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We present a general theoretical framework, incorporating both aggregation of states into classes and time interval omission, for stochastic modeling of the dynamic aspects of single channel behavior. Our semi-Markov models subsume the standard continuous-time Markov models, diffusion models and fractal models. In particular our models allow for quite general distributions of state sojourn times and arbitrary correlations between successive sojourn times. Another key feature is the invariance of our framework with respect to time interval omission: that is, properties of the aggregated process incorporating time interval omission can be derived directly from corresponding properties of the process without it. Even in the special case when the underlying process is Markov, this leads to considerable clarification of the effects of time interval omission. Among the properties considered are equilibrium behavior, sojourn time distributions and their moments, and auto-correlation and cross-correlation functions. The theory is motivated by ion channel mechanisms drawn from the literature, and illustrated by numerical examples based on these.

Original publication




Journal article


Biophys J

Publication Date





357 - 374


Biophysical Phenomena, Biophysics, Ion Channel Gating, Ion Channels, Kinetics, Markov Chains, Mathematics, Models, Biological