Equivalence of aggregated Markov models of ion-channel gating

One cannot always distinguish different Markov models of ion-channel kinetics solely on the basis of steady-state kinetic data. If two generator (or transition) matrices are related by a similarity transformation that does not combine states with different conductances, then the models described by these generator matrices have the same observable steady-state statistics. This result suggests a procedure for expressing the model in a unique form, and sometimes reducing the number of parameters in a model. I apply the similarity transformation procedure to a number of simple models. When a model specifies the dependence of the rates of transition on an experimentally variable parameter such as the concentration of a ligand or the membrane potential, the class of equivalent models may be further restricted, but a model is not always uniquely determined even under these conditions. Voltage-step experiments produce non-stationary data that can also be used to distinguish models.     

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.     

Hidden Markov analysis of mechanosensitive ion channel gating

Patch clamp data from the large conductance mechanosensitive channel (MscL) in E. coli was studied with the aim of developing a strategy for statistical analysis based on hidden Markov models (HMMs) and determining the number of conductance levels of the channel, together with mean current, mean dwell time and equilibrium probability of occupancy for each level. The models incorporated state-dependent white noise and moving average adjustment for filtering, with maximum likelihood parameter estimates obtained using an EM (expectation-maximisation) based iteration. Adjustment for filtering was included as it could be expected that the electronic filter used in recording would have a major effect on obviously brief intermediate conductance level sojourns. Preliminary data analysis revealed that the brevity of intermediate level sojourns caused difficulties in assignment of data points to levels as a result of over-estimation of noise variances. When reasonable constraints were placed on these variances using the better determined noise variances for the closed and fully open levels, idealisation anomalies were eliminated. Nevertheless, simulations suggested that mean sojourn times for the intermediate levels were still considerably over-estimated, and that recording bandwidth was a major limitation; improved results were obtained with higher bandwidth data (10 kHz sampled at 25 kHz). The simplest model consistent with these data had four open conductance levels, intermediate levels being approximately 20%, 51% and 74% of fully open. The mean lifetime at the fully open level was about 1 ms; estimates for the three intermediate levels were 54-92 micros, probably still over-estimates.     

Adjustments for the display of quantized ion channel dwell times in histograms with logarithmic bins

Dwell-time histograms are often plotted as part of patch-clamp investigations of ion channel currents. The advantages of plotting these histograms with a logarithmic time axis were demonstrated by, J. Physiol. (Lond.). 378:141-174), Pflügers Arch. 410:530-553), and, Biophys. J. 52:1047-1054). Sigworth and Sine argued that the interpretation of such histograms is simplified if the counts are presented in a manner similar to that of a probability density function. However, when ion channel records are recorded as a discrete time series, the dwell times are quantized. As a result, the mapping of dwell times to logarithmically spaced bins is highly irregular; bins may be empty, and significant irregularities may extend beyond the duration of 100 samples. Using simple approximations based on the nature of the binning process and the transformation rules for probability density functions, we develop adjustments for the display of the counts to compensate for this effect. Tests with simulated data suggest that this procedure provides a faithful representation of the data.     

Identifying kinetic gating mechanisms for ion channels by using two-dimensional distributions of simulated dwell times.

Ion channels are integral membrane proteins that regulate ionic flux through cell membranes by opening and closing (or gating) their pores. The gating can be monitored by observing step changes in the current flowing through single channels. Analysis of the durations of the open and closed intervals and of the correlations among the interval durations can give insight into the gating mechanism. Although it is well known that the correlation information can be essential to distinguish among possible gating mechanisms, it has been difficult to use this information because it has not been possible to correct the predicted correlations for the distortion of the single-channel data because of filtering and noise. To overcome this limitation we present a method based on a comparison of simulated and experimental two-dimensional dwell-time distributions constructed by analysing simulated and experimental single-channel currents in an identical manner. The simulated currents incorporate the true effects of filtering and noise, the two-dimensional distributions retain the correlation information, and the identical analysis allows direct maximum-likelihood comparison of the simulated and experimental two-dimensional distributions. We show that the two-dimensional simulation method has a greatly increased ability to distinguish among models, compared with methods that use one-dimensional distributions.     

