Mg2+ Enhances Voltage Sensor/Gate Coupling in BK Channels

BK (Slo1) potassium channels are activated by millimolar intracellular Mg2+ as well as micromolar Ca2+ and membrane depolarization. Mg2+ and Ca2+ act in an approximately additive manner at different binding sites to shift the conductance–voltage (GK-V) relation, suggesting that these ligands might work through functionally similar but independent mechanisms. However, we find that the mechanism of Mg2+ action is highly dependent on voltage sensor activation and therefore differs fundamentally from that of Ca2+. Evidence that Ca2+ acts independently of voltage sensor activation includes an ability to increase open probability (PO) at extreme negative voltages where voltage sensors are in the resting state; 2 μM Ca2+ increases PO more than 15-fold at −120 mV. However 10 mM Mg2+, which has an effect on the GK-V relation similar to 2 μM Ca2+, has no detectable effect on PO when voltage sensors are in the resting state. Gating currents are only slightly altered by Mg2+ when channels are closed, indicating that Mg2+ does not act merely to promote voltage sensor activation. Indeed, channel opening is facilitated in a voltage-independent manner by Mg2+ in a mutant (R210C) whose voltage sensors are constitutively activated. Thus, 10 mM Mg2+ increases PO only when voltage sensors are activated, effectively strengthening the allosteric coupling of voltage sensor activation to channel opening. Increasing Mg2+ from 10 to 100 mM, to occupy very low affinity binding sites, has additional effects on gating that more closely resemble those of Ca2+. The effects of Mg2+ on steady-state activation and IK kinetics are discussed in terms of an allosteric gating scheme and the state-dependent interactions between Mg2+ and voltage sensor that may underlie this mechanism.

BK (Slo1) potassium channels are activated by millimolar intracellular Mg 2+ as well as micromolar Ca 2+ and membrane depolarization. Mg 2+ and Ca 2+ act in an approximately additive manner at different binding sites to shift the conductance-voltage (G K -V) relation, suggesting that these ligands might work through functionally similar but independent mechanisms. However, we fi nd that the mechanism of Mg 2+ action is highly dependent on voltage sensor activation and therefore differs fundamentally from that of Ca 2+ . Evidence that Ca 2+ acts independently of voltage sensor activation includes an ability to increase open probability (P O ) at extreme negative voltages where voltage sensors are in the resting state; 2 μM Ca 2+ increases P O more than 15-fold at −120 mV. However 10 mM Mg 2+ , which has an effect on the G K -V relation similar to 2 μM Ca 2+ , has no detectable effect on P O when voltage sensors are in the resting state. Gating currents are only slightly altered by Mg 2+ when channels are closed, indicating that Mg 2+ does not act merely to promote voltage sensor activation. Indeed, channel opening is facilitated in a voltage-independent manner by Mg 2+ in a mutant (R210C) whose voltage sensors are constitutively activated. Thus, 10 mM Mg 2+ increases P O only when voltage sensors are activated, effectively strengthening the allosteric coupling of voltage sensor activation to channel opening. Increasing Mg 2+ from 10 to 100 mM, to occupy very low affi nity binding sites, has additional effects on gating that more closely resemble those of Ca 2+ . The effects of Mg 2+ on steady-state activation and I K kinetics are discussed in terms of an allosteric gating scheme and the state-dependent interactions between Mg 2+ and voltage sensor that may underlie this mechanism.

I N T R O D U C T I O N
The activity of BK-type Ca 2+ -activated K + channels is important for the function of nerve, muscle, and secretory cells (Vergara et al., 1998). BK channels respond to two primary signals, membrane voltage and intracellular Ca 2+ , but are also sensitive to a variety of regulatory ligands, including intracellular Mg 2+ (Shi and Cui, 2001;Zhang et al., 2001), heme , and protons (Avdonin et al., 2003), that can infl uence channel gating under physiological and/or pathophysiological condi tions. Therefore, to understand how BK channel activity is regulated it is important to understand how these signaling and regulatory factors affect channel gating and interact with each other. Previous studies suggest that millimolar [Mg 2+ ] i and micromolar [Ca 2+ ] i activate BK channels through similar functional mechanisms (Shi and Cui, 2001;Zhang et al., 2001). Here we show that the action of Mg 2+ differs from that of Ca 2+ in its dependence on voltage sensor activation. The results highlight that multiple pathways of communication exist between the transmembrane and cytosolic domains of BK channels.
That Ca 2+ and voltage act independently to regulate opening is evident from BK channel function and consistent with channel structure. BK channels can be fully activated by membrane depolarization in the absence of Ca 2+ , and Ca 2+ facilitates opening in a nearly voltage-independent manner, indicating that Ca 2+ and voltage sensor activation have energetically additive effects on opening (Cui and Aldrich, 2000;Horrigan and Aldrich, 2002). The transmembrane domain of BK channels, like voltage-gated K + (Kv) channels, includes an S5-S6 pore domain and a charged S1-S4 voltage sensor (Ma et al., 2006). Coupling between the voltage sensor and pore can occur presumably as in Kv channels, via interactions within the transmembrane domain ( Fig. 1 A) (Lu et al., 2002;Long et al., 2005b). In addition, BK channels contain a large C-terminal cytosolic domain that interacts with intracellular ligands and is directly attached to the S6 activation gate. The cytosolic domain of each α-subunit contains two putative RCK (regulator of K + conductance) homology domains (RCK1, RCK2), like those in the prokaryotic channel MthK (Jiang et al., 2001). In MthK, RCK domains from different subunits assemble into a gating ring structure that expands upon Ca 2+ binding to pull open the gate (Fig. 1 B) (Jiang et al., 2002;Ye et al., 2006), suggesting that Ca 2+ opens BK channels by increasing tension on the RCK1-S6 linker (Niu et al., 2004). Thus models of BK channel structure support that voltage and Ca 2+ sensors may act independently on the gate (Fig. 1 C) to exert additive effects on steady-state activation.
Given the example of MthK, one might suppose that any ligand binding to the cytosolic domain of the BK Correspondence to Frank T. Horrigan: horrigan@mail.med.upenn.edu Abbreviations used in this paper: Kv, voltage-gated K + ; RCK, regulator of K + conductance. channel must act, like Ca 2+ , to open the gate by pulling on the RCK1-S6 linker. However this is not necessarily the case. Voltage sensors in BK channels, which are absent from MthK, are potentially available to interact with the cytosolic domain, providing an alternative pathway of communication between ligand binding sites and the transmembrane domain. The cytosolic domain of MthK lies close to the transmembrane domain, presents a relatively fl at surface toward the membrane, and with a diameter of >70 Å extends laterally far beyond the pore (Fig. 1 B). All of these features, which are presumably necessary to exert lateral force on the gate via short peptide linkers, would also tend to bring the periphery of the gating ring close to the voltage sensor domain in BK channels (Fig. 1 D). Several observations suggest that interactions between cytosolic and voltage sensor domains occur. Ca 2+ has effects on gating current, consistent with a weak interaction between Ca 2+ sensors and voltage sensors (Horrigan and Aldrich, 2002). Intracellular heme has a pronounced effect on voltage gating, weakening the allosteric coupling between voltage sensor and gate . Likewise, chemical modifi cation of a native cysteine residue (C430) in the cytosolic domain alters voltage sensor/gate coupling (Zhang and Horrigan, 2005).
Mg 2+ has effects on BK channel gating that resemble those of Ca 2+ : increasing P O , slowing I K deactivation, and shifting the G K -V relation to more negative voltages with little change in shape (Shi and Cui, 2001;Zhang et al., 2001). Effects of Mg 2+ and Ca 2+ on the half-activation voltage (V 0.5 ) are approximately additive and involve different binding sites (Shi et al., 2002;Xia et al., 2002). Consequently, Mg 2+ has been proposed to act independently of Ca 2+ but through a similar functional mechanism. Effects of Mg 2+ on the G K -V relation can be reproduced by gating schemes that assume Mg 2+ acts, like Ca 2+ , to enhance channel opening independent of voltage sensor activation (Shi and Cui, 2001;Zhang et al., 2001;Hu et al., 2006). However, several lines of evidence suggest that Mg 2+ and Ca 2+ do not act through identical functional mechanisms. First, high concentrations of Ca 2+ (>100 μM), which are known to occupy Mg 2+ binding sites, have effects that differ qualitatively from those of low [Ca 2+ ] i . While 0-100 μM Ca 2+ produces a nearly voltage-independent increase in log(P O ), 100-1,000 μM Ca 2+ increases log(P O ) in a voltage-dependent manner (Horrigan and Aldrich, 2002). In addition, mutations in the voltage sensor, including R213Q in S4, disrupt Mg 2+ sensitivity but not Ca 2+ sensitivity, suggesting that the voltage sensor plays a critical role in Mg 2+ -dependent activation (Hu et al., 2003). Indeed, recent evidence indicates that Mg 2+ bound to the cytosolic domain interacts electrostatically with R213 (Yang et al., 2007).
This study examines the functional interaction between Mg 2+ -and voltage-dependent activation of mSlo1 BK channels. Effects of 0-100 mM Mg 2+ on the steady-state and kinetic properties of ionic and gating currents were examined over a wide voltage range in WT channels and channels where either the voltage sensor or a putative Mg 2+ -binding site in the RCK1 domain were mutated. The results show that Mg 2+ at the RCK1 site acts primarily to strengthen the allosteric coupling of voltage sensor activation to channel opening. Only at high concentrations (>10 mM), involving an additional very low affi nity binding site, does Mg 2+ have direct effects on opening, like Ca 2+ . Our results together with the proposed electrostatic nature of Mg 2+ /voltage sensor interaction suggest that Mg 2+ and the voltage sensor must interact in multiple states to account for the effects of Mg 2+ on voltage sensor/gate coupling.

