Differential Effects of β1 and β2 Subunits on BK Channel Activity

The high conductance, voltage- and calcium-activated potassium channel (BK or MaxiK channel) is widely expressed among mammalian tissues (Toro et al., 1998). Its open probability increases on membrane depolarization or an increase in intracellular calcium concentration. As its activation leads to membrane hyperpolarization, it serves as a negative-feedback mechanism for the excitatory events that lead to increases in calcium concentration or raise the membrane potential. In vascular smooth muscle cells, BK channels play a key role in regulating the contractile tone (Nelson and Quayle, 1995; Brenner et al., 2000b); in chromaffin cells these channels help terminate the action potential and thus they modulate secretion (Neely and Lingle, 1992; Solaro et al., 1995), and in neurons BK channels colocalize with voltage-dependent calcium channels and are involved in the control of neurosecretion (Sah and Davies, 2000).

The BK channel is a homotetramer of its pore-forming α subunit, which is coded by the gene Slo1 (KNCMA1) and is a member of the voltage-dependent potassium (Kv) channels superfamily. As in all other Kv channels, the S4 transmembrane segment is (or is part of) an intrinsic voltage sensor (Diaz et al., 1998; Cui and Aldrich, 2000). Gating and ionic currents in BK channels can be elicited by membrane depolarization in the absence of calcium, suggesting that this is a voltage-dependent channel (Stefani et al., 1997). The divalent cation acts as a modulator able to decrease the necessary energy to open the channel, promoting a leftward shift in the open probability (P_(O)) vs. voltage relationships. Therefore, as calcium is increased, less voltage is needed to activate the channel. The channel is also activated by intracellular Mg²⁺, whose binding sites are also low affinity binding sites for Ca²⁺ (Shi and Cui, 2001; Zhang et al., 2001).

Although only one gene codes for the BK channel (Slo1, KCNMA1), a wide variety of channel phenotypes has been found. These phenotypes differ in their calcium and voltage sensitivity as well as in their macroscopic kinetics. This variety is explained by several regulatory factors, including alternative splicing, protein phosphorylation, and, most importantly, coexpression of tissue-specific accessory proteins, called β subunits (for review see Orio et al., 2002). Four β subunits have been cloned, termed β1–β4 (Knaus et al., 1994; Wallner et al., 1999; Xia et al., 1999; Behrens et al., 2000; Brenner et al., 2000a; Meera et al., 2000; Uebele et al., 2000). The effects of the β1 subunit coexpression include an increase of the apparent calcium sensitivity, a decrease of the voltage dependence of the channel, and a slow down of the macroscopic kinetics (McManus et al., 1995; Meera et al., 1996; Cox and Aldrich, 2000; Nimigean and Magleby, 2000; Qian and Magleby, 2003). The β1 subunit also modulates the interaction of the channel with toxins and other activators (Dworetzky et al., 1996; Hanner et al., 1997; Valverde et al., 1999). The β2 subunit has an N-type or fast inactivation motif in its NH₂ terminus, causing BK currents to inactivate when this subunit is present (Wallner et al., 1999; Xia et al., 1999). Besides this, the β2 subunit also enhances BK channel's calcium sensitivity, in a very similar way to that of the β1 subunit. The β3 subunit has four splice variants. Three of them confer some degree of inactivation to BK currents, and they also produce an outward rectification of the open channel currents (Brenner et al., 2000a; Xia et al., 2000; Zeng et al., 2001). None of the β3 subunits have any effect on the apparent calcium sensitivity of the channel. Finally, the β4 subunit has the opposite effect as the β1 subunit, decreasing the apparent calcium sensitivity of the BK channel (Brenner et al., 2000a; but see Ha et al., 2004). This subunit also slows down the macroscopic kinetics of the channel.

Detailed studies of the modulation of the apparent calcium sensitivity have been carried on for the β1 subunit. These studies strongly suggest that the β1 subunit affects the functional coupling between calcium binding and channel opening rather than the calcium binding affinity (Cox and Aldrich, 2000; Nimigean and Magleby, 2000). It is unclear at present whether or not the other β subunits modify the apparent Ca²⁺ sensitivity through a mechanism similar to that proposed for the β1 subunit.

The synergistic activation of the BK channel by two different stimuli (increases in intracellular calcium and membrane depolarization) has been shown to be rather complex, leading to the proposition of different kinetic schemes of up to 250 states and even more (for review see Magleby, 2003). One of the key features that define the behavior of BK channel is that neither calcium nor voltage is strictly necessary for channel activation. In the virtual absence of calcium, the channel is activated by large depolarization (Meera et al., 1996; Cui et al., 1997; Stefani et al., 1997), and when all voltage sensors are resting, the open probability of the channel can be increased by augmenting intracellular calcium (Horrigan and Aldrich, 2002). Even in the absence of both stimuli (in very low calcium and at very negative potentials), the channel still opens with a very low, but measurable, open probability (∼10⁻⁶) and a low voltage dependence not related to the voltage sensor (Horrigan et al., 1999; Horrigan and Aldrich, 2002). This, and several other observations, including macroscopic and single channel kinetics, led to the idea that both calcium and voltage increase the open probability by an allosteric mechanism (Horrigan et al., 1999; Rothberg and Magleby, 1999, 2000; Cui and Aldrich, 2000; Talukder and Aldrich, 2000; Horrigan and Aldrich, 2002). In this type of mechanism, neither voltage sensor movement nor calcium binding are strictly coupled to channel opening; these three processes are independent equilibriums that interact allosterically with each other. The existence of at least two and maybe three calcium binding sites with different affinities (Zhang et al., 2001; Xia et al., 2002) makes the picture more complicated and raises, almost exponentially, the number of states in a model.

