Statistical analysis of channel current from a membrane patch. II. A stochastic theory of a multi-channel system in the steady-state

A general stochastic theory is presented for analysis of current records of a patch containing an arbitrary number (N) of independent homologous channels in the steady-state. We give the "basic theorem" that at the instant of any open (or shut) transition of a channel, the other N-1 channels are located in each state with a probability equal to those in the steady-state, if enough transitions are observed. Using the "basic theorem", we derived: (a) the time-dependent open and shut frequencies after a definite type of transition, and (b) the probability density functions (pdf) of the duration of any period between two successive transitions. Briefly, the main results obtained were: (1) The time-dependent open (or shut) transition frequency after every shut (or open) transition at t = 0 in an N-channel patch, fJSh,Op(t)(N) (or fJOp,Sh(t)(N)), is the same as that of a one-channel patch except for the value of the constant. (2) In the all-shut (or all-open) period of a patch, the average duration of the period is 1/N, and the slowest exponential decay constant contained in the pdf is N times those of a single channel patch, respectively. (3) An example calculation for small N showed that the stochastic properties of a single channel can be obtained even when N is uncertain, if the channel open probability is small and exponential decay constants are separated. (4) When the channels are in equilibrium, the pdf of duration of every type of period in the patch is described by a sum of exponential terms with positive coefficients. This also holds for fJSh,Op(t)(N) and fJOp,Sh(t)(N).     

Kinetic time constants independent of previous single-channel activity suggest Markov gating for a large conductance Ca-activated K channel

Models for the gating of ion channels usually assume that the rate constants for leaving any given kinetic state are independent of previous channel activity. Although such discrete Markov models have been successful in describing channel gating, there is little direct evidence for the Markov assumption of time-invariant rate constants for constant conditions. This paper tests the Markov assumption by determining whether the single-channel kinetics of the large conductance Ca-activated K channel in cultured rat skeletal muscle are independent of previous single-channel activity. The experimental approach is to examine dwell-time distributions conditional on adjacent interval durations. The time constants of the exponential components describing the distributions are found to be independent of adjacent interval duration, and hence, previous channel activity. In contrast, the areas of the different components can change. Since the observed time constants are a function of the underlying rate constants for transitions among the kinetic states, the observation of time constants independent of previous channel activity suggests that the rate constants are also independent of previous channel activity. Thus, the channel kinetics are consistent with Markov gating. An observed dependent (inverse) relationship between durations of adjacent open and shut intervals together with Markov gating indicates that there are two or more independent transition pathways connecting open and shut states. Finally, no evidence is found to suggest that gating is not at thermodynamic equilibrium: the inverse relationship was independent of the time direction of analysis.     

Adaptive processing techniques based on hidden Markov models for characterizing very small channel currents buried in noise and deterministic interferences.

Techniques for characterizing very small single-channel currents buried in background noise are described and tested on simulated data to give confidence when applied to real data. Single channel currents are represented as a discrete-time, finite-state, homogeneous, Markov process, and the noise that obscures the signal is assumed to be white and Gaussian. The various signal model parameters, such as the Markov state levels and transition probabilities, are unknown. In addition to white Gaussian noise, the signal can be corrupted by deterministic interferences of known form but unknown parameters, such as the sinusoidal disturbance stemming from AC interference and a drift of the base line owing to a slow development of liquid-junction potentials. To characterize the signal buried in such stochastic and deterministic interferences, the problem is first formulated in the framework of a Hidden Markov Model and then the Expectation Maximization algorithm is applied to obtain the maximum likelihood estimates of the model parameters (state levels, transition probabilities), signals, and the parameters of the deterministic disturbances. Using fictitious channel currents embedded in the idealized noise, we first show that the signal processing technique is capable of characterizing the signal characteristics quite accurately even when the amplitude of currents is as small as 5-10 fA. The statistics of the signal estimated from the processing technique include the amplitude, mean open and closed duration, open-time and closed-time histograms, probability of dwell-time and the transition probability matrix. With a periodic interference composed, for example, of 50 Hz and 100 Hz components, or a linear drift of the baseline added to the segment containing channel currents and white noise, the parameters of the deterministic interference, such as the amplitude and phase of the sinusoidal wave, or the rate of linear drift, as well as all the relevant statistics of the signal, are accurately estimated with the algorithm we propose. Also, if the frequencies of the periodic interference are unknown, they can be accurately estimated. Finally, we provide a technique by which channel currents originating from the sum of two or more independent single channels are decomposed so that each process can be separately characterized. This process is also formulated as a Hidden Markov Model problem and solved by applying the Expectation Maximization algorithm. The scheme relies on the fact that the transition matrix of the summed Markov process can be construed as a tensor product of the transition matrices of individual processes.     