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.     

Bayesian restoration of ion channel records using hidden Markov models

Hidden Markov models have been used to restore recorded signals of single ion channels buried in background noise. Parameter estimation and signal restoration are usually carried out through likelihood maximization by using variants of the Baum-Welch forward-backward procedures. This paper presents an alternative approach for dealing with this inferential task. The inferences are made by using a combination of the framework provided by Bayesian statistics and numerical methods based on Markov chain Monte Carlo stochastic simulation. The reliability of this approach is tested by using synthetic signals of known characteristics. The expectations of the model parameters estimated here are close to those calculated using the Baum-Welch algorithm, but the present methods also yield estimates of their errors. Comparisons of the results of the Bayesian Markov Chain Monte Carlo approach with those obtained by filtering and thresholding demonstrate clearly the superiority of the new methods.     

On identification of Na+ channel gating schemes using moving-average filtered hidden Markov models

 Transitions between distinct kinetic states of an ion channel are described by a Markov process. Hidden Markov models (HMM) have been successfully applied in the analysis of single ion channel recordings with a small signal-to-noise ratio. However, we have recently shown that the anti-aliasing low-pass filter misleads parameter estimation. Here, we show for the case of a Na⁺ channel recording that the standard HMM do neither allow parameter estimation nor a correct identification of the gating scheme. In particular, the number of closed and open states is determined incorrectly, whereas a modified HMM considering the anti-aliasing filter (moving-average filtered HMM) is able to reproduce the characteristic properties of the time series and to perform gating scheme identification. Received: 11 February 1999 / Revised version: 18 June 1999 / Accepted: 21 June 1999     

Applying hidden Markov models to the analysis of single ion channel activity

Hidden Markov models have recently been used to model single ion channel currents as recorded with the patch clamp technique from cell membranes. The estimation of hidden Markov models parameters using the forward-backward and Baum-Welch algorithms can be performed at signal to noise ratios that are too low for conventional single channel kinetic analysis; however, the application of these algorithms relies on the assumptions that the background noise be white and that the underlying state transitions occur at discrete times. To address these issues, we present an "H-noise" algorithm that accounts for correlated background noise and the randomness of sampling relative to transitions. We also discuss three issues that arise in the practical application of the algorithm in analyzing single channel data. First, we describe a digital inverse filter that removes the effects of the analog antialiasing filter and yields a sharp frequency roll-off. This enhances the performance while reducing the computational intensity of the algorithm. Second, the data may be contaminated with baseline drifts or deterministic interferences such as 60-Hz pickup. We propose an extension of previous results to consider baseline drift. Finally, we describe the extension of the algorithm to multiple data sets.     

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.     

Analysis of fractal ion channel gating kinetics: kinetic rates, energy levels, and activation energies

Ion channels in the cell membranes of the corneal endothelium, hippocampal neurons, and fibroblasts, and gramicidin channels in lipid bilayers have open and closed times that can be fit, in whole or part, by power law distributions. That is, the gating is self-similar when viewed at different time scales. Hence, kinetic processes at slow and fast time scales are not independent but rather are interrelated. To study how such a relationship can arise we analyze a closed in equilibrium open channel with the fractal dimension for leaving the closed state DCO approximately 2 and the fractal dimension for leaving the open state DOC approximately 1. This special case can be analyzed because it can be represented by equivalent Markov processes. We show that it is equivalent to Markov chains with forward and backward kinetic rate constants approximately equal at each stage, and forming an approximate geometric progression along the different stages. These kinetic rates determine the energy levels and activation energy barriers separating those levels. We find that there are many conformational states (substates) separated by high activation energy barriers. This is similar to the energy structure found for globular proteins such as myoglobin. However, the novel feature reported here is that the activation energy barriers are not independent but are interrelated and form an arithmetic progression. Because of this relationship the fast processes across the low activation energy barriers are linked to slow processes across the high activation energy barriers.     

Statistical discrimination of fractal and Markov models of single-channel gating.

