## Abstract

The properties of the small fraction of tetrodotoxin (TTX)-sensitive Na channels that remain open in the steady state were studied in internally dialyzed voltage clamped squid giant axons. The observed Ussing flux ratio exponent (*n*′) of 0.97 ± 0.03 (calculated from simultaneous measurements of TTX-sensitive current and ^{22}Na efflux) and nonindependent behavior of Na current at high internal [Na] are explained by a one-site (“*1s*”) permeation model characterized by a single effective binding site within the channel pore in equilibrium with internal Na ions (apparent equilibrium dissociation constant *K*_{Nai}(0) = 0.61 ± 0.08 M). Steady-state open probability of the TTX-sensitive channels can be modeled by the product *p*_{a}*p*_{∞}, where *p*_{a} represents voltage-dependent activation described by a Boltzmann distribution with midpoint *V*_{a} = −7 mV and effective valence *z*_{a} = 3.2 (Vandenberg, C.A., and F. Bezanilla. 1991. *Biophys. J.* 60:1499–1510) coupled to voltage-independent inactivation by an equilibrium constant (Bezanilla, F., and C.M. Armstrong. 1977. *J. Gen. Physiol.* 70:549–566) *K*_{eq} = 770. The factor *p*_{∞} represents voltage-dependent inactivation with empirical midpoint *V*_{∞}= −83 ± 5 mV and effective valence *z*_{∞} = 0.55 ± 0.03. The composite *p*_{a}*p*_{∞}*1s* model describes the steady-state voltage dependence of the persistent TTX-sensitive current well.

## Introduction

In response to step depolarization, current through voltage-gated Na channels increases rapidly and then declines. The classical description of this gating behavior (Hodgkin and Huxley, 1952b,c) postulated two independent underlying processes, “activation” and “inactivation.” Much work during the ensuing half-century has focused on these rapid (millisecond) gating events that underlie the generation and propagation of action potentials, and on the mechanisms of open channel permeation. The understanding gained has informed the study of other voltage-gated channels important in excitable cell function (Hille, 1992). Several early studies suggested the possibility of a second type of Na-channel conductance at positive membrane potentials (Adelman and Palti, 1969; Chandler and Meves, 1970; Bezanilla and Armstrong, 1977). Others demonstrated “slow” inactivation of the Na channel (Rudy, 1978; Almers et al., 1983; Kirsch and Anderson, 1986; Simoncini and Stühmer, 1987; Ruben et al., 1992). The existence of an alternative open Na-channel state arising from the inactivated state was established by Correa and Bezanilla (1994a)(b) using macroscopic, gating, and single-channel current measurements. The importance for nerve function of such slower gating processes and of the very small persistent Na-channel current is now widely recognized (Crill, 1996; Cummins and Sigworth, 1996). Indeed, anticonvulsant and post-ischemia protective effects of drugs like lidocaine, carbamazepine, and phenytoin (for review see Taylor, 1993) may reflect their reduction of sustained Na-channel current.

The purpose of our experiments was twofold. First, we investigated the permeation properties of tetrodotoxin (TTX)*-sensitive Na channels that remain open in the steady state, exploiting a sensitive and stable voltage clamp in combination with isotopic flux measurements (Rakowski, 1989; Rakowski, Gadsby and De Weer, 1989). We determined the persistent Na current as the change in holding current caused by a saturating concentration of TTX. The concurrently measured change in ^{22}Na efflux allowed computation of both unidirectional fluxes and, hence, the Ussing flux ratio exponent (*n*′). Measurements of *n*′ for K ion channels (Hodgkin and Keynes, 1955) provided strong evidence for a multi-ion (about three) pore that was recently confirmed by crystallography (Doyle et al., 1998; Morais-Cabral et al., 2001). Because Na channels undergo extensive inactivation both fast and slow, comparable data for these channels have been difficult to obtain. Our method offers the requisite stability and resolution, and our findings suggest occupancy of the Na-channel pore by, on average, a single Na ion.

Our second objective was to analyze the voltage dependence of the open probability (gating) of the few Na channels that remain open in the steady state. The product *m*^{3}(∞)*h*(∞) of the Hodgkin-Huxley (Hodgkin and Huxley, 1952c) analysis, which predicts the steady-state population of open Na channels of squid axon in the absence of slow inactivation, peaks at −47 mV with 0.5% of Na channels open. Incorporating steady-state “slow” inactivation (Rudy, 1978) yields the product *m*^{3}(∞)*h*(∞)*s*(∞), which peaks at −51 mV with 0.4% of Na channels remaining open. We find that neither model accurately predicts steady-state open probability or its voltage dependence. A simple alternative model that combines voltage-independent inactivation coupled to activation (Bezanilla and Armstrong, 1977) with an empirical Boltzmann equation encompassing all other voltage-dependent inactivation processes adequately accounts for our findings.

