Two general model types have been proposed to account for the voltage dependence of ion channel block by charged blockers. One model assumes that the blocker itself traverses (a portion of) the transmembrane electric field to reach its binding site in the pore (

Woodhull, 1973), and the voltage dependence is thus a property intrinsic to blocker binding. Characteristic of this intrinsic model is that the fraction of current not blocked (I/I

_{o}) by a nonpermeant blocker is described over the entire membrane voltage range by a single Boltzmann function. Thus, for a positively charged intracellular blocker (, gray curve):

where

^{app}K

_{d} is the apparent equilibrium dissociation constant for the blocker-binding reaction in the absence of a membrane potential, [B] is the concentration of blocker, Z is the effective valence (sometimes denoted as zδ), V is the membrane voltage, and F, R, and T have their usual meaning. As shown here, the voltage dependence of CNGA1 channel block by extracellular PhTx can be well accounted for by this type of mechanism, as equilibrium channel block varies with voltage from none to complete block according to single Boltzmann functions ( and ).

The other model posits that the blocker itself does not bind within the electric field, but that the apparent voltage dependence reflects phenomena extrinsic to blocker binding per se, namely, displacement by the blocker of permeant ions across the field (

Armstrong, 1971). We show below that block by intracellular QAs too bulky to enter the channel’s selectivity filter (

Goulding et al., 1993) is entirely accounted for by the extrinsic model. A prediction of this ion displacement model is that the voltage dependence of blocker–channel interaction should vanish if permeant ions were removed, a prediction not testable electrophysiologically because removing all permeant ions would abolish current. Fortunately, an alternative approach exists, as argued with the following example. According to the model, an intracellular cationic blocker binds at a site, outside the field, normally occupied by a permeant cation about to enter the electric field. Increasingly, strong depolarizations will lower the permeant ion occupancy of that site until eventually a blocker could occupy it without encountering and displacing a permeant ion. In the limit when blocker binding is no longer coupled to the movement of permeant ions, channel block must lose its extrinsic voltage dependence. As a consequence, the blocking curve characteristically deviates from a pure Boltzmann function at extreme positive voltages to reach a nonzero current plateau (, black curve). We now proceed to a quantitative examination of these two types of voltage-dependent block using PhTx and QAs as examples.

Analysis of PhTx block with the intrinsic voltage dependence model

PhTx blocks the channel from either side of the membrane in a strongly voltage-dependent manner ( and ). The voltage-dependent blocking curves for extracellular () or intracellular PhTx () are well described by true Boltzmann functions, so that current vanishes completely at sufficiently strong potentials of the polarity that favors blocker binding. Such behavior is expected for a blocker that binds within the electric field.

In kinetic terms, block by extracellular PhTx () appears at first sight to be a simple one-step bimolecular reaction. Although voltage jump–induced blocking and unblocking transients follow single-exponential time courses, the ratio k_{off}/k_{on} at 0 mV is only 34% of the apparent equilibrium dissociation constant ^{app}K_{d} (0 mV), and z_{on} + z_{off} is 75% of the effective valence of the voltage-sensitive apparent equilibrium ^{app}K_{d}. Additionally, the voltage dependence associated with ^{app}K_{d} is entirely attributable to k_{off}, whereas k_{on} is essentially voltage independent. As we will discuss below, these phenomena betray more complex kinetics.

The characteristics of CNGA1 block by intracellular PhTx are similar to those of block by extracellular PhTx. As in the case of block by extracellular PhTx, the sum of z

_{off} (1.60) and z

_{on} (0.23) from kinetic measurements of intracellular PhTx block is also noticeably smaller than the Z value (2.67) obtained at equilibrium. This discrepancy indicates that the blocking process is not a one-step bimolecular reaction. At least one additional transition must exist, which we have not been able to measure directly. For example, there may be a blocked state with such low affinity that it is insignificantly populated at the blocker concentrations we used. Higher PhTx concentrations might reveal a potential low affinity state, but this is impractical given the high cost of PhTx. An alternative approach is to raise the apparent blocker affinity by lowering the permeant ion concentration to reduce competition and/or trans knock-off by permeant ions (

