured in the preceding paper (equivalent to Case A). τgSlow is faster than τ(IK) for V > ∼+100 mV and is slower than τ(IK) at negative voltages. However, the voltage dependence of τgSlow can be fit (Fig. 10 F, solid line) by adjusting the model parameters as specified for Case B where the equilibrium constant L is increased 12-fold. Both τgSlow and τ(IK) are weakly voltage dependent from −80 to 0 mV, consistent with the idea that the slow relaxation of QOFF is limited by channel closing.

Taken together, the data in Fig. 10 support the hypothesis that the properties of slow charge movement can be accounted for by the allosteric voltage-gating scheme (Scheme I), provided we assume that Po is increased under the conditions where gating currents are measured. Coordinated changes in all three components of QOFF are observed with pulse voltage, consistent with the assumption that their amplitudes depend upon the Po–V relationship. The relative amplitudes of these components are also consistent with their proposed source in terms of the allosteric scheme and with the charges assigned to various transitions in the model. The voltage dependence of the Medium OFF component suggests that the Po–V relationship may be similar in shape to that measured with ionic currents (Case A) but is shifted to more negative voltages (Case C). The Fast and Slow component data are consistent with this hypothesis but are inadequate to test the precise voltage dependence of Po. The data are also insufficient to specify how the model parameters should be altered to account for a change in Po. Case B, assuming a 12-fold increase in the equilibrium constant L, provides a reasonable first approximation that can account for both a shift in the Po–V relationship as well as the observed kinetics of slow charge movement.

Simulations of the modified allosteric scheme.

When the C–O transition rates in the allosteric scheme (Scheme I) are modified, as specified by Case B, improved fits to the gating currents are generated. Fig. 11A and Fig. B, compares simulated currents to Ig evoked at +140 and +224 mV in response to pulses of different duration. The model accurately reproduces the amplitudes of ON and OFF currents, including the decrease in IgOFF amplitude that occurs with increased pulse duration at +224 mV (Fig. 11 B). The model also fits the time course of IgOFF and accounts for the slowing of decay kinetics that accompanies increased pulse duration (Fig. 11A and Fig. B). The time course of OFF charge relaxation (QOFFQOFFss) after +140 mV pulses are plotted on a semilog scale in Fig. 11 C, and are well fit at all pulse durations. Thus, the model accurately reproduces the kinetics and amplitudes of the three OFF components. The model can account for the slow time constants of both ON and OFF charge movement at all voltages (Fig. 10 F); however, the amplitude of QpSlow is underestimated at low voltages (Fig. 10 E, Case B). This point is illustrated in Fig. 11 D, which compares Qp(t) at different voltages to QON(t) generated by the model. Both the time course and amplitude of Qp are well fit at V ≥ +140 mV; however, the slow component predicted by the model at lower voltages is reduced in comparison to the data.

Simulation of gating admittance.

To further test the above conclusions, gating currents were simulated in response to a sinusoidal voltage command and compared with admittance analysis results. The CgV relationship is compared with the simulations for Cases A and B (solid lines) in Fig. 11 E. Dashed lines indicate the QOV and QC′–V relationships specified by the model. These relationships are the main determinants of Cg–V since they reflect fast voltage-sensor movement. At voltages below +100 mV where channels are closed, Cg approximates QC′. At higher voltages, Cg represents an average of QC′ and QO′ weighted by Po. Thus, Cg decreases at positive voltages (approaching QO′) when Po is increased (compare Cases A and B). Case A overestimates Cg, suggesting that it underestimates Po. However, as the Po–V relationship is shifted (Case B), the model better approximates the peak amplitude and peak voltage of Cg. The effect of Po on the shape of the Cg–V relationship explains why the mean peak voltage of Cg (+127 mV) is more negative than the half-activation voltage of the Qfast–V relationship (+155 mV).

Examination of gating currents evoked from mSlo Ca2+-activated K+ channels in the absence of Ca2+ has revealed several components of charge movement associated with voltage-dependent gating. We have shown that these results are consistent with an allosteric voltage-gating scheme (Scheme I) that was proposed in the preceding article to account for the kinetic and steady-state properties of mSlo IK in 0 Ca2+ (Horrigan et al. 1999). Indeed, many of our experiments were designed to test this model. But before discussing these conclusions concerning the allosteric scheme, it is useful to review our results from a more general perspective. The gating current data lead to several model-independent conclusions and allow many alternative gating schemes to be ruled out.

Previous Models of BK Channel Gating

BK channel gating has been extensively studied at the single channel level (Barrett et al. 1982; Magleby and Pallotta 1983a; Magleby and Pallotta 1983b; McManus and Magleby 1991; Moczydlowski and Latorre 1983; Rothberg and Magleby 1998). Kinetic analysis reveals complex dwell-time distributions indicating the presence of multiple open and closed states. Based on such analysis, gating schemes have been proposed that contain a parallel arrangement of open and closed states (McManus and Magleby 1991), superficially resembling the architecture of our allosteric voltage-gating scheme (Scheme I). However, it is important to recognize that these previous studies were performed in the presence of Ca2+, and that the gating schemes used to describe these data therefore contain Ca2+-bound states and Ca2+-dependent transitions. Thus, the kinetic complexity revealed by the single channel data isn't necessarily related to the mechanism of voltage-dependent gating. Indeed, most schemes derived from single-channel analysis fail to account for the ability of BK channels to open in the absence of Ca2+ binding. By examining mSlo channel gating in the absence of Ca2+, we have characterized this voltage-dependent pathway, thereby defining a boundary condition that must be satisfied by any complete model of BK channel gating and representing a subset of the states that are accessible in the presence of Ca2+.

