Geoffrey F. Yeo

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

We study a bivariate stochastic process {X(t)}= {(XE(t), Z(t))}, where {XE(t)} iS a continuous-time Markov chain describing the environment and {Z(t)} is the process of primary interest. In the context which motivated this study, {Z(t)} models the gating behaviour of a single ion channel. It is assumed that given {XE(t)}, the channel process {Z(t)} is a continuous-time Markov chain with infinitesimal generator at time t dependent on XE(t), and that the environment process {XE(t)} jS not dependent on {Z(t)}. We derive necessary and sufficient conditions for {X(t)} to be time reversible, showing that then its equilibrium distribution has a product form which reflects independence of the state of the environment and the state of the channel. In the special case when the environment controls the speed of the channel process, we derive transition probabilities and sojourn time distributions for {Z(t)} by exploiting connections with Markov reward processes. Some of these results are extended to a stationary environment. Applications to problems arising in modelling multiple ion channel systems are discussed. In particular, we present ways in which a multichannel

Superposition of interacting aggregated continuous-time Markov chains

Consider a system of interacting finite Markov chains in continuous time, where each subsystem is aggregated by a common partitioning of the state space. The interaction is assumed to arise from dependence of some of the transition rates for a given subsystem at a specified time on the states of the other subsystems at that time. With two subsystem classes, labelled 0 and 1, the superposition process arising from a system counts the number of subsystems in the latter class. Key structure and results from the theory of aggregated Markov processes are summarized. These are then applied also to superposition processes. In particular, we consider invariant distributions for the level m entry process, marginal and joint distributions for sojourn-times of the superposition process at its various levels, and moments and correlation functions associated with these distributions. The distributions are obtained mainly by using matrix methods, though an approach based on point process methods and conditional probability arguments is outlined. Conditions under which an interacting aggregated Markov chain is reversible are established. The ideas are illustrated with simple examples for which numerical results are obtained using Matlab. Motivation for this study has come from stochastic modelling of the behaviour of ion channels; another application is in reliability modelling.

On characterizations of beta and gamma distributions

Characterizations based on a product of independent random variables are presented for gamma and beta distributions, thereby extending recent results due to Kotz and Steutel and the authors. Moment methods are useful in some proofs. Links are made to multivariate analogues.

Superposition of Spatially Interacting Aggregated Continuous Time Markov Chains

A system s{ X(t)} = {X1(t),X2(t),..., XN(t)} of N interacting time reversible continuous time Markov chains is considered. The state space of each of the processes {Xi(t)} (i = 1, 2,...,N) is partitioned into two aggregates. Interaction between the processes {Xi(t)},{X2(t)},...,{XN(t)} is introduced by allowing the transition rates of an individual process at time t to depend on the configuration of aggregates occupied by the other N - 1 processes at that time. The motivation for this work comes from ion channel modeling, where {(X}(t)} describes the gating mechanisms of N channels and the partitioning of the state space of {Xi(t)} correspond to whether the channel is conducting or not. Let S(t) denote the number of conducting channels at time t. For a time-reversible class of such processes, expressions are derived for the mean and probability density function of the sojourns of {S(t)} at its different levels when {X(t)} is in equilibrium. Particular attention is paid to the situation when the N channels are located on a circle with nearest neighbor interaction. Necessary and sufficient conditions for a general co-operative multiple channel system to be time reversible are derived.

DURATION OF A SECRETARY PROBLEM

The distribution of the number of items drawn in a secretary problem, with an order s selection role and a success if any of the best s items is selected, is obtained by a probabilistic argument. Moments and asymptotics readily follow.

Modulatory drug action in an allosteric Markov model of ion channel behaviour: biphasic effects with access-limited binding to either a stimulatory or an inhibitory site

Concentration-dependent biphasic effects of drugs on ion channel activity have been reported in a variety of preparations, usually with stimulatory effects seen at low concentrations followed by increasingly dominant inhibition at higher levels. Such behaviour is often interpreted as evidence for the existence of separate modulatory drug binding sites. We demonstrate in this paper that it is possible for biphasic effects to be produced in an allosteric model of a ligand-activated ion channel, where diffusion-limited binding of the modulatory drug is restricted to either a stimulatory or an inhibitory site (but not both) because of steric overlap. The possibility of such an interaction mechanism should be kept in mind when interpreting experimental data if stoichiometric evidence from complementary techniques suggests that only one drug molecule is bound per receptor/ion channel complex.

Numerical evaluation of observed sojourn time distributions for a single ion channel incorporating time interval omission

The dynamical aspects of single ion channel gating can be modelled by a semi-Markov process. There is aggregation of states, corresponding to the receptor channel being open or closed, and there is time interval omission, brief sojourns in either the open or closed classes of states not being detected. This paper is concerned with the computation of the probability density functions of observed open (closed) sojourn-times incorporating time interval omission. A system of Volterra integral equations is derived, whose solution governs the required density function. Numerical procedures, using iterative and multistep methods, are described for solving these equations. Examples are given, and in the special case of Markov models results are compared with those obtained by alternative methods. Probabilistic interpretations are given for the iterative methods, which also give lower bounds for the solutions.

Interview Costs in the Secretary Problem

The effect of interview costs on the optimal selection strategy and on the chance of success in secretary problems with order k selection rules, both for a finite number of applicants and in the limiting case, is examined. Probabilistic reasoning is used and numerical examples given.

Marked Continuous-Time Markov Chain Modelling of Burst Behaviour for Single Ion Channels

Patch clamp recordings from ion channels often show bursting behaviour, that is, periods of repetitive activity, which are noticeably separated from each other by periods of inactivity. A number of authors have obtained results for important properties of theoretical and empirical bursts when channel gating is modelled by a continuous-time Markov chain with a finite-state space. We show how the use of marked continuous-time Markov chains can simplify the derivation of (i) the distributions of several burst properties, including the total open time, the total charge transfer, and the number of openings in a burst, and (ii) the form of these distributions when the underlying gating process is time reversible and in equilibrium.

Explicit asymptotic results for open times in ion channel models

The dynamical aspects of single channel gating can be modelled by a Markov renewal process, with states aggregated into two classes corresponding to the receptor channel being open or closed, and with brief sojourns in either class not detected. This paper is concerned with the relation between the amount of time, for a given record, in which the channel appears to be open compared to the amount in which it is actually open and the difference in their proportions; this may be used to obtain information on the unobserved actual process from the observed one. Results, with extensions, on exponential families have been applied to obtain relevant generating functions and asymptotic normal distributions, including explicit forms for the parameters. Numerical results are given as illustration in special cases.