Potassium-selective block of barium permeation through single KcsA channels

The ability of K⁺ channels to strongly favor permeation of K⁺ over Na⁺ lies at the heart of innumerable physiological processes. The appearance almost 15 years ago of the x-ray crystal structure of KcsA, the first high-resolution view of a K⁺ channel (Doyle et al., 1998), elevated discourse on ion selectivity mechanisms from prescient cartoons (Hodgkin and Keynes, 1955; Bezanilla and Armstrong, 1972; Neyton and Miller, 1988a) to the close-up chemistry of ion coordination. KcsA, a bacterial protein, contains a canonical K⁺ selectivity filter identical in sequence and structure to those now known in many molecular families of K⁺ channels throughout the biological universe. This structure, formed along the central axis of the homotetrameric protein, is essentially a narrow tunnel 15 Å in length, lined by 20 oxygen atoms that form four K⁺-binding sites, numbered S1–S4 from extracellular to intracellular (Fig. 1). Under physiological conducting conditions, two of these sites on average are occupied by K⁺ (Zhou and MacKinnon, 2003). The striking geometrical match of these sites to the diameter of a dehydrated K⁺ ion, 2.7 Å, immediately suggested that K⁺ selectivity emerges from “snug-fit” of the preferred ion (Doyle et al., 1998; Lockless et al., 2007), a mechanism proposed long ago (Bezanilla and Armstrong, 1972), when K⁺ channels were viewed merely as bio-electronic circuit elements of unknown molecular character. This mechanism posits a thermodynamic tug-of-war for cations between protein and aqueous solution; while the protein mimics the hydration shell of K⁺, its presumed rigidity prevents the filter from collapsing inward to similarly embrace the smaller Na⁺ ion as solvent water is free to do. Thus, a K⁺ ion diffusing up to the channel perceives the selectivity filter as an isoenergetic, aquomimetic diffusion pathway, whereas a Na⁺ ion is confronted by large kinetic barriers to entry and transit.

Despite the simplicity, elegance, and aesthetic appeal of the snug-fit picture, the mechanism of K⁺ channel selectivity has elicited a many-voiced debate over the past few years (see Andersen, 2011, and related papers for a recent perspective on this subject). With snug-fit in a rigid selectivity filter at one extreme versus a loose, dynamic filter whose selectivity reflects intrinsic chemistry of the carbonyl group at the other (Noskov and Roux, 2007; Roux, 2010), additional proposals based on ionic coordination numbers (Varma et al., 2008) or simply the number of ion-binding sites along the selectivity filter (Derebe et al., 2011) have been offered. Matters are further confounded by a gaggle of computational conclusions that have variously assigned K⁺-selective electrophysiological characteristics of KcsA to different sites in the filter, or have even claimed certain sites to be Na⁺ selective (Åqvist and Luzhkov, 2000; Bernèche and Roux, 2001; Thompson et al., 2009; Egwolf and Roux, 2010; Cheng et al., 2011).

A fundamental limitation in distinguishing these mechanisms is the scarcity of experimental selectivity measurements at thermodynamic equilibrium for specific ion-binding sites. Such data are needed to constrain computations by known values of Gibbs free energy along the filter. Conventional electrophysiological metrics of selectivity, such as single-channel conductance or bi-ionic reversal potential, are straightforward and directly relevant to biological function but problematic in linking experiment to computation, as they are kinetic properties reflecting complex mixtures of energies involving the entire collection of sites in the filter and the transition-state barriers between them. Direct equilibrium ion binding to KcsA by isothermal calorimetry (Lockless et al., 2007) shows good correspondence with electrophysiological measurements but does not distinguish individual sites within the filter, and also includes effects of linked conformational rearrangements. A recent solid-state NMR study providing a first view of site-specific K⁺ binding in KcsA (Bhate et al., 2010) concluded that K⁺ begins to occupy the filter in the 10-µM concentration range and provides an important experimental basis for future selectivity studies.

This study is aimed at gaining an enhanced view of ion binding to a specific location—the extracellular end—of the KcsA selectivity filter. We apply a method previously introduced with mammalian Ca²⁺-activated K⁺ (BK) channels for determining ion binding to pore sites at thermodynamic equilibrium (Neyton and Miller, 1988a,b). This approach exploits block of single channels by Ba²⁺ ion; with a crystal radius precisely equal to that of K⁺, Ba²⁺ masquerades as a divalent K⁺ ion and thus, in natural agreement with the snug-fit proposal (Lockless et al., 2007), specifically and strongly blocks K⁺ channels. As we show here, Ba²⁺ block allows analysis of ionic interactions within the ion-occluded selectivity filter to quantify equilibrium binding constants of K⁺ and its analogues. We focus on the extracellular region of KcsA and demonstrate the extreme K⁺ selectivity of the S1/S2 sites, with exceptionally strong selectivity for binding of K⁺, Rb⁺, Cs⁺, and NH₄⁺ over Na⁺ and Li⁺.

Escherichia coli polar lipid (EPL), 1-palmitoyl,2-oleoyl phosphatidylethanolamine (POPE), and 1-palmitoyl-2-oleoyl-sn-glycero-3-phospho-(1′-rac-glycerol) (POPG) were purchased from Avanti Polar Lipids, Inc. Decylmaltoside (DM) was purchased from Affymetrix. Ultra-pure LiCl, NaCl, RbCl, CsCl, and NH₄Cl were purchased from Alfa Aesar. N-methylglucamine (NMG) contributed <5 µM K⁺ to the recording solutions.