Channel Expression and Molecular Biology
Experiments were performed with the mbr5 clone of the mouse Slo1 gene (mSlo1) (Butler et al., 1993) expressed in Xenopus oocytes (for I K ) or HEK 293T cells (for gating current). The clone was propagated and cRNA transcribed as described previously . Oocytes were injected with ‫5-5.0ف‬ ng of cRNA, incubated at 18°C, and studied 3-7 d after injection. HEK cells were transfected with mSlo1 in an SRα vector using Lipofectamine (GIBCO BRL/Life Technologies Inc.) 2-3 d before recording as described previously (Horrigan and Aldrich, 2002). Site-directed mutagenesis was performed with the QuikChange XL Site-Directed Mutagenesis Kit (Stratagene) and confi rmed by sequencing.
Data were acquired with an Axopatch 200B amplifi er (Molecular Devices Corp.) in patch mode with the Axopatch's fi lter set at 100 kHz. Currents were fi ltered by an 8-pole Bessel fi lter (Frequency Device, Inc.) at 50 kHz (I K ) or 20 kHz (I g ) and sampled at 200 kHz with an 18-bit A/D converter (Instrutech ITC-18). For gating currents, the voltage command was also fi ltered at 20 kHz. A p/4 protocol was used for leak subtraction (Armstrong and Bezanilla, 1974) with a holding potential of −80 mV. Electrodes were made from thick-walled 1010 glass (World Precision Instruments, Inc.) and their tips coated with wax (KERR Sticky Wax). The electrode's resistance in the bath solution (0.5-1.5 MΩ) was used as an estimate of series of resistance (R S ) for correcting the voltage at which macroscopic I K was recorded. Series resistance error was <15 mV for all data presented. A Macintosh-based computer system was used in combination with Pulse Control acquisition software (Herrington and Bookman, 1995) and Igor Pro for graphing and data analysis (WaveMetrics, Inc.). A Levenberg-Marquardt algorithm was used to perform nonlinear least-squared fi ts. Data are presented as mean ± SEM.
Open probability (P O ) was estimated by recording macroscopic I K when P O was high (≥0.005-0.05) and unitary currents in the same patch when P O was low (≤0.01-0.1). Macroscopic conductance (G K ) was determined from tail currents at −80 mV following 50-ms voltage pulses, and was normalized by G Kmax to estimate P O . G Kmax was measured in 5-100 mM Mg 2+ (0 Ca 2+ ) at V > 200 mV from tail currents and was estimated at [Mg 2+ ] < 5 mM as is the single channel conductance at −80 mV and is reduced slightly ‫)%5ف(‬ in 10 Mg 2+ due to Mg 2+ block. At more negative voltages, NP O was determined from steady-state recordings of 1-60 s duration that were digitally fi ltered at 5 kHz. NP O was determined from all-points amplitude histograms by measuring the fraction of time spent (P K ) at each open level (K) using a half-amplitude criteria and summing their contributions NP o = ΣkP k . P O was then determined by estimating N from G Kmax (N = G Kmax /γ K , where γ K is the single channel conductance at −80 mV). Mean activation charge displacement (q a = kT d(ln(P O )/dV) was measured from the slope of the ln(P O )-V relation by linear regression over 60-mV intervals and plotted against mean voltage. Fits of the q a -V relation were similarly determined by linear regression of simulated data (Ma et al., 2006). Patch-to-patch variation in V 0.5 of the G-V and Q-V relationships are observed for Slo1 channels (Stefani et al., 1997;. To compensate for the effect of such variation on mean P o -V and Q C -V relations, V 0.5 was determined for each patch and individual relations were shifted along the voltage axis by ∆V 0.5 = (<V 0.5 > − V 0.5 ) before averaging, where <V 0.5 > is the mean for all experiments at the same [Mg 2+ ]. Patch to patch variation in P O at extreme negative voltages is also observed . Therefore, mean log(P O )-V and τ K -V relations for 0, 10, and 100 mM Mg 2+ were constructed using only experiments in which both Mg 2+ data and matching 0 Mg 2+ controls exist. In this way, mean relations accurately refl ect effects of Mg 2+ that were observed in individual experiments.
Gating capacitance (C g ) was measured using admittance analysis as previously described . In brief, gating currents were recorded in response to 0.5-s voltage ramps upon which a sinusoidal voltage command (781 hz, 60 mV peak to peak) was superimposed. Admittance was calculated for each cycle of the sinewave, and capacitance was determined after correcting for phase shifts due to instrumentation.

Mg 2+ and Ca 2+ Act through Different Mechanisms
The effects of millimolar [Mg 2+ ] i and micromolar [Ca 2+ ] i on steady-state activation are compared in Fig. 2. In addition to increasing open probability, intracellular Mg 2+ rapidly blocks the pore in a voltage-dependent manner that effectively reduces single channel conductance at depolarized voltages (Ferguson, 1991). Both of these effects are evident by comparing macroscopic potassium currents (I K ) in the presence and absence of 10 mM Mg 2+ (Fig. 2 A). I K was evoked by 50-ms pulses to different voltages in inside-out patches from Xenopus oocytes (A) Kv1.2, where the charged S4 segment (light blue) in the voltage sensor is proposed to interact with the S6 activation gate (dark blue) in the pore via the S4-S5 linker (arrows). The outlines of the S1-S4 voltage sensor domain and S5-S6 pore domain for two subunits from the Kv1.2 crystal structure (Long et al., 2005a) are shown in dark and light gray, respectively, and are truncated at S1 (protein data bank code 2A79). (B) MthK, where Ca 2+ binding to the cytosolic domain is proposed to cause a conformational change that pulls open the gate through short peptide linkers (dashed lines) that are unresolved in the crystal structure (Jiang et al., 2002) (protein data bank code 1LNQ). The attachment points of the linker on the cytosolic domain (in foreground and background) have been altered to better illustrate that linkers are pulled away from the pore axis. (C) A hypothetical BK channel structure formed by appending the Kv1.2 voltage sensors to MthK shows how Ca 2+ and voltage may both interact with the gate to have independent effects on activation. The position of the voltage sensors was determined by superimposing the pore domains of Kv1.2 (residues 341-406) and MthK (residues 26-91) in PyMol (http://www.pymol.org). (D) The hypothetical BK channel structure also illustrates the possibility that the large cytosolic domain may interact directly with voltage sensors. expressing mSlo1 BK channels. Mg 2+ decreases outward current at the most positive voltages owing to pore block, but increases tail currents at −80 mV, indicating that P O is increased. Normalized G K -V relations, determined from tail currents exhibit a −67.2 ± 4.9 mV shift in halfactivation voltage (V 0.5 ) in response to 10 mM Mg 2+ , with little change in shape, similar to the effect of 2 μM Ca 2+ (Fig. 2 B).
Despite similar effects of Mg 2+ and Ca 2+ on the G K -V relation, fundamental differences in the action of these ligands are evident when P O is plotted on a log-scale versus voltage (Fig. 2 C). P O was measured to values as low as ‫01ف‬ −7 by recording unitary current activity in macropatches containing hundreds of channels. Log(P O ) is increased by 2 μM Ca 2+ in a nearly voltage-independent manner, shifting the entire log(P O )-V curve upward by 1.2 log units (a 15-fold increase in P O ) until channels are maximally activated at positive voltages. P O is increased 15-fold even at extreme negative voltages (≤−100 mV) where voltage sensors are not activated and log(P O ) is weakly voltage dependent (Fig. 2 C, dotted lines) (Horrigan and Aldrich, 2002). By contrast, 10 mM Mg 2+ has no effect on steady-state activation at extreme negative voltages and increases log(P o ) in a voltage-dependent manner from −80 to +40 mV such that the effects of Mg 2+ and Ca 2+ are similar only at more positive voltages.
Why Is the Effect of Mg 2+ Voltage Dependent?
Two different mechanisms could potentially account for the voltage dependence of Mg 2+ action; the effect of Mg 2+ could depend on the activation state of the voltage sensor, or Mg 2+ binding could be intrinsically voltage dependent. Voltage-dependent binding must be considered because if Mg 2+ interacts with the voltage sensor it could also lie within the membrane electric fi eld; and a model assuming that Mg 2+ traverses a fraction of the electric fi eld (γ = 0.2) to reach its binding site can reproduce the effect of 10 mM Mg 2+ (Fig. 2, B and C, dotted curves, see legend for details). However, the effects of Mg 2+ on two voltage sensor mutants (R207Q, R167E) Figure 2. Effects of Mg 2+ on steady-state activation. (A) Effect of 10 mM Mg 2+ on macroscopic I K evoked from mSlo1 channels in response to 50-ms pulses to different voltages in 0 Ca 2+ from a holding potential of −80 mV. Mg 2+ decreases outward current at high voltages owing to rapid pore block. However tail currents indicate an increase in P O and slowing of channel deactivation. (B) Mean normalized G K -V relations (P o = G K /G Kmax ) determined from tail currents in 10 mM Mg 2+ (O), 2 μM Ca 2+ (◆) or the absence of divalent cations (control, •). (C) Mean log(P O )-V relations corresponding to B achieve a weakly voltage-dependent limiting slope at extreme negative voltages, indicating that voltage sensors are in the resting state. Dotted lines are exponential fi ts to control and Ca 2+ data with partial charge z L = 0.3e (Ma et al., 2006). (D) Voltage sensor mutations R207Q (△, ▲) and R167E (□, ■) shift both the voltage dependence of activation and Mg 2+ sensitivity (0 Mg: fi lled symbols; 10 mM Mg 2+ : open symbols) relative to the WT (O, •), implying that Mg 2+ action depends on voltage sensor activation and is not intrinsically voltage dependent. (E) Voltagegating mechanism (Scheme 1). Channels undergo a closed-open (C-O) conformational change that is allosterically regulated by four independent and identical voltage sensors that can undergo a resting-activated (R-A) transition. L and J are voltage-dependent equilibrium constants (L = L 0 exp(z L V/kT), J = J 0 exp(z J V/kT) = exp(z J [V − V hC ]/kT)) and D is an allosteric factor. (F) Scheme 2 includes a ligand-binding transition (X-X·M 2+ ) in each subunit that allosterically regulates channel opening and voltage sensor activation as described by factors C and E, respectively. Solid curves in B and C represent a fi t of this model to Ca 2+ -dependent activation in 0-100 μM Ca 2+ (Horrigan and Aldrich, 2002) (V hC = 156 mV, z J = 0.58e, L 0 = 1.06 × 10 −6 , z L = 0.3e, K D (Ca 2+ ) = 11 μM, C = 8, D = 24.4, E = 2.4). The same model with identical values of V hC , z J , L 0 , z L , D, and E can fi t the Mg 2+ data (dotted curves) if Mg 2+ binding is assumed to be voltage dependent (i.e., K D (Mg 2+ ) = K D (0)exp(−Vγ2e/kT)) (K D (0) = 9.35 mM, γ = 0.2, C = 1.66) (G) Scheme 3 assumes that ligand binding allosterically regulates voltage/ sensor gate coupling and voltage sensor activation as described by allosteric factors F and E, respectively. in Fig. 2 D indicate that voltage sensor activation is critical to the effect of Mg 2+ and probably suffi cient to account for the voltage dependence of Mg 2+ action.
R207Q in S4 and R167E in S2 shift voltage sensor activation to more negative and positive voltages, respectively (Ma et al., 2006), such that log(P O )-V relations for these mutants are shifted along the voltage axis relative to the WT in 0 Mg 2+ (Fig. 2 D, fi lled symbols). R207Q also shifts the Q-V relation by more than −250 mV, based on gating current measurements, such that even at −200 mV a signifi cant fraction ‫)2.0ف(‬ of voltage sensors are activated (Ma et al., 2006). Both R207Q and R167E are readily activated by 10 mM Mg 2+ at depolarized voltages ( Fig. 2 D), suggesting these mutations do not disrupt Mg 2+ binding nor interactions of Mg 2+ with the voltage sensor (Hu et al., 2003;Yang et al., 2007). However the voltage dependence of Mg 2+ -dependent activation for the mutants is greatly altered relative to the WT, suggesting that Mg 2+ action depends on the activation state of the voltage sensor. In the case of R167E, Mg 2+ has no effect on P O at voltages up to +60 mV where the WT is strongly activated by Mg 2+ . Conversely, R207Q exhibits a robust 30-fold increase in P O at negative voltages (−120 to −200 mV) where the WT appears insensitive to Mg 2+ . The response of R207Q indicates that Mg 2+ is capable of occupying its binding site even at extreme negative voltages. Thus, it is unlikely that Mg 2+ binding is intrinsically voltage dependent; and the inability of the WT to be activated at negative voltages likely refl ects that Mg 2+ is incapable of increasing P O when voltage sensors are in the resting state. This hypothesis is supported by comparison of the WT and R167E data that show that channels are Mg 2+ insensitive when P O is weakly voltage dependent.

