Voltage-dependent capacitance in lipid bilayers made from monolayers.

Electrocompression has been measured in lipid bilayers made by apposition of two monolayers. The capacitance C(V), as a function of membrane potential, V, was found to be well described by C(V) = C(O) [1 + alpha(V + delta psi)2] where C(O) is the capacitance at V = O, alpha is the fractional increase in capacitance per square volt, and delta psi is the surface potential difference. In lipid bilayers made from monolayers alpha has a value of 0.02 V-2, which is ca. 500-fold smaller than the value found in solvent containing membranes. In asymmetric bilayers made of one neutral and one negatively charged monolayer, delta psi values were found to be those expected from independent measurements of surface charge density. If the fractional increase in capacitance found here is a good approximation to that of biological membranes, nonlinear capacitative charge displacement derived from electrostriction is expected to be less than 1% of the total gating charge displacement found in squid axons.     

Continuum Gating Current Models Computed with Consistent Interactions

The action potential of nerve and muscle is produced by voltage-sensitive channels that include a specialized device to sense voltage. The voltage sensor depends on the movement of charges in the changing electric field as suggested by Hodgkin and Huxley. Gating currents of the voltage sensor are now known to depend on the movements of positively charged arginines through the hydrophobic plug of a voltage sensor domain. Transient movements of these permanently charged arginines, caused by the change of transmembrane potential V, further drag the S4 segment and induce opening/closing of the ion conduction pore by moving the S4-S5 linker. This moving permanent charge induces capacitive current flow everywhere. Everything interacts with everything else in the voltage sensor and protein, and so it must also happen in its mathematical model. A Poisson-Nernst-Planck (PNP)-steric model of arginines and a mechanical model for the S4 segment are combined using energy variational methods in which all densities and movements of charge satisfy conservation laws, which are expressed as partial differential equations in space and time. The model computes gating current flowing in the baths produced by arginines moving in the voltage sensor. The model also captures the capacitive pile up of ions in the vestibules that link the bulk solution to the hydrophobic plug. Our model reproduces the signature properties of gating current: 1) equality of ON and OFF charge Q in integrals of gating current, 2) saturating voltage dependence in the Q(charge)-voltage curve, and 3) many (but not all) details of the shape of gating current as a function of voltage. Our results agree qualitatively with experiments and can be improved by adding more details of the structure and its correlated movements. The proposed continuum model is a promising tool to explore the dynamics and mechanism of the voltage sensor.     

FPL 64176 modification of Ca(V)1.2 L-type calcium channels: dissociation of effects on ionic current and gating current

FPL 64176 (FPL) is a nondihydropyridine compound that dramatically increases macroscopic inward current through L-type calcium channels and slows activation and deactivation. To understand the mechanism by which channel behavior is altered, we compared the effects of the drug on the kinetics and voltage dependence of ionic currents and gating currents. Currents from a homogeneous population of channels were obtained using cloned rabbit Ca(V)1.2 (alpha1C, cardiac L-type) channels stably expressed in baby hamster kidney cells together with beta1a and alpha2delta1 subunits. We found a striking dissociation between effects of FPL on ionic currents, which were modified strongly, and on gating currents, which were not detectably altered. Inward ionic currents were enhanced approximately 5-fold for a voltage step from -90 mV to +10 mV. Kinetics of activation and deactivation were slowed dramatically at most voltages. Curiously, however, at very hyperpolarized voltages (< -250 mV), deactivation was actually faster in FPL than in control. Gating currents were measured using a variety of inorganic ions to block ionic current and also without blockers, by recording gating current at the reversal potential for ionic current (+50 mV). Despite the slowed kinetics of ionic currents, FPL had no discernible effect on the fundamental movements of gating charge that drive channel gating. Instead, FPL somehow affects the coupling of charge movement to opening and closing of the pore. An intriguing possibility is that the drug causes an inactivated state to become conducting without otherwise affecting gating transitions.     

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.     