The E71A KcsA mutation was constructed with the Quickchange Site-Directed Mutagenesis kit (Agilent Technologies) and confirmed by DNA sequencing. N-terminal hexahistidine-tagged KcsA-E71A protein was expressed in E. coli BL21 (DE3) cells, extracted in 100 mM KCl, 50 mM DM, and 20 mM tris-HCl, pH 7.5, and purified by cobalt-affinity (Talon; BD) and size-exclusion chromatography (Superdex 200; GE Healthcare). The final protein preparation was collected at 2–4 mg/ml in 100 mM KCl, 5 mM DM, and 50 mM tris-HCl, pH 7.5.

Liposomes reconstituted with KcsA-E71A were formed from a micellar solution of 35 mM CHAPS, 20 mg/ml EPL, and 10–40 µg protein/ml in reconstitution buffer (200 mM KCl and 10 mM MOPS, pH 7.0). The low protein/lipid ratio favors incorporation of single channels into the electrical recording system. The protein–lipid mix was dialyzed against reconstitution buffer 18–20 h at room temperature with three changes of solution, and the resulting suspension of channel-bearing liposomes was frozen at −80°C in 50-µl aliquots. For a day’s work, an aliquot was diluted with 50 µl of 1.5 M KCl, refrozen, thawed, and sonicated for 5–10 s in a cylindrical bath sonicator.

Single channels were recorded at 21 ± 1°C in a horizontal planar bilayer system with two aqueous chambers separated by a 80-µm-thick partition hand-crafted from overhead transparency film (Heginbotham et al., 1999); bilayers were formed from POPE/POPG (7.5:2.5 mg/ml in n-decane) on an ∼50-µm hole punched with a conical-tip butterfly-mounting pin (BioQuip Products, Inc.) and shaved smooth with a surgical scalpel. Unless specified otherwise, the “internal” aqueous chamber contained 195 mM KCl and 5 mM KOH, adjusted to pH 4.0 with succinic acid, and the “external” chamber contained the test monovalent cation (Cl⁻ salt) and 10 mM MOPS, adjusted to pH 7.0 with NMG base, as well as NMG-Cl at a concentration to maintain ionic strength at 105 mM. The pH gradient used here ensures that channels appear in the correct biological orientation, with cytoplasmic and extracellular sides of the protein facing the internal and external solutions, respectively (Heginbotham et al., 1999). Voltage, corrected for junction potential (∼5 mV), is reported according to the electrophysiological convention, with the external side defined as ground. Current was recorded via an amplifier (Axopatch-200B; Molecular Devices), with signals sampled at 10 kHz after low-pass filtering (eight-pole Bessel, 0.75–1-kHz corner frequency).

Single-channel recordings were obtained by squirting 0.5 µl of a liposome suspension onto the internal side of the bilayer and waiting 2–5 min, according to the patience of the investigator. Channel insertion was monitored and voltage controlled remotely by iPhone. Upon appearance of a channel (at 45–50 mV) as an ∼6-pA current step exhibiting submillisecond closures, chambers were perfused with fresh solutions. After recording for ∼3 min to confirm proper channel characteristics, 30–60 µM BaCl₂ was stirred into the internal solution. Recordings were checked for stationarity and idealized using Clampfit software, and dwell-time distributions were fit to multiple exponential components (after correction for an egregious Clampfit 9.2 bug in log-binning). Open-time distributions fit well to single exponentials. Spontaneous closings contributed a submillisecond component to the triple-exponential distributions for nonconducting times in the presence of Ba²⁺; correction of these distributions for missed events because of the rapid spontaneous closings was unnecessary, as the analysis here is concerned only with the Ba²⁺-induced block events. Bayesian analysis of dwell times (described in Appendix A) was performed with home-written software.

We encountered several experimental complications in obtaining blocks under the special condition of external Ba²⁺ and zero external K⁺. In this condition, the on-rate for Ba²⁺ from the external side is low (∼100 M⁻¹s⁻¹), and so high concentrations (1 mM) must be used to obtain a practicable frequency of block events; Ba²⁺ cannot be increased further, because above this concentration, the channels become intractably noisy, and even with 1 mM Ba²⁺, it was necessary to filter the records at 300 Hz to accurately analyze the blocked intervals. A further complication arose from spontaneous inactivation events, which, although extremely rare in the mutant used here, are not entirely absent (Chakrapani et al., 2011); these events contribute a small fraction of nonconducting times that, with ∼50-ms average duration, are easily separated from the Ba²⁺ blocks (∼5 ms). These spontaneous inactivation events are noticeable only with external Ba²⁺; they are so infrequent (∼0.05 s⁻¹) that they do not taint any of the recordings with internal Ba²⁺.

Despite the high purity and chromatographic homogeneity of our protein preparations, we routinely observed KcsA-E71A channels with three distinct behaviors, denoted “major,” “lazy,” and “hyper” (Fig. 2). The major channel, of ∼6.5 pA amplitude at 50 mV and excess open-channel noise, exhibits high open probability (0.95–0.99), as originally described for this mutant (Cordero-Morales et al., 2006), with closings on the 0.1–1-ms timescale. The lazy channel, of identical amplitude and open-channel noise, displays an additional set of 100-ms-timescale closings leading to open probability below 0.8. The hyper channel amplitude is 8–9 pA, with high open probability and low open-state noise. We have observed channels switch among these behaviors over the minutes-to-hours duration of our experiments, but such events are extremely rare. With proteoliposome preparations stored for less than 1 month, the major channel is observed ∼80% of the time, but this frequency decreases, and the lazy and hyper channels become more prevalent, as preparations age in the freezer. We have no explanation for this diversity in channel personality, which recalls a recent report of similar behaviors in wild-type KcsA (Chakrapani et al., 2011). The major channel was selected for all experiments reported here.

