Stochastic models for systems of interacting ion channels.

We consider a variety of Markov based models for systems of ion channels exhibiting dependence between channels. It is shown how many useful properties which may be calculated for an aggregated single-channel model, or a system of independent channels, can be extended to various types of interacting channel systems. Key structure and results from the theory of aggregated Markov processes are summarized in a convenient matrix form. These are then applied to the superposition of independent and dependent channels, including a patch of channels in a random environment, and a system of channels with spatial interactions. Calculations based on the resultant matrix expressions and intensity arguments can be implemented straightforwardly in a matrix-oriented package such as Matlab. The role of reversibility is also studied. A number of examples illustrate the strengths of the methods and enable numerical comparisons between the different types of systems.

Burst properties of a supergated double-barrelled chloride ion channel.

The chloride selective channel from Torpedo electroplax, ClC-0, is the prototype of a large gene family of chloride channels that behave as functional dimers, with channel currents exhibiting two non-zero conductance levels. Each pore has the same conductance and is controlled by a subgate, and these have seemingly identical fast gating kinetics. However, in addition to the two subgates there is a single slower 'supergate' which simultaneously affects both channels. In the present paper, we consider a six state Markov model that is compatible with these observations and develop approximations as well as exact results for relevant properties of groupings of openings, known as bursts. Calculations with kinetic parameter values typical of ClC-0 suggest that even simple approximations can be quite accurate. Small deviations from the assumption of independence within the model lead to marked changes in certain predicted burst properties. This suggests that analysis of these properties may be helpful in assessing independence/non-independence of gating in this type of channel. Based on simulations of models of both independent and non-independent gating, tests using binomial distributions can lead to false conclusions in each situation. This is made more problematic by the difficulty of selecting an appropriate critical time in defining a burst empirically.

Membrane recycling due to low and high rates of nerve stimulation at release sites in the amphibian (Bufo marinus) neuromuscular junction.

The activity-dependent labelling of motor nerve terminals with the dye FM1-43 has been used to estimate the relative levels of membrane recycling (due to synaptic vesicle exocytosis and recovery) at release sites in response to 1,200 nerve stimulations delivered at either low (0.5 Hz) or high (30 Hz) frequency. Dye in terminals appears as fluorescent spots distributed along the terminal branches; each spot is thought to be a cluster of labelled vesicles associated with a release site. Relative fluorescence in spots was quantified from images obtained with a confocal microscope. Spot intensities varied widely within branches following labelling at both frequencies, but the distribution was highly skewed towards lower intensities at low frequency stimulation; at high frequency, more spots had stronger fluorescence. Both weak and strongly stained spots were uniformly distributed along the length of terminal branches after low frequency stimulation; however, there was a gradual decline in all spot intensities towards the distal end of branches loaded with dye at high frequency stimulation. Antibody staining for synaptic vesicles was, on average, uniformly distributed along the branches. The increase in number of more strongly FM1-43-labelled spots in terminal branches stimulated at high compared with low frequency suggests that more release sites are active at high rates of nerve stimulation. This "recruitment" of release sites at high frequency stimulation occurs mostly in the proximal half of terminal branches and is not related to the abundance of synaptic vesicles in the terminal.

Model properties underlying non-identifiability in single channel inference.

Models of ion channel kinetics subserve inferential methods applied to patch clamp data. For Markov models the density function of a sojourn time in a class of states is a mixture of exponentials. Determination of kinetic parameters from density functions may be complicated by non-uniqueness of solutions. This non-identifiability is investigated analytically for a class of two states, assuming detailed balance; relations between model properties, observable density parameters, and non-uniqueness are presented. The results are further developed in terms of similarity transform methods. Additional information provided by joint distributions is discussed. An example is given where identifiability of a model can be demonstrated explicitly. Attention is drawn to instances where the number of components in a density function may be misleading when used to infer the number of underlying states.

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.

Graphs, random sums, and sojourn time distributions, with application to ion-channel modeling.

