Isomorphism on a physical system of the Hodgkin-Huxley equations for sodium conductance: toxin allosteric effects and gating currents

An isomorphism on a physical system of the Hodgkin-Huxley equations for sodium ion conductance in the nerve membrane is derived. The physical system consists of 8 states. It shows that the voltage dependence of the sodium conductance arises from a change in ionization of the molecule that provides the ion-selective conductance channels. It associates reversibly with singly charged (H+?) and doubly charged (Ca++?) ions. The inactivation process is the result of the associating of an ionized particle by half of the states. The effect of toxins and narcotics in blocking or inactivating sodium conductance can be understood as an enzyme or allosteric change of the standard free energy difference of the molecule that provides the sodium channels. The effect of changing pH and Ca++ substrate concentration on the sodium conductance is predicted. The gating charge current is predicted. The time constant predicted is in agreement with experiment.     

Multiple-state model of ionic channel in reproducing the time course of electrophysiological events.

BACKGROUND: The predictions of the Hodgkin-Huxley model do not accurately fit all the measurements of voltage-clamp currents, gating charge and single-channel currents. There are many quantitative differences between the predicted and measured characteristics of the sodium and potassium channels. For example, the two-state gate model has exponential onset kinetics, whereas the sodium and potassium conductances show S-shaped activation and the sodium conductance shows an exponential inactivation. In this paper we shall examine a more general channel model that can more faithfully represent the measured properties of ionic channels in the membrane of the excitable cell. METHODS: The model is based on the generalisation of the notion of a channel with a discrete set of states. Each state has state attributes such as the state conductance, state ionic current and state gating charge. These variables can have quite different waveforms in time, in contrast with a two-state gate channel model, in which all have the same waveforms. RESULTS: The kinetics of all variables are equivalent: gating and ionic currents give equivalent information about channel kinetics; both the equilibrium values of the current and the time constants are functions of membrane potential. The results are in almost perfect concordance with the experimental data regarding the characteristics of nerve impulse. CONCLUSIONS: The expected values of the gating charge and the ionic conductance are weighted sums of the state occupancy probabilities, but the weights differ: for the expected value of the gating charge the weights are the state gating charges and for the expected value of the ionic conductance the weights are the state conductances. Since these weights are different, the expected values of the gating charge and the ionic conductance will differ.     

Computation of axon gating currents from dipole moment changes in channel subunits.

The gating polarizational currents were computed on the basis of the dipole moment changes occurring in nerve membrane ionic channel subunits. Membrane thickness and surface density of channels were the only parameters used in addition to the Hodgkin-Huxley model. The gating currents computed for membrane potentials where the Hodgkin-Huxley empirical formulae are reliable were found to be in good agreement with the available experimental data. It is demonstrated that the gating currents of the n and h subunits are responsible for the late slowly decaying gating currents.     

Tracking S4 movement by gating pore currents in the bacterial sodium channel NaChBac

Voltage-gated sodium channels mediate the initiation and propagation of action potentials in excitable cells. Transmembrane segment S4 of voltage-gated sodium channels resides in a gating pore where it senses the membrane potential and controls channel gating. Substitution of individual S4 arginine gating charges (R1–R3) with smaller amino acids allows ionic currents to flow through the mutant gating pore, and these gating pore currents are pathogenic in some skeletal muscle periodic paralysis syndromes. The voltage dependence of gating pore currents provides information about the transmembrane position of the gating charges as S4 moves in response to membrane potential. Here we studied gating pore current in mutants of the homotetrameric bacterial sodium channel NaChBac in which individual arginine gating charges were replaced by cysteine. Gating pore current was observed for each mutant channel, but with different voltage-dependent properties. Mutating the first (R1C) or second (R2C) arginine to cysteine resulted in gating pore current at hyperpolarized membrane potentials, where the channels are in resting states, but not at depolarized potentials, where the channels are activated. Conversely, the R3C gating pore is closed at hyperpolarized membrane potentials and opens with channel activation. Negative conditioning pulses revealed time-dependent deactivation of the R3C gating pore at the most hyperpolarized potentials. Our results show sequential voltage dependence of activation of gating pore current from R1 to R3 and support stepwise outward movement of the substituted cysteines through the narrow portion of the gating pore that is sealed by the arginine side chains in the wild-type channel. This pattern of voltage dependence of gating pore current is consistent with a sliding movement of the S4 helix through the gating pore. Through comparison with high-resolution models of the voltage sensor of bacterial sodium channels, these results shed light on the structural basis for pathogenic gating pore currents in periodic paralysis syndromes.     

