Disrupted coupling of gating charge displacement to Na⁺ current activation for DIIS4 mutations in hypokalemic periodic paralysis

Defective coupling between gating charge displacement and channel activation may contribute to reduced muscle excitability in hypokalemic periodic paralysis.

Abstract

Missense mutations at arginine residues in the S4 voltage-sensor domains of Na_V1.4 are an established cause of hypokalemic periodic paralysis, an inherited disorder of skeletal muscle involving recurrent episodes of weakness in conjunction with low serum K⁺. Expression studies in oocytes have revealed anomalous, hyperpolarization-activated gating pore currents in mutant channels. This aberrant gating pore conductance creates a small inward current at the resting potential that is thought to contribute to susceptibility to depolarization in low K⁺ during attacks of weakness. A critical component of this hypothesis is the magnitude of the gating pore conductance relative to other conductances that are active at the resting potential in mammalian muscle: large enough to favor episodes of paradoxical depolarization in low K⁺, yet not so large as to permanently depolarize the fiber. To improve the estimate of the specific conductance for the gating pore in affected muscle, we sequentially measured Na⁺ current through the channel pore, gating pore current, and gating charge displacement in oocytes expressing R669H, R672G, or wild-type Na_V1.4 channels. The relative conductance of the gating pore to that of the pore domain pathway for Na⁺ was 0.03%, which implies a specific conductance in muscle from heterozygous patients of ∼10 µS/cm² or 1% of the total resting conductance.

Unexpectedly, our data also revealed a substantial decoupling between gating charge displacement and peak Na⁺ current for both R669H and R672G mutant channels. This decoupling predicts a reduced Na⁺ current density in affected muscle, consistent with the observations that the maximal dV/dt and peak amplitude of the action potential are reduced in fibers from patients with R672G and in a knock-in mouse model of R669H. The defective coupling between gating charge displacement and channel activation identifies a previously unappreciated mechanism that contributes to the reduced excitability of affected fibers seen with these mutations and possibly with other R/X mutations of S4 of Na_V, Ca_V, and K_V channels associated with human disease.

INTRODUCTION

Hypokalemic periodic paralysis (HypoPP) is a dominantly inherited disorder of skeletal muscle with recurrent episodes of weakness in association with low serum K⁺ (Lehmann-Horn et al., 2004; Cannon, 2006). Missense mutations of Na_V1.4 and Ca_V1.1 are established causes of HypoPP. A final common pathway for the mechanistic basis of weakness is paradoxical depolarization of muscle fibers in low K⁺ (often <3 mM), which inactivates Na_V1.4 and renders fibers refractory and inexcitable (Rüdel et al., 1984). Remarkably, all 10 HypoPP mutations reported in Na_V1.4 are missense substitutions at arginine residues in the outer end of S4 segments in domains I–III of the voltage sensors (Matthews et al., 2009). Six of the Ca_V1.1 HypoPP mutations are also at arginine residues in S4 segments. Expression studies in oocytes (Sokolov et al., 2007; Struyk and Cannon, 2007; Struyk et al., 2008; Francis et al., 2011) and knock-in mutant mouse models (Wu et al., 2011, 2012) have implicated a common functional defect, wherein the R/X missense mutations in S4 segments produce an aberrant ion conduction pathway in the voltage-sensor domain that allows an influx of current at the resting potential. This so-called omega current or gating pore current, I_GP, is very small compared with the ionic current conducted by the pore domain. Computer simulations have demonstrated that even small gating pore currents may increase the susceptibility to paradoxical depolarization at low K⁺, where the outward current contribution for the inward rectifier (V_rest > E_K) becomes overwhelmed by the inward gating pore leakage current (Struyk and Cannon, 2008; Jurkat-Rott et al., 2010).

