Superposition of Spatially Interacting Aggregated Continuous Time Markov Chains |
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Authors: | Frank Ball Geoffrey Yeo |
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Institution: | (1) Mathematics and Statistics, DSE, Murdoch University, Murdoch, 6150, Australia |
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Abstract: | A system s{ X(t)} = {X
1(t),X
2(t),..., X
N(t)} of N interacting time reversible continuous time Markov chains is considered. The state space of each of the processes {X
i(t)} (i = 1, 2,...,N) is partitioned into two aggregates. Interaction between the processes {X
i(t)},{X
2(t)},...,{X
N(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 {X
i(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. |
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Keywords: | ion channel modeling spatial process time reversibility equilibrium behavior sojourn time distributions |
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