The best compromise between simplicity and reproduction of the voltage and calcium dependence in a wide range of voltages and calcium concentrations, including very low open probabilities, is probably achieved by a 70-state allosteric activation model (Horrigan and Aldrich, 2002). This model (also explained in Fig. 1) takes into account two voltage dependences, one for the channel opening itself and one for the voltage sensor movement, and only the high-affinity calcium dependence. It also includes some allosteric interaction between calcium binding and voltage sensor movement. The open probability at any voltage and calcium concentration is given by the equation

where

V_h(J) is the voltage for J = 1 (the half-activation voltage of each voltage sensor) for the closed channel, with no calcium bound and at V = 0. L₀ is the equilibrium constant for channel opening with all voltage sensors in resting state, with no calcium bound and at V = 0. K_d is the dissociation constant of a single calcium binding site, for the closed channel and with all voltage sensors in resting state. z_J and z_L are the voltage dependences for J and L, respectively. C, D, and E are the allosteric factors.

In this paper, we present a detailed study of the effects of the β1 subunit and the β2 subunit without the inactivation domain (β2IR) on the calcium and voltage dependence of the BK channel. We focus on how the β subunits affect the voltage dependence of channel activation, as no previous study has shown whether the β1 subunit affects either the voltage sensor–associated or the voltage sensor–independent voltage dependence of the BK channel. Our results show that the β1 subunit, but not the β2IR subunit, has an important effect on the voltage sensor–associated voltage dependence of the channel. This effect is very likely to be a reduction in the number of effective gating charges per voltage sensor or the voltage dependence of the sensors activation (from 0.51 to 0.3). Contrary to a previous report (Cox and Aldrich, 2000), only this reduction of the voltage dependence can account for most of the β1 subunit effects on the macroscopic steady-state behavior of the channel, including the enhanced apparent calcium sensitivity. In the case of the β2IR subunit, an increase of the allosteric coupling factors is needed to account for its effects, as well as some change in the calcium binding affinity. We also found that both β subunits reduce the intrinsic equilibrium constant for channel opening (L₀).

cDNAs coding for BK channel α subunit (KCNMA) from myometrium (GenBank/EMBL/DDBJ accession no. U11058), human β1 and β2 subunits (KCNMB1 and KCNMB2, GenBank/EMBL/DDBJ accession no. U25138 and AF099137) and β2 subunit without inactivation domain (β2IR) (Wallner et al., 1999) were provided by L. Toro (University of California at Los Angeles [UCLA], Los Angeles, CA). mMESSAGE mMACHINE (Ambion) in vitro transcription kit was used to obtain mRNAs. Channels were expressed in Xenopus oocytes using standard techniques (Stühmer and Parekh, 1995). For macroscopic current recordings, 0.75–2 ng of α subunit cRNA or a mixture of 0.5–1.5 ng (α subunit) and 2–3 ng (β subunit) was injected per oocyte. For limiting voltage dependence experiments, 30 ng (α subunit) or 12.5 ng (α subunit) + 25 ng (β subunit) was injected. The approximate molar ratio was at least 6:1 (β:α), ensuring saturation of β subunit.

1–5 d after cRNA injection, currents were recorded using the inside-out configuration of the patch-clamp technique. All recordings were performed at room temperature (20–22°C). Intracellular (bath) and extracellular (pipette) solutions contained (in mM) 110 KOH, 10 HEPES, and 2 KCl, and were adjusted to pH 7.4 with methanosulfonic acid. Depending on the desired free calcium concentration, 1.8–3.5 mM CaCl₂ and 5 mM EGTA (5–200 nM), HEDTA (0.68–12 μM), or NTA (22–150 μM) was added. Free calcium concentrations were calculated using the WinMaxChelator Software (http://www.stanford.edu/~cpatton/maxc.html) and checked with a calcium electrode (World Precision Instruments). For the solution designated as 5 nM, this is an upper estimation based on contaminating calcium. Depending on the magnitude of currents, and in order to minimize voltage drops due to series resistance, some experiments were done with solutions containing 36 mM KOH and 74 mM NMDG. The pipette solution contained 100 nM or lower free calcium.

Pipettes were pulled in a horizontal pipette puller (Sutter Instruments) from Corning 7740 (Pyrex) or Custom 8250 (Warner Instruments) borosilicate capillary glass. Pipette resistance was 0.8–2 MΩ in 110 K⁺, and series resistance error was always <10 mV.

Data were acquired with an Axopatch 200B (Axon Instruments) or an EPC-7 (List Medical) amplifier, filtered with an 8-pole Bessel filter (Frequency Devices) at 1/5 of the acquisition rate and sampled with a 16-bit A/D converter (NI-6036; National Instruments). Typical acquisition rates for macroscopic current recordings ranged from 66.6 to 100 kHz (α subunit alone) or from 16.6 to 40 kHz (α + β subunits). For unitary events quantification, recordings were sampled at 50 kHz for α and 20 kHz for α + β channels. A homemade acquisition software was developed in the LabView programming environment (National Instruments). Primary data analysis was performed with Analysis (UCLA) and Clampfit 9 (Axon Instruments) software.

