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Clustering of Bursts of Openings in Markov and Semi-Markov Models of Single Channel Gating

Part of: Special processes Markov processes Physiological, cellular and medical topics

Published online by Cambridge University Press:  01 July 2016

Frank Ball  and
Sue Davies
Frank Ball*
Affiliation:
The University of Nottingham
Sue Davies*
Affiliation:
The University of Nottingham
*
∗ Postal address: Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
∗ Postal address: Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
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Abstract

The gating mechanism of a single ion channel is usually modelled by a continuous-time Markov chain with a finite state space. The state space is partitioned into two classes, termed ‘open’ and ‘closed’, and it is possible to observe only which class the process is in. In many experiments channel openings occur in bursts. This can be modelled by partitioning the closed states further into ‘short-lived’ and ‘long-lived’ closed states, and defining a burst of openings to be a succession of open sojourns separated by closed sojourns that are entirely within the short-lived closed states. There is also evidence that bursts of openings are themselves grouped together into clusters. This clustering of bursts can be described by the ratio of the variance Var (N(t)) to the mean [N(t)] of the number of bursts of openings commencing in (0, t]. In this paper two methods of determining Var (N(t))/[N(t)] and limt→∝ Var (N(t))/[N(t)] are developed, the first via an embedded Markov renewal process and the second via an augmented continuous-time Markov chain. The theory is illustrated by a numerical study of a molecular stochastic model of the nicotinic acetylcholine receptor. Extensions to semi-Markov models of ion channel gating and the incorporation of time interval omission are briefly discussed.

Keywords

CONTINUOUS-TIME MARKOV CHAINMARKOV RENEWAL PROCESSAGGREGATED PROCESSSINGLE ION CHANNEL MODELLINGFUNDAMENTAL MATRIXSPECTRAL REPRESENTATION

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60K15: Markov renewal processes, semi-Markov processes 92C05: Biophysics
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 1 , March 1997 , pp. 92 - 113
Copyright
Copyright © Applied Probability Trust 1997 

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