## Materials And Methods

### Measurement of TTX-sensitive Current and Flux

TTX-sensitive current and ^{22}Na efflux were measured simultaneously in squid (*Loligo pealei*) giant axons as described previously (Rakowski, 1989; Rakowski et al., 1989). In brief, a 23-mm-long segment of squid giant axon cannulated at both ends was internally dialyzed via a cellulose acetate capillary made porous to low molecular weight solutes. A blackened platinum wire for passing current and a glass capillary electrode filled with 3 M KCl were inserted into the axon. The membrane potential was measured as the voltage difference between this internal electrode and an external flowing 3-M KCl reference electrode (to minimize junction potentials) placed near the solution outflow. A stable, low noise voltage clamp (Rakowski, 1989) controlled the membrane potential. The superfusion chamber, kept at 17–18°C, comprised two lateral pools that contained the cannulated axon ends, physically separated by grease seals from the central, superfused, experimental pool. Two ancillary voltage-clamp circuits kept the end and central pools at the same potential to prevent current flow between them. Efflux of ^{22}Na across the axon membrane in the experimental region was measured by collecting the central pool superfusate for later assay by either liquid scintillation or gamma counting. Data are presented as mean ± SEM. Log-normal statistics are applied to ratios.

### Solutions

Internal and external solutions were the same as in Rakowski et al. (1989). The standard artificial seawater contained the following (in mM): 425 NaCl, 10 CaCl_{2}, 25 MgCl_{2}, 25 MgSO_{4}, 0.05 EDTA, and 5 Tris-HEPES, pH 7.7. 0.2 μM TTX was added from a 1-mM aqueous stock solution. In Na-free artificial seawaters, 425 mM NMG, TMA, or choline replaced Na; intermediate external Na concentrations were obtained by mixing. The high [Na] dialysate contained the following (in mM): 12.5 MgSO_{4}, 5 MgEGTA, 5 Na_{2}ATP, and 190 NaOH, pH 7.4 with ∼500 mM HEPES. The osmolality of the dialysate was adjusted to within 1% of that of the extracellular solution (typically 940 mOsm kg^{−1}). In Na-free dialysate, 10 Tris and 190 NMG replaced Na. Intermediate internal Na concentrations also were achieved by mixing. In experiments to assess the inertness of internal NMG, it was replaced by equimolar TMA. In some early experiments, the internal solution also contained 5 mM potassium phosphoarginine.

## Results

Our experimental approach demanded certain conditions: (1) membrane current and isotopic flux must be measured simultaneously from the same area of cell surface; (2) such measurements must be repeatable, reproducibly, on a preparation that has reached true steady state; (3) any spontaneous drift or channel rundown must be carefully monitored and corrected for; (4) inert nonpermeant Na replacements must be available; and (5) membrane potential must be measured accurately. We address each of these conditions before turning to the permeation properties and open probability of TTX-sensitive Na channels open in the steady state.

### Simultaneous TTX-sensitive Current and Flux Measurements

Simultaneous measurement of TTX-sensitive current (ΔI) and ^{22}Na efflux (ΔΦ* _{out}*) is illustrated in Fig. 1, which also emphasizes the need to allow sufficient time to achieve steady state after membrane potential shifts. To enhance TTX-sensitive current in this experiment, the 50-mM Na internal dialysate contained 250 μM veratridine (Ohta et al., 1973). The current traces (Fig. 1, insets, a and b) have been scaled by Faraday's constant to allow comparison of the amplitudes of changes in current (in μA cm

^{−2}) and flux (in pmol cm

^{−2}s

^{−1}) which, on addition of TTX (Fig. 1 b), were similar. Our previous simultaneous measurements of TTX-sensitive Na efflux and outward current under the same zero-trans conditions (Na-free external solution) gave a mean flux/current ratio (FΔΦ

*/ΔI) of 1.00 ± 0.04 (Rakowski et al., 1989), which demonstrates that TTX-sensitive current and*

_{out}^{22}Na efflux both derive from the same membrane area. A similar ratio, 1.05 ± 0.11, was obtained in five axons not exposed to veratridine (see Fig. 8). Veratridine was omitted in all subsequent experiments presented here.

Since many of our experiments required repeated exposure to TTX, it was important to establish the rate of recovery from block by TTX. In both axons of Fig. 2, the time constant for the exponential part of the recovery was 3.2 min. Since essentially complete recovery from TTX is achieved within 20 min, this was the minimum time allowed for TTX washout in all experiments.

### Correction for Channel Rundown

Fig. 3 shows the outward shifts of holding current produced by 21 consecutive 5-min exposures of a single axon to TTX, at holding potentials between −90 and +10 mV. A semilog plot of all eight TTX-induced current shifts at −30 mV (Fig. 4 A, closed circles), which served as controls for rundown, shows a roughly exponential loss of functional TTX-sensitive channels with time. To estimate the initial magnitude of TTX-sensitive current that would have been measured at −30 mV without rundown, an exponential fit to the first three TTX-induced shifts under those reference conditions (425 mM external [Na], internal Na-free, −30 mV) was extrapolated to the start of dialysis (Fig. 4 A, closed triangle on the ordinate). Thus, the initial TTX-induced shift estimated for this axon was 5.7 ± 0.6 μA cm^{−2}, and the rundown time constant was 230 min. For 10 axons, the mean estimates were 3.3 ± 0.3 μA cm^{−2} and 340 ± 40 min, respectively; the cause of this rundown is unknown. The absolute magnitude of the holding current near the end of each exposure to TTX in the experiment of Fig. 3 is plotted against membrane potential in Fig. 4 B. The reproducibility of the repeated measurements at −30 mV over the 10-h period confirms the remarkable stability of the background current, in contrast to the TTX-sensitive current.