A model of BK channel gating has been proposed by Cox, Cui, and Aldrich 1997 to account for the effects of voltage and Ca2+ on macroscopic mSlo ionic currents, including their ability to activate in the absence of Ca2+. The model assumes that mSlo channels undergo a single voltage-dependent transition between a closed and open conformation and that Ca2+ binding regulates this transition allosterically. This scheme is essentially a version of the MWC model (Monod et al. 1965; Scheme II) where channel opening represents an allosteric transition that alters the affinity of Ca2+-binding sites and is also voltage dependent.

(Scheme II)

As in the McManus and Magleby model (1991), Scheme II contains many states representing different Ca2+-bound versions of the closed and open conformation. However, in the absence of Ca2+, Scheme II reduces to a two-state model with a single voltage-dependent transition between a unliganded closed and open state (highlighted above). By assuming that voltage-dependent activation can be described by a two-state mechanism, Scheme II implies that channel opening, voltage-sensor movement, and changes in Ca2+ binding-site affinity all occur during a concerted allosteric transition. Our results demonstrate that a more complicated scheme is required to explain voltage-dependent gating and therefore imply that the interaction of Ca2+ with the channel may also be more complicated than proposed in Scheme II (see Horrigan et al. 1999). In particular, the voltage-dependent C to O transition in Scheme II does not consist of a completely cooperative (concerted) step, although considerable cooperativity, as formulated by the allosteric voltage-gating model, is present.

The preceding paper examines several properties of mSlo IK that are inconsistent with a two-state model of voltage gating (Horrigan et al. 1999). Single channel analysis of mSlo in 0 Ca2+ also provides evidence for multiple closed and open states (Talukder and Aldrich 1998). The gating current analysis presented here supports the conclusion that mSlo gating is a multistate process even in the absence of Ca2+. A two-state model of voltage-dependent activation requires that charge movement and channel opening occur simultaneously and therefore exhibit identical kinetic and steady-state properties. In other words, Ig should relax with the same near-exponential kinetics of IK, and the voltage dependence of steady-state charge movement (Q) and open probability (Po) should be identical. Instead, we observe multiple kinetic components of ON and OFF charge movement with major components of both preceding the relaxation of IK. In addition, the normalized Q–V and Po–V relationships are not superimposable (see also Stefani et al. 1997). These results indicate that mSlo channel opening cannot be represented by a concerted transition, and that the MWC model (Scheme II) is therefore an oversimplification in this regard, although it captures many of the major features of mSlo behavior.

Fast Ig: Evidence for a Two-state Model of Voltage-sensor Movement

Although the overall response of mSlo channels to voltage is complex, gating currents suggest that the movement of individual voltage sensors can be described by a simple two-state model when channels are closed.

Ig evoked during a voltage step exhibits a prominent fast component (IgFast) representing a majority of ON charge. This fast charge is also detected as a voltage-dependent component of membrane capacitance measured in response to a sinusoidal voltage command, thereby ruling out the possibility that leak subtraction or voltage clamp artifacts contribute to rapid current transients measured in response to large voltage steps. Both admittance analysis and the response to voltage steps indicate that fast gating charge can move at voltages where Po is normally low, and relaxes roughly 100-fold faster than the time constant of IK activation. IgFast decays with exponential kinetics during a time when few channels have opened. The relaxation of OFF current is also fast and single-exponential after brief pulses that open few channels. These results demonstrate that closed unliganded mSlo channels can undergo rapid voltage-dependent transitions.

Because the majority of ON charge moves rapidly, we assume that IgFast can be attributed to voltage-sensor movement. The exponential kinetics of IgFast and lack of a rising phase are consistent with a two-state model in which voltage sensors undergo a transition between a resting (R) and activated (A) conformation. The observation that the Qfast–V relationship is fit by a single Boltzmann function also supports a two-state model. In addition, the time constant of fast Ig relaxation (τgFast) exhibits a bell-shaped voltage dependence that can be fit by the inverse sum of two exponential functions, as predicted for a two-state model in which forward and backward rate constants are voltage dependent.

mSlo channels assemble as homotetramers (Shen et al. 1994) and are therefore presumed to contain identical voltage sensors in each subunit. Thus, the simple behavior of IgFast is consistent not only with a two-state model of voltage-sensor movement but also with the idea that voltage sensors act independently. However, interactions between voltage sensors cannot be ruled out simply based on the kinetic and steady-state properties of fast charge movement. While it is true that such interactions could lead to multiexponential IgFast kinetics and a non-Boltzmann Q–V, more subtle effects are also possible. For instance, a model that assumes four voltage sensors move in a concerted manner would also predict two-state behavior, the difference being that the Qfast–V would be fit by a Boltzmann function with equivalent charge (zFast) of 4zJ for a concerted model versus zJ for an independent scheme. To distinguish these two possibilities requires an independent estimate of the fast charge per channel (qfast). Stefani et al. 1997 have reported a total charge (qT) of 4.4 ± 0.8e per channel (mean ± SD, n = 3) for hSlo based on measurements of Ig and ionic current density in different patches from the same oocyte. Although this estimate is not precise and includes both fast and slow charge, its magnitude argues against a concerted model, since zFast = 0.59 e determined for mSlo is much smaller than qT. An independent model would predict a fast charge of 4zFast = 2.36 e, much closer to qT. The relationship between fast charge movement and channel activation, discussed below, also argues against a concerted model of voltage sensor movement and is consistent with an independent scheme. However, uncertainty in some of these measurements, such as the estimate of qfast, prevents us from completely ruling out interaction between voltage sensors.