The Solver function of Microsoft Excel was used to fit instantaneous tail currents to a Boltzmann function of the form

where I_max is the maximum obtained tail current, z is the voltage dependency of activation, V_0.5 is the half-activation voltage, T is the absolute temperature (typically 295 K), F is the Faraday's constant, and R is the universal gas constant. When [Ca²⁺] was <500 nM, only low conductances were achieved and the estimation of I_max became unreliable. In those cases, G_max was fixed using the value obtained with higher calcium concentrations in the same patch. All the analyses that follow were performed using I/I_max values.

ΔG values were calculated for each experiment as −zFV_0.5. Mean ΔG values were plotted against calcium concentration and fitted to a concentration-effect sigmoid equation (Eq. 6). As we used calcium concentrations uniformly distributed in a logarithmic scale, the fit was performed using log([Ca²⁺]).

To test whether a fit parameter was statistically different between two (or more) datasets, a global fit (null hypothesis, in which the parameter of interest was restricted to be equal among all datasets) was compared against an independent fit of each dataset (alternative hypothesis). Comparison between independent and global fits was done by an extra sum-of-squares F test, using the formula

where SS and DF are the sum-of-squares and the degrees of freedom of the fit, respectively. glb and ind denote the global and the independent fit, respectively. All the analysis and the conversion of the F value to a P value were done with the GraphPad Prism software (GraphPad Software Inc.).

All-points amplitude histograms were obtained from 20–45 s recordings. For each histogram, NP_(O) was calculated by measuring the fraction of time (P_k) spent at each open level (k) using a half-amplitude criteria and summing their contributions NP_(O) = ΣkP_k. In some experiments, it was checked that the half-amplitude criteria method yields a similar result (<5% difference in NP_(O)) than a Gaussian fit of the histograms. For voltages >50 mV, NP_(O) was calculated from the macroscopic tail current measurement, dividing by the unitary conductance (220 pS) and the tail voltage. In some experiments, the number of channels in the patch (N) was calculated by raising the calcium concentration and performing a nonstationary noise analysis (Sigworth, 1980; Alvarez et al., 2002) or measuring total patch conductance between −10 and 10 mV when all channels were open. In this way, an absolute P_(O) was obtained and was used to normalize and average the experiments.

For each calcium concentration, a mean V_0.5 (<V_0.5>) value was obtained. All the corresponding G/G_max vs. V curves were displaced in the voltage axis by V_0.5 − <V_0.5>, allowing us to construct an average curve that preserved the shape of the individual curves at every [Ca²⁺]. Each of these curves comprised three to eight sets of 30 to 35 points that were converted to a 40-point set by a smoothing function in Sigmaplot 8.0. The families of G/V curves were simultaneously fitted to Eq. 1 by minimizing least-squares with a mixture of Marquardt-Levenverg and Simplex iterations in Origin Pro (OriginLabs). Several fits were attempted with different initial parameters and the fits with lower chi-squared were chosen.

Fig. 2 summarizes the voltage- and calcium dependence of the BK channel and how it is affected by the β1 and β2 subunits. We studied a truncated β2 subunit (Δ2-19β2, β2IR) that lacks the inactivation peptide. In the presence of the β2IR subunit, BK currents are sustained (compare Fig. 2 A, α+β2wt and α+β2IR) and therefore the effects of this subunit in the activation parameters of the BK channel are best studied.

The effect of voltage on the steady-state open probability is measured by a fit of the tail current vs. voltage plots to a Boltzmann distribution (Eq. 2; Fig. 2 B). Increasing the calcium concentration lowers the half-activation voltage (V_0.5; Fig. 2 C). In the presence of both the β1 and β2IR subunits, the effect of calcium is enhanced, as the V_0.5 values for α+β1 and α+β2IR are lower than for α at [Ca²⁺] ≥ 1 μM. There are, however, noticeable differences between the β subunits. First, the V_0.5 values for α+β2IR are lower than for α+β1 at all calcium concentrations (P < 0.0001 in a two-way ANOVA test). Second, at [Ca²⁺] ≤ 100 nM, α+β1 channels have higher values than either α or α+β2IR channels (P = 0.003 compared with α, and P < 0.0001 compared with α+β2IR) while α+β2IR is not different from α (P = 0.2).

As the calcium concentration is increased >200 nM, the voltage dependence of the BK channel increases in the 0.1–10 μM range and then decreases again as the calcium concentration is further increased (Fig. 2 D). In the presence of the β1 and β2IR subunits, this behavior is maintained, but the β1 subunit shows lower z at all calcium concentrations (P < 0.0001 when compared with α with a two-way ANOVA test; P = 0.008 for [Ca²⁺] ≤ 100 nM). On the contrary, the β2IR subunit does not modify the voltage sensitivity of the channel (P = 0.27).

The changes in the V_0.5/[Ca²⁺] relationships may be interpreted as a change in the calcium binding affinity of the channel. It has been already proposed, however, that the β1 subunit acts by a mechanism different than enhancing the calcium binding to the channel (Cox and Aldrich, 2000; Nimigean and Magleby, 2000). The data we present below suggest that this is true for the β2IR subunit as well.

