A model for conductance changes in the squid giant axon. II. Channel kinetics

The kinetics of the sodium and potassium channels in voltage clamped squid giant axon following a relaxation of the membrane subunits are examined and compared with the Hodgkin-Huxley equations. Mechanisms are suggested for the turn-off of the sodium conductance and a set of kinetic states are proposed for the potassium channel which are consistent with the experimental observations. Determination of the rate constants for relaxation of the surface subunits which triggers the subsequent changes within the independent channels provide information on the equilibrium constant and free energy for this process. The free energy is observed to approach zero as the depolarizing voltage of the clamp approaches $\text{E}{}_{\mathrm{<mtext>Na</mtext>}}$, the voltage for zero sodium current in voltage clamped axons. Analysis of the final rate constants in the kinetic sequence for potassium indicates a symmetry of the channel when it is in its steady-state configuration during clamp in the absence of external gradients.     

A sodium channel gating model based on single channel, macroscopic ionic, and gating currents in the squid giant axon.

Sodium channel gating behavior was modeled with Markovian models fitted to currents from the cut-open squid giant axon in the absence of divalent cations. Optimum models were selected with maximum likelihood criteria using single-channel data, then models were refined and extended by simultaneous fitting of macroscopic ionic currents, ON and OFF gating currents, and single-channel first latency densities over a wide voltage range. Best models have five closed states before channel opening, with inactivation from at least one closed state as well as the open state. Forward activation rate constants increase with depolarization, and deactivation rate constants increase with hyperpolarization. Rates of inactivation from the open or closed states are generally slower than activation or deactivation rates and show little or no voltage dependence. Channels tend to reopen several times before inactivating. Macroscopic rates of activation and inactivation result from a combination of closed, open and inactivated state transitions. At negative potentials the time to first opening dominates the macroscopic current due to slow activation rates compared with deactivation rates: channels tend to reopen rarely, and often inactivate from closed states before they reopen. At more positive potentials, the time to first opening and burst duration together produce the macroscopic current.     

Multiple kinetic states underlying macroscopic M-currents in bullfrog sympathetic neurons.

M-current is a time- and voltage-dependent potassium current which is suppressible by muscarinic receptor activation. We have used curve fitting and noise analysis to determine if macroscopic M-currents deviate from a previously predicted simple two-state kinetic scheme. The M-current was best described by three kinetically distinct components: 'fast' (tau 0), 'intermediate' (tau 1) and 'slow' (tau 2) time constants. The 'fast' (tau 0) and 'intermediate' (tau 1) components were identified from the spectra of M-current noise at potentials positive to the cells' resting membrane potential. The 'intermediate' (tau 1) and 'slow' (tau 2) components were seen by curve fitting M-current deactivation currents. The 'intermediate' (tau 1) time constant was voltage dependent (decreasing e-fold in 23 mV), but voltage dependence of the 'fast' (tau 0) and 'slow' (tau 2) components was not obvious. All kinetic components were sensitive to muscarine, with the 'intermediate' (tau 1) and 'slow' (tau 2) being equally so. These data suggest that all components may derive from the same channel population, and that the M-channel may have at least four kinetic states.     

Optimal estimation of ion-channel kinetics from macroscopic currents

Markov modeling provides an effective approach for modeling ion channel kinetics. There are several search algorithms for global fitting of macroscopic or single-channel currents across different experimental conditions. Here we present a particle swarm optimization(PSO)-based approach which, when used in combination with golden section search (GSS), can fit macroscopic voltage responses with a high degree of accuracy (errors within 1%) and reasonable amount of calculation time (less than 10 hours for 20 free parameters) on a desktop computer. We also describe a method for initial value estimation of the model parameters, which appears to favor identification of global optimum and can further reduce the computational cost. The PSO-GSS algorithm is applicable for kinetic models of arbitrary topology and size and compatible with common stimulation protocols, which provides a convenient approach for establishing kinetic models at the macroscopic level.     