A critical question in establishing the gating pore current as a major contributor to the pathogenesis of transient weakness in HypoPP has been, what is the relative conductance or current density of I_GP compared with the normal ionic conductances that regulate excitability of skeletal muscle? Extrapolation of oocyte expression data, based on a few fortuitous measurements of both ionic Na⁺ current, I_Na, and I_GP in an oocyte expressing Na_V1.4-R672G, led to the conclusion that I_GP at the resting potential (−90 mV) was ∼1.25% of the I_Na peak during an action potential (Sokolov et al., 2007). Our estimate, based on a comparison of maximal gating charge displacement to I_GP for oocytes expressing Na_V1.4-R669H, was ∼10-fold smaller (Struyk and Cannon, 2007). Was this difference intrinsic to the particular mutation? The R672G mutation creates a nonselective cation conductance permeable to Na⁺, K⁺, and even NMDG⁺, whereas R669H produces a proton-selective conduction pathway. Or, is the discrepancy the result of different methods used to extrapolate the oocyte data to the situation in mammalian skeletal muscle? We sought to resolve this discrepancy and to gain an improved estimate for the predicted gating pore conductance in HypoPP muscle fibers by measuring the Na⁺ current, maximal on-gating charge displacement, and gating pore current all from the same oocyte expressing R669H, R672G, or WT Na_V1.4 channels. An analysis of these measurements provided a consensus for quantifying the relative magnitude of I_GP. The predicted specific conductance of the gating pore in HypoPP muscle was comparable to experimental values obtained in voltage-clamp studies of our knock-in mouse model of Na_V1.4-R669H HypoPP. Unexpectedly, we also observed a substantial decoupling between gating charge displacement and peak ionic current for both HypoPP mutant channels. This decoupling identifies a new mechanism by which excitability may be reduced in HypoPP fibers and accounts for the experimental observation that peak dV/dt and action potential overshoot in muscle are reduced in human HypoPP fibers (Jurkat-Rott et al., 2000) and in the Na_V1.4-R669H knock-in mouse (Wu et al., 2011).

MATERIALS AND METHODS

Expression of Na_V1.4 channels in oocytes

The rat adult skeletal muscle Na⁺ channel α subunit, rNa_V1.4, was coexpressed with the human β1 subunit in Xenopus laevis oocytes. WT and mutant α-subunit constructs were created in the vector pGEMHE, as described previously (Struyk and Cannon, 2007; Struyk et al., 2008), to promote high expression levels in oocytes. To facilitate comparisons with other studies on HypoPP in humans, we named the mutant constructs in this paper for their disease-associated orthologues in human Na_V1.4. The rNa_V1.4-R663H mutation in the outermost arginine of S4 in domain II is denoted as R669H, and rNa_V1.4-R666G is denoted as R672G. cRNAs were synthesized by in vitro transcription using the mMessage mMachine kit (Ambion), and oocytes were injected with ∼50 ng of cRNA encoding either WT or mutant Na_V1.4, together with ∼50 ng of β1-subunit transcript (approximately fourfold molar excess). Oocytes were either obtained commercially (EcoCyte) or harvested by partial oophorectomy. Frogs, Xenopus, were housed in an Association for Assessment and Accreditation of Laboratory Animal Care–accredited facility, and all experiments were performed within guidelines established by the University of Texas Southwestern Medical Center Institutional Animal Care and Use Committee. Injected oocytes were maintained at 18°C in 0.5× Leibovitz’s L-15 medium (HyClone) supplemented with 1% horse serum, 100 U/ml penicillin, 100 µg/ml streptomycin, and 100 µg/ml amikacin.

Electrophysiology

In all experiments, oocytes were voltage clamped in the cut-open configuration, with active clamping of the upper and guard chambers using an amplifier (CA-1B; Dagan Corporation) in headstage clamp mode. A perfusion manifold was built to allow complete exchange of the bath in the upper and lower chambers. Low resistance access to the intracellular compartment was obtained by permeabilizing the oocyte membrane in the lower chamber with 0.04% saponin. An equilibration period of >30 min was used to dialyze the oocyte intracellular contents with the lower chamber solution before recording currents. Computer control of command potentials and data acquisition were performed with an interface (ITC-1600; HEKA) using custom software developed in our laboratory.

The voltage-sensing electrode was fabricated from thin-walled borosilicate glass, filled with 3 M KCl, and positioned just under the dome of oocyte membrane in the upper chamber. Electrical connections to the three chambers were made with agarose-filled capillary bridges containing 110 mM Na⁺ methanesulfonate and 10 mM HEPES, pH 7.4, and threaded with platinum wire. For experiments to measure gating charge displacement currents, compensation for linear capacitance transients was made with the amplifier’s analogue circuitry. No analogue compensation for linear leakage current was used. Current signals were low-pass filtered at 5 kHz and sampled at 50 kHz.