The Boltzmann distribution used to obtain the V_0.5 and z values is based on the simplifying assumption of a two-state model. In this model, the difference in free energy (ΔG) between the open and closed states of the channel is given by

Although the BK channel has more than two states, we can use Eq. 3 to obtain the overall energy difference between open and closed states. The V_0.5 and z values from Fig. 2 (C and D), as well as Eq. 3, were used to obtain the ΔG values plotted in Fig. 3 against calcium concentration (symbols). The ΔG/[Ca²⁺] relationship shows that calcium affects this energy difference in a simple concentration-dependent fashion. Therefore, the ΔG/[Ca²⁺] relationship can be fitted to an equation of the form

where ΔG⁰ is ΔG in absence of calcium, ΔΔG_Ca is the change in ΔG produced by calcium binding to the channel (in saturating calcium, ΔG = ΔG⁰ + ΔΔG_Ca), EC₅₀ is the half-effect concentration, and n is the slope or concentration dependence of the change. As our data span several orders of magnitude of calcium concentration and are evenly distributed in a logarithmic scale, it is more appropriate to express the calcium concentration and EC₅₀ as log[Ca²⁺] and logEC₅₀, respectively. Then

which can be reordered to

The continuous lines in Fig. 3 represent the best fit to Eq. 6, and the parameters for the best fit are listed in Table I. Clearly the β subunits only modify significantly the ΔΔG_Ca parameter, while the rest of the parameters remain comparatively unaffected. In other words, the β1 or β2IR subunit coexpression only changes the effect of calcium on the energy difference, while neither the energy difference in absence of calcium nor the calcium binding parameters of BK channel are severely affected. The EC₅₀ and n values come from an empirical measurement of the calcium effect; therefore, they cannot be related directly to the K_d or C parameters of the allosteric activation model. As will be shown below (see discussion), changes in parameters other than K_d and C can affect the EC₅₀ of the ΔG/[Ca²⁺] plot. The obtained EC₅₀ value will be regarded as a measurement of the concentration range for the calcium effect and it is clear that neither the β1 nor the β2IR subunit affect it significantly. It has been already proposed that the β1 subunit acts by a mechanism different than enhancing the calcium binding to the channel (Cox and Aldrich, 2000; Nimigean and Magleby, 2000), and these data may suggest that this is true for the β2IR subunit as well.

The ΔG analysis presented in Fig. 3 and Table I also suggests that the β subunits do not affect ΔG⁰, the energy difference between closed and open states when the channel is in absence of calcium. Therefore, the changes in V_0.5 that the β1 subunit produces in this condition must be due to a change in the voltage sensitivity, evidenced by the change in the z parameter. As the gating of BK channel has been shown to have two voltage-dependent processes, the movement of the voltage sensors and the opening of the pore gate (Horrigan and Aldrich, 1999; Horrigan et al., 1999), in the following experiments, we addressed whether the β1 and the β2IR subunits modulate any (or both) of these voltage dependences.

In voltage-dependent channels, the lnP_(O)/V relationship becomes linear and reaches a maximum slope (known as limiting slope) at very low open probabilities. It has been shown (Almers, 1978; Sigg and Bezanilla, 1997) that for any linear voltage-dependent model

where Δq is the voltage sensor charge effectively coupled to channel activation and F, R, and T have their usual meanings. The BK channel departs from this behavior; it reaches a maximum slope of ∼2 at P_(O) ∼ 10⁻³, but when the P_(O) is further lowered, a lower slope of ∼0.3 is reached at P_(O) < 10⁻⁶. This is one of many observations that led to the proposition of an allosteric mechanism for the voltage-dependent activation of the BK channel (Horrigan et al., 1999; Rothberg and Magleby, 1999). In this model, the opening of the channel has an intrinsic voltage dependence unrelated to the movement of the voltage sensors. The allosteric model is depicted in Fig. 1 B, where both the L (for channel opening) and J (for voltage sensor activation) equilibrium constants are voltage dependent. In the absence of calcium and at very negative potentials, all voltage sensors are in the resting state. In this condition, the model is confined to the C₀–O₀ equilibrium, governed by the equilibrium constant L. Then,

As L << 1, we can approximate P_(O) ≈ L, and then

Therefore, for the BK channel the limiting slope is actually given by

where z_L is the voltage dependence for the equilibrium constant L or the voltage dependence for channel opening.

We studied whether the β1 or the β2IR subunit have any effect on this behavior of the BK channel. To determine the limiting slope of the voltage dependence, we quantified single unitary opening events in patches containing hundreds and even thousands of channels, in conditions of very low open probabilities. Fig. 4 A shows the single channel activity of a patch containing ∼10,000 BK channels (α subunit alone) at three holding voltages. The opening events become less frequent at negative potentials as evidenced in the all-points histograms in Fig. 4 B (compare the ∼200 pS peaks at −60 vs. −120 mV). In the presence of the β1 and β2IR subunits (Fig. 4, C–F), a similar behavior was observed.

Several single channel recordings as well as some macroscopic recordings were averaged (see materials and methods) to construct the semilogarithmic P_(O)/V plots shown in Fig. 5 (A–C). All three plots tend to a limiting slope at very negative potentials that represents the z_L parameter in the allosteric activation model. The value for α alone channels is very similar to that previously published by others (Horrigan et al., 1999; Horrigan and Aldrich, 2002). The presence of the β1 subunit seems to make the limiting slope higher (Fig. 5 B), however a derivative plot (Fig. 5 B, inset) shows that the actual limiting slope is not actually reached at −120 mV and that the actual value must be closer to 0.3. In the case of the β2IR subunit, the value is not statistically different to the z_L of the α alone channel (P = 0.6 for different slopes). All plots have a maximum slope in the 0–100 mV range, which is related to the voltage sensor–associated activation of the channels and represents the limiting slope value that is achieved by macroscopic current methods (Diaz et al., 1998). The slope is similar for α and α+β2IR channels (P = 0.44), but for α+β1 it is lower (P = 0.001). These observations suggest that only the β1 subunit affects the voltage sensor–associated activation of the channel and none of the studied β subunits affects the intrinsic voltage dependence of the channel opening transition. To analytically check this assumption, we performed a fit of the P_(O) relationships to the allosteric model for voltage dependence activation.