Biological effects of oscillating electric fields: Role of voltage-sensitive ion channels

An alternating component of potential across the membrane of an excitable cell may change the membrane conductance by interacting with the voltagesensing charged groups of the protein macromolecules that form voltage-sensitive ion channels. Because the probability that a voltage sensor is in a given state is a highly nonlinear function of the applied electric field, the average occupancy of a particular state will change in an oscillating electric field of sufficient magnitude. This “rectification” at the level of the voltage sensors could result in conformational changes (gating) that would modify channel conductance. A simplified two-state model is examined where the relaxation time of the voltage sensor is assumed to be considerably faster than the fastest changes of ionic conductance. Significant changes in the occupancy of voltage sensor states in response to an applied oscillating electric field are predicted by the model.     

Gating of the bacterial sodium channel, NaChBac: voltage-dependent charge movement and gating currents

The bacterial sodium channel, NaChBac, from Bacillus halodurans provides an excellent model to study structure-function relationships of voltage-gated ion channels. It can be expressed in mammalian cells for functional studies as well as in bacterial cultures as starting material for protein purification for fine biochemical and biophysical studies. Macroscopic functional properties of NaChBac have been described previously (Ren, D., B. Navarro, H. Xu, L. Yue, Q. Shi, and D.E. Clapham. 2001. Science. 294:2372-2375). In this study, we report gating current properties of NaChBac expressed in COS-1 cells. Upon depolarization of the membrane, gating currents appeared as upward inflections preceding the ionic currents. Gating currents were detectable at -90 mV while holding at -150 mV. Charge-voltage (Q-V) curves showed sigmoidal dependence on voltage with gating charge saturating at -10 mV. Charge movement was shifted by -22 mV relative to the conductance-voltage curve, indicating the presence of more than one closed state. Consistent with this was the Cole-Moore shift of 533 micros observed for a change in preconditioning voltage from -160 to -80 mV. The total gating charge was estimated to be 16 elementary charges per channel. Charge immobilization caused by prolonged depolarization was also observed; Q-V curves were shifted by approximately -60 mV to hyperpolarized potentials when cells were held at 0 mV. The kinetic properties of NaChBac were simulated by simultaneous fit of sodium currents at various voltages to a sequential kinetic model. Gating current kinetics predicted from ionic current experiments resembled the experimental data, indicating that gating currents are coupled to activation of NaChBac and confirming the assertion that this channel undergoes several transitions between closed states before channel opening. The results indicate that NaChBac has several closed states with voltage-dependent transitions between them realized by translocation of gating charge that causes activation of the channel.     

Voltage-dependent membrane capacitance in rat pituitary nerve terminals due to gating currents

We investigated the voltage dependence of membrane capacitance of pituitary nerve terminals in the whole-terminal patch-clamp configuration using a lock-in amplifier. Under conditions where secretion was abolished and voltage-gated channels were blocked or completely inactivated, changes in membrane potential still produced capacitance changes. In terminals with significant sodium currents, the membrane capacitance showed a bell-shaped dependence on membrane potential with a peak at approximately -40 mV as expected for sodium channel gating currents. The voltage-dependent part of the capacitance showed a strong correlation with the amplitude of voltage-gated Na+ currents and was markedly reduced by dibucaine, which blocks sodium channel current and gating charge movement. The frequency dependence of the voltage-dependent capacitance was consistent with sodium channel kinetics. This is the first demonstration of sodium channel gating currents in single pituitary nerve terminals. The gating currents lead to a voltage- and frequency-dependent capacitance, which can be well resolved by measurements with a lock-in amplifier. The properties of the gating currents are in excellent agreement with the properties of ionic Na+ currents of pituitary nerve terminals.     