This study requires long steady-state recordings of single KcsA channels blocked by Ba²⁺ ions. However, a robust inactivation process (Gao et al., 2005; Cordero-Morales et al., 2006) renders wild-type KcsA unsuitable for such work. For this reason, we exploit an inactivation-removed mutant, E71A, that retains the channel’s acid-activated, voltage-dependent gating and K⁺-selective conduction properties (Cordero-Morales et al., 2006; Thompson et al., 2008). When incorporated into planar lipid bilayers under maximal activation conditions (internal 200 mM [K⁺]_in, pH 4, external 20 mM [K⁺]_ex, pH 7), single KcsA-E71A channels display open probability of ∼0.98, stable up to 3 h at 50 mV, with submillisecond closings (Fig. 3 A), which Chakrapani et al. (2011) have argued reflect conformational fluctuations of the selectivity filter.

Ba²⁺ added to either side of the membrane induces a novel set of nonconducting events 10–1,000-fold longer lived than the spontaneous closings (Fig. 3 A). We refer to these events as “Ba²⁺ blocks,” which are easily distinguished from the brief closings. Block times follow a double-exponential probability density function (PDF), with fast and slow time constants (τ_f, τ_s) of ∼5 and 300 ms under these recording conditions. Block-time distributions are independent of Ba²⁺ concentration, whereas conducting dwell times follow single-exponential distributions whose inverse time constants vary linearly with [Ba²⁺] (Fig. 3 C). These characteristics are expected for a bimolecular mechanism in which each nonconducting event represents a single Ba²⁺ ion entering the pore and physically blocking K⁺ current while it resides there (Vergara and Latorre, 1983; Miller, 1987). Moreover, the block times are the same regardless of the side of Ba²⁺ addition; this fact establishes that Ba²⁺ accesses the same blocking sites from either side of the channel and hence is able to traverse the entire pore, as is well-known for Ba²⁺ block of other K⁺ channels. Block is specific to Ba²⁺; neither Sr²⁺ nor Ca²⁺ produces discernible blocking events (not depicted).

Ba²⁺-soaked crystal structures of wild-type KcsA in the absence of K⁺ show two distinct Ba²⁺ densities within the selectivity filter: an inner site coincident with the S4 K⁺ site near the internal end of the selectivity filter, and a deeper site close to S2 (Lockless et al., 2007). We suppose that the double-exponential block-time distributions observed with the E71A mutant reflect these two sites. Furthermore, the strictly bimolecular blocking behavior confirms the crystallographic inference that Ba²⁺ ions do not simultaneously occupy both sites. Thus, once inside the selectivity filter, a single Ba²⁺ ion alternates back and forth between S4 and S2 before leaving the channel to terminate the block event. This picture with two sequential Ba²⁺ block sites is embodied in Fig. 4, which explicitly represents Ba²⁺ entering the channel from the internal side to initiate block at S4. The scheme allows that Ba²⁺ may leave the channel by either of two routes: by dissociation back to the internal side from which it entered (“b route”), or by permeation through the channel to the external Ba²⁺-free side (“z route”). Thus, Ba²⁺ is formally a conducting ion like its monovalent isostere K⁺, but with ∼10⁷-fold slower dissociation kinetics. Similar two-site schemes are well established for Ba²⁺ block of BK and K_v channels (Sohma et al., 1996; Harris et al., 1998).

The kinetic scheme above, which includes the submillisecond closings, demands a triple-exponential distribution for the nonconducting dwell times, from which the Ba²⁺-induced blocks—the part of the distribution in which we are interested—may be readily dissected. These follow a double-exponential PDF, with three independent parameters: the fast and slow time constants τ_f and τ_s, and either of the corresponding amplitudes a_f and a_s, as has been treated previously in detail (Colquhoun and Hawkes, 2009a):

Also note the useful relations:

Henceforth, we will refer solely to block-time distributions, with amplitudes renormalized as in Eqs. 3a and 3b after removing the submillisecond component because of brief closings.

The channel’s multi-ion single-filing property dictates that Ba²⁺ block kinetics should be sensitive to external K⁺ concentration, with internal K⁺ held fixed. At low [K⁺]_ex, both b and z routes are available for Ba²⁺ exit. At very high [K⁺]_ex, however, the z route is occluded because the S1 site, immediately adjacent to the external solution, is always occupied by a K⁺ ion. Blocks should therefore lengthen as [K⁺]_ex increases and the z route closes down according to K⁺ occupancy at S1. The recordings of Fig. 5 A confirm this expectation. At zero [K⁺]_ex (with ionic strength maintained by NMG, an impermeant organic cation), Ba²⁺ blocks are brief (on the order of 10 ms), and the addition of external K⁺ (5 mM) lengthens them ∼30-fold. This Ba²⁺ trapping, or “lock-in,” effect has been previously used with BK and K_v channels (Neyton and Miller, 1988a,b; Harris et al., 1998; Vergara et al., 1999) as a quantitative metric of equilibrium binding of K⁺ and other cations to specific locations in the selectivity filter. This analysis, which we adopt here, assumes that the kinetics of K⁺ equilibration with the Ba²⁺-blocked pore is much faster than the kinetics of Ba²⁺ exit; because K⁺ conduction operates on an ∼10-ns timescale, while Ba²⁺ blocks span milliseconds, we consider this a safe supposition.

