Statistical properties of single sodium channels

Single channel currents were obtained from voltage-activated sodium channels in outside-out patches of tissue-cultured GH3 cells, a clonal line from rat pituitary gland. In membrane patches where the probability of overlapping openings was low, the open time histograms were well fit by a single exponential. Most analysis was done on a patch with exactly one channel. We found no evidence for multiple open states at -25 and -40 mV, since open times, burst durations, and autocorrelation functions were time independent. Amplitude histograms showed no evidence of multiple conductance levels. We fit the gating with 25 different time-homogeneous Markov chain models having up to five states, using a maximum likelihood procedure to estimate the rate constants. For selected models, this procedure yielded excellent predictions for open time, closed time, and first latency density functions, as well as the probability of the channel being open after a step depolarization, the burst duration distribution, autocorrelation, and the distribution of number of openings per record. The models were compared statistically using likelihood ratio tests and Akaike's information criterion. Acceptable models allowed inactivation from closed states, as well as from the open state. Among the models eliminated as unacceptable by this survey were the Hodgkin-Huxley model and any model requiring a channel to open before inactivating.     

Estimation of ion channel kinetics from fluctuations of macroscopic currents

For single channel recordings, the maximum likelihood estimation (MLE) of kinetic rates and conductance is well established. A direct extrapolation of this method to macroscopic currents is computationally prohibitive: it scales as a power of the number of channels. An approximated MLE that ignored the local time correlation of the data has been shown to provide estimates of the kinetic parameters. In this article, an improved approximated MLE that takes into account the local time correlation is proposed. This method estimates the channel kinetics using both the time course and the random fluctuations of the macroscopic current generated by a homogeneous population of ion channels under white noise. It allows arbitrary kinetic models and stimulation protocols. The application of the proposed algorithm to simulated data from a simple three-state model on nonstationary conditions showed reliable estimates of all the kinetic constants, the conductance and the number of channels, and reliable values for the standard error of those estimates. Compared to the previous approximated MLE, it reduces by a factor of 10 the amount of data needed to secure a given accuracy and it can even determine the kinetic rates in macroscopic stationary conditions.     

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.     

How Powerful is the Dwell-Time Analysis of Multichannel Records?

Exact algorithms for the kinetic analysis of multichannel patch-clamp records require hours to days for a single record. Thus, it may be reasonable to use a fast but less accurate method for the analysis of all data sets and to use the results for a reanalysis of some selected records with more sophisticated approaches. For the first run, the tools of single-channel analysis were used for the evaluation of the single-channel rate constants from multichannel dwell-time histograms. This could be achieved by presenting an ensemble of single channels by a ``macrochannel' comprising all possible states of the ensemble of channels. Equations for the calculations of the elements of the macrochannel transition matrix and for the steady-state concentrations for individual states are given. Simulations of multichannel records with 1 to 8 channels with two closed and one open states and with 2 channels with two open and two closed states were done in order to investigate under which conditions the one-dimensional dwell-time analysis itself already provides reliable results. Distributions of the evaluated single-channel rate constants show that a bias of the estimations of the single-channel rate constants of 10 to 20% has to be accepted. The comparison of simulations with signal-to-noise ratios of SNR = 1 or SNR = 25 demonstrates that the major problem is not the convergence of the fitting routine, but failures of the level detector algorithm which creates the dwell-times distributions from noisy time series. The macrochannel presentation allows the incorporation of constraints like channel interaction. The evaluation of simulated 4-channel records in which the rate-constant of opening increased by 20% per already open channel could reveal the interaction factor. Received: 9 June 1997/Revised: 28 April 1998     

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.     

Inversion of Markov processes to determine rate constants from single-channel data.

The determination of rate constants from single-channel data can be very difficult, in part because the single-channel lifetime distributions commonly analyzed by experimenters often have a complicated mathematical relation to the channel gating mechanism. The standard treatment of channel gating as a Markov process leads to the prediction that lifetime distributions are exponential functions. As the number of states of a channel gating scheme increases, the number of exponential terms in the lifetime distribution increases, and the weights and decay constants of the lifetime distributions become progressively more complicated functions of the underlying rate constants. In the present study a mathematical strategy for inverting these functions is introduced in order to determine rate constants from single-channel lifetime distributions. This inversion is easy for channel gating schemes with two or fewer states of a given conductance, so the present study focuses on schemes with more states. The procedure is to derive explicit equations relating the parameters of the lifetime distribution to the rate constants of the scheme. Such equations can be derived using the equality between symmetric functions of eigenvalues of a matrix and sums over principle minors, as well as expressions for the moments, derivatives, and weights of a lifetime distribution. The rate constants are then obtained as roots to this system of equations. For a gating scheme with three sequential closed states and a single gateway state, exact analytical expressions were found for each rate constant in terms of the parameters of the three-exponential closed-time distribution. For several other gating schemes, systems of equations were found that could be solved numerically to obtain the rate constants. Lifetime distributions were shown to specify a unique set of real rate constants in sequential gating schemes with up to five closed or five open states. For kinetic schemes with multiple gating pathways, the analysis of simulated data revealed multiple solutions. These multiple solutions could be distinguished by examining two-dimensional probability density functions. The utility of the methods introduced here are demonstrated by analyzing published data on nicotinic acetylcholine receptors, GABA(A) receptors, and NMDA receptors.     