Correlative studies of gating in Cx46 and Cx50 hemichannels and gap junction channels

Transjunctional voltage (V(j)) gating of gap junction (GJ) channels formed of connexins has been proposed to occur by gating of the component hemichannels. We took advantage of the ability of Cx46 and Cx50 to function as unapposed hemichannels to identify gating properties intrinsic to hemichannels and how they contribute to gating of GJ channels. We show that Cx46 and Cx50 hemichannels contain two distinct gating mechanisms that generate reductions in conductance for both membrane polarities. At positive voltages, gating is similar in Cx46 and Cx50 hemichannels, primarily showing increased transitioning to long-lived substates. At negative voltages, Cx46 currents deactivate completely and the underlying single hemichannels exhibit transitions to a fully closed state. In contrast, Cx50 currents do not deactivate completely at negative voltages and the underlying single hemichannels predominantly exhibit transitions to various substates. Transitions to a fully closed state occur, but are infrequent. In the respective GJ channels, both forms of gating contribute to the reduction in conductance by V(j). However, examination of gating of mutant hemichannels and GJ channels in which the Asp at position 3 was replaced with Asn (D3N) showed that the positive hemichannel gate predominantly closes Cx50 GJs, whereas the negative hemichannel gate predominantly closes Cx46 GJs in response to V(j). We also report, for the first time, single Cx50 hemichannels in oocytes to be inwardly rectifying, high conductance channels (gamma = 470 pS). The antimalarial drug mefloquine, which selectively blocks Cx50 and not Cx46 GJs, shows the same selectivity in Cx50 and Cx46 hemichannels indicating that the actions of such uncoupling agents, like voltage gating, are intrinsic hemichannel properties.     

Pharmacological and kinetic analysis of K channel gating currents

We have measured gating currents from the squid giant axon using solutions that preserve functional K channels and with experimental conditions that minimize Na channel contributions to these currents. Two pharmacological agents were used to identify a component of gating current that is associated with K channels. Low concentrations of internal Zn2+ that considerably slow K channel ionic currents with no effect on Na channel currents altered the component of gating current associated with K channels. At low concentrations (10-50 microM) the small, organic, dipolar molecule phloretin has several reported specific effects on K channels: it reduces K channel conductance, shifts the relationship between channel conductance and membrane voltage (Vm) to more positive potentials, and reduces the voltage dependence of the conductance-Vm relation. The K channel gating charge movements were altered in an analogous manner by 10 microM phloretin. We also measured the dominant time constants of the K channel ionic and gating currents. These time constants were similar over part of the accessible voltage range, but at potentials between -40 and 0 mV the gating current time constants were two to three times faster than the corresponding ionic current values. These features of K channel function can be reproduced by a simple kinetic model in which the channel is considered to consist of two, two-state, nonidentical subunits.     

Dipole moment changes and voltage dependent membrane capacity of squid axon

Changes in the membrane capacity of squid axons during hyper- and depolarizations are measured between ?160 and +40 mV. After corrections for the series resistance and fringe effect, we found that the membrane capacity increased from 0.68 to 1.2 μF/cm² with depolarization. It was further observed that tetrodotoxin in the external medium eliminated the change in membrane capacity without affecting the conductivity. The voltage-dependent membrane conductivity is, in turn, greatly reduced by the internal cesium ion. These observations clearly indicate that the voltage-dependent membrane capacity and conductivity are closely related to ionic channels. Particularly, the increase in membrane capacity with depolarizations may be due to sodium channels. The change in the dipole moment associated with sodium sites was determined using values of α_{m and}β_m at various depolarizations. We found, based on voltage clamp measurements, that the increase in the dipole moment of the sodium site between ?40 and ?5 mV is 1230 Debye units (D.U.) and 930 D.U. between ?5 and +60 mV, indicating that the depolarization of sodium channels may consist of two different steps.     

Gating currents of sodium channels in neurons of the rat trigeminal ganglia

Using a voltage-clamp technique and intracellular dialysis, gating currents of sodium channels were first recorded and studied in neurons of the rat trigeminal ganglia. The rising phase of gating currents lasted 30 to 70 µsec; these currents decayed in a monoexponential manner with a time constant equal to that for activation of the sodium current. Voltage dependences for the gating charge and sodium conductance were also nearly identical. Analysis of the activation of sodium conductance demonstrated that the power n of the activation variable in the equation used changed from more than 6 to 3 at test potentials of –30 mV and 0 mV, respectively. It is hypothesized that, with a change in the test potential within this voltage range, the cooperativity of activation undergoes a twofold decrease. In the presence of 2 mM caffeine or theophylline in the external solution, curves of the voltage dependence of the gating charge and sodium conductance shifted toward more negative values of the test potential, by 5.4 ± 0.7 mV, the maximum gating charge increased by 8.4 ± 3.2%, and the slope factor for both curves decreased by 9.2 ± 3.4%. Since the above effects were identical for both xanthines and developed under conditions of constant intracellular dialysis, i.e., under conditions where the effect of a change in the intracellular calcium concentration was ruled out, the most probable reason for these effects is a direct action of the tested agents on sodium ion channels, which facilitates the movement of gating charges.Neirofiziologiya/Neurophysiology, Vol. 36, Nos. 5/6, pp. 370–376, September–December, 2004.This revised version was published online in April 2005 with a corrected cover date and copyright year.     

