Simultaneous Binding of Basic Peptides at Intracellular Sites on a Large Conductance Ca²⁺-activated K⁺ Channel : Equilibrium and Kinetic Basis of Negatively Coupled Ligand Interactions

Large conductance (∼250 pS) Ca²⁺-activated K⁺ channels (BK or maxi-K_Ca channels)¹ are an intriguing class of ion channels that are activated in a synergistic fashion by intracellular Ca²⁺ and depolarizing voltage (Latorre, 1994). In Drosophila, maxi-K_Ca channels are encoded by the Slowpoke gene, part of which is homologous to the family of voltage-gated K⁺ channels (K_V channels), including Shaker (Atkinson et al., 1991; Adelman et al., 1992). Both K_V and Slowpoke channels contain a characteristic motif of six membrane-spanning segments called S1–S6. This motif includes the S4 segment involved in voltage sensing and the extracellular P-region between S5 and S6 that determines ionic selectivity. Besides these common structural elements, Slowpoke maxi-K_Ca channels have an additional transmembrane segment at the NH₂ terminus, called S0, plus a unique sequence of ∼800 residues at the COOH terminus (Meera et al., 1997). Substantial evidence supports the notion that the COOH-terminal portion of the maxi-K_Ca channel protein corresponds to a large cytoplasmic domain that contains Ca²⁺-binding site(s) involved in channel activation (Wei et al., 1994; Schreiber and Salkoff, 1997). To understand the mechanism of Ca²⁺ activation and biochemical regulation of maxi-K_Ca channels, it is necessary to define the structure and function of this cytoplasmic domain. An underlying objective of the work described in this paper is to characterize intracellular binding sites for peptide ligands that can serve as reference points in studies of the cytoplasmic architecture of the maxi-K_Ca channel.

The charybdotoxin family of small protein toxins is a set of natural ligands that has been effectively used to identify structural determinants in the extracellular pore region of K⁺ channels (Miller, 1995; MacKinnon et al., 1998). By analogy, protein ligands that bind to intracellular sites on maxi-K_Ca channels may similarly be useful for mapping this part of the channel protein. One class of such ligands is the Kunitz family of proteins that includes bovine pancreatic trypsin inhibitor (BPTI) and dendrotoxin-I (DTX-I). This family of ∼60-residue proteins shares six conserved Cys residues (see Fig. 1 A) and a common protein fold, but its members exhibit diverse inhibitory activities. BPTI and related Kunitz molecules are potent inhibitors of various members of the chymotrypsin/trypsin family of serine proteinases (Dufton, 1985; Capasso et al., 1997). DTX-I and several homologues from mamba snake venom block various K_V channels from the extracellular side, but are poor inhibitors of serine proteinases (Hollecker et al., 1993; Smith et al., 1997). Calcicludine, another Kunitz homologue from mamba venom, blocks high-threshold voltage-activated Ca²⁺ channels from the extracellular side (Schweitz et al., 1994). Yet another activity exhibited by some Kunitz proteins such as DTX-I and BPTI is a rather distinctive inhibitory effect on maxi-K_Ca channels from the intracellular side (Lucchesi and Moczydlowski, 1990, 1991). This latter effect resembles a slow blocking reaction, but differs in that the individual blocking events mimic subconductance states at the single-channel level. Apparent subconductance events induced by BPTI and DTX-I are actually due to a fast closing and opening reaction of the maxi-K_Ca channel in the ligand-bound state (Moss and Moczydlowski, 1996). For descriptive purposes, this behavior is referred to here simply as “substate behavior” or “substate events.” In contrast to the substate-inducing behavior of Kunitz inhibitors, an NH₂-terminal inactivation domain of Shaker K_V channels called the “ball peptide” is known to directly block maxi-K_Ca channels by binding within the internal vestibule or pore (Foster et al., 1992; Toro et al., 1992).

To further develop Kunitz inhibitors as structural probes of the intracellular domain of maxi-K_Ca channels, it is necessary to localize the binding site of these inhibitors with respect to the primary sequence of the channel. One line of evidence that such ligands do not occlude or bind directly in the pore is that the blocking affinity of internal Ba²⁺ and internal tetraethylammonium (TEA) is not significantly affected by BPTI and DTX-I, respectively (Lucchesi and Moczydlowski, 1991). Such results lead to the hypothesis that BPTI and DTX-I bind to an intracellular site outside the pore and induce substate behavior by an allosteric mechanism. A potential candidate for the BPTI/DTX-I binding site on the K_Ca channel protein is a region of ∼250 residues near the COOH terminus that exhibits similarity to the chymotrypsin family of serine proteinases (Moss et al., 1996a,b). However, serious consideration of this hypothesis leads to an interesting paradox. Since maxi-K_Ca channels are probably formed as homotetramers of four identical α-subunits (Shen et al., 1994), how can BPTI and DTX-I bind outside the central pore or common fourfold subunit interface and yet exhibit binding kinetics characteristic of a single site? The allosteric model of substate production invoked above predicts that four binding sites for BPTI/DTX-I must exist on each tetrameric channel complex. Such a model further predicts that the kinetics of BPTI/DTX-I association and dissociation ought to reflect multiple sites of ligand interaction.

To resolve this question, we investigated the interaction of BPTI and DTX-I with rat muscle K_Ca channels inserted into planar lipid bilayers. The titration behavior of single K_Ca channels with BPTI in the presence of DTX-I deviated significantly from simple binding competition of two ligands at a single site. The data are consistent with a model that includes states of simultaneous occupancy by one BPTI and one DTX-I molecule on different subunits of a tetrameric complex. Binding of DTX-I to one subunit lowers the binding affinity for BPTI on another subunit by ∼11-fold, and vice versa. To further probe the location of the Kunitz inhibitor site with respect to the pore, we analyzed the single-channel blocking kinetics of a homologue of Shaker ball peptide (BP) in the absence and presence of DTX-I. The results indicate that BP binds to the K_Ca channel with an approximately eightfold lower affinity when DTX-I is simultaneously bound. Detailed analysis revealed that the lower affinity for BP is solely due to a 44-fold reduction in its association rate. Unexpectedly, we found that DTX-I actually stabilizes BP on its binding site by slowing its rate of dissociation by a factor of ∼5.6. Overall, these results provide direct evidence of negatively coupled ligand–ligand interactions among sites for Kunitz proteins and ball peptide that take place on the intracellular surface of maxi-K_Ca channels. The unusual asymmetry of energetic effects on rate constants for the interaction between DTX-I and BP can be explained by structural features inherent to a channel-like architecture.

Planar bilayers were formed on a 0.2-mm diameter hole drilled in a cylindrical cup made of polystyrene or delrin as supplied by Warner Instrument Corp. The membrane was cast from a solution of 25 mg/ml phospholipids in decane. Bilayer formation was monitored by capacitance measurement. The lipid composition was a mixture of 80% 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphoethanolamine and 20% 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (Avanti Polar Lipids). Plasma membrane vesicles were prepared from adult rat skeletal muscle using the procedure described in Favre et al. (1999). The final recording solution on both sides of the bilayer contained 50 mM KCl, 10 mM MOPS-KOH, pH 7.4. Incorporation of maxi-K_Ca channels from membrane vesicles into planar bilayers followed methods described previously (Lucchesi and Moczydlowski, 1991). Only bilayers containing single maxi-K_Ca channels were used for these experiments. The intracellular or internal side of the bilayer is defined as the side that responds to activation of K_Ca channels by Ca²⁺ and positive voltage. For all experiments in this work, the holding voltage was +30 mV using the physiological convention of extracellular ground. The internal Ca²⁺ concentration was usually 200 μM, but was increased to 500 μM in some experiments to maintain the open state probability of intrinsic K_Ca channel gating at >0.9.

Single channel currents were recorded at room temperature (20–23°C) using a patch clamp amplifier (3900A; Dagan Corp.) with an expander (3910; Dagan Corp.) for bilayer capacitance measurements. The patch clamp headstage was connected to the bilayer chamber via Ag/AgCl electrodes and agar-KCl bridges. Single channel recordings were stored on VCR tape using a digital data recorder (VR-10; Instrutech Corp.). Data were filtered at a corner frequency of 100–500 Hz with an eight-pole low pass Bessel filter (902LPF; Frequency Devices Inc.) and sampled at five times the filter frequency or greater for subsequent analysis.

Bovine pancreatic trypsin inhibitor was purchased from Sigma Chemical Co. Dendrotoxin I (or toxin I from Dendroaspis polylepis) was obtained from Alomone Labs. A peptide homologue (BP) of the Shaker ball peptide with the primary sequence MAAVAVLYVLGKKRQHRKKQ was synthesized by the W.M. Keck Biotechnology Resource Center at Yale University Medical School (New Haven, CT). The purity of the peptide was confirmed by reverse phase HPLC and mass spectroscopy (mol wt, 2,325). Concentrated stock solutions of 500 μM BP were prepared in H₂O. Problems associated with adsorption of BP to the plastic chambers were minimized by adding vesicles and BP to the front chamber, composed of black delrin, rather than the rear cup chamber, composed of polystyrene. Peptide concentrations are based on dry weight and supplier information. Therefore, derived k_on and K_d values should be regarded as relative rather than absolute measurements.

These experiments involved long recordings (∼1–5 h) of single maxi-K_Ca channels exposed to the following inhibitors: DTX-I, BPTI, BP, TEA, and charybdotoxin (ChTX). After addition of a particular concentration of an inhibitor that induced discrete events, channel activity was continuously recorded for a time sufficient to collect ∼100 events when possible. Then, the next concentration of inhibitor was added, channel activity was recorded, etc., until the bilayer broke. Substate events induced by DTX-I (mean dwell time ∼38 s) were the longest events that were measured and analyzed. Durations of these latter events were measured directly from chart records with the aid of a digitizing tablet (TG1017; Houston Instrument). Substate events induced by BPTI (mean dwell time ≃ 0.41 s) and BP blocking events (mean dwell time ≃ 1.7 s) were automatically measured from digitized records using an LSI 1173 computer system (Indec). For this purpose, single-channel data was filtered at 100 Hz and sampled at 1 kHz. Transitions were detected using a 50% threshold criterion set between the substate current level and the open state for BPTI or between the zero current (blocked state) and open state for BP. To exclude brief closures due to intrinsic channel gating, a cutoff limit of α = 0.1 s was used for the shortest acceptable substate closure or blocked state event. This cutoff limit excludes >95% of Ca²⁺- and voltage-dependent closures due to channel gating under these conditions (200 μM internal Ca²⁺ and +30 mV).