To correct TTX-induced current shifts recorded at other holding potentials (Fig. 3) for rundown, they must be scaled to the response predicted for −30 mV at the time of each measurement. These predictions (Fig. 4 A, open circles) were obtained by exponential interpolation (extrapolation for the final test) between successive measurements at −30 mV. TTX-sensitive currents normalized in this way for this axon are plotted against holding potential in Fig. 4 C. The theoretical curve shown was calculated using parameters obtained from a global least squares fit (see Table I in discussion) to all relevant data obtained in this work. The good agreement of this theoretical curve with the data in Fig. 4 C shows this one axon to be representative.

### NMG, Choline, and TMA Are Impermeant Substitutes for Na

Several experiments required inert substitutes for Na. No steady-state TTX-sensitive current was detected (Fig. 5 A) when NMG replaced all internal Na, and 425 mM choline (Fig. 5 A, b), TMA (Fig. 5 A, c), or NMG (Fig. 5 A, e) replaced all external Na (Fig. 5 A, a, d, and g), which signifies that all three cation substitutes are impermeant. Fig. 5 B shows records when external Na was only partially replaced by choline or TMA. At 0.5 mole fraction external Na (Fig. 5 B, i and k) the TTX-sensitive current was about half that seen in the nearest test with 425 mM Na (mole fraction 1.0), which suggests that TMA and choline are not only impermeant, but also inert, i.e., do not hinder Na permeation. In contrast to these findings with choline or TMA, partial replacement of external Na by NMG (Fig. 5 C) reduced TTX-sensitive current further than expected if NMG were inert. For example, the size of the TTX-sensitive current at 0.5 mole fraction Na (Fig. 5 C, e and n) was about one quarter, rather than the anticipated one half, of that of the bracketing determinations at 425 mM Na. Average normalized data from 17 axons in which TMA, choline, or NMG replaced external Na are summarized in Fig. 6. With choline or TMA as substitutes, TTX-sensitive current was proportional to external Na concentration (Fig. 6, dashed line) as if the channel's apparent affinity for external Na were very low: (1)where [Na] is in mM. It can be shown that a linear relationship between current and the concentration of Na replacing another ion also would obtain if that ion were a competitive blocker, and if its apparent affinity were identical to that for Na. But the likelihood of TMA and choline sharing this special property must be small, and so we conclude that, instead, external TMA and choline are impermeant, nonblocking, substitutes for external Na and that the linear relationship (up to at least 425 mM Na_{o}) reflects a very low apparent affinity for external Na.

The curved line fitted to the NMG substitution data (Fig. 6) reflects a model in which external NMG binds to a site that, when occupied, blocks the channel: (2)where *K*_{NMG} is the dissociation constant for NMG at that site. The low affinity for external Na precludes experimental distinction between competitive and noncompetitive inhibition by NMG. Nevertheless, other data discussed below suggest that the NMG binding site, unlike that for Na, lies at the external mouth of the channel outside the membrane electrical field. The overall least- squares fit (which also includes data from Fig. 10 A to be discussed below) gave an estimate for *K*_{NMG} of 209 ± 15 mM (see Table I). A similar inhibitory effect of external NMG was seen on ^{22}Na efflux (experiments not shown). Because NMG was our standard replacement for internal Na, we checked for block of outward TTX-sensitive current by internal NMG. TTX-sensitive outward current, measured at −45 mV in Na-free (choline) external solutions with a dialysate containing 105 mM Na and 105 mM of either NMG or TMA, averaged −0.046 ± 0.008 (*n* = 3) and −0.049 ± 0.006 (*n* = 2) μA cm^{−2}, respectively. On the assumption that internal TMA is an inert substitute for internal Na (but see Horn et al., 1981), as established above for external TMA, this result implies that internal NMG does not inhibit TTX-sensitive channels. The alternative possibility is rendered unlikely by the close agreement between the currents measured with NMG or TMA, which would require the apparent *K*_{i} for any putative block by internal NMG and TMA at −45 mV to be identical.

We also examined whether small amounts of contaminating internal K, or the 5-mM internal K added as potassium phosphoarginine in early experiments, support a detectable outward TTX-sensitive current. Although, with 200 mM Na in the dialysate, we observed the usual, substantial, TTX-sensitive outward current in Na-free (choline) external solution, after exchanging the dialysate for one containing 5 mM K, but no Na (replacement: NMG), we could detect no TTX-sensitive current in Na-free external solution containing either choline or TMA (two measurements on one axon). Thus, up to 5 mM contaminating internal K carries no measurable TTX-sensitive steady-state current even in the absence of competing internal or external Na.