In the absence of calcium, the allosteric model for voltage activation predicts that the open probability as a function of voltage is given by the equation

where L = L₀exp(z_LFV/RT) and J = exp(z_JF(V − V_h(J))/RT). The parameters to be fitted are L₀, V_h(J), z_L, z_J, and D (see introduction).

P_(O) values from Fig. 5 (A–C) were fitted to Eq. 11. Before allowing the parameters to vary freely during the fit, we noted that previous works (Horrigan and Aldrich, 1999, 2002) showed that the half-activation voltage for voltage sensors (V_h(J)) is around 150 mV. This assumption is confirmed in our study by the maximum of the τ_act/V plot (see Fig. 9 B), which is highly dependent on V_h(J) (Horrigan et al., 1999). Moreover, this maximum is not affected by β subunit coexpression (Fig. 9 B). Therefore, we restricted V_h(J) to be between 140 and 160 mV in all the fits. Based on the measured limiting slope, z_L was restricted to be >0.25. Two fits were performed: one in which z_L was free to vary in each dataset (α, α+β1, and α+β2IR) and a second one in which z_L was restricted to be equal for the three datasets. An extra sum-of-squares F test yielded that a common z_L cannot be discarded (P = 0.22) and was preferred for the following analysis. Fig. 5 D (lines) shows the best simultaneous fit with the same z_L value and the parameters are listed in Table II.

The L₀ parameter is greatly reduced in the presence of both β subunits, related to the lower P_(O) values obtained at negative voltages. The reduction of L₀ is accompanied by an increase of the allosteric factor D, the functional coupling factor between the activation of the voltage sensors and channel opening. The effect on L₀ and D is a common feature of both β subunits, but a striking difference between them is that the β1 subunit has a noticeable effect on the z_J parameter. Simulations performed with the allosteric model for voltage activation show that only a change in the z_J parameter dramatically reduces the maximum slope of the ln(P_(O))/V relationship (Fig. 6 B, top left). z_J is the voltage dependence for the activation of the voltage sensors and it has been suggested to be equal to the number of apparent gating charges per sensor (Horrigan and Aldrich, 2002). In fact, this is a strict prediction of the allosteric model since it assumes that the voltage sensors can be either in a resting or in an active state (two-state model). In conclusion, the β1 subunit, but not the β2IR subunit, promotes a reduction of the voltage dependence of the voltage sensor movement.

Voltage sensor–associated and voltage sensor–independent processes can also be dissociated by analyzing the voltage dependence of the macroscopic kinetics of BK channel. To avoid contaminations arising from calcium-dependent processes, we studied the effect of the β subunits on the macroscopic kinetics of the BK channel in the virtual absence of calcium (10 nM or lower).

Fig. 7 A shows representative current traces for channel deactivation at −60 mV after a 200-mV depolarizing pulse. Both β subunits slow down the deactivation process of the channel, and the time course of the current induced by α+β1 channels is slower than that promoted by α+β2IR channels. Current traces are well fitted by a single exponential time course, and Fig. 7 B shows the deactivation time constants (τ_deact) obtained in a wide range of voltages. In the voltage range from −220 to 0 mV, the deactivation time (τ_deact) constant for α+β1 channels is higher (slower) than that for α+β2IR. At ∼0 mV, this tendency is inverted. Moreover, the semilogarithmic τ_deact/V plots for α and α+ β2IR clearly show two slopes, while the α+β1 plot apparently shows only one slope.

The observed behavior of the τ_deact/V curve for the α subunit has already been reported and interpreted in terms of the allosteric model for voltage activation in the absence of calcium (Horrigan et al., 1999). Fig. 8 A shows the same allosteric model as in Fig. 1 B but with the kinetic rate constants in place of the equilibrium constants. As the voltage sensor movement is much faster than the opening and closure of the channel, with kinetic constants up to 100 times larger (Horrigan and Aldrich, 1999; Horrigan et al., 1999), the scheme can be abbreviated to the one in Fig. 8 B, which considers the voltage sensors in equilibrium. Upon a square voltage pulse, the time constant for current relaxation will be given by the γ_n and δ_n kinetic constants weighed by the relative probabilities of the O_n and C_n states, respectively. This is expressed by the equation

At low voltages, γ_n >> δ_n, and the expression can be approximated to

At very negative potentials, all voltage sensors will be in their resting state and therefore the only (or prevailing) open state will be O₀. Then,

In other words, the τ_deact/V plot will exhibit an exponential dependence to voltage (linear in a semilogarithmic plot) whenever the voltage sensors for the open channel are all in the resting state. Thus, the limiting voltage dependence of the macroscopic kinetics reveals a process unrelated to voltage sensor movements.

