The propagation of the nerve impulse.

A partial differential equation for the propagated action potential is derived using symmetry, charge conservation, and Ohm's law. Charge conservation analysis explicitly includes the gating charge when applied in the laboratory frame. When applied in the system of reference in which capacitive currents are zero, it yields a relation between orthogonal components of the ionic current allowing us to express the nonlinear ionic current in terms of the voltage-dependent membrane capacitance C(V) and the axial current that satisfies Ohm's law. The ionic current is shown to behave as C(V)V[C(V)V2]' at the foot of the action potential while the gating current behaves as C(V)V[Cg(V)V]' where Cg(V) is the capacitance associated with gating. Improved knowledge of the nonlinear current makes it possible to describe the propagated action potential in an approximated way with quasilinear partial differential equations. These equations have analytical solutions that travel with constant velocity, retain their shape, and account for other properties of the action potential. Furthermore, the quasilinear approximation is shown to be equivalent to the FitzHugh-Nagumo equation without recovery making apparent its physical content.     

Hill coefficient for estimating the magnitude of cooperativity in gating transitions of voltage-dependent ion channels

A frequently used measure for the extent of cooperativity in ligand binding by an allosteric protein is the Hill coefficient, obtained by fitting data of initial reaction velocity (or fractional binding saturation) as a function of substrate concentration to the Hill equation. Here, it is demonstrated that the simple two-state Boltzmann equation that is widely used to fit voltage-activation data of voltage-dependent ion channels is analogous to the Hill equation. A general empiric definition for a Hill coefficient (n(H)) for channel gating transitions that is analogous to the logarithmic potential sensitivity function of Almers is derived. This definition provides a novel framework for interpreting the meaning of the Hill coefficient. In considering three particular and simple gating schemes for a voltage-activated cation channel, the relation of the Hill coefficient to the magnitude and nature of cooperative interactions along the reaction coordinate of channel gating is demonstrated. A possible functional explanation for the low value of the Hill coefficient for gating transitions of the Shaker voltage-activated K(+) channel is suggested. The analogy between the Hill coefficients for ligand binding and for channel gating transitions further points to a unified conceptual framework in analyzing enzymes and channels behavior.     

Estimation of ionic channel surface density from relaxation ionic and gating currents

It is shown that excitable membrane surface density of channels can be estimated from ionic and gating current relaxations. The gating currents are determined thermodynamically from a multistate kinetic model. The parameters of the kinetic model are derived from ionic current relaxations. The assumptions regarding the gating process made here are the same as those made in fluctuation analysis previously regarded as the only method that may yield channel density from membrane currents.     

Charge immobilization linked to inactivation in the aggregation model of channel gating

Expected gating currents are derived analytically from a continuous, time-homogenous Markov process formulation of the random behavior of a single aggregation gating site. The concept of aggregation gating involves a voltage-dependent reversible conformational change and a voltage-independent reversible aggregation process. A site is assumed to consist of four hypothetical protein subunits. Based on these assumptions the model is defined by the scheme of transitions between 12 possible site configurations. The model can account for the phenomenon of charge immobilization in asymmetry current data of the voltage-clamped sodium conductance system. It predicts gating currents without a rising phase. A rising phase is obtained, however, if the model is subjected to conventional symmetrical pulse protocols for measuring asymmetry currents in the axon. Novel pulse protocols are given that do not result in a rising phase if applied to the aggregation model. Simplified transition schemes that describe the basic kinetic behavior of the potassium and the sodium conductance system are derived by eliminating transitions of negligible probability from the original scheme.     

Continuum molecular simulation of large conformational changes during ion-channel gating

A modeling framework was developed to simulate large and gradual conformational changes within a macromolecule (protein) when its low amplitude high frequency vibrations are not concerned. Governing equations were derived as alternative to Langevin and Smoluchowski equations and used to simulate gating conformational changes of the Kv7.1 ion-channel over the time scale of its gating process (tens of milliseconds). The alternative equations predict the statistical properties of the motion trajectories with good accuracy and do not require the force field to be constant over the diffusion length, as assumed in Langevin equation. The open probability of the ion-channel was determined considering cooperativity of four subunits and solving their concerted transition to the open state analytically. The simulated open probabilities for a series of voltage clamp tests produced current traces that were similar to experimentally recorded currents.     