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 characteristics of a steeply voltage-dependent gap junction channel in rat Schwann cells

The gating properties of macroscopic and microscopic gap junctional currents were compared by applying the dual whole cell patch clamp technique to pairs of neonatal rat Schwann cells. In response to transjunctional voltage pulses (Vj), macroscopic gap junctional currents decayed exponentially with time constants ranging from < 1 to < 10 s before reaching steady-state levels. The relationship between normalized steady-state junctional conductance (Gss) and (Vj) was well described by a Boltzmann relationship with e-fold decay per 10.4 mV, representing an equivalent gating charge of 2.4. At Vj > 60 mV, Gss was virtually zero, a property that is unique among the gap junctions characterized to date. Determination of opening and closing rate constants for this process indicated that the voltage dependence of macroscopic conductance was governed predominantly by the closing rate constant. In 78% of the experiments, a single population of unitary junctional currents was detected corresponding to an unitary channel conductance of approximately 40 pS. The presence of only a limited number of junctional channels with identical unitary conductances made it possible to analyze their kinetics at the single channel level. Gating at the single channel level was further studied using a stochastic model to determine the open probability (Po) of individual channels in a multiple channel preparation. Po decreased with increasing Vj following a Boltzmann relationship similar to that describing the macroscopic Gss voltage dependence. These results indicate that, for Vj of a single polarity, the gating of the 40 pS gap junction channels expressed by Schwann cells can be described by a first order kinetic model of channel transitions between open and closed states.     

Estimation of kinetic rate constants from multi-channel recordings by a direct fit of the time series.

The maximum-likelihood technique for the direct estimation of rate constants from the measured patch clamp current is extended to the analysis of multi-channel recordings, including channels with subconductance levels. The algorithm utilizes a simplified approach for the calculation of the matrix exponentials of the probability matrix from the rate constants of the Markov model of the involved channel(s) by making use of the Kronecker sum and product. The extension to multi-channel analysis is tested by the application to simulated data. For these tests, three different channel models were selected: a two-state model, a three-state model with two open states of different conductance, and a three-state model with two closed states. For the simulations, time series of these models were calculated from the related first-order, finite-state, continuous-time Markov processes. Blue background noise was added, and the signals were filtered by a digital filter similar to the anti-aliasing low-pass. The tests showed that the fit algorithm revealed good estimates of the original rate constants from time series of simulated records with up to four independent and identical channels even in the case of signal-to-noise ratios being as low as 2. The number of channels in a record can be determined from the dependence of the likelihood on channel number. For large enough data sets, it takes on a maximum when the assumed channel number is equal to the "true" channel number.     

Effects of physostigmine on the voltage dependent ionic conductances of skeletal muscle fibres

The voltage dependent ionic conductances were studied by analysing the phase plane trajectories of action potentials evoked by electrical stimulation of the sartorius muscles of the frog (Rana esculenta). The delayed outward potassium current was measured also under voltage clamp conditions on muscle fibres of either the frog (Rana esculenta) or Xenopus laevis. On analysing the effect of physostigmine decreasing the peak amplitude, the rate of both the rising and falling phases of the action potentials, it was revealed that the alkaloid at a concentration of 1 mmol/l reduced significantly both the delayed potassium conductance and the outward ionic current values during the action potentials. The inhibition of sodium conductance and inward ionic current was less expressed. The maximum value of delayed potassium conductance measured under voltage clamp conditions was decreased by 1 mmol/l physostigmine. The time constant determined from the development of delayed potassium conductance was increased at a given membrane potential. The voltage vs. n relationship describing the membrane potential dependence of the delayed rectifier was not influenced by physostigmine. It has been concluded that physostigmine changes the time course of the action potentials by decreasing the value of both voltage dependent ionic conductances and by slowing down their kinetics. It is discussed that results obtained from the phase plane analysis of complex pharmacological effects can only be accepted with some restrictions.     

Nonstationary fluctuation analysis of the delayed rectifier K channel in cardiac Purkinje fibers. Actions of norepinephrine on single-channel current.

We have studied the large increase in macroscopic potassium channel current caused by catecholamines in mammalian cardiac cells. An increase in macroscopic K current could result from either an increase in the single-channel current or by an increase in the number of channels that are open. Therefore, we have measured nonstationary potassium current fluctuations under voltage clamp conditions to determine whether norepinephrine increases the current through this channel. The single-channel current (at a potential of -30 mV in 4 mM external [K]) was estimated to be 3.7 pA and was not altered by concentrations of norepinephrine up to 2 microM. The spectral density of the current fluctuations were fitted well by a sum of 2 Lorentzians with corner frequencies that correspond with the measured time constants for deactivation of the macroscopic K current tails. We conclude that the increase in macroscopic K current caused by norepinephrine in these cells is not the result of an increase in single-channel conductance and therefore must involve an increase in the number of open K channels.     

