Measurements of the BK_Ca Channel's High-Affinity Ca²⁺ Binding Constants: Effects of Membrane Voltage

Abstract

It has been established that the large conductance Ca²⁺-activated K⁺ channel contains two types of high-affinity Ca²⁺ binding sites, termed the Ca²⁺ bowl and the RCK1 site. The affinities of these sites, and how they change as the channel opens, is still a subject of some debate. Previous estimates of these affinities have relied on fitting a series of conductance–voltage relations determined over a series of Ca²⁺ concentrations with models of channel gating that include both voltage sensing and Ca²⁺ binding. This approach requires that some model of voltage sensing be chosen, and differences in the choice of voltage-sensing model may underlie the different estimates that have been produced. Here, to better determine these affinities we have measured Ca²⁺ dose–response curves of channel activity at constant voltage for the wild-type mSlo channel (minus its low-affinity Ca²⁺ binding site) and for channels that have had one or the other Ca²⁺ binding site disabled via mutation. To accurately determine these dose–response curves we have used a series of 22 Ca²⁺ concentrations, and we have used unitary current recordings, coupled with changes in channel expression level, to measure open probability over five orders of magnitude. Our results indicate that at −80 mV the Ca²⁺ bowl has higher affinity for Ca²⁺ than does the RCK1 site in both the opened and closed conformations of the channel, and that the binding of Ca²⁺ to the RCK1 site is voltage dependent, whereas at the Ca²⁺ bowl it is not.

INTRODUCTION

Large-conductance Ca²⁺-activated potassium (BK_Ca) channels are important for the modulation of many physiological processes, such as neuronal firing, smooth muscle contraction, and neurotransmitter release (Storm, 1987; Roberts et al., 1990; Sah and McLachlan, 1992; Robitaille et al., 1993; Nelson and Quayle, 1995; Brenner et al., 2000; Hu et al., 2001; Wang et al., 2001; Semenov et al., 2006). They are uniquely suited to regulate these processes because they are sensitive to both intracellular Ca²⁺ and membrane voltage. This is seen as a leftward shift in the BK_Ca channel's conductance–voltage (G-V) relation as the internal Ca²⁺ concentration is increased. Biophysical studies have shed considerable light on the mechanisms by which voltage influences channel opening (Cui et al., 1997; Stefani et al., 1997; Diaz et al., 1998; Horrigan and Aldrich, 1999, 2002; Horrigan et al., 1999; Rothberg and Magleby, 2000; Bao and Cox, 2005); however, the mechanisms by which Ca²⁺ influences channel opening remain less well understood.

Unlike K⁺ channels gated solely by voltage, the BK_Ca channel's pore-forming α subunit (four per functional channel) contains a large intracellular domain that confers Ca²⁺ sensitivity on a voltage-gated structure (Wei et al., 1994; Schreiber and Salkoff, 1997; Schreiber et al., 1999). The structure of this domain remains a matter of debate; however, it is generally agreed that within this domain there are three distinct Ca²⁺ binding sites, one of low affinity (millimolar dissociation constants) and two of high affinity (micromolar dissociation constants) (Bao et al., 2002; Shi et al., 2002; Xia et al., 2002; Magleby, 2003). Mutations at these sites together eliminate the BK_Ca channel's characteristic Ca²⁺-dependent G-V shift (Xia et al., 2002). The first of these sites to be identified, the Ca²⁺ bowl, is an aspartate-rich region near the carboxy terminus (Schreiber and Salkoff, 1997). Considered high affinity, this site contributes to the channel's Ca²⁺ sensitivity in the micromolar range (Bao et al., 2002; Xia et al., 2002). Mutations within the Ca²⁺ bowl such as D897-901N (referred to as D5N5) or D898A/D900A (referred to here as D2A2) can eliminate the contribution of this site to the channel's Ca²⁺ sensitivity (Bian et al., 2001; Bao et al., 2004). The second high-affinity site, termed here the RCK1 site, resides in a domain thought to be similar in structure to the ligand binding RCK domains of bacterial potassium channels and transporters (Schreiber and Salkoff, 1997; Jiang et al., 2001; Bao et al., 2002; Jiang et al., 2002; Xia et al., 2002; Zeng et al., 2005). Although the residues that coordinate Ca²⁺ at this site have yet to be determined, the mutation D367A has been shown to eliminate the contribution of this second high-affinity site to Ca²⁺ sensing (Xia et al., 2002). The BK_Ca channel's low-affinity Ca²⁺ binding site is also thought to reside in the channel's RCK1 domain, and its influence can be eliminated by the mutation E399N (Shi et al., 2002; Xia et al., 2002).

The binding properties of the BK_Ca channel's two high-affinity Ca²⁺ binding sites are uncertain. Bao et al. (2002) estimated the Ca²⁺ bowl's Ca²⁺ dissociation constant to be 3.5 μM when the channel is closed (K_C) and 0.8 μM when it is open (K_O) (Bao et al., 2002), whereas Xia et al. (2002) estimated K_C to be 4.5 μM and K_O to be 2.0 μM (Xia et al., 2002). These numbers may seem similar, but according to allosteric theory the ratio K_C/K_O is equivalent to the factor by which Ca²⁺ binding at a given site alters the equilibrium constant for channel opening. The estimates of Bao et al. (2002) yield a K_C /K_O value of 4.4, whereas those of Xia et al. (2002) yield a K_C /K_O value of 2.2. Thus, for a single binding event the two groups predict effects of Ca²⁺ on the equilibrium constant for channel opening that differ by a factor of two, and if there are four Ca²⁺ bowl–related sites—as there appears to be (Niu and Magleby, 2002)—then when all four sites are occupied, the difference is 14-fold. Further, there are larger differences between the two groups' estimates of K_C and K_O for the channel's other type of high-affinity Ca²⁺ binding site, the RCK1 site. The estimates of Bao et al. (2002) are considerably smaller than those of Xia et al. (2002) and more like those of the Ca²⁺ bowl (Bao et al. [2002]: K_C = 3.8 μM and K_O = 0.9 μM; Xia et al. [2002]: K_C = 17.2 μM and K_O = 4.6 μM).

One likely reason for these discrepancies is that both groups made their estimates by fitting gating models to a series of G-V relations determined for a series of [Ca²⁺], and to make these estimates they necessarily had to assume some model of the voltage-sensing mechanism of the channel. The two groups used different voltage-sensing models. More generally, however, a better way to estimate the affinity constants of the channel's Ca²⁺ binding sites would be to study the effect of Ca²⁺ on channel opening at many [Ca²⁺] but at a single voltage, such that the effect of voltage on channel opening can be treated as a constant.

Here, we have taken this approach. We have used mutations at each type of Ca²⁺ binding site and high-resolution Ca²⁺ dose-response curves to characterize the binding properties of each of the BK_Ca channel's high-affinity Ca²⁺ binding sites at both −80 and 0 mV. Our results indicate that the two sites have substantially different affinities, as suggested by Xia et al. (2002), at both these potentials, and that Ca²⁺ binding at the RCK1 site is voltage dependent, whereas at the Ca²⁺ bowl it is not.