The Coupling of Voltage-sensor Movement to Channel Activation

Since the decay of IgFast is much faster than the activation of IK, we considered the possibility that fast charge movement might be unrelated to channel activation. An early component of charge movement has been described in Shaker K channels (Sigg et al. 1999; Stefani and Bezanilla 1996) and squid Na channels (Forster and Greeff 1992) that relaxes rapidly (Shaker: τ < 10 μs, Na channel: τ < 25 μs) and represent <10% of the total gating charge. The speed and small magnitude of this early charge movement suggest it could represent transitions that are not important for channel activation. IgFast described for mSlo is only severalfold slower than these early components and exhibits a similar equivalent charge. However, in the case of mSlo, several lines of evidence support the idea that fast charge movement is coupled to channel activation.

In contrast to the “early” charge movement in Shaker and Na channels, IgFast represents a majority of ON charge (Fig. 6 B). In addition, the estimated fast charge per channel Qfast = 4zJ = 2.36 e (assuming independent voltage sensors) is similar to the equivalent charge that characterizes the maximum voltage dependence of Po in 0 Ca (z(Po) = 2.0 e; Horrigan et al., 1999). Thus, the magnitude of IgFast is consistent with the idea that fast charge movement is important for mSlo channel activation.

The kinetic relationship between IgFast and IK also argues that fast charge movement reflects transitions in the activation pathway. Fast charge movement and the delay in IK activation occur on similar time scales. An example in Fig. 3 D shows that IgON decays at the same time that IK achieves an exponential time course. Thus, the achievement of a maximal rate of IK activation appears correlated with equilibration of fast gating charge. IK also exhibits a multiexponential rate of increase during the delay (Horrigan et al. 1999), supporting the idea that voltage-sensor transitions are not concerted. If the delay in IK depends only on the transitions that give rise to IgFast, then we have previously argued that the delay duration (Δt) should be roughly proportional to τgFast (Horrigan et al. 1999). Consistent with this prediction, τgFast (Fig. 4) and Δt (Horrigan et al. 1999) exhibit similar bell-shaped voltage dependencies that can be characterized by an equivalent charge of 0.55e and peak voltages of 136 and 153 mV, respectively.

Finally, the Qfast–V and Po–V relationships, defining the voltage dependence of fast charge movement and IK activation, respectively, activate over a similar voltage range, consistent with the idea that these two processes are coupled. We have also shown that the normalized GK–V relationship can be approximated by raising the Qg–V relationship to the 4th power (Fig. 1 B). As discussed below, an approximate 4th power relationship between Qfast–V and Po–V is predicted by many schemes that assume Po is enhanced by the activation of four voltage sensors. The relationship between Q–V and G–V is an important test of any voltage-dependent model, but experimental factors limit the interpretation of these data in the case of mSlo. The precise relationship because Q–V and G–V is unclear, owing to the likelihood that gating is altered under the conditions where gating currents are measured.

Sequential Voltage-gating Schemes

Taken together, the above observations indicate that the conformational changes underlying fast charge movement are involved in mSlo channel activation. Therefore, any plausible gating scheme must include a pathway that allows rapid voltage-dependent transitions to occur before channels open. The properties of fast charge movement are consistent with these closed-state transitions, arising from the activation of four independent and identical voltage sensors. Two sequential gating schemes incorporating such a mechanism are considered below (Scheme I and Scheme II) and can reproduce many features of fast charge movement, but can be ruled out based on their failure to account for slow charge movement. These arguments parallel those in the preceding paper based on IK measurements (Horrigan et al. 1999), and lead to similar conclusions as to the requirement for an allosteric model.

One of the simplest schemes that can account for the properties of IgFast is the Hodgkin-Huxley (HH) model (Scheme III).

(Scheme III)

The HH scheme assumes channels are open when all four voltage sensors are activated and predicts a 4th power relationship between the Q–V and G–V relationships:

\begin{equation*}P_{{\mathrm{o}}}=Q^{4}\end{equation*}
19

where Q represents the normalized charge distribution defined by the equilibrium constant for voltage-sensor activation (Q = J/(1 + J)). As noted previously, the observed relationship between Qg–V and GK–V appears consistent with this prediction. However, Scheme III is inadequate because it cannot account for the presence of both fast and slow components of mSlo charge movement. Similarly, the HH scheme cannot reproduce both the brief delay and slow exponential relaxation that characterize IK activation kinetics (Horrigan et al. 1999).

Models that assume voltage-sensor activation is followed by a distinct opening transition have proven useful in describing the behavior of channels that deviate from the predictions of the HH scheme (Koren et al. 1990; Perozo et al. 1992; Schoppa and Sigworth 1998; Sigworth 1994; Zagotta and Aldrich 1990; Zagotta et al. 1994a). Such models can account for the presence of fast and slow components of gating current as well as nonsigmoidal IK activation kinetics (Horrigan et al. 1999; Ledwell and Aldrich 1999; Smith-Maxwell et al. 1998a; Smith-Maxwell et al. 1998b). Scheme IV (below) assumes that channels can undergo a rate-limiting C–O transition after four independent and identical voltage sensors are activated.

(Scheme IV)

Scheme IV predicts an approximate 4th power relationship between G–V and Q–V described by the expression

\begin{equation*}P_{{\mathrm{o}}}=\frac{Q_{{\mathrm{C}}}^{4}}{\displaystyle\frac{1}{{\mathrm{{\epsilon}}}}+Q_{{\mathrm{C}}}^{4}}\end{equation*}
20

where QC, the closed channel charge distribution, defines the voltage dependence of fast charge movement (QC = J/(1 + J)). As illustrated in Fig. 12 A, this model can approximate the observed relationship between the Qfast–V and GK–V for mSlo. Scheme IV can also account for a slow component of ON charge movement (Fig. 12 B) but, as discussed below, cannot reproduce some important aspects of gating current behavior. Similarly, Scheme IV can approximate the time course of mSlo IK but does not account for the complex voltage dependence of IK relaxation kinetics and open probability (Horrigan et al. 1999).