Lifetimes of BPTI-substate, BP-blocked, ChTX-blocked, or unblocked state events were generally measured from populations of dwell times (n > 100 events) that were binned, plotted as cumulative probability distributions, and fit to an exponential function of time. These distributions were invariably well described by a single exponential (Lucchesi and Moczydlowski, 1991). Measured lifetimes of the unoccupied open state or unblocked state were corrected for a small amount of artificial lengthening due to the minimum cutoff limit for closures of α = 0.1 s using the following formula (Colquhoun and Sigworth, 1983): τ_corrected = τ_observed exp(−α/τ_closed). Certain types of event samples such as DTX-I substates, burst lengths, or BP-blocked events in the presence of DTX-I were measured as simple arithmetical means. Arithmetical means were used in these latter cases, either because the sample size was often <100 events and not well suited to histogram fitting, or because the analysis focused on the behavior of the actual mean dwell time of more complex distributions (e.g., see Eq. 19). Dwell-time histograms obtained in the presence of DTX-I and BP exhibited two exponential components. These data were also plotted as probability density functions and fit to a sum of two exponentials using PCLAMP analysis software from Axon Instruments.

Single-channel block by external TEA was manifested as a “fast block” corresponding to an apparent decrease in unitary current (e.g., Moczydlowski, 1992). Unitary current as a function of [TEA] was measured with the use of all-points amplitude histograms to determine the mean value of the closed and apparent open state current levels. Kinetic parameters directly measured from single-channel data are reported as the mean ± SD or SEM

Previous studies of DTX-I– and BPTI-induced substate events suggested that these ligands bind reversibly to a single intracellular site or homogeneous class of sites on maxi-K_Ca channels that is located outside the pore (Lucchesi and Moczydlowski, 1990, 1991). However, this conclusion appears to be in conflict with the known tetrameric structure of K⁺ channels and the prevailing idea that the functional unit of maxi-K_Ca channels is a tetramer of ∼130 kD α-subunits (Shen et al., 1994; Doyle et al., 1998). A nonpore binding site on a homotetrameric channel structure implies that there should be four symmetrically located sites on the cytoplasmic side of the protein. To pursue this question, we re-examined the kinetics of BPTI-induced substate events as a function of BPTI concentration to ascertain whether one-site behavior is strictly obeyed.

Fig. 1 B shows typical results from an experiment in which a single K_Ca channel recorded at a high open-state probability is consecutively exposed to 1, 2, 4, and 8 μM internal BPTI. These records show that BPTI induces a noisy but well-defined substate at ∼27% of the unitary current and that the channel's probability of residing in this substate increases with BPTI concentration. The records also show a few examples of well resolved closures to the zero current level, plus other subconductance levels that are occasionally observed under these conditions. However, these latter events are quite rare compared with the time the channel spends in the open state and the major substate. They are ignored for this analysis since they occur too infrequently for quantitative evaluation.

A possible model for binding of BPTI to four equivalent sites on a homotetramer is shown in Fig. 2, Scheme 1. For the sake of discussion, this model assumes that the microscopic association rate constants, k₁, for binding to a site on each of the four subunits of the unliganded U state are equivalent. With this assumption, the observed macroscopic association rate constant for binding of one BPTI molecule to the U state would be 4k₁. Similar statistical factors apply to the other association and dissociation steps involving binding to or unbinding from multiply occupied states. Fig. 2, Scheme 1, is only one possible example of a sequential reaction that could describe ligand binding to a homotetrameric protein. Many other ligand binding schemes are possible, including mechanisms that involve conformational changes of individual subunits or the whole complex. The essential point is that ligand binding to multisubunit complexes can potentially display complex behavior including positive and negative cooperativity.

If more than one BPTI molecule can bind to a single tetrameric K_Ca channel complex, one would expect to observe some indication of the existence of more than one class of ligand-bound states in single channel records. In the experiment of Fig. 1 B, increasing BPTI concentration from 0 to 8 μM decreases the probability of the channel residing in the fully open conductance state from 1.0 to ∼0.15. However, evidence for different states of BPTI occupancy (e.g., states S₁–S₄ of Fig. 2, Scheme 1) is not obvious from the single-channel record. One explanation may be that the binding of one, two, three, or four BPTI molecules produces a channel with exactly the same conductance behavior, corresponding to the ∼27% substate level. However, if this were the case, the probability distribution for the dwell time in the substate would correspond to the distribution of a group of four states, S1, S2, S3, and S4, within the framework of Fig. 2, Scheme 1. The predicted theoretical form for this distribution would be a sum of four exponential components. Furthermore, Fig. 2, Scheme 1 predicts that the distribution of substate dwell times would be dependent on BPTI concentration. Thus, the mean substate dwell time should lengthen with BPTI concentration if Fig. 2, Scheme 1 is applicable.

To search for such behavior, dwell time histograms of BPTI-substate events were compiled for eight different single channels at 1, 2, 4, and 8 μM BPTI, as described in materials and methods. Consistent with previous work (Lucchesi and Moczydlowski, 1991), these histograms conformed to a single exponential distribution at all BPTI concentrations (not shown). The mean lifetime of the exponentially distributed BPTI substate (τ_s) was equal to 0.411 ± 0.022 s (Fig. 2 A). Although there is a slight tendency for τ_s to increase with [BPTI] in Fig. 2 A, the data points at 1, 2, 4, and 8 μM BPTI are not statistically different by the usual criterion of P ≤ 0.05. Thus, there is no hard evidence that the mean duration of the BPTI-induced substate lengthens significantly with increasing [BPTI] or that there is a kinetically distinct population containing >5% of the total substate events. These results confirm that the kinetics of BPTI-induced substate events are well described by a one-site process such as that represented by Fig. 2, Scheme 1A.

Fig. 2, Scheme 1A, predicts the following relationships for the lifetime, τ_S, of the S1 BPTI substate, and for the lifetime, τ_U, of the unliganded U state:

The dissociation rate constant, k₋₁, calculated according to Eq. 1 from the reciprocal of the mean of four τ_S measurements in Fig. 2 A is k₋₁ = 2.43 ± 0.13 s⁻¹ (±SEM). The observed association rate constant, 4k₁, can be obtained from the reciprocal relationship between τ_U and [BPTI] according to Eq. 2. Probability histograms of unliganded U-state events between adjacent BPTI substates were also uniformly well described by a single exponential function (not shown). The mean lifetime of τ_U measured for eight different single channels at each BPTI concentration is plotted in Fig. 2 B and fit to Eq. 2. This fit gives an estimate of 4k₁ = 1.63 ± 0.07 × 10⁶ s⁻¹M⁻¹ for the apparent BPTI-association rate constant. The kinetic ratio, k₋₁/4k₁, or K₁ = 1.49 ± 0.10 μM, is the equilibrium dissociation constant for BPTI in Fig. 2, Scheme 1A.

The K₁ equilibrium constant can also be estimated from the time-averaged probability of the unliganded state, P_U. Fig. 2, Scheme 1A, predicts that P_U = K₁/(K₁ + [BPTI]). A fit of the data in Fig. 2 C to this latter function yields K₁ = 1.28 ± 0.07 μM, in agreement with the kinetic ratio of k₋₁/4k₁. For models such as Fig. 2, Scheme 1, where there is a substantial population of multi-liganded states, Hill coefficients (n) higher than n = 1 are also predicted. The fact that the data of Fig. 2 C is closely fit using an n of 1.0 is additional evidence that Fig. 2, Scheme 1A, or an equivalent one-site model describes the interaction of BPTI with the maxi-K_Ca channel. Thus, if the K_Ca channel does contain four structurally homologous BPTI binding sites in the tetrameric complex, the kinetic data imply that the binding of BPTI to one of the sites essentially precludes the binding of a second BPTI molecule to any of the remaining three sites.

The sequence alignment of Fig. 1 A illustrates that BPTI (58 residues) and DTX-I (60 residues) are 32% identical, including the presence of six conserved Cys residues. It is also known that these two small proteins share a common fold at the tertiary structural level (Berndt et al., 1993; Lancelin et al., 1994; Skarzynski, 1992). Like BPTI, DTX-I also induces discrete substate events when added to the intracellular side of maxi-K_Ca channels, but the average duration of the DTX-I– induced substate is ∼38 s compared with ∼0.41 s for BPTI. Fig. 3 (top) shows a segment of a typical single-channel recording taken after addition of 460 nM DTX-I. Long-lived substate events induced by DTX-I typically exhibit less excess noise and a higher mean current level (∼60% of unitary current at +30 mV) than the shorter duration BPTI substate (Fig. 1 B). Previous studies have also established that the kinetics of interaction of DTX-I with single K_Ca channels is well described by a one-site process equivalent to Fig. 2, Scheme 1A, discussed above for BPTI (Lucchesi and Moczydlowski, 1990, 1991). To investigate the physical relationship of binding sites for BPTI and DTX-I, we examined the effect of titrating single maxi-K_Ca channels with increasing concentrations of BPTI in the presence of a nearly saturating concentration of DTX-I.

In the presence of 460 nM DTX-I alone (Fig. 3, top), single K_Ca channels exhibit two distinct states or modes of activity: (a) long-lived substates at ∼60% of the unitary current that correspond to residence times of DTX-I on the channel, and (b) shorter intervals between adjacent DTX-substates defined as “inter-DTX periods.” The latter periods correspond to waiting times of the unoccupied channel between the dissociation of one DTX-I molecule and association of another. If BPTI can bind to a channel that is already occupied by DTX-I, one might expect to observe discrete interruption of the long-lived DTX substate with short BPTI-like substate events. Alternatively, if binding of BPTI to the unoccupied channel is greatly favored over binding to the DTX-occupied channel, then brief BPTI-substates should occur only within inter-DTX periods.

