ABSTRACT 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 forwardbackward 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.
INTRODUCTION
Recordings of single ion-channel currents provide a wealth of information about the activity of single allosteric protein molecules. The open-closed behavior of ion channels has generally been described in terms of continuous-time Markov models (Colquhoun and Hawkes, 1995) in which model states are taken to correspond to distinct states of protein conformation or ligand binding. Finding the best Markov model description of a channel's behavior is therefore taken to be equivalent to a complete elucidation of the kinetic behavior of the channel protein with the Markov transition probabilities corresponding directly to rate constants of ligand binding and unbinding and of conformational changes.
Finding the best Markov model involves two steps. First, the general topology of the model must be chosen, specifying the number of states and the connectivity that specify the allowable transitions among states. The second step is the optimization of Markov model parameters. Given a Markov model A with N states, the parameters are the current levels tLi corresponding to each state qi, i = 1, . . . , N, the initial state probability pi, and the transition rates contained in an (N X N) matrix Q. Maximum-likelihood techniques are typically used to optimize these parameters, and the likelihood-ratio test is commonly used to identify the best model topologies.
Several methods have been used to compute likelihoods and estimate model parameters from single-channel data. Most commonly, threshold detection is used to identify channel-open and channel-closed intervals; the distributions of these dwell times are then fitted to the predictions of Markov models by maximum-likelihood techniques (Magleby and Weiss, 1990; Colquhoun and Sigworth, 1995). Alternatively, the likelihoods of models are computed on the basis of the entire sequence of open and closed dwell times (Horn and Lange, 1983; Ball and Sansom, 1989; Qin et al., 1997). An improved approach to the identification of open and closed intervals has been introduced through the use of the Viterbi algorithm (Fredkin and Rice, 1992a). Discussed in the present paper is an approach that does not require the identification of open and closed intervals at all but makes use of the raw single-channel recording in the form of sampled time course of membrane current. This application of signal processing based on hidden Markov models (HMMs) has already been demonstrated to be particularly useful in characterizing channel behavior when the signal-to-noise ratios are low and when multiple subconductance levels exist (Chung et al., 1990; Fredkin and Rice, 1992b; Chung and Gage, 1998). An excellent overview of the HMM approach to single channel analysis is given by Qin et al. (2000a,b). Various implementations of the HMM algorithms have been applied to experimental single channel data (Becker et al., 1994; Milburn et al., 1995; Michalek et al., 1999; Farokhi et al., 2000). The algorithms described here have been used in the recent studies by Sunderman and Zagotta (1999a,b), Wang et al. (2000), and Zheng et al. (2001).
We thank Daniel Brown and Lin Ci Brown (Bruxton Corporation) for integrating our algorithms into the TAC ion-channel analysis software. We also thank Profs. R. Kuc and P. Schultheiss (Yale University) for helpful discussions. This work was supported by NIH grants NS35282 and NS21501.
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[Author Affiliation]
L. Venkataramanan and F. J. Sigwortht
[Author Affiliation]
Schlumberger-Doll Research, Ridgefield, Connecticut 06877 USA; and tDepartment of Cellular and Molecular Physiology, Yale University, New Haven, Connecticut 06520 USA
[Author Affiliation]
Aud Ast 22, 2001, and accepted for publication December 12,
[Author Affiliation]
Address reprint requests to F. J. Sigworth, Department of Cellular and Molecular Physiology, Yale University School of Medicine, 333 Cedar Street, New Haven, CT 06520-8026. Tel.: 203-785-5773; Fax: 203-7854951; E-mail: fred.sigworth@yale.edu.
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