The distributions of the apparent open times and shut times in a single channel record when brief events cannot be detected

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/or shut times, are too short to be detected reliably. If a fixed dead-time r 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 r followed by any number of openings and shuttings, all the shut times being shorter than r ; the e-opening ends when a shut time longer than r occurs. A similar definition is used for e-shut times. Several authors have derived approximations to the distribution of durations of e-openings and e-shuttings. In this paper the exact distributions are derived. They are defined piecewise over the intervals r to 2r, 2r to 3 r ,..., etc., the distribution in each interval being a sum of products of polynomials in t with exponential terms. The number of terms is finite, but increases as intervals get further from t = r. An asymptotic form for large t (for which the exact solution becomes difficult to compute) is given for the two state case. The exact solution is compared with several approximations, some of which are shown to be good enough for use in most practical applications.