At the high K⁺ extreme, z = 0 by assumption, and so in this condition, the three unknown rate constants b, c, and d may be determined from the three independent parameters derived from the experimental block-time distribution. As will be shown below, at external K⁺ concentrations >1 mM, the z route is fully shut down. At such saturating concentrations, the block-time distribution of the dataset of Fig. 5 B yields b = 75 s⁻¹, c = 87 s⁻¹, and d = 7 s⁻¹, and Table I reports values derived from multiple independent datasets, along with values of the underlying rate constants estimated by a Bayesian analysis. Bayesian analysis also confirms the assumption that z ≈ 0 at high K⁺. In this fully locked-in condition at +50 mV, with K⁺ mainly occupying S1, Ba²⁺ occupancy prefers S2 over S4 by ∼1.5 kcal/mol; this estimate is approximate, as the derived values of c and d show substantial channel-to-channel variation, but the preference for S2 is clear.

At the opposite extreme of zero external K⁺, with the z route unimpeded, block times are short-lived, as seen above (Fig. 5 A). This condition reveals a novel phenomenon: a nonmonotonic block-time distribution with a clear rising or plateau phase (Fig. 6 A; note linear binning of histogram to reveal this feature). Here, the fast-fraction amplitude is negative, as is expected when z becomes sufficiently large (Eq. 3a); a negative PDF amplitude is a rigorous indicator of irreversible steps in the kinetic scheme (Kijima and Kijima, 1987; Colquhoun and Hawkes, 2009a; Csanády et al., 2010), as occurs here in Ba²⁺ exit to the Ba²⁺-free external solution. Intuitively, the rising phase appears because with S1 unoccupied, most of the blocks terminate via the z route, requiring a delay between the initial binding of Ba²⁺ to S4 and its dissociation from S2 and subsequent passage through the channel. This delay produces a deficit of short blocks—a rising or plateau phase of the distribution. The fast and slow time constants under this condition are 5 and 15 ms, respectively (Table I).

If we would assume that b, c, and d at zero [K⁺]_ex maintain the values determined above at high K⁺, the block-time distribution at zero K⁺ would lead immediately to a value for z. But this assumption is unjustified because Ba²⁺ occupancy on S2 and S4 may differ depending on K⁺ occupancy at S1. Indeed, if b, c, and d are assumed fixed at their high K⁺ values, impossible solutions for z emerge from the measured PDF parameters at [K⁺]_ex = 0 (Eqs. 2a and 2b). We are therefore confronted with an apparently intractable problem: four unknown rate constants to be determined from three measured quantities at zero K⁺. In an attempt to obtain rough estimates of the rate constants, we performed a Bayesian analysis of the dwell-time data using a maximum entropy prior distribution that exploits a known upper limit for the rate constants, as described in Appendix A. This analysis produces probability distributions for all four rate constants (Fig. 6 B). The distributions are wide, and the rate constant values are correlated with each other (not depicted), because they are underdetermined by the data. The unsatisfactory nature of the analysis is further indicated by the poor fit to the measured dwell-time histogram obtained from the mean values of these distributions (Fig. 6 A, dotted curve).

The situation, however, is not so bleak; we can finesse the difficulty and gain a fourth independent measurement by switching Ba²⁺ to the external side of the channel. Under this condition, all blocks begin in and most terminate from S2, a circumstance that is expected to render both PDF amplitudes positive (and the block-time distribution monotonic). The time constants, however, must be identical according to the kinetic model (Fig. 4) to those obtained from the peaked distribution with internal Ba²⁺. Either of the PDF amplitudes, which are readily derived from symmetry considerations, thus becomes the fourth independent measurement:

where C_f is the fraction of fast blocks in the cumulative distribution function.

These expectations are fulfilled, despite experimental difficulties arising under this special external Ba²⁺ zero-K⁺ condition (see Materials and methods). The unavoidably low block frequency produces statistically noisy block-time distributions, and consequently the exponential fits are less well defined than with the internal Ba²⁺ data. Nevertheless, the distributions are, as expected, monotonic falling and dominated by the fast fraction of ∼6 ms, which makes up at least 90% of the blocks; indeed, the amplitude of the expected 15-ms slow fraction is not significantly different from zero, and the block-time distribution is effectively single exponential under these conditions (Fig. 7 A). For this situation, z ≈ 1/τ_f, whose value, 175 s⁻¹, is accurately known from the internal Ba²⁺ data. This allows solution for all four rate constants, b = 25 s⁻¹, c = 49 s⁻¹, d = 18 s⁻¹, and z =175 s⁻¹, as reported in Table I. Bayesian analysis was also applied to the external Ba²⁺ datasets to produce a probability distribution of z. Using this as a prior distribution in a repeat Bayesian analysis of the internal Ba²⁺ data above, we arrive at far cleaner estimates of all four rate constants than before (Fig. 6 B), and derive rigorous estimates of their uncertainties (Fig. 7 B and Table I).

The above experiments validate a two-site sequential block model and provide a rationale for Ba²⁺ trapping by external K⁺. By examining block times as [K⁺]_ex varies at fixed [K⁺]_in, we can estimate the ion’s equilibrium dissociation constant for the external trapping site (Neyton and Miller, 1988b). Block-time distributions at several K⁺ concentrations (Fig. 8 A) are shown for data collected using internal Ba²⁺, and parameters of these distributions for all K⁺ concentrations are reported in Table I. As [K⁺]_ex increases, block times lengthen noticeably at concentrations as low as 10 µM, and the rising phase of the distribution disappears by 25 µM, as K⁺ begins to occlude the z route. In the range of 25–500 µM, block times become single exponential as the fast-fraction amplitude rises from its initially negative value through zero to take on negligible values, thus leaving the slow fraction alone to dominate the PDF (Fig. 7 B, dashed region). Above ∼1 mM [K⁺]_ex, the distribution is again double exponential but now monotonic falling, as a_f grows increasingly positive. The sign reversal of a_f dictates that Ba²⁺ trapping is best monitored by the time constant of the slow fraction (Fig. 8 B), which dominates the distribution over the entire K⁺ concentration range.