The data of Fig. 3 illustrate that the system is best described by the latter alternative. Long recordings from such experiments reveal that titration of BPTI in the presence of DTX-I exhibits a pattern of activity that may be roughly described as bursts of BPTI-substate activity punctuated by DTX substates. In particular, records of Fig. 3 obtained in the presence of both ligands show that two adjacent DTX-substate events are nearly always flanked on both the exit and entry side by transitions to the fully open current level. This implies that DTX-I usually dissociates before a burst of shorter BTPI-binding events begins and that BPTI usually dissociates before DTX-I can bind to produce a long-lived substate. To quantitate this latter observation, 458 inter-DTX periods were examined for the type of first transition flanking the exit from the DTX-substate current level and the entry back to the next DTX substate. This analysis revealed that 96.1% of all DTX substates were terminated by an opening to the fully open current level and 95.6% of all inter-DTX periods were terminated by a transition from the fully open current level to the DTX-substate level. These measurements confirm the visual impression from the single-channel records that DTX-I preferentially dissociates before BPTI can bind to the channel, and vice versa.

Such behavior is expected for a mechanism like that shown in Fig. 4, Scheme 2A. This model is an extension of Fig. 2, Scheme 1A, for formation of the BPTI substate (S_BPTI) that includes binding of DTX-I to the unoccupied channel (U) to produce the DTX substate (S_DTX). The implicit assumption of Fig. 4, Scheme 2A, is that, by virtue of structural homology, BPTI and DTX-I bind to the same site(s) on a K_Ca channel tetramer, but that some mechanism strongly inhibits binding of more than one of these molecules at a time. Although there are four potentially available sites on the tetrameric complex, Fig. 4, Scheme 2A, is formally equivalent to simple, mutually exclusive competition of two ligands at a single site and may be referred to as “pseudo one-site competition.”

The cursory description of data from experiments of Fig. 3 outlined so far generally supports Fig. 4, Scheme 2A. However, there are indications that the system is actually more complex. Particularly at higher concentrations of BPTI, current records of Fig. 3 (8 μM BPTI) display a new type of event. Many of the inter-DTX periods exhibit one or more states of very low conductance or “nearly blocked” events that last several seconds in duration. These latter events begin with a decrease in current from the BPTI-substate level to a level near zero current and terminate with an increase in current directly back to the BPTI-substate level. Such events are quite rare in the presence of BPTI alone (Fig. 1 B), but they are noticeably more frequent in the presence of DTX-I. These latter observations lead to the hypothesis that the BPTI substate is susceptible to conversion to a different type of low-conductance state in the presence of DTX-I. With BPTI alone, doubly bound states (Fig. 2, Scheme 1, S2) may occur with very low probability such that their contribution to the time-averaged behavior is hard to rigorously detect (e.g., on the order of 1% or less). However, in the presence of DTX-I plus BPTI, a state in which both ligands are bound simultaneously may occur with a probability that is high enough to measure a significant deviation from purely one-site behavior.

To pursue this latter conjecture, we analyzed whether the detailed kinetic behavior obeys the strict predictions of Fig. 4, Scheme 2A. The first step in this analysis requires determination of the dissociation and apparent association rate constants (j₋₁, 4j₁, respectively) for DTX-I in the absence of BPTI from relationships analogous to those of Eqs. 1 and 2:

The mean dwell time of the DTX substate, τ_S-DTX, was measured as 37.6 ± 2.1 s (±SEM) from a population of 358 events pooled from 18 different single K_Ca channels. Likewise, the mean dwell time of the unoccupied channel measured in the presence of 460 nM DTX-I, τ_U, was 1.35 ± 0.09 s (±SEM). These values were used to calculate the rate constants, j₋₁ = 0.0266 ± 0.0015 s⁻¹ and 4j₁ = 1.61 ± 0.11 × 10⁶ s⁻¹M⁻¹ from Eqs. 3 and 4, respectively. The apparent equilibrium dissociation constant for DTX-I is obtained as follows: J₁ = j₋₁/4j₁ = 16.5 ± 1.4 nM.

For a system that strictly adheres to the pseudo one-site competition mechanism of Fig. 4, Scheme 2A, the dwell time of the apparent DTX substate is expected to be independent of BPTI concentration. In contrast, the dwell time between adjacent DTX substates is expected to increase with BPTI concentration, as the channel spends more time in the combined burst state of U and S_BPTI. These relationships can be derived from Fig. 4, Scheme 2A, as follows:

Durations of the apparent DTX substate corresponding to the lifetime predicted by Eq. 5 and inter-DTX periods or BPTI “bursts” corresponding to the lifetime predicted by Eq. 6 can be readily identified and measured from current records of such experiments. Fig. 4 A shows a plot of the lifetime of the apparent DTX substate at 460 nM DTX-I as [BPTI] is varied over the range of 0–8 μM. The dashed line in Fig. 4 A is the prediction of Scheme 2A as stated by Eq. 5. In contrast to predicted independence with respect to [BPTI], the data reveal a significant lengthening of τ_S-DTX with increasing BPTI concentration. In quantitative terms, there is a 47% increase in the mean value of τ_S-DTX over the range of 0–8 μM BPTI. The underlying basis for the increase in τ_S-DTX with increasing [BPTI] appears to be an increased frequency of discrete low conductance events of several seconds in duration that occasionally interrupt DTX substates (not shown). Fig. 4 B shows a plot of the measured lifetime of inter-DTX periods or BPTI bursts as a function of [BPTI]. The dashed line in Fig. 4 B is the prediction for the simple competition mechanism of Scheme 2A according to Eq. 6, using previous derived values for 4j₁ and K₁. The results of Fig. 4 B reveal a large deviation from the mechanism of Scheme 2A since the slope of the line described by the data (3.25 s/μM) is 3.6-fold greater than the slope of the dashed line predicted from Eq. 6 (0.91 s/μM). Thus, although the current records from experiments of Fig. 3 exhibit the type of behavior that is qualitatively expected for the pseudo one-site model of Fig. 4, Scheme 2A, the results of Fig. 4, A and B, show that there is a significant quantitative discrepancy.

A possible model to explain this discrepancy is shown in Fig. 4, Scheme 2B. This scheme assumes that one DTX-I molecule can bind with a certain affinity (equilibrium dissociation constant, J₂) to an unoccupied site on a channel that is already occupied by one BPTI molecule on a different subunit. We hypothesize that these latter DTX-binding events result in a doubly occupied mixed state (S_BPTI/DTX) that functionally corresponds to the very low conductance or “nearly blocked” states within BPTI bursts that were noted above in the description of Fig. 3. Fig. 4, Scheme 2B, predicts additional lengthening of apparent BPTI bursts as compared with Scheme 2A due to the low conductance interruptions caused by S_BPTI/DTX states. In accord with Fig. 4, Scheme 2B, we assume that BPTI burst events represent dwell times in the compound state consisting of U, S_BPTI, and S_BPTI/DTX. Such burst events are defined as beginning with the appearance of the U state after dissociation of DTX from the S_DTX state, followed by prolonged sojourn in the U ↔ S_BPTI ↔ S_BPTI/DTX equilibria, and ending with a transition from the U state back to the S_DTX state. With this definition, the following relationship applies:

Eq. 7 still requires that τ_BURST is a linear function of [BPTI] as observed in Fig. 4 B; however, it predicts a larger slope due to the additional factor of (1 + [DTX]/J₂). Within this interpretive framework, Eq. 7 can be used to calculate the J₂ equilibrium constant for DTX binding to the S_BPTI state from the slope of τ_BURST vs. [BPTI] in Fig. 4 B (slope = 3.25 s/μM) and the values of 4j₁ and K₁ derived previously. This calculation yields a value of 178 ± 17 nM for J₂. This estimate for J₂ implies that DTX binds to a channel that is occupied by BPTI with an affinity (J₂ = 178 nM) that is 10.8-fold weaker than that for DTX binding to the unoccupied channel (J₁ = 16.5 nM).

Fig. 4, Scheme 2B, can also be used to explain the results of Fig. 4 A. If we define the apparent DTX substate as beginning with an entry to the S_DTX state from the U state, followed by a possible sojourn in the S_DTX ↔ S_BPTI/DTX equilibria, and ending with a transition from the S_DTX state back to the U state, then the following relation gives the predicted lifetime of this apparent DTX substate:

Comparison of Eqs. 8 and 5 shows that the apparent DTX substate, defined for Fig. 4, Scheme 2B, to include closures to the mixed S_BPTI/DTX state, is lengthened relative to the DTX substate of Scheme 2A by the K₂ equilibrium. A fit of the data of Fig. 4 A to Eq. 8 yields a value of K₂ = 17.5 ± 2.9 μM for K₂. This result implies that the affinity for BPTI binding to the DTX-occupied channel (K₂ = 17.5 μM) is 11.7-fold weaker than that for the unoccupied channel (K₁ = 1.49 μM).

Although Fig. 4, Scheme 2B, seems to explain aspects of the bursting behavior of the two-ligand system of DTX-I and BPTI, there are indications of even further complexity. Specifically, the behavior of the nearly blocked events, assumed to represent the doubly occupied S_BPTI/DTX state, appears to depend on which ligand binds first. Careful inspection of many current records reveals that such blocked states virtually always end with a transition back to the BPTI substate if they first begin with a transition from the BPTI substate. Similarly, such nearly blocked states virtually always end with a transition back to the DTX substate if they first begin with a transition from the DTX substate. This implies that the last ligand that binds to the channel is always the first ligand to come off the channel. For a simple cyclic equilibrium such as Fig. 4, Scheme 2B, the allowable routes of leaving a state should not depend on how the state was entered. This suggests that there is more than one type of doubly liganded state, S_BPTI/STX, and that the system is far more complex than that represented by the four-state equilibrium of Fig. 4, Scheme 2B. Nevertheless, we can evaluate whether Fig. 4, Scheme 2B, approximates a cyclic equilibrium by checking whether the following relationship holds:

Using the estimated values for K₁, K₂, J₁, and J₂ obtained above, the ratio of the left side of Eq. 9 is 17.5/1.49 μM = 11.7 ± 2.1, and the ratio of the right side of Eq. 9 is 178/16.5 nM = 10.8 ± 1.4. The fact that these ratios are equivalent within the propagated error of our measurements supports the hypothesis that a coupled ligand interaction like that of Fig. 4, Scheme 2B, underlies the deviation from the pseudo one-site competition model of Scheme 2A.

