Application of the ensemble nonequilibrium response spectroscopy to shaker potassium channel gating

Standard electrophysiology techniques study relaxation transients in voltage-gated ion channels generated by discrete voltage steps. The nonequilibrium response spectroscopy involves analyzing responses to fluctuating potentials. We apply the ensemble NRS method to gating kinetics of Shaker potassium ion channels. We evaluate various proposed Markov models of channel gating from the nonequilibrium response viewpoint. These new NRS protocols can be used to test otherwise indistinguishable models or improve estimates for parameters of channel kinetics models.     

Application of oscillating potentials to the Shaker potassium channel

Nonequilibrium response spectroscopy (NRS) has been proposed recently to complement standard electrophysiological techniques used to investigate ion channels. It involves application of rapidly oscillating potentials that drive the ion channel ensemble far from equilibrium. It is argued that new, so far undiscovered features of ion channel gating kinetics may become apparent under such nonequilibrium conditions. In this paper we explore the possibility of using regular, sinusoidal voltages with the NRS protocols to facilitate Markov model selection for ion channels. As a test case we consider the Shaker potassium channel for which various Markov models have been proposed recently. We concentrate on certain classes of such models and show that while some models might be virtually indistinguishable using standard methods, they show marked differences when driven with an oscillating voltage. Model currents are compared to experimental data obtained for the Shaker K+ channel expressed in mammalian cells (tsA 201).     

Fractal analysis of a voltage-dependent potassium channel from cultured mouse hippocampal neurons.

The kinetics of ion channels have been widely modeled as a Markov process. In these models it is assumed that the channel protein has a small number of discrete conformational states and the kinetic rate constants connecting these states are constant. In the alternative fractal model the spontaneous fluctuations of the channel protein at many different time scales are represented by a kinetic rate constant k = At1-D, where A is the kinetic setpoint and D the fractal dimension. Single-channel currents were recorded at 146 mM external K+ from an inwardly rectifying, 120 pS, K+ selective, voltage-sensitive channel in cultured mouse hippocampal neurons. The kinetics of these channels were found to be statistically self-similar at different time scales as predicted by the fractal model. The fractal dimensions were approximately 2 for the closed times and approximately 1 for the open times and did not depend on voltage. For both the open and closed times the logarithm of the kinetic setpoint was found to be proportional to the applied voltage, which indicates that the gating of this channel involves the net inward movement of approximately one negative charge when this channel opens. Thus, the open and closed times and the voltage dependence of the gating of this channel are well described by the fractal model.     

Applications of nonequilibrium response spectroscopy to the study of channel gating. Experimental design and optimization

A novel experimental technique known as non-equilibrium response spectroscopy (NRS) based on ion channel responses to rapidly fluctuating voltage waveforms was recently described (Millonas & Hanck, 1998a). It was demonstrated that such responses can be affected by subtle details of the kinetics that are otherwise invisible when conventional stepped pulses are applied. As a consequence, the kinetics can be probed in a much more sensitive way by supplementing conventional techniques with measurements of the responses to more complex voltage waveforms. In this paper we provide an analysis of the problem of the design and optimization of such waveforms. We introduce some methods for determination of the parametric uncertainty of a class of kinetic models for a particular data set. The parametric uncertainty allows for a characterization of the amount of kinetic information acquired through a set of experiments which can in turn be used to design new experiments that increase this information. We revisit the application of dichotomous noise (Millonas & Hanck, 1998a, b), and further consider applications of a more general class of continuous wavelet -based waveforms. A controlled illustration of these methods is provided by making use of a simplified "toy" model for the potassium channel kinetics.     

Gating kinetics of potassium channel in rat dorsal root ganglion neurons analyzed with fractal model

The kinetics of ion channels have been widely modeled as a Markov process. In these models it is assumed that the channel protein has a small number of discrete conformational states and kinetic rate constants connecting these states are constant. To study the gating kinetics of voltage-dependent K(+) channel in rat dorsal root ganglion neurons, K(+) channel current were recorded using cell-attached patch-clamp technique. The K(+) channel characteristic of kinetics were found to be statistically self-similar at different time scales as predicted by the fractal model. The fractal dimension D for the closed times and for the open times depend on the pipette potential. For the open and closed times of kinetic setpoint, it was found dependent on the applied pipette potential, which indicated that the ion channel gating kinetics had nonlinear kinetic properties. Thus, the open and closed durations, which had the voltage dependence of the gating of this ion channel, were well described by the fractal model.     

