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.     

Selective modification of sodium channel gating in lobster axons by 2, 4, 6-trinitrophenol: Evidence for two inactivation mechanisms

Trinitrophernol (TNP) selectively alters the sodium conductance system of lobster giant axons as measured in current clamp and voltage clamp experiments using the double sucrose gap technique. TNP has no measurable effect on potassium currents but reversibly prolongs the time-course of sodium currents during maintained depolarizations over the full voltage range of observable currents. Action potential durations are increased also. Tm of the Hodgkin-Huxley model is not markedly altered during activation of the sodium conductance but is prolonged during removal of activation by repolarization, as observed in sodium tail experiments. The sodium inactivation versus voltage curve is shifted in the hyperpolarizing direction as is the inactivation time constant curve, measured with conditioning voltage steps. This shift speeds the kinetics of inactivation over part of the same voltage range in which sodium currents are prolonged, a contradiction incompatible with the Hodgkin-Huxley model. These results are interpreted as support for a hypothesis of two inactivation processes, one proceeding directly from the resting state and the other coupled to the active state of sodium conductance.     

Inactivation of the Sodium Current in Myxicola Giant Axons : Evidence for coupling to the activation process

Experiments were conducted on Myxicola giant axons to determine if the sodium activation and inactivation processes are coupled or independent. The main experimental approach was to examine the effects of changing test pulses on steady-state inactivation curves. Arguments were presented to show that in the presence of a residual uncompensated series resistance the interpretation of the results depends critically on the manner of conducting the experiment. Analytical and numerical calculations were presented to show that as long as test pulses are confined to an approximately linear negative conductance region of the sodium current-voltage characteristic, unambiguous interpretations can be made. When examined in the manner of Hodgkin and Huxley, inactivation in Myxicola is quantitatively similar to that described by the h variable in squid axons. However, when test pulses were increased along the linear negative region of the sodium current-voltage characteristic, steady-state inactivation curves translate to the right along the voltage axis. The shift in the inactivation curve is a linear function of the ratio of the sodium, conductance of the test pulses, showing a 5.8 mv shift for a twofold increase in conductance. An independent line of evidence indicated that the early rate of development of inactivation is a function of the rise of the sodium conductance.     

Barbiturates Block Sodium and Potassium Conductance Increases in Voltage-Clamped Lobster Axons

Sodium pentobarbital and sodium thiopental decrease both the peak initial (Na) and late steady-state (K) currents and reduce the maximum sodium and potassium conductance increases in voltage-clamped lobster giant axons. These barbiturates also slow the rate at which the sodium conductance turns on, and shift the normalized sodium conductance vs. voltage curves in the direction of depolarization along the voltage axis. Since pentobarbital (pK_a = 8.0) blocks the action potential more effectively at pH 8.5 than at pH 6.7, the anionic form of the drug appears to be active. The data suggest that these drugs affect the axon membrane directly, rather than secondarily through effects on intermediary metabolism. It is suggested that penetration of the lipid layer of the membrane by the nonpolar portion of the barbiturate molecules may cause the decrease in membrane conductances, while electrostatic interactions involving the anionic group on the barbiturate, divalent cations, and "fixed charges" in the membrane could account for the slowing of the rate of sodium conductance turn-on and the shift of the normalized conductance curves along the voltage axis.     

Comparison of heterologously expressed human cardiac and skeletal muscle sodium channels.

In this study we have expressed and characterized recombinant cardiac and skeletal muscle sodium channel alpha subunits in tsA-201 cells under identical experimental conditions. Unlike the Xenopus oocyte expression system, in tsA-201 cells (transformed human embryonic kidney) both channels seem to gate rapidly, as in native tissue. In general, hSkM1 gating seemed faster than hH1 both in terms of rate of inactivation and rate of recovery from inactivation as well as time to peak current. The midpoint of the steady-state inactivation curve was approximately 25 mV more negative for hH1 compared with hSkM1. In both isoforms, the steady-state channel availability relationships ("inactivation curves") shifted toward more negative membrane potentials with time. The cardiac isoform showed a minimal shift in the activation curve as a function of time after whole-cell dialysis, whereas hSkM1 showed a continued and marked negative shift in the activation voltage dependence of channel gating. This observation suggests that the mechanism underlying the shift in inactivation voltage dependence may be similar to the one that is causing the shift in the activation voltage dependence in hSkM1 but that this is uncoupled in the cardiac isoform. These results demonstrate the utility and limitations of measuring cardiac and skeletal muscle recombinant Na+ channels in tsA-201 cells. This baseline characterization will be useful for future investigations on channel mutants and pharmacology.     

Effects of proteolytic enzymes on ionic conductances of squid axon membranes.

