×

Empirical clustering of bursts of openings in Markov and semi-Markov models of single channel gating incorporating time interval omission. (English) Zbl 0903.60058

Summary: The gating mechanism of a single ion channel is usually modelled by a continuous-time Markov chain with a finite state space, partitioned into two classes termed ‘open’ and ‘closed’. It is possible to observe only which class the process is in. A burst of channel openings is defined to be a succession of open sojourns separated by closed sojourns all having duration less than \(t_0\). Let \(N(t)\) be the number of bursts commencing in \((0,t]\). Then \(\text{Var}(N(t))/\mathbb{E}[N(t)]\) and \(\lim_{t\to\infty} (\text{Var}(N(t))/\mathbb{E}[N(t)])\) are measures of the degree of temporal clustering of bursts. We develop two methods for determining the above measures. The first method uses an embedded Markov renewal process and remains valid when the underlying channel process is semi-Markov and/or brief sojourns in either the open or closed classes of state are undetected. The second method uses a ‘backward’ differential-difference equation.
The observed channel process when brief sojourns are undetected can be modelled by an embedded Markov renewal process, whose kernel is shown, by exploiting connections with bursts when all sojourns are detected, to satisfy a differential-difference equation. This permits a unified derivation of both exact and approximate expressions for the kernel, and leads to a thorough asymptotic analysis of the kernel as the length of undetected sojourns tends to zero.

MSC:

60J27 Continuous-time Markov processes on discrete state spaces
60K15 Markov renewal processes, semi-Markov processes
92C05 Biophysics
15A99 Basic linear algebra
PDFBibTeX XMLCite
Full Text: DOI