### Accuracy of Membrane Potential Measurement: Reversal of the TTX-sensitive Current

To observe both inward and outward steady-state TTX-sensitive currents in an axon under fixed ionic conditions, we chose appropriate Na concentrations (184 mM inside and 111 mM outside, confirmed by flame photometry). Fig. 7 A shows records of repeated exposures to TTX alternately at −30 or 0 mV (Fig. 7 A, a–h), and at −15 mV (Fig. 7 A, i). The magnitudes of the TTX-sensitive currents (Fig. 7 B) gave (by linear interpolation) an estimate for the reversal potential of −14.4 mV, close to the calculated value of E* _{Na}* (−12.7 mV). A second axon treated similarly gave a reversal potential of −14.6 mV. These findings corroborate the accuracy of our membrane potential measurements (i.e., within 2 mV).

### Flux Ratio Exponent

TTX-sensitive ^{22}Na efflux and net current were simultaneously measured (Fig. 8) in axons held at −30 mV, internally dialyzed with 200 mM Na, and exposed alternately to 425 mM external Na (b, d) or choline (a, c). Five axons in the Na-free solution gave mean values of 0.198 ± 0.009 μA cm^{−2} (*n* = 17 trials) for TTX-sensitive current and 2.15 ± 0.22 pmol cm^{−2} s^{−1} (*n* = 15 trials) for TTX-sensitive ^{22}Na efflux, yielding a ratio FΔΦ* _{out}*/ΔI = 1.05 ± 0.11. This good agreement with the expected ratio of 1.0 and with our findings on veratridine-treated axons (Fig. 1) validates our comparisons of simultaneously measured flux and current. The magnitudes of TTX-induced changes in net current and unidirectional efflux in 425 mM external and 200 mM internal Na solution (Fig. 8, episodes b and d) can be used to calculate the Ussing flux ratio exponent (Ussing, 1949; Hodgkin and Keynes, 1955). Thus, the TTX-sensitive net current (ΔI) represents, via Faraday's constant, a net flux (ΔΦ) from which the simultaneously measured unidirectional

^{22}Na efflux component (ΔΦ

*) can be subtracted to yield unidirectional influx (ΔΦ*

_{out}*). The flux ratio exponent (*

_{in}*n*′) then follows from Eq. 3 (3)

For eight paired measurements of ΔI and ΔΦ* _{out}* on four axons at a holding potential (

*V*) of −30 mV, we find

*n*′ = 0.97 ± 0.03. For visual corroboration of this result, we show in Fig. 8 C (smoother downward traces in b and d) the hypothetical TTX-induced current shift expected for abolition of the outward component alone, which is computed from the actually observed net current shift assuming a flux ratio exponent

*n*′ = 1.

### Comparison of TTX-sensitive Currents with and without trans Na Present: Evidence for Nonindependence

According to the independence principle, “the chance that any individual ion will cross the membrane in a specified interval of time is independent of the other ions which are present” (Hodgkin and Huxley, 1952a). To test whether Na fluxes are influenced by Na present on the trans side, we measured TTX-sensitive currents first under zero-trans conditions, i.e., with Na present either on the outside of the axon only (Fig. 9, left column) or on the inside only (Fig. 9, middle column), and then with Na on both sides (Fig. 9, right column). The measurements were made in the (alphabetical) order shown, and all were subject to rundown, so that comparison required interpolation between adjacent pairs of like measurements. The traces marked by asterisks in the right column are hypothetical and were constructed as the sum of the two zero-trans currents (Fig. 9, left and middle columns), both interpolated to the time of the actual measurement (Fig. 9, right column, no asterisk) with Na present on both sides. Evidently, the in- and outward movements of Na did not obey independence, as the net current measured with Na present on both sides (Fig. 9, right column) was significantly and consistently smaller than the sum of the corresponding zero-trans currents. The observed net current averaged 0.81 ± 0.01 (*n* = 7 on two axons) of the sum of the corresponding zero-trans currents. A measurement artifact is highly unlikely since (1) each comparison was made on a single axon at constant holding potential and only the solutions were varied; (2) liquid junction potentials at the external reference electrode were minimized by means of a flowing 3-M KCl junction; and (3) any liquid junction potential difference at the internal electrode between Na-containing and Na-free dialysates would be expected to exaggerate, not lessen, the discrepancy depicted in the right-hand column of Fig. 9.

Furthermore, the observed zero-trans TTX-sensitive outward current (inward shifts, middle column) is significantly (17%) smaller than calculated (assuming independence) for the same potential from the (appropriately interpolated) corresponding zero-trans inward current (outward shifts, left column): the observed ratio of zero-trans currents averaged −0.123 ± 0.003 (*n* = 12 from 2 axons) compared with −0.149 predicted at −30 mV for independent zero-trans fluxes at 425 mM external and at 200 mM internal [Na]. This observation furthers the evidence against independence. Although a sizeable constant error in our measurement of membrane potential (−35 mV instead of −30 mV) would be able to account for the discrepancy and, thus, weaken it as evidence, such a large error seems unlikely in view of the good agreement we find between computed and measured reversal potentials (Fig. 7). Moreover, this smaller-than-expected zero-trans outward Na current cannot be attributed to block by external choline, which we have shown (Figs. 5 and 6) to be an inert substitute for Na. Finally, any channel block by internal NMG, which was present only during the measurements of zero-trans inward Na current in these experiments (Fig. 9, left column, 425/0), could only cause us to underestimate the size of that inward current: in turn, this would cause us to also underestimate the degree of nonindependence.

