MULTIPLICATIVELY CLOSED MARKOV MODELS MUST FORM LIE ALGEBRAS

We prove that the probability substitution matrices obtained from a continuous-time Markov chain form a multiplicatively closed set if and only if the rate matrices associated with the chain form a linear space spanning a Lie algebra. The key original contribution we make is to overcome an obstruction, due to the presence of inequalities that are unavoidable in the probabilistic application, which prevents free manipulation of terms in the Baker–Campbell–Haursdorff formula.

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Clustering of Bursts of Openings in Markov and Semi-Markov Models of Single Channel Gating

Advances in Applied Probability ◽

10.1017/s0001867800027804 ◽

1997 ◽

Vol 29 (01) ◽

pp. 92-113 ◽

Cited By ~ 2

Author(s):

Frank Ball ◽

Sue Davies

Keyword(s):

Markov Chain ◽

Ion Channel ◽

State Space ◽

Continuous Time ◽

Markov Models ◽

Numerical Study ◽

Channel Gating ◽

Markov Renewal Process ◽

Time Interval ◽

Continuous Time Markov Chain

The gating mechanism of a single ion channel is usually modelled by a continuous-time Markov chain with a finite state space. The state space is partitioned into two classes, termed ‘open’ and ‘closed’, and it is possible to observe only which class the process is in. In many experiments channel openings occur in bursts. This can be modelled by partitioning the closed states further into ‘short-lived’ and ‘long-lived’ closed states, and defining a burst of openings to be a succession of open sojourns separated by closed sojourns that are entirely within the short-lived closed states. There is also evidence that bursts of openings are themselves grouped together into clusters. This clustering of bursts can be described by the ratio of the variance Var (N(t)) to the mean[N(t)] of the number of bursts of openings commencing in (0, t]. In this paper two methods of determining Var (N(t))/[N(t)] and limt→∝Var (N(t))/[N(t)] are developed, the first via an embedded Markov renewal process and the second via an augmented continuous-time Markov chain. The theory is illustrated by a numerical study of a molecular stochastic model of the nicotinic acetylcholine receptor. Extensions to semi-Markov models of ion channel gating and the incorporation of time interval omission are briefly discussed.

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Clustering of Bursts of Openings in Markov and Semi-Markov Models of Single Channel Gating

Advances in Applied Probability ◽

10.2307/1427862 ◽

1997 ◽

Vol 29 (1) ◽

pp. 92-113 ◽

Cited By ~ 8

Author(s):

Frank Ball ◽

Sue Davies

Keyword(s):

Markov Chain ◽

Ion Channel ◽

State Space ◽

Continuous Time ◽

Markov Models ◽

Numerical Study ◽

Channel Gating ◽

Markov Renewal Process ◽

Time Interval ◽

Continuous Time Markov Chain

The gating mechanism of a single ion channel is usually modelled by a continuous-time Markov chain with a finite state space. The state space is partitioned into two classes, termed ‘open’ and ‘closed’, and it is possible to observe only which class the process is in. In many experiments channel openings occur in bursts. This can be modelled by partitioning the closed states further into ‘short-lived’ and ‘long-lived’ closed states, and defining a burst of openings to be a succession of open sojourns separated by closed sojourns that are entirely within the short-lived closed states. There is also evidence that bursts of openings are themselves grouped together into clusters. This clustering of bursts can be described by the ratio of the variance Var (N(t)) to the mean [N(t)] of the number of bursts of openings commencing in (0, t]. In this paper two methods of determining Var (N(t))/[N(t)] and limt→∝ Var (N(t))/[N(t)] are developed, the first via an embedded Markov renewal process and the second via an augmented continuous-time Markov chain. The theory is illustrated by a numerical study of a molecular stochastic model of the nicotinic acetylcholine receptor. Extensions to semi-Markov models of ion channel gating and the incorporation of time interval omission are briefly discussed.

Download Full-text