Armstrong and Binstock, 1965). Having lowered the Na

^{+} concentration on both sides of the membrane from 130 to 30 mM, we find that at negative potentials (which minimize intrinsic voltage-sensitive blocker binding), PhTx indeed blocks current in a dose-dependent but voltage-independent manner (). Therefore, binding of intracellular PhTx to the channel evidently produces at least two blocked states: a voltage-dependent state plus a voltage-independent state.

The two blocked states (ChB

_{1} and ChB

_{2}) could in principle be formed via either sequential or parallel transitions (). For either model, the fraction of current not blocked is given by the ratio of the nonblocked state to the sum of all states:

For the sequential model (), the equilibrium constants in the absence of an electric field are defined as K

_{1} = [Ch][B]/[ChB

_{1}] and K

_{2} = [ChB

_{1}]/[ChB

_{2}]. In the presence of an applied electric field, the fraction of current not blocked is then given by:

where Z

_{1} and Z

_{2} are the effective valences associated with K

_{1} and K

_{2}, respectively. For the parallel model (), the fraction of current not blocked is given by:

where K

_{1} = [Ch][B]/[ChB

_{1}], K

^{’}_{2} = [Ch][B]/[ChB

_{2}].

Eqs. 3 and

4 are of identical form, but K

_{2} in the sequential model (

Eq. 3) and K

^{’}_{2} in the parallel model (

Eq. 4) have different physical meanings. The ΔG of the blocked states with respect to the unblocked state is ΔG

_{ChB2} = −RT × ln (K

_{1}K

_{2}) for the sequential model and ΔG

_{ChB2} = −RT × ln K

^{’}_{2} in the parallel model; for both models, ΔG

_{ChB1} = −RT × ln K

_{1}. As discussed in the next section, a sequential model has important practical implications for attempts to deduce channel gate properties from channel block kinetics. For this reason, we will discuss in detail the sequential two-step model as we analyze block by intracellular PhTx (under low Na

^{+} conditions to reveal the low affinity blocker-binding step).

A blocker is likely to transiently contact parts of the pore with some affinity on its way to the energetically most stable site. That is, a realistic blocking scheme likely consists of sequential interactions (regardless of whether there are additional parallel blocking transitions). In principle, the first blocked state may reflect the engagement of a blocker with a part of the pore as shallow at its innermost (or outermost) end. Indeed, as previously demonstrated with inward-rectifier K

^{+} channels (

Shin and Lu, 2005;

Shin et al., 2005;

Xu et al., 2009), a sequential model is physically plausible given the significant depth of ion channel pores.

We fitted the steady-state blocking curves in with

Eq. 3 to obtain equilibrium constants K

_{1} = 1.04 ± 0.03 × 10

^{−5} M and K

_{2} = 2.03 ± 0.14 × 10

^{−3}, and valence Z

_{2} = 2.08 ± 0.04. Z

_{1} was set to 0 because formation of the first (shallow) blocked state is of little voltage sensitivity (witness the voltage-insensitive but dose-dependent asymptotes at extreme negative voltages). As for the individual rate constants at 0 mV, analysis of the blocking and unblocking kinetics () yields k

_{on} = 8.42 × 10

^{8} M

^{−1}s

^{−1} with z

_{on} = 0.14, and k

_{off} = 27 s

^{−1} with z

_{off} = 1.10. As a first approximation, the experimental blocking rate k

_{on} primarily reflects k

_{1} in the model (and thus z

_{1} = z

_{on}), and the model’s k

_{–1} can then be calculated as K

_{1}k

_{1} = 8.76 × 10

^{3} s

^{−1}. This value of k

_{–1} is ~300-fold larger than the experimental unblocking rate k

_{off}. The latter must therefore approximate the model’s OFF-rate–limiting k

_{–2} (and z

_{off} must approximate z

_{–2}). Finally, we can estimate the model’s k

_{2} as k

_{–2}/K

_{2} = 1.33 × 10

^{4} s

^{−1} with z

_{2} as Z

_{2} − z

_{–2} = 0.98. Our kinetic analysis in terms of a sequential two-step blocking model shows that most of the voltage sensitivity of channel block by intracellular PhTx resides in the transition between the two blocked states, with k

_{2} and k

_{–2} exhibiting comparable voltage dependence.

