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.     

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.     

Kinetic analysis of channel gating. Application to the cholinergic receptor channel and the chloride channel from Torpedo californica.

Identification of the minimum number of ways in which open and closed states communicate is a crucial step in defining the gating kinetics of multistate channels. We used certain correlation functions to extract information about the pathways connecting the open and closed states of the cation channel of the purified nicotinic acetylcholine receptor and of the chloride channel of Torpedo californica electroplax membranes. Single channel currents were recorded from planar lipid bilayers containing the membrane channel proteins under investigation. The correlation functions are conveniently computed from single channel current records and yield information on E, the minimum number of entry/exit states into the open or closed aggregates. E gives a lower limit on the numbers of transition pathways between open and closed states. For the acetylcholine receptor, the autocorrelation analysis shows that there are at least two entry/exit states through which the open and closed aggregates communicate. The chloride channel fluctuates between three conductance substates, here indentified as C, M, and H for closed, intermediate, and high conductance, respectively. Correlation analysis shows that E is greater than or equal to 2 for the M aggregate, indicating that there are at least two distinct entry/exit states in the M aggregate. In contrast, there is no evidence for the existence of more than one entry/exit state in the C or H aggregates. Thus, these correlation functions provide a simple and general strategy to extract information on channel gating kinetics.     

Existence of memory in membrane channels: analysis of ion current through a voltage‐dependent potassium single channel

Ion current fluctuation of voltage‐dependent potassium channel in LβT2 cells has been investigated by autocorrelation function and DFA (detrended fluctuation analysis) methods. The calculation of the autocorrelation function exponent and DFA exponent of the sample was based on the digital signals or the 0–1 series corresponding to closing and opening of channels after routine evolution, rather than the sequence of sojourn times. The persistent character of the correlation of the time series was evident from the slow decay of the autocorrelation function. DFA exponent α was significantly greater than 0.5. The main outcome has been the demonstration of the existence of memory in this ion channel. Thus, the ion channel current fluctuation provided information about the kinetics of the channel protein. The result suggests the correlation character of the ion channel protein non‐linear kinetics indicates whether the channel is open or not.     

Noise analysis predicts at least four states for channels closed by glutamate.

For ion channels that are opened by neurotransmitters, analysis of current noise has given valuable information on the kinetics of synaptic channel gating. In depolarizing bipolar cells of the vertebrate retina, we have recently characterized a synaptic current for which the neurotransmitter glutamate closes channels, and for which the channel open probability is low even in the absence of glutamate. We present here predictions for the current noise spectrum expected for various models of glutamate's action on the ion channels. Comparison of these theoretical predictions with experimental data allows us to rule out several simple kinetic schemes for the action of glutamate, and to conclude that the channels closed by glutamate must be able to exist in at least four different states.     

Fractal and Markov behavior in ion channel kinetics

Kinetic analysis of ion channel recordings attempts to distinguish the number and lifetimes of channel molecular states. Most kinetic analysis assumes that the lifetime of each state is independent of previous channel history, so that open and closed durations are Markov processes whose probability densities are sums of exponential decays. An alternative approach assumes that channel molecules have many configurtions with widely varying lifetimes. Rates of opening and closing then vary with the time scale of observation, leading to fractal kinetics. We have examined kinetic behavior in two types of channels from human and avian fibroblasts, using a maximum likehood method to test the dependence of rates on observational time scale. For both channels, openings showed mixed fractal and Markov behavior, while closings gave mainly fractal kinetics.     

Open channel block by gadolinium ion of the stretch-inactivated ion channel in mdx myotubes.