MATERIALS AND METHODS

Heterologous Expression of BK_Ca Channels in TSA 201 Cells

TSA201 cells (modified human embryonic kidney cells) were transiently transfected with expression vectors (pcDNA 3; Invitrogen) encoding the α subunit of the BK_Ca channel from mouse (mSlo-mbr5) (Butler et al., 1993), enhanced green fluorescent protein (eGFP-N1; BD Biosciences), and the empty pcDNA 3.1+ vector (Invitrogen) to control for total amount of transfected DNA. Cells were transiently transfected using the Lipofectamine 2000 reagent (Invitrogen). The enhanced green fluorescent protein was used to monitor successfully transfected cells. For transfection, cells at 80–90% confluence in 35-mm falcon dishes were incubated with a mixture of the plasmids (total of 4 μg DNA) Lipofectamine and Optimem (Invitrogen) according to the manufacturer's instructions. In brief, the mixture was left on the cells 4–8 h after which the cells were replated into recording Falcon 3004 dishes in standard tissue culture media: DMEM with 1% fetal bovine serum, 1% l-glutamine, and 1% penicillin-streptomycin solution (all from Invitrogen). The cells were patch-clamped 1–3 d after transfection.

Electrophysiology

All recordings were performed in the inside-out patch-clamp configuration (Hamill et al., 1981). Patch pipettes were made of borosilicate glass (VWR micropipettes) with 0.8–5-MΩ resistances that were varied for different recording purposes. The tips of the patch pipettes were coated with sticky wax (KerrLab) and fire polished. Data were acquired using an Axopatch 200B patch clamp amplifier and a Macintosh-based computer system equipped with an ITC-16 hardware interface (InstruTECH) and Pulse acquistion software (HEKA electronik). For macroscopic current recordings, data were sampled at 50 kHz and filtered at 10 kHz. In most macroscopic current recordings, capacity and leak current were subtracted using a P/5 subtraction protocol with a holding potential of −120 mV and leak pulses in opposite polarity to the test pulse, but with BK_Ca currents recorded with >100 μM Ca²⁺, no leak subtraction was performed.

Unitary current recordings acquired at −80 mV were sampled at 100 KHz and filtered at 10 KHz. Unitary current recordings acquired at 0 mV were sampled at 100 kHz and filtered at 2 kHz. All experiments were performed at room temperature, 22–24°C.

Solutions

The pipette solution for macroscopic current recordings contained the following: 118 mM KMeSO₃, 20 mM N-methyl-glucamine-MeSO₃, 2 mM KCl, 2 mM MgCl₂, 2 mM HEPES, pH 7.20. The pipette solution for current recordings at 0 mV contained the following: 3 mM KMeSO₃, 135 mM N-methyl-glucamine-MeSO₄, 2 mM KCl, 2 mM MgCl₂, 2 mM HEPES, pH 7.20. 10 μM GdCl₃ was added to both pipette solutions to block endogenous stretch-activated channels. GdCl₃ did not block BK_Ca currents (not depicted) (Yang and Sachs, 1989; Qian and Magleby, 2003). The bath solution for all recordings contained the following: 118 mM KMeSO₃, 20 mM N-methyl-glucamine-MeSO₃, 2 mM KCl, 2 mM HEPES, pH 7.20. 1 mM EGTA (Fluka) was used as the Ca²⁺ buffer for solutions containing 3–500 nM free [Ca²⁺]. 1 mM HEDTA (Sigma-Aldrich) was used as the Ca²⁺ buffer for solutions containing 0.8–20 μM free [Ca²⁺], and no Ca²⁺ chelator was used in solutions containing between 20 μM and 2.5 mM free Ca²⁺. 50 μM (+)-18-crown-6-tetracarboxylic acid (18C6TA) was added to all internal solutions to prevent contaminant Ba²⁺ block at high voltages. Both internal and external solutions were brought to pH 7.20.

The appropriate amount of total Ca²⁺ (100 mM CaCl₂ standard solution; Orion Research, Inc.) to add to the buffered solutions to yield the desired approximate free Ca²⁺ concentrations of 3 nM to 2.5 mM was calculated using the program MaxChelator (see Online Supplemental Material), and the solutions were prepared as described previously (Bao et al., 2002). The Ca²⁺ concentrations reported are averages of three independent measurements determined with an Orion Ca²⁺-sensitive electrode. The solutions bathing the intracellular side of the patch were changed by means of a DAD valve–controlled pressurized superfusion system (ALA Scientific Instruments).

Data Analysis

All data analysis was performed with Igor Pro graphing and curve-fitting software (WaveMetrics Inc.), and the Levenberg-Marquardt algorithm was used to perform nonlinear least-square curve fitting. Values in the text are given ± SEM.

G-V Curves

G-V relations were determined from the amplitude of tail currents measured 200 μs after repolarizations to −80 mV following voltage steps to the test voltage. Each G-V relation was fitted with a Boltzmann function,

and normalized to the maximum of the fit.

Single-Channel Analysis

Under conditions where the open probability (Popen) is small (<10⁻²), single-channel openings were observed in patches containing hundreds of channels and I_K was measured from steady-state recordings 30 s in duration. NPopen was determined from all-points histograms by measuring the fraction of time spent (P_K) at each open level (k) using a half-amplitude criteria and summing their contributions, NPopen = Σ kP_K., where N is the number of channels in the patch.

Popen Versus Ca²⁺ Curves

The effect of Ca²⁺ on Popen was determined from the ratio of NPopen at a given Ca²⁺ to NPopen at 5.3 μM Ca²⁺ for all Ca²⁺ concentrations tested on a given patch. The (NPopen/NPopen_{5.3 μM}) versus [Ca²⁺] relation from each patch was then plotted and averaged across many patches. The mean (NPopen/NPopen_{5.3 μM}) versus [Ca²⁺] relation was then adjusted such that at 3 nM Ca²⁺ log (NPopen/NPopen_min) = 0. In some cases Popen, rather than being normalized (NPopen/NPopen_min), was reported as a function of [Ca²⁺]. This was done by determining Popen for each channel type at a single [Ca²⁺] in separate experiments, and then adjusting the average log (NPopen/NPopen_min) versus log [Ca²⁺] curve vertically, such that Popen was correct at the [Ca²⁺] at which Popen was known. At 0 mV, the calibrated Popen was determined at 2.5 mM from patches whose channel content was apparent (n = 1–4). At −80 mV, the calibrating Popen was determined at 5.3 μM [Ca²⁺] from unitary current measurements of NPopen from a series of patches in which N was calculated from the maximal current measured at +80 mV and separate Popen measurements taken at +80 mV for single channels.

Online Supplemental Material

The amount of Ca²⁺ to add to internal solutions to yield the desired free Ca²⁺ concentrations was calculated using the program MaxChelator, which was downloaded from http://www.stanford.edu/∼cpatton/maxc.html and is included as executable files here. The online supplemental material is available at http://www.jgp.org/cgi/content/full/jgp.200810094/DC1.

RESULTS

The Effect of Ca²⁺ with Both Sites Intact

The BK_Ca channel is both Ca²⁺ and voltage sensitive, and the effects of these stimuli are often displayed as a series of G-V relations determined at several Ca²⁺ concentrations (Barrett et al., 1982). Such a series, determined from BK_Ca channels exogenously expressed in TSA-201 cells, is shown in Fig. 1 B. The data are from excised inside-out macropatches (Fig. 1 A). Increasing intracellular Ca²⁺ shifts the channel's G-V curve leftward, an effect that is known under wild-type conditions to be due to three types of Ca²⁺ binding sites, two of high affinity and one of low affinity. The channels in the patches of Fig. 1, however, contained the mutation E399N, which eliminates low-affinity Ca²⁺ sensing (Shi et al., 2002; Xia et al., 2002) and thereby allows one to examine high-affinity Ca²⁺ sensing in isolation. We refer to the mouse Slo (mSlo) channel carrying this mutation as ΔE. Increasing Ca²⁺ from 3 nM to 2.5 mM shifts the ΔE G-V relation ∼−200 mV.