Statistical properties of ion channel records. Part II: estimation from the macroscopic current

Macroscopic ion channel current is the summation of the stochastic records of individual channel currents and therefore relates to their statistical properties. As a consequence of this relationship, it may be possible to derive certain statistical properties of single channel records or even generate some estimates of the records themselves from the macroscopic current when the direct measurement of single channel currents is not applicable. We present a procedure for generating the single channel records of an ion channel from its macroscopic current when the stochastic process of channel gating has the following two properties: (I) the open duration is independent of the time of opening event and has a single exponential probability density function (pdf), (II) all the channels have the same probability to open at time t. The application of this procedure is considered for cases where direct measurement of single channel records is difficult or impossible. First, the probability density function (pdf) of opening events, a statistical property of single channel records, is derived from the normalized macroscopic current and mean channel open duration. Second, it is shown that under the conditions (I) and (II), a non-stationary Markov model can represent the stochastic process of channel gating. Third, the non-stationary Markov model is calibrated using the results of the first step. The non-stationary formulation increases the model ability to generate a variety of different single channel records compared to common stationary Markov models. The model is then used to generate single channel records and to obtain other statistical properties of the records. Experimental single channel records of inactivating BK potassium channels are used to evaluate how accurately this procedure reconstructs measured single channel sweeps.     

Statistical properties of ion channel records. Part I: relationship to the macroscopic current

Macroscopic ion channel current can be derived by summation of the stochastic records of individual channel currents. In this paper, we present two probability density functions of single channel records that can uniquely determine the macroscopic current regardless of other statistical properties of records or the stochastic model of channel gating (presented often with stationary Markov models). We show that H(t), probability density function of channel opening events (introduced explicitly in this paper), and D(t), probability density function of the open duration (sometimes has named dwell time distribution as well), determine the normalized macroscopic current, G(t), through G(t) = P(t) - H(t) * Q(t) where P(t) is the cumulative density function of H(t), Q(t) is the cumulative density function of D(t), * is the symbol of convolution integral and G(t) is the macroscopic current divided by the amplitude of single channel current and the number of single channel sweeps. Compared to other equations for the macroscopic current, here the macroscopic current is expressed only in terms of the statistical properties of single channel current and not the stochastic model of ion channel gating or a conditioned form of macroscopic current. Single channel currents of an inactivating BK channel were used to validate this relationship experimentally too. In this paper, we used median filters as they can remove the unwanted noise without smoothing the transitions between open and closed states (compare to low pass filters). This filtering leads to more accurate measurement of transition times and less amount of missed events.     

Kinetic properties of a voltage-dependent junctional conductance

We have proposed that the gap junctions between amphibian blastomeres are comprised of voltage-sensitive channels. The kinetic properties of the junctional conductance are here studied under voltage clamp. When the transjunctional voltage is stepped to a new voltage of the same polarity, the junctional conductance changes as a single exponential to a steady-state level. The time constant of the conductance change is determined by the existing transjunctional voltage and is independent of the previous voltage. For each voltage polarity, the relations between voltage, time constant, and steady-state conductance are well modeled by a reversible two-state reaction scheme in which the calculated rate constants for the transitions between the states are exponential functions of voltage. The calculated rate constant for the transition to the low-conductance state is approximately twice as voltage dependent as that for the transition to the high-conductance state. When the transjunctional voltage polarity is reversed, the junctional conductance undergoes a transient recovery. The polarity reversal data are well modeled by a reaction scheme in which the junctional channel has two gates, each with opposite voltage sensitivity, and in which an open gate may close only if the gate in series with it is open. A simple explanation for this contingent gating is a mechanism in which each gate senses only the local voltage drop within the channel.     