The Slow Component of ON Charge Movement

A slow component of ON charge movement (QpSlow) is detected as an increase in QOFF after pulses of increasing duration. QpSlow develops with the exponential kinetics of IK activation at depolarized voltages, suggesting that activation and slow charge movement are limited by the same transitions. We have shown that the allosteric voltage-gating scheme (Scheme I) can reproduce both the kinetics and voltage dependence of QpSlow (Fig. 10E and Fig. F). An important conclusion of this analysis is that QpSlow represents not only charge moved during the C–O conformational change but a reequilibration of voltage sensors that is limited by channel opening. The allosteric model predicts that transitions among open states (O–O) can contribute to slow charge movement, since voltage sensors reequilibrate after channels have opened. However, a contribution of voltage-sensor activation to slow charge movement does not require a model with multiple open states.

Scheme IV provides an example of a mechanism by which closed-state transitions contribute to both fast and slow charge movement. Fast Ig is evoked as voltage sensors initially equilibrate between resting (R) and activated (A) while the channel is closed. As channels open, this equilibrium is perturbed because channels can only open when all four voltage sensors are activated. In other words, opening stabilizes the activated voltage sensor, as in the allosteric model. However, in the case of Scheme IV, the establishment of a new voltage-sensor equilibrium can only involve transitions between R and A while the channel is closed.

Scheme IV can reproduce the time course of Qp measured at +140 mV, including a large slow component (Fig. 12 B). In addition, Scheme IV predicts a QpSlow–V relationship (Fig. 12 C, solid lines) similar to that produced by the allosteric model (Fig. 12 C, dashed lines). As with the allosteric scheme (Scheme I), a portion of QpSlow represents the charge assigned to the C–O transition (z = 0.32 e). The bell-shaped voltage dependence of QpSlow predicted by Scheme IV demonstrates that closed-state transitions also contribute to slow charge movement.

In summary, the presence of fast and slow components of ON charge, and their relationship to the time course of IK activation, suggest that the activation pathway must, at minimum, contain a rate-limiting step that is preceded by one or more rapid voltage-dependent transitions. The kinetics and voltage dependence of IgFast and the delay in IK activation, the voltage dependence of Qfast and GK together with the tetrameric structure of the channel further suggest that the rapid transitions may be described by the movement of four independent and identical voltage sensors. Therefore, Scheme IV provides the simplest model that can account for these basic features of the ionic and ON gating current data. However, as discussed below, the properties of OFF charge movement are inconsistent with Scheme IV and indicate that the activation pathway must include multiple open states.

OFF Charge Movement: Evidence for Multiple Open States

OFF currents, recorded after brief voltage pulses, decay with a single-exponential time course. Such a response is predicted by Scheme IV and is consistent with activated voltage sensors relaxing rapidly back to a resting state when channels are closed. However, Scheme IV also predicts that, once channels open, the decay of IgOFF will be limited by the speed of channel closing (Zagotta et al. 1994b). Therefore, as pulse duration increases, a slow component of OFF charge relaxation should be observed that develops with the time course of IK activation and decays with the kinetics of IK deactivation. At the same time that the slow component increases, the fast component of OFF current should decrease as the number of channels that are closed at the end of the pulse is reduced.

Contrary to the prediction of Scheme IV, we observed three components of OFF charge movement. The Fast and Slow components are analogous to those predicted by Scheme IV. However, the Medium component, representing a majority of QOFF when channels are maximally activated, provides evidence that channels can undergo transitions among open states. In response to pulses of increasing duration, QOFFfast decreases with approximately the same time course as IK activation while the two slower components, QOFFmed and QOFFslow, increase in parallel. QOFFfast is essentially eliminated under conditions that maximally activate mSlo channels (20 ms at +160 mV in 60 μM Ca2+; Fig. 8 B), implying that the Slow and Medium components reflect the relaxation of open channels back to the resting closed state.

We have proposed that the relaxation of the Slow OFF component is limited by the speed of channel closing and, at −80 mV, primarily represents charge moved during the O–C conformational change. QOFFslow represents a minority of the total OFF charge (Fig. 10 B), consistent with the notion that the O–C transition is weakly voltage dependent. Similarly, the time constants of slow charge movement (τgSlow) and IK deactivation (τ(IK)) are weakly voltage dependent at negative voltages (Fig. 10 F). However, the decay of the Slow component is approximately threefold slower than that of potassium tail currents. To account for this difference, we have suggested that channel closing is slowed under the ionic conditions that are used to measure gating currents. The τgSlow–V relationship is similar in shape to the τ(IK)–V relationship and can be fit by the allosteric model if the forward rate constants from C to O are increased while the backward rates are decreased relative to those used to describe IK. Such a change requires a 12-fold increase in the C–O equilibrium constant (ΔΔG = 2.48 kT), producing a change in the Po–V relationship that appears consistent with the observed voltage dependence of QOFF components and QpSlow (Fig. 10, Case B).

The Medium component of OFF charge relaxes ninefold faster than the Slow component and threefold faster than IK tail currents. Thus, regardless of the effect of ionic conditions on channel gating, the Medium component appears to relax faster than channel closing, implying that voltage sensors can move when channels are open. The similar voltage dependence of τM and τF (Fig. 4 B2 and 9 C) supports the idea that the Medium component represents voltage-sensor movement. Thus, any plausible gating scheme must include multiple open states with rapid voltage-dependent transitions between them. The voltage dependence of Po leads to the same conclusion (Horrigan et al. 1999). The parallel development of Slow and Medium components indicate that once a channel is open, OFF charge relaxation can be described by a constant ratio of QOFFmed and QOFFslow. This behavior is consistent with the idea that equilibration of channels among different open states is fast relative to the speed of IK activation.