Understanding the Action of Mg 2+ in Terms of Allosteric Mechanisms
To determine how Mg 2+ and voltage sensors interact mechanistically we interpreted our results in terms of a well-established allosteric gating scheme that accounts for the effects voltage on steady-state activation of BK channels ( Fig. 2 E, Scheme 1) . Scheme 1 asserts that the opening of the gate can be described as a concerted, weakly voltage-dependent transition between an open and closed conformation (C-O) with zero-voltage equilibrium constant L 0 and partial charge z L . In addition, voltage sensors in each of four independent and identical subunits can undergo a transition between a resting and activated conformation (R-A) characterized by equilibrium constant J 0 and partial charge z J . The coupling between voltage sensor and gate is described by an allosteric factor D, such that the C-O equilibrium constant increases D-fold for each voltage sensor activated and the voltage sensor equilibrium increases D-fold when the channel opens.
Based on Scheme 1 there are four distinct mechanisms by which a ligand might alter P o . Ligand binding could (a) infl uence the intrinsic stability of the gate (L 0 ), (b) perturb the voltage sensor equilibrium (J 0 ), (c) alter voltage sensor/gate coupling (D-factor), or (d) alter the gating charge associated with voltage sensor activation (z J ) or channel opening (z L ). The action of Ca 2+ is mainly accounted for by an effect on the gate (L 0 ) with a minor impact on voltage sensor activation (J 0 ). This mechanism can be represented by an allosteric model (Fig. 2 F, Scheme 2) that successfully reproduces the effects of Ca 2+ (Fig. 2, B and C, solid lines) in terms of a ligand-binding equilibrium (X-X·M 2+ ) and allosteric factors C and E that describe the coupling of Ca 2+ sensors to the gate and to voltage sensors, respectively (Horrigan and Aldrich, 2002). Direct coupling between Ca 2+ sensors and gate is evident from the ability of Ca 2+ to increase P O whether or not voltage sensors are activated. Thus the insensitivity of P O to 10 mM Mg 2+ when voltage sensors are in the resting state rules out a similar interaction between Mg 2+ sensors and gate and indicates that L 0 is unchanged, but leaves open the possibility that Mg 2+ affects voltage sensor activation or coupling ( Fig. 2 G, Scheme 3) or charge.

Mg 2+ Has Little Effect on Voltage Sensor Activation when Channels Are Closed
To characterize effects of Mg 2+ on the voltage sensor we recorded gating current from WT and mutant channels (Fig. 3). The results indicate that Mg 2+ has little effect on the voltage sensor equilibrium (J 0 ) or gating charge (z J ).
Voltage sensor activation in BK channels is much faster than channel opening (Stefani et al., 1997;. Therefore ON gating current (I gON ) refl ects primarily voltage sensor activation when channels are closed. Fig. 3 A shows the effects of Mg 2+ on I gON evoked by brief 0.5-ms depolarizations to +200 mV. 10 mM Mg 2+ causes an ‫%02ف‬ increase in peak I gON with little change in kinetics ( Fig. 3 A, top). Application of 100 mM Mg 2+ , to activate very low affi nity binding sites , produces a greater increase in peak current ( Fig. 3 A, bottom). Mean charge-voltage relations for closed channels (Q C -V), determined from I gON (see Fig. 3 legend) are shifted to more negative voltages by −16.7 ± 2.3 mV (n = 5) or −29.6 ± 2.6 mV (n = 3) in 10 or 100 mM Mg 2+ , respectively (Fig. 3 B). This shift in half-activation voltage (V hC ) occurs without appreciable change in the shape of the Q C -V relation, suggesting that Mg 2+ does not alter gating charge. Similarly, effects of 10 mM Mg 2+ on the time constant of I gON (τ gFast ) for a single experiment (Fig. 3 C) refl ect a small −21 mV shift in the τ gFast -V relation to more negative voltages.
To better quantify small changes in V hC (∆V hC ) by Mg 2+ we examined gating capacitance (C g ) (Fig. 3, D-F). C g is the voltage-dependent component of membrane capacitance and refl ects the derivative of gating charge with respect to voltage. The C g -V relation was determined from admittance analysis by measuring I g evoked by a sinusoidal voltage command superimposed on a 500-ms voltage ramp from −200 to +200 mV (see Materials and methods) Aldrich, 1999, 2002). At voltages where steady-state P O is low (<0.1), the C g -V relation approximates the derivative of Q C -V . Thus ∆V hC can be determined directly from shifts in the foot of C g -V relation ( Fig. 3 D). Because C g -V is determined rapidly and with high voltage resolution, small V hC shifts are detectable, the effects of applying and washing out Mg 2+ can be assessed rapidly, and errors in ∆V hC due to slow changes in V hC from channel oxidation or electrode potential drift are minimized. C g -V analysis for the WT (Fig. 3 D) yielded ∆V hC values (10 Mg: 17.2 ± 1.5 mV [n = 6], 100 Mg: 36.1 ± 1.6 mV [n = 6]) similar to those determined from Q C -V relations.
The effect of Mg 2+ on V hC is comparable to that of micromolar Ca 2+ (Horrigan and Aldrich, 2002), consistent with a weak interaction between Mg 2+ binding and voltage sensor activation in closed channels that increases the voltage sensor equilibrium constant J 0 (e.g., E-factor in Schemes 2 and 3, Fig. 2, F and G). However, millimolar Mg 2+ may also shift voltage-dependent parameters by nonspecifi c mechanisms such as screening of membrane surface charge . Therefore, to assess the impact of the RCK1 Mg 2+ site on J 0 we compared effects of Mg 2+ on the WT channel to those of a mutant (E374A/E399N) whose RCK1 site is disrupted (Shi et al., 2002;Hu et al., 2006). This mutation of putative Mg 2+ -coordinating residues, which reduces the G K -V shift in 10 mM Mg 2+ by 80% to −14.0 ± 2.0 mV, also reduces effects of 10 or 100 mM Mg 2+ on the C g -V relation (Fig. 3, E and F). The difference in ∆V hC between the WT and mutant (∆∆V hC = ∆V hC (E374A/E399N) − ∆V hC (WT)) was −9.5 ± 2.8 mV and −11.6 ± 2.8 mV for 10 and 100 Mg 2+ , respectively, likely representing the effect of RCK1 site occupancy on J 0 . The similar impact of mutation in 10 and 100 mM Mg 2+ is consistent with the expectation that the RCK1 site is nearly saturated in 10 mM Mg 2+ .
The small effect of Mg 2+ on gating current suggests that Mg 2+ enhances voltage sensor/gate coupling (D). Effects of Mg 2+ on gating current were measured in the absence of permeant ions and presence of extracellular tetraethyl ammonium (TEA) to block the pore. (A) I g evoked by 0.5-ms pulses to +200 mV in the presence (dotted traces) or absence (solid traces) of Mg 2+ , from two different patches testing the effects 10 or 100 mM Mg 2+ . (B) Mean Q C -V relations in 0 (•), 10 (△), and 100 (◆) mM Mg 2+ are fi t by Boltzmann functions with identical equivalent charge (z J = 0.58e) (Horrigan and Aldrich, 2002) and were normalized by the fi t amplitude (Q cMax ). Q C represents the charge distribution for closed channels, determined by fi tting the initial 50-100-μs decay of I gON with an exponential function with time constant τ gFast and integrating the area under the rising phase and fi t . (C) τ gFast -V relations for the patch in A are shifted by −21 mV in 10 mM Mg 2+ . Relations are fi t by functions of the form τ gFast = [α(V) + β(V)] −1 where α and β are forward and backward rate constants for voltage sensor activation (0 Mg: Representative C g -V relations obtained with admittance analysis for (D) WT or (E) E374A/E399N channels in the presence (black curves) and absence (gray curves) of 10 or 100 mM Mg 2+ are fi t by the derivative of a Boltzmann function with respect to voltage (dotted curves) with identical amplitude and charge (z J = 0.58e). Shifts along the voltage axis indicate the change in V hC (∆V hC ). Arrows indicate the upper voltage limit of the fit that was determined over a range where most channels are closed (P o < 0.1). (F) ∆V hC = [V hC (Mg 2+ ) − V hC (0 Mg 2+ )] for WT (◆, n = 6) and E374A/E399N (◇, n = 4) determined from C g -V relations.
The −17.2 mV ∆V hC by 10 mM Mg 2+ (Fig. 3 F) can account for a similar G K -V shift but is much too small to explain the observed −67-mV shift (Fig. 2 B). By contrast, an increase in the coupling factor D can shift V 0.5 without affecting V hC . In addition, enhanced coupling should facilitate channel opening when voltage sensors are fully activated and facilitate voltage sensor activation when channels are open. These predictions are confi rmed below by results in Figs. 4 and 5.