All recordings were performed in Cl⁻-free solutions to minimize the contribution from anion conductances endogenous to the oocyte. The oocyte internal solution was equilibrated with the lower guard chamber containing (mM): 120 KOH, 10 EGTA, and 10 HEPES, pH 7.4 with methane sulfonate. In some experiments, KOH was replaced by NMDG. The upper chamber solution was initially high Na⁺ to record ionic Na⁺ currents conducted through the conventional pore (mM): 115 NaOH, 3 KOH, 1.5 Ca(OH)₂, 2.5 Ba(OH)₂, and 10 HEPES, pH 7.4 with methane sulfonate. Gating pore (omega) currents were then recorded from the same oocyte by replacing the external solution for which Na⁺ was substituted with an equimolar amount of K⁺ plus 1 µM tetrodotoxin (TTX). For some experiments, to isolate the gating change displacement current, an NMDG-based external bath was used (1 µM TTX plus [mM]: 113 NMDG, 1.5 Ca(OH)₂, 2.5 Ba(OH)₂, and 10 HEPES, pH 7.4 with methane sulfonate).

Data analysis

Peak and steady-state current levels were extracted from current tracings using custom software. Gating pore ionic currents were isolated in oocytes expressing mutant channels by subtracting the average current–voltage response for oocytes expressing WT channels in the presence of TTX, as described previously (Struyk and Cannon, 2007). Gating currents were quantified as the nonlinear charge displacement, Q_on, in response to step depolarization from a holding potential of −100 mV. Q_on was calculated by integration of the leak-subtracted current transient elicited by a 50-msec step depolarization. The leakage current was measured for a small 10-mV depolarization from a holding potential of −100 mV, for which the gating charge contribution was miniscule. This leakage current was digitally scaled and subtracted from raw current records to obtain P/N leak-subtracted traces. Additional compensation for nonlinear contributions from incomplete block of ionic Na⁺ currents or in conditions with substantial gating pore currents was performed by offline subtraction (see Online supplemental material below). The Q_on–V relation for a series of test voltages was then fit with a Boltzmann distribution to determine the maximal gating charge displacement, Q_{on_max}, as channels transitioned from fully closed to fully open:

$$Q_{on}=Q_{on\_max}/\left(1+e^{-\left(V-V_{1/2}\right)/k}\right),$$

where V_1/2 is the voltage of half-maximal charge movement, and k is a steepness factor. In all figures, error bars represent ± SEM.

Online supplemental material

The supplemental text describes baseline correction (see also Fig. S1) for persistent gating pore currents that may have caused a modest overestimate of on-gating charge displacement from the integrated current, especially for R672G mutant channels, but the error is <15%. The online supplemental material is available at http://www.jgp.org/cgi/content/full/jgp.201411199/DC1.

RESULTS

A primary objective of these experiments was to measure the conventional ionic Na⁺ current (I_Na), the nonlinear gating charge displacement current (I_g), and for mutant channels, the ionic gating pore current (I_GP), all in the same oocyte. The motivation for these experiments was to gain a quantitative method to translate the magnitude of the anomalous gating pore currents observed for mutant Na_V1.4 channels expressed in oocytes to a predicted current density or specific conductance for the gating pore current in human HypoPP muscle fibers. Optimization of the experimental conditions was challenging because the high expression level required for detecting both the nonlinear charge displacement currents as well as gating pore currents resulted in space-clamp and series-resistance distortion of the ionic Na⁺ currents, which were enormous. Low Na⁺ solutions were not used to reduce voltage-clamp errors because permeant ion concentration has a marked effect on peak open probability (Townsend et al., 1997) and slow inactivation gating of Na_V1.4 (Townsend and Horn, 1997), and we did not want to assume that the peak I_Na would scale linearly with Na⁺ concentration. Nor did we want to use partial block with TTX, which would introduce another scale factor for relating I_GP to measured I_Na.

Fig. 1 shows a montage of representative I_Na, I_g, and I_GP currents, all recorded from one oocyte expressing WT, R669H, or R672G Na_V1.4 channels. Oocytes with similar peak I_Na current amplitudes were selected as a common basis for visual comparison. Display scales are identical across different channel types to facilitate a direct comparison of current amplitudes. The I_Na and I_g traces are P/N leak subtracted, whereas no digital subtraction has been performed on the I_GP traces in the bottom row. The current traces in a 115-mM Na⁺ bath without TTX (Fig. 1, top row) show that for oocytes with peak inward I_Na amplitudes of 10–20 µA, an early outward transient was pronounced for the mutant channels compared with WT. We interpret these transients to be nonlinear gating charge displacement currents that were unexpectedly larger in mutant channels compared with WT. Unlike our previous studies in HEK cells (Struyk et al., 2000), the decay of I_Na for R669H mutants expressed in oocytes was markedly slower than WT. This slower rate of macroscopic inactivation was accompanied by slower kinetics for the on-gating charge (Fig. 1, middle row).

