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.     

Optimal estimation of ion-channel kinetics from macroscopic currents

Markov modeling provides an effective approach for modeling ion channel kinetics. There are several search algorithms for global fitting of macroscopic or single-channel currents across different experimental conditions. Here we present a particle swarm optimization(PSO)-based approach which, when used in combination with golden section search (GSS), can fit macroscopic voltage responses with a high degree of accuracy (errors within 1%) and reasonable amount of calculation time (less than 10 hours for 20 free parameters) on a desktop computer. We also describe a method for initial value estimation of the model parameters, which appears to favor identification of global optimum and can further reduce the computational cost. The PSO-GSS algorithm is applicable for kinetic models of arbitrary topology and size and compatible with common stimulation protocols, which provides a convenient approach for establishing kinetic models at the macroscopic level.     

A numerical approach to ion channel modelling using whole-cell voltage-clamp recordings and a genetic algorithm

The activity of trans-membrane proteins such as ion channels is the essence of neuronal transmission. The currently most accurate method for determining ion channel kinetic mechanisms is single-channel recording and analysis. Yet, the limitations and complexities in interpreting single-channel recordings discourage many physiologists from using them. Here we show that a genetic search algorithm in combination with a gradient descent algorithm can be used to fit whole-cell voltage-clamp data to kinetic models with a high degree of accuracy. Previously, ion channel stimulation traces were analyzed one at a time, the results of these analyses being combined to produce a picture of channel kinetics. Here the entire set of traces from all stimulation protocols are analysed simultaneously. The algorithm was initially tested on simulated current traces produced by several Hodgkin-Huxley–like and Markov chain models of voltage-gated potassium and sodium channels. Currents were also produced by simulating levels of noise expected from actual patch recordings. Finally, the algorithm was used for finding the kinetic parameters of several voltage-gated sodium and potassium channels models by matching its results to data recorded from layer 5 pyramidal neurons of the rat cortex in the nucleated outside-out patch configuration. The minimization scheme gives electrophysiologists a tool for reproducing and simulating voltage-gated ion channel kinetics at the cellular level.     

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.     

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.     

Fractal model of ion-channel kinetics

Markov models with discrete states, such as closed in equilibrium with closed in equilibrium with open have been widely used to model the kinetics of ion channels in the cell membrane. In these models the transition probabilities per unit time (the kinetic rate constants) are independent of the time scale on which they are measured. However, in many physical systems, a property, L, depends on the scale, epsilon, at which it is measured such that L(epsilon) alpha epsilon 1-D where D is the fractal dimension. Such systems are said to be 'fractal'. Based on the assumption that the kinetic rates are given by k(t) alpha t1-D we derive a fractal model of ion-channel kinetics. This fractal model has fewer adjustable parameters, is more consistent with the dynamics of protein conformations, and fits the single-channel recordings from the corneal endothelium better than the discrete-state Markov model.     

Simulating complex ion channel kinetics with IonChannelLab

In silico simulation based on Markov chains is a powerful way to describe and predict the activity of many transport proteins including ion channels. However, modeling and simulation using realistic models of voltage- or ligand-gated ion channels exposed to a wide range of experimental conditions require building complex kinetic schemes and solving complicated differential equations. To circumvent these problems, we developed IonChannelLab a software tool that includes a user-friendly Graphical User Interface and a simulation library. This program supports channels with Ohmic or Goldman-Hodgkin-Katz behavior and can simulate the time-course of ionic and gating currents, single channel behavior and steady-state conditions. The program allows the simulation of experiments where voltage, ligand and ionic concentration are varied independently or simultaneously.     

Simulating complex ion channel kinetics with IonChannelLab

In silico simulation based on Markov chains is a powerful way to describe and predict the activity of many transport proteins including ion channels. However, modeling and simulation using realistic models of voltage- or ligand-gated ion channels exposed to a wide range of experimental conditions require building complex kinetic schemes and solving complicated differential equations. To circumvent these problems, we developed IonChannelLab a software tool that includes a user-friendly Graphical User Interface and a simulation library. This program supports channels with Ohmic or Goldman-Hodgkin-Katz behavior and can simulate the time-course of ionic and gating currents, single channel behavior and steady-state conditions. The program allows the simulation of experiments where voltage, ligand and ionic concentration are varied independently or simultaneously.     