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.     

Estimating kinetic constants from single channel data.

The process underlying the opening and closing of ionic channels in biological or artificial lipid membranes can be modeled kinetically as a time-homogeneous Markov chain. The elements of the chain are kinetic states that can be either open or closed. A maximum likelihood procedure is described for estimating the transition rates between these states from single channel data. The method has been implemented for linear kinetic schemes of fewer than six states, and is suitable for nonstationary data in which one or more independent channels are functioning simultaneously. It also provides standard errors for all estimates of rate constants and permits testing of smoothly parameterized subhypotheses of a general model. We have illustrated our approach by analysis of single channel data simulated on a computer and have described a procedure for analysis of experimental data.     

Variance analysis of gamma-aminobutyric acid (GABA)-ergic inhibitory postsynaptic currents from melanotropes of Xenopus laevis.

We have studied the variance in the decay of large spontaneous gamma-aminobutyric acid (GABA)-ergic inhibitory postsynaptic currents (IPSCs) in melanotropes of Xenopus laevis to obtain information about the number of GABAA receptor channels that bind GABA during the IPSCs. The average decay of the IPSCs is well described by the sum of two exponential functions. This suggests that a three-state Markov model is sufficient to describe the decay phase, with one of the three states being an absorbing state, entered when GABA dissociates from the GABAA receptor. We have compared the variance in the decay of large spontaneous IPSCs with the variance calculated for two different three-state models: a model with one open state, one closed state, and one absorbing state (I), and a model with two open states and one absorbing state (II). The data were better described by the more efficient model II. This suggests that the efficacy of GABA at synaptic GABAA receptor channels is high and that only a small number of channels are involved in generating the GABA-ergic IPSCs.     

MCMC estimation of Markov models for ion channels

Ion channels are characterized by inherently stochastic behavior which can be represented by continuous-time Markov models (CTMM). Although methods for collecting data from single ion channels are available, translating a time series of open and closed channels to a CTMM remains a challenge. Bayesian statistics combined with Markov chain Monte Carlo (MCMC) sampling provide means for estimating the rate constants of a CTMM directly from single channel data. In this article, different approaches for the MCMC sampling of Markov models are combined. This method, new to our knowledge, detects overparameterizations and gives more accurate results than existing MCMC methods. It shows similar performance as QuB-MIL, which indicates that it also compares well with maximum likelihood estimators. Data collected from an inositol trisphosphate receptor is used to demonstrate how the best model for a given data set can be found in practice.     

Improved hidden Markov models for molecular motors, part 1: basic theory

Hidden Markov models (HMMs) provide an excellent analysis of recordings with very poor signal/noise ratio made from systems such as ion channels which switch among a few states. This method has also recently been used for modeling the kinetic rate constants of molecular motors, where the observable variable—the position—steadily accumulates as a result of the motor's reaction cycle. We present a new HMM implementation for obtaining the chemical-kinetic model of a molecular motor's reaction cycle called the variable-stepsize HMM in which the quantized position variable is represented by a large number of states of the Markov model. Unlike previous methods, the model allows for arbitrary distributions of step sizes, and allows these distributions to be estimated. The result is a robust algorithm that requires little or no user input for characterizing the stepping kinetics of molecular motors as recorded by optical techniques.     

Global parameter optimization for cardiac potassium channel gating models.

