CONTINUOUS-TIME MARKOV-CHAINS IN A RANDOM ENVIRONMENT, WITH APPLICATIONS TO ION-CHANNEL MODELING

We study a bivariate stochastic process {X(t)}={(X(E)(t))}, Z(t))}, wh ere {X(E)(t)} is a continuous-time Markov chain describing the environ ment and {Z(t)} is the process of primary interest. In the context whi ch motivated this study, {Z(t)} models the gating behaviour of a singl e ion channel. It is assumed that given {X,(t)}, the channel process { Z(t)} is a continuous-time Markov chain with infinitesimal generator a t time t dependent on X(E)(t), and that the environment process {X(E)( t)} is not dependent on {Z(t)}. We derive necessary and sufficient con ditions for {X(t)} to be time reversible, showing that then its equili brium distribution has a product form which reflects independence of t he state of the environment and the state of the channel. In the speci al case when the environment controls the speed of the channel process , we derive transition probabilities and sojourn time distributions fo r {Z(t)} by exploiting connections with Markov reward processes. Some of these results are extended to a stationary environment. Application s to problems arising in modelling multiple ion channel systems are di scussed. In particular, we present ways in which a multichannel model in a random environment does and does not exhibit behaviour identical to a corresponding model based on independent and identically distribu ted channels.