Ionic processes in excitable membranes

The dynamical theory of ionized media is applied to the semi-electrolyte component of an excitable cell membrane, and the adjacent electrolytes. The equations of conservation of charge and momentum for the ions, and Poisson's equation for the electrostatic potential, are applied first to investigate the steady states of the membrane, and then transient effects in the membrane. A dispersion equation is derived, and the characteristic modes of relaxation within the membrane are determined. Among these are oscillating modes whose frequencies and amplitudes are of the correct order of magnitude to explain the observed excitation phenomena.A pair of coupled non-linear equations in the ionic potentials, with action potential solutions, is derived from the time-dependent electrodiffusion equations, and calculations are presented which model the behaviour of the excitable membrane during the voltage clamp. It is not necessary to postulate large changes in the ionic permeabilities in the course of the action potential and the voltage clamp to account for the large transient membrane conductances. It is suggested that the sodium hypothesis be replaced by one which attributes the action potential to non-linear plasma oscillations.     

Capacitance fluctuations causing channel noise reduction in stochastic Hodgkin-Huxley systems

Voltage-dependent ion channels determine the electric properties of axonal cell membranes. They not only allow the passage of ions through the cell membrane, but also contribute to an additional charging of the cell membrane resulting in the so-called capacitance loading. The switching of the channel gates between an open and a closed configuration is intrinsically related to the movement of gating charge within the cell membrane. At the beginning of an action potential, the transient gating current is opposite to the direction of the current of sodium ions through the membrane. Therefore, the excitability is expected to become reduced due to the influence of a gating current. Our stochastic Hodgkin-Huxley-like modeling takes into account both the channel noise-i.e. the fluctuations of the number of open ion channels-and the capacitance fluctuations that result from the dynamics of the gating charge. We investigate the spiking dynamics of membrane patches of a variable size and analyze the statistics of the spontaneous spiking. As a main result, we find that the gating currents yield a drastic reduction of the spontaneous spiking rate for sufficiently large ion channel clusters. Consequently, this demonstrates a prominent mechanism for channel noise reduction.     

Distribution and kinetics of membrane dielectric polarization. II. Frequency domain studies of gating currents

We have studied the admittance of the membrane of squid giant axon under voltage clamp in the absence of ionic conductances in the range of 0-12 kHz for membrane potentials (V) between --130 and 70 mV. The admittance was measured at various holding potentials (HP) or 155 ms after pulsing from a given holding potential. Standard P/4 procedure was used to study gating currents in the same axons. We found that the membrane capacity Cm (omega) is voltage as well as frequency dependent. For any given V, the voltage-dependent part of the membrane capacitance has a maximum as the frequency approaches zero and requires at least a two-time constant equivalent circuit to be described. When the holding potential is varied, the voltage-dependent capacitance follows a bell- shaped curve with a maximum change of 0.15 muF/cm2 at about --60 mV. With the pulse method, the maximum is at --40 mV for HP = --70 and it shifts to --70 mV for HP = 0. The shift in the maximum of the voltage- dependent capacitance is consistent with the shift in the charge (Q) vs. V curve observed in our experiments with regular P/4 procedure when the HP is varied. Our data can be explained qualitatively by a four- state model for the sodium channel gating, where a charged particle can move within the field and interact with another particle not affected by the field.     

Nonlinear charge movement in mammalian cardiac ventricular cells. Components from Na and Ca channel gating [published erratum appears in J Gen Physiol 1989 Aug;94(2):401]

Intramembrane charge movement was recorded in rat and rabbit ventricular cells using the whole-cell voltage clamp technique. Na and K currents were eliminated by using tetraethylammonium as the main cation internally and externally, and Ca channel current was blocked by Cd and La. With steps in the range of -110 to -150 used to define linear capacitance, extra charge moves during steps positive to approximately -70 mV. With holding potentials near -100 mV, the extra charge moving outward on depolarization (ON charge) is roughly equal to the extra charge moving inward on repolarization (OFF charge) after 50-100 ms. Both ON and OFF charge saturate above approximately +20 mV; saturating charge movement is approximately 1,100 fC (approximately 11 nC/muF of linear capacitance). When the holding potential is depolarized to -50 mV, ON charge is reduced by approximately 40%, with little change in OFF charge. The reduction of ON charge by holding potential in this range matches inactivation of Na current measured in the same cells, suggesting that this component might arise from Na channel gating. The ON charge remaining at a holding potential of -50 mV has properties expected of Ca channel gating current: it is greatly reduced by application of 10 muM D600 when accompanied by long depolarizations and it is reduced at more positive holding potentials with a voltage dependence similar to that of Ca channel inactivation. However, the D600-sensitive charge movement is much larger than the Ca channel gating current that would be expected if the movement of channel gating charge were always accompanied by complete opening of the channel.     