Optimal-Sensitivity Analysis of Ion Channel Gating Kinetics

We propose a new approach to analysis of kinetic models for ion channel gating, based on application of fluctuating voltages through a voltage clamp, in addition to conventional techniques. We show that the channel kinetics can be probed in a much more sensitive way, leading to more efficient model selection and more reliable estimates of model parameters. We use wavelet transform as an analytic tool for fluctuating currents and parametric dispersion plots as a measure of model compatibility with experimental data.This revised version was published online in August 2005 with a corrected cover date.     

Nonequilibrium voltage fluctuations in biological membranes. I. General framework of charge transport in discrete systems and related voltage noise

This paper continues our work on the theory of nonequilibrium voltage noise generated by electric transport processes in membranes. Introducing the membrane voltage as a further variable, a system of kinetic equations linearized in voltage is derived by which generally the time-dependent behaviour of charge-transport processes under varying voltage can be discussed. Using these equations, the treatment of voltage noise can be based on the usual master equation approach to steady-state fluctuations of scalar quantities. Thus, a general theoretical approach to nonequilibrium voltage noise is presented, completing our approach to current fluctuations which had been developed some years ago. It is explicitly shown that at equilibrium the approach yields agreement with the Nyquist relation, while at nonequilibrium this relation is not valid. A further general property of voltage noise is the reduction of low-frequency noise with increasing number of transport units as a consequence of the interactions via the electric field. In a second paper, the approach will be applied for a number of special transport mechanisms, such as ionic channels, carriers or electrogenic pumps.     

Wavelet analysis of nonequilibrium ionic currents in human heart sodium channel (hH1a)

Nonequilibrium response spectroscopy (NRS), the technique of using rapidly fluctuating voltage pulses in the study of ion channels, is applied here. NRS is known to drive an ensemble of ion channels far from equilibrium where, it has been argued, new details of ion channel kinetics can be studied under nonequilibrium conditions. In this paper, a single-pulse NRS technique with custom-designed waveforms built from wavelets is used. The pulses are designed to produce different responses from two competing models of a human heart isoform of the sodium channel (hH1a). Experimental data using this new type of pulses are obtained through whole-cell recordings from mammalian cells (HEK 293). Wavelet analysis of the model response and the experimental data is introduced to show how these NRS pulses can aid in distinguishing the better of the two models and thus introduces another important application of this new technique.     

Kinetic time constants independent of previous single-channel activity suggest Markov gating for a large conductance Ca-activated K channel

Models for the gating of ion channels usually assume that the rate constants for leaving any given kinetic state are independent of previous channel activity. Although such discrete Markov models have been successful in describing channel gating, there is little direct evidence for the Markov assumption of time-invariant rate constants for constant conditions. This paper tests the Markov assumption by determining whether the single-channel kinetics of the large conductance Ca-activated K channel in cultured rat skeletal muscle are independent of previous single-channel activity. The experimental approach is to examine dwell-time distributions conditional on adjacent interval durations. The time constants of the exponential components describing the distributions are found to be independent of adjacent interval duration, and hence, previous channel activity. In contrast, the areas of the different components can change. Since the observed time constants are a function of the underlying rate constants for transitions among the kinetic states, the observation of time constants independent of previous channel activity suggests that the rate constants are also independent of previous channel activity. Thus, the channel kinetics are consistent with Markov gating. An observed dependent (inverse) relationship between durations of adjacent open and shut intervals together with Markov gating indicates that there are two or more independent transition pathways connecting open and shut states. Finally, no evidence is found to suggest that gating is not at thermodynamic equilibrium: the inverse relationship was independent of the time direction of analysis.     

Wanderlust kinetics and variable Ca(2+)-sensitivity of Drosophila, a large conductance Ca(2+)-activated K+ channel, expressed in oocytes.

