Alan G. Hawkes

On the Stochastic Properties of Single Ion Channels

It is desirable to be able to predict, from a specified mechanism, the appearance of currents that flow through single ion channels (a) to enable interpretation of experiments in which single channel currents are observed, and (b) to allow physical meaning to be attached to the results observed in kinetic (noise and relaxation) experiments in which the aggregate of many single channel currents is observed. With this object, distributions (and their means) are derived for the length of the sojourn in any specified subset of states (e. g. all shut states). In general these are found to depend not only on the state in which the sojourn starts, but also on the state that immediately follows the sojourn. The methods described allow derivation of the distribution of, for example, (a) the number of openings, and total length of the burst of openings, that may occur during a single occupancy, and (b) the apparent gap between such bursts. The methods are illustrated by their application to two simple theories of agonist action. The Castillo-Katz (non-cooperative) mechanism predicts, for example, that the number of openings per occupancy, and the apparent burst length, are independent of agonist concentration whereas a simple cooperative mechanism predicts that both will increase with agonist concentration.

Relaxation and Fluctuations of Membrane Currents that Flow through Drug-Operated Channels

The theoretical background is presented for (a) the relaxation towards equilibrium of drug-induced membrane currents, and (b) the fluctuations of membrane current about its equilibrium value that originate in the opening and closing of membrane ion channels. General expressions are given that relate the relaxation current, autocovariance function, spectral density function, fluctuation variance and mean open channel lifetime to the rate constants and single channel conductances for any theory of drug action based on the law of mass action. The question of how much can be validly inferred from experimental spectra that appear to have only one component is discussed. The equations are illustrated by their application to some simple theories of drug action that are currently under consideration.

The Distributions of the Apparent Open Times and Shut Times in a Single Channel Record when Brief Events Cannot Be Detected

The openings and shuttings of individual ion channel molecules can be described by a Markov process with discrete states in continuous time. The predicted distributions of the durations of open times, shut times, bursts of openings, etc., are all described, in principle, by mixtures of exponential densities. In practice it is usually found that some of the open times, and/or shut times, are too short to be detected reliably. If a fixed dead-time r is assumed then it is possible to define, as an approximation to what is actually observed, an ‘ extended opening ’ or e-opening which starts with an opening of duration at least r followed by any number of openings and shuttings, all the shut times being shorter than r ; the e-opening ends when a shut time longer than r occurs. A similar definition is used for e-shut times. Several authors have derived approximations to the distribution of durations of e-openings and e-shuttings. In this paper the exact distributions are derived. They are defined piecewise over the intervals r to 2r, 2r to 3 r ,..., etc., the distribution in each interval being a sum of products of polynomials in t with exponential terms. The number of terms is finite, but increases as intervals get further from t = r. An asymptotic form for large t (for which the exact solution becomes difficult to compute) is given for the two state case. The exact solution is compared with several approximations, some of which are shown to be good enough for use in most practical applications.

The quality of maximum likelihood estimates of ion channel rate constants

Properties of maximum likelihood estimators of rate constants for channel mechanisms are investigated, to see what can and cannot be inferred from experimental results. The implementation of the HJCFIT method is described; it maximises the likelihood of an entire sequence of apparent open and shut times, with the rate constants in a specified reaction mechanism as free parameters. The exact method for missed brief events is used. Several methods for testing the quality of the fit are described. The distributions of rate constants, and correlations between them, are investigated by doing sets of 1000 fits to simulated experiments. In a standard nicotinic receptor mechanism, all nine free rate constants can be estimated even from one single channel recording, as long as the two binding sites are independent, even when the number of channels in the patch is not known. The estimates of rate constants that apply to diliganded channels are robust; good estimates can be obtained even with erroneous assumptions (e.g. about the value of a fixed rate constant or the independence of sites). Rate constants that require distinction between the two sites are less robust, and require that an EC50 be specified, or that records at two concentrations be fitted simultaneously. Despite the complexity of the problem, it appears that there exist two solutions with very similar likelihoods, as in the simplest case. The hazards that result from this, and from the strong positive correlation between estimates of opening and shutting rates, are discussed.

A Q-Matrix Cookbook

It is clear from the examples in Chapter 18 (this volume) that the algebra involved in kinetic arguments can be quite lengthy, even for simple mechanisms with only three states. For more complex mechanisms it becomes rapidly worse. Furthermore, this complicated algebra would have to be carried out separately for every kinetic mechanism that was of interest. On the other hand, the use of matrix notation allows perfectly general solutions to be written down. Not only are the results general, but they are also compact and simple-looking. They do not result in pages of complicated-looking algebra. For example, once a solution has been obtained for a quantity such as the distribution of the burst length, this result can be applied to any kinetic mechanism that is postulated. There is no need for further algebra (or for further programming) when a new mechanism is considered. With a general computer program, all that is needed is to supply the program with a definition of the states and the values of the transition rates for the mechanism you wish to study.