Fast Inactivation in Shaker K+ Channels : Properties of Ionic and Gating Currents

Fast inactivating Shaker H4 potassium channels and nonconducting pore mutant Shaker H4 W434F channels have been used to correlate the installation and recovery of the fast inactivation of ionic current with changes in the kinetics of gating current known as “charge immobilization” (Armstrong, C.M., and F. Bezanilla. 1977. J. Gen. Physiol. 70:567–590.). Shaker H4 W434F gating currents are very similar to those of the conducting clone recorded in potassium-free solutions. This mutant channel allows the recording of the total gating charge return, even when returning from potentials that would largely inactivate conducting channels. As the depolarizing potential increased, the OFF gating currents decay phase at −90 mV return potential changed from a single fast component to at least two components, the slower requiring ∼200 ms for a full charge return. The charge immobilization onset and the ionic current decay have an identical time course. The recoveries of gating current (Shaker H4 W434F) and ionic current (Shaker H4) in 2 mM external potassium have at least two components. Both recoveries are similar at −120 and −90 mV. In contrast, at higher potentials (−70 and −50 mV), the gating charge recovers significantly more slowly than the ionic current. A model with a single inactivated state cannot account for all our data, which strongly support the existence of “parallel” inactivated states. In this model, a fraction of the charge can be recovered upon repolarization while the channel pore is occupied by the NH₂-terminus region.     

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.     

Membrane tension directly shifts voltage dependence of outer hair cell motility and associated gating charge.

The unique electromotility of the outer hair cell (OHC) is believed to promote sharpening of the passive mechanical vibration of the mammalian basilar membrane. The cell also presents a voltage-dependent capacitance, or equivalently, a nonlinear gating current, which correlates well with its mechanical activity, suggesting that membrane-bound voltage sensor-motor elements control OHC length. We report that the voltage dependence of the gating charge and motility are directly related to membrane stress induced by intracellular pressure. A tracking procedure was devised to continuously monitor the voltage at peak capacitance (VpkCm) after obtaining whole cell voltage clamp configuration. In addition, nonlinear capacitance was more fully evaluated with a stair step voltage protocol. Upon whole cell configuration, VpkCm was typically near -20 mV. Negative patch pipette pressure caused a negative shift in VpkCm, which obtained a limiting value near the normal resting potential of the OHC (approximately -70 mV) at the point of cell collapse. Positive pressure in the pipette caused a positive shift that could reach values greater than 0 mV. Measures of the mechanical activity of the OHC mirrored those of charge movement. Similar membrane-tension dependent peak shifts were observed after the cortical cytoskeletal network was disrupted by intracellular dialysis of trypsin from the patch pipette. We conclude that unlike stretch receptors, which may sense tension through elastic cytoskeletal elements, the OHC motor senses tension directly. Furthermore, since the voltage dependence of the OHC nonlinear capacitance and motility is directly regulated by intracellular turgor pressure, we speculate that modification of intracellular pressure in vivo provides a mechanism for controlling the gain of the mammalian "cochlear amplifier".     

Hysteresis in the voltage dependence of HCN channels: conversion between two modes affects pacemaker properties

Hyperpolarization-activated, cyclic nucleotide-gated (HCN) ion channels are important for rhythmic activity in the brain and in the heart. In this study, using ionic and gating current measurements, we show that cloned spHCN channels undergo a hysteresis in their voltage dependence during normal gating. For example, both the gating charge versus voltage curve, Q(V), and the conductance versus voltage curve, G(V), are shifted by about +60 mV when measured from a hyperpolarized holding potential compared with a depolarized holding potential. In addition, the kinetics of the tail current and the activation current change in parallel to the voltage shifts of the Q(V) and G(V) curves. Mammalian HCN1 channels display similar effects in their ionic currents, suggesting that the mammalian HCN channels also undergo voltage hysteresis. We propose a model in which HCN channels transit between two modes. The voltage dependence in the two modes is shifted relative to each other, and the occupancy of the two modes depends on the previous activation of the channel. The shifts in the voltage dependence are fast (tau approximately 100 ms) and are not accompanied by any apparent inactivation. In HCN1 channels, the shift in voltage dependence is slower in a 100 mM K extracellular solution compared with a 1 mM K solution. Based on these findings, we suggest that molecular conformations similar to slow (C-type) inactivation of K channels underlie voltage hysteresis in HCN channels. The voltage hysteresis results in HCN channels displaying different voltage dependences during different phases in the pacemaker cycle. Computer simulations suggest that voltage hysteresis in HCN channels decreases the risk of arrhythmia in pacemaker cells.     