Inactivation viewed through single sodium channels

Recordings of the sodium current in tissue-cultured GH3 cells show that the rate of inactivation in whole cell and averaged single channel records is voltage dependent: tau h varied e-fold/approximately 26 mV. The source of this voltage dependence was investigated by examining the voltage dependence of individual rate constants, estimated by maximum likelihood analysis of single channel records, in a five-state kinetic model. The rate constant for inactivating from the open state, rather than closing, increased with depolarization, as did the probability that an open channel inactivates. The rate constant for closing from the open state had the opposite voltage dependence. Both rate constants contributed to the mean open time, which was not very voltage dependent. Both open time and burst duration were less than tau h for voltages up to -20 mV. The slowest time constant of activation, tau m, was measured from whole cell records, by fitting a single exponential either to tail currents or to activating currents in trypsin-treated cells, in which the inactivation was abolished. tau m was a bell-shaped function of voltage and had a voltage dependence similar to tau h at voltages more positive than -35 mV, but was smaller than tau h. At potentials more negative than about -10 mV, individual channels may open and close several times before inactivating. Therefore, averaged single channel records, which correspond with macroscopic current elicited by a depolarization, are best described by a convolution of the first latency density with the autocorrelation function rather than with 1 - (channel open time distribution). The voltage dependence of inactivation from the open state, in addition to that of the activation process, is a significant factor in determining the voltage dependence of macroscopic inactivation. Although the rates of activation and inactivation overlapped greatly, independent and coupled inactivation could not be statistically distinguished for two models examined. Although rates of activation affect the observed rate of inactivation at intermediate voltages, extrapolation of our estimates of rate constants suggests that at very depolarized voltages the activation process is so fast that it is an insignificant factor in the time course of inactivation. Prediction of gating currents shows that an inherently voltage-dependent inactivation process need not produce a conspicuous component in the gating current.     

Fast inactivation causes rectification of the IKr channel

The mechanism of rectification of HERG, the human cardiac delayed rectifier K+ channel, was studied after heterologous expression in Xenopus oocytes. Currents were measured using two-microelectrode and macropatch voltage clamp techniques. The fully activated current- voltage (I-V) relationship for HERG inwardly rectified. Rectification was not altered by exposing the cytoplasmic side of a macropatch to a divalent-free solution, indicating this property was not caused by voltage-dependent block of outward current by Mg2+ or other soluble cytosolic molecules. The instantaneous I-V relationship for HERG was linear after removal of fast inactivation by a brief hyperpolarization. The time constants for the onset of and recovery from inactivation were a bell-shaped function of membrane potential. The time constants of inactivation varied from 1.8 ms at +50 mV to 16 ms at -20 mV; recovery from inactivation varied from 4.7 ms at -120 mV to 15 ms at -50 mV. Truncation of the NH2-terminal region of HERG shifted the voltage dependence of activation and inactivation by +20 to +30 mV. In addition, the rate of deactivation of the truncated channel was much faster than wild-type HERG. The mechanism of HERG rectification is voltage-gated fast inactivation. Inactivation of channels proceeds at a much faster rate than activation, such that no outward current is observed upon depolarization to very high membrane potentials. Fast inactivation of HERG and the resulting rectification are partly responsible for the prolonged plateau phase typical of ventricular action potentials.     

Hodgkin-Huxley type ion channel characterization: an improved method of voltage clamp experiment parameter estimation

The Hodgkin-Huxley formalism for quantitative characterization of ionic channels is widely used in cellular electrophysiological models. Model parameters for these individual channels are determined from voltage clamp experiments and usually involve the assumption that inactivation process occurs on a time scale which is infinitely slow compared to the activation process. This work shows that such an assumption may lead to appreciable errors under certain physiological conditions and proposes a new numerical approach to interpret voltage clamp experiment results. In simulated experimental protocols the new method was shown to exhibit superior accuracy compared to the traditional least squares fitting methods. With noiseless input data the error in gating variables and time constants was less than 1%, whereas the traditional methods generated upwards of 10% error and predicted incorrect gating kinetics. A sensitivity analysis showed that the new method could tolerate up to approximately 15% perturbation in the input data without unstably amplifying error in the solution. This method could also assist in designing more efficient experimental protocols, since all channel parameters (gating variables, time constants and maximum conductance) could be determined from a single voltage step.     