According to

Eq. 3 derived for the sequential two-step blocking model, the apparent equilibrium constant at 0 mV is given by:

Because in the present example k

_{–2} (= 27 s

^{−1})

k

_{2} (= 1.3 × 10

^{4} s

^{−1}),

^{app}K

_{d} reduces to

and because in this instance the values of k

_{–1} and k

_{2} differ by less than twofold,

^{app}K

_{d} is approximately equal to k

_{–2}/k

_{1}. As current is blocked from the instant the first blocked state is reached, k

_{1} must effectively determine the experimental ON rate of channel block. Given the low PhTx concentrations we used, the pseudo first-order rate constants k

_{1}[B] were resolvable by our recording system. As for k

_{–2}, we saw above that it is vastly slower than k

_{–1} in this example; therefore, it effectively determines the experimental OFF rate for recovery from PhTx block. The finding that in this instance the equilibrium

^{app}K

_{d} approximates the experimental k

_{off}/k

_{on} ratio is then merely a coincidence and, by no means, indicative of a one-step process. A clear indication that the blocking reaction involves more than one step is the discrepancy between the equilibrium valence, Z, and the sum of the valences z

_{on} and z

_{off} associated with kinetic constants k

_{on} and k

_{off}. Not only for intracellular, but also in the case of extracellular PhTx block, is experimental k

_{off} voltage dependent and k

_{on} independent. Within the theoretical framework of a simple one-step blocking model, almost the entire voltage dependence of equilibrium block would have to be attributed to the sole dissociation reaction—an unlikely scheme. In the sequential two-step model, however, blocker binding to and unbinding from the first (shallow) site are voltage insensitive, but the transitions to and from the deep site are sensitive in such a manner that the voltage dependence is nearly evenly allocated between the forward and backward reactions—a physically more plausible scheme. Even though experimental demonstration has thus far succeeded only for intracellular PhTx, it is likely that extracellular PhTx also interacts with the channel in more than one step as it penetrates into the selectivity filter.

Implications of a sequential blocking mechanism for attempts to probe channel gating by analyzing blocking kinetics

Mindful of the possibility that channel-blocking kinetics encompass several binding steps, we now examine how the existence of multiple, sequential blocked states may preclude inferences being drawn regarding channel-gating behavior from the dependence of blocking kinetics on channel open probability. The apparent rate of channel inhibition caused by the binding of an open-channel blocker (or by, for example, a cysteine residue modifier) that travels through the channel gate region is expected to vary with the gating state. Such variation has been successfully exploited to delineate the channel gate, e.g., in voltage-gated K

^{+} channels (

Liu et al., 1997). To our surprise, the apparent rate of channel block by either extracellular or intracellular PhTx exhibits little dependence on channel open probability, even though block is strongly voltage dependent in both cases. This unexpected finding may help us stipulate the necessary conditions under which properties of the channel gate may be inferred from the dependence of blocking kinetics on gating status.

Kinetics of CNGA1 channel block or unblock are essentially unaltered by lowering the cGMP concentration from 2 mM (saturating) to 20 µM (). This holds for both extracellular and intracellular PhTx. There are two possible explanations for these unexpected results. One possibility is that the gate location, e.g., in the selectivity filter (

Sun et al., 1996;

Bucossi et al., 1997;

Becchetti et al., 1999;

Becchetti and Roncaglia, 2000;

Liu and Siegelbaum, 2000;

Flynn and Zagotta, 2001;

Contreras et al., 2008), is inaccessible to PhTx from either side of the membrane. This would mean, in the present example, that the ~20-Å long spermine portion of the PhTx molecule cannot reach deep into the ~12-Å long filter, even though spermine itself readily permeates the channel from either direction. This possibility appears unlikely.