Sequence of Gating Charge Movement and Pore Gating in hERG Activation and Deactivation Pathways

K_V11.1 voltage-gated K⁺ channels are noted for unusually slow activation, fast inactivation, and slow deactivation kinetics, which tune channel activity to provide vital repolarizing current during later stages of the cardiac action potential. The bulk of charge movement in human ether-a-go-go-related gene (hERG) is slow, as is return of charge upon repolarization, suggesting that the rates of hERG channel opening and, critically, that of deactivation might be determined by slow voltage sensor movement, and also by a mode-shift after activation. To test these ideas, we compared the kinetics and voltage dependence of ionic activation and deactivation with gating charge movement. At 0 mV, gating charge moved ∼threefold faster than ionic current, which suggests the presence of additional slow transitions downstream of charge movement in the physiological activation pathway. A significant voltage sensor mode-shift was apparent by 24 ms at +60 mV in gating currents, and return of charge closely tracked pore closure after pulses of 100 and 300 ms duration. A deletion of the N-terminus PAS domain, mutation R4AR5A or the LQT2-causing mutation R56Q gave faster-deactivating channels that displayed an attenuated mode-shift of charge. This indicates that charge movement is perturbed by N- and C-terminus interactions, and that these domain interactions stabilize the open state and limit the rate of charge return. We conclude that slow on-gating charge movement can only partly account for slow hERG ionic activation, and that the rate of pore closure has a limiting role in the slow return of gating charges.     

Glutamate 268 regulates transport probability of the anion/proton exchanger ClC-5

The Cl(-)/H(+) exchange mediated by ClC transporters can be uncoupled by external SCN(-) and mutations of the proton glutamate, a conserved residue at the internal side of the protein. We show here for the mammalian ClC transporter ClC-5 that acidic internal pH led to a greater increase in currents upon exchanging extracellular Cl(-) for SCN(-). However, transport uncoupling, unitary current amplitudes, and the voltage dependence of the depolarization-induced activation were not altered by low pH values. Therefore, it is likely that an additional gating process regulates ClC-5 transport. Higher internal [H(+)] and the proton glutamate mutant E268H altered the ratio between ClC-5 transport and nonlinear capacitance, indicating that the gating charge movements in ClC-5 arise from incomplete transport cycles and that internal protons increase the transport probability of ClC-5. This was substantiated by site-directed sulfhydryl modification of the proton glutamate mutant E268C. The mutation exhibited small transport currents together with prominent gating charge movements. The charge restoration using a negatively charged sulfhydryl reagent reinstated also the WT phenotype. Neutralization of the charge of the gating glutamate 211 by the E211C mutation abolished the effect of internal protons, showing that the increased transport probability of ClC-5 results from protonation of this residue. S168P (a mutation that decreases the anion affinity of the central binding site) reduced also the internal pH dependence of ClC-5. These results support the idea that protonation of the gating glutamate 211 at the central anion-binding site of ClC-5 is mediated by the proton glutamate 268.     

Computation of Impulse Initiation and Saltatory Conduction in a Myelinated Nerve Fiber

A mathematical model of the electrical properties of a myelinated nerve fiber is given, consisting of the Hodgkin-Huxley ordinary differential equations to represent the membrane at the nodes of Ranvier, and a partial differential cable equation to represent the internodes. Digital computer solutions of these equations show an impulse arising at a stimulating electrode and being propagated away, approaching a constant velocity. Action potential curves plotted against distance show discontinuities in slope, proportional to the nodal action currents, at the nodes. Action potential curves plotted against time, at the nodes and in the internodes, show a marked difference in steepness of the rising phase, but little difference in peak height. These results and computed action current curves agree fairly accurately with published experimental data from frog and toad fibers.     