Description of interacting channel gating using a stochastic Markovian model

Single-channel recordings from membrane patches frequently exhibit multiple conductance levels. In some preparations, the steady-state probabilities of observing these levels do not follow a binomial distribution. This behavior has been reported in sodium channels, potassium channels, acetylcholine receptor channels and gap junction channels. A non-binomial distribution suggests interaction of the channels or the presence of channels with different open probabilities. However, the current trace sometimes exhibits single transitions spanning several levels. Since the probability of simultaneous transitions of independent channels is infinitesimally small, such observations strongly suggest a cooperative gating behavior. We present a Markov model to describe the cooperative gating of channels using only the all-points current amplitude histograms for the probability of observing the various conductance levels. We investigate the steady-state (or equilibrium) properties of a system ofN channels and provide a scheme to express all the probabilities in terms of just two parameters. The main feature of our model is that lateral interaction of channels gives rise to cooperative gating. Another useful feature is the introduction of the language of graph theory which can potentially provide a different avenue to study ion channel kinetics. We write down explicit expressions for systems of two, three and four channels and provide a procedure to describe the system ofN channels.     

Mechanism of allosteric modulation of rod cyclic nucleotide-gated channels

The cyclic nucleotide-gated (CNG) channel of retinal rod photoreceptor cells is an allosteric protein whose activation is coupled to a conformational change in the ligand-binding site. The bovine rod CNG channel can be activated by a number of different agonists, including cGMP, cIMP, and cAMP. These agonists span three orders of magnitude in their equilibrium constants for the allosteric transition. We recorded single-channel currents at saturating cyclic nucleotide concentrations from the bovine rod CNG channel expressed in Xenopus oocytes as homomultimers of alpha subunits. The median open probability was 0.93 for cGMP, 0.47 for cIMP, and 0.01 for cAMP. The channels opened to a single conductance level of 26-30 pS at +80 mV. Using signal processing methods based on hidden Markov models, we determined that two closed and one open states are required to explain the gating at saturating ligand concentrations. We determined the maximum likelihood rate constants for two gating schemes containing two closed (denoted C) and one open (denoted O) states. For the C left and right arrow C left and right arrow O scheme, all rate constants were dependent on cyclic nucleotide. For the C left and right arrow O left and right arrow C scheme, the rate constants for only one of the transitions were cyclic nucleotide dependent. The opening rate constant was fastest for cGMP, intermediate for cIMP, and slowest for cAMP, while the closing rate constant was fastest for cAMP, intermediate for cIMP, and slowest for cGMP. We propose that interactions between the purine ring of the cyclic nucleotide and the binding domain are partially formed at the time of the transition state for the allosteric transition and serve to reduce the transition state energy and stabilize the activated conformation of the channel. When 1 microM Ni2+ was applied in addition to cyclic nucleotide, the open time increased markedly, and the closed time decreased slightly. The interactions between H420 and Ni2+ occur primarily after the transition state for the allosteric transition.     

Superposition properties of interacting ion channels.

Quantitative analysis of patch clamp data is widely based on stochastic models of single-channel kinetics. Membrane patches often contain more than one active channel of a given type, and it is usually assumed that these behave independently in order to interpret the record and infer individual channel properties. However, recent studies suggest there are significant channel interactions in some systems. We examine a model of dependence in a system of two identical channels, each modeled by a continuous-time Markov chain in which specified transition rates are dependent on the conductance state of the other channel, changing instantaneously when the other channel opens or closes. Each channel then has, e.g., a closed time density that is conditional on the other channel being open or closed, these being identical under independence. We relate the two densities by a convolution function that embodies information about, and serves to quantify, dependence in the closed class. Distributions of observable (superposition) sojourn times are given in terms of these conditional densities. The behavior of two channel systems based on two- and three-state Markov models is examined by simulation. Optimized fitting of simulated data using reasonable parameters values and sample size indicates that both positive and negative cooperativity can be distinguished from independence.     