Another parameter that is easily measured is the time-averaged probability of residing in the DTX substate, P_S-DTX. This probability was computed by dividing the sum of the durations of all recognized DTX substates at the 60% current level by the total time of the records. This measurement was made for data pooled from 6–18 single K_Ca channels for experiments of Fig. 3 at each tested concentration of BPTI. The resulting probability was normalized to obtain the probability ratio, P_S-DTX/P  ⁰_S-DTX, by dividing P_S-DTX by the probability of the DTX substate measured at 460 nM DTX in the absence of BPTI, P  ⁰_S-DTX. The theoretical dependence on [BPTI] of this latter normalized probability can be derived for Fig. 4, Schemes 2A and 2B, as follows:

Both Schemes 2A and 2B of Fig. 4 predict an apparent “relief” of the normalized probability of the DTX substate with increasing [BPTI] that follows a titration curve given by Eq. 10. The parameter, K_App, of Eq. 10 is an apparent equilibrium constant that is given by Eq. 11 for Fig. 4, Scheme 2A, and Eq. 12 for Scheme 2B. The data points in Fig. 4 C are the values of the normalized probability (P_S-DTX/P  ⁰_S-DTX) of the DTX substate measured in our experiments at 1, 2, 4, and 8 μM BPTI. The data shows that there is ∼30% relief of the DTX substate at 8 μM BPTI. The theoretical curve, shown as a dashed line in Fig. 4 C, was calculated from Eqs. 10 and 11, where the K_App = 43.0 μM for Fig. 4, Scheme 2A, using the known values of K₁, J₁, and [DTX]. The fact that the measured data points clearly lie to the left of this curve with a significantly lower value for K_App is an independent piece of evidence that the system does not strictly obey the model of Scheme 2A. A nonlinear fit of the data in Fig. 4 C to Eq. 10 gives a best-fit value of K_App = 17.8 ± 1.7 μM. This latter estimate for K_App is close to the value of K_App = 12 ± 1.8 μM predicted for Fig. 4, Scheme 2B, using Eq. 12. The underlying reason that K_App is lower for Fig. 4, Scheme 2B, than Scheme 2A is that the reaction corresponding to the J₂ equilibrium constant provides additional relief of the 60% DTX substate (S_DTX) due to a significant level of DTX binding within the BPTI burst.

In summary, detailed analysis of single K_Ca channels shows that the ligand interaction between DTX-I and BPTI is not consistent with the pseudo one-site competition mechanism of Fig. 4, Scheme 2A. Most likely, additional state(s) of simultaneous binding of these two different Kunitz inhibitor molecules contribute to the observed behavior. At minimum, the results implicate the simultaneous binding of one DTX-I and one BPTI molecule on the tetrameric channel as represented in Fig. 4, Scheme 2B. It is also clear that there is a strongly antagonistic interaction between DTX-I and BPTI that lowers the affinity for binding of each ligand by a factor of ∼11 in the doubly occupied state. This evidence for simultaneous but antagonistic ligand occupancy by BPTI and DTX-I provides an explanation for the mechanistic quandary posed at the beginning of the results; i.e., how can the interaction kinetics of BPTI with a nonpore site on a homotetrameric complex approximate one-site behavior. Despite this satisfying resolution, our experience with the practical limitations in collecting such data sets and the formidable kinetic complexities that would be faced in attempting to analyze the interaction of BPTI and DTX-I in greater detail dissuaded us from pursuing this system. Instead, we focused next on ligand interactions between DTX-I and a homologue of the Shaker ball peptide.

The evidence for simultaneous binding of BPTI and DTX-I presented above is compatible with the idea that the binding sites for these inhibitors are off-center from the pore, as it is difficult to envision that both of these large molecules with approximate dimensions of ∼19 × 19 × 29 Å could simultaneously fit inside the central channel. As mentioned in the introduction, previous work showed that internal Ba²⁺ readily blocks the BPTI-occupied channel, and internal TEA readily blocks the DTX-occupied channel (Lucchesi and Moczydlowski, 1991). The latter findings were actually the first indication that the binding sites for BPTI/DTX-I are not located within the pore. However, we wondered whether it might be possible to detect interactions between Kunitz inhibitors and pore blockers by using a larger blocking molecule. The Shaker ball peptide and its homologues are currently the largest molecules known to block K_V channels by binding within the pore. Fig. 5 A shows the sequence of the 20-residue NH₂ terminus of the Shaker K⁺ channel of Drosophila that mediates rapid inactivation by a ball-and-chain type of mechanism (Hoshi et al., 1990; Zagotta et al., 1990). Even though most maxi-K_Ca channels do not exhibit rapid inactivation and do not have a ball peptide sequence at the NH₂ terminus, the Shaker ball peptide is known to block maxi-K_Ca channels by a pore-occlusion mechanism (Foster et al., 1992; Toro et al., 1992). This finding has been explained by the idea that K_Ca channels possess a vestigial or surrogate receptor site for the ball peptide that confers sensitivity to block by such peptides. To investigate the relationship between the binding sites for Kunitz inhibitor proteins and the internal pore region, we studied the interaction between DTX-I and a ball peptide homologue.

The sequence of the particular ball peptide homologue that we chose for this study is shown in Fig. 5 A. The BP homologue has four amino acid substitutions of the native peptide, G6V, G9V, E12K, and D13K, that result in increased hydrophobicity and net positive charge (z = +6) compared with that of native BP (z = +2). This particular BP homologue was first described by Toro et al. (1994), who showed it has ∼200-fold greater blocking affinity than the native Shaker ball peptide for a maxi-K_Ca channel from pig. Fig. 5 B shows the effect of titrating a single rat muscle K_Ca channel with BP homologue at +30 mV and 50 mM symmetrical KCl. The BP homologue induces the appearance of discrete blocking events of several seconds duration in the single channel record and the probability of this blocked state progressively increases with BP concentration over the range of 0–160 nM.

The kinetics of interaction of BP homologue with single K_Ca channels were analyzed by compiling dwell time histograms of blocked events and unblocked waiting times between adjacent blocked states. The probability histograms for these two types of events are well described by single-exponential functions. For example, Fig. 10 A shows a dwell time histogram of BP-blocked events plotted in the format of square root of events vs. log time. The binned dwell time data exhibits one predominant exponential component as shown by the superimposed fit. Fig. 6, A and B, plot mean lifetimes of the blocked state, τ_B, and the unblocked state, τ_U, as a function of the concentration of BP homologue. Fig. 6 A shows that τ_B is essentially independent of blocker concentration over the range of 10–160 nM peptide with a mean value of 1.71 ± 0.07 s (±SEM, n = 6). Fig. 6 B shows that τ_U is inversely related to the blocker concentration. The results of Figs. 6, A and B, are consistent with a simple reversible blocking reaction like that of Fig. 6, Scheme 3. Fig. 6, Scheme 3, demands the following relationships:

The mean lifetime of the BP-blocked events obtained from the data in Fig. 6 A yields a value of b₋₁ = 0.585 ± 0.025 s⁻¹ for the dissociation rate constant of BP according to Eq. 13. The data of Fig. 6 B were fit to Eq. 14 to obtain a value of b₁ = 8.17 ± 0.73 × 10⁷ s⁻¹M⁻¹ for the bimolecular association rate constant of BP. The kinetic ratio of b₋₁/b₁ gives a value of B₁ = 7.16 ± 0.71 nM for the equilibrium dissociation constant of BP under these conditions. The assumption of a single class of binding sites for BP was also tested by plotting the time-averaged probability of the unblocked U state, P_U, vs. [BP] as shown in Fig. 6 C. A nonlinear fit of these data to Eq. 15 gives a value of B₁ = 8.85 ± 0.71 nM, which closely agrees with the corresponding ratio of rate constants. Thus, the blocking behavior of the particular high-affinity BP homologue that we have studied is well described by a one-site blocking reaction. This blocking site is assumed to be located directly within the inner pore because the smaller pore-blocking molecule, TEA, inhibits ball peptide block of the maxi-K_Ca channel in a strictly competitive fashion (Toro et al., 1992). External K⁺ has also been found to relieve the block by internal BP (Foster et al., 1992; Toro et al., 1992). We have also confirmed that internal TEA decreases the association rate of BP homologue in a competitive fashion under our conditions (data not shown). This characterization of a BP homologue as a pore blocker thus provides the groundwork for investigating ligand–ligand interactions between the Kunitz inhibitor site and the BP blocking site.

If the binding site for BP overlaps with that of DTX-I, then BP-blocking events should only interrupt dwell time durations in the open state when the channel is unoccupied by DTX-I. Alternatively, if the binding site for BP is physically distinct from that of DTX-I, then BP-blocking events should also interrupt the long-lived DTX substate. The current records of Fig. 7 demonstrate that the latter alternative prevails. In this experiment, 460 nM DTX-I induced long-lived DTX-substate events that comprised ∼97% of the time-averaged activity of a single K_Ca channel. Subsequent addition of 20 nM BP to the internal side resulted in a complex pattern of blocking activity consisting of four easily recognized types of discrete blocking events that can be defined on the basis of how they begin and end. Some of the discrete blocking events are flanked at both the entry and exit transitions by the DTX substate. This type of event would be expected to occur if BP binds to the DTX-occupied channel and dissociates again before DTX-I has a chance to dissociate. A second type of discrete blocking event is initiated from a DTX substate and is terminated by a transition to a fully open current level. This may correspond to an event in which BP first binds to a DTX-occupied channel, followed by electrically silent dissociation of DTX-I, and then by dissociation of BP. A third class of discrete blocking events are initiated from a fully open state and terminated by a transition to a DTX sublevel. This may correspond to a blocking event that starts with BP binding to an unoccupied channel, followed by the electrically silent binding of DTX-I to the fully blocked channel, and terminated by dissociation of BP from the DTX-occupied channel. The fourth type of blocking event begins and ends with the fully open state. This class of events would be expected to occur when BP binds and dissociates from the channel during periods when DTX is never bound. The latter events could also occur when BP binds to an unoccupied channel and DTX-I binds and then dissociates one or more times before BP finally dissociates. If the binding of another ligand to the BP-blocked state is presumed to be electrically silent, the various classes of blocked states described above could also reflect multiple numbers of silent DTX-I– binding events. This preliminary discussion of the single-channel behavior in the presence of DTX-I and BP points to a mechanism that involves simultaneous occupancy and unordered binding of these two ligands.