The effects of proteolytic enzymes on ionic conductances of squid axon membranes have been studied by means of the voltage clamp technique. When perfused internally alpha-chymotrypsin (1 mg/ml) increased and prolonged the depolarizing after-potential. Sodium inactivation was partially inhibited causing a prolonged sodium current, and peak sodium and steady-state potassium currents were suppressed. The time for sodium current to reach its peak was not affected. Leakage conductance increased later. On the other hand, carboxypeptidases A and B, both at 1mg/ml, suppressed the sodium and potassium conductance increases with little or no change in sodium inactivation. The mechanism that controls sodium inactivation appears to be associated with the structure of membrane proteins which is modified by alpha-chymotrypsin but not by carboxypeptidases and is located in a position accessible to alpha-chymotrypsin only from inside the membrane.     

Interaction of n-alkylguanidines with the sodium channels of squid axon membrane

The effects of n-alkylguanidine derivatives on sodium channel conductance were measured in voltage clamped, internally perfused squid giant axons. After destruction of the sodium inactivation mechanism by internal pronase treatment, internal application of n-amylguanidine (0.5 mM) or n-octylguanidine (0.03 mM) caused a time-dependent block of sodium channels. No time-dependent block was observed with shorter chain derivatives. No change in the rising phase of sodium current was seen and the block of steady-state sodium current was independent of the membrane potential. In axons with intact sodium inactivation, an apparent facilitation of inactivation was observed after application of either n-amylguanidine or n-octylguanidine. These results can be explained by a model in which alkylguanidines enter and occlude open sodium channels from inside the membrane with voltage-independent rate constants. Alkylguanidine block bears a close resemblance to natural sodium inactivation. This might be explained by the fact that alkylguanidines are related to arginine, which has a guanidino group and is thought to be an essential amino acid in the molecular mechanism of sodium inactivation. A strong correlation between alkyl chain length and blocking potency was found, suggesting that a hydrophobic binding site exists near the inner mouth of the sodium channel.     

A Model of sodium channel-inactivation based upon the modulated blocker

An attempt is made to model sodium channel inactivation based upon real physical processes. The principle involved, which is supported by calculation and by direct appeal to experimental results, is that the gating dipole reversal or gating charge transfer that occurs when the channel is activated, markedly modulates the electrical properties of charged groups at the channel ends. Four examples of possible mechanisms that lead to channel inactivation are described. The simple four-state model that results is able to predict: (a) the steep voltage dependence of the equilibrium inactivation characteristic without the presence of any appreciable displacement current associated with inactivation; (b) the negative shift in membrane voltage of the equilibrium inactivation characteristic relative to the activation characteristic; (c) the bell-shaped dependence of inactivation time constant on membrane voltage; (d) the similarity of the membrane voltage dependence of the time constant of recovery from inactivation, to that of inactivation itself. A brief discussion of a model for sodium channel activation based upon the same physical principle is included.     

A transient excited state model for sodium permeability changes in excitable membranes.

In this paper we explore the properties of a mathematical model for the passive sodium permeability system of excitable membranes. This model is distinguished by the explicit inclusion of a rate constant which depends not on instantaneous voltage, but on rate of voltage change. Actually, the model is a rather modest modification of the Hodgkin-Huxley model, but displays some behaviors which the H-H model does not. Among these behaviors are a pronounced inactivation shift (for certain parameter values), a difference between inactivation time constant as measured by turning off a sodium current under sustained depolarization and as measured by double pulse experiments, skip runs under sustained current stimulation, and accommodation to slowing rising currents.     

Inactivation in HCN channels results from reclosure of the activation gate: desensitization to voltage

Hyperpolarization-activated HCN channels are modulated by direct binding of cyclic nucleotides. For HCN2 channels, cAMP shifts the voltage dependence for activation, with relatively little change in the maximal conductance. By contrast, in spHCN channels, cAMP relieves a rapid inactivation process and produces a large increase in maximum conductance. Our results suggest that these two effects of cAMP represent the same underlying process. We also find that spHCN inactivation occurs not by closure of a specialized inactivation gate, as for other voltage-dependent channels, but by reclosure of the same intracellular gate opened upon activation. Effectively, the activation gate exhibits a "desensitization to voltage," perhaps by slippage of the coupling between the voltage sensors and the gate. Differences in the initial coupling efficiency could allow cAMP to produce either the inactivation or the shift phenotype by strengthening effective coupling: a shift would naturally occur if coupling is already strong in the absence of cAMP.     

Possible reduction of surface charge by a mutation inParamecium tetraurelia

Summary Under voltage clamp, a mutant ofParamecium tetraurelia (teaB) shows a shift in the positive direction of the voltage sensitivity of the Ca conductance and the depolarization inactivation curve by 10 mV with no change in the total conductance. This effect can be mimicked in the wild type by the addition of external Ca²⁺ or Mg²⁺. The mutation also shifts the resting potential and the voltage sensitivities of the delayed rectification (depolarization-sensitive) K conductance and the anomalous rectification (hyperpolarization-sensitive) K conductance in the positive direction to a similar extent. This systematic shift of channel voltage sensitivities is best explained by the reduction of the surface negative charges of the membrane due to the mutation.     