Figure 1.

Currents, recorded under different conditions for the same oocyte, to measure I_Na, I_g, and I_GP. First, Na⁺ currents were recorded in a 115-mM Na⁺ bath with analogue compensation for linear capacitance and digital P/N subtraction of residual linear leak (top row). Responses are shown for test potentials of −20 to 30 mV. The external solution was then exchanged with K⁺/TTX (R672G) or NMDG⁺/TTX (R669H or WT), and P/N leak subtraction was performed to reveal the nonlinear gating charge displacement current, I_g (middle row). Finally, all analogue and digital P/N leak subtraction was omitted to reveal the gating pore current for mutant channels and the residual leakage current for oocytes expressing WT channels (bottom row). Each column shows the responses recorded from a single oocyte (black, WT; blue, R669H; red, R672G).

The external solution was then exchanged with an NMDG⁺/TTX solution for R669H or a K⁺/TTX solution for R672G and WT to suppress I_Na and record I_g and I_GP. The nonlinear charge displacement currents revealed by P/N leak subtraction (Fig. 1 middle row) were indeed substantially larger for oocytes expressing R669H and R672G channels that had I_Na amplitudes comparable to WT (Fig. 1, top row). For the sample I_g traces shown in Fig. 1, the maximal on-gating charge was Q_on = 0.90 nC for WT, 7.9 nC for R669H, and 2.8 nC for R672G. Pilot experiments with NMDG⁺ substituted for Na⁺, but without TTX, revealed very large nonlinear current transients in response to depolarization, with a similar appearance to on-gating charge displacement currents. We attribute this large current transient to egress of NMDG⁺ from within the transmembrane electric field of Na_V1.4, most likely the outer pore because it was abolished by the addition of 1 µM TTX (apparent Q_{on_max} decreased fivefold with TTX; not depicted). Capes et al. (2012) reported this same effect and also noted that the addition of TTX reduced the off-gating charge by 24%. The exceptionally large nonlinear current transients for R669H (Fig. 1, center of middle row) are not attributable to this phenomenon because comparably large Ig currents for R669H were recorded in control conditions with K⁺/TTX external solution. The steady-state inward current for R672G traces (Fig. 1, middle row) was caused by the gating pore current, which is permeable to NMDG⁺ and not blocked by TTX. The paradoxical appearance of larger inward steady-state current with more positive test-pulse depolarizations was caused by the leak-subtraction method. The gating pore was open during measurement of the “leak” response at −100 mV, and then closed with depolarization, thereby resulting in overcompensation for the P/N leak subtraction. The contribution from the gating pore current, I_GP, is shown more clearly by the steady-state values for traces in which no leak or offset subtraction has been performed (Fig. 1, bottom row). The traces for WT show a small nonspecific linear leak compared with the inward-rectifying I_GP currents observed in HypoPP mutant channels.

The coupling of gating charge displacement to channel opening was quantified by normalizing peak I_Na to Q_{on_max}, both measured from the same oocyte, and then computing the population behavior across many experiments in separate oocytes. Fig. 2 shows the mean and SEM for the ratio of I_Na/Q_{on_max} over a series of test potentials from −80 to 40 mV. This ratio was reduced twofold at all voltages for R672G and fourfold for R669H compared with WT. The data for WT channels showed evidence for poor voltage clamp with a left shift of the voltage at which the inward peak I_Na was maximal and there was a steepened increase in voltage dependence for opening. Partial compensation for these effects was obtained by rejecting oocytes in which the peak I_Na occurred at a test potential more negative than −30 mV. The reductions in normalized current amplitude for mutant channels from WT in Fig. 2 cannot be accounted for by a bias from poorer quality voltage clamp of WT channels alone. First, the maximal raw current amplitudes spanned a comparable range ([in µA] WT: mean of −25.8, range of −5.6 to −49; R669H: mean of −14.7, range of −1.1 to −36; R672G: mean of −25.6, range of −5.4 to −42), and the main cause for the difference in the ratio was the large Q_{on_max} for mutants (mean values of, [in nC] WT: 1.4, range of 0.22 to 2.5; R669H: 4.4, range of 1.1 to 10.5; R672G: 3.6, range of 1.8 to 7.4). Second, a bias from a greater loss of voltage-clamp control for WT channels would have caused a greater underestimate of I_Na amplitude compared with better clamped mutant channels, but just the opposite is shown in Fig. 2. Third, the reduced coupling ratio for mutant channels was prominent at strongly depolarized potentials (10–30 mV), where the voltage-clamp errors were much smaller because the peak amplitude of inward I_Na was much smaller.