Alternatively, constancy of the apparent channel-blocking rate regardless of ligand concentration may reflect the fact that the rate-limiting transition in the PhTx-blocking sequence is not affected by channel gating. In the initial step of the blocking sequence, PhTx associates with the channel, docking at a site which may, in principle, be as shallow as the innermost part of the pore. It then travels deeper along the pore to form an additional blocked state (or more). If a gate were located between the shallow and the deep sites (or at the deep site), formation of only the deep, not the shallow, blocked state would depend on the gate’s open probability. The second-order rate constant for PhTx associating with the channel to form the first blocked state is very large and arguably near diffusion limited: k_{on} = 8.42 × 10^{8} M^{−1}s^{−1} in symmetric 30 mM Na^{+} or 4.23 × 10^{7} M^{−1}s^{−1} in 130 mM Na^{+}. However, because we used very low PhTx concentrations to achieve experimentally resolvable ON rates, the resulting pseudo first-order rate constant (k_{1}[PhTx]) for formation of the first blocked state in fact became rate limiting, being much lower than the (first-order) rate constant k_{2} for PhTx transit from the shallow to the deep binding site. As a result, any gating effect on k_{2} would be obscured. Although k_{2} might well be sensitive to the gating state, the reverse rate constant k_{–2} clearly (in the present example) is not because recovery from PhTx block, which is rate limited by k_{–2} (see above), is insensitive to cGMP concentration. Now, if only k_{2} is gating sensitive and k_{–2} is not, this sensitivity will still be reflected in K_{2} (). Indeed, lowering cGMP from 2 mM to 20 µM, which decreases the macroscopic current (that is, P_{o}) by 8.6 ± 1.2–fold (*n* = 8), causes a 7.5-fold increase in K_{2} for PhTx block (). Therefore, insensitivity of the apparent PhTx-blocking rate to cGMP concentration may simply be a consequence of the multistep character of channel block. In any case, whereas probing the gate by examining the kinetics of channel block (or cysteine modification) is conceptually straightforward for a single-step blocking reaction, this approach is possible for a reaction involving multiple sequential steps only if the observed effect is rate limited by a gating-dependent transition. The mere demonstration that a second-order association rate constant k_{on} is (nearly) diffusion limited does not ensure that the association step is not in fact rate limiting under the actual experimental conditions.

Analysis of QA block with the extrinsic voltage dependence model

We next discuss extrinsic voltage dependence of channel block by QAs in the framework of an ion displacement model. The most economical version of this model exhibiting the features described in Results () considers three states: one blocked plus two states of the ion conduction cycle that are blocker free (). In this model, the narrow external part of the channel (Ch) can only be occupied by permeant ions, whereas the wide internal part can accommodate either a permeant (e.g., Na^{+}) or a blocking (B) ion. The transmembrane electric field exists exclusively across the narrow part of the pore. The two states of the steady-state Na^{+} conduction cycle are represented on the left. Extracellular, but not intracellular, Na^{+} must traverse the narrow part of the pore to reach the internal site in a process that is voltage sensitive. The upper blocking transition is voltage independent as the empty blocker site is outside the field. The left and upper transitions together represent competition between Na^{+} and intracellular blocker for the internal site. In the limit when the empty species (Ch) vanishes and Na^{+} and blocker move in a concerted manner, these two processes collapse into a single (electrostatic) displacement reaction as described by the lower transition. This “concerted” blocking transition is voltage sensitive as the blocker site is already occupied by a permeant Na^{+} ion that must then enter the field.