To analyze this two-ligand system, we first measured the durations of the following two types of events: (a) unconditional BP-blocked events defined as any uninterrupted residence time in the fully blocked state, and (b) unconditional DTX-substate events defined as any uninterrupted residence time in the ∼60% DTX-substate current level. Samples of these two types of events containing 50–350 dwell times were manually identified and pooled from five different single channel bilayers each at seven different concentrations of BP ranging from 5 to 320 nM. The simple arithmetical mean of these samples is plotted as a function of BP concentration in Fig. 8, A and B. Fig. 8 A shows that the mean unconditional dwell time of the BP-blocked state, τ_BP, is independent of [BP] and has a value of 4.62 ± 0.13 s (±SEM, n = 7) averaged over all BP concentrations. This result is inconsistent with one-site competition. Such a mechanism (Fig. 8, Scheme 4A) predicts that the lifetime of the BP-blocked state is equal to the reciprocal of the BP dissociation rate constant, 1/b₋₁, which was previously measured as 1.71 ± 0.07 s from the mean blocked time in the absence of DTX-I (Fig. 6 A). This comparison shows that average duration of the BP-blocked state is increased by a factor of 4.62/1.71 s = 2.7 in the presence of 460 nM DTX-I vs. the absence of DTX-I. Fig. 8, Scheme 4A, also predicts that the unconditional lifetime of the DTX substate is equal to the reciprocal of the DTX-I dissociation rate constant, 1/j₋₁, and is independent of [BP]. This is inconsistent with the results of Fig. 8 B, which show that the mean unconditional dwell time of the DTX substate varies inversely with [BP]. Thus, two major predictions of Fig. 8, Scheme 4A, are clearly violated, ruling out an overlapping binding site for BP and DTX-I.

Fig. 8, Scheme 4B, is a cyclic four-state model that allows for the unordered and simultaneous binding of BP and DTX-I to distinct sites. Since there is only one state with a current level at the DTX-substate, S_DTX, this model predicts that the unconditional mean dwell time of the DTX substate is equal to the reciprocal of the sum of the rates for leaving this state:

Eq. 16 describes the observed shortening of the mean duration of the DTX substate with increasing [BP]. A nonlinear fit of the data in Fig. 8 B to Eq. 16 using the previously established value of j₋₁ = 0.0266 s⁻¹, yields 1.85 ± 0.30 × 10⁶ s⁻¹M⁻¹ for the parameter, b₂, which is the bimolecular association rate constant for binding of BP to the DTX-occupied substate. Comparison of b₂ with b₁ measured above (see Table I) shows that the association rate constant for binding of BP to the DTX-occupied channel is 44-fold slower than that for binding of BP to the unoccupied channel.

Additional information on the interaction of BP and DTX-I can be obtained from measurements of the time-averaged probability of the BP-blocked state. As described above for experiments in the absence of DTX-I, the probability that the channel is not blocked by BP follows a titration curve with an apparent dissociation constant of 8.85 ± 0.71 nM, which is an estimate of the equilibrium dissociation constant, B₁. This data and fit is reproduced (Fig. 8 C, •). We similarly computed the probability that the channel is not blocked by BP for single channel records in the presence of 460 nM DTX and is plotted (Fig. 8 C, ○). The rightward shift of this data to higher BP concentrations provides another quantitative measurement of the interaction between BP and DTX-I. The following equations derived from Fig. 8, Scheme 4B, give the form of theoretical dependence of the probability, P_{not BP}, that the channel is not in either of the two BP-blocked states, B_BP and B_BP/DTX, as a function of BP concentration:

The data of Fig. 8 C corresponding to BP titrations in the presence of 460 nM DTX-I (○) were fit to Eq. 17 to obtain a value of K_App = 45.9 ± 1.5 nM. Since J₁, B₁, and [DTX] are already known from calculations presented above, Eq. 18 can be solved to obtain an estimate for the equilibrium constant, B₂. This calculation yields B₂ = 57.0 ± 8.6 nM, which indicates that BP binds to the DTX substate with 8.0 ± 1.6-fold lower affinity than to the unoccupied channel (see Table I).

Since the values of the bimolecular rate constant b₂ and the equilibrium constant B₂ have now been obtained, the dissociation rate constant b₋₂ can be estimated from the relation, b₋₂ = b₂B₂. This calculation yields a value of b₋₂ = 0.105 ± 0.023 s⁻¹ and reveals the surprising result that BP dissociates 5.6 ± 1.2-fold more slowly from the DTX-bound channel than from the unoccupied channel (see Table I). In other words, our analysis shows that, although DTX-I lowers the affinity of BP by a factor of ∼8 at equilibrium, it actually stabilizes the BP-bound state of the channel by slowing the rate of BP dissociation. The inhibition of BP binding by DTX-I at equilibrium is solely due to the 44-fold inhibitory effect of DTX-I on the association rate of BP.

If the rate of dissociation of BP from the DTX-occupied channel is calculated to be 5.6-fold slower than that in the absence of DTX-I, why is the average dwell time of the BP-blocked state only increased by 2.7-fold over that in the absence of DTX-I (i.e., Fig. 8 A)? Fig. 8, Scheme 4B, actually has two BP-blocked states, B_BP and B_BP/DTX, that are functionally indistinguishable by unitary current. According to principles of Markov processes, the reciprocal mean dwell time of the group of these two indistinguishable states, τ_BP, is given by the sum of the two rate constants, b₋₁ and b₋₂, for leaving this group, with each of these rate constants weighted by the fraction of the group in the respective exit state, B_BP or B_BP/DTX, at equilibrium. This relationship leads to the following formula for τ_BP:

Eq. 19 predicts the experimental result that the mean dwell time of the grouped BP-blocked state is independent of BP concentration. Since numerical estimates of the rate constants, b₋₁ and b₋₂, are available from calculations already described above, the value of τ_BP = 4.63 ± 0.35 s (Fig. 8 A) may be used to compute J₃, the equilibrium dissociation constant for binding of DTX-I to the BP-blocked state, directly from Eq. 19. This calculation yields J₃ = 139 ± 34 nM. Since the equilibrium dissociation constants B₁, B₂, J₁, and J₃ have now all been independently obtained from various single-channel measurements on the system, the microscopic reversibility of Fig. 8, Scheme 4B, can be checked from the following relationship:

Using the values for J₁, J₃, B₁, and B₂ listed in Table I, the left hand side of Eq. 20, J₃/J₁, is equal to 8.4 ± 2.2, and the right hand side of Eq. 20, B₂/B₁, is equal to 8.0 ± 1.4. These ratios are equivalent within the propagated error of the measurements. This further supports the use of Fig. 8, Scheme 4B, as an appropriate model for this ligand–ligand interaction. According to this model, the 2.7-fold increase in the average lifetime of the apparent BP-blocked state (Fig. 8 A) originates from the slower rate of BP dissociation from the DTX-bound state (b₋₂) compared with the unoccupied channel (b₋₁), and the DTX-binding equilibrium constant, J₃, which determines the fraction of the grouped BP-blocked state that exists as B_BP or B_BP/DTX.

The single-channel records of Fig. 7 indicate that the BP-blocked state can either be entered from the unoccupied state of the channel or directly from the DTX substate. Fig. 8, Scheme 4B, can be used to calculate the predicted time-averaged probability of these two types of events. P_entry,U may be defined as the probability that the grouped BP-blocked state is entered directly from the unoccupied U state versus the S_DTX substate. This probability is computed as a ratio of the rate that a channel will transit from the U state to the B_BP state over the sum of the rates for transiting from state U to B_BP and from state S_DTX to B_BP/DTX. Following from Fig. 8, Scheme 4B, the numerator of this ratio is equal to the product of the association rate constant (b₁), the concentration of BP ([BP]), and the equilibrium fraction f_U of states U and S_DTX that is in the U state, where f_U = J₁/(J₁ + [DTX]). Similarly, the denominator of this ratio is equal to b₁[BP]f_U + b₂[BP]f_S-DTX, where f_S-DTX is equal to [DTX]/(J₁ + [DTX]). Algebraic simplification of the expression for this ratio leads to the following formula for P_entry,U:

Using the estimated values of b₁, b₂, and J₁ listed in Table I and [DTX] = 460 nM, P_{entry, U} as calculated from Eq. 21 is equal to 0.61 ± 0.13. Fig. 9 A plots experimentally determined values of P_{entry, U} measured from single channel records in the presence of 460 nM DTX at different BP concentrations in the range of 10–320 nM. The figure shows that the measured probability of entering the BP-blocked state from the unoccupied U state is independent of BP concentration, as expected from Eq. 21. The average measured value of P_{entry, U} = 0.58 ± 0.06 is not significantly different from the value of 0.61 ± 0.13 predicted from the relationship of Eq. 21.

The single-channel records of Fig. 7 also show that the channel can exit from the BP-blocked state either directly to the fully open state or to the DTX substate. Using similar reasoning as that described for the conditional probability of entering the BP-blocked state, Fig. 8, Scheme 4B, can be used to derive the following theoretical formula for P_{exit, U}, which is defined as the time-averaged probability for exiting the grouped BP-blocked state directly to the unoccupied U state vs. the S_DTX substate:

Substitution of the values of b₋₁, b₋₂, and J₃ from Table I, and using [DTX] = 460 nM, we find that P_{exit, U} = 0.63 ± 0.24 as calculated from Eq. 22. In Fig. 9 B, we have plotted experimental values of P_{exit, U} measured at various concentrations of BP ranging from 10 to 320 nM. Again, we see that this probability is independent of BP concentration, as expected from Eq. 22. We also find that the mean experimental value of P_{exit, U} = 0.61 ± 0.03 is indistinguishable from the predicted value of 0.63 ± 0.24.