Figure 1.

Macroscopic currents and normalized conductance versus voltage curves (G-V) determined for the mSlo mutant E399N (ΔE). (A) Shown are averaged macroscopic current families. Each family displayed is the average of three consecutive families recorded from a single TSA 201 inside-out macropatch exposed to 1.4, 5.3, and 113 μM [Ca²⁺]. Membrane voltages were as follows: For 1.4 μM [Ca²⁺] and 5.3 μM [Ca²⁺], V_H was −80 mV, test potentials were to between −80 and +200 mV, and tail potentials were −80 mV; for 113 μM [Ca²⁺], V_H was −150 mV, test potentials were to between −100 and +150 mV in 10-mV steps, and tail potentials were −80 mV. (B) G-V relations were determined from data like that in A at the following [Ca²⁺]: 0.003, 0.36, 1.4, 5.3, 113, and 2,500 μM. Each curve represents the average between 7 and 13 individual curves. Error bars indicate SEM. The data have been fitted (solid lines) with a Boltzmann function (G/G_max = 1/(1+e ^q-F(V-V_1/2^)/RT). The Boltzmann fits have the following parameters: 3 nM Ca²⁺: Q = 1.21 e, V_1/2 = 183 mV; 360 nM Ca²⁺: Q = 1.18 e, V_1/2 = 151 mV; 1.4 μM Ca²⁺: Q = 1.47 e, V_1/2 = 101 mV; 5.3 μM Ca²⁺: Q = 1.38 e, V_1/2 = 59 mV; 113 μM Ca²⁺: Q = 1.00 e, V_1/2 = 13 mV; 2.5 mM Ca²⁺: Q = 1.04 e, V_1/2 = −5.7 mV.

Previous estimates of the affinities of the BK_Ca channel's high-affinity Ca²⁺ binding sites have been based on fitting gating models to data like that shown in Fig. 1 B. This necessarily requires that one assume some model of voltage-dependent gating, and the resulting Ca²⁺ binding parameters produced by the fits are then dependent on this choice (Bao et al., 2002; Xia et al., 2002). To circumvent this necessity, and thereby more directly determine the BK_Ca channel's high-affinity Ca²⁺ binding constants, here we have studied the effects of Ca²⁺ at constant voltage.

Fig. 2 A shows unitary ΔE currents recorded from a single membrane patch at −80 mV and four different [Ca²⁺]. Corresponding amplitude histograms are shown in Fig. 2 B. Although the patch contained hundreds of channels, each channel's open probability (Popen) is low in the absence of Ca²⁺, such that activity is observed as the infrequent and brief opening of single channels. Application of Ca²⁺ then caused a large increase in Popen that resulted in multi-channel openings. From data like these we derived the ΔE channel's Popen versus [Ca²⁺] relation (Fig. 2 C). So that all parts of the curve could be well determined, Popen was measured over five orders of magnitude with 22 Ca²⁺ concentrations. To do this, many patches were used and normalized by their values of NPopen at 5.3 μM, where N is the number of channels in a given patch. The data were then averaged at each [Ca²⁺], and the whole curve was adjusted vertically to match the BK_Ca channel's Popen at 5.3 μM and −80 mV, which was determined in separate experiments (see Materials and methods).

Figure 2.

The Ca²⁺ dependence of Popen for mutant ΔE. (A) Inward potassium currents recorded at −80 mV and filtered at 10 kHz from a macropatch exposed to the indicated [Ca²⁺] demonstrate that Popen increases in a Ca²⁺-dependent manner when voltage is constant. The corresponding all-points amplitude histograms are plotted in B on a semi-log scale and were constructed from 30-s recordings at each [Ca²⁺]. The dose–response relation for the effect of Ca²⁺ on Popen at negative voltage (−80 mV) is shown in C. For determination of Popen see Materials and methods. Each point represents the average of between 7 and 17 patches at each Ca²⁺ concentration tested. Error bars represent SEM.

These data were then analyzed as follows. If one assumes that there are four of each type of Ca²⁺ binding site and that each site influences channel opening by altering the equilibrium constant of a single conformational change between closed and open—as much evidence suggests (McManus and Magleby, 1991; Cox et al., 1997; Cui et al., 1997; Horrigan and Aldrich, 1999, 2002; Horrigan et al., 1999; Rothberg and Magleby, 1999, 2000; Cox and Aldrich, 2000)—and that there are no interactions between binding sites, then at constant voltage the channel's open probability as a function of voltage can be written as:

(1)

where K_C1 and K_C2 represent the dissociation constants of binding sites 1 and 2 in the closed conformation. K_O1 and K_O2 represent the dissociation constants of binding sites 1 and 2 in the open conformation, and M represents the closed-to-open equilibrium constant when no Ca²⁺ is bound. As relates to the BK_Ca channel, M is voltage dependent and incorporates all effects of voltage on opening.

In the absence of Ca²⁺, Eq. 1 reduces to:

(2)

which can be rearranged to:

(3)

Thus, M can be determined directly from Popen in the absence of Ca²⁺. From Fig. 2 C we can estimate M to be ∼2.5 × 10⁻⁶. However, better still, when Popen is small over the entire Ca²⁺ dose-response curve, as is the case here, Eq. 1 reduces to Eq. 4 below (Horrigan and Aldrich, 2002):

(4)

and then dividing through by Popen at 0 [Ca²⁺] yields:

(5)

This eliminates M and leaves a curve whose properties are determined solely by the channel's Ca²⁺ binding constants (Horrigan and Aldrich, 2002). Thus, the curve in Fig. 2 C was normalized by its minimum value to yield a Popen (Ca²⁺)/Popen (0) versus [Ca²⁺] curve (Fig. 3) and then fitted with Eq. 5. Properties of this curve of note are: (1) Ca²⁺ increases Popen by a factor of 2.8 × 10⁴; (2) Popen saturates at high [Ca²⁺], ∼100 μM; and (3) the curve has a shallow quality suggestive of multiple binding sites with differing affinities. Indeed the fit (solid line) yielded the following dissociation constants: SITE 1, K_C1 = 3.7 ± 2.1 μM, K_O1 = 0.7 ± 0.14 μM; SITE 2, K_C2 = 51 ± 42 μM, K_O2 = 21 ± 24 μM.

Figure 3.

The Ca²⁺ binding affinity of mutant ΔE at −80 mV. The mean log ratio of NPopen in the presence and absence of Ca²⁺ determined from the data shown in Fig. 2. Log (NPopen / NPopen_min) spans the entire [Ca²⁺] range and is fit (solid line) by Eq. 5 yielding values of K_O1 = 0.7 μM, K_C1 = 3.7 μM, K_O2 = 21 μM, and K_C2 = 51 μM. Also shown is the fit (dashed line) assuming both types of binding sites have the same affinity for Ca²⁺ (K_O = 1.9 μM and K_C = 6.4 μM). Error bars represent SEM.

Further, when we forced both types of binding sites to have the same affinities, a substantially worse fit was obtained (dashed line, K_C = 6.1 ± 0.4 μM; K_O = 1.9 ± 0.31 μM). Thus, this analysis suggests that one of the BK_Ca channel's high-affinity Ca²⁺ binding sites has substantially higher affinity for Ca²⁺ than the other, both in the open and closed conformations, although noise in the data introduces some uncertainty about the fitted values.

