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Aggregated Markov processes incorporating time interval omission

Published online by Cambridge University Press:  01 July 2016

Frank Ball  and
Mark Sansom
Frank Ball*
Affiliation:
University of Nottingham
Mark Sansom*
Affiliation:
University of Nottingham
*
∗Postal address: Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
∗∗Postal address: Department of Zoology, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
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Abstract

We consider a finite-state-space, continuous-time Markov chain which is time reversible. The state space is partitioned into two sets, termed ‘open' and ‘closed', and it is only possible to observe which set the process is in. Further, short sojourns in either the open or closed sets of states will fail to be detected. We show that the dynamic stochastic properties of the observed process are completely described by an embedded Markov renewal process. The parameters of this Markov renewal process are obtained, allowing us to derive expressions for the moments and autocorrelation functions of successive sojourns in both the open and closed states. We illustrate the theory with a numerical study.

Keywords

TIME REVERSIBLE PROCESSMARKOV RENEWAL PROCESSEMBEDDED PROCESSSINGLE CHANNEL GATING KINETICS
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 3 , September 1988 , pp. 546 - 572
Copyright
Copyright © Applied Probability Trust 1988 

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References

Ball, F. G. and Sansom, M. S. P. (1987) Temporal clustering of ion channel openings incorporating time interval omission. IMA J. Maths Appl in Medicine and Biol. 4, 333–361.Google Scholar
Bellman, R. (1960) Introduction to Matrix Analysis. McGraw-Hill, New York.Google Scholar
Çinlar, E. (1969) Markov renewal theory. Adv. Appl. Prob. 1, 123–187.CrossRefGoogle 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. (1981) On the stochastic properties of single ion channels. Proc. R. Soc. London B 211, 205–235.Google ScholarPubMed
Colquhoun, D. and Hawkes, A. G. (1983) The principles of stochastic interpretation of ion-channel mechanisms. In Single-Channel Recording, ed. Sakmann, B. and Neher, E., Plenum Press, New York, 135–175.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., Montal, M. and Rice, J. A. (1985) Identification of aggregated Markovian models: application to the nicotinic acetylcholine receptor. Proc Berkeley Conf. in Honor of Jerzy Neyman and Jack Kiefer 1, ed. Le Cam, L. and Olshen, R., Wadsworth, Belmont, CA, 269–290.Google Scholar
Horn, R. and Lange, K. (1983) Estimating kinetic constants from single channel data. Biophys. J. 43, 207–223.CrossRefGoogle ScholarPubMed
Karlin, A. (1967) On the application of a plausible model of allosteric proteins to the receptor for acetylcholine. J. Theoret. Biol. 16, 306–320.Google Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, Chichester.Google Scholar
Kerry, C. J., Kits, K. S., Ramsey, R. L., Sansom, M. S. P. and Usherwood, P. N. R. (1987) Single channel kinetics of a glutamate receptor. Biophys. J. 51, 137–144.Google Scholar
Kerry, C. J., Ramsey, R. L., Sansom, M. S. P. and Usherwood, P. N. R. (1988) Glutamate receptor-channel kinetics: the effect of glutamate concentration. Biophys. J. 53, 39–59.Google Scholar
Labarca, P., Rice, J. A., Fredkin, D. R. and Montal, M. (1985) Kinetic analysis of channel gating: application to the cholinergic receptor channel and the chloride channel from Torpedo californica . Biophys. J. 47, 469–478.Google Scholar
Monod, J., Wyman, J. and Changeux, J. P. (1965) On the nature of allosteric transitions: a plausible model. J. Mol. Biol. 12, 88–118.CrossRefGoogle ScholarPubMed
Pyke, R. (1961) Markov renewal processes: definitions and preliminary properties. Ann. Math. Statist. 32, 1231–1242.Google Scholar
Roux, B. and Sauve, R. (1985) A general solution to the time interval omission problem applied to single channel analysis. Biophys. J. 48, 149–158.CrossRefGoogle Scholar
Sakmann, B. and Neher, E. (1983) Single-Channel Recording. Plenum Press, New York.Google Scholar
Seneta, E. (1973) Non-Negative Matrices. George Allen and Unwin, London.Google Scholar