In analyzing the preceding time-averaged transition probabilities, we have purposely measured probabilities for two types of transitions that involve the same step (U ↔ B_BP) in Fig. 8, Scheme 4B, but in opposite directions, namely the entry probability (P_{entry, U}) for moving from state U to state B_BP and the exit probability (P_{exit, U}) for moving from state B_BP to state U. If the system is in thermodynamic equilibrium, then for any step in the cycle, the probability for moving in the forward direction must be equal the probability for moving in the reverse direction (Colquhoun and Hawkes, 1995). This is a hallmark feature of purely reversible Markov processes. The theoretical expressions of Eqs. 21 and 22 can be readily demonstrated to be mathematically equivalent by algebraic manipulation and use of the microscopic reversibility relationship of Eq. 20. The actual data presented in Fig. 9, A and B, show that P_{entry, U} equals P_{exit, U} within statistical limits of uncertainty. Thus, our analysis of BP transition probabilities is fully consistent with the expectations of Fig. 8, Scheme 4B.

Thus far, our analysis has provided estimates for all of the individual rate constants of Fig. 8, Scheme 4B, except 4j₃ and j₋₃, which are the respective association and dissociation rate constants of DTX-I binding to the BP-blocked state. These parameters can be determined from the complex dwell time distribution of all BP-blocked events. Since the BP-blocked state (B_BP) is indistinguishable from the BP-blocked state occupied by DTX-I (B_BP/DTX) on the basis of unitary current, the distribution of all blocked events represents sojourns in both of these states and the probability distribution of all BP-blocked states for Fig. 8, Scheme 4B, is predicted to be a sum of two exponentials. Since these two blocked states are in direct communication and can undergo interconversion, the observed lifetimes of the two exponentials will be complex functions of j_−3, 4j₃, and [DTX], parameters that govern the interconversion of B_BP and B_BP/DTX, plus the two rate constants for leaving the grouped block state, b₋₁ and b₋₂. Theoretical expressions for the blocked state probability distribution of Fig. 8, Scheme 4B, can be derived by mathematical methods previously described in the channel literature (Pietrobon et al., 1989). The resulting sum-of-two-exponentials function is characterized by two time constants, τ₁ and τ₂, that must satisfy the following two equations:

Fig. 10 shows dwell time histograms of BP-blocked events compiled from single maxi-K_Ca channels in the absence (A) and presence (B) of 460 nM DTX-I. As discussed earlier, the histogram of BP-blocked events in the absence of DTX-I is well described by a single exponential component with a time constant of 1.73 s, which is equal to the mean dwell time of BP-blocked events measured over a wide range of BP concentration (Fig. 6 A). In contrast, the histogram compiled in the presence of DTX-I is described by a sum-of-two-exponentials with a short time constant of 0.90 s and a long time constant of 6.22 s. If average determinations of these latter two values are substituted for τ₁ and τ₂ in Eqs. 23 and 24, and other measurements (Table I) are substituted for b₋₁ and b₋₂, this leaves us with two equations in two unknowns that may be solved simultaneously for 4j₃ and j₋₃. The results of this calculation are: 4j₃ = 1.2 ± 0.38 × 10⁶ s⁻¹M⁻¹ and j₋₃ = 0.11 ± 0.22 s⁻¹. Comparison of these latter values with 4j₁ and j₋₁ (Table I), respectively, reveals that occupation of the channel by the ball peptide homologue only slows the apparent association rate of DTX-I by a factor of ∼1.3-fold and increases the dissociation rate of DTX-I by a factor of ∼4.1-fold.

Unfortunately, the use of Eqs. 23 and 24 to compute j₋₃ gives rise to a rather large propagated error on the value of this parameter, because it is calculated as the difference of two larger numbers, each with a substantial error. Thus, j₋₃ is the least certain parameter of the various rate constants determined for Fig. 8, Scheme 4B. Since the values of J₃ and 4j₃ are known with greater precision, we have also calculated j₋₃ from the relationship, j₋₃ = (J₃)(4j₃). This calculation gives j₋₃ = 0.17 ± 0.07 s⁻¹ and leads to the revised statement that BP increases the dissociation rate of DTX-I by ∼6.3-fold (rather than ∼4.1-fold).

The results presented thus far reveal significant negatively coupled interactions between BPTI, DTX-I, and a ball peptide homologue that all bind at the intracellular face of maxi-K_Ca channels. To examine whether such interactions also occur between a peptide ligand on the intracellular side of the K_Ca channel and another ligand on the extracellular side, we measured block by external TEA or external ChTX in the absence and presence of internal DTX-I. Fig. 11 A shows the effect of increasing the concentration of TEA on the external side of a single K_Ca channel from 0 to 800 μM. The progressive decrease in amplitude of the unitary current observed as a function of TEA is consistent with previous studies that have described the fast blocking effect of this cation. At the single-channel level, “fast block” is observed whenever the dissociation rate of a blocking molecule is too brief to be resolved by the recording system (e.g., Moczydlowski, 1992). If the apparent single-channel current is normalized to the unblocked current and plotted as a function of external TEA concentration, the results are well described by a single-site titration curve with a K_d of 190 ± 37 μM (Fig. 11 C). Fig. 11 B shows records of a similar experiment on a different maxi-K_Ca channel in the presence of 460 nM DTX-I. If TEA binds with lower affinity to the DTX- occupied channel, then one would have expected the DTX substate to be less sensitive to the fast blocking effect of external TEA. The normalized apparent unitary current of the DTX substate is also plotted in Fig. 11 C as a function of external [TEA]. The data is well described by a single-site titration curve with a K_d of 257 ± 10 μM. Although there is a small increase in the K_d for TEA for the DTX substate vs. the unmodified channel, the difference is not statistically significant (P = 0.1). Thus, there is no detectable binding interaction between internal DTX-I and external TEA.

A similar experiment was performed with internal DTX-I and external ChTX. ChTX is a slow-blocking K⁺ channel toxin with a binding site localized to the outer vestibule and pore entrance (MacKinnon et al., 1998). Fig. 12 (top three traces) shows current records of a single K_Ca channel in the presence of 10 nM ChTX. Previous studies have shown that the long-lived blocking events correspond to individual dwell times of ChTX on the channel and the unblocked events correspond to waiting times between blocking events (Miller et al., 1985; Anderson et al., 1988). Such data can be analyzed in the manner described above for BP to obtain dissociation (k_off) and association (k_on) rate constants for ChTX. The results of this analysis are: k_off = 0.0369 ± 0.0004 s⁻¹ and k_on = 2.77 ± 0.42 × 10⁷ s⁻¹M⁻¹. The ratio of k_off/k_on gives a calculated K_d of 1.33 ± 0.20 nM for ChTX in the absence of DTX-I. Typical current records from the same K_Ca channel in the presence of 460 nM internal DTX are also presented in Fig. 12 (bottom three traces). Although the unblocked periods between ChTX blocking events have a lower unitary amplitude due to the DTX substate, the mean durations of the blocked and unblocked dwell times are very similar to those recorded in the absence of DTX-I. Quantitative analysis of the ChTX-blocked and unblocked dwell times gives the following rate constants for ChTX in the presence of DTX-I: k_off = 0.0350 ± 0.0003 s⁻¹ and k_on = 2.89 ± 0.51 × 10⁷ s⁻¹M⁻¹. The corresponding ratio of k_off/k_on gives a value of 1.21 ± 0.21 nM for the K_d of ChTX binding to the DTX-occupied channel. Comparison of these data with those cited above for ChTX alone shows that there is no significant difference between the rate constants for ChTX binding in the absence and presence of DTX-I. Thus, there is no detectable binding interaction between internal DTX-I and external ChTX.

In this study, we used planar bilayer recording and single-channel analysis to investigate the physical relationship of binding sites for two Kunitz inhibitor proteins (BPTI and DTX-I) and a homologue of the Shaker ball peptide on the intracellular side of a maxi-K_Ca channel. At the membrane voltage (+30 mV) and intracellular Ca²⁺ concentration (200 μM) used for these experiments, the K_Ca channel gating equilibrium is poised strongly toward the open state. Under these conditions, practically all of the channel closures due to normal gating are briefer than ∼100 ms. The much slower association and dissociation kinetics of three different peptide ligands can be readily monitored by discrete changes in the unitary current record. Binding of BPTI and DTX-I produce distinct types of “substate” events and binding of BP produces discrete blocked states. The fact that each of these three ligands produces a different type of current signal upon binding to the channel allows us to identify events that are likely to arise from the simultaneous binding of two different ligands. Thus, the microscopic kinetics of complex ligand– ligand interactions are directly accessible in a single experiment. Analysis of such data allows us to discriminate a simple competitive interaction based on binding to physically overlapping sites from more complex types of interactions that involve simultaneous binding.

In this work, we presented two new lines of evidence that BPTI and DTX-I do not bind within the internal ion conduction pathway or directly occlude the pore: (a) simultaneous binding of two Kunitz inhibitors, and (b) block of the DTX-occupied channel by a ball peptide homologue.

The evidence for simultaneous binding of BPTI and DTX-I is discussed below in more detail. Because BPTI and DTX-I have approximate dimensions of 19 × 19 × 29 Å, it is a priori unlikely that two of these molecules could simultaneously occupy distinct sites within the pore. Based on current knowledge of K⁺ channel pore structure and tetrameric design (Doyle et al., 1998), our finding of a negatively coupled ligand–ligand interaction between BPTI and DTX-I is best explained by a model that involves one potentially available binding site for Kunitz inhibitors located on each of the four monomers of the K_Ca channel tetramer.

The first 20 NH₂-terminal residues of the Shaker B K⁺ channel are directly involved in an N-type inactivation process that is referred to as a “ball and chain” mechanism. Much evidence supports the hypothesis that the tethered amino terminus or “ball domain” binds to the internal pore of the open Shaker channel and directly blocks K⁺ current resulting in inactivation (Hoshi et al., 1990; Zagotta et al., 1990). Synthetic peptides corresponding to such ball domains of various voltage-gated K⁺ channels and β subunits of K⁺ channels have been found to mimic this inactivation process when added to channels that have been rendered defective in normal N-type inactivation (Demo and Yellen, 1991; Ruppersberg et al., 1991; Murrell-Lagnado and Aldrich, 1993b; Rettig et al., 1994). The primary evidence that such peptides directly block the Shaker pore is that ball peptide block is competitive with internal TEA and that external K⁺ antagonizes binding of internal ball peptide (Murrell-Lagnado and Aldrich, 1993a,b).

