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

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

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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References

Anderson, W. J. and Mcdunnough, P. M. (1990) On the representation of symmetric transition functions. Adv. Appl. Prob. 22, 548–563.Google Scholar
Ball, F. G. and Rice, J. A. (1992) Stochastic models for ion channels: introduction and bibliography. Math. Biosci. 112, 189–206.Google Scholar
Ball, F. G. and Sansom, M. S. P. (1988) Aggregated Markov processes incorporating time interval omission. Adv. Appl. Prob. 20, 546–572.Google Scholar
Ball, F. G. and Yeo, G. F. (1993) Lumpability and marginalisability for continuous-time Markov chains. J. Appl. Prob. 30, 518–528.Google Scholar
Ball, F. G., Milne, R. K. and Yeo, G. F. (1991) Aggregated semi-Markov processes incorporating time interval omission. Adv. Appl. Prob. 23, 772–797.Google Scholar
Beaudry, M. D. (1978) Performance-related reliability measures for computing systems. IEEE Trans. Comp. 27, 540–547.Google Scholar
Bellman, R. (1960) Introduction to Matrix Analysis. McGraw-Hill, New York.Google Scholar
Branford, A. J. (1985) A self-excited migration process. J. Appl. Prob. 22, 58–67.Google Scholar
Colquhoun, D. and Hawkes, A. G. (1977) Relaxation and fluctuations of membrane currents that flow through drug-operated channels. Proc. R. Soc. London B 199, 231–262.Google Scholar
Colquhoun, D. and Hawkes, A. G. (1982) On the stochastic properties of bursts of single ion channel openings and of clusters of bursts. Phil. Trans. R. Soc. London B 300, 1–59.Google Scholar
Cox, D. R. and Isham, V. (1980) Point Processes. Chapman and Hall, London.Google Scholar
Dabrowski, A. R. and Mcdonald, D. (1992) Statistical analysis of multiple ion channel data. Ann. Statist. 20, 1180–1202.Google Scholar
Dabrowski, A. R., Mcdonald, D. and Rösler, U. (1990) Renewal properties of ion channels. Ann. Statist. 18, 1091–1115.Google Scholar
Foley, R. D., Klutke, G.-A. and König, D. (1991) Stationary increments of accumulation processes in queues and generalized semi-Markov schemes. J. Appl. Prob. 28, 864–872.Google Scholar
Fredkin, D. and Rice, J. A. (1986) On aggregated Markov processes. J. Appl. Prob. 23, 208–214.Google Scholar
Fredkin, D. and Rice, J. A. (1987) Correlation functions of a function of a finite-state Markov process with application to channel kinetics. Math. Biosci. 87, 161–172.Google Scholar
Fredkin, D. and Rice, J. A. (1991) On the superposition of currents from ion channels. Phil. Trans. R. Soc. London B 334, 347–356.Google Scholar
Fredkin, D. R., Montal, M. and Rice, J. A. (1985) Identification of aggregated Markovian models: application to the nicotinic acetylcholine receptor. In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, ed. Le Cam, Lucien and Olshen, Richard, vol. 1, pp. 269–289. Wadsworth, Belmont CA.Google Scholar
Gaver, D. P., Jacobs, P. A. and Latouche, G. (1984) Finite birth-and-death models in randomly changing environments. Adv. Appl. Prob. 16, 715–731.Google Scholar
Karlin, S. and Mcgregor, J. L. (1957a) The differential equations of birth and death processes, and the Stieltjes moment problem. Trans. Amer. Math. Soc. 85, 489–546.Google Scholar
Karlin, S. and Mcgregor, J. L. (1957b) The classification of birth and death processes. Trans. Amer. Math. Soc. 86, 366–400.Google Scholar
Keilson, J. (1979) Markov Chain Models–Rarity and Exponentiality. Springer-Verlag, New York.Google Scholar
Keilson, J. and Subba Rao, S. (1970) A process with chain dependent growth rate. J. Appl. Prob. 7, 699–711.Google Scholar
Keilson, J. and Subba Rao, S. (1971) A process with chain dependent growth rate. Part II: The ruin and ergodic problems. Adv. Appl. Prob. 3, 315–338.Google Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, Chichester UK.Google Scholar
Kijima, S. and Kijima, H. (1987) Statistical analysis of channel current from a membrane patch II. A stochastic theory of a multi-channel system in the steady-state. J. Theoret. Biol. 128, 435–455.Google Scholar
Kulkarni, V. G. (1989) A new class of multivariate phase type distributions. Operat. Res. 37, 151–158.Google Scholar
Läuger, P. (1983) Conformational transitions of ionic channels. In Single-Channel Recording, ed. Sakmann, B. and Neher, E., pp. 177–189, Plenum, New York.Google Scholar
Lefevre, C. and Michaletzky, G. (1990) Interparticle dependence in a linear death process subjected to a random environment. J. Appl. Prob. 27, 491–498.Google Scholar
Masuda, Y. and Sumita, U. (1991) A multivariate reward process defined on a semi-Markov process. J. Appl. Prob. 28, 360–373.Google Scholar
Rao, C. R. (1973) Linear Statistical Inference and Its Applications, 2nd ed. Wiley, New York.Google Scholar
Sumita, U. and Masuda, Y. (1987) An alternative approach to the analysis of finite semi-Markov processes. Stoch. Models 3, 67–87.Google Scholar
Yeo, G. F., Edeson, R. O., Milne, R. K. and Madsen, B. W. (1989) Superposition properties of independent ion channels. Proc. R. Soc. London B 238, 155–170.Google Scholar
Yeramian, E., Trautmann, A. and Claverie, P. (1986) Acetylcholine receptors are not functionally independent. Biophys. J. 50, 253–263.Google Scholar