Single ion channel models incorporating aggregation and time interval omission.

We present a general theoretical framework, incorporating both aggregation of states into classes and time interval omission, for stochastic modeling of the dynamic aspects of single channel behavior. Our semi-Markov models subsume the standard continuous-time Markov models, diffusion models and fractal models. In particular our models allow for quite general distributions of state sojourn times and arbitrary correlations between successive sojourn times. Another key feature is the invariance of our framework with respect to time interval omission: that is, properties of the aggregated process incorporating time interval omission can be derived directly from corresponding properties of the process without it. Even in the special case when the underlying process is Markov, this leads to considerable clarification of the effects of time interval omission. Among the properties considered are equilibrium behavior, sojourn time distributions and their moments, and auto-correlation and cross-correlation functions. The theory is motivated by ion channel mechanisms drawn from the literature, and illustrated by numerical examples based on these.     

Two-dimensional probability density analysis of single channel currents from reconstituted acetylcholine receptors and sodium channels

Two-dimensional probability density analysis of single channel current recordings was applied to two purified channel proteins reconstituted in planar lipid bilayers: Torpedo acetylcholine receptors and voltage-sensitive sodium channels from rat brain. The information contained in the dynamic history of the gating process, i.e., the time sequence of opening and closing events was extracted from two-dimensional distributions of transitions between identifiable states. This approach allows one to identify kinetic models consistent with the observables. Gating of acetylcholine receptors expresses "memory" of the transition history: the receptor has two channel open (O) states; the residence time in each of them strongly depends on both the preceding open time and the intervening closed interval. Correspondingly, the residence time in the closed (C) states depends on both the preceding open time and the preceding closed time. This result confirms the scheme that considers, at least, two transition pathways between the open and closed states and extends the details of the model in that it defines that the short-lived open state is primarily entered from long-lived closed states while the long-lived open state is accessed mainly through short-lived closed states. Since ligand binding to the acetylcholine-binding sites is a reaction with channel closed states, we infer that the longest closed state (approximately 19 ms) is unliganded, the intermediate closed state (approximately 2 ms) is singly liganded and makes transitions to the short open state (approximately 0.5 ms) and the shortest closed state (approximately 0.4 ms) is doubly liganded and isomerizes to long open states (approximately 5 ms). This is the simplest interpretation consistent with available data. In contrast, sodium channels modified with batrachotoxin to eliminate inactivation show no correlation in the sequence of channel opening and closing events, i.e., have no memory of the transition history. This result is, therefore, consistent with any kinetic scheme that considers a single transition pathway between open and closed states, and confirms the C-C-O model previously inferred from one-dimensional distribution analysis. The strategy described should be of general validity in the analysis of single channel events from channel proteins in both natural and reconstituted membranes.     

Modeling nonhomogeneous Markov processes via time transformation

Summary .   Longitudinal studies are a powerful tool for characterizing the course of chronic disease. These studies are usually carried out with subjects observed at periodic visits giving rise to panel data. Under this observation scheme the exact times of disease state transitions and sequence of disease states visited are unknown and Markov process models are often used to describe disease progression. Most applications of Markov process models rely on the assumption of time homogeneity, that is, that the transition rates are constant over time. This assumption is not satisfied when transition rates depend on time from the process origin. However, limited statistical tools are available for dealing with nonhomogeneity. We propose models in which the time scale of a nonhomogeneous Markov process is transformed to an operational time scale on which the process is homogeneous. We develop a method for jointly estimating the time transformation and the transition intensity matrix for the time transformed homogeneous process. We assess maximum likelihood estimation using the Fisher scoring algorithm via simulation studies and compare performance of our method to homogeneous and piecewise homogeneous models. We apply our methodology to a study of delirium progression in a cohort of stem cell transplantation recipients and show that our method identifies temporal trends in delirium incidence and recovery.     