When the points from −220 to −190 of each plot are fitted to Eq. 15, a curve is defined (dotted lines in Fig. 7 C) that overlaps very well the points up to −35 mV for α, up to −160 mV for α+β1, and up to −120 mV for α+β2IR. These curves have the same value for z_γ, 0.13 (P = 0.17 for different values). This indicates that the β1 and β2IR subunits do not affect the voltage dependence of the kinetic rate for channel closure (z_γ). Since z_L = z_γ − z_δ, it is expected that β subunits do not affect the voltage dependence of the kinetic rate for channel opening, z_δ.

The fit of the τ_deact/V plots to Eq. 15 also evidences that the α+β1 plot has two slopes. This second slope of the ln(τ_deact)/V relationship obeys to a more complex relation than the first slope, involving all the γ_n (n > 0) kinetic constants (Eq. 13), and it is highly dependent on the voltage dependence of the voltage sensor activation. As expected, the z value for the second slope in α+β1 is much lower (0.23) compared with α (0.46) and α+β2IR (0.47).

Fig. 9 A shows representative current traces for BK channel activation at 200 mV in the absence and presence of the β1 and β2IR subunits. Both β subunits slow down channel activation, though the effect of the β2IR subunit is more pronounced. Like the deactivation time course, all traces are well fitted by a single exponential rise, and Fig. 9 B shows the activation time constant (τ_act) obtained at several activation voltages. The τ_act for channels formed by α+β1 and α+β2IR subunits are always higher than the α subunit alone, and the τ_act for α+β2IR currents are higher than for α+β1. In the voltage range presented here, the limiting voltage dependence is not reached and the only approximation that can be made from Eq. 12 is δ_n >> γ. Thus, the linear range in this plot obeys the expression

which involves voltage sensor equilibriums within closed states. The fit of this linear range to a simple exponential equation is shown in Fig. 9 C. As expected for a voltage sensor–related relationship, the z value for α+β1 (0.11) is lower than for α (0.19) and α+β2IR (0.24).

It is noteworthy that all three τ_act/V plots have more or less the same shape and a maximum around +140 mV. As mentioned before, this strongly suggests that the β subunits do not affect the half-activation voltage of the voltage sensors (see Horrigan et al., 1999).

In summary, the study of the voltage dependence of the macroscopic channel kinetics is consistent with our previous conclusions: (a) only the β1 subunit affects the voltage sensor–associated voltage dependence of BK channel, presumably decreasing the voltage dependence of the voltage sensor activation, (b) none of the studied β subunit affects the voltage dependence for the C–O transition, and (c) both β subunits enhance the allosteric coupling between voltage sensor activation and channel opening (allosteric factor D).

We have established how the voltage sensitivity of the BK channel is affected by the β1 and β2IR subunits by looking at the channel behavior in the absence of calcium. Our next step is to evaluate the impact of these calcium-independent changes on the apparent calcium sensitivity of the channel, i.e., the behavior of the channel when the calcium concentration is increased. To do so, we fitted the P_(O)/V relationships in the whole 5 nM–150 μM range to Eq. 1. P_(O)/V curves for α, α+β1, and α+β2IR channels and at different calcium concentrations were averaged in a way that preserved the average V_0.5 and shape of the individual data obtained (see materials and methods). It is important to note that we and others (Horrigan and Aldrich, 2002) have noticed that a simple least-squares fit of the data to Eq. 1 can yield parameters far from those obtained by direct measurements. Therefore, before attempting such a fit, it is important to collect as much data as possible about realistic parameter values and introduce them as restrains in the fitting procedure. Also, we will refrain from drawing conclusions about the absolute values obtained with this procedure and will focus on the qualitative comparison between the different sets of parameters.

First, we obtained a satisfactory fit of the P_(O)/V curves for α channels. Based on the maximum of the τ_act/V plots, as well as on previous data by others (Horrigan et al., 1999; Horrigan and Aldrich, 2002), we restricted V_h(J) to be between 140 and 160 mV. After the results of the fit to the voltage-dependent activation in the absence of calcium (Table II), we applied the restrictions 0.5 < z_J < 0.6, 0.25 < z_L < 0.35, and 10⁻⁶ < L₀ < 10⁻⁵. K_d, C, D, and E were only restricted to be higher than 0. The best fit obtained under these conditions is shown on Fig. 10 A, together with the best-fit parameters. The parameters obtained are in a reasonable agreement with those obtained in the absence of calcium (Table II) and with previous fits (Horrigan and Aldrich, 2002). The predicted V_0.5 and z values are plotted together with the experimental values in Fig. 10 (C and D) (continuous line and filled triangles, respectively), showing that these parameters reproduce very well the voltage- and calcium-dependent steady-state activation of the BK channel.

Our previous analysis demonstrated a significant reduction of the z_J parameter in the presence of the β1 subunit. Therefore, we assessed the importance of this reduction in the apparent calcium sensitivity of the BK channel. Fig. 10 B, left, shows the result of fitting the α+β1 P_(O)/V curves to Eq. 1 with all the parameters fixed at the same value as in the α fit, with the exception of z_J, which was allowed to vary freely between 0.25 and 0.6. Surprisingly, a reduction of z_J from 0.5 to 0.3 is enough to induce both the shift to the left of the G/V curves along the voltage axis as well as the reduction of their voltage dependence. This is best appreciated in Fig. 10 (C and D), where the fit (gray lines) accounts for the experimental V_0.5/[Ca²⁺] and z/[Ca²⁺] relationships (open triangles). A better fit is obtained when L₀ is set to 3 × 10⁻⁷ (to be in agreement with the results in Table II) and the allosteric factors C and D are left free to vary (Fig. 10 B, right). As in the fit to the model in absence of calcium (Table II), a higher D value is obtained compared with the fit parameters obtained for α channels. The z_J value is still 0.3 and the allosteric factor C does not change, even though many iterations with different initial values were performed.