Mg 2+ Facilitates Opening When Voltage Sensors Are Constitutively Activated
Voltage sensor/gate coupling defi nes the extent to which P O is increased by voltage sensor activation. According to Scheme 1, the C-O equilibrium constant increases from L 0 when voltage sensors are in the resting state to L 0 D 4 when all four voltage sensors are activated. If 10 mM Mg 2+ enhances coupling (D) without affecting L 0 , then the C-O equilibrium constant when voltage sensors are fully activated should increase by a factor of δD 4 where δD = D(Mg 2+ )/D(control). This predicted facilitation of opening cannot be tested for the WT channel because in 0 Mg 2+ P O is already nearly saturated ‫)1ف(‬ at potentials >250 mV where voltage sensors are fully activated (compare G K -V in Fig. 2 B and Q c -V in Fig. 3 B). However, facilitated opening by Mg 2+ can be observed in mutants with enhanced voltage sensor activation (R207Q, R210C; Fig. 4). Fig. 4 A compares the log(P O )-V relations for WT, R207Q, and R210C channels in 0 Mg 2+ . R207Q shifts Q C -V, as noted above, such that this relation is saturated and voltage sensors are fully activated at V ≥ −20 mV (Ma et al., 2006) (Fig. 4 A, labeled arrow). Over this voltage range, P O is weakly voltage dependent and well described by a Boltzmann function (dotted curve) as expected for channels with fully activated voltage sensors undergoing a two-state C-O transition with equilibrium constant L 0 D 4 (Horrigan and Aldrich, 2002). Importantly, the weak voltage dependence of L 0 reduces P O to nonsaturating levels at potentials where the mutant voltage sensors are still fully activated (e.g., P O = 0.04 at −20 mV). The P O -V relation for R210C is indistinguishable from that of R207Q at V ≥ −20 but extends the weakly voltagedependent phase of P O to less than −200 mV, suggesting that R210C voltage sensors are constitutively activated (Ma et al., 2006). The difference between WT and R207Q in Fig. 4 A is consistent with a 100-fold change in the voltage sensor equilibrium constant J 0 Ma et al., 2006) and illustrates qualitatively the effect expected if Mg 2+ acts merely to promote voltage sensor activation. That is, an increase in J 0 can shift the steepest part of the log(P O )-V relation to more negative voltages but cannot increase P O above the limiting relation defi ned by L 0 D 4 (Fig. 4 A, dotted curve). By contrast, 10 mM Mg 2+ markedly increases P O for R207Q (Fig. 4 B) and R210C channels (Fig. 4 C) when voltage sensors are fully activated, indicating that L 0 D 4 is increased. This increase presumably refl ects enhanced voltage sensor/gate coupling (increased D) because L 0 , although it cannot be measured directly for R207Q or R210C channels, is insensitive to 10 mM Mg 2+ in WT and R167E (Fig. 2 D). Dotted curve represents a Boltzmann fi t to R210C with partial charge z = 0.46e that describes the weak voltage dependence of channel opening in both mutants when voltage sensors are fully activated (Ma et al., 2006). (B) Mean log(P O )-V relations for R207Q show that 10 mM Mg 2+ (△) increases P O even when voltage sensors are fully activated. Dotted Boltzmann fi ts from −20 to +240 mV have identical charge (0.46e) but V 0.5 is shifted by −91.6 mV in Mg 2+ , consistent with a 8.2-fold increase in the C-O equilibrium constant. (C) Mean log(P O )-V relations for R210C show that 10 mM Mg 2+ (▽) increases the C-O equilibrium constant in a voltage-independent manner when voltage sensors are constitutively activated. Boltzmann fi ts have identical charge (0.46e) but V 0.5 is shifted by −122 mV in Mg 2+ , consistent with a 16.6-fold increase in the C-O equilibrium constant and a twofold increase in the allosteric coupling factor D.
Analysis of the mutant data indicate that 10 mM Mg 2+ produces an approximate twofold increase in the coupling factor D. Fig. 4 C shows that R210C channels are activated by 10 mM Mg 2+ in a voltage-independent manner, consistent with the assumption that voltage sensors are constitutively activated. Mean log(P 0 )-V relations in the presence and absence of Mg 2+ were fi t by Boltzmann functions with identical charge (Fig. 4 C, dotted curves), as expected if Mg 2+ increases the C-O equilibrium constant by a voltage-independent factor (δD 4 ) and Mg 2+ binding is not voltage dependent. A similar response is observed for R207Q at V ≥ −20 mV (Fig. 4 B, dotted curves). The fi ts to R207Q and R210C yield δD 4 values of 8.2 and 16.6, respectively, implying a 1.7-2.0-fold increase in D. A more direct estimate of δD 4 is obtained from the Mg 2+ -dependent increase in P O for R210C at extreme negative voltage. Over a range of voltages (−200 to −140 mV) where P O is small (<0.1) and therefore approximates the C-O equilibrium constant, Mg 2+ increases P O by an average of 21.6 ± 6.0-fold (δD 4 ), indicating a 2.16 ± 0.13-fold increase in D, consistent with the Boltzmann fi ts.

Different Mg 2+ Sites Have Distinct Effects on Gating
To further characterize the action of Mg 2+ at both RCK1 and very low affi nity binding sites we examined the effects on P O of different [Mg 2+ ] i , up to 100 mM, in WT and mutant channels (Fig. 5). The RCK1 site with an Figure 5. Effects of Mg 2+ on different binding sites were determined by measuring P O for WT and mutant channels in 0-100 mM Mg 2+ (0, blue circle; 2, □; 5, ▽; 10, blue triangle; 21, ; 100, blue diamond, mM Mg 2+ ). (A) Normalized mean G K -V relations for the WT in 0-100 mM Mg 2+ . (B) Mean log(P O )-V relations for WT in 0, 10, and 100 mM Mg 2+ reveal that 100 Mg 2+ increases P O at extreme negative voltages. These plots represent the same data as in A but were averaged in 20 mV rather than 10-mV bins to match the voltage interval for low P O measurements. Dotted lines are exponential fi ts to P O at extreme negative voltages in 0 and 100 Mg 2+ with partial charge z L = 0.3e. (C) Mean log(P O )-V relations in 0, 10, and 100 mM Mg 2+ for E374A/E399N show that mutation of the putative Mg 2+ site in RCK1 almost abolishes effects of 10 mM Mg 2+ but leaves the P O increase by 100 Mg 2+ (dotted lines) intact. (D) Mean log(P O )-V relations in 0, 10, and 100 mM Mg 2+ for R167E achieve a limiting slope at more positive voltages than the WT, confi rming that P O is increased by 100 Mg 2+ when voltage sensors are in the resting state (dotted lines). Inset graph plots the ratio of NP O in the presence and absence of Mg 2+ at −80 mV from a larger set of experiments. (E) Mean q a -V relations for the WT in 0, 10, and 100 mM Mg 2+ were derived from the slope of log(P O )-V in B (see text). Solid curves are estimates of the Q O -V relation determined by fi tting the foot of the q a -V relation (to ‫%08ف‬ of peak q a ) with a function q a = z L + 4z J [1 + exp([V hO − V]z J /kT)] −1 where z L = 0.3e, z J = 0.58e, and V hO = 26.5 ± 2.1, −33.6 ± 2.4, −65.6 ± 3.7 mV in 0, 10, and 100 mM Mg 2+ , respectively. (F) Plot of ∆∆V CO = ∆V hC − ∆V hO for 10 and 100 mM Mg 2+ in WT (blue circle) and E374A/E399N (red square) channels where ∆V hO = [V hO (Mg 2+ ) − V hO (0 Mg 2+ )] was estimated from the q a -V relation as in E and ∆V hC was determined from gating current (Fig. 3 F). Nonzero values of ∆∆V CO indicate that Mg 2+ alters voltage sensor/ gate coupling in the WT. However, effects on coupling are abolished by the E374A/E399N mutation. (G) Alternative representation of Scheme 2 (Fig. 2 F) shows the four states defi ned by the voltage sensor and gate conformation in each subunit and the equilibrium constants defi ned by Scheme 1 (Fig. 2 E). Boxes denote the states stabilized by ligand binding, as defi ned by allosteric factors C and E. (H) In Scheme 3, the allosteric factor F defi nes a stabilization of the OA state that alters voltage sensor/gate coupling (allosteric factor D). Dotted curves in A-E represent simultaneous fi ts to the P o -V, log(P O )-V, and q a -V relations in 0, 10, and 100 mM Mg 2+ using Scheme 1 (Fig. 2 E) (Table I parameters WT A , E374A/E399N). Solid black curves in A and B represent an alternative fi t to the WT data (WT B parameters). Red curves in A represent fi ts to a two-site model (Eq. 2) that assumes each subunit in the WT contains a very low affi nity and an RCK1 binding site described by Schemes 2 and 3, respectively (Table II parameters). estimated K D of 2.2-5.5 mM should be nearly saturated at 10 mM Mg 2+ , whereas a putative very low affi nity site (K D : 40-136 mM) is not signifi cantly affected until concentrations >10 mM . The effects of high and low concentrations of Mg 2+ appear similar in that G K -V relations are progressively shifted to more negative voltages with little change in shape (Fig. 5 A) (Shi and Cui, 2001;Zhang et al., 2001;Hu et al., 2006). Increasing Mg 2+ from 10 to 100 mM shifted the G-V by an additional −34 mV (Fig. 5 A). However, log(P O )-V relations reveal that 100 mM Mg 2+ also increases P O signifi cantly at extreme negative voltages (Fig. 5 B). Thus it appears that occupancy of very low affi nity Mg 2+ sites, unlike the RCK1 site, infl uence the gate (L 0 ) to increase P O when voltage sensors are not activated.
The log(P O )-V relation in 100 mM Mg 2+ appears to achieve a limiting slope between −160 and −140 mV, where P O is increased 2.9-fold relative to the control (Fig. 5 B, dotted lines), implying a 2.9-fold increase in L 0 . However measurements below −140 mV were difficult to obtain. Therefore, to confi rm that 100 mM Mg 2+ increases L 0 we also compared log(P O )-V relations in 0, 10, and 100 mM Mg 2+ for two mutant channels (Fig. 5, C and D).
Mutation of the RCK1 site (E374A/E399N) largely eliminates the response to 10 mM Mg 2+ but leaves the P O increase by 100 mM Mg 2+ intact (Fig. 5 C). The mutation reduces the G K -V shift (∆V 0.5 ) by 10 and 100 Mg 2+ to −14.0 ± 2.0 mV and −34.6 ± 1.8 mV, respectively, similar to previous reports (Shi and Cui, 2001;Hu et al., 2006). The reduced shift allows log(P O )-V to achieve a limiting slope at more positive voltages than the WT (Fig. 5 C, dotted lines), indicating a 2.9-fold increase in L 0 with 100 mM Mg 2+ , like the WT.
The response of E374A/E399N is important not only in verifying that L 0 is increased by 100 mM Mg 2+ but also in ruling out that Mg 2+ occupancy of either the RCK1 site or Ca 2+ -binding sites contribute to L 0 . The insensitivity of L 0 to 10 mM Mg 2+ in the WT could conceivably refl ect an ability of Mg 2+ at the RCK1 site to decrease L 0 that is masked by the action of Mg 2+ at other sites. However, this possibility is unlikely since L 0 is also insensitive to 10 mM Mg 2+ in E374A/E399N. In addition this result indicates that high affi nity Ca 2+ -binding sites, which are known to bind Mg 2+ at millimolar concentrations, are not responsible for the L 0 increase in 100 mM Mg 2+ because such sites should already be nearly saturated in 10 mM Mg 2+ (Shi and Cui, 2001;Zhang et al., 2001;Hu et al., 2006). Thus the E374A/ E399N results confi rm that occupancy of a very low affi nity Mg 2+ site is required to increase L 0 .
The R167E mutant was used to better quantify the effects of Mg 2+ on L 0 (Fig. 5, D and E). R167E is activated strongly by 100 mM Mg 2+ but achieves a limiting slope at voltages ‫001ف‬ mV more positive than the WT (Fig.  5 D, dotted lines). Thus L 0 can be determined at voltages that are suffi ciently negative to achieve a limiting slope but not too extreme to prevent steady-state recording. The effects on L 0 of 10 and 100 mM Mg 2+ were determined from the change in NP O at −80 mV in patches where Mg 2+ was repeatedly applied and washed out. The results confi rm that 10 mM Mg 2+ has little or no effect on L 0 (1.06 ± 0.05-fold change, n = 10), whereas 100 mM Mg 2+ increased L 0 by a factor of 2.24 ± 0.23 (n = 5) (Fig. 5 D,   . However, Q o can be assessed in a model-independent fashion from the logarithmic slope of the P o -V relation (Horrigan and Aldrich, 2002). The mean activation charge displacement (q a = kTd(ln[P o ])/dV) exhibits a bell-shaped dependence on voltage that is plotted in Fig. 5 E for 0, 10, and 100 mM Mg 2+ . q a is expected to approximate Q o at voltages where both P O and Q c are small (Horrigan and Aldrich, 2002;Ma et al., 2006). Thus the Q o -V relation is approximated by the foot of the q a -V relation at negative voltages, and is markedly left-shifted by Mg 2+ , indicating that voltage sensor activation is enhanced when channels are open.
To estimate the increase in voltage sensor/gate coupling by Mg 2+ we fi t the foot of the mean q a -V relations with a function predicted by Scheme 1 to describe Q o : Q o = z l + 4z J B(V), where B(V) is a Boltzmann function with charge z J and half activation voltage V hO (Fig. 5 E,  solid curves). The fi ts indicate that V hO is shifted by −60 ± 3 mV and −92 ± 4 mV in 10 Mg 2+ and 100 Mg 2+ , respectively. Changes in the coupling factor D can be determined by comparing the shift in Q O (∆V hO ) to that in Q C (∆V hC ). Q O depends on both D and J 0 , whereas Q C depends only on J 0 . Therefore the quantity ∆∆V CO = [∆V hC − ∆V hO ], plotted in Fig. 5 F for WT and E374A/ E399N channels, should depend only on D (i.e., δD = exp(z J ∆∆V CO /kT), based on Scheme 1). Fig. 5 F indicates ∆∆V CO = 43 ± 3 mV in 10 mM Mg 2+ , corresponding to a 2.7-fold increase in D. 100 mM Mg 2+ produces a larger ∆∆V CO (56 ± 4 mV), corresponding to a 3.6fold increase in D, that probably represents the effect of saturating the RCK1 site. ∆∆V CO for the E374A/E399N mutant is not signifi cantly different from 0 (P < 0.05, t test) in 10 or 100 Mg 2+ (Fig. 5 F), implying that all changes in voltage sensor/gate coupling can be attributed to the RCK1 site.