Currents flowing through single stretch-inactivated ion channels were recorded from cell-attached patches on myotubes from mdx mice. Adding micromolar concentrations of gadolinium to patch electrodes containing normal saline produced rapid transitions in the single-channel current between the fully open and closed states. The kinetics of the current fluctuations followed the predictions of a simple model of open channel block in which the transitions in the current arise from the entry and exit of Gd from the channel pore: histograms of the open and closed times were well fit with single exponentials, the blocking rate depended linearly on the concentration of gadolinium in the patch electrode, and the unblocking rate was independent of the concentration of gadolinium. Hyperpolarizing the patch increased the rate of unblocking (approximately e-fold per 85 mV), suggesting the charged blocking particle can exit the channel into the cell under the influence of the applied membrane field. The rate of blocking was rapid and was independent of the patch potential, consistent with the rate of ion entry into the pore being determined by its rate of diffusion in solution. When channel open probability was reduced by applying suction to the electrode, the blocking kinetics were independent of the extent of inactivation, suggesting that mechanosensitive gating does not modify the structure of the channel pore.     

A model of ion channel kinetics using deterministic chaotic rather than stochastic processes.

Models of ion channel kinetics have previously assumed that the switching between the open and closed states is an intrinsically random process. Here, we present an alternative model based on a deterministic process. This model is a piecewise linear iterated map. We calculate the dwell time distributions, autocorrelation function, and power spectrum of this map. We also explore non-linear generalizations of this map. The chaotic nature of our model implies that its long-term behavior mimics the stochastic properties of a random process. In particular, the linear map produces an exponential probability distribution of dwell times in the open and closed states, the same as that produced by the two-state, closed in equilibrium open, Markov model. We show how deterministic and random models can be distinguished by their different phase space portraits. A test of some experimental data seems to favor the deterministic model, but further experimental evidence is needed for an unequivocal decision.     

Probing kinetic drug binding mechanism in voltage-gated sodium ion channel: open state versus inactive state blockers

The kinetics and nonequilibrium thermodynamics of open state and inactive state drug binding mechanisms have been studied here using different voltage protocols in sodium ion channel. We have found that for constant voltage protocol, open state block is more efficient in blocking ionic current than inactive state block. Kinetic effect comes through peak current for mexiletine as an open state blocker and in the tail part for lidocaine as an inactive state blocker. Although the inactivation of sodium channel is a free energy driven process, however, the two different kinds of drug affect the inactivation process in a different way as seen from thermodynamic analysis. In presence of open state drug block, the process initially for a long time remains entropy driven and then becomes free energy driven. However in presence of inactive state block, the process remains entirely entropy driven until the equilibrium is attained. For oscillating voltage protocol, the inactive state blocking is more efficient in damping the oscillation of ionic current. From the pulse train analysis it is found that inactive state blocking is less effective in restoring normal repolarisation and blocks peak ionic current. Pulse train protocol also shows that all the inactive states behave differently as one inactive state responds instantly to the test pulse in an opposite manner from the other two states.     

Covariance of nonstationary sodium current fluctuations at the node of Ranvier.

A theory is presented which relates the nonstationary autocovariance (covariance) function to the kinetics of independently-gated ionic channels. The experimental covariance was calculated from ensembles of 256--504 current records elicited from single, voltage-clamped, frog myelinated nerve fibers. Analysis of the covariance shows that the decay of channels from conducting to nonconducting states proceeds more slowly late in a depolarization to near 0 mV, as compared with early in the same depolarization. This behavior is inconsistent with there being only one kinetic state corresponding to the open channel. The behavior can be explained by the existence of multiple kinetic states corresponding to the open channel, or, alternatively, by the existence of multiple, kinetically distinct populations of channels.     

Probing kinetic drug binding mechanism in voltage-gated sodium ion channel: open state versus inactive state blockers

The kinetics and nonequilibrium thermodynamics of open state and inactive state drug binding mechanisms have been studied here using different voltage protocols in sodium ion channel. We have found that for constant voltage protocol, open state block is more efficient in blocking ionic current than inactive state block. Kinetic effect comes through peak current for mexiletine as an open state blocker and in the tail part for lidocaine as an inactive state blocker. Although the inactivation of sodium channel is a free energy driven process, however, the two different kinds of drug affect the inactivation process in a different way as seen from thermodynamic analysis. In presence of open state drug block, the process initially for a long time remains entropy driven and then becomes free energy driven. However in presence of inactive state block, the process remains entirely entropy driven until the equilibrium is attained. For oscillating voltage protocol, the inactive state blocking is more efficient in damping the oscillation of ionic current. From the pulse train analysis it is found that inactive state blocking is less effective in restoring normal repolarisation and blocks peak ionic current. Pulse train protocol also shows that all the inactive states behave differently as one inactive state responds instantly to the test pulse in an opposite manner from the other two states.     