This paper considers the distribution of a sojourn time in a class of states of a stochastic process having finite discrete state space where sojourn times in any individual state are independent and identically distributed, and transitions between states follow a Markov chain. The state space and possible transitions of the process are represented by a graph. Class sojourn time distributions are derived by modifying this graph using 'composition' of states, defining a new Markov chain on the modified graph, and expressing the sojourn time in a composition state as a random sum. Appropriate compositions are chosen according to the possible "cores" of sojourns in the particular class, where a core describes the structure of a sojourn in terms of a single state or a chain in the original graph. Graph methods provide an algorithmic basis for the derivation, which can be simplified by using symmetry results. Models of ion-channel kinetics are used throughout for illustration; class sojourn time distributions are important in such models because individual states are often indistinguishable experimentally. Markov processes are the special case where sojourn times in individual states are exponentially distributed. In this case kinetic parameter estimation based on the observed class sojourn time distribution is briefly discussed; explicit estimating equations applicable to sequential models of nicotinic receptor kinetics are given.

Superposition properties of independent ion channels.

Membrane patches usually contain several ion channels of a given type. However, most of the stochastic modelling on which data analysis (in particular, estimation of kinetic constants) is currently based, relates to a single channel rather than to multiple channels. Attempts to circumvent this problem experimentally by recording under conditions where channel activity is low are restrictive and can introduce bias; moreover, possibly important information on how multichannel systems behave will be missed. We have extended existing theory to multichannel systems by applying results from point process theory to derive some distributional properties of the various types of sojourn time that occur when a given number of channels are open in a system containing a specified number of independent channels in equilibrium. Separate development of properties of a single channel and the superposition of several such independent channels simplifies the presentation of known results and extensions. To illustrate the general theory, particular attention is given to the types of sojourn time that occur in a two channel system; detailed expressions are presented for a selection of models, both Markov and non-Markov.

Estimation of single channel kinetic parameters from data subject to limited time resolution.

The limited responsiveness of single-channel recording systems results in some brief events not being detected, and if this is ignored parameter estimation from the observed data will be biased. Statistical methods of correcting for this limited time resolution in a two-state Markov model have been proposed by Neher (1983. J. Physiol. (Lond.). 339:663-678) and by Colquhoun and Sigworth (1983. Single Channel Recording. 191-263). However, a numerical study by Blatz and Magleby (1986. Biophys. J. 49:967-980) indicated differences of 3-40% in the corrected values given by the two techniques. Here we explain why Neher's method produces biased results and the Colquhoun and Sigworth approach, which is no more difficult, provides reasonably accurate estimates.

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.

Verapamil, neuromuscular transmission and the nicotinic receptor.

The effect of verapamil on neuromuscular transmission was studied in the frog by analysing ionophoretic endplate current (iEPC) trains and the growth and decay phases of miniature endplate currents (mepcs). In addition, single channel data on the interaction of verapamil with the nicotinic acetylcholine receptor were obtained from cultured embryonic chick skeletal muscle cells. Verapamil caused both open and closed channel blockade in the iEPC trains. Mepc amplitude was decreased at low micromolar concentrations, and at higher concentrations there was also accelerated mepc decay indicating open channel blockade. The latter effect could not be explained by a sequential channel occlusion mechanism. Analysis of the mepc rising phase showed that low micromolar concentrations of the drug decreased the pool of receptors which could be activated. Single channel data confirmed the specific interaction of verapamil with the nicotinic receptor, showing closed channel blockade at low concentrations, and at higher levels the shortening of open channel lifetime. It is suggested that both forms of blockade may involve desensitization processes.

Stochastic modelling of a single ion channel: an alternating renewal approach with application to limited time resolution.

Stochastic models of ion channels have been based largely on Markov theory where individual states and transition rates must be specified, and sojourn-time densities for each state are constrained to be exponential. This study presents an approach based on random-sum methods and alternating-renewal theory, allowing individual states to be grouped into classes provided the successive sojourn times in a given class are independent and identically distributed. Under these conditions Markov models form a special case. The utility of the approach is illustrated by considering the effects of limited time resolution (modelled by using a discrete detection limit, xi) on the properties of observable events, with emphasis on the observed open-time (xi-open-time). The cumulants and Laplace transform for a xi-open-time are derived for a range of Markov and non-Markov models; several useful approximations to the xi-open-time density function are presented. Numerical studies show that the effects of limited time resolution can be extreme, and also highlight the relative importance of the various model parameters. The theory could form a basis for future inferential studies in which parameter estimation takes account of limited time resolution in single channel records. Appendixes include relevant results concerning random sums and a discussion of the role of exponential distributions in Markov models.