Structural stability,alternate descriptions and information

The inactivation properties of a model of the nerve membrane are examined. The inactivation kinetics are closely first order and may be characterized by Hodgkin-Huxley (H-H) parameters h_∞ and τ_h which depend on potential in agreement with experiments. Some differences from the H-H equations are identified. The forms predicted for τ_h variation with hyper-polarization and change of external [K⁺] agree with available data. While the inactivation time delay predicted by the model is too small to be detected experimentally, there are grounds for expecting that it may be larger in other tissues, as observed in Myxicola giant axons. The variation of the delay with test potential is predicted to be exponential. Although the model is coupled in the sense defined by Hoyt, it gives rise to an inactivation shift of negligible magnitude. However, introducing a simple variability in one physical parameter leads to the observed form of both the peak transient current voltage relation and the inactivation shift. Inactivation shift thus does not unambiguously indicate coupling; that it results from parametric heterogeneity may be a better hypothesis, and is readily testable. The inactivation shift dependence on current ratio, from experimental data, can be used to correct for the effects of parametric heterogeneity and obtain the value of a previously predicted fundamental parameter of excitable membranes. It is suggested that the effects of parametric heterogeneity must be considered in interpreting experiments and designing models for excitable systems.     

Dynamical characterization of inactivation path in voltage-gated Na+ ion channel by non-equilibrium response spectroscopy

Inactivation path of voltage gated sodium channel has been studied here under various voltage protocols as it is the main governing factor for the periodic occurrence and shape of the action potential. These voltage protocols actually serve as non-equilibrium response spectroscopic tools to study the ion channel in non-equilibrium environment. In contrast to a lot of effort in finding the crystal structure based molecular mechanism of closed-state(CSI) and open-state inactivation(OSI); here our approach is to understand the dynamical characterization of inactivation. The kinetic flux as well as energetic contribution of the closed and open- state inactivation path is compared here for voltage protocols, namely constant, pulsed and oscillating. The non-equilibrium thermodynamic quantities used in response to these voltage protocols serve as improved characterization tools for theoretical understanding which not only agrees with the previously known kinetic measurements but also predict the energetically optimum processes to sustain the auto-regulatory mechanism of action potential and the consequent inactivation steps needed. The time dependent voltage pattern governs the population of the conformational states which when couple with characteristic rate parameters, the CSI and OSI selectivity arise dynamically to control the inactivation path. Using constant, pulsed and continuous oscillating voltage protocols we have shown that during depolarization the OSI path is more favored path of inactivation however, in the hyper-polarized situation the CSI is favored. It is also shown that the re-factorisation of inactivated sodium channel to resting state occurs via CSI path. Here we have shown how the subtle energetic and entropic cost due to the change in the depolarization magnitude determines the optimum path of inactivation. It is shown that an efficient CSI and OSI dynamical profile in principle can characterize the open-state drug blocking phenomena.     

Block and inactivation of sodium channels in nerve by amino acid derivatives. I. Dependence on voltage and sodium concentration.

The side chain of arginine, n-propylguanidinium (nPG), reversibly decreases peak sodium conductance and increases the speed of sodium current decay, when perfused internally. Effects are voltage dependent and are more pronounced at high depolarizations. Results are also dependent on the sodium concentration gradient. Both the decline in peak conductance and the speeding of inactivation are greater if the sodium concentration gradient is reversed from the normal. The decrease in peak sodium current is too large to be due solely to the faster decay kinetics. The difference is not due to a change in slow inactivation of the channel. Sodium current inactivation has also been studied with a double pulse procedure. Results show that at - 70 mV, nPG leaves sodium channels rapidly (less than 500 microseconds) in normal sodium gradient, but more slowly (greater than 1 ms) in reversed sodium gradient. Several structural analogs of nPG have been tested. Shortening the alkyl chain weakens effects significantly. Arginine itself, which contains extra charged groups, is also less effective. n-Propylammonium is active but with an apparent affinity only one-fifth that of arginine. We conclude that nPG acts within the sodium channel, and has at least two modes of action.     

Independence of the Sodium and Potassium Conductance Channels: A Kinetic Argument

Tightly coupled models for the sodium and potassium conductance changes, in which the potassium “on” process is intimately related to the sodium “on” and “off” processes, are studied. It is shown that such coupled models are incapable of simultaneously showing the observed effects of conditioning potentials on sodium inactivation and on the translation of the potassium conductance in time. It is concluded that the primary mechanisms for the sodium and potassium channels are probably independent.     

