A direct optimization approach to hidden Markov modeling for single channel kinetics

Hidden Markov modeling (HMM) provides an effective approach for modeling single channel kinetics. Standard HMM is based on Baum's reestimation. As applied to single channel currents, the algorithm has the inability to optimize the rate constants directly. We present here an alternative approach by considering the problem as a general optimization problem. The quasi-Newton method is used for searching the likelihood surface. The analytical derivatives of the likelihood function are derived, thereby maximizing the efficiency of the optimization. Because the rate constants are optimized directly, the approach has advantages such as the allowance for model constraints and the ability to simultaneously fit multiple data sets obtained at different experimental conditions. Numerical examples are presented to illustrate the performance of the algorithm. Comparisons with Baum's reestimation suggest that the approach has a superior convergence speed when the likelihood surface is poorly defined due to, for example, a low signal-to-noise ratio or the aggregation of multiple states having identical conductances.     

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.     

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.     

Characterization of single channel currents using digital signal processing techniques based on Hidden Markov Models.

Techniques for extracting small, single channel ion currents from background noise are described and tested. It is assumed that single channel currents are generated by a first-order, finite-state, discrete-time, Markov process to which is added 'white' background noise from the recording apparatus (electrode, amplifiers, etc). Given the observations and the statistics of the background noise, the techniques described here yield a posteriori estimates of the most likely signal statistics, including the Markov model state transition probabilities, duration (open- and closed-time) probabilities, histograms, signal levels, and the most likely state sequence. Using variations of several algorithms previously developed for solving digital estimation problems, we have demonstrated that: (1) artificial, small, first-order, finite-state, Markov model signals embedded in simulated noise can be extracted with a high degree of accuracy, (2) processing can detect signals that do not conform to a first-order Markov model but the method is less accurate when the background noise is not white, and (3) the techniques can be used to extract from the baseline noise single channel currents in neuronal membranes. Some studies have been included to test the validity of assuming a first-order Markov model for biological signals. This method can be used to obtain directly from digitized data, channel characteristics such as amplitude distributions, transition matrices and open- and closed-time durations.     

Relationship between membrane excitability and single channel open-close kinetics.

We have developed a novel technique for simulating the influence of the effects of single channel kinetics on the voltage changes associated with membrane excitability. The technique uses probability distribution functions for the durations of channel open- and closed-state lifetimes, which can be calculated for any model of the ion conductance process. To illustrate the technique, we have used the Hodgkin and Huxley model of nerve membrane ion conductances to simulate channel kinetics during predetermined voltage changes, such as a voltage jump and an action potential. We have also simulated the influence of channels on voltage changes in a free running, non-voltage-clamped patch of membrane 1 micron2 or less in area. The latter results provide a direct illustration of the relationship between fluctuations of membrane excitability and fluctuations in channel open- and closed-state lifetimes.     

Statistical inference from single channel records: two-state Markov model with limited time resolution

Though stochastic models are widely used to describe single ion channel behaviour, statistical inference based on them has received little consideration. This paper describes techniques of statistical inference, in particular likelihood methods, suitable for Markov models incorporating limited time resolution by means of a discrete detection limit. To simplify the analysis, attention is restricted to two-state models, although the methods have more general applicability. Non-uniqueness of the mean open-time and mean closed-time estimators obtained by moment methods based on single exponential approximations to the apparent open-time and apparent closed-time distributions has been reported. The present study clarifies and extends this previous work by proving that, for such approximations, the likelihood equations as well as the moment equations (usually) have multiple solutions. Such non-uniqueness corresponds to non-identifiability of the statistical model for the apparent quantities. By contrast, higher-order approximations yield theoretically identifiable models. Likelihood-based estimation procedures are developed for both single exponential and bi-exponential approximations. The methods and results are illustrated by numerical examples based on literature and simulated data, with consideration given to empirical distributions and model control, likelihood plots, and point estimation and confidence regions.     

The use of quartz patch pipettes for low noise single channel recording.

Quartz has a dissipation factor of approximately 10(-4), which is an order of magnitude less than that of the best glasses previously used to fabricate patch pipettes; it's dielectric constant of 3.8 is also lower than that of other glasses. On the basis of these electrical characteristics it is expected that patch pipettes pulled from quartz tubing will produce significantly less noise than pipettes made from other glasses. Our work confirms these expectations and we describe theoretical and practical aspects of the use of quartz pipettes for single channel patch voltage clamp measurements. Methods for pulling quartz pipettes with a laser-based puller and coating them with low-loss elastomers are discussed, as are precautions that are necessary to achieve low noise recordings. We have shown that quartz pipettes can be pulled from tubing with outer diameter to inner diameter ratios as large as 3 and a method of applying heavy elastomer coatings all the way to the tip of pipettes is presented. Noise sources arising from the pipette and its holder are described theoretically, and it is shown that measured noise is in good agreement with such predictions. With low noise capacitive feedback electronics, small geometry holders, and thick-walled quartz pipettes coated with low-loss elastomers we have been routinely able to achieve noise of 100 fA rms or less in a 5-kHz bandwidth with real cell patches and a pipette immersion depth of approximately 2 mm. On occasion we have achieved noise as low as 60 fA rms in this bandwidth.     

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.     