Figure 2.

Voltage dependence for peak I_Na, normalized by maximal on-gating charge displacement, Q_{on_max}. Symbols show mean and SEM for WT (black; n = 15), R669H (blue; n = 34), and R672G (red; n = 18). Oocytes for which the maximal I_Na occurred at a voltage of less than −30 mV were excluded to reduce the effect of inadequate voltage-clamp control. Error bars represent ± SEM.

The variability of the peak I_Na to Q_{on_max} relationship is shown in the scatter plot of Fig. 3 A, where each symbol is from a separate oocyte. The data for WT channels were tightly clustered to the expected linear relation (17.1 ± 1.1 µA/nC; r² = 0.93). For mutant channels, a linear relation was also found; however, the scatter was substantially greater (R669H: 2.81 ± 0.36 µA/nC, r² = 0.63; R672G: 6.46 ± 0.73 µA/nC, r² = 0.77). We interpret this variability for mutant channels as a stable disruption of charge displacement coupling to channel opening. Repeated measurements from the same oocyte showed a consistent coupling ratio over tens of minutes without evidence for rapid fluctuation. Perhaps this stable variability is caused by uncontrolled differences in separate oocytes, but there was no clustering of weak versus strong coupling within a single harvest of oocytes. Moreover, the coupling variability was absent from WT channels expressed in the same batch of oocytes. This suggests the possibility for a range of coupling states for HypoPP mutant channels, and that the rate of switching between these states is very slow. Another source of variability was rundown, with I_Na and Q_{on_max} both slowly decreasing over tens of minutes, but the ratio was stable. A statistical test for the difference in coupling efficiency of mutant channels from WT was made by ANOVA for the I_Na/Q_{on_max} ratio at the voltage where peak I_Na was maximal. The spread of these data are shown in the box plot of Fig. 3 B. The coupling of charge displacement to channel opening was reduced for both HypoPP mutant channels compared with WT (P < 0.0001), with R669H likely being more severe than R672G (P = 0.027).

Figure 3.

Coupling of gating charge displacement to peak ionic current is diminished and more variable for HypoPP mutants than for WT channels. (A) Each symbol represents the maximal peak I_Na and associated Q_{on_max} for a separate oocyte. Lines show the best fit constrained through the origin. (B) The ensemble coupling behavior for oocytes among each channel type is summarized in a box plot. For each oocyte, the ratio I_Na/Q_{on_max} was computed. Box plots show the mean (circle), 50% level (bar), 25–75% range (box), and 10–90% range (whiskers). The coupling ratio was lower for HypoPP mutant channels than in WT (P < 0.0001). Error bars represent ± SEM.

The steady-state amplitudes of the gating pore currents recorded from mutant channels are shown in Fig. 4. The current–voltage relation in Fig. 4 A shows an apparent inward rectification from the voltage dependence of the S4 displacement being in the permissive position for current flow, with an inward bias at negative membrane potentials. Currents are displayed as normalized I_GP/_{Qon_max} so that responses from separate oocytes with different expression levels can be combined. The normalized current density for the cation-nonselective R672G mutant was larger than for the proton-selective R669H mutant. The limiting slope conductance at the normal resting potential for muscle of −90 mV was 230 nS/nC for R669H and 560 nS/nC for R672G. The variability of I_GP amplitude with Q_{on_max} is shown in the scatter plot of Fig. 4 B. A linear relation was observed for both mutants (R669H: 25.6 ± 1.2 nA/nC, r² = 0.93; R672G: 99.1 ± 5.5 nA/nC, r² = 0.95). Importantly, these data are from the same oocytes for which the I_Na–Q_{on_max} relation is shown in Fig. 3. The coupling between gating charge movement and I_GP amplitude was highly correlated (Fig. 4), whereas the coupling between Q_{on_max} and channel opening for mutant channels showed substantial variability (Fig. 3). This observation suggests that every mutant channel at the surface membrane with a functioning voltage-sensor domain contributes to the gating pore leakage current, whereas not all mutant channels with functioning voltage sensors result in channel opening with depolarization.

