The effects of non-identifiability on testing for detailed balance in aggregated Markov models for ion-channel gating

Aggregated Markov models are a widely used tool to model patch clamp data measured from single ion channels. These channels must obey the principle of detailed balance in thermodynamic equilibrium; otherwise, the channel is driven by an external source of energy. We investigate the power of a likelihood ratio test for detailed balance for a number of data points which is in the order of magnitude of patch clamp experiments. We show that for certain models with nearly equal dwell times, a test for detailed balance suffers from a loss of power to detect violations of detailed balance which is due to the non-identifiability of the transition rates for models with equal dwell times.     

Sodium channels in planar lipid bilayers. Channel gating kinetics of purified sodium channels modified by batrachotoxin

Single channel currents of sodium channels purified from rat brain and reconstituted into planar lipid bilayers were recorded. The kinetics of channel gating were investigated in the presence of batrachotoxin to eliminate inactivation and an analysis was conducted on membranes with a single active channel at any given time. Channel opening is favored by depolarization and is strongly voltage dependent. Probability density analysis of dwell times in the closed and open states of the channel indicates the occurrence of one open state and several distinct closed states in the voltage (V) range-120 mV less than or equal to V less than or equal to +120 mV. For V less than or equal to 0, the transition rates between stages are exponentially dependent on the applied voltage, as described in mouse neuroblastoma cells (Huang, L. M., N. Moran, and G. Ehrenstein. 1984. Biophysical Journal. 45:313-322). In contrast, for V greater than or equal to 0, the transition rates are virtually voltage independent. Autocorrelation analysis (Labarca, P., J. Rice, D. Fredkin, and M. Montal. 1985. Biophysical Journal. 47:469-478) shows that there is no correlation in the durations of successive open or closing events. Several kinetic schemes that are consistent with the experimental data are considered. This approach may provide information about the mechanism underlying the voltage dependence of channel activation.     

Gating kinetics of batrachotoxin-modified sodium channels in neuroblastoma cells determined from single-channel measurements

We have observed the opening and closing of single batrachotoxin (BTX)-modified sodium channels in neuroblastoma cells using the patch-clamp method. The conductance of a single BTX-modified channel is approximately 10 pS. At a given membrane potential, the channels are open longer than are normal sodium channels. As is the case for normal sodium channels, the open dwell times become longer as the membrane is depolarized. For membrane potentials more negative than about -70 mV, histograms of both open-state dwell times and closed-state dwell times could be fit by single exponentials. For more depolarized potentials, although the open-state histograms could still be fit by single exponentials, the closed-state histograms required two exponentials. This data together with macroscopic voltage clamp data on the same system could be accounted for by a three-state closed-closed-open model with transition rates between these states that are exponential functions of membrane potential. One of the implications of this model, in agreement with experiment, is that there are always some closed BTX-modified sodium channels, regardless of membrane potential.     

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.     

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.     

Gating kinetics of batrachotoxin-modified Na+ channels in the squid giant axon. Voltage and temperature effects.

The gating kinetics of batrachotoxin-modified Na+ channels were studied in outside-out patches of axolemma from the squid giant axon by means of the cut-open axon technique. Single channel kinetics were characterized at different membrane voltages and temperatures. The probability of channel opening (Po) as a function of voltage was well described by a Boltzmann distribution with an equivalent number of gating particles of 3.58. The voltage at which the channel was open 50% of the time was a function of [Na+] and temperature. A decrease in the internal [Na+] induced a shift to the right of the Po vs. V curve, suggesting the presence of an integral negative fixed charge near the activation gate. An increase in temperature decreased Po, indicating a stabilization of the closed configuration of the channel and also a decrease in entropy upon channel opening. Probability density analysis of dwell times in the closed and open states of the channel at 0 degrees C revealed the presence of three closed and three open states. The slowest open kinetic component constituted only a small fraction of the total number of transitions and became negligible at voltages greater than -65 mV. Adjacent interval analysis showed that there is no correlation in the duration of successive open and closed events. Consistent with this analysis, maximum likelihood estimation of the rate constants for nine different single-channel models produced a preferred model (model 1) having a linear sequence of closed states and two open states emerging from the last closed state. The effect of temperature on the rate constants of model 1 was studied. An increase in temperature increased all rate constants; the shift in Po would be the result of an increase in the closing rates predominant over the change in the opening rates. The temperature study also provided the basis for building an energy diagram for the transitions between channel states.     

Evidence for interactions between batrachotoxin-modified channels in hybrid neuroblastoma cells.

Current records from voltage-clamped membrane patches containing two batrachotoxin-modified sodium channels were analyzed to determine whether these channels are identical and independent. In most two-channel patches, the experimentally observed probabilities that zero, one, or two channels are open differ from the binomial distribution, demonstrating that the two channels are nonidentical or nonindependent or both. From the same current records, we also determined the rate for the transition from two open channels to one open channel and for the transition from one open channel to zero open channels. These data are consistent with closing rates for the two channels that are equal and independent. Both probability and closing rate data can be fit by a model wherein the channels are identical, the closing rates are independent, and the opening rate is greater when the other channel is closed than when it is open. The implications of this model for analyzing noise spectra and current variance are examined.     

