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Superposition of Interacting Aggregated Continuous-Time Markov Chains

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

Published online by Cambridge University Press:  01 July 2016

Frank Ball ,
Robin K. Milne ,
Ian D. Tame  and
Geoffrey F. Yeo
Frank Ball*
Affiliation:
The University of Nottingham
Robin K. Milne*
Affiliation:
The University of Western Australia
Ian D. Tame*
Affiliation:
The University of Nottingham
Geoffrey F. Yeo*
Affiliation:
Murdoch University
*
∗ Postal address: Department of Mathematics, The University of Nottingham, Nottingham, NG7 2RD, UK. E-mail address: [email protected]
∗∗ Postal address: Department of Mathematics, The University of Western Australia, Nedlands, 6907, Australia. E-mail address: [email protected]
∗∗∗ Postal address: Area of Humanities and Science, Bridgwater College, Bath Road, Bridgwater, Somerset, TA6 4PZ, UK.
∗∗∗∗ Postal address: School of Physical Sciences, Engineering and Technology, Murdoch University, Murdoch, 6150, Australia. E-mail address: [email protected]
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Abstract

Consider a system of interacting finite Markov chains in continuous time, where each subsystem is aggregated by a common partitioning of the state space. The interaction is assumed to arise from dependence of some of the transition rates for a given subsystem at a specified time on the states of the other subsystems at that time. With two subsystem classes, labelled 0 and 1, the superposition process arising from a system counts the number of subsystems in the latter class. Key structure and results from the theory of aggregated Markov processes are summarized. These are then applied also to superposition processes. In particular, we consider invariant distributions for the level m entry process, marginal and joint distributions for sojourn-times of the superposition process at its various levels, and moments and correlation functions associated with these distributions. The distributions are obtained mainly by using matrix methods, though an approach based on point process methods and conditional probability arguments is outlined. Conditions under which an interacting aggregated Markov chain is reversible are established. The ideas are illustrated with simple examples for which numerical results are obtained using Matlab. Motivation for this study has come from stochastic modelling of the behaviour of ion channels; another application is in reliability modelling.

Keywords

AGGREGATED MARKOV CHAINCONTINUOUS-TIME MARKOV CHAINEQUILIBRIUM BEHAVIOURINTERACTING SUBSYSTEMSION CHANNEL MODELLINGPHASE-TYPE DISTRIBUTIONSRELIABILITY MODELLINGREVERSIBILITYSOJOURN-TIME DISTRIBUTIONSSUPERPOSITION PROCESS

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60G55: Point processes 60K15: Markov renewal processes, semi-Markov processes 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) 92C05: Biophysics
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 1 , March 1997 , pp. 56 - 91
Copyright
Copyright © Applied Probability Trust 1997 