Nonequilibrium response spectroscopy of voltage-sensitive ion channel gating.

We describe a new electrophysiological technique called nonequilibrium response spectroscopy, which involves application of rapidly fluctuating (as high as 14 kHz) large-amplitude voltage clamp waveforms to ion channels. As a consequence of the irreversible (in the sense of Carnot) exchange of energy between the fluctuating field and the channel protein, the gating response is exquisitely sensitive to features of the kinetics that are difficult or impossible to adequately resolve by means of traditional stepped potential protocols. Here we focus on the application of dichotomous (telegraph) noise voltage fluctuations, a broadband Markovian colored noise that fluctuates between two values. Because Markov kinetic models of channel gating can be embedded within higher-dimensional Markov models that take into account the effects of the voltage fluctuations, many features of the response of the channels can be calculated algebraically. This makes dichotomous noise and its generalizations uniquely suitable for model selection and kinetic analysis. Although we describe its application to macroscopic ionic current measurements, the nonequilibrium response method can also be applied to gating and single channel current recording techniques. We show how data from the human cardiac isoform (hH1a) of the Na+ channel expressed in mammalian cells can be acquired and analyzed, and how these data reveal hidden aspects of the molecular kinetics that are not revealed by conventional methods.     

Kinetics of the Opening and Closing of Individual Excitability-Inducing Material Channels in a Lipid Bilayer

The kinetics of the opening and closing of individual ion-conducting channels in lipid bilayers doped with small amounts of excitability-inducing material (EIM) are determined from discrete fluctuations in ionic current. The kinetics for the approach to steady-state conductance during voltage clamp are determined for lipid bilayers containing many EIM channels. The two sets of measurements are found to be consistent, verifying that the voltage-dependent conductance of the many-channel EIM system arises from the opening and closing of individual EIM channels. The opening and closing of the channels are Poisson processes. Transition rates for these processes vary exponentially with applied potential, implying that the energy difference between the open and closed states of an EIM channel is linearly proportional to the transmembrane electric field. A model incorporating the above properties of the EIM channels predicts the observed voltage dependence of ionic conductance and conductance relaxation time, which are also characteristic of natural electrically excitable membranes.     

Properties of amphotericin B channels in a lipid bilayer

Properties of individual ionic channels formed by polyene antibiotic Amphotericin B were studied on brain phospholipid membranes containing cholesterol. The ionic channels have a closed state and an open one (with conductance of about 6.5 pS in 2 M KCl). The conductance value of an open channel is independent of cholesterol concentration in the membrane and of pH in the range from 3.5 to 8.0. The voltage-current characteristics of a single channel are superlinear. Zero current potential value in the case of different KCl concentrations in the two solutions indicates preferential but not ideal anionic selectivity of a single channel. Channel conductivity grows as the electrolyte concentration is increased and tends to a limiting value at high concentrations. A simple model having only one site for an ion was shown to represent satisfactorily an open channel behaviour under different conditions. An individual ionic channel performs a large number of transitions between the open and closed states during its life-time of several minutes. Rate constants of these transitions depend on the kind and concentration of salt in aqueous solutions. The switching system functioning is not influenced by an ion situated inside the pore.     

Tetanus toxin channel in phosphatidylserine planar bilayers: conductance states and pH dependence

Tetanus toxin (TeTx) forms ionic channel in phosphatidylserine bilayers. TeTx channels exhibit different modes of channel bursting activity, from a closed state to well defined open states of different amplitudes. At positive applied voltages, TeTx channels flicker continuously between a closed state and the various distinct open states. Furthermore, fast transitions into subconductance states are discernible within the bursts of channel activity. Elementary conductance steps submultiple of the open states were not identified in single channel records owing to rapid transitions between different states. However, statistical analysis shows that conductances cluster with amplitudes multiple of an elementary value: e.g. 25–30 pS at neutral pH. Single channel current amplitudes decrease with the pH of the bulk electrolyte solution. Conductance decrements can be accounted for by the relative decrease of permeant cation concentration at the membrane-water interface, by a relative enrichment of protons that block the channel or by the stabilization of a conformational state of the channel protein. Offprint requests to: F. Gambale     

Effect of protein kinase A-induced phosphorylation on the gating mechanism of the brain Na+ channel: model fitting to whole-cell current traces.