From the analysis of the data in the absence of calcium, we concluded that the β1 subunit reduces the voltage dependence of the activation of the voltage sensors. The fit to the allosteric model for the voltage- and calcium-dependent activation shows that this reduction can account for most of the characteristic increase of the apparent calcium sensitivity that the β1 subunit induces on the channel. The model predicts that an effect completely unrelated to the calcium-dependent activation parameters is almost sufficient to affect the apparent calcium sensitivity.

The fit of the P_(O)/V curves for α+β2IR channels to Eq. 1 was first attempted with the following constraints: 0.25 < z_J < 0.6; 0.25 < z_L < 0.4; V_h(J) =140; 10⁻⁸ < L₀ < 10⁻⁶ and K_d, C, D, and E > 0. These are similar to the constraints applied to the fit for α except for a different range for L₀ (see Table II) and a wider range for z_J. Fig. 11 A, left, plots the predictions of the best fit (lines) together with the experimental averaged P_(O)/V curves (symbols) and the list of the best-fit parameters (fit 1). This fit predicts a z_J value similar to that of α+β1 and much lower than for α, suggesting that the β2IR subunit also reduces the voltage dependence for voltage sensor activation. This result is incompatible with the maximum slope measurements (Fig. 6) and its deviation from the experimental data is clearly reflected in the failure of fit 1 to reproduce properly the z/[Ca²⁺] relationship (Fig. 11 C, continuous line). We ultimately restricted conditions so as to match best our previous experimental results. Therefore, we applied the following restrictions: from the fit of the P_(O)/V curve in the absence of calcium we fixed V_h(J) = 140 mV, 10⁻⁷ < L₀ < 10⁻⁶, z_L = 0.3, and D = 29. From the same result, z_J was restricted to be only slightly lower than the value for α, therefore the restriction was 0.45 < z_J < 0.6. Finally, K_d, C, and E > 0. One of the best fits obtained in this way is shown in Fig. 11 A, right (fit 2). Though fit 2 does not accurately characterize the P_(O)/V relationships at all calcium concentrations, it does predict a higher voltage dependence (higher z values) than α+β1 channels (Fig. 11 C, dotted line). This is the key difference between β1 and β2IR subunits and should not be disregarded. Moreover, in the absence of calcium, the parameters from fit 2 predict a maximum dln(P_(O))/dV value similar to that of the α fit and higher than the α+β1 fits (not depicted). Unlike the β1 subunit, this fit predicts the increase of the allosteric coupling factors for both calcium- and voltage-dependent activation (C and D, respectively). Remarkably, fit 2 also predicts an increase of the allosteric factor that relates the activation of voltage sensors with calcium binding (E). The above analysis suggests that the β2IR subunit enhances the calcium sensitivity of the BK channel by a different mechanism than the β1 subunit, involving changes in the allosteric coupling factors.

In this study we have addressed in detail the effects that the β2IR subunit has on the calcium and voltage dependence of the BK channel. By comparing the effects of the β1 and β2IR subunits, we draw three principal conclusions: (1) The reduction of the voltage dependence of the BK channel observed in the presence of the β1 subunit is strictly related to a reduction of the voltage dependence of the voltage sensor activation process (z_J); (2) In the allosteric model for voltage- and calcium-dependent activation, this reduction of z_J produces an increase of the apparent calcium sensitivity and is able to account for all the effects of the β1 subunit on the steady-state properties of BK channel, and (3) The β2IR subunit enhances the calcium sensitivity of the BK channel by a different mechanism than the β1 subunit, very likely to involve severe changes to the allosteric coupling factors.

Though previous studies showed a reduction of the BK channel voltage dependence in the presence of the β1 subunit (Cox and Aldrich, 2000), we probed further by determining whether this effect was related to a modification in the voltage sensor activation mechanism. The reduction of the maximum dln(P_(O))/dV relationship or maximum slope in the presence of the β1 subunit is a clear indication that the voltage dependence of the voltage sensor activation is the parameter affected by this subunit. Surprisingly, when introduced into the allosteric activation model, this reduction in the voltage dependence provokes an increase of the apparent calcium sensitivity, almost identical to the effect of the β subunit on the BK channel. The similarity to the steady-state effects of the β1 subunit becomes more striking when ΔG values as a function of calcium concentration are predicted from the model. Fig. 12 A shows ΔG values corresponding to the fits in Fig. 10 plotted against calcium concentration, together with the corresponding fits to Eq. 6. Just like with the fit of the experimental data (Table I), reducing the z_J value is enough to provoke a slight decrease on the EC₅₀ for the effect of calcium on channel energetic. While in the experimental data (Table I) this shift was from 3 to 1.5 μM (not significant because of the data dispersion), in the model, a reduction of z_J produces an almost identical shift from 3.4 to 1.9 μM. This also shows that the EC₅₀ value for the ΔG/[Ca²⁺] relationship is related to more than just the K_d and C parameters.

