Asymptotic distributions of apparent open times and shut times in a single channel record allowing for the omission of brief events

Abstract

The openings and shuttings of individual ion channel molecules can be described by a Markov process with discrete states in continuous time. The predicted distributions of the durations of open times, shut times, bursts of openings, etc. are all described, in principle, by mixtures of exponential densities. In practice it is usually found that some of the open times, and the shut times, are too short to be detected reliably. If a fixed dead-time τ is assumed then it is possible to define, as an approximation to what is actually observed, an ‘extended opening’ or e-opening which starts with an opening of duration at least τ followed by any number of openings and shuttings, all the shut times being shorter than τ ; the e-opening ends when a shut time longer than τ occurs. A similar definition is used for e-shut times. The probability densities, f(t) , of these extended times have previously been obtained as expressions which become progressively more complicated, and numerically unstable to compute, as t→∞. In this paper we present, for the two-state model, an alternative representation as an infinite series of which a small number of terms gives a very accurate approximation of f(t) for large t . For the general model we present an asymptotic representation as a mixture of exponentials which is accurate for all except quite small values of t . Some simple model-independent corrections for missed events are discussed in relationship to the exact solutions.