### Voltage Dependence of Steady-state TTX-sensitive Current

We have already illustrated (Figs. 3 and 4) our method for examining the voltage dependence of steady-state TTX-sensitive inward current in axons exposed to 425 mM Na and dialyzed with Na-free solution. We similarly examined the voltage dependence of outward persistent TTX-sensitive currents in axons dialyzed with 200 mM Na and exposed to Na-free (choline) solutions. To permit direct comparison of normalized outward currents with normalized inward currents, we exploited data from experiments like that in Fig. 9 in which both types of TTX-sensitive current were measured in a single axon at a constant holding potential of −30 mV. Two further conditions were also examined: (1) with 50 mM Na inside and 425 outside; and (2) with 266 mM Na plus 159 mM NMG outside and Na-free solution inside.

All of our data concerning the steady-state voltage dependence of TTX-sensitive currents obtained under these four experimental conditions, annotated as (external [Na]/internal [Na], both in mM) 425/0, 425/50, 266/0, or 0/200, are summarized in Fig. 10 A. The theoretical curves through the data represent the currents calculated for each of the four ionic conditions, using a single set of parameters obtained from a simultaneous least-squares fit to all relevant data of an economical steady-state gating and permeation model that postulates a single binding site for Na in the pore (the *p*_{a}*p*_{∞}*1s* model, to be presented in the discussion). In the global least-squares fit, the estimation of individual model parameters is governed by various data subsets: (1) the gating parameters are governed by the voltage dependence data (Fig. 10 A); (2) the apparent dissociation constant for binding of internal Na, *K*_{Nai}, by the ratios of current amplitudes at −30 mV in the 425/200 and 0/200 conditions to those in 425/0 (Fig. 9) as well as by the in- versus outward current pairs over a range of voltages (Fig. 10 A); and (3) the *K*_{NMG} for channel block by external NMG by data of Fig. 6 together with the 266/0 curve of Fig. 10 A. The assumption of voltage-independent block by external NMG is validated here by the demonstration that inward currents measured in the presence of 159 mM external NMG are adequately fit (266/0 curve; Fig. 10 A) by an equation that is scaled uniformly across the entire voltage range by a single factor appropriate for block by that concentration of NMG. The theoretical curves in Fig. 10 B will be described in the discussion. The dashed line in Fig. 10 A represents the outward current expected in 0/200 conditions for a model (Fig. 10 B, curve *p*_{a}) that lacks voltage-dependent inactivation (i.e., *p*_{a}*1s*, without *p*_{∞}). Its poor fit to the 0/200 data demonstrates (see discussion) the need to include voltage-dependent inactivation (Fig. 10 B, curve *p*_{a}*p*_{∞}) in our *p*_{a}*p*_{∞}*1s* model.

## Discussion

The presumption seems justified that the extremely small fraction of TTX-sensitive channels that remain open in the steady state represent “classical” voltage-sensitive Na-selective channels: the experimental reversal potential of the currents they carry was close to the theoretical value for *E*_{Na}; choline, TMA, and NMG were impermeant; and recovery from TTX inhibition had a time course similar to that found for classical Na channels. Flux ratio measurements revealed an Ussing flux ratio exponent close to 1.0. However, inward and outward Na movements were not independent of one another, a property that requires at least one saturable, if low affinity, binding site for Na ions within the pore. Moreover, as discussed in the section on persistent gating, voltage dependence of channel open probability was consistent with literature values of the steady-state gating parameters that describe voltage-dependent activation and fast voltage-independent inactivation of Na channels, but necessitated an additional parameter describing steady-state voltage-dependent inactivation.

### Minimal Model for Nonindependent Sodium Permeation that Exhibits an Ussing Flux Ratio Exponent (n*′*) of Unity

From eight paired measurements of TTX-sensitive current and ^{22}Na efflux including those in Fig. 8 with 425 mM external and 200 mM internal Na, we find an Ussing flux ratio exponent (*n*′) of 0.97 ± 0.03. This removes the need to postulate a multi-ion pore to account for the ion flux behavior under our experimental conditions. A flux ratio exponent of unity implies ion movements that either obey the independence principle or are mediated by (effectively) one-site channels; multi-site channels with very low occupancy (rarely exceeding one ion per channel) also yield *n*′ values close to one (Hille and Schwarz, 1978; Begenisich, 1987). However, Fig. 9 shows that ion movements through persistent Na channels are not independent, as net current measured with Na both inside (200 mM) and outside (425 mM) is less than the sum of the corresponding zero-trans currents. Nonindependence requires that permeating ions interact with, i.e., “bind” to a “site” within, the channel. Since the Ussing flux ratio exponent of unity does not require a multi-ion pore, this observed deviation from independence can be explained by a simple permeation model (Scheme I):

in which both external Na (*Na*_{o}) and internal Na (*Na*_{i}) have direct access to a single effective binding site in the pore of the channel protein (*E*). This effective binding site could, in reality, be distributed over several physically distinct points in space as long as occupancy of the “site” precludes concomitant binding of a second ion in the pore. At this site, the permeating ion senses a fraction δ* _{Na}* (arbitrarily set here at 0.5) of the electrical field (from the internal solution), so that the rate coefficients

