Joint distributions of apparent open and shut times of single-ion channels and maximum likelihood fitting of mechanisms

The openings and shuttings of individual ion channel molecules can be modelled in terms of an underlying Markov process with discrete states in continuous time. In practice, some of the open times, and/or shut times, are too short to be detected reliably, making the durations of some of these intervals appear to be longer than they really are. Under certain assumptions about how this happens, the probability densities of these apparent times have previously been obtained. It has been shown that the ability to distinguish between alternative postulated reaction mechanisms can be greatly improved by considering bivariate distributions. In this paper we obtain joint distributions, and hence conditional distributions, of adjacent apparent open and shut times. Numerical examples illustrate what insight these conditional distributions may provide about the underlying mechanism. Bivariate distributions are readily generalized to multivariate distributions which enable the likelihood for an entire single-channel recording to be computed, and hence efficient maximum likelihood estimates for the mechanism's rate constants can be obtained. Numerical examples of such fitting are given.