Modeling the Effects of Mg 2+ on Steady-State Activation
The above results suggest that 10 mM Mg 2+ acts primarily at the RCK1 binding site to enhance voltage sensor/ gate coupling (D) with a minor effect on the voltage equilibrium (J 0 ), and that 100 mM Mg 2+ causes additional increases in D, J 0 , and L 0 that refl ect action at both RCK1 and very low affi nity binding sites. To confi rm that these parameter changes are suffi cient to describe the effects of Mg 2+ on steady-state activation and to better quantify the changes in D we fi t simultaneously the log(P O )-V, q a -V, and P O -V relations for 0, 10, and 100 mM Mg 2+ using Scheme 1, as defi ned by Eq. 1: The 0 Mg 2+ data were fi t by setting charge parameters to values determined previously (z J = 0.58 e, z L = 0.3 e) (Horrigan and Aldrich, 2002) and allowing all other parameters (J 0 , L 0 , D) to vary, yielding results (Table I, WT 0 Mg 2+ ) similar to previous reports (Horrigan and Aldrich, 2002;Ma et al., 2006). To fi t the Mg 2+ data, the change in J 0 relative to 0 Mg 2+ was set by gating current measurements (∆V hC , Fig. 3 F), while L 0 and D were allowed to vary. Excellent fi ts to the log(P O )-V and q a -V relations were obtained in this way (Fig. 5, B and E, dotted curves), with 10 Mg 2+ increasing D by a factor of 2.14, and 100 Mg 2+ causing a greater increase in D (2.83-fold) together with a 2.43-fold increase in L 0 (Table I,

WT A parameters).
A similar procedure was used to fi t the E374A/E399N data (Fig. 5 C, dotted curves), yielding values of D (Table I, E374A/E399N parameters) that vary by <15% relative to the control, which is within the resolution of our measurement (Ma et al., 2006). Thus, shifts in V hc determined from gating currents together with an increase in L 0 in 100 Mg 2+ appear suffi cient to account for the changes in log(P O )-V when the RCK1 site is disrupted.
Although the fi ts from Scheme 1 reproduced the effects of Mg 2+ on log(P O ) and q a , they overestimated the shift in V 0.5 especially in 100 Mg 2+ (Fig. 5 A, (Table I, WT B parameters). These parameters reasonably approximate the 10 Mg 2+ P O -V and log(P O )-V relations but overestimate the steepness of P O -V in 100 Mg 2+ (Fig. 5 A). This discrepancy refl ects a decrease in the slope of P O -V as Mg 2+ is increased from 10 to 100 mM Mg 2 , a feature of the data that is not reproduced by Scheme 1.
To account explicitly for the effects of Mg 2+ binding on the parameters in Scheme 1 we modeled the effects of very low affi nity and RCK1 binding sites in terms of Schemes 2 and 3, respectively (Fig. 2, F and G). These allosteric mechanisms assume that Mg 2+ can bind to any state of the channel defi ned by Scheme 1 and that effects of Mg 2+ on the relative stability of different states therefore refl ect the relative affi nity of Mg 2+ for these states. This principal is illustrated in Fig. 5 (G and H) by alternative representations of Schemes 2 and 3 that include the four states (CR, OR, CA, OA) and equilibrium constants defi ned by Scheme 1 for a single subunit, where boxes and associated allosteric factors indicate the states that are stabilized relative to the CR state by ligand binding. In Scheme 2, the action of Mg 2+ is defi ned by its dissociation constant for the CR state (K D ) and two allosteric factors (C and E). C defi nes the increase in affi nity associated with channel opening, such that K D is reduced C-fold for the OR and OA states (Fig. 5 G, labeled box). Likewise, E defi nes the change in K D associated with voltage sensor activation (Fig. 5 G, CA and OA states). These changes in affi nity can account for effects of Mg 2+ on the equilibria for channel opening (L 0 ) and voltage sensor activation (J 0 ). However, voltage sensor/ gate coupling (D) is unchanged in Scheme 2 because the effects of C and E are independent and neither alters binding in a manner that depends on the state of both the voltage sensor and gate. By contrast, in Scheme 3, D Parameters for Eq. 1 that were varied to fi t WT and E374A/E399N data in 0-100 mM Mg 2+ (Fig. 5, A, B, C, and E). V hC in 10 and 100 mM were determined as V hC (0) + ∆V hC (Mg 2+ ) where V hC (0) was determined from the 0 Mg 2+ fi t and ∆V hC (Mg 2+ ) from gating current (Fig. 3 F). J 0 is determined from V hC (J 0 = exp(−V hC z J /kT)). Charge parameters were fi xed (z J = 0.58 e, z J = 0.30 e).
is increased F-fold upon ligand binding by assuming that the affi nity of the open-activated state (OA) is increased relative to all other states by an allosteric factor F. Scheme 3 also includes an independent effect of voltage sensor activation on Mg 2+ affi nity (E-factor) as in Scheme 2, to account for effects on J 0 . The data in Fig. 5 (A and B) were fit (red curves, Table II parameters) by a two-site model that assumes the channel contains a single RCK1 (site 1, Scheme 3) and a very low affi nity (site 2, Scheme 2) binding site in each of four identical subunits, yielding Eq. 2 (see below), where L, J, and D are defi ned as in Eq. 1; E 1 , F 1 and C 2 , E 2 are allosteric factors for sites 1 and 2, respectively; and K i = [Mg 2+ ]/K Di (i = 1,2), where K Di is dissociation constant for each binding site in the CR state.
The fi ts of Eq. 2 to P O -V and log(P O )-V relations in 0, 10, and 100 mM Mg 2+ were almost identical to those obtained with Scheme 1 (Fig. 5, A and B) and predict shifts in V hC for both WT and E374A/E399N (Table II) similar to those measured from gating current (Fig.  3 F). Parameters for site 1 (RCK1) were well constrained by fi tting P O -V relations in 2, 5, 10, and 21 mM Mg 2+ (Fig. 5 A) and yielded a K D for the CR state of 5.0 mM (the K D of other states are indicated in Table II). A small E 1 -factor (1.35) was suffi cient to explain the ∆V hC attributed to the RCK1 site, whereas a larger F 1 -factor of 2.0 was required to explain the enhancement of voltage sensor/gate coupling. The site 2 (very low affi nity) parameters were not as well constrained because few concentrations >10 mM were tested and the effect of saturating this site was not determined.