The kinetics of monazomycin-induced voltage-dependent conductance. II. Theory and a demonstration of a form of memory

The empirical differential equation that describes the kinetics of monazomycin-induced voltage-dependent conductance is derived using a standard chemical kinetic formulation and the assumption that monazomycin entry into and exit from the membrane is autocatalytic. The predicted form of gating currents is shown and numerical calculations for this process are made using a range of values for two unmeasured variables. A form of "memory" is then demonstrated, along with the ability of the theoretical equation to explain the nature of the memory.     

The kinetics of monazomycin-induced voltage-dependent conductance. I. Proof of the validity of an empirical rate equation

Monazomycin (a positively-charged, polyene-like antibiotic) induces a strongly voltage-dependent conductance in thin lipid membranes when added to one of the bathing solutions. We show here that the kinetics of conductance changes after a step of membrane potential are only superficially similar to the kinetics of the potassium gating system of squid giant axons, in that the beginning of conductance increases are growth functions of the time, as opposed to power functions of the time. We find that the rate constant (reciprocal of the time constant) of the growth varies with the approximately 2.6 power of the monazomycin concentration. The rate constant also varies exponentially with membrane potential such that an e-fold change is associated with a 10-11 mV change of membrane potential. We show that solutions of a simple differential equation are able to reproduce the actual conductance changes almost exactly. In the accompanying paper (Muller and Peskin. 1981. J. Gen. Physiol. 78:201-229), we derive the differential equation from a molecular model and use the theoretical equation so obtained to investigate the gating current of this system and to predict an interesting form of memory.     

Single channel gating events in tracer flux experiments I. Theory

Tracer ion flux measurements are a commonly used method for studying ion transport through membranes of cellular systems, where the rate of ion flow is determined by gating processes which control the opening and closing of transmembrane channels. Due to recent advances in the theoretical analysis of tracer flux from or into closed membrane structures (CMS), the mechanism of gating reactions can, in principle, be derived from flux data. A physically well founded analysis is presented for the dependence of the total tracer ion content of a collection of CMS on the gating processes. For functionally uncoupled gating units a mean single channel flux contribution [equation, see text] can be defined, where k is the intrinsic single channel flux coefficient, t the time over which flux is measured, and p(tau,t) is the probability that a given channel was open for a total period tau during t. This quantity reflects the mean time course of the tracer content due to flux through a single channel. Expressions for are derived that explicitly take into account a distribution in the lifetime of open channels. On the basis of the results, kinetic and thermodynamic parameters of multiphasic gating reactions can be determined from the time course of the overall tracer content in a colleciion of CMS.     

Current noise spectrum and capacitance due to the membrane motor of the outer hair cell: theory

The voltage-dependent motility of the outer hair cell is based on a membrane motor densely distributed in the lateral membrane. The gating charge of the membrane motor is manifested as a bell-shaped membrane potential dependence of the membrane capacitance. In this paper it is shown that movements of the gating charge should produce a high-pass current noise described by an inverse Lorentzian similar to the one shown by Kolb and Läuger for ion carriers. The frequency dependence of the voltage-dependent capacitance is also derived. These derivations are based on membrane motor models with two or three states. These two models lead to similar predictions on the capacitance and current noise. It is expected that the examination of the spectral properties of these quantities would be a useful means of determining the relaxation time for conformational transitions of the membrane motor.     

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.     

Gating current harmonics. III. Dynamic transients and steady states with intact sodium inactivation gating.

Internally perfused squid giant axons with intact sodium inactivation gating were prepared for gating current experiments. Gating current records were obtained in sinusoidally driven dynamic steady states and as dynamic transients as functions of the mean membrane potential and the frequency of the command sinusoid. Controls were obtained after internal protease treatment of the axons that fully removed inactivation. The nonlinear analysis consisted of determining and interpreting the harmonic content in the current records. The results indicate the presence of three kinetic processes, two of which are associated with activation gating (the so-called primary and secondary processes), and the third with inactivation gating. The dynamic steady state data show that inactivation gating does not contribute a component to the gating current, and has no direct voltage-dependence of its own. Rather, the inactivation kinetics appear to be coupled to the primary activation kinetics, and the coupling mechanism appears to be one of reciprocal steric hindrance between two molecular components. The mechanism allows the channel to become inactivated without first entering the conducting state, and will do so in about 40 percent of depolarizing voltage-clamp steps to 0 mV. The derived model kinetics further indicate that the conducting state may flicker between open and closed with the lifetime of either state being 10 microseconds. Dynamic transients generated by the model kinetics (i.e., the behavior of the harmonic components as a function of time after an instantaneous change in the mean membrane potential from a holding potential of -80 mV) match the experimental dynamic transients in all details. These transients have a duration of 7-10 ms (depending on the level of depolarization), and are the result of the developing inactivation following the discontinuous voltage change. A detailed hypothetical molecular model of the channel and gating machinery is presented.     