*k*

_{1}…

*k*

_{−2}of Scheme I become functions

*k*

_{1}(V)…

*k*

_{−2}(V) (defined in Eqs. 8 and 9) of membrane potential. In the absence of membrane potential (

*V*= 0), microscopic reversibility constrains as follows: (4)

The expressions for zero-trans Na influx (inward flux is negative) and efflux rates (i.e., ions traversing each pore per unit time) are as follows: (5)

Our finding that TTX-sensitive inward current is a linear function of external [Na] (Fig. 6) while a finite apparent affinity for internal Na is required to account for the observed nonindependence (Fig. 9) permits a further simplification. These features are satisfied over practical voltage ranges by setting the rate coefficients for transitions between binding site and internal solution, *k*_{2} and *k*_{−2}, very large compared with those for the transition between binding site and external solution, *k*_{1} and *k*_{−1}:

Zero-trans influx (Eq. 5, top) will then be linearly proportional to [Na]_{o} as required by Fig. 6, whereas *K*_{0.5}(Na_{i}) (given by *(k*_{−1} + *k*_{2})/k_{−2} [Eq. 5, bottom]) assumes a finite value as required of the saturable binding site that underlies nonindependence. The simplification yields:

in which equilibrium prevails between binding site and internal solution, governed by the equilibrium dissociation constant for internal sodium binding, *K*_{Nai}(V)= *k*_{2}(V)/k_{−2}(V). The net flux is then: (6)

From Boltzmann's law the equilibrium constant varies with membrane voltage, decreasing with depolarization (steepness being determined by δ* _{Na}*) as given by Eq. 7: (7)

The sole determinants of site occupancy, therefore, are internal [Na] and membrane potential. For simplicity, we further assume symmetry in the effect of membrane potential on the rate coefficients *k*_{1} and *k*_{−1}: (8)(9)

As our fluxes are normalized to a reference value, and because of Eq. 4, only a single adjustable parameter (*K*_{Nai}) is required to account for the open-channel flux properties in the absence of NMG. The least-squares value for *K*_{Nai}(0) (see Table I) yields a *K*_{Nai}(*V*) of 1.12 M at −30 mV. From measurements of peak Na-channel current as a function of internal ion activity, Begenisich and Cahalan (1980a)(b) found a *K*_{Nai} of 0.86 M at +25 mV. Our value for *K*_{Nai} at −30 mV correctly predicts the nonindependence of unidirectional fluxes we observed: net flux under 425/200 conditions is calculated to be only 0.82 of the sum of fluxes measured under the corresponding zero-trans conditions 425/0 and 0/200 (observed current ratio was 0.81 ± 0.01; Fig. 9). Furthermore, the model predicts a ratio of currents under zero-trans (0/200 and 425/0) conditions of −0.125, which is compatible with the observed current ratio of −0.123 ± 0.003.

The *1s* model has an obligatory Ussing flux ratio exponent of unity (we observed 0.97 ± 0.03). Begenisich and Busath (1981) also found a value (1.03 ± 0.04) not significantly different from 1, and independent of voltage between −22 and +33 mV. Though a one-site model suffices to account for these data, multi-site models that have low occupancy could also explain them, but need more parameters than can be uniquely determined. Such a multi-site (but single ion occupancy) model for the Na channel has been invoked to describe Na current-voltage relationships (Hille, 1975). Begenisich (1987) concluded that, although two sites are needed to explain the concentration dependence of the *P*_{K}/P_{Na} permeability ratio, both sites are rarely simultaneously filled. We conclude that the Na channel either has one effective binding site with relatively low affinity or is a multi-site channel rarely occupied by more than one Na ion at a time. It remains to be seen whether other models of ion permeation such as the Poisson-Nernst-Planck formulation (Nonner et al., 1998; Levitt, 1999) can also explain the observed deviations from independence.

### Gating Model for the Persistent Na Channels

Three processes govern the theoretical curves in Figs. 4 and 10 A: channel gating, open channel permeation, and (for the data at 266 mM external [Na]) block by external NMG. Our least-squares fit to the data uses two independent (i.e., multiplicative) factors for steady-state channel gating: one (*p*_{a}) reflecting activation coupled to voltage independent inactivation, and a second (*p*_{∞}) encompassing all other (voltage dependent) inactivation processes. The first factor (*p*_{a}) is the steady-state open probability function for Scheme III

in which activation and voltage-independent inactivation are coupled (Goldman and Schauf, 1972; Bezanilla and Armstrong, 1977) rather than independent: (10)where *a*(∞) describes the steady-state voltage dependence of channel activation (⇄),
and *K*_{eq} is a voltage-independent equilibrium constant governing inactivation
(⇄).
For the steady-state activation function, *a*(∞*)* we use a Boltzmann distribution (11)with midpoint *V*_{a} = −7 mV and effective valence *z*_{a} = 3.2, as determined by Vandenberg and Bezanilla (1991) in *L. pealei* axons under ionic conditions comparable to ours. For the voltage-dependent steady-state inactivation factor (*p*_{∞}), a Boltzmann distribution function (12)with empirical midpoint V_{∞} = −83 mV and effective valence z_{∞} = 0.55 was found to be sufficient. The five parameters (*K*_{eq}, *z*_{a}, *V*_{a}, *z*_{∞}, and *V*_{∞}) that describe the channel's open probability (Eqs. 10–12) are constrained by the experimental open probability (13)for which we have an estimate at −30 mV (see *Absolute Magnitude*…) so that effectively only four parameters are independent. For algebraic convenience, we chose to calculate *K*_{eq} from the others: (14)

