Discrete-state Markov Models Research Articles

A User-Friendly Computational Tool for Markov Modelling Channel Gating and Transport Cycling

The behavior of ion channels and transporters is often modelled using discrete-state continuous-time Markov models. Such models are helpful for the interpretation of experimental data and can guide the design of experiments by testing specific predictions. Here, we present a computational tool that allows to create Markov models of chosen complexity, and to calculate their predictions on a macroscopic as well on a single-molecule scale. The program (https://github.com/mikpusch/MarkovEditor) calculates steady-state properties (current, state-probabilities, cycle frequencies), deterministic macroscopic and stochastic time courses, gating currents, dwell time histograms, and power spectra of channels and transporters. In addition, a visual simulation mode allows to follow the time-dependent stochastic behavior of a single channel or transporter. After a basic introduction into the concept of Markov models, real-life examples are discussed, including a model of a simple K+ channel, a voltage-gated sodium channel, a three-state ligand-gated channel, and an electrogenic uniporter. The MarkovEditor program can serve students to explore Markov models at a basic level, but is also suited for research scientists to test and develop models on the mechanisms of protein function.

Detection of atrial fibrillation using discrete-state Markov models and Random Forests

In this paper, we present a fully automated technique for robust detection of Atrial Fibrillation (AF) episodes in single-lead electrocardiogram (ECG) signals using discrete-state Markov models and Random Forests. Methods: The ECG signal is first preprocessed using Stationary Wavelet Transforms (SWT) for noise suppression, signal quality assessment and subsequent R-peak detection. Discrete-state Markov probabilities modelling transitions between successive RR intervals along with other statistical quantities derived from the RR-interval series constitute the feature set to perform AF classification using Random Forests. Further enhancement in AF detection is achieved by using a post-processing false positive suppression algorithm based on autocorrelation analysis of the RR-interval series. Datasets: The AF classifier was trained using the Physionet/Computing in Cardiology 2017 AF Challenge dataset and the Atrial Fibrillation Termination Database (AFTDB). The test datasets consist of the MIT-BIH Atrial Fibrillation Database (AFDB) and the MIT-BIH Arrhythmia Database (MITDB). Results: Our algorithms achieved sensitivity, specificity and F-score values of 97.4%, 98.6% and 97.7% respectively on the AFDB dataset and 96.3%, 97.0% and 85.6% respectively on the MITDB dataset. It was also observed that inclusion of the false positive suppression step resulted in a 1.1% increase in specificity and a 4.0% increase in F-score for the MITDB dataset without any decrease in sensitivity. Conclusion: The proposed method of AF detection, combining Markov models and Random Forests, achieves high accuracy across multiple databases and demonstrates comparable or superior performance to several other state-of-the-art algorithms.

Event-based state estimation of discrete-state hidden Markov models

The state estimation problem for hidden Markov models subject to event-based sensor measurement updates is considered in this work, using the change of probability approach. We assume the measurement updates are transmitted through wired or wireless communication networks. For the scenarios with reliable and unreliable communication channels, analytical expressions for the probability distributions of the states conditioned on all the past point- and set-valued measurement information are obtained. Also, we show that the scenario with a lossy channel, but without the event-trigger, can be treated as a special case of the reliable channel results. Based on these results, closed-form expressions for the estimated communication rates under the original probability measure are presented, which are shown to be the ratio between a weighted 1-norm and the 1-norm of the unnormalized conditional probability distributions of the states under the new probability measures constructed. Implementation issues are discussed, and the effectiveness of the results is illustrated by numerical examples and comparative simulations.

Single-channel kinetics of BK (Slo1) channels.

