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Continuous-time Markov chains in a random environment, with applications to ion channel modelling

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

Published online by Cambridge University Press:  01 July 2016

Frank Ball ,
Robin K. Milne  and
Geoffrey F. Yeo
Frank Ball*
Affiliation:
The University of Nottingham
Robin K. Milne*
Affiliation:
The University of Western Australia
Geoffrey F. Yeo*
Affiliation:
Murdoch University
*
* Postal address: Department of Mathematics, The University of Nottingham, Nottingham, NG7 2RD, UK.
** Postal address: Department of Mathematics, The University of Western Australia, Nedlands, WA 6009, Australia.
*** Postal address: School of Mathematical and Physical Sciences, Murdoch University, Murdoch, WA 6150, Australia.
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Abstract

We study a bivariate stochastic process {X(t)} = Z(t))}, where {XE(t)} is a continuous-time Markov chain describing the environment and {Z(t)} is the process of interest. In the context which motivated this study, {Z(t)} models the gating behaviour of a single ion channel. It is assumed that given {XE(t)}, the channel process {Z(t)} is a continuous-time Markov chain with infinitesimal generator at time t dependent on XE(t), and that the environment process {XE{t)} is not dependent on {Z(t)}. We derive necessary and sufficient conditions for {X(t)} to be time reversible, showing that then its equilibrium distribution has a product form which reflects independence of the state of the environment and the state of the channel. In the special case when the environment controls the speed of the channel process, we derive transition probabilities and sojourn time distributions for {Z(t)} by exploiting connections with Markov reward processes. Some of these results are extended to a stationary environment. Applications to problems arising in modelling multiple ion channel systems are discussed. In particular, we present ways in which a multichannel model in a random environment does and does not exhibit behaviour identical to a corresponding model based on independent and identically distributed channels.

Keywords

REVERSIBILITYEQUILIBRIUM BEHAVIOURMARKOV REWARD PROCESSRANDOMTIME TRANSFORMATIONSPECTRAL REPRESENTATIONSTATIONARY PROCESSESMULTIPLE ION CHANNEL MODELLINGPOINT PROCESS

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60G55: Point processes 92C05: Biophysics
Type
Research Article
Information
Advances in Applied Probability , Volume 26 , Issue 4 , December 1994 , pp. 919 - 946
Copyright
Copyright © Applied Probability Trust 1994 

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