For this simple three-state model, the fraction of current not blocked is given by:

The apparent equilibrium blocker dissociation constant is

where the equilibrium constants for the blocker-binding transitions without and with concomitant Na

^{+} displacement are K

_{B} = [Ch][B]/[ChB] and K

_{B-Na} = [ChNa][B]/([ChB][Na

^{+}]), respectively (

Spassova and Lu, 1998).

Eq. 7.1 parses how the two blocking transitions contribute to the overall blocking process, for the experimentally simplest case where the Na

^{+} concentrations on both sides of the membrane are equal.

At sufficiently hyperpolarized voltages where

[Na

^{+}] >> K

_{B},

Eq. 7 reduces to:

an equation dominated by the voltage-dependent blocking transition that involves Na

^{+} displacement. Conversely, at highly depolarized potentials,

approaches 0 and

Eq. 7 reduces to

which describes voltage-insensitive block not accompanied by Na

^{+} displacement.

Eq. 9 accounts for the voltage-independent plateau at depolarized potentials whose amplitude varies with blocker concentration. Indeed, the blocking curve for all seven intracellular QAs tested becomes voltage independent at strongly depolarized potentials. Using bis-QA

_{C10} as an example, we show below that this ion displacement model accounts well for channel block by QAs that are too bulky to enter the selectivity filter (compare

Huang et al., 2000).

As shown in for bis-QA

_{C10}, the blocking curve clearly deviates from a Boltzmann function (dotted curve) at depolarized potentials but is well fitted by the ion displacement model (solid curve). As already discussed for PhTx, the voltage-insensitive block at very negative potentials points to an additional blocked state. A detailed treatment of this additional blocked state will be presented in the next section. To limit the present discussion to the properties of a three-state ion displacement model, we simply scaled the fraction of current not blocked at hyperpolarized potentials () to unity ().

Eq. 7 derived for our three-state model makes the following experimental prediction: increasing the blocker concentration would not only shift the blocking curve to the left but also lower the plateau seen at positive potentials. This is consistent with our experimental observations (; compare black with blue symbols). Mathematically, at higher blocker concentrations, half-maximal block will occur at a lower voltage (

Eq. 7) and, consequently, the blocking curve shifts leftward. Thermodynamically, the total energy needed to achieve a given amount of block remains unchanged, and thus the relative energy contribution of electric or chemical origin varies reciprocally. At highly depolarized potentials,

Eq. 7 reduces to

Eq. 9, meaning that the extent of channel block in this voltage range is independent of voltage and determined solely by the blocker concentration. Therefore, the plateau of the blocking curve at very depolarized potentials decreases with increasing blocker concentration. These predictions make intuitive sense in that strong depolarization reduces the permeant ion occupancy at the shared site, so the blocker will be able to occupy the site without the need to displace permeant Na

^{+}. In the limit, binding becomes a standard bimolecular chemical reaction, where the extent of block increases with blocker concentration.

Another prediction is that decreasing the permeant ion concentration will shift the blocking curve leftward without affecting the plateau level at positive potentials, as in fact we observed when we lowered the Na

^{+} concentration from 130 to 65 mM (; compare filled with open symbols).

Eq. 7 shows that for a given blocker concentration, at lower Na

^{+} concentrations lower voltage suffices to produce 50% block because less electric energy is needed to generate the same probability of vacancy for the blocker. It follows that block of a channel with lower affinity for permeant ions should lose its extrinsic voltage dependence at lower voltages.

Eq. 9, on the other hand, indicates that the extent of channel block at extreme positive potentials is independent of Na

^{+} concentration because at sufficiently strong depolarization, Na

^{+} occupancy will vanish.