Restoration of single-channel currents using the segmental k-means method based on hidden Markov modeling

Patch-clamp recording provides an unprecedented means for study of detailed kinetics of ion channels at the single molecule level. Analysis of the recordings often begins with idealization of noisy recordings into continuous dwell-time sequences. Success of an analysis is contingent on accuracy of the idealization. I present here a statistical procedure based on hidden Markov modeling and k-means segmentation. The approach assumes a Markov scheme involving discrete conformational transitions for the kinetics of the channel and a white background noise for contamination of the observations. The idealization is sought to maximize a posteriori probability of the state sequence corresponding to the samples. The approach constitutes two fundamental steps. First, given a model, the Viterbi algorithm is applied to determine the most likely state sequence. With the resultant idealization, the model parameters are then empirically refined. The transition probabilities are calculated from the state sequences, and the current amplitudes and noise variances are determined from the ensemble means and variances of those samples belonging to the same conductance classes. The two steps are iterated until the likelihood is maximized. In practice, the algorithm converges rapidly, taking only a few iterations. Because the noise is taken into explicit account, it allows for a low signal/noise ratio, and consequently a relatively high bandwidth. The approach is applicable to data containing subconductance levels or multiple channels and permits state-dependent noises. Examples are given to elucidate its performance and practical applicability.     

Identification of structures for ion channel kinetic models

Markov models of ion channel dynamics have evolved as experimental advances have improved our understanding of channel function. Past studies have examined limited sets of various topologies for Markov models of channel dynamics. We present a systematic method for identification of all possible Markov model topologies using experimental data for two types of native voltage-gated ion channel currents: mouse atrial sodium currents and human left ventricular fast transient outward potassium currents. Successful models identified with this approach have certain characteristics in common, suggesting that aspects of the model topology are determined by the experimental data. Incorporating these channel models into cell and tissue simulations to assess model performance within protocols that were not used for training provided validation and further narrowing of the number of acceptable models. The success of this approach suggests a channel model creation pipeline may be feasible where the structure of the model is not specified a priori.     

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.     

Simple, fast and accurate implementation of the diffusion approximation algorithm for stochastic ion channels with multiple states

BACKGROUND: The phenomena that emerge from the interaction of the stochastic opening and closing of ion channels (channel noise) with the non-linear neural dynamics are essential to our understanding of the operation of the nervous system. The effects that channel noise can have on neural dynamics are generally studied using numerical simulations of stochastic models. Algorithms based on discrete Markov Chains (MC) seem to be the most reliable and trustworthy, but even optimized algorithms come with a non-negligible computational cost. Diffusion Approximation (DA) methods use Stochastic Differential Equations (SDE) to approximate the behavior of a number of MCs, considerably speeding up simulation times. However, model comparisons have suggested that DA methods did not lead to the same results as in MC modeling in terms of channel noise statistics and effects on excitability. Recently, it was shown that the difference arose because MCs were modeled with coupled gating particles, while the DA was modeled using uncoupled gating particles. Implementations of DA with coupled particles, in the context of a specific kinetic scheme, yielded similar results to MC. However, it remained unclear how to generalize these implementations to different kinetic schemes, or whether they were faster than MC algorithms. Additionally, a steady state approximation was used for the stochastic terms, which, as we show here, can introduce significant inaccuracies. MAIN CONTRIBUTIONS: We derived the SDE explicitly for any given ion channel kinetic scheme. The resulting generic equations were surprisingly simple and interpretable--allowing an easy, transparent and efficient DA implementation, avoiding unnecessary approximations. The algorithm was tested in a voltage clamp simulation and in two different current clamp simulations, yielding the same results as MC modeling. Also, the simulation efficiency of this DA method demonstrated considerable superiority over MC methods, except when short time steps or low channel numbers were used.     

ESTIMATION OF THE PHYSIOLOGICAL PARAMETERS OF A MEMBRANE CHANNEL EXPOSED TO ELECTROMAGNETIC FIELDS USING THE GPOF APPROACH

A method is proposed for the estimation of kinetic parameters of ionic channels in the cell membrane. The method is based on the generalized pencil-of-function approach, which exploits transient current signals from single channels to derive the frequency of the system poles. The proposed approach is validated for the well-known potassium channel by comparing the estimated values with the theoretical values given by Hodgkin and Huxley. The approach is superior to previous spectral approaches, both for its accuracy and for its robustness. It is especially useful for parameter estimation when the channel is exposed to electromagnetic fields. Results are given for exposure to 200-Hz and 915-MHz signals, to demonstrate the effect of fields on the kinetic parameters of the channel.     

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.     