Maximum likelihood estimation of ion channel kinetics from macroscopic currents

We describe a maximum likelihood method for direct estimation of rate constants from macroscopic ion channel data for kinetic models of arbitrary size and topology. The number of channels in the preparation, and the mean and standard deviation of the unitary current can be estimated, and a priori constraints can be imposed on rate constants. The method allows for arbitrary stimulation protocols, including stimuli with finite rise time, trains of ligand or voltage steps, and global fitting across different experimental conditions. The initial state occupancies can be optimized from the fit kinetics. Utilizing arbitrary stimulation protocols and using the mean and the variance of the current reduce or eliminate problems of model identifiability (Kienker, 1989). The algorithm is faster than a recent method that uses the full autocovariance matrix (Celentano and Hawkes, 2004), in part due to the analytical calculation of the likelihood gradients. We tested the method with simulated data and with real macroscopic currents from acetylcholine receptors, elicited in response to brief pulses of carbachol. Given appropriate stimulation protocols, our method chose a reasonable model size and topology.     

Inward-rectifier potassium channels in basolateral membranes of frog skin epithelium

Inward-rectifier K channel: using macroscopic voltage clamp and single- channel patch clamp techniques we have identified the K+ channel responsible for potassium recycling across basolateral membranes (BLM) of principal cells in intact epithelia isolated from frog skin. The spontaneously active K+ channel is an inward rectifier (Kir) and is the major component of macroscopic conductance of intact cells. The current- voltage relationship of BLM in intact cells of isolated epithelia, mounted in miniature Ussing chambers (bathed on apical and basolateral sides in normal amphibian Ringer solution), showed pronounced inward rectification which was K(+)-dependent and inhibited by Ba2+, H+, and quinidine. A 15-pS Kir channel was the only type of K(+)-selective channel found in BLM in cell-attached membrane patches bathed in physiological solutions. Although the channel behaves as an inward rectifier, it conducts outward current (K+ exit from the cell) with a very high open probability (Po = 0.74-1.0) at membrane potentials less negative than the Nernst potential for K+. The Kir channel was transformed to a pure inward rectifier (no outward current) in cell- attached membranes when the patch pipette contained 120 mM KCl Ringer solution (normal NaCl Ringer in bath). Inward rectification is caused by Mg2+ block of outward current and the single-channel current-voltage relation was linear when Mg2+ was removed from the cytosolic side. Whole-cell current-voltage relations of isolated principal cells were also inwardly rectified. Power density spectra of ensemble current noise could be fit by a single Lorentzian function, which displayed a K dependence indicative of spontaneously fluctuating Kir channels. Conclusions: under physiological ionic gradients, a 15-pS inward- rectifier K+ channel generates the resting BLM conductance in principal cells and recycles potassium in parallel with the Na+/K+ ATPase pump.     

Movement of the S4 segment in the hERG potassium channel during membrane depolarization

Abstract

The hERG potassium channel is a member of the voltage gated potassium (Kv) channel family, comprising a pore domain and four voltage sensing domains (VSDs). Like other Kv channels, the VSD senses changes in membrane voltage and transmits the signal to gates located in the pore domain; the gates open at positive potentials (activation) and close at negative potentials, thereby controlling the ion flux. hERG, however, differs from other Kv channels in that it is activated slowly but inactivated rapidly – a property that is crucial for the role it plays in the repolarization of the cardiac action potential. Voltage-gating requires movement of gating charges across the membrane electric field, which is accomplished by the transmembrane movement of the fourth transmembrane segment, S4, of the VSD containing the positively charged arginine or lysine residues. Here we ask if the functional differences between hERG and other Kv channels could arise from differences in the transmembrane movement of S4. To address this, we have introduced single cysteine residues into the S4 region of the VSD, expressed the mutant channels in Xenopus oocytes and examined the effect of membrane impermeable para-chloromercuribenzene sulphonate on function by the two-electrode voltage clamp technique. Our results show that depolarization results in the accessibility of seven consecutive S4 residues, including the first two charged residues, K525 and R528, to extracellularly applied reagent. These data indicate that the extent of S4 movement in hERG is similar to other Kv channels, including the archabacterial KvAP and the Shaker channel of Drosophila.     