Shaker ball peptide also blocks other types of ion channels that are more distantly related to voltage-gated K⁺ channels, such as maxi-K_Ca and cyclic nucleotide–gated channels (Foster et al., 1992; Toro et al., 1992; Kramer et al., 1994). Although maxi-K_Ca channels do not exhibit N-type inactivation, block of these latter channels by ball peptide is functionally analogous to block of voltage-gated K⁺ channels with respect to competition by internal TEA and antagonism by external K⁺. This observation has been used to argue that maxi-K_Ca channels possess a similar kind of receptor site for BP as that of the Shaker K⁺ channel. The particular BP homologue used in this study was chosen for its high affinity conferred by replacement of two acidic residues (Glu12 and Asp13) with Lys and replacement of two Gly residues (Gly6 and Gly9) with Val (Fig. 5 A). Although the native Shaker BP does not have a well-defined conformation in solution, there is evidence that residues 3–15 tend to adopt an α-helical structure that is further stabilized by replacement of Gly residues with Val (Murrell-Lagnado and Aldrich, 1993b; Schott et al., 1998). It is possible that the BP homologue used in our experiments may form a short α helix with an extended and highly charged COOH terminus (net charge, +6). The finding that this BP homologue readily blocks the DTX-bound form of the maxi-K_Ca channel is another strong piece of evidence that DTX-I itself does not bind within the internal conduction pathway or directly occlude the pore. Otherwise, the binding of DTX-I would be expected to completely prevent the binding of BP.

By the usual criteria, the solitary interaction of BPTI or DTX-I with single K_Ca channels behaves like a typical slow blocking reaction mediated by binding to a single class of sites, with the exception that a discrete substate, rather than a fully blocked state, is produced. These criteria include single-exponential dwell time histograms, an independent relationship between mean substate duration and peptide concentration, and a reciprocal relationship between the mean intersubstate dwell time and peptide concentration. Given that BPTI and DTX-I bind outside the central pore of a homotetrameric complex, this raises the question: why does the single-channel kinetics of these two molecules apparently conform to one-site behavior when each inhibitor is looked at alone?

Our work points to an answer that involves mutually antagonistic or repulsive ligand–ligand interactions. As presented in the results, we do find a state of mixed occupancy when these ligands are added together. The most compelling and direct evidence for this latter state is the lengthening of inter-DTX periods (τ_BURST in Fig. 4 B) as a function of BPTI. This measurement is a classic test for simple binding competition between two ligands at a common site. If there is only one site in common for both of the two ligands, τ_BURST should increase according to the term {1 + ([BPTI]/K₁)}, where K₁ equals the dissociation equilibrium constant of BPTI (Eq. 6). Instead, we find that τ_BURST is a much steeper function of [BPTI] than predicted by simple competition (Fig. 4 B). This “extra lengthening” can be explained by the existence of a doubly occupied S_BPTI/DTX state. If DTX-I binds reversibly to a BPTI-occupied channel, the contribution of such events will result in extra lengthening of τ_BURST by the factor (1 + [DTX]/ J₂), where J₂ equals the equilibrium dissociation constant for DTX-I binding to the S_BPTI state (Eq. 7). An antagonistic binding interaction between BPTI and DTX-I is demonstrated by the finding that J₂ (178 nM) is 11-fold greater than the equilibrium constant for DTX-I binding alone (J₁ = 16.5 nM).

In our interpretation, the state of double occupancy by BPTI and DTX-I corresponds to short, nearly blocked (∼1 s) dwell times within inter-DTX periods, since the frequency of these events is greatly increased in the presence of DTX-I plus BPTI, as compared with BPTI alone. However, these particular blocked events commence from a BPTI substate and virtually always end with a transition back to the BTPI-substate level. The fact that these events almost never terminate by a recognizable transition directly to the DTX sublevel implies that the order of binding of the two ligands is important. This suggests that a doubly occupied state reached by binding DTX-I as the last event is not structurally equivalent to one that is reached by binding BPTI as the last event. The explanation for this phenomenon is not obvious. It is possible that this behavior is due to structural changes of the channel protein that accompany the binding of these ligands. Complementary to these findings, there is also a significant lengthening of the apparent DTX substate as a function of BPTI concentration (Fig. 4 A). This behavior can also be accounted for by a state of double occupancy with an 11-fold lower affinity for BPTI in the S_DTX state vs. the unoccupied channel.

We can imagine two different types of mechanisms that would restrict the binding of more than one Kunitz inhibitor molecule to four presumably equivalent sites on a tetrameric channel complex. One possibility is that binding of one Kunitz molecule physically interferes with the binding of another Kunitz molecule. A precedent for this latter situation occurs in the crystal structure of human β tryptase (Barbosa Pereira et al., 1998). Tryptase is a serine proteinase from mast cells that forms a ring-like homotetrameric complex in which the four active sites face a central pore. This central pore on the β-tryptase tetramer is oval shaped with dimensions of roughly 30 × 50 Å. The geometry of this complex is such that the binding of only two molecules of an atypical Kazal-type proteinase inhibitor (leech-derived-tryptase-inhibitor, LDTI) to this complex is possible. Docking analysis indicates that it is only possible to simultaneously bind two LDTI molecules to two sites located on diagonally opposite corners of the tetrameric pore (Barbosa Pereira et al., 1998). This physical arrangement is analogous to the schematic representation of the S_BPTI/DTX state shown in Fig. 4. This particular example of self-inhibition of ligand binding is also interesting with respect to our hypothesis that the COOH-terminal portion of Slowpoke K_Ca channels is homologous to the serine proteinase family (Moss et al., 1996a,b). If binding of BPTI and DTX-I is actually mediated by this COOH-terminal domain, then a tetrameric complex analogous to that formed by β tryptase could be the underlying structural basis of apparently antagonistic ligand–ligand interactions among Kunitz inhibitors.

Protein electrostatics provides another plausible explanation for a negatively coupled interaction. Both BPTI and DTX-I have a large excess of positively charged residues with a net charge of +6 and +10, respectively (Fig. 1 A). If four equivalent binding sites on a tetrameric channel complex are arranged within close proximity, repulsive electrostatic forces could be responsible for inhibiting the binding of more than one protein ligand to these sites. The magnitude and direction of such repulsive forces would depend on the distance between the sites and the ionic strength, as well as the location and distribution of charged residues on the Kunitz inhibitors. At the ionic strength of these experiments (50 mM KCl), the Debye length is ∼14 Å. Under these conditions, there could be significant through-space electrostatic interactions between Kunitz inhibitor proteins at distances up to 30 Å. The fact that states of double occupancy apparently occur more readily in the presence of a mixture of BPTI and DTX, in comparison to either inhibitor alone, could be due to the unique surface charge distribution of these two molecules (Lancelin et al., 1994) and the particular docking trajectories that they assume in the course of their respective binding reactions. Further experiments involving alterations of the charge distribution on Kunitz proteins and/or ionic strength will be required to test this electrostatic hypothesis.

The data of Figs. 5 B and 6 show that BP exhibits a high affinity blocking interaction with an equilibrium dissociation constant of B₁ = 7.2 nM. The single-exponential dwell time distribution of discrete BP-blocking events (Fig. 10 A) is consistent with a kinetically homogeneous site for BP located somewhere within the internal conduction pathway. The lifetime of this BP-blocked state has a mean value of 1.7 s in the absence of other peptide ligands (Fig. 6 A). DTX-I has a striking effect on BP-blocking kinetics. Comparison of Figs. 5 B and 7 shows that the average duration of BP-blocked states is dramatically lengthened in the presence of 460 nM DTX-I. Quantitatively, at this concentration of DTX, there is a 2.7-fold increase in the mean BP-blocked time (Fig. 8 A vs. 6 B). Intuitively, this implies that the binding of DTX-I must somehow stabilize the BP-blocked state of the K_Ca channel by slowing the rate of BP dissociation. This is exactly the opposite of what would normally be expected for an allosteric ligand– ligand interaction that is negatively coupled. If two ligands that bind to different sites exhibit mutual inhibition of binding at equilibrium, then each ligand would ordinarily be expected to enhance the dissociation rate of the other ligand. This unexpected subversion of a common kinetic assumption is perhaps the most interesting aspect of our results.

In essence, Fig. 8, Scheme 4B, is a classic model of a cyclic equilibrium describing two ligands binding to nonoverlapping sites on a protein. The only extra assumption is that the K_Ca channel is a homotetramer that can only bind one molecule of DTX-I at a time to four otherwise equivalent sites. Although this supposition invokes an extreme example of “negative cooperativity,” it is supported by previous analysis of the kinetics of DTX-I-substate events and the negatively coupled interaction observed between DTX-I and BPTI discussed above.

While the biochemical literature contains many examples of ligand–ligand interactions described by cyclic models formally equivalent to Fig. 8, Scheme 4B, the present example of DTX-I and BP is one of the few cases (if not the only case) where all eight rate constants are experimentally accessible. The routes of analysis used to obtain values for the eight rate constants of Fig. 8, Scheme 4B, are briefly summarized in Table I. The rate constants b₁, b₋₁, 4j₁, and j₋₁ are directly measurable from the single-channel kinetics of BP and DTX-I alone. The rate constant, b₂, for association of BP to the DTX substate is measured directly from the shortening of the mean duration of the DTX substate as a function of BP concentration (Eq. 16 and Fig. 8 B). The value of the equilibrium constant, B₂, for BP binding to the DTX substate is obtained from titration of the probability of the unconditional BP-blocked state in the presence of DTX-I (Eq. 18 and Fig. 8 C). The dissociation rate constant b₋₂ is directly obtained from the known values of B₂ and b₂. The equilibrium constant J₃ for binding of DTX-I to the BP-blocked state is independently obtained from the unconditional mean dwell time of the BP-blocked state in the presence of DTX-I (Eq. 19 and Fig. 8 A). The rate constants 4j₃ and j₋₃ can be extracted from the measured time constants of the biexponential distribution of BP-blocked dwell times (Eqs. 23 and 24, and Fig. 10 B). A similar, but more precise estimate of j₋₃ can be directly calculated from the relationship j₋₃ = (J₃)(4j₃).