How should we expect τ_s to vary as external K⁺ increases from zero to concentrations at which its effect on block times saturates? We cannot answer this question quantitatively, because we have no model for the K⁺ dependence of each individual rate constant in the two-site blocking scheme. A purely empirical analysis, however, can be used to capture the effect of external K⁺, as in previous work with BK channels (Neyton and Miller, 1988b; Vergara et al., 1999). We note (Fig. 8 B) that 1/τ_s, the slow-fraction Ba²⁺ escape rate, decreases in a graded fashion according to a simple, although theoretically unjustified, single-site binding function:

where K_o is an empirical “observed” dissociation constant. In general, K_o is not equal to the true K_d but should roughly approximate it, as it directly reflects K⁺ occupancy of the trapping site. Calculations based on Eq. 2b show, for instance, that if z alone would decrease with K⁺ as above, 1/τ_s would follow a curve similar in shape but with an observed dissociation constant ∼1.5-fold larger than the true K_d. Our experimental results (Fig. 8 B) indicate a high K⁺ affinity of the lock-in site, with K_o ≈ 30 µM, close to the 20-µM value found for the BK channel of mammalian skeletal muscle (Neyton and Miller, 1988b; Vergara et al., 1999).

The equilibrium binding selectivity of this site is now straightforwardly determined from analogous experiments with other monovalent cations in the external solution. Recordings in the presence of 100 µM Cs⁺, Rb⁺, or NH₄⁺ (Fig. 9 A) show that these ions also lengthen Ba²⁺ block times. Full Ba²⁺-trapping curves based on slow-fraction time constants are shown for all ions tested (Fig. 9 B). The pattern of binding selectivity at the external end of the KcsA pore is reminiscent of electrophysiological selectivity metrics among the K⁺-like cations for many K⁺ channels: Rb⁺ > Cs⁺ > K⁺ > NH₄⁺. Fig. 9 also illustrates the complete absence of a lock-in effect by Na⁺. The addition of 5 mM Na⁺, a saturating concentration for K⁺, fails to lengthen the block times, and the addition 0.1 mM K⁺ on top of this exerts its effect as if no Na⁺ were present. Similar experiments with the addition of Na⁺ or Li⁺ showed no Ba²⁺-trapping effect whatsoever up to 200 mM (not depicted); this result places a lower limit of ∼1.5 M on K_o for Na⁺ and Li⁺ at the lock-in site, equivalent to a K⁺-over-Na⁺ selectivity free energy of ΔΔG^o > 7 kcal/mol.

Two main conclusions emerge from these experiments: Ba²⁺ acts as a slow permeant blocker of KcsA, and the extracellular side of the KcsA selectivity filter binds K⁺ with extremely high selectivity over Na⁺ or Li⁺. These are unsurprising inferences, as it is known from electrophysiological work extending over a half-century that most K⁺ channels conduct K⁺ with high specificity over other biologically pertinent cations—Na⁺, H⁺, Mg²⁺, and Ca²⁺—and that Ba²⁺, uniquely among divalent cations, strongly blocks K⁺ channels. The utility of the present experiments is that they quantify the standard-state Gibbs free energy of cation transfer from aqueous solution to a specific region within the ion channel’s conduction pathway, and thus provide direct experimental links between high-resolution structures and ion-interaction energetics proposed from computational studies. Although our results are factually straightforward, several questions and ambiguities surrounding them warrant discussion.

Mechanistic investigations of K⁺ channels have frequently required the use of mutations that suppress the various “inactivation” processes necessary for their physiological purposes. The KcsA channel is no exception, as it inactivates strongly after opening (Gao et al., 2005), so that under maximally activated steady-state conditions, it is open <10% of the time (Heginbotham et al., 1999). The Perozo group (Cordero-Morales et al., 2006) described a set of KcsA mutations that suppress inactivation to various degrees, and we chose one of these, E71A, for the experiments here. This inactivation-suppressed mutant displays basic gating and conduction behavior very similar to that of the wild-type channel: activation by low pH and depolarized voltage, K⁺ selectivity, and unitary K⁺ conductance (Cordero-Morales et al., 2006; Thompson et al., 2008). Moreover, crystal structures of wild-type KcsA and E71A are virtually identical throughout the protein (Chakrapani et al., 2011; Cheng et al., 2011). In our view, these fundamental mechanistic correspondences between wild-type and E71A certify the mutant as a useful model for ion permeation in K⁺-selective pores.

However, a few differences between this mutant and wild-type KcsA have been reported. E71A occasionally crystallizes with its selectivity filter in a perplexing “flipped” conformation (Cordero-Morales et al., 2006) that has never been observed in wild-type crystals. In this structure, the backbone carbonyl of Val76 is rotated ∼180° away from the pore, and the amide group of Gly77 points inwards, a circumstance that places electropositive moieties at the level of S2. For this reason, we consider this unusual pore configuration to be nonconducting, although Cheng et al. (2011) have proposed it to be K⁺ conducting. A second oddity reported for the E71A mutant is more alarming: permeation by Na⁺. Cheng et al. (2011) recently reported ²²Na⁺ transport mediated by E71A channels in liposomes. Based on those flux-rate estimates, E71A channels prefer K⁺ over Na⁺ by a factor of only ∼20, a much lower selectivity than found for wild-type KcsA by ionic fluxes or electrophysiological methods (LeMasurier et al., 2001). However, electrical recording of E71A failed to detect Na⁺ current under any conditions. Our own experiments (Appendix B; Fig. B1) demonstrate that the K⁺/Na⁺ permeability ratio for the E71A mutant, derived by a high-accuracy reversal potential method, is >250. Resolution of these conflicting results awaits future work, but the wild-type–like electrical behavior of E71A gives us confidence in using this mutant as a faithful representative of K⁺ channel selectivity filters.