The use of automated parameter searches to improve ion channel kinetics for neural modeling

The voltage and time dependence of ion channels can be regulated, notably by phosphorylation, interaction with phospholipids, and binding to auxiliary subunits. Many parameter variation studies have set conductance densities free while leaving kinetic channel properties fixed as the experimental constraints on the latter are usually better than on the former. Because individual cells can tightly regulate their ion channel properties, we suggest that kinetic parameters may be profitably set free during model optimization in order to both improve matches to data and refine kinetic parameters. To this end, we analyzed the parameter optimization of reduced models of three electrophysiologically characterized and morphologically reconstructed globus pallidus neurons. We performed two automated searches with different types of free parameters. First, conductance density parameters were set free. Even the best resulting models exhibited unavoidable problems which were due to limitations in our channel kinetics. We next set channel kinetics free for the optimized density matches and obtained significantly improved model performance. Some kinetic parameters consistently shifted to similar new values in multiple runs across three models, suggesting the possibility for tailored improvements to channel models. These results suggest that optimized channel kinetics can improve model matches to experimental voltage traces, particularly for channels characterized under different experimental conditions than recorded data to be matched by a model. The resulting shifts in channel kinetics from the original template provide valuable guidance for future experimental efforts to determine the detailed kinetics of channel isoforms and possible modulated states in particular types of neurons.     

TOMOBFLOW: feature-preserving noise filtering for electron tomography

Background  

Noise filtering techniques are needed in electron tomography to allow proper interpretation of datasets. The standard linear filtering techniques are characterized by a tradeoff between the amount of reduced noise and the blurring of the features of interest. On the other hand, sophisticated anisotropic nonlinear filtering techniques allow noise reduction with good preservation of structures. However, these techniques are computationally intensive and are difficult to be tuned to the problem at hand.     

A Sufficient Condition for Reducing Recursions in Hidden Markov Models

In hidden Markov models, the probability of observing a set of strings can be computed using recursion relations. We construct a sufficient condition for simplifying the recursion relations for a certain class of hidden Markov models. If the condition is satisfied, then one can construct a reduced recursion where the dependence on Markov states completely disappears. We discuss a specific example—namely, statistical multiple alignment based on the TKF-model—in which the sufficient condition is satisfied.     

MCMC for hidden Markov models incorporating aggregation of states and filtering

This paper is concerned with the statistical analysis of single ion channel records. Single channels are modelled by using hidden Markov models and a combination of Bayesian statistics and Markov chain Monte Carlo methods. The techniques presented here provide a straightforward generalization to those in Rosales et al. (2001, Biophys. J., 80, 1088–1103), allowing to consider constraints imposed by a gating mechanism such as the aggregation of states into classes. This paper also presents an extension that allows to consider correlated background noise and filtered data, extending the scope of the analysis toward real experimental conditions. The methods described here are based on a solid probabilistic basis and are less computationally intensive than alternative Bayesian treatments or frequentist approaches that consider correlated data.     

Algorithms for variable length Markov chain modeling

We present a general purpose implementation of variable length Markov models. Contrary to fixed order Markov models, these models are not restricted to a predefined uniform depth. Rather, by examining the training data, a model is constructed that fits higher order Markov dependencies where such contexts exist, while using lower order Markov dependencies elsewhere. As both theoretical and experimental results show, these models are capable of capturing rich signals from a modest amount of training data, without the use of hidden states. AVAILABILITY: The source code is freely available at http://www.soe.ucsc.edu/~jill/src/     

affylmGUI: a graphical user interface for linear modeling of single channel microarray data

SUMMARY: affylmGUI is a graphical user interface (GUI) to an integrated workflow for Affymetrix microarray data. The user is able to proceed from raw data (CEL files) to QC and pre-processing, and eventually to analysis of differential expression using linear models with empirical Bayes smoothing. Output of the analysis (tables and figures) can be exported to an HTML report. The GUI provides user-friendly access to state-of-the-art methods embodied in the Bioconductor software repository. AVAILABILITY: affylmGUI is an R package freely available from http://www.bioconductor.org. It requires R version 1.9.0 or later and tcl/tk 8.3 or later and has been successfully tested on Windows 2000, Windows XP, Linux (RedHat and Fedora distributions) and Mac OS/X with X11. Further documentation is available at http://bioinf.wehi.edu.au/affylmGUI CONTACT: keith@wehi.edu.au.     

Dicarboxylic acid analogs of Gramicidin A: Dimerization kinetics and single channel properties

Summary According to the model of Urry, the cation-permeable gramicidin channel is a dimeric helix formed by association of two peptide monomers linked at their amino ends. In this paper the channel properties of gramicidin analogs are described which have been obtained by chemical modification at the coupling site of the two half-channels. In these analogs the amino terminal-CHO group is replaced by-CO(CH₂) _n COOH(n=2, 3, 4, 5, 6). All analogs form conducting channels in black lipid membranes with the same general properties as found for gramicidin A. The observation that the channel-forming activity decreases with increasing pH is consistent with the notion that the half-channels are linked at the amino terminus. The channel lifetime of the different analogs varies between 2 msec and 50 sec, the longest lifetime being found for the compound withn=3. The single-channel conductance is always smaller than that of gramicidin A, but the reduction of depends on the nature of the permeable ion. Ion specificity was studied at 1m electrolyte by measuring the conductance for different permeable ions (Na⁺, K⁺, Cs⁺). The conductance ratio(Cs⁺)/(Na⁺) was found to vary between 2 and 10.5 for the different analogs.