Extracellular pH selectively modulates recovery from sodium inactivation in frog myelinated nerve.

The Hodgkin-Huxley kinetic parameters, alpha h and beta h, which govern the rate of recovery from and development of sodium channel inactivation, respectively, have been measured as a function of membrane potential and external pH using a three-pulse protocol. alpha h but not beta h is substantially accelerated by reducing external pH from 7.4 to 6.4. The alpha h vs. voltage curve appears to be selectively shifted in the depolarizing direction by approximately 12 mV for this pH change, giving an apparent, h infinity curve shift of approximately 6 mV in the same direction (less inactivation).     

Intracellular pH and ionic channels in the Loligo vulgaris giant axon.

Squid giant axons were used to investigate the reversible effects of intracellular pH(pHi) on the kinetic properties of ionic channels. The pharmacologically separated K+ and Na+ currents were measured under: (a) internal perfusion, (b) enzymatic Pronase treatment, and (c) continuous estimate of periaxonal ion accumulation. Variation of internal pH from 4.8 to 11 resulted in: (a) a decrease of steady-state sodium inactivation at positive potentials similar to the effect of the proteolytic enzyme Pronase, (b) a shift of the h infinity (E) curve toward depolarizing voltages, and (c) a decrease of the time constant of inactivation for potentials below -20 mV (an increase above). A plot of the steady-state sodium conductance at E = +40 mV as a function of pHi suggests that two groups with pKa 10.4 and 5.6 affect respectively the inactivation gate and the rate constants for the transition from the inactivated to the second open state (h2) (Chandler and Meves, 1970b). The voltage shifts of the kinetic parameters predicted by the Gouy-Chapman-Stern theory are well satisfied at high pHi and less at low. Once corrected for voltage shifts, the forward rate constants for channel opening were found to be slowed with the acidity of the internal or external solution.     

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.     

Characterization of the inward-rectifying potassium current in cat ventricular myocytes

Whole-cell membrane currents were measured in isolated cat ventricular myocytes using a suction-electrode voltage-clamp technique. An inward-rectifying current was identified that exhibited a time-dependent activation. The peak current appeared to have a linear voltage dependence at membrane potentials negative to the reversal potential. Inward current was sensitive to K channel blockers. In addition, varying the extracellular K+ concentration caused changes in the reversal potential and slope conductance expected for a K+ current. The voltage dependence of the chord conductance exhibited a sigmoidal relationship, increasing at more negative membrane potentials. Increasing the extracellular K+ concentration increased the maximal level of conductance and caused a shift in the relationship that was directly proportional to the change in reversal potential. Activation of the current followed a monoexponential time course, and the time constant of activation exhibited a monoexponential dependence on membrane potential. Increasing the extracellular K+ concentration caused a shift of this relationship that was directly proportional to the change in reversal potential. Inactivation of inward current became evident at more negative potentials, resulting in a negative slope region of the steady state current-voltage relationship between -140 and -180 mV. Steady state inactivation exhibited a sigmoidal voltage dependence, and recovery from inactivation followed a monoexponential time course. Removing extracellular Na+ caused a decrease in the slope of the steady state current-voltage relationship at potentials negative to -140 mV, as well as a decrease of the conductance of inward current. It was concluded that this current was IK1, the inward-rectifying K+ current found in multicellular cardiac preparations. The K+ and voltage sensitivity of IK1 activation resembled that found for the inward-rectifying K+ currents in frog skeletal muscle and various egg cell preparations. Inactivation of IK1 in isolated ventricular myocytes was viewed as being the result of two processes: the first involves a voltage-dependent change in conductance; the second involves depletion of K+ from extracellular spaces. The voltage-dependent component of inactivation was associated with the presence of extracellular Na+.     

Early development of voltage-dependent sodium currents in cultured mouse spinal cord neurons

Spinal cord neurons were dissociated from 13-day embryonic mice and grown in culture for 1-28 days. Sodium currents of neurons in culture for 1-2 days were compared with those in culture for 2-4 weeks, using the whole-cell voltage clamp method. Rapid neurite outgrowth created space clamp limitations so that unclamped neuritic sodium action potentials prevented accurate analysis of sodium current properties. Therefore neurons were bathed in sodium-free solution and brief puffs of sodium were delivered to the cell soma so that only somatic sodium currents were recorded. Sodium currents of neurons at 1-2 days in culture had voltage-dependent activation and inactivation characteristic of these channels, both in mature cultured spinal neurons and in other preparations. However, the estimated channel density on the soma of neurons 1-2 days in culture was less than two channels per micron2. Since the available sodium conductance (as measured by action potential rise rates) increases during development of spinal cord neurons in culture (Westbrook and Brenneman, 1984), we suggest that changes in channel density and/or distribution, rather than in channel kinetics, may underlie the increase in sodium conductance.