Sodium ionic and gating currents in mammalian cells

Ionic and gating currents from voltage-gated sodium channels were recorded in mouse neuroblastoma cells using the path-clamp technique. Displacement currents were measured from whole-cell recordings. The gating charge displaced during step depolarizations increased with the applied membrane potential and reached saturating levels above 20 mV Prolonged large depolarizations produced partial immobilization of the gating charge, and only about one third of the displaced charge was quickly reversed upon return to negative holding potentials. The activation and inactivation properties of macroscopic sodium currents were characterized by voltage-clamp analysis of large outside-out patches and the single-channel conductance was estimated from nonstationary noise analysis. The general properties of the sodium channels in mouse neuroblastoma cells are very similar to those previously reported for various preparations of invertebrate and vertebrate nerve cells. Offprint requests to: O. Moran     

Rate of opening of a population of Na channels in frog skeletal muscle fibers

Linear Systems convolution analysis of muscle sodium currents was used to predict the opening rate of sodium channels as a function of time during voltage clamp pulses. If open sodium channel lifetimes are exponentially distributed, the channel opening rate corresponding to a sodium current obtained at any particular voltage, can be analytically obtained using a simple equation, given single channel information about the mean open-channel lifetime and current.Predictions of channel opening rate during voltage clamp pulses show that sodium channel inactivation arises coincident with a decline in channel opening rate.Sodium currents pharmacologically modified with Chloramine-T treatment so that they do not inactivate, show a predicted sustained channel opening rate.Large depolarizing voltage clamp pulses produce channel opening rate functions that resemble gating currents.The predicted channel opening rate functions are best described by kinetic models for Na channels which confer most of the charge movement to transitions between closed states.Comparisons of channel opening rate functions with gating currents suggests that there may be subtypes of Na channel with some contributing more charge movement per channel opening than others.Na channels open on average, only once during the transient period of Na activation and inactivation.After transiently opening during the activation period and then closing by entering the inactivated state, Na channels reopen if the voltage pulse is long enough and contribute to steady-state currents.The convolution model overestimates the opening rate of channels contributing to the steady-state currents that remain after the transient early Na current has subsided.     

The representation of membrane admittance.

An integral representation for the membrane admittance in terms of its known current response to a voltage step function is presented. It is demonstrated that the frequency-dependent terms in the contribution to the membrane admittance by the ion-selective conductance of the nerve membrane are proportional to the static conductances. The additional information contained in the real and imaginary parts of the membrane admittance should allow the parameters of the ion conductance to be determined. Eventually, these measurements should also give information about the electric dipole displacement currents of the conductance systems themselves, and about the metabolically supported active ion transport currents that maintain the ion concentration gradients.     

Gating of Shaker K+ channels: I. Ionic and gating currents.

Ionic and gating currents from noninactivating Shaker B K+ channels were studied with the cut-open oocyte voltage clamp technique and compared with the macropatch clamp technique. The performance of the cut-open oocyte voltage clamp technique was evaluated from the electrical properties of the clamped upper domus membrane, K+ tail current measurements, and the time course of K+ currents after partial blockade. It was concluded that membrane currents less than 20 microA were spatially clamped with a time resolution of at least 50 microseconds. Subtracted, unsubtracted gating currents with the cut-open oocyte voltage clamp technique and gating currents recorded in cell attached macropatches had similar properties and time course, and the charge movement properties directly obtained from capacity measurements agreed with measurements of charge movement from subtracted records. An accurate estimate of the normalized open probability Po(V) was obtained from tail current measurements as a function of the prepulse V in high external K+. The Po(V) was zero at potentials more negative than -40 mV and increased sharply at this potential, then increased continuously until -20 mV, and finally slowly increased with voltages more positive than 0 mV. Deactivation tail currents decayed with two time constants and external potassium slowed down the faster component without affecting the slower component that is probably associated with the return between two of the closed states near the open state. In correlating gating currents and channel opening, Cole-Moore type experiments showed that charge moving in the negative region of voltage (-100 to -40 mV) is involved in the delay of the conductance activation but not in channel opening. The charge moving in the more positive voltage range (-40 to -10 mV) has a similar voltage dependence to the open probability of the channel, but it does not show the gradual increase with voltage seen in the Po(V).     