Quantitative ion channel model evaluation requires the estimation of voltage dependent rate constants. We have tested whether a unique set of rate constants can be reliably extracted from nonstationary macroscopic voltage clamp potassium current data. For many models, the rate constants derived independently at different membrane potentials are not unique. Therefore, our approach has been to use the exponential voltage dependence predicted from reaction rate theory (Stevens, C. F. 1978. Biophys. J. 22:295-306; Eyring, H., S. H. Lin, and S. M. Lin. 1980. Basic Chemical Kinetics. Wiley and Sons, New York) to couple the rate constants derived at different membrane potentials. This constrained the solution set of rate constants to only those that also obeyed this additional set of equations, which was sufficient to obtain a unique solution. We have tested this approach with data obtained from macroscopic delayed rectifier potassium channel currents in voltage-clamped guinea pig ventricular myocyte membranes. This potassium channel has relatively simple kinetics without an inactivation process and provided a convenient system to determine a globally optimized set of voltage-dependent rate constants for a Markov kinetic model. The ability of the fitting algorithm to extract rate constants from the macroscopic current data was tested using "data" synthesized from known rate constants. The simulated data sets were analyzed with the global fitting procedure and the fitted rate constants were compared with the rate constants used to generate the data. Monte Carlo methods were used to examine the accuracy of the estimated kinetic parameters. This global fitting approach provided a useful and convenient method for reliably extracting Markov rate constants from macroscopic voltage clamp data over a broad range of membrane potentials. The limitations of the method and the dependence on initial guesses are described.     

Activation and inactivation of single calcium channels in snail neurons

Activation and inactivation properties of Ca currents were investigated by studying the behavior of single Ca channels in snail neurons. The methods described in the previous paper were used. In addition, a zero- phase digital filter has been incorporated to improve the analysis of latencies to first opening, or waiting times. It was found that a decrease in the probability of single channel opening occurred with time. This was especially marked at 29 degrees C and paralleled the inactivation observed in macroscopic currents. The fact that a single channel was observed means that there is a significant amount of reopening from the "inactivated" state. Small depolarizations at 18 degrees C showed little inactivation. From these measurements, histograms of single channel open, closed, and waiting times were analyzed to estimate the rate constants of a three-state model of activation. Two serious discrepancies with the model were found. First, waiting time distributions at -20 mV were slower than those predicted by parameters obtained from an analysis of the single channel closed times. Second, it was shown that the time and the magnitude of the peak of the waiting time histogram were inconsistent with a three-state model. It is concluded that a minimum of four states are involved in activation. Some four-state models may be eliminated from further consideration. However, a comprehensive model of Ca channel kinetics must await further measurements.     

Efficient maximum likelihood estimation of kinetic rate constants from macroscopic currents

A new method is described that accurately estimates kinetic constants, conductance and number of ion channels from macroscopic currents. The method uses both the time course and the strength of correlations between different time points of macroscopic currents and utilizes the property of semiseparability of covariance matrix for computationally efficient estimation of current likelihood and its gradient. The number of calculation steps scales linearly with the number of channel states as opposed to the cubic dependence in a previously described method. Together with the likelihood gradient evaluation, which is almost independent of the number of model parameters, the new approach allows evaluation of kinetic models with very complex topologies. We demonstrate applicability of the method to analysis of synaptic currents by estimating accurately rate constants of a 7-state model used to simulate GABAergic macroscopic currents.     

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.     

Evaluation and Estimation of Various Markov Models with Applications to Membrane Channel Kinetics

Hidden Markov modelling is a powerful and efficient digital signal processing strategy for extracting the maximum likelihood model from a finite length sample of noisy data. Assuming the number of states in the model is known, then the state levels, transition probabilities, initial state distribution and the noise variance can be estimated. We investigate the applicability of this technique in membrane channel kinetics not only as a parameter estimator, but also as an aid to discriminating between various model types according to their statistical likelihood. We survey three representative classes of channel dynamics, namely: aggregated Markov models, semi-Markov models (with asymptotically convergent transition probabilities), and coupled Markov models; reformulating each within a discrete-time hidden Markov model framework. We then provide numerical evidence of the effectiveness of the procedure using simulated channel data and hence show that the correct model, as well as the model parameters, can be discerned. We also demonstrate that the model likelihood can be used to indicate the approximate number of states in the model.     

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.     

The anomalous mole fraction effect in Chara: gating at the edge of temporal resolution

The anomalous mole fraction effect (AMFE) of the K(+) channel in excised patches of the tonoplast of Chara showed a minimum of apparent open-channel current at 20 mM Tl(+) and 230 mM K(+). Time series obtained at a sampling rate of 100 kHz (filter 25 kHz) were analyzed by three methods to find out whether the AMFE results from an effect on gating or on the conductivity of the open state. Fitting the amplitude histograms by a superposition of gaussians showed a broadening in the presence of Tl(+). Dwell-time analysis based on an O-O-C-C-C model failed to evaluate rate constants above the filter frequency. Thus, the absence of any reduction of apparent open-channel current in time series simulated with the evaluated rate constants could not be taken as evidence against the hypothesis of gating. Finally, a direct fit of the measured time series using five different 5-state Hidden Markov models revealed that the presence of Tl(+) changed the rate constants in such a way that the number of transitions into the short-lived open state (30 micros) increased strongly compared to those in the absence of Tl(+). These models explain 25% reduction of apparent single-channel current amplitude through a rapid gating mechanism.