Figure 4.

Gating pore currents in HypoPP mutant channels. (A) The gating pore current was measured as the steady-state current at the end of a 150-ms pulse that was leak corrected by subtraction of the average steady-state current recorded from oocytes expressing WT Na_V1.4. The gating pore current from each oocyte was normalized by Q_{on_max} for the same egg and plotted as a function of test potential. Symbols show mean and SEM for R669H mutants (blue; n = 32) and for R672G mutants (red; n = 18). Error bars represent ± SEM. (B) The correlation between the gating pore current measured at −140 mV and Q_{on_max} is shown as a scatter plot. Each symbol represents the response from a separate oocyte. Lines show the best fit constrained through the origin. The correlation for I_GP–Q_{on_max} had much less variance than for I_Na–Q_{on_max} measured from the same set of oocytes (Fig. 3 A).

DISCUSSION

Our efforts to measure I_Na, Q_{on_max}, and I_GP all from the same oocyte, for the purpose equating I_GP current density in the oocyte to a predicted density in human muscle fibers, led to the unexpected finding of a dissociation between I_Na amplitude and Q_{on_max} for both HypoPP mutant channels. The data convincingly show that the ratio I_Na/Q_{on_max} is reduced for HypoPP mutants. The question is whether I_Na is smaller than expected or Q_{on_max} is too large. We favor the view that I_Na is smaller than normal. This hypothesis is consistent with the observation that dV/dt during the upstroke of the action potential is reduced in R672G human muscle (Jurkat-Rott et al., 2000). The basis for this apparent reduction in Na⁺ current density was unknown, as the only recognized gating defect at that time was enhanced inactivation, and this should have been compensated by the protocol to hyperpolarize fibers before eliciting the action potential. The oocyte data in our study now show that for equivalent levels of expression in the membrane (as quantified by Q_{on_max}), the Na⁺ current density would be reduced about twofold for R672G compared with WT. A similar finding was observed in our knock-in mutant mouse model of R669H (Wu et al., 2011). Heterozygous mutant mice had comparable transcript levels of WT and R669H alleles by RT-PCR, and yet dV/dt was reduced by 45% and the action potential amplitude was reduced by 29 mV. We interpret this relative reduction of I_Na compared with Q_{on_max} as a decoupling between voltage-sensor displacement to channel opening. The coupling defect appears to be noisy, with substantial variability from oocyte to oocyte (Fig. 3 A), but in the aggregate it is consistently reduced compared with WT channels. This defect had not been appreciated in prior expression studies of R669H or R672G in HEK cells (Jurkat-Rott et al., 2000; Struyk et al., 2000), because charge displacement currents were not measured in those studies. The observed mismatch of I_Na relative to Q_{on_max} for HypoPP mutant channels cannot be fully attributed to an overestimate of Q_{on_max} caused by the technical challenge of a superimposed I_GP current, because this error was at most 15% (supplemental text), whereas the ratio I_Na/Q_{on_max} was reduced 200% for R672H and 400% for R669H (Fig. 2). Our finding of a coupling defect could be a general mechanism by which mutations of arginine residues in S4 voltage sensors, as occurs in 14 of 15 published HypoPP mutations (Cannon, 2010), aggravate the susceptibility to failure of action potential propagation during an attack of periodic paralysis.

The absolute magnitude of I_Na/Q_{on_max} observed in our oocyte studies was smaller than that anticipated from first principles of channel biophysics, even for WT channels. Starting with the assumptions that (a) the transition from fully closed to open is associated with the translocation of 12 elementary charges per channel (Hirschberg et al., 1995), (b) the peak open probability for Na_V1.4 is ∼0.2 (Cannon and Strittmatter, 1993), and (c) the unitary current is ∼1 pA, then the predicted ratio is:

$$\frac{I_{Na}}{Q_{on\_\max }}=\frac{0.2*\left(1pA/channel\right)*\left({10}^{-6}\mu A/pA\right)}{\left(12e/channel\right)*\left(1.6\times {10}^{-10}nC/e\right)}=\hspace{0.17em}104\hspace{0.17em}\mu A/nC.$$