Mutations That Eliminate Ca²⁺ Sensing

To measure the affinities of each type of high-affinity Ca²⁺ binding site individually, we used mutations that selectively eliminate the effect of Ca²⁺ at each type of site. D367A eliminates Ca²⁺ sensing via RCK1 sites (Xia et al., 2002), and D897N/D898N/D899N/D900N/D901N (D5N5) or D898A/D900A (D2A2) eliminate Ca²⁺ sensing via the Ca²⁺ bowls (Schreiber and Salkoff, 1997; Bian et al., 2001; Bao et al., 2004). Before using these mutations, however, it was important to confirm that in conjunction with E399N they eliminate all Ca²⁺ sensing. Shown in Fig. 4 A are currents recorded at various [Ca²⁺] from a patch expressing the triple mutant (E399N)(D367A)(D897N/D898N/D899N/D900N/D901N), which we refer to as ΔEΔRΔB_(D5N5). Corresponding amplitude histograms are superimposed in Fig. 4 B, and in Fig. 4 C the ΔEΔRΔB_(D5N5) channel's Ca²⁺ dose-response relation is plotted at −80 mV. As is evident, the triple mutant shows virtually no response to Ca²⁺, which demonstrates that the three sites targeted by these mutations can together account for all of the channel's Ca²⁺ sensing.

Figure 4.

Mutation of all three types of Ca²⁺ binding sites eliminates the Ca²⁺ dependence of Popen. (A) Inward K⁺ currents recorded for mutant ΔEΔRΔB_(D5N5) at −80 mV and filtered at 10 kHz from a macropatch in the indicated [Ca²⁺] demonstrate that Popen does not increase in a Ca²⁺-dependent manner when voltage is constant. The corresponding all-points amplitude histograms are plotted in B on a semi-log scale and were constructed from 30-s recordings. (C) Dose–response relations for the effect of Ca²⁺ on Popen at negative voltage (−80 mV) obtained by plotting the mean log ratio of NPopen in the presence and absence of Ca²⁺. For both mutant ΔEΔRΔB_(D5N5) (filled circles) and mutant ΔEΔRΔB_(D2A2), log (NPopen/NPopen_min) spans the entire [Ca²⁺] range but cannot be fitted because Popen does not vary with [Ca²⁺]. Each point represents the average of between 6 and 8 patches at each [Ca²⁺] tested. Error bars represent SEM.

Ca²⁺ Binding at the Ca²⁺ Bowl

We then used the mutant (E399N)(D367A), which we refer to as ΔEΔR, to examine Ca²⁺ sensing via the Ca²⁺ bowl. Fig. 5 A compares BK_Ca currents at various [Ca²⁺] recorded from a single ΔEΔR patch at −80 mV. The corresponding amplitude histograms are shown in Fig. 5 B. As with ΔE, Popen is low in the absence of Ca²⁺, and activity is observed as the infrequent and brief opening of single channels. Application of Ca²⁺ then increases Popen, but the increase is not as great (∼10²-fold) as it is with the ΔE channel (∼10⁴-fold), presumably because the ΔEΔR channel has lost half of its high-affinity binding sites. A Ca²⁺ dose–response relation for the ΔEΔR channel at −80 mV is shown in Fig. 5 C. The affinities of the intact Ca²⁺ bowl site were then determined from a fit (solid line) with Eq. 6 below, which is analogous to Eq. 5, but represents the case where there is only one type of Ca²⁺ binding site (Horrigan and Aldrich, 2002).

(6)

Figure 5.

The Ca²⁺ binding affinities of the Ca²⁺ bowl site at −80 mV. (A) Inward K⁺ currents recorded from mutant ΔEΔR at −80 mV and filtered at 10 kHz from a macropatch exposed to the indicated [Ca²⁺] demonstrate that Popen increases in a Ca²⁺-dependent manner when voltage is constant. The corresponding all-point amplitude histograms are plotted in B on a semi-log scale and were constructed from 30-s recordings as in Fig. 2. The dose–response relation for the effect of [Ca²⁺] on Popen (left axis) and NPopen/NPopen_min (right axis) at negative voltage (−80 mV) is shown in C. Each point represents the average of between 6 and 11 patches at each [Ca²⁺] tested. Log (NPopen/NPopen_min) spans the entire [Ca²⁺] range and is fit (solid line) by Eq. 6 yielding values of K_O = 0.88 μM and K_C = 3.13 μM. Error bars represent SEM.

Of importance here, Eq. 6 contains only the channel's open- and closed-state Ca²⁺ dissociation constants as free parameters, and in the limit of high [Ca²⁺], Eq. 6 becomes:

(7)

or equivalently:

(8)

Thus, the change in log (Popen) from 0 to saturating [Ca²⁺], which is the distance along the vertical axis spanned by the data in Fig. 5 C, depends only on the ratio of the open and closed conformation Ca²⁺ dissociation constants. This means that measuring Popen precisely at both the top and the bottom of the curve—as we have done here with unitary current recordings—places an important constraint on the fitting. Indeed, because the amplitude of the curve is determined by C, this leaves only one parameter free to determine the shape of the curve, either K_C or K_O. Thus, in fitting with Eq. 6, the fit is highly constrained, and it is therefore remarkable how well Eq. 6 fits the data (solid line). The fit yields the following Ca²⁺ dissociation constants for the Ca²⁺ bowl (K_O = 0.88 ± 0.06 μM; K_C = 3.13 ± 0.22 μM; C = 3.55)(see also Table I), and it is of note that they are similar to the K_C1 and K_O1 values estimated from the ΔE data in Fig. 3.

TABLE I.

BK_Ca Channel Ca²⁺ Binding Parameters

Binding Site	Membrane Potential (mV)	KC (μM)	KO (μM)	M	f	C
Ca2+ Bowl
ΔEΔR	−80	3.1 ± 0.2	0.88 ± 0.06			3.55
ΔEΔR	0	3.1	0.88			3.55
RCK1
ΔEΔB(D2A2)	−80	23.2 ± 2.6	4.9 ± 0.6			4.75
ΔEΔB(D5N5)	−80	26.8 ± 3.8	5.6 ± 0.8			4.75
ΔEΔB(D2A2) + Cooperativity	−80	13.7 ± 2.3	2.8 ± 0.5		0.45 ± 0.1
ΔEΔB(D5N5)+ Cooperativity	−80	9.4 ± 1.8	1.8 ± 0.2		0.27 ± 0.05
ΔEΔB(D2A2)	0	15.8 ± 3.1	2.10 ± 0.4	1.8 × 107		7.52