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.     

Measuring kinetics of complex single ion channel data using mean-variance histograms.

The measurement of single ion channel kinetics is difficult when those channels exhibit subconductance events. When the kinetics are fast, and when the current magnitudes are small, as is the case for Na+, Ca2+, and some K+ channels, these difficulties can lead to serious errors in the estimation of channel kinetics. I present here a method, based on the construction and analysis of mean-variance histograms, that can overcome these problems. A mean-variance histogram is constructed by calculating the mean current and the current variance within a brief "window" (a set of N consecutive data samples) superimposed on the digitized raw channel data. Systematic movement of this window over the data produces large numbers of mean-variance pairs which can be assembled into a two-dimensional histogram. Defined current levels (open, closed, or sublevel) appear in such plots as low variance regions. The total number of events in such low variance regions is estimated by curve fitting and plotted as a function of window width. This function decreases with the same time constants as the original dwell time probability distribution for each of the regions. The method can therefore be used: 1) to present a qualitative summary of the single channel data from which the signal-to-noise ratio, open channel noise, steadiness of the baseline, and number of conductance levels can be quickly determined; 2) to quantify the dwell time distribution in each of the levels exhibited. In this paper I present the analysis of a Na+ channel recording that had a number of complexities. The signal-to-noise ratio was only about 8 for the main open state, open channel noise, and fast flickers to other states were present, as were a substantial number of subconductance states. "Standard" half-amplitude threshold analysis of these data produce open and closed time histograms that were well fitted by the sum of two exponentials, but with apparently erroneous time constants, whereas the mean-variance histogram technique provided a more credible analysis of the open, closed, and subconductance times for the patch. I also show that the method produces accurate results on simulated data in a wide variety of conditions, whereas the half-amplitude method, when applied to complex simulated data shows the same errors as were apparent in the real data. The utility and the limitations of this new method are discussed.     

Model selection in non-nested hidden Markov models for ion channel gating

An important task in the application of Markov models to the analysis of ion channel data is the determination of the correct gating scheme of the ion channel under investigation. Some prior knowledge from other experiments can reduce significantly the number of possible models. If these models are standard statistical procedures nested like likelihood ratio testing, provide reliable selection methods. In the case of non-nested models, information criteria like AIC, BIC, etc., are used. However, it is not known if any of these criteria provide a reliable selection method and which is the best one in the context of ion channel gating. We provide an alternative approach to model selection in the case of non-nested models with an equal number of open and closed states. The models to choose from are embedded in a properly defined general model. Therefore, we circumvent the problems of model selection in the non-nested case and can apply model selection procedures for nested models.     

Superposition properties of interacting ion channels.

Quantitative analysis of patch clamp data is widely based on stochastic models of single-channel kinetics. Membrane patches often contain more than one active channel of a given type, and it is usually assumed that these behave independently in order to interpret the record and infer individual channel properties. However, recent studies suggest there are significant channel interactions in some systems. We examine a model of dependence in a system of two identical channels, each modeled by a continuous-time Markov chain in which specified transition rates are dependent on the conductance state of the other channel, changing instantaneously when the other channel opens or closes. Each channel then has, e.g., a closed time density that is conditional on the other channel being open or closed, these being identical under independence. We relate the two densities by a convolution function that embodies information about, and serves to quantify, dependence in the closed class. Distributions of observable (superposition) sojourn times are given in terms of these conditional densities. The behavior of two channel systems based on two- and three-state Markov models is examined by simulation. Optimized fitting of simulated data using reasonable parameters values and sample size indicates that both positive and negative cooperativity can be distinguished from independence.     

Correlation character of ionic current fluctuations: analysis of ion current through a voltage-dependent potassium single channel

The gating of ion channels has widely been modeled by assuming the transition between open and closed states is a memoryless process. Nevertheless, the statistical analysis of an ionic current signal recorded from voltage dependence K(+) single channel is presented. Calculating the sample auto-correlation function of the ionic current based on the digitized signals, rather than the sequence of open and closed states duration time. The results provide evidence for the existence of memory. For different voltages, the ion channel current fluctuation has different correlation attributions. The correlations in data generated by simulation of two Markov models, on one hand, auto-correlation function of the ionic current shows a weaker memory, after a delayed period of time, the attribute of memory does not exist; on the other hand, the correlation depends on the number of states in the Markov model. For V(p)=-60 mV pipette potential, spectral analysis of ion channel current was conducted, the result indicates that the spectrum is not a flat spectrum, the data set from ionic current fluctuations shows considerable variability with a broad 1/f -like spectrum, alpha=1.261+/-0.24. Thus the ion current fluctuations give information about the kinetics of the channel protein, the results suggest the correlation character of ion channel protein nonlinear kinetics regardless of whether the channel is in open or closed state.     