Our observation for WT channels expressed in oocytes is about sixfold less, 17 µA/nC (Fig. 3). Comparative numbers are not widely available in the literature because for most studies optimized to measure Ig, the ionic current density is too high to maintain voltage-clamp control and therefore I_Na is not reported. In neuroblastoma cells, a whole-cell peak I_Na of −11 nA was recorded in 160 mM Na⁺, after which block with TTX revealed a Q_{on_max} of 46 fC (Moran and Conti, 1990). These values are equivalent to a ratio of I_Na/Q_{on_max} = 240 µA/nC. In the squid giant axon, a Na⁺ current density of 3.8 mA/cm² in 350 mM NaCl was associated with a charge displacement of 1,600 e/µm² for a voltage step from −70 to 20 mV (Bezanilla and Armstrong, 1976). The equivalent Q_{on_max} is 25.6 nC/cm², and assuming that the Na⁺ current density scales linearly with Na⁺ concentration, it implies a ratio of I_Na/Q_{on_max} = 50 µA/nC in 120 mM Na⁺. This latter value for an invertebrate preparation is within threefold of the value we measured for WT Na_V1.4 expressed in oocytes. The reason for the discrepancy is not immediately apparent; perhaps a mammalian Na⁺ channel does not function efficiently in an amphibian expression system. Whatever the reason, this comparison to the behavior of endogenous channels illustrates the possible pitfall of estimating the I_GP density for HypoPP mutants expressed in mammalian fibers based on measurements in oocytes and assumptions about the ratio of I_Na to Q_{on_max}.

The measurement of I_Na, Q_{on_max}, and I_GP all from the same oocyte provides the information needed to predict the gating pore current density likely to occur in human HypoPP fibers, and also helps to resolve conflicting reports in the literature about the relative magnitude of I_Na and I_GP (Sokolov et al., 2007, 2010; Struyk and Cannon, 2007; Struyk et al., 2008). The preferred method for comparing the relative ease with which ions pass through the gating pore to that of Na⁺ through the conventional central pore is by a ratio of unitary conductances, γ. This method avoids the influences of driving force and open probability. Assuming Ohmic behavior for both ion-conducting pathways and that the number of active channels in the membrane is proportional to Q_{on_max} (valid for I_Na of WT channels [Fig. 3 A, black circles] and for I_GP conducted by HypoPP mutants [Fig. 4 B]), we can write the ratio as:

$$\frac{{\gamma }_{GP\_mut}}{{\gamma }_{Na\_WT}}=\frac{I_{GP\_mut}/Q_{on\_\text{max}\_mut}}{I_{\text{Na}\_WT}/Q_{on\_\text{max}\_WT}}\times \frac{P_{o\_Na\_WT}}{P_{o\_GP\_mut}}\times \frac{V_{Na}-E_{Na}}{V_{GP}-E_{GP}},$$

where the subscripts on V indicate the voltage at which the Na⁺ current or gating pore current were measured, V_Na = −25, and V_GP = −140 mV. The reversal potentials were E_Na = 45 mV and E_GP = 0 mV. We assumed P_o__GP__mut = 1 at −140 mV, and used the published value of peak P_o__Na__WT = 0.2 (Cannon and Strittmatter, 1993). The results of these calculations are summarized in Table 1. The relative conductance of the gating pore in HypoPP mutant channels to the conventional pore of a WT channel is ∼0.03% (see fourth row in Table 1). The unitary conductance of Na_V1.4 in 140 mM Na⁺ is ∼20 pS (Cannon and Strittmatter, 1993), which implies that the unitary gating pore conductance is remarkably small, on the order of 10 fS.

Table 1.

Parameter estimates for the gating pore conductance

Parameter	Units	WT	R669H	R672G
INa_max/Qon_max	nA/nC	17,400 ± 110	2,810 ± 360	6,460 ± 730
IGP/Qon_max	nA/nC	NA	25.6 ± 1.2	99.1 ± 5.5
IGP/INa_mut	—	NA	0.0091	0.0155
γGP/γNa_WT	—	NA	0.000174	0.000571
γGP: γNa_WT		NA	1:5, 750	1:1, 750
γGPa	fS	NA	2.95	11.4
Predicated GP conductance in HypoPP muscle
GGPb	µS/cm2	NA	4.9	19.0
GGP/Gtotal	—	NA	0.005	0.019

^aAssumes γ_{Na_WT} = 20 pS.