A statistical comparison is presented of Markov and fractal models of ion channel gating. The analysis is based on single-channel data from two types of ion channels: open times from a 90 pS Ca-activated K channel from GH3 pituitary cells, and closed times from a nonselective channel from rabbit corneal endothelium (Liebovitch et al., 1987a). Maximum likelihood methods were used to fit the data. For both data sets the best Markov model had three exponential components. The best Markov model had a higher likelihood than the fractal model, and the Asymptotic Information Criterion favored the Markov model for each data set. A more detailed analysis, using the Monte Carlo methods described in Horn (1987), showed that the Markov model was not significantly better than the fractal model for the corneal endothelium channels. The inability to discriminate the models definitively in this case was shown to be due in part to the small size of the data set.     

Hidden Markov models for settings with interval-censored transition times and uncertain time origin: application to HIV genetic analyses

The simplicity and flexibility of Markov models make them appealing for investigations of the acquisition of HIV drug-resistance mutations, whose presence can define specific Markov states. Because the exact time of acquiring a mutation is not observed during clinical research studies on HIV infection, it is important that methods for fitting such models accommodate interval-censored transition times. Furthermore, many such studies include patients with extensive treatment experience prior to the onset of the studies. Therefore, the virus in these patients may have already acquired resistance mutations by study entry. Retrospective data regarding the time on treatment, which is often known or known with error, provide information about the acquisition rates before the start of a study. Finally, variability in the genetic sequences of circulating HIV creates uncertainty in the Markov states. This paper considers approaches to fitting Markov models to data with interval-censored transition times when the time origin and the Markov states are known with error. The methods were applied to AIDS Clinical Trial Group protocol 398, a randomized comparison of mono- versus dual-protease inhibitor use in heavily pretreated patients. We found that the estimated rates of acquiring certain classes of mutations depended on the presence of others, and that the precision of these estimates can be considerably improved by inclusion of retrospective data.     

Comparison of ion current noise predicted from different models of the sodium channel gating mechanism in nerve membrane

Summary The theoretical power density spectrumS(f) of ion current noise is calculated from several models of the sodium channel gating mechanism in nerve membrane. Sodium ion noise experimental data from the frog node of Ranvier [Conti, F.,et al. (1976),J. Physiol. (London) 262:699] is used as a test of the theoretical results. The motivation for recent modeling has been evidence for a coupling between sodium activation and inactivation from voltage clamp data. The two processes are independent of one another in the Hodgkin and Huxley (HH) model [Hodgkin, A.L., Huxley, A.F. (1952),J. Physiol. (London) 117:500] The noise data is consistent with HH, as noted by Contiet al. (1976). The theoretical results given here appear to indicate that only one case of coupling models is also consistent with the noise data.     

Understanding ion channel selectivity and gating and their role in cellular signalling

Ion channels play an essential role in the communication between and within cells. Here some of the different ion channel proteins and the roles they perform are introduced, before a discussion of the mechanisms by which they discriminate between different ion types and open and close to allow the passage of ions at the appropriate times.     

Ion binding to KcsA: implications in ion selectivity and channel gating

Binding of K+ and Na+ to the potassium channel KcsA has been characterized from the stabilization observed in the heat-induced denaturation of the protein as the ion concentration is increased. KcsA thermal denaturation is known to include (i) dissociation of the homotetrameric channel into its constituent subunits and (ii) protein unfolding. The ion concentration-dependent changes in the thermal stability of the protein, evaluated as the Tm value for thermal-induced denaturation of the protein, may suggest the existence of both high- and low-affinity K+ binding sites of KcsA, which lend support to the tenet that channel gating may be governed by K+ concentration-dependent transitions between different affinity states of the channel selectivity filter. We also found that Na+ binds to KcsA with a KD similar to that estimated electrophysiologically from channel blockade. Therefore, our findings on ion binding to KcsA partly account for K+ over Na+ selectivity and Na+ blockade and argue against the strict “snug fit” hypothesis used initially to explain ion selectivity from the X-ray channel structure. Furthermore, the remarkable effects of increasing the ion concentration, K+ in particular, on the Tm of the denaturation process evidence that synergistic effects of the metal-mediated intersubunit interactions at the channel selectivity filter are a major contributor to the stability of the tetrameric protein. This observation substantiates the notion of a role for ions as structural “effectors” of ion channels.