How we can explain that a change completely unrelated to the effect of calcium produces such a change in the apparent calcium sensitivity? The z_J parameter defines how J changes with voltage, as J = exp[z_JF(V − V_h(J))/RT]. Fig. 12 B shows an important effect of changing z_J: at any voltage lower than V_h(J), J will be higher when z_J is lower. As the P_(O) of the BK channel increases with J, the effect of the β1 subunit would be of increasing the P_(O) at any voltage <140 mV. This explains that when the V_0.5 of the channel is >140 mV (at [Ca²⁺] < 200 nM) the β1 subunit appears with a double effect: it increases the P_(O) at V < 140 mV and decreases it at V > 140 mV, with the consequent increase in V_0.5. On the contrary, when the V_0.5 for the α subunit is <140 mV ([Ca²⁺] > 200 nM), the β1 subunit appears only to increase the P_(O) at any voltage with the consequent decrease of V_0.5.

Qian et al. (2002), showed that the β1 subunit can still enhance the calcium sensitivity of BK channels with a disrupted Ca²⁺ bowl, but not when the entire “tail” domain is replaced with the tail from the highly related, pH-dependent, Slo3 channel. Our interpretation of the data offers an explanation for those findings. In the case of the disrupted Ca²⁺ bowl, the channel is partially calcium dependent, and thus the β1 subunit still enhances the apparent calcium sensitivity by activating it whenever the V_0.5 is <∼140 mV. The Slo1 core/Slo3 tail construct is completely calcium independent and is only activated by voltage (Schreiber et al., 1999). Therefore, the only effect of the β1 subunit that can be detected on this channel is a reduction of the voltage dependence and the consequent increase of V_0.5 just as it does for the wild-type channel in the absence of calcium.

Previous works (Cox and Aldrich, 2000; Nimigean and Magleby, 2000) reported an increase of the intrinsic equilibrium constant for channel opening, represented in the allosteric model by the parameter L₀. This was proposed to explain the increase of the open probability that the β1 subunit produces between 0 and 100 mV in the absence of calcium. The appropriate condition to measure L₀ is, however, in the absence of calcium and at lower voltages so that the voltage sensors are in the resting position (see Eq. 9). Under these conditions, we found that β1 actually decreases the intrinsic open–close equilibrium constant, L₀. Fig. 5 clearly shows that the β subunits reduce, rather than increase, the open probability of the BK channel at 0 Ca²⁺ and voltages lower than −50 mV. We conclude that the β1 subunit increases the P_(O) in the 0–100 mV range because the reduced z_J increases the equilibrium constant for voltage sensor activation. When the calcium and voltage dependences are studied in terms of the macroscopic P_(O)/V curves (Cox and Aldrich, 2000; see Fig. 10 and 11), a change in L₀ is unnecessary to reproduce the effects of the β1 subunit. This is because when all other parameters are kept constant, any decrease of L₀ will be compensated by an increase of the allosteric factor D.

Both β1 and β2IR subunits have a dramatic effect on channel kinetics, slowing down the macroscopic current relaxation as well as increasing the mean open and closed times of single channel activity (Nimigean and Magleby, 2000; and current traces in Fig. 4, C and E). This implies that both the γ and δ kinetic constants are decreased due to the presence of the β subunits. As L₀ = δ₀₍₀₎/γ₀₍₀₎, it follows that the effect on these kinetic rate constants must be asymmetric otherwise L₀ would not change. δ₀₍₀₎, the kinetic rate constant for the opening of the channel, has to be the more affected (decreased) by these regulatory subunits.

Unlike the β1 subunit, we were unable to identify a simple parameter change that would account for the effects of the β2IR subunit on the voltage and calcium sensitivity of the BK channel. The best fit to the data using the allosteric model indicates an increase in the allosteric coupling factor (C, D, and E), together with slight decreases of the calcium binding affinity and the voltage dependence of the voltage sensor activation. What is the molecular nature of an increase of the allosteric coupling factors? It has been proposed that the intracellular domains of each BK α subunit contributes two RCK (regulatory of K⁺ conductance) domains and that these eight RCK domains form a gating ring similar to the gating ring of MthK channels (Jiang et al., 2001, 2002). From the structure of the MthK channel, it is thought that this gating ring is in charge of extracting free energy from calcium binding, thus providing a molecular basis for the functional coupling between calcium binding and pore gate opening. Recently, the linker between the sixth (S6) transmembrane domain and the first RCK domain of the BK channel has also been shown to play a role in this molecular coupling, acting as a passive spring (Niu et al., 2004). We expect that a change in the allosteric factor C would be related to some interaction of any (or both) of these domains with the β subunits, stabilizing an “active” conformation that in turn stabilizes the open state of the pore. In the case of the molecular coupling between voltage sensor activation and channel opening, the S4–S5 linker as well as the COOH-terminal half of the S6 segment have been postulated to play this role in other voltage-gated channels (Chen et al., 2001; Lu et al., 2002; Tristani-Firouzi et al., 2002). Thus, a change of the allosteric factor D would be related to some interaction with those intracellular domains.

We thank Dr. Ligia Toro for providing the cDNAs used in this work and Mr. Timothy Warren for his helpful revision of the manuscript.

This work was supported by grants from the Fondo Nacional de Investigación Científica y Tecnológica (Fondecyt 103-0830) (R. Latorre) and the Human Frontiers in Science Program (R. Latorre). The Centro de Estudios Científicos is a Millennium Science Institute and is funded in part by a grant from Fundación Andes.

Lawrence G. Palmer served as editor.