Evidence for cooperativity between nicotinic acetylcholine receptors in patch clamp records

It is often assumed that ion channels in cell membrane patches gate independently. However, in the present study nicotinic receptor patch clamp data obtained in cell-attached mode from embryonic chick myotubes suggest that the distribution of steady-state probabilities for conductance multiples arising from concurrent channel openings may not be binomial. In patches where up to four active channels were observed, the probabilities of two or more concurrent openings were greater than expected, suggesting positive cooperativity. For the case of two active channels, we extended the analysis by assuming that 1) individual receptors (not necessarily identical) could be modeled by a five-state (three closed and two open) continuous-time Markov process with equal agonist binding affinity at two recognition sites, and 2) cooperativity between channels could occur through instantaneous changes in specific transition rates in one channel following a change in conductance state of the neighboring channel. This allowed calculation of open and closed sojourn time density functions for either channel conditional on the neighboring channel being open or closed. Simulation studies of two channel systems, with channels being either independent or cooperative, nonidentical or identical, supported the discriminatory power of the optimization algorithm. The experimental results suggested that individual acetylcholine receptors were kinetically identical and that the open state of one channel increased the probability of opening of its neighbor.     

Gating of the squid sodium channel at positive potentials: II. Single channels reveal two open states.

Single-channel recordings from squid axon Na+ channels were made under conditions of reverse sodium gradient. In the range of potentials studied, +40-(+)120 mV, channels opened promptly after depolarization, closed and reopened several times during the pulse. In patches containing only one channel, the distributions of open dwell times showed two components showing the existence of a second open state. The ensemble average of single-channel records showed incomplete inactivation that became more pronounced at more positive potentials, showing that the maintained phase of the current is the result of only one type of sodium channel with two open states. Analysis of bursts indicated that the dwell times of the events at the onset of the depolarization are longer than those later in the pulse. The dwell open times of the first events could be fitted with a single exponential. This indicated that the channels open preferentially through the first open state, the access to the second open state happening subsequently. Maximum likelihood analysis was used to evaluate several possible kinetic schemes incorporating a second open state. The best model to fit the data from single channels, and consistent with the data from macroscopic and gating currents, has a second open state evolving from the inactivated state. A kinetic model is proposed that incorporates information obtained from dialyzed axons.     

Glutamate receptor-channel gating. Maximum likelihood analysis of gigaohm seal recordings from locust muscle.

Gigaohm recordings have been made from glutamate receptor channels in excised, outside-out patches of collagenase-treated locust muscle membrane. The channels in the excised patches exhibit the kinetic state switching first seen in megaohm recordings from intact muscle fibers. Analysis of channel dwell time distributions reveals that the gating mechanism contains at least four open states and at least four closed states. Dwell time autocorrelation function analysis shows that there are at least three gateways linking the open states of the channel with the closed states. A maximum likelihood procedure has been used to fit six different gating models to the single channel data. Of these models, a cooperative model yields the best fit, and accurately predicts most features of the observed channel gating kinetics.     

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.     

A model of the gating of ion channels

The gating of ion channels in biological membranes has usually been described in terms of Markov transitions between a few discrete open or closed states. Such models predict that the distributions of open and closed durations decay as a sum of exponential terms. Recent experimental data have indicated that certain channels are not easily described by these models. We show that distributions of open and closed times similar to those seen experimentally are predicted by a model that involves only one open and closed state but that assumes the activation energy of the gating process to be stochastic. This model involves only a few parameters and these have direct physical interpretations. Measurements of the correlation between the durations of successive open or closed events is shown to provide an experimental method for distinguishing between this and other models.     

Interactions of protons with single open L-type calcium channels. pH dependence of proton-induced current fluctuations with Cs+, K+, and Na+ as permeant ions

We studied the pH dependence of the proton-induced current fluctuations that appear in single open L-type Ca channels when monovalent ions are the charge carriers. We used different methods of analysis to obtain kinetic measurements even under conditions where the individual transitions were too fast to be resolved directly as discrete current steps between two conductance levels. The reciprocal of the dwell times at the high conductance level increased linearly with the pipette proton activity, with a slope that was similar for Cs, K, and Na as permeant ions. Contrary to the expectation for a simple model in which the high and low conductances represent the unprotonated and protonated states of the channel, respectively, the dwell times at the low conductance level were also pH dependent and lengthened with increasing proton activity. At all pH values the dwell times at the low conductance level were longest with Cs as permeant ion and shortened in the order Cs greater than K greater than Na. We introduce a more general model of the protonation cycle in which the channel is represented by four states and can be protonated and deprotonated both at the high and low conductance levels. The conductance change is represented by a conformational change of the channel protein. We discuss the validity of this model and its implications for the mechanism by which protons interact with ion permeation through L-type Ca channels.     

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.