Numerical evaluation of observed sojourn time distributions for a single ion channel incorporating time interval omission

The dynamical aspects of single ion channel gating can be modelled by a semi-Markov process. There is aggregation of states, corresponding to the receptor channel being open or closed, and there is time interval omission, brief sojourns in either the open or closed classes of states not being detected. This paper is concerned with the computation of the probability density functions of observed open (closed) sojourn-times incorporating time interval omission. A system of Volterra integral equations is derived, whose solution governs the required density function. Numerical procedures, using iterative and multistep methods, are described for solving these equations. Examples are given, and in the special case of Markov models results are compared with those obtained by alternative methods. Probabilistic interpretations are given for the iterative methods, which also give lower bounds for the solutions.

[1]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[2]  Frank Kelly,et al.  Reversibility and Stochastic Networks , 1979 .

[3]  A G Hawkes,et al.  Relaxation and fluctuations of membrane currents that flow through drug-operated channels , 1977, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[4]  R O Edeson,et al.  Statistical inference from single channel records: two-state Markov model with limited time resolution , 1988, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[5]  R O Edeson,et al.  Estimation of single channel kinetic parameters from data subject to limited time resolution. , 1989, Biophysical journal.

[6]  Alan G. Hawkes,et al.  The distribution of apparent occupancy times in a two-state Markov process in which brief events cannot be detected , 1992, Advances in Applied Probability.

[7]  R O Edeson,et al.  Stochastic modelling of a single ion channel : an alternating renewal approach with application to limited time resolution , 1988, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[8]  Statistical Inference for a Two‐State Markov Model of a Single Ion Channel, Incorporating Time Interval Omission , 1995 .

[9]  E. Çinlar.  Markov renewal theory , 1969, Advances in Applied Probability.

[10]  Larry S. Liebovitch,et al.  Ion channel kinetics: a model based on fractal scaling rather than multistate Markov processes , 1987 .

[11]  Alan G. Hawkes,et al.  The distributions of the apparent open times and shut times in a single channel record when brief events cannot be detected , 1990, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[12]  M S Sansom,et al.  Single ion channel models incorporating aggregation and time interval omission. , 1993, Biophysical journal.

[13]  Fred J. Sigworth,et al.  Fitting and Statistical Analysis of Single-Channel Records , 1983 .

[14]  Graphs, random sums, and sojourn time distributions, with application to ion-channel modeling. , 1990, Mathematical biosciences.

[15]  Erhan Çinlar,et al.  Introduction to stochastic processes , 1974 .

[16]  R. Pyke.  Markov renewal processes: Definitions and preliminary properties , 1961 .

[17]  E. Salpeter,et al.  Diffusion models of ion-channel gating and the origin of power-law distributions from single-channel recording. , 1988, Proceedings of the National Academy of Sciences of the United States of America.

[18]  F G Ball,et al.  Single-channel data and missed events , 1990, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[19]  Frank Ball,et al.  Aggregated Markov processes incorporating time interval omission , 1988, Advances in Applied Probability.

[20]  Alan G. Hawkes,et al.  The Principles of the Stochastic Interpretation of Ion-Channel Mechanisms , 1983 .

[21]  A. Hawkes,et al.  On the stochastic properties of bursts of single ion channel openings and of clusters of bursts. , 1982, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[22]  Asymptotic theory for multiple solutions of likelihood equations, with application to a single ion channel model , 1993 .

[23]  A G Hawkes,et al.  Asymptotic distributions of apparent open times and shut times in a single channel record allowing for the omission of brief events. , 1992, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[24]  Alan G. Hawkes,et al.  GENERALISED EIGENPROBLEMS ARISING IN AGGREGATED MARKOV PROCESSES ALLOWING FOR TIME INTERVAL OMISSION , 1992 .

[25]  B. Sakmann,et al.  Single-Channel Recording , 1995, Springer US.

[26]  K L Magleby,et al.  Correcting single channel data for missed events. , 1986, Biophysical journal.

[27]  Frank Ball,et al.  On the exact distribution of observed open times in single ion channel models , 1993, Journal of Applied Probability.