Effects of Mg 2+ on I g Kinetics and the Deactivation Pathway
Although gating current is relatively insensitive to Mg 2+ when channels are closed (Fig. 3), the ability of Mg 2+ to shift the Q o -V relation implies that I g kinetics must be altered when channels are open. Such an effect is evident from OFF gating currents (I gOFF ) following pulses of different duration (0.05-20 ms, +200 mV) in 0 and 10 mM Mg 2+ (Fig. 6 A). Normalized I gOFF traces (Fig. 6 B) show that OFF kinetics following brief pulses (≤0.1 ms, red traces), that are comparable to the delay in I K activation  and open few channels, are similar in 0 or 10 Mg 2+ . However, as pulse duration increases and channels open, voltage sensor deactivation is slowed, refl ecting that Q o -V is shifted to more negative voltages than Q c -V . The slowing of I gOFF occurs in 0 Mg 2+ but is more prominent in 10 Mg 2+ (Fig. 6 B) because Mg 2+ increases the difference between Q O and Q C (Fig. 5 F) and the fraction of channels that open at +200 mV (Fig. 2 B).
To distinguish effects of Mg 2+ on voltage sensor deactivation kinetics and P O , OFF kinetics were analyzed by plotting the time course of OFF charge movement (Q OFF ) on a semi-log scale (Fig. 6 C) and comparing the effect of brief ("closed") and prolonged ("open") depolarizations. The decay of Q OFF can be described by three exponential components (τ Fast , τ Med , τ Slow ) reflecting the predicted pathways of voltage sensor deactivation . Scheme 1* (Fig. 6 D) is a kinetic scheme that depicts the 10 states and transitions defi ned by Scheme 1, representing the O and C conformation with zero to four activated voltage sensors. Q OFF following a brief depolarization is well described by a single exponential function (τ Fast ) representing the deactivation of voltage sensors in closed channels (C 4 to C 0 transitions in Scheme 1*). Q OFF for open channels is characterized by a component representing the deactivation of voltage sensors when channels are open (Fig.  6 D, Med) and another that is rate limited by channel closing (Fig. 6 D, Slow). Therefore, Q OFF following prolonged depolarizations to +200 mV ( Fig. 6 C, dotted red lines) is slowed (2.3-fold) by 10 mM Mg 2+ . In addition τ Slow (Fig. 6 C, solid red lines) is increased 1.5-fold, consistent with a slowing of channel closing by Mg 2+ .
Further insight into the mechanism of Mg 2+ action is provided by plotting the amplitude of the three OFF TA B L E I I Parameters that were varied to fi t Mg 2+ data (Fig. 5, A and B) for WT (Eq. 2) and E374A/E399N (Eq. 2 with K 1 = 0). Remaining parameters (z J , z L , J 0 , L 0 , D) were identical to those for 0 Mg 2+ in Table I components (Q Fast , Q Med , Q Slow ) versus pulse duration (Fig. 6 E). As pulse duration increases, Q Fast is reduced while Q Med and Q Slow increase, following the exponential time course of channel opening (solid curves Fig.  6 E) Aldrich, 1999, 2002). These kinetics are similar in the presence and absence of Mg 2+ because 10 mM Mg 2+ has little effect on activation kinetics (Shi and Cui, 2001;Zhang et al., 2001). However the relative amplitudes of Q Med and Q Slow are reversed by Mg 2+ . The change in relative amplitude of Q Med and Q Slow by Mg 2+ is qualitatively similar to the effect of recording I gOFF at a more positive voltage in 0 Mg 2+ , supporting that Mg 2+ shifts Q o to more negative voltages. In the absence of Mg 2+ , Q Med is greater than Q Slow because voltage sensors can completely deactivate at −80 before channels close (Fig. 6 D Fig. 6 D, 10 Mg 2+ ), and voltage sensor deactivation becomes rate limited by closing (Fig. 6 D, 10 Mg 2+ pink arrow) such that the ratio of Q Slow to Q Med is increased.  Fig. 5 E). The fi gure illustrates one possible pathway, through the O 2 state. After channels close, voltage sensors can completely deactivate in 10 mM Mg 2+ (Q C , Fig. 3 B). This charge movement is limited by the closing rate and therefore contributes to the Slow component of Q OFF . (E) The amplitude of the three OFF components (Fast, ▲; Med, light blue square; Slow, pink circle) for the experiment in A are plotted vs. pulse duration and fi t by exponential functions (0 Mg 2+ , τ = 2.8 ms; 10 Mg 2+ , τ = 2.6 ms).

Effects of Mg 2+ on I K Kinetics
The ability to slow channel closing is a characteristic of Mg 2+ action at the RCK1 site that is evident by comparing I K tail currents at −80 mV in 0 and 10 mM Mg 2+ (Fig. 2 A). Time constants of I K relaxation (τ K ) plotted versus voltage in Fig. 7 A for 0, 10, and 100 mM Mg 2+ indicate that 10 mM Mg 2+ slows I K deactivation approximately twofold over a wide voltage range (−380 to +100 mV) with little impact on activation kinetics at very positive voltages (+160 to +280 mV). By contrast, increasing Mg 2+ from 10 to 100 mM to occupy very low affi nity sites speeds activation with no additional effect on deactivation at extreme negative voltages (Fig. 7 A) (Zhang et al., 2001;Hu et al., 2006).
That 10 and 100 mM Mg 2+ have distinct effects on τ K is not surprising given their action at different binding sites. However, the effects of Mg 2+ on deactivation kinetics are remarkable in comparison to the effects on steady-state activation. Macroscopic I K kinetics are rate limited by the C-O transition  and at suffi ciently negative voltages by the O 0 to C 0 rate constant (Fig. 6 D, top). Thus, the ability of 10 Mg 2+ to slow deactivation at extreme negative voltages is surprising given that 10 mM Mg 2+ has no detectable effect on the equilibrium constant for the C 0 -O 0 transition (L 0 ). Similarly, the failure of 100 Mg 2+ to slow deactivation relative to 10 Mg 2+ is notable because 100 Mg 2+ increases L 0 .
Although previous studies have shown that Mg 2+ slows deactivation at voltages as low as -200 mV, measurements at more negative voltage were required to confi rm that the O 0 to C 0 transition rate is altered. Because Mg 2+ shifts the Q o -V relation, channels can close by pathways involving different C-O transitions in the presence and absence of Mg 2+ (e.g., Fig. 6 D). This difference contributes to the slowing of tail currents in Mg 2+ at −80 mV. However at suffi ciently negative voltages, τ K should be limited only by the O 0 to C 0 transition whether or not Mg 2+ is present. Consistent with this prediction, at voltages less than −240 mV the difference in τ K between 10 and 100 mM Mg 2+ is almost eliminated and τ K achieves a limiting voltage dependence that is identical in the presence or absence of Mg 2+ (Fig. 7 A, dotted lines). Normalized τ K -V relations (Fig. 7 B) illustrate that a major effect of Mg 2+ is to shift the τ K -V relation along the voltage axis, consistent with effects on Q o . However, the ability of 10 mM Mg 2+ to increase τ K 1.53-fold at V < −240 mV confi rms that the O 0 to C 0 rate is also reduced. That the kinetics of the C 0 -O 0 transition are affected by 10 mM Mg 2+ without altering the equilibrium constant suggests that both the forward and reverse rate constants for this transition must be slowed. To test this hypothesis we analyzed unitary current kinetics from a macropatch containing R167E channels at −80 mV where voltage sensors should be in the resting state. Although the patch contained an estimated 10,000 channels, >99% of openings were to the single channel level because P O at this voltage is low (4 × 10 −7 in 0 Mg 2+ , Fig.  5 D). Both open and closed dwell times histograms were reasonably fi t by single exponential components as illustrated for 0 Mg 2+ in Fig. 7 C, consistent with a twostate C 0 -O 0 transition. Changes in NP O (Fig. 7 D) and mean open and closed times (Fig. 7 E) are plotted as [Mg 2+ ] was changed in sequence from 100 mM to 0, 10, and back to 0 Mg 2+ . The results show that the change from 0 to 10 mM Mg 2+ increases both open and closed times, consistent with a similar slowing of both forward and backward rates of channel opening that leaves NP O unchanged. Open times were brief but comparable to the ‫-001ف‬μs time constant of macroscopic I K measured for this patch at −80 mV in 0 Mg 2+ . The frequency of channel openings was reversibly reduced by 10 mM Mg 2+ as expected from the effect on mean closed times. In a patch with N independent channels and brief openings the mean closed time should equal N/(opening rate), implying that the mean closed time of individual channels was ‫4-3ف‬ min. Increasing Mg 2+ from 10 to 100 mM only slightly increased open times but markedly decreased closed times, consistent with the ability of 100 Mg 2+ to increase NP O without changing τ K relative to 10 mM Mg 2+ at negative voltages. These conclusions about the C 0 -O 0 transition from single-channel analysis are consistent with effects of Mg 2+ on rate constants in Scheme 1* (δ 0 , γ 0 ) derived by fi tting the macroscopic data ( Fig. 7 A, solid curves, Table III parameters).
The effect of Mg 2+ on the kinetics of the C 0 -O 0 transition is important in providing mechanistic information that is not evident from the steady-state data. In particular, the ability of 10 mM Mg 2+ to slow opening and closing rates demonstrates that Mg 2+ remains bound to the RCK1 site and can infl uence channel gating when voltage sensors are in the resting state even though P O is insensitive to Mg 2+ under these conditions.

D I S C U S S I O N
Our results show that intracellular Mg 2+ activates Slo1 BK channels through a fundamentally different mechanism from Ca 2+ . While micromolar [Ca 2+ ] i promotes channel opening independent of voltage sensor activation, millimolar [Mg 2+ ] i acting at a putative binding site in the cytosolic RCK1 domain has effects on opening and voltage sensor activation that are highly interdependent. Mg 2+ promotes opening only when voltage sensors are activated and facilitates voltage sensor activation mainly when channels are open. Because Mg 2+ action depends on the conformation of both voltage sensor and gate, the primary effect on gating is to enhance voltage sensor/gate coupling such that open-activated states are stabilized. A two-site allosteric model that incorporates this mechanism reproduces the major effects of Mg 2+ on steady-state activation. As discussed below, these results support that multiple pathways of communication exist between the cytosolic and transmembrane domains of BK channels. Our data also constrain the possible physical mechanisms that underlie Mg 2+ -dependent activation and voltage sensor/gate coupling, and suggest that the physiological impact of Mg 2+ could differ qualitatively from that of Ca 2+ .