Single potassium channels of human glioma cells.

Single potassium channels in the membrane of human malignant glioma cells U-118MG were studied using the technique of patch clamp in cell-attached and inside-out configurations. Three types of potassium channels were found which differed from each other under conditions close to physiological in their conductance and gating characteristics. The lowest-conductance channel (20 pS near the reversal potential) showed a mild outward rectification up to 45 pS at positive voltages and spontaneous modes of high and low activity. At extreme values of potentials its activity was generally low. The intermediate conductance channel had an S-shaped I-V curve, giving a conductance of 63 pS at reversal, and a low and voltage independent opening probability. The high-conductance (215 pS) channel was found to be activated by both membrane potential and Ca2+ ions and blocked by internal sodium at high voltages. The current-voltage curves of all three channel types displayed saturation.     

Action potential-like responses due to the inward rectifying potassium channel

Summary This paper describes experiments carried out in the absence of sodium and calcium in the external solution. Frog atrial trabeculae were stimulated in current clamp with the double sucrose gap technique. The voltage responses looked like slow action potentials with a clear threshold. These responses were not suppressed in the presence of EGTA, in the presence of sodium or calcium channel blockers, or when sulfate ions replaced chloride. Guinea pig isolated ventricular myocytes were studied in whole cell clamp mode with a pathch pipette. Under current clamp, they displayed also voltage responses with a threshold. These responses were resistant to cadmium (5mm), and were suppressed by barium (0.5mm). A negative slope conductance is required to take into account these results. The membrane current in current clamp can be estimated by plotting the response in the phase plane. This analysis shows that on both types of preparations, the current responsible for the negative slope is not time dependent. This current is suppressed by barium. It can be concluded that it is the outward current flowing through the inward rectifying potassium channels. To confirm this hypothesis, data obtained in voltage clamp on the same preparations were introduced into a computer model to predict the response in current clamp. The results were in agreement with the experiments. Similar responses could be recorded and analyzed on skeletal muscle in isotonic potassium solution. These results show that the inward rectifier can induce by itself properties looking like excitability on different preparations. The physiological significance of this effect in normal conditions is discussed. The voltage responses described in this paper look similar to the slow action potentials on heart, which are sensitive to modifications of the calcium channels, but also of the potassium channels. Some implications in cardiac pharmacology are discussed.     

Exponentiated exponential model (Gompertz kinetics) of Na+ and K+ conductance changes in squid giant axon

The conductance changes, gK(t) and gNa(t), of squid giant axon under voltage clamp (Hodgkin and Huxley, 1952) may be modeled by exponentiated exponential functions (Gompertz kinetics) from any holding potential VO to any membrane clamp potential V. The equation constants are set by the membrane potential V, and include, for any voltage step in the case of gK, the initial conductance, gO, the asymptote conductance g, and rate constant k: gK = g exp(-be-kt) where b = 1n g/gO. Equations of similar form relate g and k to the voltage V, and govern the corresponding parameters of the gNa system. For the gNa, the fast phase y = y exp (-be-kt) is cut down in proportion to a slow process p = (1 - p)e-k't + p, and thus gNa = py. The expo-exponential functions involve fewer constants than the Hodgkin-Huxley model. In particular, the role of the n, m, h parameters appears to be filled largely by 1n (g/gO) in the case of gK and by 1n (y/yO) in the case of gNa. Membrane action potentials during current clamp may be computed from the conductances generated by use of the appropriate differential forms of the equations; diverse other membrane behaviors may be predicted.     

The effect of phloretin on single potassium channels in myelinated nerve

The effect of phloretin, a dipolar organic compound, on single potassium channel currents of myelinateed nerve fibres of Xenopus laevis has been investigated, using inside-out patches prepared by the method of Jonas et al. (1989). The I channel, a potential dependent K channel with intermediate deactivation kinetics, was reversibly blocked by 20 µM phloretin applied on the inside; the block was strongest at negative membrane potentials and less pronounced at positive potentials. Phloretin shifted the curve relating open probability to membrane potential towards more positive potentials and reduced its slope and maximum. This confirms previous findings on the effect of phloretin on the voltage dependence of the fast macroscopic K conductance. Single channel conductance and deactivation kinetics were not altered by phloretin. Offprint requests to: Correspondence to: H. Meves