^bAssumes G_{Na_max} = 65 mS/cm² for WT Na_V1.4 in mammalian skeletal muscle.

Apparent discrepancies in the reported values for the relative amplitude of the gating pore and central Na⁺ currents are primarily caused by different methods to describe the relative permeation of the two pathways. Sokolov et al. (2007) characterized the relative amplitude of the gating pore conductance for R672G as 1.2% for the central pore Na⁺ current, based on currents recorded before and after the addition of TTX in one favorable oocyte where voltage-clamp control was sufficient. This value was calculated from a consideration of two physiological anchor points in skeletal muscle: the resting potential of −90 mV where for this oocyte, I_GP was 50 nA, and the peak of an action potential of 30 mV where in this case, I_Na was 4 µA. We initially estimated the ratio of the gating pore current to the peak Na⁺ current for R669H to be ∼0.1% (Struyk and Cannon, 2007), based on the observation that the amplitude of I_GP (−140 mV) was ∼1/20th of the peak amplitude for the charge displacement current, Ig, which in turn is ∼50-fold smaller than the peak I_Na (Armstrong and Bezanilla, 1974). A similar conclusion of 0.1% was made for subsequent measures of I_GP for three other HypoPP mutations at R672H/S/C, now based on normalization to Q_{on_max} and the assumptions that each channel contributes 12 elementary charges and has a unitary current of 1 pA (Struyk et al., 2008). Subsequently, Sokolov et al. (2010) commented that including a peak P_{o_max} of 0.2 for Na_V1.4 in our calculation, and considering the possibility of partial I_GP block by divalent cations, may account for the 10-fold difference. With the extensive set of data in the present study where I_Na, Q_{on_max}, and I_GP were all measured in the same oocyte, there is now agreement that the relative amplitude of I_GP (−140) to the maximal peak I_Na for that same mutant channel is ∼1% (0.9–1.6%; Table 1, row 3). Our data also identify a defect in coupling between Q_{on_max} and I_Na for HypoPP mutant channels (Fig. 3), which would have caused the experimentally measured ratio to be about two to four times larger than predicted based on Q_{on_max} and coupling for WT channels. In our view, the more physiologically relevant measure is the conductance ratio of γ_GP in a HypoPP mutant to γ_Na in a WT channel. This unitary conductance ratio is ∼0.03% (Table 1, row 4). The 30-fold smaller value relative to the ratio of I_GP/I_{Na_mut} is accounted for by the multiplicative effects of peak P_o (fivefold), decoupling of charge displacement to channel opening (two- to fourfold), and the ratio of the driving forces at the two test potentials (twofold).

Knowing the relative amplitudes for I_Na, Q_{on_max}, and I_GP in the same oocyte, we have greater confidence in estimating the magnitude of the anomalous gating pore conductance in human HypoPP muscle fibers. The predicted value is 5 µS/cm² for R669H and 20 µS/cm² for R672G. This calculation assumes that in heterozygous patients, WT and HypoPP Na_V1.4 alleles are expressed at the same level, as we have confirmed experimentally in a knock-in mouse model (Wu et al., 2011), and that the peak Na⁺ conductance in mammalian skeletal muscle is 65 mS/cm² (Fu et al., 2011). This prediction of 5 µS/cm² for the R669H gating pore conductance in heterozygous patients is remarkably similar to the value of 7 µS/cm² we observed in homozygous R669H mutant mice (Wu et al., 2011). The measurement of I_GP in voltage-clamped mouse fibers was extremely challenging because of the difficulty in adequately blocking all of the normal ionic currents, and the concordance between the oocyte and fiber studies is an important confirmation. In HypoPP fibers, this gating pore conductance will be active at the resting potential and would account for ∼0.5–2% of the total resting conductance. Although this may appear to be a small disturbance to the stability of the resting potential, computer simulation has shown that the resulting inward current is sufficient to create a susceptibility to paradoxical depolarization in low-normal extracellular K⁺ levels (Struyk and Cannon, 2008; Jurkat-Rott et al., 2010). The depolarized shift of the resting potential and inactivation of Na⁺ channels cause loss of excitability and flaccid paralysis during transient attacks of weakness in HypoPP (Rüdel et al., 1984).