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A G Hawkes

Publications and source records attributed to A G Hawkes.

10 recordsLinked to original sources
PubMed2003

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.

Binding Sites↗
PubMed1997

Stochastic behavior of ion channels in varying conditions.

The opening and closing of a single ion channel can be described as the aggregation into two sets of states, open and shut, of an underlying Markov process with a finite number of states. In this paper we consider an inhomogeneous process due to changing physical conditions, in particular a pulse of activity in which the transition rates during the pulse differ from those obtaining both before and after the pulse. Outside the pulse the chain contains an absorbing shut subset, so that no activity is observed before the pulse and almost surely, at most a finite number of openings are generated by the pulse. We consider, in particular, the probability distribution of first latency (the time to first opening) and the distribution of the total activation time. We take into account the phenomenon of time-interval omission, when short open or shut times fail to be observed.

Animals↗
PubMed1995

Desensitization of N-methyl-D-aspartate receptors: a problem of interpretation.

The phenomenon of desensitization is universal, but its mechanism is still ill-understood and controversial. A recently published study [Lin, F. & Stevens, C. F. (1994) J. Neurosci, 14, 2153-2160] attempted to cast light on the mechanism of desensitization of N-methyl-D-aspartate (NMDA) receptors, in particular the vexed question of whether the channel must open before it can desensitize. During the desensitizing preexposure to agonist in those experiments, more desensitization was produced when channel openings were observed than when no openings were observed. The conclusion that "desensitization occurs more rapidly from the open state" unfortunately was based on a stochastic fallacy, and we present here a theoretical treatment and illustration showing that the observed behavior is predicted by a simple mechanism in which desensitization can occur only from a shut state.

Animals↗
PubMed1992

Asymptotic distributions of apparent open times and shut times in a single channel record allowing for the omission of brief events.

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 the shut times, are too short to be detected reliably. If a fixed dead-time tau 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 tau followed by any number of openings and shuttings, all the shut times being shorter than tau; the e-opening ends when a shut time longer than tau occurs. A similar definition is used for e-shut times. The probability densities, f(t), of these extended times have previously been obtained as expressions which become progressively more complicated, and numerically unstable to compute, as t-->infinity. In this paper we present, for the two-state model, an alternative representation as an infinite series of which a small number of terms gives a very accurate approximation of f (t) for large t. For the general model we present an asymptotic representation as a mixture of exponentials which is accurate for all except quite small values of t. Some simple model-independent corrections for missed events are discussed in relationship to the exact solutions.

Ion Channel Gating↗
PubMed1990

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.

Animals↗
PubMed1987

A note on correlations in single ion channel records.

General expressions are derived for the correlation coefficients between the length of an opening and that of the nth subsequent opening for a single ion channel. Analogous results are given for the correlation between shut times, and between an open time and subsequent shut times. An alternative derivation of the results of Fredkin et al. (in Proc. Berkeley Conf. in honor of Neyman & Kiefer, vol. 1, pp. 269-289 (1985] is given, and their results are extended to the case where openings occur in bursts. Expressions are given for the correlation between the first and nth opening in a burst, between the lengths of bursts, and between the number of openings per burst. Each of these sorts of correlation can give information about the connections that exist between the various states of the system; interpretations of the correlations are discussed. Expressions are derived for the distributions of the nth open time, shut time, burst length, etc. following the application of a perturbation (e.g. a voltage jump or a concentration jump). It is shown that these distributions will all be the same (namely the equilibrium distribution) only in the case where the openings, burst lengths, etc. are not correlated. Certain reaction schemes predict a component in the distribution of the number of openings per burst that has a unit mean (i.e. a component of isolated single openings). For some schemes this component is predicted to have zero amplitude, in principle, whereas in others it may be quite prominent. The presence or absence of this component can give information about the way in which the various states of the system are connected. The interpretation in terms of mechanism is discussed.

Animals↗
PubMed1981

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 the 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.

Animals↗