Our analysis rests on the presumption that Ba²⁺ blocks E71A at the two sites observed in Ba²⁺-soaked crystals of wild-type KcsA under K⁺-free conditions (Lockless et al., 2007). That structure reveals prominent Ba²⁺ densities of roughly equal magnitude near K⁺ sites S2 and S4. Of course, Ba²⁺ must transiently visit S3 and S1 during its journey through the selectivity filter, but occupancy on these sites would not, we suppose, be sufficiently favorable to be crystallographically evident. To represent our own recording conditions, it would be desirable to have a crystal structure of E71A in the presence of both Ba²⁺ and K⁺. However, we have failed to obtain satisfactory crystal structures of E71A in the presence of Ba²⁺, with or without K⁺, despite the ease of solving Ba²⁺-free structures of this mutant (not depicted).

The K⁺-selective trapping site must lie on the extracellular side of Ba²⁺, which itself preferentially occupies S2. This is a powerful reason to assign S1 alone for Ba²⁺ trapping, as proposed in analogous work on the BK channel (Vergara et al., 1999). But we should highlight a slight ambiguity on this point. According to the rate constants determined for the two-site blocking scheme (Table I), Ba²⁺ alternates between S2 and S4 on the 10–100-ms timescale during a block interval. Whenever the blocker occupies S2, external K⁺ equilibrates exclusively with S1; but when Ba²⁺ resides on S4, K⁺ has access to S1, S2, and S3. Because Ba²⁺ tends to favor S2 over S4, the free energy of K⁺ binding will be dominated by the chemical properties of S1, with those of S2 and S3 contributing to a lesser degree. Despite this uncertainty, we can confidently assert that S1 must be highly K⁺ selective; if it were nonselective, as deduced in a recent computational study (Egwolf and Roux, 2010), external Na⁺ should produce significant Ba²⁺ trapping at concentrations comparable to those at which K⁺ acts. Clearly, that is not the case. Moreover, it is known (Heginbotham et al., 1999) that although internal Na⁺ blocks K⁺ current though KcsA, external Na⁺ does not, as it would if it had significant affinity in the vicinity of S1. We also note a counterintuitive result regarding the Ba²⁺ sites for which we have no explanation: that the preference of Ba²⁺ for S2 over S4 is even greater at high than at zero K⁺, as is evident from the ratio of c to d at the two K⁺ concentrations (Table I).

The Ba²⁺-trapping maneuver used here on KcsA was originally developed to quantify binding of conducting ions to BK channels from rat muscle (Neyton and Miller, 1988a,b), and was also applied to Shaker-type K_v channels (Harris et al., 1998) before any K⁺ channel structures were known. The present results on KcsA generally recapitulate those on BK channels. The external trapping site of BK binds K⁺ in the 10-µM range, as with KcsA, and shows the same pattern of ion selectivity for equilibrium affinity: Rb⁺ > Cs⁺ > K⁺ > NH⁺₄ >> Na⁺, Li⁺, a pattern that also appears in conventional electrophysiological selectivity measurements in many classes of K⁺ channels (Hille, 2001), including KcsA (LeMasurier et al., 2001). In BK channels, however, Na⁺ and Li⁺ produce robust Ba²⁺ trapping, with K_d on the order of 10–50 mM, a 1,000-fold selectivity for K⁺ over Na⁺. The absence, in KcsA, of Ba²⁺ trapping by Na⁺ or Li⁺, even at concentrations as high as 0.2 M, means that selectivity here is even higher, >10⁵-fold. Recent work on the MthK channel (Ye et al., 2010) also finds high K⁺/Na⁺ selectivity at S1 (as well as at S3).

The appearance of the first KcsA structure evoked many computational studies on the ion selectivity of this prototypical pore. From the perspective of experimentalists, these computations have introduced much confusion. Varied theoretical approaches to the KcsA selectivity filter show wildly discrepant results. For example, two molecular dynamics computations from the same research group reported high K⁺ selectivity for the S1 site using free energy perturbation methods (Noskov et al., 2004), but little selectivity for the same site using an umbrella sampling approach (Egwolf and Roux, 2010). The difference between these studies is likely a result of sampling density, such that the former examined ions placed inside the carbonyl cage, whereas the latter allowed ions to visit the plane of the carbonyls, a binding configuration also investigated in the S4 region of filter (Thompson et al., 2009). A detailed understanding of the energetic basis of K⁺-selective conduction must ultimately rely upon results from molecular dynamics, but until the various computational approaches converge upon consistent results and basic consensus among its practitioners, their utility toward this end will remain doubtful.

Finally, we confess to perplexity regarding a fundamental result: the high affinity of K⁺ binding to the selectivity filter, as indicated by Ba²⁺ trapping in the 10-µM range for K⁺ and Rb⁺. At first blush, this appears to be inconsistent with the channel’s electrophysiological behavior. The apparent problem arises from the relationship among rate and dissociation constants for ligand binding to a single site:

Given an experimental equilibrium dissociation constant (∼10⁻⁵ M for K⁺), this relationship places an upper limit on the dissociation rate of a bound ligand, as the association rate cannot exceed diffusion limitation (k_diff ∼10⁹ M⁻¹s⁻¹):

Thus, if K⁺ binds to the selectivity filter with micromolar affinity, it must linger there for ∼100 µs before dissociating. However, this is 10⁴-fold longer than the nanosecond-timescale residence times for K⁺ manifested in typical ∼10-pA single-channel currents. This well-known discrepancy is usually resolved by pointing out that K⁺ channels are occupied by several conducting ions simultaneously, so that an ion on a site of intrinsically high affinity becomes destabilized and thus speeded in its exit rate by a second ion nearby in the pore (Hille and Schwarz, 1978; Levitt, 1978; Neyton and Miller, 1988a). Indeed, current models of K⁺ channels envision the selectivity filter almost always doubly and occasionally triply occupied at physiological K⁺ concentrations on the order of 0.1 M (Morais-Cabral et al., 2001). This idea was corroborated by cunning experiments demonstrating a very slow dissociation rate, 10⁴ s⁻¹, for the “last” K⁺ ion to leave K⁺-depleted (presumably singly occupied) Shaker channels (Baukrowitz and Yellen, 1996).