A resonance model gives the response to membrane potential for an ion channel: II. Simplification of the calculation, and prediction of stochastic resonance

In a previous communication (Green, 1998), the initial step in ion channel gating for voltage-gated channels was attributed to the tunneling of a proton between groups with similar p K values, under the influence of an electric field. This is in contrast to the standard thermally activated model, which leads to a "Boltzmann equation" for the gating current. In the paper that introduced the present model, the current-voltage curve was determined from a resonance effect, in which gating began when the local voltage crossed a threshold, causing a proton to tunnel to a new location. We have therefore investigated further the consequences of tunneling as the first step in gating; we find a method of improving the previous calculation. We also calculate a consequence of our model that has yet to be experimentally looked for, stochastic resonance. With gating a threshold process, one expects that such an effect should exist. Only a small effect is predicted by our calculation, but it may be detectable. If it is it would make possible the determination of important characteristics of the initiation of gating. For this reason it is worth determining the nature of the stochastic resonance to be expected. In addition, we have investigated further the possible ways of understanding our resonance model itself. The model assumes that not all channels have the same threshold, as local perturbations in the potential interfere. We therefore assume a Gaussian distribution of the thresholds, which is simpler than in the previous paper, in which a Gaussian gave inadequate results with the method used there. In this paper, we have reduced the number of parameters to two, and obtained the current-voltage curve, gating current, the response to a large sine wave (in the previous paper, the model was more complex), and stochastic resonance.     

Charge Movement Associated with the Opening and Closing of the Activation Gates of the Na Channels

The sodium current (I_Na) that develops after step depolarization of a voltage clamped squid axon is preceded by a transient outward current that is closely associated with the opening of the activation gates of the Na pores. This "gating current" is best seen when permeant ions (Na and K) are replaced by relatively impermeant ones, and when the linear portion of capacitative current is eliminated by adding current from positive steps to that from exactly equal negative ones. During opening of the Na pores gating current is outward, and as the pores close there is an inward tail of current that decays with approximately the same time-course as I_Na recorded in Na-containing medium. Both outward and inward gating current are unaffected by tetrodotoxin (TTX). Gating current is capacitative in origin, the result of relatively slow reorientation of charged or dipolar molecules in a suddenly altered membrane field. Close association with the Na activation process is clear from the time-course of gating current, and from the fact that three procedures that reversibly block I_Na also block gating current: internal perfusion with Zn²⁺, prolonged depolarization of the membrane, and inactivation of I_Na with a short positive prepulse.     

Charge dissociation and the rising phase of gating currents

Gating currents from voltage-sensitive channels are generally attributed to the translocation or redistribution of ionic charge associated with the channel molecule. Such charge moves in the direction of the applied field to produce a decreasing current in the external circuit. An early rising phase for the gating current is observed for a number of channel systems and might be either some special kinetic redistribution of charge or an experimental artifact. A model that produces net charge in the channel through sequential molecular dissociation of a charged channel segment gives a rising phase for the gating current. Translocation of the charged segment produces the decay phase for a biphasic gating current. This kinetic molecular explanation constitutes a physical explanation for the biphasic gating currents that is consistent with present views of channel structure.     