Neurotransmission parameters estimated from miniature endplate current growth phase.

A numerical model of miniature endplate current (mepc) generation was fitted to the rising phase of individual mepcs recorded at the frog neuromuscular junction, and estimates of 6 transmission parameters were obtained. Model fitting was enabled by assuming literature values for geometric parameters and determining single channel current by noise analysis, the channel closing rate constant from the mepc decay, and acetylcholine hydrolysis parameters from mepcs recorded in esterase-blocked endplates. Under control conditions, mean estimates were: number of molecules in a quantum = 29,000, diffusion coefficient = 2.8 X 10(-6) cm2s-1, endplate receptor density = 8500 micron-2, forward binding rate constant = 7.6 X 10(8) M-1s-1, equilibrium dissociation constant = 58 microM and channel opening rate constant = 8100 s-1.

Stochastic modelling of a single ion channel: interdependence of burst length and number of openings per burst.

Previous modelling of single channel behaviour based on Markov processes has been concerned mainly with means and marginal distributions of particular quantities. The present study derives the joint distribution, conditional distributions, and associated mean values for the burst length (T) and the number (N) of openings per burst in two simple three-state models in which bursting is possible, one for an agonist-only and one for a channel blocking mechanism. In both models the conditional mean burst length (E(T/N = r)) increases linearly as a function of the number of openings per burst, while the conditional mean number of openings per burst (E(N/T = x)) is a nonlinear strictly increasing function of burst length, which is asymptotically linear for large burst length. The asymptotic intercept for each model is shown to be less than, equal to, or greater than unity according as mean channel closed-time is less than, equal to, or greater than mean open-time. For parameter values typical of the nicotinic receptor, this intercept is less than unity for the agonist-only model and greater than unity for the blocking model. As a result of the dependence between the number of openings per burst and burst length, it is shown that experimental estimates of the unconditional mean number of openings per burst may be biased if bursts of only short duration are collected.

Distribution of exponentiality in miniature endplate current decay.

Exponential functions are commonly used to describe miniature endplate current (MEPC) decay; under control conditions a monoexponential is usually regarded as sufficient, whereas in the presence of some drugs a biexponential may be necessary. Using an automated fitting procedure which estimated exponential parameters and the period of decay from peak to baseline, a unimodal distribution of curvature was found for control MEPCs recorded in frog sartorius muscle. The majority of MEPCs were of monoexponential form, while the remainder were biexponential with either less or greater curvature than expected for a simple exponential (hypoexponential or hyperexponential, respectively). The proportion of MEPCs in each of the 3 groups was constant for a given endplate but varied between endplates. A possible explanation for this phenomenon could be differences in synaptic geometry within and between endplates. The increased curvature of individual MEPCs in the hyperexponential group was analyzed by assuming a sequential model for agonist blockade or desensitization, and calculating closing and reopening rate constants. These rate constants were altered by procaine and verapamil (100 microM) in a manner consistent with blockade of the acetylcholine receptor by enhancement of agonist-induced desensitization.

Verapamil alters the amplitude and time course of miniature endplate current.

The effect of verapamil on neuromuscular transmission was examined by recording miniature endplate currents (mepcs) in voltage-clamped frog sartorius muscle fibres. In the presence of 100 microM verapamil, the amplitude and time constant of decay of the mepcs (tau D) were reduced to 68% and 55% of control respectively, and the normal voltage dependency of tau D was decreased. Part of the decrease in amplitude of the mepc was independent of changes in tau D because, on washout, recovery of tau D was more rapid than that of amplitude, and in some cells smaller concentrations of verapamil (1 and 10 microM) decreased amplitude without affecting tau D. Evidence of open and closed channel blockade by verapamil (5-20 microM) was obtained from ionophoretic end-plate current trains and it is proposed that these effects are mediated via an allosteric mechanism.

Numerical simulation of miniature endplate currents.

A FORTRAN computer program is described which enables solution of a set of partial differential equations describing synaptic transmission, including neurotransmitter diffusion, binding to receptors on the postsynaptic membrane with subsequent formation of open ion-conducting channels, and hydrolysis. Integration is performed with a widely available package (PDETWO), and flexibility is achieved through use of 6 user-specified routines. The reliability of solution was checked by comparison with an analytic solution for a diffusion-only model. Application of the program is illustrated with an analysis of miniature endplate current rise times.