Mutual compatibility of the various measured parameters with Fig. 8, Scheme 4B, can be checked in three different ways: (a) the derived values of J₁, J₃, B₁, and B₂ satisfy the equilibrium relationship for microscopic reversibility (Eq. 20), (b) the derived rate and equilibrium parameters predict the measured conditional probabilities for entry into and exit from the BP-blocked state (Eqs. 21 and Eq. 22, and Fig. 9), (c) the measured rate constants satisfy the following kinetic microscopic reversibility relationship:

Eq. 25 is simply the well known thermodynamic rule that the product of rate constants for traveling around a reversible cycle in one direction must equal the same product in the other direction (Colquhoun and Hawkes, 1995). Using the best estimates of Table I, the left side of Eq. 25 is equal to 2.74 × 10¹¹ s⁻⁴M⁻², and the right hand side is equal to 2.96 × 10¹¹ s⁻⁴M⁻². The discrepancy between these two values is only 8%, which is quite low considering the errors associated with the individual determinations. Thus, by at least three criteria, the system consisting of single K_Ca channels and two ligands (BP and DTX-I) is very well described by Fig. 8, Scheme 4B, with the values for all eight rate constants summarized in Table I.

Aside from the aesthetics of such calculations, it is pertinent to ask what new insights have been gained from this strenuous exercise in single-channel analysis. The essential revelations are summarized in Fig. 13. This cartoon illustrates how the kinetic results may be interpreted in the context of a plausible structural model of the BP/DTX-I interaction. The cartoon depicts the BP homologue as a pore blocker binding somewhere inside the internal ion conduction pathway. The Kunitz protein, DTX-I, is depicted as binding to an external site on one monomer that is close to, but not occluding, the pore entrance. Considering the likelihood of protein–protein electrostatic interactions discussed above, we propose that the unusual asymmetry in rate effects between the two ligands arises from the positioning of these ligands on a channel architecture. For example, one could imagine that the 44-fold decrease in association rate of BP binding to the DTX- occupied channel could be the result of a strong, transient repulsive interaction experienced by the positively charged BP molecule (net charge, +6) as it enters the pore by passing within close proximity to the positively charged DTX-I molecule (net charge, +10). If the BP molecule is located at some distance deep inside the inner mouth of the channel in its bound state, then DTX-I would not have to pass as close to BP in the course of its own association step and, thus, it would not experience a similar repulsive barrier to association with the channel. This asymmetry in the trajectories of the respective association and docking steps would then explain the smaller antagonistic effect for DTX association to the BP-occupied state (1.3-fold reduction) versus the 44-fold reduction in the association rate for BP binding to the DTX-occupied state.

Similar electrostatic considerations could also apply to the ligand dissociation reactions. If the BP-binding site is located some distance inside the pore, then BP may transiently experience strong electrostatic repulsion in passing by the bound DTX-I molecule as it attempts to dissociate from its site and return to the solution. Thus, the charged DTX-I molecule could create a local electrostatic barrier to BP dissociation at the mouth of the channel that would lower the effective BP dissociation rate constant and essentially act to lock BP inside the channel. In contrast, DTX bound to a site outside the pore would not be confined by such a local electrostatic barrier and its dissociation rate would simply be enhanced by the positively charged BP molecule located some distance inside the pore.

Devising an unambiguous test of this electrostatic hypothesis is difficult, but such a model does make definite predictions for relatively simple experimental manipulations such as variation of ionic strength and decreasing the net charge of the ball peptide. It should also be noted that there is an alternative, but less parsimonious, explanation of the effects. One could argue that the observed rate effects are due to particular conformational changes in the structure of the channel protein induced by binding of the Kunitz peptides. Since binding of DTX-I does induce some kind of rapid flickering of the open channel, which is the basis of the apparent subconductance events (Moss and Moczydlowski, 1996), it is possible that this effect contributes to observed changes in BP rate constants. However, it is not clear why such rapid conformational changes of the pore should not also be sensed by smaller and less charged blocking molecules, Ba²⁺ and TEA⁺. The fact that these latter blockers of the internal conduction pathway do not exhibit a detectable ligand interaction with BPTI or DTX-I (Lucchesi and Moczydlowski, 1991) suggests to us that the electrostatic mechanism for the antagonism between DTX-I and BP is more likely to underlie this phenomenon.

The theoretical basis of free energy changes associated with allosteric binding of two different ligands to two distinct sites on a protein molecule has been lucidly analyzed by Gregorio Weber (Weber, 1975, 1992). According to this formulation, cyclic reactions such as Fig. 8, Scheme 4B, are characterized by a coupling free energy, ΔΔG⁰_coup, defined as the standard free energy for binding a ligand X to a protein that already has another ligand Y bound, ΔG⁰_X(Y), minus the standard free energy for binding the same ligand X to the unliganded protein, ΔG⁰_X. Conservation of energy requires that the coupling free energies for the two possible pairs of binding energies are equivalent. For the two ligands of Fig. 8, Scheme 4B, these energy-coupling relationships may be expressed as:

The standard free energy, ΔG⁰, for a reversible ligand dissociation reaction with an equilibrium constant, K_d, is equal to RTlnK_d, where R is the gas constant, T is absolute temperature, and K_d is the equilibrium dissociation constant. Using the respective values for K_d given as B₂, B₁, J₂, J₁ in Table I, the coupling free energy calculated for both the left and right hand side of Eq. 26 is equal to +1.3 kcal/mol. The absolute errors on these calculated free energies are ∼0.1 kcal/mol. The fact that the two ways of calculating the coupling free energy in Eq. 26 yield identical results is simply a verification of microscopic reversibility discussed above. A positive value of coupling free energy corresponds to a negative or mutually destabilizing ligand–ligand interaction, whereas a negative coupling free energy corresponds to a positive or mutually stabilizing ligand– ligand interaction.

The value of +1.3 kcal/mol found for the antagonistic interaction between DTX-I and BP binding to the maxi-K_Ca channel falls within the range of ±2 kcal/mol typical of the magnitude of coupling free energies that have been previously measured for various ligand– ligand interactions on proteins (Weber, 1975, 1992). As noted above, our study is rather unique in that all eight rate constants for this ligand interaction have been experimentally determined. This allows us to analyze the relationships that govern the free energy coupling of the transition states for the various ligand-binding steps. The formal relationship between the equilibrium coupling free energy, ΔΔG⁰_coup, and the change in transition state energy for the respective dissociation and association reactions for BP and DTX-I can be readily derived by substituting the appropriate ratio of rate constants for the various standard free energies of Eq. 26 [e.g., ΔG⁰_BP = RTlnB₁ = RTln(b₋₁/b₁)]. Rearrangement of this substituted form of Eq. 26 leads to:

In Eq. 27, each of the respective changes for dissociation or association transition state energies, ΔΔG^‡_diss or ΔΔG^‡_ass, is defined as the relative change in free energy for the respective reaction of a given ligand for the doubly occupied protein complex relative to the singly occupied complex. Thus, we have the following definitions:

A positive value of the ΔΔG^‡_diss means that the other ligand exerts an antagonistic (or repulsive) influence on dissociation rate of the reference ligand, whereas a negative value for this parameter means that the other ligand exerts a stabilizing influence and slows the dissociation rate of the reference ligand. Since a decrease in association rate constant is an antagonistic effect, the opposite interpretations apply to the sign of ΔΔG^‡_ass: a positive value is stabilizing, a negative value is destabilizing.

Fig. 14 depicts the respective change in transition state energies calculated from the rate constants of Table I in the form of a bar graph where the energies for the two ligands are plotted according to Eq. 27. This graph shows that the change in transition state energies for the two ligands appropriately subtract from each other to give a value very close to the equilibrium-coupling free energy of +1.3 kcal/mol, as expected from Eq. 27 and energy conservation. Fig. 14 also highlights the asymmetry in the change in transition state energies for the two ligands. DTX-I has a positive (destabilizing) change in transition state energy for dissociation (+1.1 kcal/mol), whereas BP has a negative (stabilizing) value for this parameter (−1.1 kcal/mol). Both of the calculated changes in transition state energies for association are negative (destabilizing), but ΔΔG^‡_ass for BP (−2.2 kcal/mol) is much stronger than that for DTX (−0.2 kcal/mol). As discussed above, these asymmetries in transition state energies provide important mechanistic insights. For purely electrostatic interactions, symmetrical effects between two ligands that bind in close proximity would be expected for situations where the binding trajectories essentially mirror each other on the protein surface. For a tetramer, such a situation would be expected for binding sites that lie in a plane of symmetry. The observation of highly asymmetric values for ΔΔG^‡_diss and ΔΔG^‡_ass in the present system is evidence for asymmetric pathways of ligand interaction such as that depicted in Fig. 13.

The molecular basis for apparent one-site binding kinetics of the positively charged Kunitz inhibitors, BPTI or DTX-I, to tetrameric maxi-K_Ca channels was explored by single-channel analysis of recordings carried out in the simultaneous presence of both ligands. Evidence was obtained for the existence of mixed states containing one DTX-I and one BPTI molecule simultaneously bound to the channel. Measurement of equilibrium binding affinity showed that bound DTX-I lowers the affinity for BPTI by 11-fold, and vice versa. Observations of the discrete block of the DTX-occupied channel by a potent BP homologue established that the binding site for DTX-I does not physically overlap with the site of internal entrance of BP to the pore. Analysis of single-channel records taken in the presence of DTX-I and BP revealed negative allosteric coupling between the binding of these two ligands to the K_Ca channel. In this interaction, the binding of one ligand decreases the equilibrium affinity of the other by eightfold, and vice versa. Mutually repulsive electrostatic interactions between basic peptide ligands binding within close proximity on the surface of a channel protein is a plausible explanation for the apparent negative cooperativity of ligand interactions between BPTI, DTX-I, and BP in this system. An interesting asymmetry in rate effects for the interaction between DTX-I and BP may be caused by an electrostatic barrier created by bound DTX-I at the channel's inner entryway that hinders both the association and dissociation reactions of BP.

BP

Shaker ball peptide homolog

BPTI

bovine pancreatic trypsin inhibitor

ChTX

charybdotoxin

DTX-I

Toxin I of Dendroaspis polylepis

maxi-K_Ca

large conductance Ca-activated K⁺ channel

TEA

tetraethylammonium