A multi-ion picture like this is quite satisfactory in accounting for K⁺ permeation, but it leaves our own results in a quandary. Our experiments focus on K⁺ binding to a pore already occupied by a coresident cation, Ba²⁺, and worse, this cation, being divalent, should be more electrostatically destabilizing than a nearby K⁺ ion. The puzzle thus poses itself: if multiple K⁺ ions act to mutually destabilize binding 10⁴-fold so as to achieve high conduction rates, why does coresidency by Ba²⁺ fail to destabilize K⁺ during trapping? Why do we not observe ∼0.1-M K_d values for Ba²⁺ trapping by K⁺? One possibility is that Ba²⁺ on its blocking sites somehow perturbs the structure or dynamics of the K⁺ site at S1, altering its chemistry to raise its intrinsic affinity to values far greater than under K⁺ conduction conditions. This idea—little more than desperate handwaving—is undermined by two experimental facts: (1) Ba²⁺ does not alter the selectivity filter structure of wild-type KcsA (at least to the extent visible at 2.7-Å resolution); and (2) regardless of absolute affinities, the selectivity among group IA cations faithfully reprises the various classical selectivity indicators determined electrophysiologically on the conducting pore.

This problem, which has been staring the field in the face for almost 25 years, thus remains unresolved. Perhaps computational approaches, once they converge to reliability, will accurately address models involving high-charge densities in narrow pores to comprehend at a deeper level the chemical underpinnings of the remarkable and biologically essential selectivity of K⁺ channels.

The object of Bayesian inference is to provide a probability distribution for the parameters of a model, given the observed data (Bernardo and Smith, 1994; Gelman, 2004; Lee, 2004). The method is based on Bayes’s theorem, which expresses the probability of the parameters given the data p(θ|x) in terms of the likelihood—the probability of the data given the parameters p(x|θ)—and additional information about the parameters expressed as a “prior” distribution p(θ):

where A is a dimensionless normalization factor.

Relative to classical and likelihood statistics, Bayesian methodology offers many favorable features, of which four of the most relevant for the present analysis are listed below.

• (1) Bayesian analysis provides a complete distribution for the parameters of a model, including direct measures of uncertainty. The posterior probability distribution provides a measure of the uncertainty of a parameter estimate, as well as the shape of that uncertainty (which may be represented by a nonsymmetric, skewed, or even multimodal distribution).

• (2) Bayesian methodology can incorporate pertinent prior information, in addition to the information given by the data at hand. Previous experiments in different systems or with different experimental setups may provide useful information for constraining realistic parameter values in the present experiment.

• (3) Bayesian methodology elegantly handles complex, multi-parameter models without invoking ad hoc techniques. In Bayesian analyses nuisance parameters can simply be integrated out to give a marginal distribution of a parameter that is independent of the others.

• (4) Bayesian methods provide solutions for underdetermined systems, which are generally intractable using classical methods. As long as valid priors are specified, Bayes’s theorem provides a valid posterior distribution, which may have large uncertainty yet still retain useful information about the unknown parameters.

Our application of Bayesian methodology to single-channel analysis differs substantially from earlier treatments (Gin et al., 2009; Rosales, 2004; Siekmann et al., 2011). Rather than using hidden Markov models to directly fit the single-channel data (comprising a specific sequence of openings and closings), we use a Bayesian Markov chain Monte Carlo algorithm to infer the rate parameters of our models from the individual distributions of conducting and nonconducting dwell times.

For the full Bayesian analysis, we generally assume a five-state scheme with an open state O, two block states B₁ and B₂, and two gating states, one extremely fast C (the spontaneous closings) and one very slow I (the residual rare inactivations). For the internal Ba²⁺ experiments, the B₂ to O transition is irreversible, whereas for the external Ba²⁺ experiments, the B₁ to O transition is irreversible.

Although the five-state scheme may at first appear rather complicated, inference is greatly simplified by performing two experiments on each channel, one without Ba²⁺ and one with Ba²⁺ added to one side. The no-Ba²⁺ experiments give an independent measurement of the f, r, g, and s rate constants involving only three states in which microscopic reversibility can be assumed. This results in four different datasets from two experiments: openings and closings without Ba²⁺, and openings and closings with Ba²⁺. The full likelihood function p(t|θ) (where t is the dwell time and θ is a vector of nine independent rate constants) is thus the product of four PDFs, each with its own independent dataset:

For a dataset of multiple dwell times (a vector t of N dwell times), the full joint likelihood function is simply the product of the likelihood for the individual dwell times:

where the product is taken over all N observed dwell times.

In the following sections, we give expressions for each of these four PDFs.