Single-channel kinetics has proven a powerful tool to reveal information about the gating mechanisms that control the opening and closing of ion channels. This introductory review focuses on the gating of large conductance Ca2+- and voltage-activated K+ (BK or Slo1) channels at the single-channel level. It starts with single-channel current records and progresses to presentation and analysis of single-channel data and the development of gating mechanisms in terms of discrete state Markov (DSM) models. The DSM models are formulated in terms of the tetrameric modular structure of BK channels, consisting of a central transmembrane pore-gate domain (PGD) attached to four surrounding transmembrane voltage sensing domains (VSD) and a large intracellular cytosolic domain (CTD), also referred to as the gating ring. The modular structure and data analysis shows that the Ca2+ and voltage dependent gating considered separately can each be approximated by 10-state two-tiered models with five closed states on the upper tier and five open states on the lower tier. The modular structure and joint Ca2+ and voltage dependent gating are consistent with a 50 state two-tiered model with 25 closed states on the upper tier and 25 open states on the lower tier. Adding an additional tier of brief closed (flicker states) to the 10-state or 50-state models improved the description of the gating. For fixed experimental conditions a channel would gate in only a subset of the potential number of states. The detected number of states and the correlations between adjacent interval durations are consistent with the tiered models. The examined models can account for the single-channel kinetics and the bursting behavior of gating. Ca2+ and voltage activate BK channels by predominantly increasing the effective opening rate of the channel with a smaller decrease in the effective closing rate. Ca2+ and depolarization thus activate by mainly destabilizing the closed states.

A numerical technique for the identification of discrete-state continuous-time Markov models

A numerical technique for the identification of discrete-state continuoustime Markov models is presented. The technique is characterized by utilizing both initial approximations derived from observed data and estimates of minimized criterion sensitivity to small variations of identified parameters. A new feature of the proposed approach is improved running time of the applied combinatorial optimization algorithm achieved through replacing, at each iteration, of enumeration of various combinations of parameter values in the neighborhood of their current estimates by enumeration of values of only those parameters to which the minimized criterion is highly sensitive.

Micro and macro views of discrete-state markov models and their application to efficient simulation with phase-type distributions

Micro and macro views of discrete-state markov models and their application to efficient simulation with phase-type distributions

Optimal use of data in parallel tempering simulations for the construction of discrete-state Markov models of biomolecular dynamics

Parallel tempering (PT) molecular dynamics simulations have been extensively investigated as a means of efficient sampling of the configurations of biomolecular systems. Recent work has demonstrated how the short physical trajectories generated in PT simulations of biomolecules can be used to construct the Markov models describing biomolecular dynamics at each simulated temperature. While this approach describes the temperature-dependent kinetics, it does not make optimal use of all available PT data, instead estimating the rates at a given temperature using only data from that temperature. This can be problematic, as some relevant transitions or states may not be sufficiently sampled at the temperature of interest, but might be readily sampled at nearby temperatures. Further, the comparison of temperature-dependent properties can suffer from the false assumption that data collected from different temperatures are uncorrelated. We propose here a strategy in which, by a simple modification of the PT protocol, the harvested trajectories can be reweighted, permitting data from all temperatures to contribute to the estimated kinetic model. The method reduces the statistical uncertainty in the kinetic model relative to the single temperature approach and provides estimates of transition probabilities even for transitions not observed at the temperature of interest. Further, the method allows the kinetics to be estimated at temperatures other than those at which simulations were run. We illustrate this method by applying it to the generation of a Markov model of the conformational dynamics of the solvated terminally blocked alanine peptide.

Probability distributions of molecular observables computed from Markov models. II. Uncertainties in observables and their time-evolution

Discrete-state Markov (or master equation) models provide a useful simplified representation for characterizing the long-time statistical evolution of biomolecules in a manner that allows direct comparison with experiments as well as the elucidation of mechanistic pathways for an inherently stochastic process. A vital part of meaningful comparison with experiment is the characterization of the statistical uncertainty in the predicted experimental measurement, which may take the form of an equilibrium measurement of some spectroscopic signal, the time-evolution of this signal following a perturbation, or the observation of some statistic (such as the correlation function) of the equilibrium dynamics of a single molecule. Without meaningful error bars (which arise from both approximation and statistical error), there is no way to determine whether the deviations between model and experiment are statistically meaningful. Previous work has demonstrated that a Bayesian method that enforces microscopic reversibility can be used to characterize the statistical component of correlated uncertainties in state-to-state transition probabilities (and functions thereof) for a model inferred from molecular simulation data. Here, we extend this approach to include the uncertainty in observables that are functions of molecular conformation (such as surrogate spectroscopic signals) characterizing each state, permitting the full statistical uncertainty in computed spectroscopic experiments to be assessed. We test the approach in a simple model system to demonstrate that the computed uncertainties provide a useful indicator of statistical variation, and then apply it to the computation of the fluorescence autocorrelation function measured for a dye-labeled peptide previously studied by both experiment and simulation.