Subthreshold oscillatory responses of the Hodgkin-Huxley cable model for the squid giant axon

The classical cable equation, in which membrane conductance is considered constant, is modified by including the linearized effect of membrane potential on sodium and potassium ionic currents, as formulated in the Hodgkin-Huxley equations for the squid giant axon. The resulting partial differential equation is solved by numerical inversion of the Laplace transform of the voltage response to current and voltage inputs. The voltage response is computed for voltage step, current step, and current pulse inputs, and the effect of temperature on the response to a current step input is also calculated.The validity of the linearized approximation is examined by comparing the linearized response to a current step input with the solution of the nonlinear partial differential cable equation for various subthreshold current step inputs.All the computed responses for the squid giant axon show oscillatory behavior and depart significantly from what is predicted on the basis of the classical cable equation. The linearization procedure, coupled with numerical inversion of the Laplace transform, proves to be a convenient approach which predicts at least qualitatively the subthreshold behavior of the nonlinear system.     

Controllability of excitable systems

Mathematical models of cell electrical activity typically consist of a current balance equation, channel activation (or inactivation) variables and concentrations of regulatory agents. These models can be thought of as nonlinear filters whose input is some applied current I (possibly zero) and output is a membrane potential V. A natural question to ask is if the applied current I can be deduced from the potential V. For a surprisingly large class of models the answer to this question is yes. To show this, we first demonstrate how many models can be embedded into higher dimensional quasilinear systems. For quasilinear models, a procedure for determining the inverse of the nonlinear filter is then described and demonstrated on two models: (1) the FitzHugh-Nagumo model and (2) the Sherman-Rinzel-Keizer (SRK) [Sherman et al., (1988, Biophysics Journal, 54, 411-425)] model of bursting electrical activity in pancreatic beta-cells. For the latter example, the inverse problem is then used to deduce model parameter values for which the model and experimental data agree in some measure. An advantage of the correlation technique is that experimental values for activation (and/or regulatory) variables need not be known to make the estimates for these parameter values.     

Digital Computer Solutions for Excitation and Propagation of the Nerve Impulse

The Hodgkin-Huxley model of the nerve axon describes excitation and propagation of the nerve impulse by means of a nonlinear partial differential equation. This equation relates the conservation of the electric current along the cablelike structure of the axon to the active processes represented by a system of three rate equations for the transport of ions through the nerve membrane. These equations have been integrated numerically with respect to both distance and time for boundary conditions corresponding to a finite length of squid axon stimulated intracellularly at its midpoint. Computations were made for the threshold strength-duration curve and for the repetitive firing of propagated impulses in response to a maintained stimulus. These results are compared with previous solutions for the space-clamped axon. The effect of temperature on the threshold intensity for a short stimulus and for rheobase was determined for a series of values of temperature. Other computations show that a highly unstable subthreshold propagating wave is initiated in principle by a just threshold stimulus; that the stability of the subthreshold wave can be enhanced by reducing the excitability of the axon as with an anesthetic agent, perhaps to the point where it might be observed experimentally; but that with a somewhat greater degree of narcotization, the axon gives only decrementally propagated impulses.     

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.     

Microscopic origin of gating current fluctuations in a potassium channel voltage sensor

Voltage-dependent ion channels open and close in response to changes in membrane electrical potential due to the motion of their voltage-sensing domains (VSDs). VSD charge displacements within the membrane electric field are observed in electrophysiology experiments as gating currents preceding ionic conduction. The elementary charge motions that give rise to the gating current cannot be observed directly, but appear as discrete current pulses that generate fluctuations in gating current measurements. Here we report direct observation of gating-charge displacements in an atomistic molecular dynamics simulation of the isolated VSD from the KvAP channel in a hydrated lipid bilayer on the timescale (10-μs) expected for elementary gating charge transitions. The results reveal that gating-charge displacements are associated with the water-catalyzed rearrangement of salt bridges between the S4 arginines and a set of conserved acidic side chains on the S1-S3 transmembrane segments in the hydrated interior of the VSD.