A Modeling Study of T-Type Ca Channel Gating and Modulation by L-Cysteine in Rat Nociceptors

L-cysteine (L-cys) increases the amplitude of T-type Ca²⁺ currents in rat T-rich nociceptor-like dorsal root ganglia neurons. The modulation of T-type Ca²⁺ channel gating by L-cys was studied by fitting Markov state models to whole-cell currents recorded from T-rich neurons. The best fitting model tested included three resting states and inactivation from the second resting state and the open state. Inactivation and the final opening step were voltage-independent, whereas transitions between the resting states and deactivation were voltage-dependent. The transition rates between the first two resting states were an order of magnitude faster than those between the second and third resting states, and the voltage-dependency of forward transitions through resting states was two to three times greater than for analogous backward transitions. Analysis with the best fitting model suggested that L-cys increases current amplitude mainly by increasing the transition rate from resting to open and decreasing the transition rate from open to inactivated. An additional model was developed that could account for the bi-exponential time course of recovery from inactivation of the currents and the high frequency of blank sweeps in single channel recordings. This model detected basically the same effects of L-cys on channel gating as the best fitting model.     

Evidence for cooperativity between nicotinic acetylcholine receptors in patch clamp records

It is often assumed that ion channels in cell membrane patches gate independently. However, in the present study nicotinic receptor patch clamp data obtained in cell-attached mode from embryonic chick myotubes suggest that the distribution of steady-state probabilities for conductance multiples arising from concurrent channel openings may not be binomial. In patches where up to four active channels were observed, the probabilities of two or more concurrent openings were greater than expected, suggesting positive cooperativity. For the case of two active channels, we extended the analysis by assuming that 1) individual receptors (not necessarily identical) could be modeled by a five-state (three closed and two open) continuous-time Markov process with equal agonist binding affinity at two recognition sites, and 2) cooperativity between channels could occur through instantaneous changes in specific transition rates in one channel following a change in conductance state of the neighboring channel. This allowed calculation of open and closed sojourn time density functions for either channel conditional on the neighboring channel being open or closed. Simulation studies of two channel systems, with channels being either independent or cooperative, nonidentical or identical, supported the discriminatory power of the optimization algorithm. The experimental results suggested that individual acetylcholine receptors were kinetically identical and that the open state of one channel increased the probability of opening of its neighbor.     

The kinetic and physical basis of K(ATP) channel gating: toward a unified molecular understanding

K(ATP) channels can be formed from Kir6.2 subunits with or without SUR1. The open-state stability of K(ATP) channels can be increased or reduced by mutations throughout the Kir6.2 subunit, and is increased by application of PIP(2) to the cytoplasmic membrane. Increase of open-state stability is manifested as an increase in the channel open probability in the absence of ATP (Po(zero)) and a correlated decrease in sensitivity to inhibition by ATP. Single channel lifetime analyses were performed on wild-type and I154C mutant channels expressed with, and without, SUR1. Channel kinetics include a single, invariant, open duration; an invariant, brief, closed duration; and longer closed events consisting of a "mixture of exponentials," which are prolonged in ATP and shortened after PIP(2) treatment. The steady-state and kinetic data cannot be accounted for by assuming that ATP binds to the channel and causes a gate to close. Rather, we show that they can be explained by models that assume the following regarding the gating behavior: 1) the channel undergoes ATP-insensitive transitions from the open state to a short closed state (C(f)) and to a longer-lived closed state (C(0)); 2) the C(0) state is destabilized in the presence of SUR1; and 3) ATP can access this C(0) state, stabilizing it and thereby inhibiting macroscopic currents. The effect of PIP(2) and mutations that stabilize the open state is then to shift the equilibrium of the "critical transition" from the open state to the ATP-accessible C(0) state toward the O state, reducing accessibility of the C(0) state, and hence reducing ATP sensitivity.     

Gating of Shaker K+ channels: II. The components of gating currents and a model of channel activation.