A Sequential Scheme with Multiple Open States

A sequential scheme, represented in general form below (Scheme V), could account for Medium and Slow components of OFF charge relaxation, provided transitions among open states are fast compared with the transition from O to C.

Scheme V

However, such a model is inconsistent with the voltage dependence of steady-state activation. Po is weakly voltage dependent at limiting negative voltages, consistent with a charge of 0.4e assigned the Co–Oo transition in the allosteric scheme (Horrigan et al. 1999). For Scheme V to reproduce this limiting voltage dependence, a total charge of 0.4e must be assigned to the transition between Co and Oo, inconsistent with the assumption that closed-state transitions from Co to Cm represent the activation of four voltage sensors carrying a total charge of 4zfast = 2.36 e.

Scheme V also appears inconsistent with the relative amplitudes of various ON and OFF charge movement components. For example, we observe that the Medium component of OFF charge is two- to threefold larger than the Slow component measured at −80 mV (Fig. 7 E and 8 A). Thus,

\begin{equation*}Q_{{\mathrm{OFFmed}}}{\geq}2Q_{{\mathrm{OFFslow}}}\end{equation*}
21

In addition, the fast component of ON charge is larger than the slow component at all voltages studied:

\begin{equation*}Q_{{\mathrm{pFast}}}>Q_{{\mathrm{pSlow}}}\end{equation*}
22

It can be shown (below) that Scheme V cannot account for these observations if and are valid at voltages where Po ≥ 1/2. This last condition cannot be verified directly, but appears reasonable since and are true at +224 to +240 mV (Fig. 6 B and 8 A), whereas the half-activation voltage of the GK–V relationship is +190 mV (Horrigan et al. 1999). Moreover, we have argued that Po may increase under the conditions where gating currents are measured.

The amplitudes of the different charge movement components for either the allosteric model or Scheme V can be expressed in terms of QC, QO, and Po as specified by . Therefore, by substituting and , can be rewritten:

\begin{equation*}P_{{\mathrm{o}}} \left \left(V\right) \right \left \left[Q_{{\mathrm{O}}} \left \left(V\right) \right -Q_{{\mathrm{O}}} \left \left(HP\right) \right \right] \right {\geq}2P_{{\mathrm{o}}} \left \left(V\right) \right \left \left[Q_{{\mathrm{O}}} \left \left(HP\right) \right -Q_{{\mathrm{C}}} \left \left(HP\right) \right \right] \right \end{equation*}
23

where V is the pulse voltage and HP is the holding potential (−80 mV). Solving for QO(V), we obtain:

\begin{equation*}Q_{O} \left \left(V\right) \right {\geq} \left \left[3Q_{{\mathrm{O}}} \left \left(HP\right) \right -2Q_{{\mathrm{C}}} \left \left(HP\right) \right \right] \right \end{equation*}
24

can also be rewritten by substituting and .

\begin{equation*} \left \left[Q_{{\mathrm{C}}} \left \left(V\right) \right -Q_{{\mathrm{C}}} \left \left(HP\right) \right \right] \right >P_{{\mathrm{o}}} \left \left(V\right) \right \left \left[Q_{{\mathrm{O}}} \left \left(V\right) \right -Q_{{\mathrm{C}}} \left \left(V\right) \right \right] \right \end{equation*}
25

Combining and :

\begin{equation*} \left \left[1+P_{{\mathrm{o}}} \left \left(V\right) \right \right] \right \left \left[Q_{{\mathrm{C}}} \left \left(V\right) \right -Q_{{\mathrm{C}}} \left \left(HP\right) \right \right] \right >3P_{{\mathrm{o}}} \left \left(V\right) \right \left \left[Q_{{\mathrm{O}}} \left \left(HP\right) \right -Q_{{\mathrm{C}}} \left \left(HP\right) \right \right] \right \end{equation*}
26

For a sequential model like Scheme V, we can further assume:

\begin{equation*}Q_{{\mathrm{O}}} \left \left(HP\right) \right >Q_{{\mathrm{C}}} \left \left(V\right) \right \end{equation*}
27

Finally, combining and :

\begin{equation*} \left \left[1+P_{{\mathrm{o}}} \left \left(V\right) \right \right] \right \left \left[Q_{{\mathrm{C}}} \left \left(V\right) \right -Q_{{\mathrm{C}}} \left \left(HP\right) \right \right] \right >3P_{{\mathrm{o}}} \left \left(V\right) \right \left \left[Q_{{\mathrm{C}}} \left \left(V\right) \right -Q_{{\mathrm{C}}} \left \left(HP\right) \right \right] \right \end{equation*}
28

reduces to Po(V) < 1/2, indicating that Scheme V cannot account for the relative amplitude of ON and OFF charge components while also assuming Po ≥ 1/2.

The Allosteric Voltage-gating Scheme

Scheme V assumes that closed- and open-state transitions occur sequentially and must therefore represent distinct conformational events. An alternative, represented by the allosteric model, is that C–C and O–O transitions represent the same conformational events, i.e., voltage-sensor movement. The kinetics and voltage dependence of the Fast and Medium components of OFF charge movement (Fig. 9 C) are consistent with both C–C and O–O transitions representing voltage-sensor movement, differing only in that the equilibrium constant for voltage-sensor activation is increased when channels open. As demonstrated in this study and in the preceding article, the allosteric model can account for many other properties of mSlo gating in 0 Ca2+.