Eqs. 10 and 12 may be combined with our one-site (*1s*) model for open channel permeation (Eq. 6) to give an overall model, *p*_{a}p_{∞}*1s* (the product of Eqs. 6, 10, and 12), for the voltage dependence of steady-state TTX-sensitive current. Although fitting this model to normalized data (Fig. 10 A) discards information on absolute current magnitude, we used the mean value of TTX-sensitive currents measured under reference conditions, corrected for rundown, to recover the lost information (see *Absolute Magnitude*…).

### Global Least-squares Fit to the Entire Data Set

We verified the ability of the *p*_{a}*p _{∞} 1s* model, augmented with noncompetitive voltage-independent channel block by external NMG, to account for our combined findings on the following: (1) current ratios reflecting nonindependent permeation (Fig. 9 and similar experiments not shown); (2) the voltage dependence of steady-state TTX-sensitive inward and outward current (Fig. 10 A); and (3) inhibition by external NMG (Figs. 6 and 10 A). To this end, we fitted the model (using a single set of parameters) simultaneously to the entire data set. The augmented

*p*

_{a}

*p*

_{∞}

*1s*model contains six independent parameters. Two (

*V*

_{a}and

*z*

_{a}) were taken from measurements of the steady-state voltage dependence of Na channel activation in squid giant axon (Vandenberg and Bezanilla, 1991). Our global fit then yields estimates (Table I) for the remaining four adjustable model parameters:

*K*

_{Nai}(0),

*K*, V

_{NMG}_{∞}, and

*z*

_{∞}.

*K*

_{eq}, calculated via Eq. 14 from our estimate (0.0003; see next section) of P

_{o}at −30 mV was 770, which is comparable to the corresponding value of κ

*/*λ = 1,000 estimated for

*L*.

*pealei*by Bezanilla and Armstrong (1977). The principal determinants of the shape of the I-V curves in Fig. 10 A are the gating parameters (

*z*

_{a},

*V*

_{a},

*z*

_{∞}, and

*V*

_{∞}) that describe the voltage dependence of the channel's open probability in the steady state. The in- versus outward current ratios (not only those obtained at −30 mV in Fig. 9 and similar experiments, but also those of the in- versus outward pairs at various voltages in Fig. 10 A) constrain the estimation of

*K*

_{Nai}(0), the cause of nonindependent behavior. The NMG inhibition data in Fig. 6 and the 266/0 curve in Fig. 10 A yield

*K*

_{NMG}(the voltage-independent equilibrium constant for NMG block). All curves in Figs. 4 C, 6, and 10 are drawn using the parameters listed in Table I. A striking feature of Fig. 10 A is the fact that the outward currents do not steadily increase with voltage at positive membrane potentials, as would be expected (Fig. 10 A, dashed line) in a model lacking

*p*

_{∞}, the voltage-dependent inactivation factor. Also indicative of voltage-dependent inactivation is the inflection in the inward I-V curve near 0 mV.

### Absolute Magnitude of the Persistent TTX-sensitive Current

After correction for rundown, we found the persistent TTX-sensitive current at −30 mV and 425 mM external [Na] to have an amplitude of 3.3 ± 0.3 μA cm^{−2} (11 axons). Baker et al. (1969) found a TTX-sensitive Na influx in resting intact *L. forbesi* axons in 10 mM K artificial seawater (and resting potential likely near −60 mV) of ∼1.6 μA cm^{−2}, which is compatible with our results at −30 mV. To calculate the fraction of Na channels that remain open in the steady state requires knowledge of the current expected at −30 mV if all Na channels were open. Using the maximum Na conductance of 130 mS cm^{−2} for *L. pealei* axons at 20°C not exposed to internal fluoride (Rosenthal and Bezanilla, 2001; and Rosenthal, J.J.C., and F. Bezanilla, personal communication), we estimate the maximum possible inward Na current at −30 mV to be ∼11 mA cm^{−2}, which implies that the persistently open fraction of the total population of channels at −30 mV averages only ∼0.03% (i.e., P_{o} = 0.0003, the value we used to constrain the other fit parameters via Eq. 13). This is more than an order of magnitude smaller than the value of 0.43% predicted by the classical product *m*^{3}(∞)*h*(∞), using the parameter values listed by Meves (1984). Even after introducing a slow inactivation factor, *s*(∞) (Rudy, 1978), the product *m*^{3}(∞)*h*(∞)*s*(∞) at −30 mV predicts a persistent open channel fraction of 0.19%, still much larger than our estimate. Fig. 10 B shows steady-state absolute open probabilities calculated for *p*_{a} alone as well as for the product *p*_{a}*p*_{∞}, using the least-squares parameters found and the value calculated for *K*_{eq}. The latter defines the amplitude of the *p*_{a} curve asymptote at positive voltages. At −30 mV, the value of *p*_{a} (i.e., the steady-state open probability resulting solely from activation coupled to voltage-independent inactivation) is 0.0013. The voltage-dependent steady-state inactivation factor *p*_{∞} of our model closes another 76% of those open channels, yielding the open probability of 0.0003 at −30 mV.