Ca²⁺ Binding at the RCK1 Site

Similarly, to determine the affinities of the RCK1 site, we examined the effect of Ca²⁺ on the open probability of the mutant (E399N)(D898A/D900A), which we refer to as ΔEΔB_(D2A2). The two D→A mutations render the Ca²⁺ bowl nonfunctional (Bao et al., 2004). Fig. 6 A shows unitary ΔEΔB_(D2A2) currents recorded at −80 mV with various [Ca²⁺] from a patch that contained hundreds of channels. Corresponding amplitude histograms are shown in Fig. 6 B, and the Ca²⁺ dose–response relation we acquired for the ΔEΔB_(D2A2) channel at −80 mV is shown in Fig. 6 C (open squares). In fact, both ΔEΔB_(D2A2) and another Ca²⁺ bowl mutation, (D897N/D898N/D899N/D900N/D901N) (ΔEΔB_(D5N5)), were analyzed (Fig. 6 C, closed squares), and both mutations behave similarly. The affinity of the RCK1 site was then estimated by fitting Eq. 6 to the two datasets in Fig. 6 C. The fits yielded similar values (K_O = 4.9 ± 0.6 μM; K_C = 23.2 ± 2.6 μM; C = 4.75) for ΔEΔB_(D2A2) and (K_O = 5.6 ± 0.8 μM; K_C = 26.8 ± 3.8 μM; C = 4.75) for ΔEΔB_(D5N5) (see Table I). Thus, the RCK1 site binds Ca²⁺ more weakly than does the Ca²⁺ bowl site, both when the channel is open and when it is closed (Ca²⁺ bowl: K_O = 0.88 ± 0.06 μM; K_C = 3.13 ± 0.22 μM; C = 3.55 from Fig. 5), but it has a 36% larger C value and thus a bigger effect on opening at saturating [Ca²⁺]. This is illustrated graphically in Fig. 7, where the ΔEΔR (closed triangles) and ΔEΔB_(D2A2) (closed squares) Ca²⁺ dose–response curves are overlaid.

Figure 6.

The Ca²⁺ binding affinities of the high-affinity RCK1 site at −80 mV. (A) Inward K⁺ currents recorded from mutant ΔEΔB_(D2A2) at −80 mV and filtered at 10 kHz from a macropatch in the indicated [Ca²⁺] demonstrate that Popen increases in a Ca²⁺-dependent manner when voltage is constant. The corresponding all-points amplitude histograms are plotted in B on a semi-log scale and were constructed from 30-s recordings. The dose–response relation for the effect of Ca²⁺ on Popen (left axis) and NPopen/NPopen_min (right axis) at negative voltage (−80 mV) is shown in C. Each point represents the average of between 6 and 16 patches at each Ca²⁺ concentration tested. For mutant ΔEΔB_(D2A2) (open squares), log (NPopen/NPopen_min) spans the entire [Ca²⁺] range and is fitted (dotted line) by Eq. 6 yielding values of K_O = 4.9 μM of K_C = 23.2 μM. For mutant ΔEΔB_(D5N5) (closed squares), the fit (dashed line) yields values of K_OB = 5.6 μM and K_CB = 26.8 μM. (D) The data were also fitted with Eq. 7, which incorporates an interaction between binding sites. For mutant ΔEΔB_(D2A2) (open squares), the fit yielded values of K_O = 2.8 μM, K_C = 13.7 μM, and f = 0.45. For mutant ΔEΔB_(D5N5) (closed squares), the fit (dashed line) yielded K_O = 1.8 μM, K_C = 9.4 μM, and f = 0.27. Error bars represent SEM.

Figure 7.

The two binding sites are less than additive. The mean log ratio of NPopen at −80 mV in the presence and absence of Ca²⁺ for mutants ΔE (solid circles), ΔEΔR (solid triangles), and ΔEΔB_(D2A2) (solid squares) are plotted versus [Ca²⁺]. Various fits of log (NPopen/NPopen_min) are superimposed on the data. The fit of ΔEΔR also displayed in Fig. 2 is shown as a short dashed curve. The fit of ΔEΔB_(D2A2) also displayed in Fig. 3 is shown as a long dashed curve. We simulated the log (NPopen/NPopen_min) relation (dark solid line) predicted by the affinities determined from each of the mutants using Eq. 5. The parameters of the fit were: K_O1 = 0.88 μM, K_C1 = 3.13 μM, K_O2 = 4.88 μM, and K_C2 = 23.2 μM. Also plotted (gray curve) is a fit that incorporates cooperativity between the binding sites. The equation for the fit was log (NPopen/NPopen_min) = ((1+(K_O1+ K_O2)+K_O1K_O2b)⁴) / ((1+(K_C1+ K_C2)+K_C1K_C2a)⁴). The parameters of the fit were: K_O1 = 0.88 μM, K_C1 = 3.13 μM, K_O2 = 4.88 μM, K_C2 = 23.2 μM, a = 1, and b = 0.75. Error bars represent SEM.

With regard to Figs. 5 and 6, however, it is interesting to note that Eq. 6 fits the data from the ΔEΔR channel (Fig. 5) better than it does those from the ΔEΔB_(D5N5) and ΔEΔB_(D2A2) channels (Fig. 6 C). That is, the idea represented by Eq. 6 does not appear to be as good an approximation of reality for the RCK1 site as it does the Ca²⁺ bowl site. To try to improve the fit, we have added a cooperativity factor by which the binding at one site influences binding at sites on adjacent subunits. If we call this factor f, and suppose for simplicity that f is the same for the opened and closed channel, then Eq. 9 below represents this idea (Cox et al., 1997, scheme III and discussion page 269).

(9)

Fitting the ΔEΔB_(D2A2) data in Fig. 6 with Eq. 9 did produce better fits (Fig. 6 D) and yielded f = 0.45 ± 0.1, K_C = 13.7 ± 2.3 μM, and K_O = 2.8 ± 0.5 μM for the ΔEΔB_(D2A2) channel and f = 0.27 ± 0.05, K_C = 9.4 ± 1.8 μM, and K_O = 1.8 ± 0.2 μM for the ΔEΔB_(D5N5) channel (see Table I). The fact that f is <1 for both fits suggests that, if this explanation is correct, Ca²⁺ binding at one RCK1 site negatively influences Ca²⁺ binding at other RCK1 sites on adjacent subunits.

To improve the fit to the ΔEΔB data we also considered a ring model. It has been proposed that the BK_Ca channel has a gating ring that hangs below the channel and expands upon Ca²⁺ binding, and that this expansion pulls open the channel (Jiang et al., 2002; Kim et al., 2008). If ring expansion and channel opening are strictly coupled, such that one does not occur without the other, this idea is mathematically equivalent to the simple MWC-like allosteric models we have used thus far. If, however, ring expansion favors opening, but is not obligate for opening, they are not equivalent. A model for this situation is as follows:

where P represents the equilibrium constant for channel opening when the ring is relaxed. H represents the equilibrium constant for ring expansion when the channel is closed, and no Ca²⁺ is bound. Q represents the factor by which ring expansion favors channel opening, and U_Ca represents the Ca²⁺-dependent factor by which Ca²⁺ binding favors ring expansion. Then, for Popen we have:

(10)

where

(11)

or

(12)

and (Popen/Popen_min) at low open probabilities is:

(13)

Thus, we fit the data in Fig. 6 with Eq. 13. In the fitting, however, we found that to get an acceptable fit, H had to be small (at most 0.001) and Q had to be large (at least 1,000), and when this is the case, the ring model becomes mathematically equivalent to the MWC-like model we used above, and it yields the same fits. Thus, we could not improve the fit via a ring model, and we view the fact that the best ring model fits were obtained with small values of H and large values of Q as evidence that, if the BK_Ca channel has a gating ring, its expansion is tightly coupled to channel opening such that one seldom occurs without the other.