Isolation of myocardial L-type calcium channel gating currents with the spider toxin omega-Aga-IIIA

The peptide omega-agatoxin-IIIA (omega-Aga-IIIA) blocks ionic current through L-type Ca channels in guinea pig atrial cells without affecting the associated gating currents. omega-Aga-IIIA permits the study of L- type Ca channel ionic and gating currents under nearly identical ionic conditions. Under conditions that isolate L-type Ca channel currents, omega-Aga-IIIA blocks all ionic current during a test pulse and after repolarization. This block reveals intramembrane charge movements of equal magnitude and opposite sign at the beginning of the pulse (Q(on)) and after repolarization (Q(off)). Q(on) and Q(off) are suppressed by 1 microM felodipine, saturate with increasing test potential, and are insensitive to Cd. The decay of the transient current associated with Q(on) is composed of fast and slow exponential components. The slow component has a time constant similar to that for activation of L-type Ca channel ionic current, over a broad voltage range. The current associated with Q(off) decays monoexponentially and more slowly than ionic current. Similar charge movements are found in guinea pig tracheal myocytes, which lack Na channels and T-type Ca channels. The kinetic and pharmacological properties of Q(on) and Q(off) indicate that they reflect gating currents associated with L-type Ca channels. omega-Aga-IIIA has no effect on gating currents when ionic current is eliminated by stepping to the reversal potential for Ca or by Cd block. Gating currents constitute a significant component of total current when physiological concentrations of Ca are present and they obscure the activation and deactivation of L-type Ca channels. By using omega- Aga-IIIA, we resolve the entire time course of L-type Ca channel ionic and gating currents. We also show that L- and T-type Ca channel ionic currents can be accurately quantified by tail current analysis once gating currents are taken into account.     

Chloride channels in toad skin

A study of the voltage and time dependence of a transepithelial Cl- current in toad skin (Bufo bufo) by the voltage-clamp method leads to the conclusion that potential has a dual role for Cl- transport. One is to control the permeability of an apical membrane Cl-pathway, the other is to drive Cl- ions through this pathway. Experimental analysis of the gating kinetics is rendered difficult owing to a contamination of the gated currents by cellular ion redistribution currents. To obtain insight into the effects of accumulation-depletion currents on voltage clamp currents of epithelial membranes, a mathematical model of the epithelium has been developed for computer analysis. By assuming that the apical membrane Cl- permeability is governed by a single gating variable (Hodgkin-Huxley kinetics), the model predicts fairly well steady-state current-voltage curves, the time course of current activations from a closed state, and the dependence of unidirectional fluxes on potential. Other predictions of the model do not agree with experimental findings, and it is suggested that the gating kinetics are governed by rate coefficients that also depend on the holding potential. Evidence is presented that Cl- transport through open channels does not obey the constant-field equation.     

External K+ modulates the activity of the Arabidopsis potassium channel SKOR via an unusual mechanism

Plant outward-rectifying K+ channels mediate K+ efflux from guard cells during stomatal closure and from root cells into the xylem for root-shoot allocation of potassium (K). Intriguingly, the gating of these channels depends on the extracellular K+ concentration, although the ions carrying the current are derived from inside the cell. This K+ dependence confers a sensitivity to the extracellular K+ concentration ([K+]) that ensures that the channels mediate K+ efflux only, regardless of the [K+] prevailing outside. We investigated the mechanism of K+-dependent gating of the K+ channel SKOR of Arabidopsis by site-directed mutagenesis. Mutations affecting the intrinsic K+ dependence of gating were found to cluster in the pore and within the sixth transmembrane helix (S6), identifying an 'S6 gating domain' deep within the membrane. Mapping the SKOR sequence to the crystal structure of the voltage-dependent K+ channel KvAP from Aeropyrum pernix suggested interaction between the S6 gating domain and the base of the pore helix, a prediction supported by mutations at this site. These results offer a unique insight into the molecular basis for a physiologically important K+-sensory process in plants.     

Gating current noise produced by Brownian models of a voltage sensor

The activation of voltage-dependent ion channels is associated with the movement of gating charges, which give rise to gating currents. Although gating currents from a single channel are too small to be detected, analysis of the fluctuations of macroscopic gating currents from a population of channels allows a good guess of their magnitude. The analysis of experimental gating current fluctuations, when interpreted in terms of a rate model of channel activation and assuming sufficiently high bandwidth, is in accordance with the presence of a main step along the activation pathway carrying a charge of 2.3–2.4 e₀. To give a physical interpretation to these results and to relate them to the known atomic structure of the voltage sensor domain, we used a Brownian model of voltage-dependent gating based on atomic detail structure, that follows the laws of electrodynamics. The model predicts gating currents and gating current fluctuations essentially similar to those experimentally observed. The detailed study of the model output, also performed by making several simplifications aimed at understanding the basic dependencies of the gating current fluctuations, suggests that in real channels the voltage sensor moves along a sequence of intermediate states separated by relatively low (<5 kT) energy barriers. As a consequence, crossings of successive gating charges through the gating pore become very frequent, and the corresponding current shots are often seen to overlap because of the relatively high filtering. Notably, this limited bandwidth effect is at the origin of the relatively high single-step charge experimentally detected.     