Assuming a Na-channel density in squid axon of ∼600 μm^{-2} (Levinson and Meves, 1975) for a P_{o} of 0.0003 there will be 1.8 × 10^{7} channels cm^{−2} open in the steady state at −30 mV, sustaining the observed zero-trans inward current of 3.3 μA cm^{−2} when [Na]_{o} = 0.425 M, or 1.1 × 10^{6} ions s^{−1} channel^{−1}. Eq. 8 then yields an estimate for the second-order rate coefficient *k*_{1}(0) = ∼2 × 10^{6} s^{-1} M^{−1}. Any correction of the 130-mS cm^{−2} maximum Na conductance value for the inactivation likely present in those experiments would tend to increase our estimate of the maximum current at −30 mV (perhaps by about one third; Stimers et al., 1985). That would, in turn, further reduce (by the same factor) our estimate of the fraction of persistently open channels, engendering (via Eq. 13) minor adjustments in our fit parameters and estimate of *k*_{1}(0).

### Voltage-dependent Inactivation, or Block?

We do not know the mechanism underlying the voltage-dependent inactivation, but one possibility is block by internal Mg^{2+}, which is well documented for K channels (for review see Stanfield, 1988). Indeed, Pusch et al. (1989) have described voltage-dependent block of outward Na current by internal Mg^{2+} in *Xenopus* oocytes injected with cRNA encoding rat brain type II Na channels; *K*_{0.5} at 0 mV was 3–4 mM. Since the free [Mg^{2+}] in our experiments was ∼12.5 mM (17.5 mM total [Mg^{2+}] with 5 mM ATP present) we may consider a model of voltage-dependent Na-channel block by internal free Mg^{2+}, with the fraction of channels not blocked given by (15)where *K*_{Mg}(0) is the dissociation constant for the putative Mg^{2+} binding site at 0 mV, δ* _{Mg}* is the fraction of the membrane potential (

*V*) at the site, and the valence of Mg

^{2+}(

*z*) is 2. Eq. 15 has the same canonical form as the empirical steady-state Eq. 12 we used to describe

*p*

_{∞}, steady-state voltage-dependent inactivation. The two formulations are identical when

Substitution of Eq. 15 for Eq. 12, thus, yields an alternative model, *p*_{a}*p*_{Mg}*1s*, indistinguishable from *p*_{a}*p*_{∞}*1s* until experiments are performed at various internal [Mg^{2+}]. The fit parameters obtained for this internal Mg^{2+} blocking model are also listed in Table I. With *K*_{Mg}(0) = 2.0 mM and δ* _{Mg}* = 0.28, the “pure” Mg

^{2+}block model yields the required 76% block at −30 mV.

### Estimated K_{d} for TTX Binding to Persistent Na Channels

Cuervo and Adelman (1970), recording the time course of recovery of (depolarization induced) peak inward current after TTX withdrawal, found a mean time constant of 8.6 min at 7°C or (taking a Q_{10} of 2.0 calculated from their data) 4.3 min at 17°C, which compares reasonably with our value of 3.2 min (Fig. 2). Keynes et al. (1975) have pointed out that recovery of current does not straightforwardly reflect the rate of dissociation of bound TTX since the toxin's escape is retarded in the restricted-diffusion Frankenhaeuser-Hodgkin (Frankenhaeuser and Hodgkin, 1956) space within which it can rebind to vacant sites. After their treatment, the time constant, τ* _{rec}*, for the latter part of the recovery is τ

*=*

_{rec}*N*/(

*P*

_{TTX}

*K*

_{d}), where

*N*is the surface concentration of TTX binding sites in moles per unit area,

*P*

_{TTX}is the apparent permeability to TTX of the virtual boundary delimiting the Frankenhaeuser-Hodgkin space, and

*K*

_{d}is the TTX dissociation constant. Assuming a site density of 600 μm

^{−2}(Levinson and Meves, 1975) and a

*P*

_{TTX}value of 2 × 10

^{−5}cm s

^{−1}(Keynes et al., 1975), we calculate a

*K*

_{d}of 5 nM for TTX block of persistent Na-channel current, which is comparable to the value of 4.7 nM obtained by Schwarz et al. (1973) for peak Na current in frog myelinated nerve.

## Acknowledgments

We thank Wolfgang Nonner for pointing out the work of Pusch et al. (1989).

This work was supported by National Institutes of Health grants NS-22979, HL-36783, and NS-11223.

## Footnotes

↵*

*Abbreviations used in this paper:*P_{o}, open probability; TTX, tetrodotoxin.Dr. Gadsby's present address is Laboratory of Cardiac/Membrane Physiology, Rockefeller University, 1230 York Avenue, New York, NY 10021. Dr. De Weer's present address is Department of Physiology, University of Pennsylvania, School of Medicine, 3700 Hamilton Walk, Philadelphia, PA 19104.

- Submitted: 13 September 2001
- Accepted: 30 January 2002
- Revision received 18 January 2002