Joint distributions of apparent open and shut times of single-ion channels and maximum likelihood fitting of mechanisms

The openings and shuttings of individual ion channel molecules can be modelled in terms of an underlying Markov process with discrete states in continuous time. In practice, some of the open times, and/or shut times, are too short to be detected reliably, making the durations of some of these intervals appear to be longer than they really are. Under certain assumptions about how this happens, the probability densities of these apparent times have previously been obtained. It has been shown that the ability to distinguish between alternative postulated reaction mechanisms can be greatly improved by considering bivariate distributions. In this paper we obtain joint distributions, and hence conditional distributions, of adjacent apparent open and shut times. Numerical examples illustrate what insight these conditional distributions may provide about the underlying mechanism. Bivariate distributions are readily generalized to multivariate distributions which enable the likelihood for an entire single-channel recording to be computed, and hence efficient maximum likelihood estimates for the mechanism’s rate constants can be obtained. Numerical examples of such fitting are given.

Stochastic Properties of Ion Channel Openings and Bursts in a Membrane Patch that Contains Two Channels: Evidence Concerning the Number of Channels Present when a Record Containing Only Single Openings is Observed

If a single ion channel record is observed in which two ion channels are never simultaneously open, then it is often of interest to know whether the observations indeed arose from the activity of only one ion channel. This question can be answered if it is possible to calculate the distribution of the duration of runs of single openings in a membrane patch that contains two active channels. If the observed run of single openings is much longer than that expected for a patch with two channels it is likely that only one channel was active. An approximate method is presented for calculating the distribution of the duration of runs of single openings in a patch with two active channels; this method has the advantage that it can be calculated from observable quantities, and requires no knowledge of the details of the ion-channel mechanism or its rate constants. The accuracy of this approximation is tested by exact calculations of the properties of runs of single openings, and of single bursts, for two specific mechanisms and a large range of rate constants. The approximation is good in all cases in which openings occur singly, or in closely spaced bursts. If, as is common in practice, openings occur in clusters that are separated by long shut periods, then overlap of clusters from two different channels may be detected, if no double opening is produced, as a period in the middle of a cluster in which the probability of being open doubles. The results derived here can be applied to such a period to test whether it results from the simultaneous activity of two channels, rather than from a change in the properties of a single channel.

A study on a single-unit Markov repairable system with repair time omission

This paper introduces a new model for a single-unit Markov repairable system in which repair times that are sufficiently short (less than some critical value) do not result in system failure. We can say that such a repair interval is omitted from the downtime record. First we suppose that the critical repair time is a constant. The model is then generalized to allow the critical repair time to be a non-negative random variable. We calculate system availability for these new models as a measure of reliability. Some numerical examples are given to illustrate the results in the paper.

Properties of single ion channel currents elicted by a pulse of agonist concentration or voltage

Experiments are often performed to study the behaviour of a single ion channel in response to a perturbation produced by a step change (‘jump’) in a variable that influences its equilibrium position, for example a voltage jump or jump in agonist concentration. It is also common to apply a rectangular pulse (consisting of an on jump followed by an off jump); for example brief concentration pulses are used to mimic synaptic transmission. Assuming a general Markov mechanism for channel dynamics, we obtain theoretical probability distributions of observable characteristics that describe the non–stationary behaviour of single ion channels which are subject to a jump, or to a pulse of finite duration. These characteristics are such things as open times, shut times, first latency, burst length and length of activation. We concentrate particularly on jumps to or from a zero level of agonist, which necessitates some modification to the usual arguments to cope with having some absorbing sets of states. Where possible, we include results which make allowance for the phenomenon of time interval omission, whereby some short intervals may be missed due to imperfect resolution of the recording method. A numerical example is studied in detail.

Modeling the evolution of system reliability performance under alternative environments

The dynamics of a system represented by a finite-state Markov process operating under two alternating regimes, for example, day/night, machine working/machine idling, etc., are modeled in this article. The transition rate matrices under the two regimes will usually be different. Also, the set of states of the system that are regarded as satisfactory may depend on the regime in operation: for example, a particular state of the system that may be regarded as satisfactory by day might not be tolerated at night (e.g., the headlights on a car not working). It is assumed that the regime durations are random variables and results are obtained for the availability of such a system and probability distributions for uptimes. Results and numerical examples are also given for two special cases: (i) when the regimes are of fixed duration; and (ii) when the regime durations have negative exponential distributions.