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References

Albertsen, A. and Hansen, U.-P. (1994) Estimation of kinetic rate constants from multi-channel recordings by a direct fit of the time series. Biophys. J. 67, 1391–1403.Google Scholar
Asmussen, S. (1987) Applied Probability and Queues. Wiley, Chichester.Google Scholar
Ball, F. G., Davies, S. S. and Sansom, M. S. P. (1997) A sequential allosteric model for receptor-gated ion channels: modelling of single channel properties. In preparation.Google Scholar
Ball, F. G., Kerry, C. J., Ramsey, R. L., Sansom, M. S. P. and Usherwood, P. N. R. (1988) The use of dwell time cross-correlation functions to study single-ion channel gating kinetics. Biophys. J. 54, 309–320.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
Ball, F. G., Milne, R. K. and Yeo, G. F. (1994) Continuous-time Markov chains in a random environment, with applications to ion channel modelling. Adv. Appl. Prob. 26, 919–946.Google Scholar
Ball, F. G. and Rice, J. A. (1989) A note on single-channel autocorrelation functions. Math. Biosci. 97, 17–26.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. and Yeo, G. F. (1994) Numerical evaluation of observed sojourn time distributions for a single ion channel incorporating time interval omission. Statist. Comp. 4, 1–12.Google Scholar
Ball, F. G., Yeo, G. F., Milne, R. K., Edeson, R. O., Madsen, B. W. and Sansom, M. S. P. (1993) Single ion channel models incorporating aggregation and time interval omission. Biophys. J. 64, 357–374.Google Scholar
Bellman, R. (1970) Introduction to Matrix Analysis. 2nd edn. McGraw-Hill, New York.Google Scholar
Burke, C. J. and Rosenblatt, M. (1958) A Markovian function of a Markov chain. Ann. Math. Statist. 29, 1112–1122.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
Colquhoun, D. and Hawkes, A. G. (1987) A note on correlations in single ion channel records. Proc. R. Soc. London B 230, 15–52.Google Scholar
Colquhoun, D. and Hawkes, A. G. (1990) 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. Proc. R. Soc. London B 240, 453–477.Google Scholar
Csenki, A. (1992) The joint distribution of sojourn times in finite Markov processes. Adv. Appl. Prob. 24, 141–160.Google Scholar
Csenki, A. (1994) Dependability for Systems with a Partitioned State Space. Markov and Semi-Markov Theory and Computational Implementation (Lecture Notes in Statistics 90). Springer, New York.Google Scholar
Dabrowski, A. R., Mcdonald, D. and Rösler, U. (1990) Renewal theory properties of ion channels. Ann. Statist. 18, 1091–1115.Google Scholar
Draber, S., Schultze, R. and Hansen, U.-P. (1993) Cooperative behaviour of K+ channels in the tonoplast of Chara corallina. Biophys. J. 65, 1553–1559.Google Scholar
Edeson, R. O., Ball, F. G., Yeo, G. F., Milne, R. K. and Davies, S. S. (1994) Model properties underlying non-identifiability in single channel inference. Proc. R. Soc. London B 255, 21–29.Google Scholar
Edeson, R. O., Yeo, G. F., Milne, R. K. and Madsen, B. W. (1990) Graphs, random sums, and sojourn time distributions, with application to ion-channel modeling. Math. Biosci. 102, 75–104.Google Scholar
Fredkin, D. R., Montal, M. and Rice, J. A. (1985) Identification of aggregated Markovian models: application to the nicotinic acetylcholine receptor. In Proc. Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, Vol. 1. ed. Le Cam, L. and Olshen, R.. Wadsworth, Belmont, CA. pp. 269–289.Google Scholar
Fredkin, D. R. and Rice, J. A. (1986) On aggregated Markov processes. J. Appl. Prob. 23, 208–214.Google Scholar
Fredkin, D. R. 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. and Rice, J. A. (1992) Maximum likelihood estimation and identification directly from single-channel recordings. Proc. R. Soc. London B 249, 125–132.Google Scholar
Horn, R. A. and Johnson, C. R. (1991) Topics in Matrix Analysis. Cambridge University Press, Cambridge.Google Scholar
Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
Keleshian, A., Yeo, G. F., Edeson, R. O., and Madsen, B. W. (1994) Superposition properties of interacting ion channels. Biophys. J. 67, 634–640.Google Scholar
Keller, A. Z. and Qamber, I. S. (1988) System availability synthesis. In Tenth Advances in Reliability Technology Symposium. ed. Libberton, G. P.. Elsevier Applied Science, London. pp. 173–188.Google Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, Chichester.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. Theor. Biol. 128, 435–455.Google Scholar
Läuger, P. (1983) Conformational transitions of ionic channels. In Single-Channel Recording. ed. Sakmann, B. and Neher, E.. Plenum, New York. pp. 177–189.Google Scholar
Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979) Multivariate Analysis. Academic Press, London.Google Scholar
Matlab, (1989) MATLAB for Macintosh Computers. Users Guide. Mathworks, South Natick, MA.Google Scholar
Morrison, D. F. (1976) Multivariate Statistical Methods. McGraw-Hill, New York.Google Scholar
Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore.Google Scholar
Riordan, J. (1958) An Introduction to Combinatorial Analysis. Wiley, New York.Google Scholar
Ross, S. M. (1983) Stochastic Processes. Wiley, New York.Google Scholar
Rubino, G. and Sericola, B. (1989) Sojourn times in finite Markov processes. J. Appl. Prob. 26, 744–756.Google Scholar
Seneta, E. (1973) Non-Negative Matrices. George Allen and Unwin, London.Google Scholar
Sericola, B. (1990) Closed-form solution for the distribution of total time spent in a subset of states of a homogeneous Markov process during a finite observation period. J. Appl. Prob. 27, 713–719.Google Scholar
Syski, R. (1992) Passage Times for Markov Chains. IOS Press, Amsterdam.Google Scholar
Tame, I. D. (1993) The modelling of multiple ion channels by product Markov Processes. M. Phil. thesis. University of Nottingham.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