Cloned large conductance Ca(2+)-activated K+ channels (BK or maxi-K+ channels) from Drosophila (dSlo) were expressed in Xenopus oocytes and studied in excised membrane patches with the patch-clamp technique. Both a natural variant and a mutant that eliminated a putative cyclic AMP-dependent protein kinase phosphorylation site exhibited large, slow fluctuations in open probability with time. These fluctuations, termed "wanderlust kinetics," occurred with a time course of tens of seconds to minutes and had kinetic properties inconsistent with simple gating models. Wanderlust kinetics was still observed in the presence of 5 mM caffeine or 50 nM thapsigargin, or when the Ca2+ buffering capacity of the solution was increased by the addition of 5 mM HEDTA, suggesting that the wanderlust kinetics did not arise from Ca2+ release from caffeine and thapsigargin sensitive internal stores in the excised patch. The slow changes in kinetics associated with wanderlust kinetics could be generated with a discrete-state Markov model with transitions among three or more kinetic modes with different levels of open probability. To average out the wanderlust kinetics, large amounts of data were analyzed and demonstrated up to a threefold difference in the [Ca2+]i required for an open probability of 0.5 among channels expressed from the same injected mRNA. These findings indicate that cloned dSlo channels in excised patches from Xenopus oocytes can exhibit large variability in gating properties, both within a single channel and among channels.     

Wanderlust kinetics and variable Ca(2+)-sensitivity of dSlo [correction of Drosophila], a large conductance CA(2+)-activated K+ channel, expressed in oocytes.

Cloned large conductance Ca2+-activated K+ channels (BK or maxi-K+ channels) from Drosophila (dSlo) were expressed in Xenopus oocytes and studied in excised membrane patches with the patch-clamp technique. Both a natural variant and a mutant that eliminated a putative cyclic AMP-dependent protein kinase phosphorylation site exhibited large, slow fluctuations in open probability with time. These fluctuations, termed "wanderlust kinetics," occurred with a time course of tens of seconds to minutes and had kinetic properties inconsistent with simple gating models. Wanderlust kinetics was still observed in the presence of 5mM caffeine or 50 nM thapsigargin, or when the Ca2+ buffering capacity of the solution was increased by the addition of 5 mM HEDTA, suggesting that the wanderlust kinetics did not arise from Ca2+ release from caffeine and thapsigargin sensitive internal stores in the excised patch. The slow changes in kinetics associated with wanderlust kinetics could be generated with a discrete-state Markov model with transitions among three or more kinetic modes with different levels of open probability. To average out the wanderlust kinetics, large amounts of data were analyzed and demonstrated up to a threefold difference in the [Ca2+]i required for an open probability of 0.5 among channels expressed from the same injected mRNA. These findings indicate that cloned dSlo channels in excised patches from Xenopus oocytes can exhibit large variability in gating properties, both within a single channel and among channels.     

Fluctuations in ion channel gating currents. Analysis of nonstationary shot noise.

Conti and Stühmer (1989. Eur. Biophys. J. 17:53-59) have measured the nonstationary shot noise in the gating current of a population of sodium channels. Here we present expressions for the autocovariance and variance of such noise from general Markov models of channel kinetics, based on the theoretical work of E. Frehland. We compare the predictions of the independent, two-state gating model used by Conti and Stühmer with a six-state model of sodium channel activation based on the work of Armstrong and Gilly (1979. J. Gen. Physiol. 74:691-711). We find that Conti and Stühmer's experiment would not be able to distinguish between these schemes. We describe experimental conditions under which better model discrimination would be possible.     

Exploring voltage-dependent ion channels in silico by hysteretic conductance

Kinetic models of voltage-dependent ion channels are normally inferred from time records of macroscopic current relaxation or microscopic single channel data. A complementary explorative approach is outlined. Hysteretic conductance refers to conductance delays in response to voltage changes, delays at either macroscopic or microscopic levels of observation. It enables complementary assessments of model assumptions and gating schemes of voltage-dependent channels, e.g. independent versus cooperative gating, and multiple gating modes. Under the Hodgkin-Huxley condition of independent gating, and under ideal measurement conditions, hysteretic conductance makes it also possible to estimate voltage-dependent rate functions. The argument is mainly theoretical, based on experimental observations, and illustrated by simulations of Markov kinetic models.     