The PDF for the Ba²⁺ blocks can be derived using the Q-matrix method as described previously (Colquhoun and Hawkes, 2009a,b), resulting in a quadruple exponential PDF:

where τ_f and τ_s are given in the main text, Eq. 2, and

For the internal Ba²⁺ scheme, the amplitudes a_f and a_s are given in the main text, Eqs. 3a and 3b, and

The PDF for openings with Ba²⁺ is a single exponential. For the internal Ba²⁺ scheme,

With some channels, the very rare inactivated state I was not observed. For these channels, a four-state model was used for inference, in which the g and s rate constants were fixed at zero, with obvious modifications to each of the PDFs. The above relations all pertain to the internal Ba²⁺ scheme. Analogous PDFs for the external Ba²⁺ scheme are obtained by exchanging appropriate rate constants (b<=>z, c<=>d, and a=>α).

Low-pass filtering effectively results in missing events in the dwell-time recordings. We treat these missed events as left-truncated data in the probability distribution for the likelihood function. Hence, all of the PDFs given above were renormalized to account for this missing data by dividing by the factor F, where t_o is the cutoff time below which all events are missed (∼0.2 ms for 1-kHz Bessel filtering):

where the sum is over all exponential components.

As described in the main text, the rate constants c, d, and z for the Ba²⁺ blocks are underdetermined by the internal Ba²⁺ experiments, as there are only two exponentials measured with three experimentally accessible unknowns: two time constants and one amplitude. This state of affairs renders the likelihood function for the Ba²⁺ blocks degenerate and in practice means that many different combinations of c, d, and z are all consistent with the data, although to differing degrees. Fortunately, Bayesian methods provide workable solutions to underdetermined problems as long as proper priors are used, although degenerate likelihoods usually result in wide posterior distributions for the parameters. For this reason, we used informative, though vague, priors for c, d, and z in the analyses of internal Ba²⁺ data. The sum of two of the eigenvalues of the blocking scheme has a simple analytic relationship to the rate constants (Eq. 4c in the main text):

From previous experimental work with internal Ba²⁺ blocks, we know that the sum of these two eigenvalues is roughly 300 s⁻¹. Hence, we conservatively assume vague exponential (maximum entropy) priors for b, c, d, and z, each with a mean of 100 s⁻¹. Conventional reference priors (Lee, 2004) were used for all other rate constants (i.e., the inverse of the rate constant).

With external Ba²⁺, z is fully identified by the sum of the block-time amplitudes a_f + a_s, according to (Eq. 6) in the main text. In a separate analysis, we therefore used the posterior distribution of z from external Ba²⁺ data as a proper data-based prior for z with the internal Ba²⁺ data. This procedure effectively turns an underdetermined problem into a determined one. For convenience, we approximated the posterior distribution for z from the external Ba²⁺ analyses as a gamma distribution characterized by the mean and variance of the posterior.

Posterior distributions of the parameters were approximated using a parallel Metropolis-coupled Markov chain Monte Carlo algorithm (Metropolis et al., 1953; Hastings, 1970; Geyer, C.J. 1991. Markov chain Monte Carlo maximum likelihood. Computing Science and Statistics Proceedings of the 23rd Symposium on the Interface; Green, 1995; Altekar et al., 2004). Because the posterior distribution dictated by the likelihood function (Eq. A3) and our assumed priors has no analytic form, we used Metropolis–Hastings moves to generate samples from the posterior distribution. Eight chains (two cold and six heated) were run in parallel using the Metropolis-coupled algorithm to efficiently explore the multimodal parameter space. Chains were generally run for one million iterations, sampling every 100 iterations. Convergence was assessed by monitoring the potential scale reduction factor statistic (Gelman, 2004) and requiring that it drop below 1.01. The first half of the iterations were discarded as “burn-in.” The joint posterior distribution of the parameters was investigated and summarized by histogram and moment analysis. Analysis of a single parameter from the joint posterior distribution samples provides an approximation of the marginal posterior distribution for that parameter.

Reversal potential measurements of selectivity under strictly bi-ionic conditions (K⁺ only intracellular, Na⁺ only extracellular) are usually impossible for K⁺ channels, as no Na⁺ current is observed even at very negative voltage, and so no reversal can be interpolated in the I-V plot. The present measurement seeks to determine K⁺/Na⁺ selectivity under “almost bi-ionic” conditions, with a large but well-defined K⁺ gradient and Na⁺ on only the external side of the channel. (These experiments use internal solution of 200 mM K⁺ and external solution of 10 mM K⁺ plus 190 mM Na⁺.) The reversal potential of the single-channel I-V curve, V_c, is then compared with the reversal potential, V_K, of an ideally K⁺-selective ionophore, valinomycin, added to the planar bilayer immediately after the single-channel measurement. The difference between these two reversal potentials, ΔV = V_c − V_K, nulls out small electrical imbalances and uncertainties regarding single-ion activity coefficients, and thereby determines the channel reversal potential with uncertainties on the order of 1 mV (Miller, 1987; LeMasurier et al., 2001). It may be easily shown that:

under our conditions, where ΔV is in millivolts and T = 21°C.

Analysis of four independent KcsA-E71A KcsA datasets resulted in ΔV = −0.8 ± 1.3 mV, showing that E71A-KcsA and valinomycin cannot be distinguished in K⁺/Na⁺ selectivity by reversal potential under these conditions. This does not mean, however, that the KcsA mutant’s selectivity is identical to the nearly perfect K⁺ selectivity of valinomycin. The uncertainty in these measurements places a lower limit (based on a 2-σ limit) of P_K/P_Na > ∼250. This lower limit is similar to that found for wild-type KcsA by this method (LeMasurier et al., 2001).

We are grateful for the copious ridicule and humiliation supplied by Miller laboratory members and Dr. C. Darwin upon scrutinizing drafts of the manuscript.

Frederick J. Sigworth served as guest editor.