Steady-state and kinetic properties of gating currents of the Shaker K+ channels were studied in channels expressed in Xenopus oocytes and recorded with the cut-open oocyte voltage clamp. The charge versus potential (Q-V) curve reveals at least two components of charge, the first moving in the hyperpolarized region (V1/2 = -63 mV) and the second, with a larger apparent valence, moving in the more depolarized region (V1/2 = -44 mV). The kinetic analysis of gating currents revealed also two exponential decaying components that corresponded in their voltage dependence with the charge components described in the steady-state. The first component was found to correlate with the effects of prepulses that produce the Cole-Moore shift of the ionic and gating currents and seems to be occurring completely within closed conformations of the channel. The second component seems to be related to the events occurring between the closed states just preceding, but not including, the transition to the open state. The ON and OFF gating currents exhibit a pronounced rising phase at potentials at which the second component becomes important, and this region corresponds to the potential range where the channel opens. The results could not be explained with simple parallel models, but the data can be fitted to a sequential model that could be related to a first rearrangement of the putative four subunits in cooperative fashion, followed by a concerted charge movement that leads to the open channel. The first series of charge movements are produced by transitions between several closed states carrying less than two electronic charges per step, while a step carrying about 3.5 electronic charges can explain the second component. This step is followed by the transition to the open state carrying less than 0.5 electronic charges. This model is able to reproduce all the kinetic and steady-state properties of the gating currents and predicts many of the properties of the ionic currents.     

Monotonic and non-monotonic single channel open time distributions with two sequential open states

Some conditions under which kinetic schemes including two sequential open states of identical conductance will display a non-monotonic (i.e. with a deficit of short open times and a maximum at t>0) distribution of single channel open times are described theoretically. Neither a closed cyclic scheme nor exclusively irreversible transitions between states are required for non-monotonic distributions. A required condition for the schemes considered here is that all openings are to a state from which closing is not possible. It is the presence of a precursor process to channel closing that produces the non-monotonic distribution. Following each channel opening some time is required for a transition into the second open state from which all closings proceed. Simple schemes of this sort cannot provide the basis of any experimental reports of non-monotonic distributions.     

Biophysical properties of gap junction channels formed by mouse connexin40 in induced pairs of transfected human HeLa cells.

A clone of human HeLa cells stably transfected with mouse connexin40 DNA was used to examine gap junctions. Two separate cells were brought into physical contact with each other ("induced cell pair") to allow insertion of gap junction channels and, hence, formation of a gap junction. The intercellular current flow was measured with a dual voltage-clamp method. This approach enabled us to study the electrical properties of gap junction channels (cell pairs with a single channel) and gap junctions (cell pairs with many channels). We found that single channels exhibited multiple conductances, a main state (gamma j(main state)), several substates (gamma j(substates)), a residual state (gamma j (residual state)), and a closed state (gamma j(closed state)). The gamma j(main state) was 198 pS, and gamma j(residual state) was 36 pS (temperature, 36-37 degrees C; pipette solution, potassium aspartate). Both properties were insensitive to transjunctional voltage, Vj. The transitions between the closed state and an open state (i.e., residual state, substate, or main state) were slow (15-45 ms); those between the residual state and a substate or the main state were fast (1-2 ms). Under steady-state conditions, the open channel probability, Po, decreased in a sigmoidal manner from 1 to 0 (Boltzmann fit: Vj,o = -44 mV; z = 6). The temperature coefficient, Q10, for gamma j(main state) and gamma j(residual state) was 1.2 and 1.3, respectively (p < 0.001; range 15-40 degrees C). This difference suggests interactions between ions and channel structure in case of gamma j(residual state). In cell pairs with many channels, the gap junction conductance at steady state, gj, exhibited a bell-shaped dependency from Vj (Boltzmann fit, negative Vj, Vj,o = -45 mV, gj(min) = 0.24; positive Vj, Vj,o = 49 mV, gj(min) = 0.26; z = 6). We conclude that each channel is controlled by two types of gates, a fast one responsible for Vj gating and involving transitions between open states (i.e., residual state, substates, main state), and a slow one involving transitions between the closed state and an open state.