Poisson sampling-based inference for single ion channel data with time interval omission.

Patch-clamp recording allows investigations of the gating kinetics of single ion channels. Statistical analysis of kinetic data can enhance our understanding of channel gating at a molecular level. Experimental channel records suffer from time interval omission, i.e. failure to detect brief channel openings and closings. It is important to incorporate this phenomenon into statistical analyses of ion channel data. When time interval omission is ignored, the method of maximum likelihood can usually be used to estimate gating parameters from a single channel record. However, it is far more difficult to apply this method when time interval omission is incorporated. We present an alternative approach to parameter estimation based on Poisson sampling. A simulated homogeneous Poisson process is superimposed onto the channel record and inference is based on the numbers of points in successive open and closed sojourns, rather than on the sojourn times themselves. We describe the method for the two-state Markov model C<-->O, although it is applicable to more general models. Computer-simulated data are used to demonstrate the efficacy of the method. Modifications of the method are discussed briefly.     

Pharmacological and kinetic analysis of K channel gating currents

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

Computing transient gating charge movement of voltage-dependent ion channels

The opening of voltage-gated sodium, potassium, and calcium ion channels has a steep relationship with voltage. In response to changes in the transmembrane voltage, structural movements of an ion channel that precede channel opening generate a capacitative gating current. The net gating charge displacement due to membrane depolarization is an index of the voltage sensitivity of the ion channel activation process. Understanding the molecular basis of voltage-dependent gating of ion channels requires the measurement and computation of the gating charge, Q. We derive a simple and accurate semianalytic approach to computing the voltage dependence of transient gating charge movement (Q–V relationship) of discrete Markov state models of ion channels using matrix methods. This approach allows rapid computation of Q–V curves for finite and infinite length step depolarizations and is consistent with experimentally measured transient gating charge. This computational approach was applied to Shaker potassium channel gating, including the impact of inactivating particles on potassium channel gating currents.     

Simulating complex ion channel kinetics with IonChannelLab

In silico simulation based on Markov chains is a powerful way to describe and predict the activity of many transport proteins including ion channels. However, modeling and simulation using realistic models of voltage- or ligand-gated ion channels exposed to a wide range of experimental conditions require building complex kinetic schemes and solving complicated differential equations. To circumvent these problems, we developed IonChannelLab a software tool that includes a user-friendly Graphical User Interface and a simulation library. This program supports channels with Ohmic or Goldman-Hodgkin-Katz behavior and can simulate the time-course of ionic and gating currents, single channel behavior and steady-state conditions. The program allows the simulation of experiments where voltage, ligand and ionic concentration are varied independently or simultaneously.     

Simulating complex ion channel kinetics with IonChannelLab

In silico simulation based on Markov chains is a powerful way to describe and predict the activity of many transport proteins including ion channels. However, modeling and simulation using realistic models of voltage- or ligand-gated ion channels exposed to a wide range of experimental conditions require building complex kinetic schemes and solving complicated differential equations. To circumvent these problems, we developed IonChannelLab a software tool that includes a user-friendly Graphical User Interface and a simulation library. This program supports channels with Ohmic or Goldman-Hodgkin-Katz behavior and can simulate the time-course of ionic and gating currents, single channel behavior and steady-state conditions. The program allows the simulation of experiments where voltage, ligand and ionic concentration are varied independently or simultaneously.     

Theory of transport noise in membrane channels with open-closed kinetics

A theoretical approach to transport noise in kinetic systems, which has recently been developed, is applied to electric fluctuations around steady-states in membrane channels with different conductance states. The channel kinetics may be simple two state (open-closed) kinetics or more complicated. The membrane channel is considered as a sequence of binding sites separated by energy barriers over which the ions have to jump according to the usual single-file diffusion model. For simplicity the channels are assumed to act independently. In the special case of ionic movement fast compared with the channel open-closed kinetics the results agree with those derived from the usual Master equation approach to electric fluctuations in nerve membrane channels.For the simple model of channels with one binding site and two energy barries the coupling between the fluctuations coming from the open-closed kinetics and from the jump diffusion is investigated.