The allosteric model is mechanistically similar to Scheme IV in that it assumes channels undergo only two types of conformational change: voltage-sensor activation and channel opening. Voltage sensors are assumed to move rapidly and independently in each subunit. Channel opening is relatively slow, weakly voltage dependent, and assumed to represent a concerted transition. Like Scheme IV, the allosteric model assumes channel opening stabilizes the activated voltage sensor. Thus, opening results in a slow component of charge movement that is limited by the speed of channel opening but largely represents voltage-sensor charge movement. Unlike Scheme IV, the coupling of voltage-sensor activation to channel opening is not an obligatory process but rather an allosteric interaction. Therefore, voltage sensors can move when channels are open, accounting for the Medium component of IgOFF, and channels can open when voltage sensors are not activated, accounting for the weak voltage dependence of Po measured at negative voltages (Horrigan et al. 1999).

Although the allosteric model allows channels to open when voltage sensors are in a resting state, it predicts that they are most likely to open when all four are activated. Consequently, channels pass through multiple closed states before opening, consistent with the presence of a delay in IK activation (Horrigan et al. 1999). Similarly, the allosteric scheme can account for an approximate 4th power relationship between Q–V and G–V. The model predicts the following relationship between Po and QC.

\begin{equation*}P_{{\mathrm{o}}}=\frac{ \left \left[1+ \left \left(D-1\right) \right Q_{{\mathrm{C}}}\right] \right ^{4}}{\displaystyle\frac{1}{L}+ \left \left[1+ \left \left(D-1\right) \right Q_{{\mathrm{C}}}\right] \right ^{4}}\end{equation*}
29

When L is small and D >> 1, as determined in the preceding article (L = 2 × 10−6, D = 17, 0 Ca2+), this expression can be approximated as:

\begin{equation*}P_{{\mathrm{o}}}=L \left \left[1+DQ_{{\mathrm{C}}}\right] \right ^{4}\end{equation*}
30

Finally, the allosteric scheme can account for the presence of three components of OFF charge movement as well as the relative amplitudes of various ON and OFF components. In contrast to Scheme V, the allosteric model predicts that pathways traversed during channel activation and deactivation are different. Activation involves fast voltage-sensor movement as channels undergo closed-state transitions before opening. Deactivation involves movement of the same voltage sensors as channels undergo open-state transitions before closing. Because the same voltage sensors are moved during open- and closed-state transitions, the rapid components of ON (Qfast) and OFF charge (QOFFfast, QOFFmed) are of similar amplitude while the slow components are smaller. Furthermore, the relative amplitudes of QOFFmed and QOFFslow change with repolarization voltage in a manner specifically predicted by the allosteric scheme (Fig. 8C and Fig. D).

Comparison with Previous Studies of BK Channel Gating

mSlo ionic currents.

The gating current data support the conclusion from the preceding paper that mSlo channel voltage gating in the absence of Ca2+ can be described by an allosteric scheme. Indeed, many of the model parameters that were derived to fit IK data required little or no adjustment to describe the gating currents (e.g., zJ, Vh(J), zL, D). One feature of the model that could not be determined accurately from IK measurements was the speed of transitions among open states. The Medium component of IgOFF provides a direct assay of these transitions and demonstrates that channel opening slows the relaxation of voltage sensors from A to R. The magnitude of this effect is consistent with our previous estimate of the allosteric factor D = 17, provided we assume channel opening almost symmetrically affects the forward and backward rate constants for the R–A transition (Fig. 9 C). However the effect of channel opening on the forward rate was not measured; therefore, the value of D cannot be directly determined from the gating current data.

Single channel studies.

Many of the properties of IK that implicate a model with multiple open states are observed only at extreme voltages (Horrigan et al. 1999). However, the Medium component of IgOFF demonstrates that the channels can undergo open-state transitions in response to moderate voltage stimuli, such as repolarization from +160 to 0 mV (Fig. 8 D). This is important because it suggests that complex open-time distributions described previously for single BK channels may to some extent reflect the occupancy of multiple open states in the voltage-dependent activation pathway in addition to different Ca2+-bound open states.

Gating currents.

BK channel gating currents have been described previously for hSlo by Stefani et al. 1997, and many of their results, obtained in 0 Ca2+, are similar to those presented here for mSlo. For example, the major component of ON current decayed with a rapid exponential time course (τ = 57 ± 10 μs, at +200 mV). A slow component of charge movement was also detected but not examined in detail. The Q–V curve determined with brief 1-ms depolarizations was well fit by a Boltzmann function (z = 0.6 e, Vh = 190 ± 15 mV). In addition, the G–V curve was reported to be steeper than the Q–V, and charge movement was observed at voltages where most channels should be closed.

In addition to these similarities, there are important differences between our results and conclusions and those of Stefani et al. 1997. The normalized G–V and Q–V for hSlo were reported to cross such that the G–V is negative to the Q–V at positive voltages. Such a cross-over, as pointed out by Stefani et al., implies that gating charge can move when channels are open. Although our results lead to a similar conclusion, we do not observe such a relationship between Q–V and G–V in the absence of Ca2+. And the allosteric model used to fit our data does not predict a cross-over even though it allows voltage-dependent open-state transitions. In addition, Stefani et al. conclude that only a small fraction of total gating charge must move before channels can open because (a) the Q–V crosses the foot of the G–V where Po is small, and (b) the Cole-Moore shift is weakly voltage dependent. We disagree with these observations and, for reasons discussed below, conclude that in the absence of Ca2+, the majority of gating charge moves before channels open. Stefani et al. also note that the onset of ionic current overlaps the decay of Ig (see Fig. 3) and suggest this provides evidence for open-state charge movement. But most models, including those with a single open state, predict some overlap in the time course of macroscopic Ig and IK. Finally, Stefani et al. conclude that open-state transitions must be weakly voltage dependent to account for the observation that the major component of charge movement is fast. In contrast, we have shown that the allosteric scheme predicts a small slow component of charge movement even when C–C and O–O transitions are assumed to be equally voltage dependent. And the large Medium component of OFF charge movement provides evidence for significant open-state charge movement.