Divalent cations regulate connexin hemichannels by modulating intrinsic voltage-dependent gating

Connexin hemichannels are robustly regulated by voltage and divalent cations. The basis of voltage-dependent gating, however, has been questioned with reports that it is not intrinsic to hemichannels, but rather is derived from divalent cations acting as gating particles that block the pore in a voltage-dependent manner. Previously, we showed that connexin hemichannels possess two types of voltage-dependent gating, termed V_j and loop gating, that in Cx46 operate at opposite voltage polarities, positive and negative, respectively. Using recordings of single Cx46 hemichannels, we found both forms of gating persist in solutions containing no added Mg²⁺ and EGTA to chelate Ca²⁺. Although loop gating persists, it is significantly modulated by changing levels of extracellular divalent cations. When extracellular divalent cation concentrations are low, large hyperpolarizing voltages, exceeding −100 mV, could still drive Cx46 hemichannels toward closure. However, gating is characterized by continuous flickering of the unitary current interrupted by occasional, brief sojourns to a quiet closed state. Addition of extracellular divalent cations, in this case Mg²⁺, results in long-lived residence in a quiet closed state, suggesting that hyperpolarization drives the hemichannel to close, perhaps by initiating movements in the extracellular loops, and that divalent cations stabilize the fully closed conformation. Using excised patches, we found that divalent cations are only effective from the extracellular side, indicative that the binding site is not cytoplasmic or in the pore, but rather extracellular. V_j gating remains essentially unaffected by changing levels of extracellular divalent cations. Thus, we demonstrate that both forms of voltage dependence are intrinsic gating mechanisms in Cx46 hemichannels and that the action of external divalent cations is to selectively modulate loop gating.     

Ion permeation and block of the gating pore in the voltage sensor of NaV1.4 channels with hypokalemic periodic paralysis mutations

Hypokalemic periodic paralysis and normokalemic periodic paralysis are caused by mutations of the gating charge–carrying arginine residues in skeletal muscle Na_V1.4 channels, which induce gating pore current through the mutant voltage sensor domains. Inward sodium currents through the gating pore of mutant R666G are only ∼1% of central pore current, but substitution of guanidine for sodium in the extracellular solution increases their size by 13- ± 2-fold. Ethylguanidine is permeant through the R666G gating pore at physiological membrane potentials but blocks the gating pore at hyperpolarized potentials. Guanidine is also highly permeant through the proton-selective gating pore formed by the mutant R666H. Gating pore current conducted by the R666G mutant is blocked by divalent cations such as Ba²⁺ and Zn²⁺ in a voltage-dependent manner. The affinity for voltage-dependent block of gating pore current by Ba²⁺ and Zn²⁺ is increased at more negative holding potentials. The apparent dissociation constant (K_d) values for Zn²⁺ block for test pulses to −160 mV are 650 ± 150 µM, 360 ± 70 µM, and 95.6 ± 11 µM at holding potentials of 0 mV, −80 mV, and −120 mV, respectively. Gating pore current is blocked by trivalent cations, but in a nearly voltage-independent manner, with an apparent K_d for Gd³⁺ of 238 ± 14 µM at −80 mV. To test whether these periodic paralyses might be treated by blocking gating pore current, we screened several aromatic and aliphatic guanidine derivatives and found that 1-(2,4-xylyl)guanidinium can block gating pore current in the millimolar concentration range without affecting normal Na_V1.4 channel function. Together, our results demonstrate unique permeability of guanidine through Na_V1.4 gating pores, define voltage-dependent and voltage-independent block by divalent and trivalent cations, respectively, and provide initial support for the concept that guanidine-based gating pore blockers could be therapeutically useful.