The Two Sites Are Less Than Additive

Can the affinities measured for each binding site in isolation, when combined, explain the effect of Ca²⁺ when both binding sites are intact? To answer this question we calculated the predicted Popen(Ca²⁺)/Popen(0) versus [Ca²⁺] curve for the ΔE channel based on the affinities measured for each high-affinity Ca²⁺ binding site in Figs. 5 C and 6 C. In Fig. 7 this prediction (dark solid curve) is compared with the ΔE data (filled circles). Of most interest, the predicted curve, although similar to the data, does not everywhere overlay the data, but rather it predicts a larger response to Ca²⁺ than is observed. We might consider two possible reasons for this outcome. The first is that one or the other of the mutations we have used is not completely selective. That is, in addition to eliminating Ca²⁺ sensing via one type of Ca²⁺ binding site, a given mutation may also affect Ca²⁺ binding at the other site. We do not favor this explanation, however, because in order for it to explain the data, the mutation would have to eliminate Ca²⁺ binding at one site while augmenting it at the other. Although this cannot be ruled out, it seems unlikely. The second potential explanation is that there is negative cooperativity between Ca²⁺ binding sites, such that their individual influences are naturally less than what is observed when they are combined. Pursuing this idea further we have calculated that a cooperativity factor between the RCK1 and Ca²⁺ bowl Ca²⁺ binding sites on the same subunit of 1, when the channel is closed (no cooperativity), and 0.75, when the channel is open (negative cooperativity), could explain this effect (Fig. 7, gray curve).

Do These Results Explain the G-V Shifts with [Ca²⁺]?

Another question of interest is do the binding affinities we have measured at a single voltage (−80 mV) explain the BK_Ca channel's sensitivity to Ca²⁺ over a range of voltages? Fig. 8 A shows the mSlo G-V relation at a series of [Ca²⁺] fit simultaneously with the BK_Ca-gating model of Horrigan and Aldrich (2002) (the HA model) but modified to include two sets of Ca²⁺ binding sites, four per set. There were no free parameters in this fit, but rather gating parameters determined from these and previous experiments (Bao and Cox, 2005) were used. The parameters were as follows: K_O1 = 0.88 μM; K_C1 = 3.13 μM; K_O2 = 4.88 μM; K_C2 = 23.2 μM; L_O = 2e-06; z_L= 0.41 e; V_hc = 151 mV; V_ho = 27 mV; and Z_J = 0.58 e. The allosteric factors E₁ and E₂ were set to 1 to simulate no interaction between voltage–sensor movement and Ca²⁺ binding at either site. The fit is poor. The model responds to Ca²⁺ less than is required to move the model G-V relation along with the data. Interestingly, however, when we let E₁ and E₂ vary freely, that is, we allowed interactions between binding sites and voltage sensors, the fit markedly improved (E_i = 1.43; E₂ = 1.73) (Fig. 8 B). This suggests that voltage-sensor movement may alter Ca²⁺ binding and vice versa.

Figure 8.

Voltage likely affects the affinity of the BK_Ca channel for Ca²⁺. Shown are a series of mSlo1α G-V relations determined at the following [Ca²⁺]: 0.003, 0.070, 0.130, 0.360, 0.8, 10, and 100 μM and fitted simultaneously with the HA model modified to include two Ca²⁺ binding sites (Bao et al., 2002; Horrigan and Aldrich, 2002). Using values determined from this and previous experiments in our laboratory (Bao and Cox, 2005), the parameters were held as follows: K_O1 = 0.88 μM, K_C1 = 3.18 μM, K_O2 = 4.88 μM, K_C2 = 23.2 μM, L_O = 2.2 × 10⁻⁶, z_L= 0.41 e, V_hc = 151 mV, V_ho = 27 mV, and z_J = 0.58 e. In A, the values of allosteric factors E₁ and E₂ were held to a value of 1 for both Ca²⁺ binding sites A and B. In B, the values of E₁ and E₂ were allowed to vary. The best fit values of E₁ and E₂ were 1.43 and 1.73, respectively.

Voltage Affects Ca²⁺ Binding

To test this hypothesis directly, we repeated the experiments so far described, but changed the voltage from −80 to 0 mV. We reasoned that at −80 mV few voltage sensors would be active (5% or less) (Stefani et al., 1997; Horrigan and Aldrich, 1999, 2002; Bao and Cox, 2005), and thus there would be very little influence of voltage-sensor movement on Ca²⁺ binding. But at 0 mV, where the channels' voltage sensors are active 35% of the time when the channels are open (Horrigan and Aldrich, 1999, 2002; Bao and Cox, 2005), if voltage-sensor movement affects Ca²⁺ binding, some influence of the change in voltage should be observed. Shown in Fig. 9 (A and C) are the Popen(Ca²⁺)/Popen(0) versus [Ca²⁺] curves derived from these experiments (open symbols) along with their counterparts determined at −80 mV (filled symbols). Examining first the ΔEΔR channel (Fig. 9 A), we see that its 0- and −80-mV Popen(Ca²⁺)/Popen(0) versus [Ca²⁺] curves superimpose. This indicates that voltage-sensor movement does not affect Ca²⁺ binding at the Ca²⁺ bowl, but rather a change in voltage simply slides the Popen versus [Ca²⁺] curve up the Popen axis (see Fig. 9 B). Conversely, there is a substantial effect of voltage on Ca²⁺ binding at the RCK1 sites (Fig. 9, C and D). The maximal effect of Ca²⁺ on the open probability of the ΔEΔB_(D2A2) channel is ∼10-fold larger at 0 mV than it is at −80 mV (Fig. 9 C), and fitting the 0 mV curve in Fig. 9 C with Eq. 6 yields Ca²⁺ dissociation constants of 15.6 ± 2.5 μM and 2.1 ± 0.3 μM (C = 7.39), as compared with 23.2 ± 2.6 μM and 4.9 ± 0.6 μM (C = 4.75) at −80 mV.

Figure 9.

The Ca²⁺ dependence of Popen for the RCK1 site is affected by voltage. However, voltage does not alter the binding at the Ca²⁺ bowl site. (A) The mean log ratio of NPopen/NPopen_min versus [Ca²⁺] for mutant ΔEΔR is shown for patches held at 0 mV (open circles) or at −80 mV (solid circles). Each point represents the average of between 6 and 13 patches at each Ca²⁺ concentration tested. Shown is the fit (dashed curve) of NPopen/NPopen_min based on Eq. 6 and previously shown in Fig. 5. The values determined from the fit are: K_O = 0.88 μM and K_C = 3.13 μM. (B) The mean log Popen versus [Ca²⁺] relation for mutant ΔEΔR at both 0 and −80 mV are well fitted with the HA model using the Ca²⁺ binding constants determined. The values for the parameters were held as follows: K_O = 0.88 μM, K_C = 3.18 μM, L_O = 6.3 × 10⁻⁶, z_L= 0.41 e, V_hc = 151 mV, V_ho = 27 mV, z_J = 0.58 e, and E = 1. (C) The NPopen/NPopen_min versus [Ca²⁺] relation for mutant ΔEΔB_(D2A2) is shown for patches held at 0 mV (open circles) or at −80 mV (solid circles). Each point represents the average of between 6 and 14 patches at each [Ca²⁺] tested. The NPopen/NPopen_min versus Ca²⁺ relations are fitted with Eq. 6. The values of the fit parameters are: ΔEΔB_(D2A2): −80 mV, K_O = 4.9 μM and K_C = 23.2 μM; ΔEΔB_(D2A2): 0 mV, K_O = 2.1 μM and K_C = 15.6 μM. (D) The mean log Popen versus [Ca²⁺] relation for mutant ΔEΔB_(D2A2) at both 0 and −80 mV. First, the 0 mV data were fitted by Eq. 14 to yield the values: K_O = 2.1 μM and K_C = 15.8 μM. The data were also fitted with the HA model. The parameters were held as follows: K_O = 4.9 μM, K_C = 23.2 μM, L_O = 1.2 × 10⁻⁶, z_L = 0.41 e, V_hc = 151 mV, V_ho = 27 mV, z_J = 0.58 e, and E = 6.03.