Communication between Cytosolic and Transmembrane Domains
Ca 2+ has no effect on voltage sensor/gate coupling (Horrigan and Aldrich, 2002) and is thought to promote channel opening by pulling on the RCK1-S6 linker (Niu et al., 2004). Therefore, the ability of Mg 2+ to enhance coupling suggests that the cytosolic domain can communicate with the transmembrane domain through pathways other than the RCK1-S6 linker, presumably involving direct or indirect interactions with the voltage sensor (e.g., Fig. 1 D). A similar conclusion was drawn from the effects of intracellular heme . Rate constants for the C i -O i transitions in Scheme 1* (Fig. 6 D) determined from fi ts to WT τ K -V relations in 0-100 mM Mg 2+ (Fig. 7 A), using equilibrium parameters from Table I (WT, WT B ). Since voltage sensor activation equilibrates rapidly, τ K was fi t by , where i represents the number of activated voltage sensors, δ i and γ i are forward and backward rate constants for the C i -O i transitions and pC i , pO i are conditional occupancies of the open and closed states (pC i = p(C i |C), pO i = p(O i |O)) . Forward rates (δ i ) were varied while backward rates were γ i = δ i /L 0 D i and partial charges associated with these rate constants were z δ = 0.165e and z γ = −0.135e (Horrigan and Aldrich, 2002).
However, the action of Mg 2+ is remarkable in its simplicity and specifi city relative to heme. While heme weakens voltage sensor/gate coupling, it also reduces the effi cacy of Ca 2+ and increases P O at extreme negative voltages (i.e., increases L 0 ). These diverse effects on gating, together with a slow onset of action (minutes), suggest that heme binding may perturb the normal conformation of the cytosolic domain such that tension in the RCK1-S6 linker is increased while multiple allosteric interactions are also disrupted . In contrast, 10 mM Mg 2+ does not alter Ca 2+ sensitivity (Shi and Cui, 2001;Zhang et al., 2001) and acts rapidly (<1 s) at the RCK1 binding site to enhance voltage sensor/gate coupling without affecting the intrinsic stability of the gate (L 0 ). The effect of Mg 2+ is complimentary to that of Ca 2+ , which increases L 0 with no effect on coupling (Horrigan and Aldrich, 2002). Thus Mg 2+ and Ca 2+ appear to independently regulate two different pathways of communication, consistent with their additive effects on channel activation. Unlike heme, the effects of Mg 2+ do not suggest a widespread structural perturbation, and can be understood in terms of stabilizing particular channel states.

The Energetic Effects of Mg 2+
Mg 2+ at the RCK1 site acts primarily to stabilize the openactivated (OA) state relative to the other three states (CR, CA, and OR) defi ned by the voltage sensor and gate conformation in each subunit (Fig. 5 H; Table II). We reached this conclusion by measuring independently each of the equilibrium constants between these states, in effect determining their relative free energies in the presence and absence of Mg 2+ , as specifi ed by three independent parameters (L 0 , J 0 , and D). The CR-OR equilibrium constant (L 0 ) is not altered. The CR-CA equilibrium constant (J 0 ) is only increased slightly. However, the CA-OA and OR-OA equilibria are substantially altered, consistent with an approximate twofold increase in the voltage sensor/gate coupling factor D. By comparing the effects of Mg 2+ on WT and mutant (E374A/ E399N) channels, we showed that all changes in voltage sensor/gate coupling and a fraction of the changes in J 0 can be attributed to the RCK1 site. Modeling the effects of Mg 2+ in terms of a two-site allosteric scheme (Table II) suggests that saturating the RKC1 site (site 1) stabilizes the OA state relative to CR by ΔΔ OA CR G = kTln(E 1 F 1 ) = 0.60 kcal mol −1 , while stabilizing the CA state by ΔΔ CA CR G = kTln(E 1 ) = 0.18 kcal mol −1 . Thus the energetic effect of Mg 2+ is weak compared with that of Ca 2+ binding or voltage sensor activation (1.25 or 1.91 kcal mol −1 per subunit) (Horrigan and Aldrich, 2002).

Parallel Pathways of Voltage Sensor/Gate Coupling
While the interactions that couple voltage sensor activation to channel opening are well defi ned energetically, they could refl ect many different physical mechanisms.
The coupling represents a differential effect of voltage sensor conformation on the free energy of closed and open states. Therefore voltage sensor and gate must interact in a state-dependent manner that depends on the conformation of both domains. Such interactions could represent a stabilizing interaction (e.g., bond formation) between voltage sensor and gate that occurs only in the OA state, or destabilizing interactions (e.g., steric or repulsive) that occur in every state but OA. Alternatively, sensor and gate could be coupled in all states by a spring-like connection whose tension varies with both voltage sensor activation and channel opening. Any of the above or similar mechanisms, alone or in combination, could contribute to the allosteric coupling factor D. Thus the ability of Mg 2+ to increase D does not require that interactions that normally mediate coupling in the absence of Mg 2+ are enhanced. Indeed, two lines of evidence suggest that the effect of Mg 2+ represents a parallel pathway of coupling, independent of that in the transmembrane domain (Fig. 8 A). First, Mg 2+ in the RCK1 binding site appears to interact with a part of the voltage sensor that is not important for voltage sensor/ gate coupling in the absence of Mg 2+ . Interactions of the S4-S5 linker near S5 with S6 are proposed to mediate coupling in Kv channels (Lu et al., 2002;Long et al., 2005b). Although mutations in the S4-S5 linker of Slo1 infl uence Mg 2+ sensitivity (Hu et al., 2003), they are at sites near S4 (E219, Q222) that are unlikely to interact directly with S6. In addition, mutations of R213 in S4 virtually abolish Mg 2+ sensitivity (Hu et al., 2003;Yang et al., 2007), but have no effect on voltage sensor/gate coupling in 0 Mg 2+ (Ma et al., 2006). Second, interactions between the voltage sensor and Mg 2+ -binding site should be suffi cient to infl uence voltage sensor/gate coupling, because the binding site is in a cytosolic domain attached to S6 and therefore is connected to the gate.

Physical Mechanism of Mg 2+ Action
Since Mg 2+ stabilizes primarily the OA state, thermodynamics require that Mg 2+ affi nity is greatest in the OA state, as in Scheme 3. However, the conclusion that Mg 2+ binding is favored in a particular state does not fully describe how and when Mg 2+ interacts with its binding site and the voltage sensor. If we suppose that the interactions of Mg 2+ with its binding site are purely stabilizing, then interactions must exist in the OA state that are stronger or more numerous than in other conformations. Such a situation could arise if both the cytosolic and voltage sensor domains contribute to the Mg 2+ binding site, but are favorably aligned with each other only when the voltage sensor is activated and the channel is open. However, this hypothesis appears too simple given recent evidence that a destabilizing electrostatic interaction exists between Mg 2+ and the voltage sensor (Yang et al., 2007); and physical models that take this into account are necessarily more complex, as discussed below.
The effects on Mg 2+ sensitivity and gating of chargealtering mutations in the voltage sensor and RCK1 domain strongly suggest that Mg 2+ activates the channel through a long-range electrostatic interaction with R213 in S4 (Yang et al., 2007). Addition of positive charge at position 397 near the RCK1 site , by either mutation (Q397K, Q397R) or chemical modifi cation, mimics the major effects of Mg 2+ on channel gating; whereas negative charge at this position (Q397E, Q397D) inhibits activation. Conversely a positive charge at position 213 appears necessary for sensing either Mg 2+ or charge at position 397. Finally, the effects on gating of Mg 2+ or Q397K but not Ca 2+ are strongly inhibited in high ionic strength solutions (1 M KCl), consistent with a long-range electrostatic interaction between Mg 2+ and the voltage sensor. Based on these results, Yang et al. propose that Mg 2+ bound to the RCK1 domain alters the local electric fi eld sensed by R213 in S4 and thereby promotes voltage sensor activation. That is, electrostatic repulsion between Mg 2+ and R213 destabilizes the resting conformation of the voltage sensor. This interaction is presumably state dependent since voltage sensor activation is enhanced by Mg 2+ primarily when the channel is open.
The observations of Yang et al. (2007) together with our functional analysis place important constraints on the physical nature and state dependence of the interactions underlying Mg 2+ -dependent activation. Most importantly, the ability of R213 mutants to virtually eliminate Mg 2+ sensitivity suggests that the electrostatic interaction between R213 and Mg 2+ accounts for the major effects of Mg 2+ on gating, and therefore must act on balance to stabilize the OA state. This conclusion has two key implications. First, the coordination of Mg 2+ at its binding site cannot be strongly state dependent. If it were, then the channel's Mg 2+ affi nity should change in a state-dependent manner even in the absence of electrostatic interaction with S4; and gating should therefore remain Mg 2+ sensitive when R213 is mutated. Although subtle effects of this sort cannot be ruled out, R213 mutations largely eliminate the G K -V shift and slowing of I K deactivation by Mg 2+ (Hu et al., 2003;Yang et al., 2007). Thus it appears unlikely that conformational changes in the binding site associated with channel opening favor Mg 2+ binding and contribute to Mg 2+ -dependent activation. This conclusion is supported by the observation that a permanent positive charge at position 397 mimics the effects of Mg 2+ (Yang et al., 2007). Second, if Mg 2+ Figure 8. Voltage sensor/gate coupling and the mechanism of Mg 2+ action. (A) Multiple pathways of voltage sensor/gate coupling may exist in BK channels. Because the cytosolic domain is connected to the gate via the RCK1-S6 linker (dashed line), interactions between cytosolic and voltage sensor domains could provide a coupling pathway (curved arrow) that is independent of that within the transmembrane domain (straight arrow). (B) A speculative model of Mg 2+ -dependent activation, based on an electrostatic mechanism of interaction between Mg 2+ in the putative RCK1 binding site and R213 in S4 (Yang et al., 2007). The BK channel is shown in the four states defi ned in each subunit by the movement of the voltage sensor from resting (R) to activated (A) and by the closed (C) to open (O) conformational change. Channel opening is accompanied by an expansion of the cytosolic domain that alters the position of Mg 2+ (in its binding site) relative to the transmembrane domain. The Mg 2+ binding site (green semicircle) in the RCK1 domain includes coordinating residues E374 and E399 and nearby Q397 . Voltage sensor activation moves R213 relative to the cytosolic domain. In this way the distance and therefore electrostatic repulsion between Mg 2+ and R213 (indicated qualitatively by the thickness of red boxes surrounding each state) depends on the conformation of both voltage sensor and gate, such that voltage sensor/gate coupling is enhanced. To account for the observed effects of Mg 2+ on both opening and voltage sensor activation requires that the OA state be stabilized (see text). This is accomplished in the model by maintaining a similar distance between Mg 2+ and R213 in all states except OA, such that all states are destabilized relative to OA. To account for a small effect of Mg 2+ on voltage sensor activation when channels are closed we also assume interaction in the CA state is slightly weaker than the CR state, owing to a small outward movement of R213 upon voltage sensor activation, consistent with the small contribution of R213 to gating charge (Ma et al., 2006). stabilizes the OA state via a destabilizing electrostatic interaction, then the interaction must be weakest in the OA state and of similar strength in all other states (CR, CA, OR). That is, Mg 2+ must destabilize all states relative to OA.
That Mg 2+ interacts with the voltage sensor in multiple states including closed states may seem counterintuitive since Mg 2+ has little effect on voltage sensor activation when channels are closed. Indeed, interaction in a single state, open resting (OR), would be sufficient to account for the ability of Mg 2+ to promote voltage sensor activation when channels are open, by destabilizing OR relative to OA. However, this simple model cannot account for the effects of Mg 2+ on channel opening. If the OR state is destabilized without also destabilizing the CR state then channel opening should be inhibited when voltage sensors are in the resting state, inconsistent with our fi nding that L 0 is unaffected by 10 mM Mg 2+ . Conversely, a lack of interaction between Mg 2+ and the activated voltage sensor would be inconsistent with the ability of Mg 2+ to facilitate opening of R210C channels when voltage sensors are constitutively activated. Thus accounting for effects of Mg 2+ on both channel opening and voltage sensor activation requires that multiple states are destabilized.
The conformational changes that allow Mg 2+ to interact with the voltage sensor in a state-dependent manner are not known in detail. However, a speculative model consistent with our results is illustrated in Fig. 8 B. The model assumes that Mg 2+ and the voltage sensor interact electrostatically in all states but that the interaction is weakest in the OA state (as indicated by red boxes). Consequently, the OA state has the highest affi nity for Mg 2+ because the voltage sensor repels Mg 2+ the least. For illustration purposes, interaction strength is assumed to vary with changes in the distance between Mg 2+ and the charged R213 sidechain, although changes in local dielectric environment could also be involved. Fig. 8 B shows how the distance between Mg 2+ and R213 could remain relatively constant in all states other than OA if the position of the Mg 2+ -binding site moves with channel opening and R213 moves with voltage sensor activation.
The insensitivity of P O to Mg 2+ at extreme negative voltages is accounted for in Fig. 8 B by assuming the Mg 2+ binding site in the closed and open conformation occupies two different locations that are equidistant from R213 in the resting state. In this way the CR and OR states are equally destabilized. Interestingly, this mechanism is also consistent with our fi nding that Mg 2+ slows both opening and closing rates when voltage sensors are in the resting state. This kinetic effect implies that Mg 2+ destabilizes a transition state between CR and OR. In Fig. 8 B, movement of the Mg 2+ binding site between closed and open positions would allow Mg 2+ to approach R213 more closely in a transition state than in either the CR or OR states, thereby destabilizing the transition state.
Although the details of Fig. 8 B are speculative, they are consistent with known features of BK and related channels. The location and movement of the Mg 2+binding site relative to R213 is based on the crystal structures of MthK and Kv1.2. Residues in MthK (E138, N158), corresponding to E374 and E399 in the putative Mg 2+ binding site of Slo1, are on the surface of the gating ring facing the membrane at a distance from the central axis of the channel ‫13ف(‬ Å) that roughly matches the radial position of the S4 segment in Kv1.2 (Yang et al., 2007). The Mg 2+ -binding site is shown to move away from the central axis upon channel opening, consistent with an expansion of the cytosolic domain as in MthK (Jiang et al., 1999;Ye et al., 2006). Voltage sensor activation is depicted as a tilt and outward translation of S4 that is qualitatively consistent with recent models of Kv channel activation (Ruta et al., 2005;Campos et al., 2007;Pathak et al., 2007). We have assumed that movement of R213 perpendicular to the membrane is small such that interaction with Mg 2+ in the CA state is not disrupted. This assumption is consistent with the fact that Slo1 is weakly voltage dependent and neutralization of R213 reduces the gating charge associated with voltage sensor activation (z J ) by only 0.3 e (Ma et al., 2006).
One possibility that was not incorporated into Fig. 8 B is that the position of R213 in the transmembrane domain may change with channel opening. Previous results suggest that R213 moves relative to the membrane electric fi eld during opening because mutations at this position reduce the charge associated with opening (z L ) (Ma et al., 2006). If the position of R213 depends on the conformation of both voltage sensor and gate then the requirement that the Mg 2+ -binding site move during channel opening may be reduced. Therefore, Fig. 8 B does not represent a unique solution, but it does illustrate the constraints that must be satisfi ed to account for the experimental data in terms of electrostatic interactions.