Gating currents in the node of Ranvier: voltage and time dependence.

Like the axolemma of the giant nerve fibre of the squid, the nodal membrane of frog myelinated nerve fibres after blocking transmembrane ionic currents exhibits asymmetrical displacement currents during and after hyperpolarizing and depolarizing voltage clamp pulses of equal size. The steady-state distribution of charges as a function of membrane potential is consistent with Boltzmanns law (midpoint potential minus 33.7 mV; saturation value 17200 charges/mum-2). The time course of the asymmetry current and the voltage dependence of its time constant are consistent with the notion that due to a sudden change in membrane potential the charges undergo a first order transition between two configurations. Size and voltage dependence of the time constant are similar to those of the activation of the sodium conductance assuming m-2h kinetics. The results suggest that the presence of ten times more sodium channels (5000/mum-2) in the node of Ranvier than in the squid giant axon with similar sodium conductance per channel (2-3 pS).     

Dipole cooperative model of ion channels of excitable biomembranes. Conductance of ion channels

An analysis of ionic channel conductance is presented in terms of dipole cooperative model. The dependence of conductance on displaced charge is found to be an S-shaped function. Basing on this function and kinetics of gating currents, the kinetic curves for the conductance are calculated. These curves are compared with Hodgkin--Huxley results on sodium channel. A good agreement may be observed for the case of positive jumps of the potential. Less accurate coincidence takes place for negative jumps of the potential.     

pH-regulated Slo3 K+ channels: properties of unitary currents

Here we have examined the voltage and pH dependence of unitary Slo3 channels and used analysis of current variance to define Slo3 unitary current properties over a broader range of voltages. Despite complexity in Slo3 channel openings that precludes simple definition of the unitary conductance, average current through single Slo3 channels varies linearly with voltage at positive activation potentials. Furthermore, the average Slo3 unitary current at a given activation potential does not change with pH. Consistent with macroscopic conductance estimates, the apparent open probability of Slo3 channel exhibits a pH-dependent maximum, with limiting values around 0.3 at the most elevated pH and voltage. Estimates of Slo3 conductance at negative potentials support a weaker intrinsic voltage dependence of gating than is observed for Slo1. For the pH-regulated Slo3 K(+) channel, the dependence of macroscopic conductance on pH suggests that the pH-sensitive mechanism regulates gating in an allosteric manner qualitatively similar to regulation of Slo1 by Ca(2+). Together, the results support the view that the regulation of macroscopic Slo3 currents by pH reflects regulation of gating equilibria, and not a direct effect of pH on ion permeation. Specifically, both voltage and pH regulate a closed-open conformational change in a largely independent fashion.     

Voltage-dependent gating of Shaker A-type potassium channels in Drosophila muscle

The voltage-dependent gating mechanism of A1-type potassium channels coded for by the Shaker locus of Drosophila was studied using macroscopic and single-channel recording techniques on embryonic myotubes in primary culture. From a kinetic analysis of data from single A1 channels, we have concluded that all of the molecular transitions after first opening, including the inactivation transition, are voltage independent and therefore not associated with charge movement through the membrane. In contrast, at least some of the activation transitions leading to first opening are considerably voltage dependent and account for all of the voltage dependence seen in the macroscopic currents. This mechanism is similar in many ways to that of vertebrate neuronal voltage-sensitive sodium channels, and together with the sequence similarities in the S4 region suggests a conserved mechanism for voltage-dependent gating among channels with different selectivities. By testing independent and coupled models for activation and inactivation we have determined that the final opening transition and inactivation are not likely to arise from the independent action of multiple subunits, each with simple gating transitions, but rather come about through their aggregate properties. A partially coupled model accurately reproduces all of the single-channel and macroscopic data. This model will provide a framework on which to organize and understand alterations in gating that occur in Shaker variants and mutants.