Temporal Clustering of Ion Channel Openings Incorporating Time Interval Omission

Abstract

Ion channel gating mechanisms can be satisfactorily modelled by a time-reversible continuous-time Markov chain on a finite state space. The complete process is not observable, but rather the state space is partitioned into 'open' and 'closed' states corresponding to the receptor channel being open or closed, and it is only possible to observe whether the process is in an open or a closed state. Previous studies of locust muscle glutamate receptor channels have revealed single channel openings to be highly clustered in time. This clustering can be described by the ratio of the variance var N(t) to the mean E[N(t)] of the number of channel openings in a time interval of length t. In this paper we obtain expressions for (formula; see text) for the above aggregated Markov process. Applications of these expressions to a model for the locust muscle glutamate receptor channel show this aspect of the model to be reasonably consistent with experimental data. In practice very short sojourns in either the open or closed states will fail to be detected, a phenomenon known as time interval omission. Using a semi-Markov approach, we outline a general theoretical framework for analysing dynamic properties of aggregated Markov processes incorporating time interval omission. We illustrate the applicability of this framework by using it to find limt----infinity [[var N(t)]/E[N(t)]] theoretically, when time interval omission is incorporated. This allows us to study the robustness of limt----infinity [[var N(t)]/E[N(t)]] to time interval omission, as a measure of temporal clustering.