Ion displacement model with sequential blocked states for QA block

When we analyzed (in the above section) intracellular QA block in the framework of an ion displacement model, we deferred discussion of voltage-independent block at very negative potentials. To account for that additional feature, we now extend the ion displacement model to include a sequential blocking scheme, analogous to the one we formulated for PhTx (). Inserting an extra blocked state in each of the two blocking transitions raises the number of states in the model to five. The additional blocked states ChB

_{1} and ChNaB

_{1} represent a blocker bound at a shallow site without and with a Na

^{+} at the deep site, respectively. For simplicity, we assume that the equilibrium between the shallow blocked state and the unblocked state is characterized by the same equilibrium dissociation constant K

_{B1}. K

_{B2-Na} and K

_{B2} are the equilibrium constants for the two subsequent transitions to the deep blocked state (ChB

_{2}) with and without Na

^{+} displacement, respectively. For this five-state model, the fraction of current not blocked is given by:

The equilibrium constant for blocker binding to the shallow site is:

Blocker movement between the shallow and the deep site is governed by:

for blocker partition to the empty deep site and

for blocker binding to the deep site occupied by Na

^{+}. The latter step, which displaces Na

^{+}, is the only one assumed to be voltage dependent; this is accounted for by the exponential term in

Eq. 10.

At sufficiently hyperpolarized voltages where

approaches 0,

Eq. 10 reduces to a simple voltage-independent blocking mechanism:

Eq. 14 accounts for voltage-independent block at hyperpolarized potentials. At these potentials, the block primarily involves the two left horizontal transitions, and the fraction of current not blocked is a function of blocker concentration.

At extreme positive voltages, the term

approaches 0 and

Eq. 10 reduces to

where the block again becomes voltage independent. At depolarized potentials, the block primarily involves the two upper transitions, and

Eq. 15 thus contains both K

_{B1} and K

_{B2}. The curves superimposed on the blocking curves of all seven QAs (, , and ) are fits of

Eq. 10 (all resulting parameters are listed in ). Thus, the five-state model fully accounts for QA block.

| **Table I.**Five-state ion displacement model parameters for QA block |

A brief comment is in order regarding how an additional, shallow blocking site may affect extrinsic voltage dependence of channel block. Recall that a reduction in ion occupancy due to a lowered permeant ion concentration shifts the voltage-dependent blocking curve in the hyperpolarized direction. Consequently, the blocking process loses its apparent voltage dependence at less depolarized potentials (). As explained below, this phenomenon can also be caused by the binding of a blocker at the shallow site. Blocker binding at the shallow site (not shared with permeant ions) would prevent intracellular ions from accessing the deep site (shared by both permeant ions and blocker) (). In this case, refilling the deep site with permeant ions from the intracellular side would no longer be possible. In addition, at very depolarized potentials, the deep site could not be effectively refilled from the extracellular side because depolarization decreases the probability of extracellular ions moving across the selectivity filter to reach the site. Thus, the existence of a shallow blocking site could lower the permeant ion occupancy of the deep site, causing the extrinsic voltage sensitivity to vanish at less depolarized potentials.

In conclusion, given that an ion channel pore has considerable depth, a blocker likely blocks the pore in sequential steps. Experimentally, at appropriate blocker concentrations, association with the shallow site may be the rate-limiting step of current block, even if the steady-state occupancy of that site is negligible. Consequently, a sequential binding mechanism could preclude inferences being drawn regarding channel-gating behavior from the dependence of blocking kinetics on channel open probability. Penetration of blocker into the pore can exhibit voltage sensitivity in two possible ways with experimentally distinguishable characteristics. In one case, the charged blocker itself enters the transmembrane electric field (with and without displacing permeant ions), and the observed voltage sensitivity is an intrinsic property of the blocker. Such voltage sensitivity arising from a direct interaction of the charged blocker with the electric field follows a Boltzmann function. Alternatively, the blocker does not bind within the electric field, and its binding indirectly derives voltage sensitivity solely from the concurrent movement of permeant ions across the electric field. Such acquired voltage sensitivity vanishes at voltages where displaceable permeant ions no longer occupy the blocker binding site, and therefore does not follow a Boltzmann function. This striking feature is the electrophysiological hallmark of the extrinsic type of voltage-dependent ion channel block.