It is not rigorously correct, however, to fit the 0 mV ΔEΔB_(D2A2) data in Fig. 9 C with Eq. 6, as an assumption underlying this equation is that Popen is never greater than ∼10⁻². Although this is the case for the ΔEΔR and ΔEΔB channels at −80 mV, as shown in Fig. 9 D, it is not the case for the ΔEΔB_(D2A2) channel at 0 mV. Thus, to determine the dissociation constants of the RCK1 sites at 0 mV, we fit the ΔEΔB_(D2A2) Popen versus [Ca²⁺] curve in Fig. 9 D with Eq. 14 below, which does not require this assumption.

(14)

This yielded (Fig. 9 D, solid curve) K_C = 15.8 ± 3.1 μM, K_O = 2.10 ± 0.4 μM (C = 7.52), and M = 1.8 × 10⁻⁵ ± 0.5 × 10⁻⁵ (see Table I). Thus, changing the voltage from −80 mV to 0 mV decreases K_C at the RCK1 Ca²⁺ binding site by a factor of 0.7 (23.2→15.8). It decreases K_O by a factor of 0.4 (4.88→2.10), and it increases C by a factor of 1.8. This increase in C makes the efficacy of the RCK1 sites an order of magnitude larger than the efficacy of the Ca²⁺ bowl sites at 0 mV. This is highlighted in Fig. 10, where the 0 mV Ca²⁺ dose-response curves for the two sites are superimposed. Also evident, at 0 mV, as we saw at −80 mV, the ΔE channel's Ca²⁺ dose-response curve spans a smaller range of open probabilities than is predicted (Fig. 10, dark solid curve) by the combination of the fits to each individual dose-response curve. And again, we can explain this effect by supposing negative cooperativity between the RCK1 and the Ca²⁺ bowl sites in each subunit. A cooperativity factor of 1 when the channel is closed (no cooperativity) and 0.65 when the channel is open (negative cooperativity) produced the best fit (Fig. 10, gray curve).

Figure 10.

The two binding sites are less than additive at 0 mV. The mean log ratio of NPopen at 0 mV in the presence and absence of Ca²⁺ for mutants ΔE (open circles), ΔEΔR (open triangles), and ΔEΔB_(D2A2) (open squares) are plotted versus [Ca²⁺]. Each point represents the average of between 6 and 14 patches at each [Ca²⁺] tested. Various fits of log (NPopen/NPopen_min) are superimposed on the data. The fit of ΔEΔR (short dashed line) with Eq. 6 yielded values of K_O = 0.63 μM and K_C = 2.28 μM. The fit of ΔEΔB_(D2A2) (long dashed curve) yielded values of K_O = 1.56 μM and K_C = 12.7 μM. Using the same equation, we simulated the log (NPopen/NPopen_min) versus Ca²⁺ relation (dark solid line) predicted by the affinities determined for each site in isolation. The parameters of the fit were: K_O1 = 0.63 μM, K_C1 = 2.28 μM, K_O2 = 1.56 μM, and K_C2 = 12.7 μM. Also plotted (gray curve) is a log (NPopen/NPopen_min) versus [Ca²⁺] fit that incorporates cooperativity between binding sites. The equation for the fit was log (NPopen/NPopen_min) = ((1+(K_O1+K_O2)+ K_O1K_O2b)⁴) / ((1+(K_C1+ K_C2)+ K_C1K_C2a)⁴). The parameters of the fit were: K_O1 = 0.63 μM, K_C1 = 2.28 μM, K_O2 = 1.56 μM, K_C2 = 12.7, a = 1, and b = 0.65. Error bars represent SEM.

We could explain the effect of voltage on Ca²⁺ binding at the RCK1 sites in terms of the HA model by supposing that as the channel's voltage sensors move to their active conformation, they alter the affinity of the channel's RCK1 sites by a factor E (Horrigan and Aldrich, 2002). To estimate E, we fit the Popen versus [Ca²⁺] curves at both −80 and 0 mV in Fig. 9 D simultaneously with the HA model. We held the voltage sensing parameters: V_hc V_ho z_J and z_L to values previously determined for the mSlo channel (Bao and Cox, 2005). We held K_C and K_O to the values we determined in this study at −80 mV. We set L(0) to the value determined by the bottom of the Popen versus [Ca²⁺] curve at −80 mV, and we allowed only E to vary. Remarkably, both RCK1 site Ca²⁺ dose-response curves, −80 and 0 mV, could be fitted fairly well with the same parameters with E equal to 6.03 (Fig. 9 D, dotted line).

The value of E estimated in this way, however, is dependent on the voltage-sensing parameters of the model (V_ho V_hc, z_L and z_J), parameters that we have taken from previous experiments with wild-type mSlo channels. As here, however, we are using the mutant E399N as our background channel. It may be that this mutation alters one or more of these parameters and thereby renders this method inaccurate. Indeed, differences between the mSlo and E399N G-V curves in the absence of Ca²⁺ (unpublished data) make us think this may be the case. Thus, another approach we have taken to determining E for the wild-type mSlo channel's RCK1 site is to fit the wild-type channel's G-V relation as a function of [Ca²⁺] with a two-Ca²⁺ binding site HA model that includes a voltage sensor–Ca²⁺ binding site interaction factor E for only one of the high-affinity binding sites, the one with lower affinities. That is:

(15)

where

We held all parameters but E to values that have been independently determined either here or previously (Bao and Cox, 2005) and allowed only E to vary. The resulting best fit from this approach is shown in Fig. 11. It shows that even with these severe constraints, the two-site HA model with our newly determined Ca²⁺ binding constants can approximate the shifting of the mSlo channel's G-V relation as a function of [Ca²⁺], and remarkably the fit yields E = 2.8, a value that is very similar to the value of E (2.4) estimated independently by Horrigan and Aldrich (2002) from measurements of the channel's gating charge movement as a function of voltage at 0 and 70 μM [Ca²⁺]. Thus, we currently favor this estimate.

Figure 11.

A modified HA model can explain the effect of Ca²⁺ on the steady-state gating properties of the BK_Ca channel. Shown are series of mSlo1α G-V relations determined at the following [Ca²⁺]: 0.003, 0.070, 0.130, 0.360, 0.8, 10, and 100 μM and fit simultaneously with the modified model that includes two types of Ca²⁺ binding sites, one type interacts with the voltage sensor, and the other type is independent. The data are the same as shown in Fig. 8. The parameters were held as follows: K_O1 = 0.88 μM, K_C1 = 3.18 μM, K_O2 = 4.88 μM, K_C2 = 23.2 μM, L_O = 2.2 × 10⁻⁶, z_L = 0.41 e, V_hc = 151 mV, V_ho = 27 mV, and z_J = 0.58 e. The value of E coupling voltage sensor activation and Ca²⁺ binding at one type of site was allowed to vary. Shown is the best fit obtained. The value of E was calculated to be 2.84 ± 0.13.