Linking Exponential Components to Kinetic States in Markov Models for Single-Channel Gating

Discrete state Markov models have proven useful for describing the gating of single ion channels. Such models predict that the dwell-time distributions of open and closed interval durations are described by mixtures of exponential components, with the number of exponential components equal to the number of states in the kinetic gating mechanism. Although the exponential components are readily calculated (Colquhoun and Hawkes, 1982, Phil. Trans. R. Soc. Lond. B. 300:1–59), there is little practical understanding of the relationship between components and states, as every rate constant in the gating mechanism contributes to each exponential component. We now resolve this problem for simple models. As a tutorial we first illustrate how the dwell-time distribution of all closed intervals arises from the sum of constituent distributions, each arising from a specific gating sequence. The contribution of constituent distributions to the exponential components is then determined, giving the relationship between components and states. Finally, the relationship between components and states is quantified by defining and calculating the linkage of components to states. The relationship between components and states is found to be both intuitive and paradoxical, depending on the ratios of the state lifetimes. Nevertheless, both the intuitive and paradoxical observations can be described within a consistent framework. The approach used here allows the exponential components to be interpreted in terms of underlying states for all possible values of the rate constants, something not previously possible.

Detecting and Exploiting Symmetry in Discrete-State Markov Models

Dependable systems are usually designed with multiple instances of components or logical processes, and often possess symmetries that may be exploited in model-based evaluation. The problem of how best to exploit symmetry in models has received much attention from the modeling community, but no solution has garnered widespread support, primarily because each solution is limited in terms of either the types of symmetry that can be exploited, or the difficulty of translating from the system description to the model formalism. We propose a new method for detecting and exploiting model symmetry in which 1) models retain the structure of the system, and 2) all symmetry inherent in the structure of the model can be detected and exploited for the purposes of state-space reduction. Composed models are constructed from models through specification of connections between models that correspond to shared state fragments. The composed model is interpreted as an undirected graph, and results from group theory, and graph theory are used to develop procedures for automatically detecting, and exploiting all symmetries in the composed model. We discuss the necessary algorithms to detect and exploit model symmetry, and provide a proof that the theory generates an equivalent model. After a thorough analysis of the added complexity, a state-space generator which implements these algorithms within Mobius is then presented.

Automatic discovery of metastable states for the construction of Markov models of macromolecular conformational dynamics

To meet the challenge of modeling the conformational dynamics of biological macromolecules over long time scales, much recent effort has been devoted to constructing stochastic kinetic models, often in the form of discrete-state Markov models, from short molecular dynamics simulations. To construct useful models that faithfully represent dynamics at the time scales of interest, it is necessary to decompose configuration space into a set of kinetically metastable states. Previous attempts to define these states have relied upon either prior knowledge of the slow degrees of freedom or on the application of conformational clustering techniques which assume that conformationally distinct clusters are also kinetically distinct. Here, we present a first version of an automatic algorithm for the discovery of kinetically metastable states that is generally applicable to solvated macromolecules. Given molecular dynamics trajectories initiated from a well-defined starting distribution, the algorithm discovers long lived, kinetically metastable states through successive iterations of partitioning and aggregating conformation space into kinetically related regions. The authors apply this method to three peptides in explicit solvent-terminally blocked alanine, the 21-residue helical F(s) peptide, and the engineered 12-residue beta-hairpin trpzip2-to assess its ability to generate physically meaningful states and faithful kinetic models.

Bayesian approaches to multiple sources of evidence and uncertainty in complex cost-effectiveness modelling.