Derivation of cable parameters for a reduced model that retains asymmetric voltage attenuation of reconstructed spinal motor neuron dendrites

Spinal motor neurons have voltage gated ion channels localized in their dendrites that generate plateau potentials. The physical separation of ion channels for spiking from plateau generating channels can result in nonlinear bistable firing patterns. The physical separation and geometry of the dendrites results in asymmetric coupling between dendrites and soma that has not been addressed in reduced models of nonlinear phenomena in motor neurons. We measured voltage attenuation properties of six anatomically reconstructed and type-identified cat spinal motor neurons to characterize asymmetric coupling between the dendrites and soma. We showed that the voltage attenuation at any distance from the soma was direction-dependent and could be described as a function of the input resistance at the soma. An analytical solution for the lumped cable parameters in a two-compartment model was derived based on this finding. This is the first two-compartment modeling approach that directly derived lumped cable parameters from the geometrical and passive electrical properties of anatomically reconstructed neurons.     

Exploring voltage-dependent ion channels in silico by hysteretic conductance

Kinetic models of voltage-dependent ion channels are normally inferred from time records of macroscopic current relaxation or microscopic single channel data. A complementary explorative approach is outlined. Hysteretic conductance refers to conductance delays in response to voltage changes, delays at either macroscopic or microscopic levels of observation. It enables complementary assessments of model assumptions and gating schemes of voltage-dependent channels, e.g. independent versus cooperative gating, and multiple gating modes. Under the Hodgkin-Huxley condition of independent gating, and under ideal measurement conditions, hysteretic conductance makes it also possible to estimate voltage-dependent rate functions. The argument is mainly theoretical, based on experimental observations, and illustrated by simulations of Markov kinetic models.     

Time-Dependent Molecular Memory in Single Voltage-Gated Sodium Channel

Excitability in neurons is associated with firing of action potentials and requires the opening of voltage-gated sodium channels with membrane depolarization. Sustained membrane depolarization, as seen in pathophysiological conditions like epilepsy, can have profound implications on the biophysical properties of voltage-gated ion channels. Therefore, we sought to characterize the effect of sustained membrane depolarization on single voltage-gated Na⁺ channels. Single-channel activity was recorded in the cell-attached patch-clamp mode from the rNa_v1.2α channels expressed in CHO cells. Classical statistical analysis revealed complex nonlinear changes in channel dwell times and unitary conductance of single Na⁺ channels as a function of conditioning membrane depolarization. Signal processing tools like weighted wavelet Z (WWZ) and discrete Fourier transform analyses attributed a “pseudo-oscillatory” nature to the observed nonlinear variation in the kinetic parameters. Modeling studies using the hidden Markov model (HMM) illustrated significant changes in kinetic states and underlying state transition rate constants upon conditioning depolarization. Our results suggest that sustained membrane depolarization induces novel nonlinear properties in voltage-gated Na⁺ channels. Prolonged membrane depolarization also induced a “molecular memory” phenomenon, characterized by clusters of dwell time events and strong autocorrelation in the dwell time series similar to that reported recently for single enzyme molecules. The persistence of such molecular memory was found to be dependent on the duration of depolarization. Voltage-gated Na⁺ channel with the observed time-dependent nonlinear properties and the molecular memory phenomenon may determine the functional state of the channel and, in turn, the excitability of a neuron.     

Real-time kinetic modeling of voltage-gated ion channels using dynamic clamp

We propose what to our knowledge is a new technique for modeling the kinetics of voltage-gated ion channels in a functional context, in neurons or other excitable cells. The principle is to pharmacologically block the studied channel type, and to functionally replace it with dynamic clamp, on the basis of a computational model. Then, the parameters of the model are modified in real time (manually or automatically), with the objective of matching the dynamical behavior of the cell (e.g., action potential shape and spiking frequency), but also the transient and steady-state properties of the model (e.g., those derived from voltage-clamp recordings). Through this approach, one may find a model and parameter values that explain both the observed cellular dynamics and the biophysical properties of the channel. We extensively tested the method, focusing on Na_v models. Complex Markov models (10-12 states or more) could be accurately integrated in real time at >50 kHz using the transition probability matrix, but not the explicit Euler method. The practicality of the technique was tested with experiments in raphe pacemaker neurons. Through automated real-time fitting, a Hodgkin-Huxley model could be found that reproduced well the action potential shape and the spiking frequency. Adding a virtual axonal compartment with a high density of Na_v channels further improved the action potential shape. The computational procedure was implemented in the free QuB software, running under Microsoft Windows and featuring a friendly graphical user interface.