Some of the discrepancies between our results and those of Stefani et al. 1997 probably reflect real differences in the gating of mSlo and hSlo, but we suggest that the cross-over of Q–V and G–V reported for hSlo may also be affected by the conditions used to measure ionic currents. The Q–Vs for mSlo and hSlo are similar in shape based on Boltzmann fits (z = 0.59 e mSlo; z = 0.6 e hSlo), and the half-activation voltage for the hSlo Q–V is 35 mV more positive than that for mSlo (155 mV mSlo). In line with this difference, the GK–V for hSlo, measured in symmetrical K+ and 0 Ca, is roughly 30 mV more positive than that for mSlo (hSlo: Vh = 220 mV [Diaz et al. 1998; Meera et al. 1996]; mSlo: Vh = 190 mV V [Horrigan et al. 1999). In contrast, the G–V described by Stefani et al. is shifted by −70 mV (Vh = 150 mV) relative to that previously reported for hSlo and is much steeper than that of mSlo based on Boltzmann fits (hSlo: z = 1.3–1.6 e [Stefani et al. 1997]; mSlo: z = 1.0 e [Cui et al., 1997; Horrigan et al. 1999]). These differences may reflect the fact that the G–V reported by Stefani et al. was obtained in symmetrical Cs+ rather than K+.

Cs+ permeates BK channels poorly, allowing Cs+ currents to be recorded in patches where IK would be immeasurably large (Stefani et al. 1997). However, Cs+ is also known to alter BK channel gating and is likely to affect the G–V. Demo and Yellen 1992 studied Cs+ block of single BK channels and concluded that Cs+ occupancy of the pore destabilizes the closed state. This destabilization shifts the Po–V relationship to more negative voltages and changes its shape because Cs+ block is voltage dependent. Similarly the GCs–V relationship for hSlo is shifted to negative voltages relative to the GK–V for hSlo and is steeper than that for mSlo, consistent with an effect of Cs+ on channel gating (Demo and Yellen 1992). An increase in steepness of the G–V could occur if Cs+ occupancy of the pore is voltage dependent. Consistent with this possibility, the instantaneous Ics–V relationship recorded for mSlo (Fig. 12 D) is highly nonlinear, indicating that Cs+ permeates more readily at very positive or negative voltages. Cs+ currents also activate slowly, failing to achieve a steady state during a 25-ms pulse (Stefani et al. 1997). These kinetics differ from those of IK measured for hSlo (Meera et al. 1996) or mSlo (Horrigan et al. 1999) in 0 Ca2+, providing additional evidence for an effect of Cs+ on gating and implying that the Gcs–V recorded with 25-ms pulses doesn't represent the steady-state G–V.

hSlo gating currents were also measured in the presence of internal Cs+, using isotonic external TEA to block the channel. Therefore, the use of Cs+ has the apparent advantage of allowing gating and ionic currents to be recorded with the same internal solution. However, the presence of internal Cs+ does not guarantee that channels gate identically when blocked by TEA or conducting Cs+. Aside from the possibility that TEA directly affects gating (see below), TEA may inhibit the effect of Cs+. Demo and Yellen 1992 found that BK channel block by either internal Cs+ or external TEA had no effect on Po, and they concluded that Cs+ could occupy at least two sites in the pore, only one of which affects gating. Thus, it is possible that internal Cs+ cannot occupy the critical gating site when the pore is blocked by TEA. This could explain why hSlo gating currents resemble those recorded for mSlo, and may help account for the observed crossing of Q–V and G–V for hSlo.

As discussed previously, mSlo gating may also be affected by the presence of external TEA, internal NMDG, or the absence of K+ in experiments measuring gating currents. Several properties of slow charge movement summarized in Fig. 10 suggest that mSlo channels open more readily under the conditions where gating currents were measured. A 10-fold reduction in internal and external K+ had no appreciable effect on the GK–V (data not shown) but we cannot rule out the possibility that gating is altered by the complete removal of K+ or its replacement with NMDG. Stefani et al. 1997 found that fast Ig, evoked at voltages where channels do not open, were unaltered by the presence of TEA. We also saw no effect of TEA on fast charge movement measured with admittance analysis in 0 K+ (data not shown). However, these experiments do not rule out an effect of TEA on slow charge movement and channel opening.

Another factor that could contribute to an apparent cross-over between Q–V and G–V is the duration of the voltage pulses used to measure gating currents. The Q–V for hSlo was determined using 1-ms pulses and is therefore similar to Qfast and not a steady-state measurement. Stefani et al. 1997 state that the Q–V determined with longer pulses (10–20 ms) in 0 Ca2+ is shifted by −20 mV relative to Q1ms, an effect that is insufficient to account for the apparent cross-over of Q–V and G–V. We also observed little difference between the normalized Qss–V and Qfast–V curves in 0 Ca2+ (Fig. 6 D). However, in the presence of 60 μM Ca2+, we observe a large 50 mV shift between Q–Vs determined with 1- or 20-ms pulses (Horrigan, F.T, and R.W. Aldrich, manuscript in preparation). Thus, a cross-over of Q–V and G–V reported by Stefani in 85 μM Ca2+ should be strongly influenced by the use of 1-ms voltage pulses.