Increasingly complex models are being used to evaluate the cost-effectiveness of medical interventions. We describe the multiple sources of uncertainty that are relevant to such models, and their relation to either probabilistic or deterministic sensitivity analysis. A Bayesian approach appears natural in this context. We explore how sensitivity analysis to patient heterogeneity and parameter uncertainty can be simultaneously investigated, and illustrate the necessary computation when expected costs and benefits can be calculated in closed form, such as in discrete-time discrete-state Markov models. Information about parameters can either be expressed as a prior distribution, or derived as a posterior distribution given a generalized synthesis of available data in which multiple sources of evidence can be differentially weighted according to their assumed quality. The resulting joint posterior distributions on costs and benefits can then provide inferences on incremental cost-effectiveness, best presented as posterior distributions over net-benefit and cost-effectiveness acceptability curves. These ideas are illustrated with a detailed running example concerning the cost-effectiveness of hip prostheses in different age-sex subgroups. All computations are carried out using freely available software for conducting Markov chain Monte Carlo analysis.

Voltage-dependent gating mechanism for single fast chloride channels from rat skeletal muscle.

1. A voltage-dependent gating mechanism for the fast Cl- channel was developed from the analysis of single-channel current records obtained with the patch clamp technique from primary cultures of rat skeletal muscle. Up to 10(6) open and shut intervals were analysed from each of five different excised patches of membrane containing a single fast Cl- channel. 2. Rate constants for a kinetic scheme with six closed and two open states (scheme I) were estimated at a given voltage by maximum likelihood fitting of open and closed dwell-time distributions obtained at that voltage. This procedure was then repeated for data obtained at each of three to eight different membrane potentials for each channel. 3. Plots of the estimated rate constants against membrane potential typically appeared linear on semilogarithmic co-ordinates, consistent with rate constants that are exponentially dependent on voltage. 4. Regression analysis of these plots yielded two parameters for each rate constant: the value of the rate constant at -50 mV (B) and its voltage sensitivity (A). The dwell-time distributions predicted with these parameters and scheme I gave a good description of the experimental dwell-time distributions at all the studied voltages, lending further support for an exponential dependence of rate constants on membrane potential. 5. Estimates of A and B were also obtained by simultaneously fitting dwell-time distributions obtained at three to eight different voltages, in order to better define these parameters. Predicted dwell-time distributions obtained with these estimates and scheme I could approach the theoretical best description of the data for discrete-state Markov models. 6. Eight to twelve of the fourteen rate constants in scheme I appeared voltage sensitive, with effective gating charges ranging from about -1.5 to +1.0 units of electronic charge. 7. The estimated rate constants and their voltage sensitivities for the five analysed channels were generally similar, but showed some heterogeneity. 8. Gating mechanisms which had fewer kinetic states than scheme I, or equal and opposite effective gating charges over each transition barrier, or four or five identical and independent voltage-dependent subunits, all gave poorer descriptions of the data than scheme I. These simpler mechanisms were also ranked below scheme I by the Schwarz criterion, which applies a heavy penalty for additional free parameters. 9. These findings indicate that the voltage dependence of the fast Cl- channel is consistent with a kinetic scheme with six closed and two open states, in which a majority of transitions among the states are voltage dependent.(ABSTRACT TRUNCATED AT 400 WORDS)

USING LOG-LINEAR TESTS OF FIT FOR A TRANSACTIONAL SYSTEMS MODEL

For the past decade the Interact System Model (ISM) of Fisher and Hawes (1971) has been used in conjunction with traditional statistical tests of fit of discrete-time, discrete-state Markov models to examine small group processes of communication. The authors argue that the ISM is an inappropriate model of interpersonal and noninterpersonal communication processes, in part because it takes no account of nonverbal messages. As an alternative, they propose a Transactional Systems Model (TSM) of person-to-person communication, and, using data on police-complainant interaction for illustration, show how log-linear statistical models provide an efficient and flexible means of testing and comparing ISM with TSM models. Log-linear tests may replace traditional tests of such models and open the way to a greater understanding of complex communication processes.