The activity of the voltage-gated Na+ channel is subjected to modulation through covalent modifications. It has been previously shown that brain Na+ currents are reduced following the activation of the protein kinase A (PKA) pathway, but the effect of the phosphorylation on the gating mechanism of the channel has not been demonstrated so far. In this study, we analyze the whole-cell Na+ current recorded in the absence or presence of forskolin, which stimulates the PKA pathway. A minimal molecular model of the gating mechanism of the Na+ channel is defined to fit the experimental data: it consists of three closed states, one open state, and two inactivated states. We experimentally demonstrate that the kinetics of inactivation from the closed states are not affected by phosphorylation. The results obtained by computer fitting indicate that, among all the kinetic parameters describing the transitions between states, only one parameter is significantly modified in the presence of forskolin, and corresponds to the acceleration of the inactivation from the open state. This conclusion is supported by the analysis of current traces obtained from cells in the presence of a phosphatase inhibitor or loaded with the PKA catalytic unit, and is in agreement with previously reported single channel records.     

Statistical properties of ion channel records. Part I: relationship to the macroscopic current

Macroscopic ion channel current can be derived by summation of the stochastic records of individual channel currents. In this paper, we present two probability density functions of single channel records that can uniquely determine the macroscopic current regardless of other statistical properties of records or the stochastic model of channel gating (presented often with stationary Markov models). We show that H(t), probability density function of channel opening events (introduced explicitly in this paper), and D(t), probability density function of the open duration (sometimes has named dwell time distribution as well), determine the normalized macroscopic current, G(t), through G(t) = P(t) - H(t) * Q(t) where P(t) is the cumulative density function of H(t), Q(t) is the cumulative density function of D(t), * is the symbol of convolution integral and G(t) is the macroscopic current divided by the amplitude of single channel current and the number of single channel sweeps. Compared to other equations for the macroscopic current, here the macroscopic current is expressed only in terms of the statistical properties of single channel current and not the stochastic model of ion channel gating or a conditioned form of macroscopic current. Single channel currents of an inactivating BK channel were used to validate this relationship experimentally too. In this paper, we used median filters as they can remove the unwanted noise without smoothing the transitions between open and closed states (compare to low pass filters). This filtering leads to more accurate measurement of transition times and less amount of missed events.     

ATP transport through a single mitochondrial channel, VDAC, studied by current fluctuation analysis.

The "molecular Coulter counter" concept has been used to study transport of ATP molecules through the nanometer-scale aqueous pore of the voltage-dependent mitochondrial ion channel, VDAC. We examine the ATP-induced current fluctuations and the change in average current through a single fully open channel reconstituted into a planar lipid bilayer. At high salt concentration (1 M NaCl), the addition of ATP reduces both solution conductivity and channel conductance, but the effect on the channel is several times stronger and shows saturation behavior even at 50 mM ATP concentration. These results and simple steric considerations indicate pronounced attraction of ATP molecules to VDAC's aqueous pore and permit us to evaluate the effect of a single ATP molecule on channel conductance. ATP addition also generates an excess noise in the ionic current through the channel. Analysis of this excess noise shows that its spectrum is flat in the accessible frequency interval up to several kilohertz. ATP exchange between the pore and the bulk is fast enough not to display any dispersion at these frequencies. By relating the low-frequency spectral density of the noise to the equilibrium diffusion of ATP molecules in the aqueous pore, we calculate a diffusion coefficient D = (1.6-3.3)10(-11) m2/s. This is one order of magnitude smaller than the ATP diffusion coefficient in the bulk, but it agrees with recent results on ATP flux measurements in multichannel membranes using the luciferin/luciferase method.