DISCUSSION

Here, we have measured the Ca²⁺ binding constants of the BK_Ca channel's two types of high-affinity Ca²⁺ binding sites. To be as accurate as possible, we used unitary current recordings from patches containing from a few hundred to just a few channels. This allowed us to determine Popen over five orders of magnitude. To be as model-independent as possible, as pioneered by Horrigan and Aldrich (2002), we have made measurements at constant voltage and low Popen, such that the amplitudes and shapes of the resulting Ca²⁺ dose-response curves were dependent only on the channel's Ca²⁺ binding parameters. Further, to prevent potential interactions between Ca²⁺ binding sites and voltage sensors from complicating our analysis, our initial experiments were done at −80 mV, where the BK_Ca channel's voltage sensors are very seldom active. The essential assumptions we made in fitting our data were as follows: (1) that there is a single conformational change between open and closed that can occur with any number of Ca²⁺ bound (this idea is consistent with a great many single-channel and macroscopic BK_Ca channel studies and all current models) (McManus and Magleby, 1991; Cox et al., 1997; Cui et al., 1997; Horrigan and Aldrich, 1999, 2002; Horrigan et al., 1999; Rothberg and Magleby, 1999, 2000; Cox and Aldrich, 2000); and (2) that there are four of each type of high-affinity site. This has been established for the Ca²⁺ bowl (Niu and Magleby, 2002), and given the fourfold symmetry of the channel, it seems likely to be the case for the RCK1 site as well.

Of primary interest, we found that the Ca²⁺ bowl's dose-response curve at −80 mV could be well fitted by supposing that each Ca²⁺ bowl independently influences opening, and that each site has an affinity of 3.13 ± 0.22 μM when the channel is closed and 0.88 ± 0.06 μM when the channel is open. These values produce a C value of 3.55, which allows us to calculate that each bound Ca²⁺ at a Ca²⁺ bowl decreases the energy difference between open and closed by 3.1 KJ/mol. These numbers may be compared with previous estimates of K_C and K_O for this site. Xia et al. (2002) estimated K_C = 4.5 ± 1.7 μM and K_O = 2.0 ± 0.7 (C = 2.25), and Bao et al. (2002) estimated K_C = 3.8 ± 0.2 and K_O = 0.94 ± 0.06 (C = 4.0). Thus, our current estimates are quite close to those of Bao et al. (2002) and similar as well to those of Xia et al. (2002), although their larger value of K_O (2.0 μM) reduces C to 2.2, too low to be compatible with our data. Further, we found no change in the binding properties of this site when the membrane voltage was changed from −80 to 0 mV.

At the RCK1 site we found the ΔEΔB_(D2A2) dose-response curve at −80 mV could be fitted with a K_C of 23.2 ± 2.6 μM and a K_O of 4.9 ± 0.55 μM (C = 4.75), which yields a change in the energy difference between open and closed per bound Ca²⁺ of −3.8 KJ/mol. However, in this case the fit was improved by supposing some negative cooperativity between sites. The best fit was achieved with a cooperativity factor of 0.45 ± 0.1, now with K_C = 13.7 ± 2.3 μM and K_O = 2.8 ± 0.5 μM. With either method of fitting, however, the RCK1 site has a substantially lower affinity for Ca²⁺ than does the Ca²⁺ bowl in both the closed and in the open conformation. These numbers may be compared with the previous estimates of Xia et al. (2002), K_C = 17.2 ± 4.0 μM and K_O = 4.6 ± 1.0 μM, C = 3.74, which are similar to what we have found, and those of Bao et al. (2002), K_O = 3.8 ± 0.2 μM and K_C = 0.94 ± 0.06 μM, C = 4.0, which are higher affinity than what we have found (but see next paragraph). Interestingly, and of relevance here, we have found that Ca²⁺ binding at the RCK1 site is voltage dependent. The Ca²⁺ binding affinities in both the closed and open conformation increased as the voltage was depolarized. Moving the membrane voltage from −80 to 0 mV decreased K_C from 23.2 ± 2.6 μM to 15.8 ± 3.1 μM and K_O from 4.9 ± 0.6 μM to 2.1 ± 0.4 μM. This increased C from 4.75 to 7.52. Thus, as the voltage is depolarized, Ca²⁺ ions bind more tightly to the RCK1 site in both the closed and open conformations of the channel, and the factor by which each binding event increases the equilibrium constant for opening increases ∼1.5-fold.

In light of this result, one might suppose that the binding properties of the BK_Ca channel's two types of high-affinity Ca²⁺ binding sites will come closer together as the membrane voltage is further depolarized and the RCK1 sites' Ca²⁺ dissociation constants become progressively smaller. Although we have not done experiments at membrane voltages more positive than 0 mV and therefore cannot here confirm this hypothesis, such an idea could explain why Qian et al. (2006), in experiments with hybrid channels containing differing numbers of functional RCK1 or Ca²⁺ bowl sites, found that channels with either just four Ca²⁺ bowl sites or just four RCK1 sites showed almost identical Ca²⁺ dose-response curves at +50 mV. Further, it may also account, at least in part, for the estimates of Bao et al. (2002) being higher affinity than what we have found here for the RCK1 site, as Bao et al.'s estimates were based on the behavior of the mutant channel's full G-V relation as a function of Ca²⁺, and therefore they necessarily took into account Popen measurements at high voltage.

We have also observed that the effects of Ca²⁺ binding at each site, when measured individually, sum to more than what is observed when both sites are intact. We are unsure of the cause of this lack of strict independence, but we can explain it by supposing negative cooperativity between the Ca²⁺ bowl and RCK1 sites within the same subunit. In fact, all that is required is weak negative cooperativity between sites when the channel is open (b = 0.75 [−80 mV] or b = 0.65 [0 mV]) and no cooperative interaction between sites when the channel is closed. Thus, perhaps as the BK_Ca channel opens, a negative interaction between binding sites develops. In contrast to this result, however, in a study of single hybrid BK_Ca channels that contained two RCK1 sites and two Ca²⁺ bowl sites either on the same or on different subunits, Qian et al. (2006) found that there was positive rather than negative cooperativity between binding sites in the same subunit. The reason for these differing conclusions is not clear to us; however, as their study was done at +50 mV and ours at lower voltages, this difference may be the most relevant factor. What we can say, however, is that the Ca²⁺ binding constants reported by Qian et al. (2006) at +50 mV are not compatible at either site with our Ca²⁺ dose-response curves recorded at −80 and 0 mV.

Finally, one might ask what is the physical mechanism by which a change in voltage influences Ca²⁺ binding at the RCK1 sites? And why does this not occur at the Ca²⁺ bowl sites. We do not yet know the answers to these questions, but our current hypothesis is that the RCK1 sites lie in close proximity to the channel's voltage-sensing domains, and that as a given voltage sensor moves, it alters the structure of its nearby RCK1 Ca²⁺ binding site, while having no such interaction at the Ca²⁺ bowl. An allosteric interaction between the BK_Ca channel's low-affinity Ca²⁺ binding sites (those disabled by the E399N mutation) and its voltage sensors has already been firmly established (Hu et al., 2001; Cui et al., 1997, Yang and Sachs, 1989; Cui et al., 1997; Horrigan and Ma, 2008), and like the high-affinity RCK1 sites we have investigated here, these low-affinity sites are also thought to reside in the channel's RCK1 domains. Alternatively, one might suppose that Ca²⁺ binding is voltage dependent because Ca²⁺ binds within the electric field of the membrane; however, the RCK1 domains of the channel are thought to be suspended below the channel and thus they are not likely within the membrane's electric field.