Despite uncertainties as to the precise relationship between the Q–V and G–V, gating currents recorded for both mSlo and hSlo show that most charge movement in 0 Ca2+ is fast, indicating that most charge moves before channels open. The kinetics and voltage dependence of the delay in IK activation are also consistent with the idea the multiple voltage-dependent closed-state transitions, accounting for the bulk of charge movement, occur before channels open. Finally, the change in delay duration with prepulse voltage (Cole-Moore shift [Cole and Moore 1960]) is, as stated by Stefani et al., less voltage dependent than that observed for Shaker. However, this is consistent with the overall weak voltage dependence and reduced gating charge of BK channels as compared with Shaker and does not imply that a small proportion of total charge moves before Slo channels open. The Cole-Moore shift for mSlo is well described by the allosteric model, which predicts most voltage sensors activate before channels open (Horrigan et al. 1999). The lack of a rising phase in Ig also shows that the earliest closed-state transitions are not weakly voltage dependent.

Detection of Allosteric Voltage Gating

In the preceding article (Horrigan et al. 1999), we discussed the possibility that the behavior of voltage-gated ion channels such as Shaker may be consistent with an allosteric voltage-gating scheme (Scheme I) like that used to describe mSlo. One reason to consider this possibility is that many, but not all, features of mSlo ionic currents can be adequately described by sequential gating schemes that have been used to describe a variety of other channels. The behaviors of mSlo that deviate from these conventional schemes and implicate the allosteric model are mainly observed under conditions of extreme voltage or low open probability. Therefore, it is possible that other channels operate through an allosteric mechanism but have not been studied under conditions that are necessary to test this model, which may be even more extreme in those channels than for mSlo.

Many of the gating current properties described here for mSlo can also be accounted for by sequential gating schemes containing a single open state such as Scheme IV. However, deviations from the prediction of Scheme IV are more obvious for gating current than for ionic current. The Medium component of OFF charge movement, in particular, provides a direct indication of open-state transitions. Many voltage-gated K+ channels such as Shaker exhibit OFF currents that become slower as pulse duration is increased and channels open (Bezanilla et al. 1991; Bezanilla et al. 1994; Chen et al. 1997; Kanevsky and Aldrich 1999; Zagotta et al. 1994b). However, unlike mSlo, the decay of OFF currents for open Shaker channels appear to be limited by the speed of channel closing as predicted by models like Scheme IV (Zagotta et al. 1994b). This observation seems to argue against a model with multiple open states. However, as discussed below, the allosteric scheme can account for such results when the speed of voltage-sensor movement or the voltage dependence of open-state transitions is altered.

Two factors allow open-channel charge movement to be detected for mSlo. First, voltage-sensor movement is much faster than channel closing. This difference allows the Medium and Slow components of OFF charge movement to be distinguished and allows open-state transitions to occur before channels close. As discussed below, the relative speed of Ig and IK in channels such as Shaker might prevent detection of open-state transitions. Another factor that is important for detecting open-state transitions is the open-state charge distribution (QO). That is, the voltage dependence of open-state transitions must be such that repolarization to the holding potential causes a redistribution of channels among open states. The effect of repolarization voltage on QOFF components in Fig. 8 shows that the Medium component is sensitive to the open-channel charge distribution. Therefore, a change in the voltage dependence of QO might alter the ability to detect open-state transitions. For example, if the allosteric factor D is increased, QO will be shifted to more negative voltages such that QOFFmed measured at −80 mV is reduced.

Fig. 13 shows that a slowing of voltage-sensor kinetics reduces the ability to detect open-channel charge movement. IK and Ig were simulated in response to a 20-ms pulse to +240 mV as the forward and backward rates for voltage-sensor movement (α, β) were both slowed 10-fold (10×) or 30-fold (30×) relative to those describing mSlo (1×). C–O transition rates and all equilibrium constants were unchanged (relative to Case B parameters). As voltage-sensor movement is slowed, the delay in IK activation increases (Fig. 13 A) and gating currents are slowed (Fig. 13 B). Under these conditions, IK and Ig resemble those evoked from a channel such as Shaker where IK activation kinetics are sigmoidal and ionic and gating currents relax on a similar time scale. Interestingly, a 30-fold slowing of voltage sensor movement also produces a “hook” in IgOFF, a feature that is also observed in Shaker Ig (Bezanilla et al. 1991; Chen et al. 1997; Perozo et al. 1992). The simulation predicts that Po approaches unity at the end of a pulse to +240 mV. Therefore, IgOFF represents the relaxation of open channels. The time course of QOFF(t) plotted on a semilog scale in Fig. 13 C is biphasic when voltage-sensor movement is fast (1×) representing the Medium and Slow components of QOFF. However, kinetically distinct components of QOFF are not evident when voltage-sensor movement is slow (Fig. 13 C, 10× and 30×). Similarly, the decay of IgOFF is much faster than that of IK when voltage sensors are fast (Fig. 13 D, 1×). However, IgOFF and IK decay with similar kinetics when voltage sensors are slow (Fig. 13 D, 10× and 30×). These results demonstrate that OFF charge movement can be limited by the speed of channel deactivation even when multiple open states are present.

This work was supported by a grant from the National Institutes of Health (NS23294) and by a National Institute of Mental Health Silvio Conte Center for Neuroscience Research grant (MH48108). R.W. Aldrich is an investigator with the Howard Hughes Medical Institute.

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1used in this paper: BK channel, large-conductance Ca2+-activated K+ channel; NMDG, N-methyl-d-glucamine; TEA, tetraethylammonium