Very Low Affi nity Mg 2+ Binding Sites
Divalent cation binding sites with lower affi nity for Mg 2+ than the RCK1 site are proposed to exist in BK channels because G K -V shifts produced by increasing [Mg 2+ ] or [Ca 2+ ] from 10 to 100 mM appear too large to represent surface charge screening (Zhang et al., 2001) and are nearly independent of mutations that eliminate responses to lower concentrations of Mg 2+ or Ca 2+ . The location, number, and physiological signifi cance of these very low affi nity sites are unknown. However, our results provide new information about their mechanism of action.
We fi nd that 100 mM Mg 2+ , unlike 10 mM Mg 2+ , increases L 0 and causes a shift in V hC that is only partially eliminated by mutating the RCK1 site. These effects can be reproduced by an allosteric mechanism (Scheme 2) similar to that used to describe effects of micromolar Ca 2+ . However, very low affi nity sites and Ca 2+ sites differ greatly in the relative magnitude of their effects on the gate and voltage sensor represented by allosteric factors C and E, respectively. The G K -V shift by Ca 2+ is dominated by a C-factor of 8, which results in a 4,000fold increase in P O at −120 mV (Horrigan and Aldrich, 2002). By contrast the very low affi nity site (C = 1.6) increases P O (−120 mV) approximately threefold in 100 mM Mg 2+ , suffi cient to account for only one third of the G-V shift from 10 to 100 mM Mg 2+ . The majority of the G-V shift in high [Mg 2+ ] refl ects a change in V hC , which was attributed to an E-factor of 2.3.
The ability of 100 mM Mg 2+ to increase L 0 supports that the effects of high [Mg 2+ ] refl ect mechanisms other than surface charge screening. However, the possibility remains that surface charge screening contributes to V hC . Hu et al. (2006) estimated, based on surface potential calculations, that such a mechanism could shift V hC by ‫9−ف‬ mV from 10 to 100 mM Mg 2+ , which is roughly half of the observed ∆V hC (−19 mV) for the WT over this concentration range. Therefore, although we were able to model the effects of Mg 2+ in terms of a single very low affi nity site per subunit this is likely an oversimplifi cation. The E-factor in our model could represent a combination of allosteric and surface charge effects. In addition, we observed a decrease in G K -V slope from 10 to 100 Mg 2+ that was not accounted for by our model.

Relation to Previous Studies and Physiological Implications
Since Mg 2+ and Ca 2+ act through distinct mechanisms, it seems surprising that previous studies were able to model effects of Mg 2+ on the G K -V relation, in the presence and absence of Ca 2+ , by assuming Mg 2+ and Ca 2+ act through the same mechanism (Shi and Cui, 2001;Zhang et al., 2001;Hu et al., 2006). This comparison raises several questions addressed below. First, can our model account for previous observations that the G K -V shift by Mg 2+ is similar in the presence or absence or presence of Ca 2+ ? Second, were conclusions of previous studies affected by the use of a Ca 2+ -like model of Mg 2+ action? Third, is the difference in mechanism of Mg 2+ and Ca 2+ action physiologically signifi cant?
Although our experiments were performed in 0 Ca 2+ , we simulated the effect of saturating Ca 2+ on our twosite model of Mg 2+ -dependent activation (Eq. 2) by increasing the values of L 0 (4096-fold) and J 0 (2.4-fold) as predicted from previous work on Ca 2+ -dependent activation (Horrigan and Aldrich, 2002). This simulation predicts that an increase in [Mg 2+ ] from 0 to 100 mM will shift V 0.5 by −105 mV and −78 mV in 0 and saturating Ca 2+ , respectively, similar to the experimental results measured from Zhang et al. (2001) in 0 Ca 2+ (−104 mV) and 300 μM Ca 2+ (−72 mV). Thus our model appears to reproduce the combined effects of Mg 2+ and Ca 2+ on steady-state activation.
Conclusions about the Mg 2+ affi nity of the RCK1 site should be relatively unaffected by the incorrect assumption that Mg 2+ acts through a Ca 2+ -like mechanism. Previous studies assumed that Mg 2+ increases the C-O equilibrium constant L 0 by binding with higher affi nity to the open rather than closed conformation (Shi and Cui, 2001;Zhang et al., 2001;Hu et al., 2006). Similarly, our model of the RCK1 site (Scheme 3) assumes the Mg 2+ affi nity of the open-activated (OA) state is greater than any other state. That is, Mg 2+ does bind better to the open conformation than the closed, so long as the probability that an open channel is in the OA state is high. This probability is indicated by the Q o -V relation, which is saturated at voltages where the G-V is usually measured (Fig. 5, compare A and E). Therefore, Scheme 3 affects the G-V relation similar to models that assume Mg 2+ binds equally well to all open states. Consistent with this similarity a previous study using a twosite Ca 2+ -like model of Mg 2+ action estimated K D values for the RCK1 site of 5.46 mM (closed) and 2.25 mM (open) , comparable to our estimates based on Eq. 2 (K D (CR) = 5 mM, K D (CA) = 3.7 mM, K D (OA) = 1.85 mM).
The physiological impact of Mg 2+ -dependent BK channel activation has yet to be defi ned. However, our results raise the possibility that Mg 2+ could have a differential impact on BK-dependent processes such as action potential shaping in neurons and the control of vascular smooth muscle tone. Both these processes require an increase in [Ca 2+ ] i to open BK channels that are poorly activated by resting [Ca 2+ ] i at physiological voltages. During an action potential, Ca 2+ infl ux through voltagedependent Ca 2+ channels together with membrane depolarization activates BK channels to regulate spike duration and the fast afterhyperpolarization (Faber and Sah, 2003). By contrast, BK channels in vascular smooth muscle can contribute to resting membrane potential and myogenic tone by activating at hyperpolarized voltages, near −40 mV, in response to local Ca 2+ release from internal stores (Ca 2+ sparks) (Brayden and Nelson, 1992;Wellman and Nelson, 2003). Since physiological concentrations of free Mg 2+ (0.4-3 mM) (Gupta et al., 1984;Romani and Scarpa, 1992) are suffi cient to shift the G K -V relation  and the effects of Mg 2+ and Ca 2+ on V 0.5 are approximately additive, one might suppose that Mg 2+ simply increases P o during any Ca 2+ transient; enhancing apparent Ca 2+ sensitivity. However, our results suggest this is not the case. The effect of Mg 2+ , unlike Ca 2+ , depends on voltage sensor activation and is greatly diminished at hyperpolarized potentials (Fig. 2 C). Therefore, Mg 2+ might be expected to have a smaller impact on vascular smooth muscle tone than on action potential shaping. Of course determining the magnitude of such differences and the overall contribution of Mg 2+ to BK channel physiology must await further investigation.
We thank Drs. Jianmin Cui and Toshinori Hoshi for helpful comments and Kin Yu Wong for excellent technical assistance. This work was supported by a grant from the National Institutes of Health (NS42901) to F.T. Horrigan. An abstract of this work was presented at the 49th Annual Meeting of the Biophysical Society (2005)