From Brownian Dynamics to Markov Chain: An Ion Channel Example

Wan Chen; Radek Erban; S. Jonathan Chapman

doi:10.1137/120882780

From Brownian Dynamics to Markov Chain: An Ion Channel Example

Wan Chen, Radek Erban, S. Jonathan Chapman

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

A discrete rate theory for multi-ion channels is presented, in which the continuous dynamics of ion diffusion is reduced to transitions between Markovian discrete states. In an open channel, the ion permeation process involves three types of events: an ion entering the channel, an ion escaping from the channel, or an ion hopping between different energy minima in the channel. The continuous dynamics leads to a hierarchy of Fokker-Planck equations, indexed by channel occupancy. From these the mean escape times and splitting probabilities (denoting from which side an ion has escaped) can be calculated. By equating these with the corresponding expressions from the Markov model, one can determine the Markovian transition rates. The theory is illustrated with a two-ion one-well channel. The stationary probability of states is compared with that from both Brownian dynamics simulation and the hierarchical Fokker-Planck equations. The conductivity of the channel is also studied, and the optimal geometry maximizing ion flux is computed. © 2014 Society for Industrial and Applied Mathematics.

Original language	English (US)
Pages (from-to)	208-235
Number of pages	28
Journal	SIAM Journal on Applied Mathematics
Volume	74
Issue number	1
DOIs	https://doi.org/10.1137/120882780
State	Published - Feb 27 2014
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-C1-013-04
Acknowledgements: This work was partially supported by award KUK-C1-013-04 from King Abdullah University of Science and Technology (KAUST) and by funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement 239870.The second author's work was partially supported by a Royal Society University Research Fellowship; by a Fulford Junior Research Fellowship of Somerville College, University of Oxford; by a Nicholas Kurti Junior Fellowship of Brasenose College, University of Oxford; and by a Philip Leverhulme Prize awarded by the Leverhulme Trust.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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title = "From Brownian Dynamics to Markov Chain: An Ion Channel Example",

abstract = "A discrete rate theory for multi-ion channels is presented, in which the continuous dynamics of ion diffusion is reduced to transitions between Markovian discrete states. In an open channel, the ion permeation process involves three types of events: an ion entering the channel, an ion escaping from the channel, or an ion hopping between different energy minima in the channel. The continuous dynamics leads to a hierarchy of Fokker-Planck equations, indexed by channel occupancy. From these the mean escape times and splitting probabilities (denoting from which side an ion has escaped) can be calculated. By equating these with the corresponding expressions from the Markov model, one can determine the Markovian transition rates. The theory is illustrated with a two-ion one-well channel. The stationary probability of states is compared with that from both Brownian dynamics simulation and the hierarchical Fokker-Planck equations. The conductivity of the channel is also studied, and the optimal geometry maximizing ion flux is computed. {\textcopyright} 2014 Society for Industrial and Applied Mathematics.",

author = "Wan Chen and Radek Erban and Chapman, {S. Jonathan}",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KUK-C1-013-04 Acknowledgements: This work was partially supported by award KUK-C1-013-04 from King Abdullah University of Science and Technology (KAUST) and by funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement 239870.The second author's work was partially supported by a Royal Society University Research Fellowship; by a Fulford Junior Research Fellowship of Somerville College, University of Oxford; by a Nicholas Kurti Junior Fellowship of Brasenose College, University of Oxford; and by a Philip Leverhulme Prize awarded by the Leverhulme Trust. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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PY - 2014/2/27

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N2 - A discrete rate theory for multi-ion channels is presented, in which the continuous dynamics of ion diffusion is reduced to transitions between Markovian discrete states. In an open channel, the ion permeation process involves three types of events: an ion entering the channel, an ion escaping from the channel, or an ion hopping between different energy minima in the channel. The continuous dynamics leads to a hierarchy of Fokker-Planck equations, indexed by channel occupancy. From these the mean escape times and splitting probabilities (denoting from which side an ion has escaped) can be calculated. By equating these with the corresponding expressions from the Markov model, one can determine the Markovian transition rates. The theory is illustrated with a two-ion one-well channel. The stationary probability of states is compared with that from both Brownian dynamics simulation and the hierarchical Fokker-Planck equations. The conductivity of the channel is also studied, and the optimal geometry maximizing ion flux is computed. © 2014 Society for Industrial and Applied Mathematics.

AB - A discrete rate theory for multi-ion channels is presented, in which the continuous dynamics of ion diffusion is reduced to transitions between Markovian discrete states. In an open channel, the ion permeation process involves three types of events: an ion entering the channel, an ion escaping from the channel, or an ion hopping between different energy minima in the channel. The continuous dynamics leads to a hierarchy of Fokker-Planck equations, indexed by channel occupancy. From these the mean escape times and splitting probabilities (denoting from which side an ion has escaped) can be calculated. By equating these with the corresponding expressions from the Markov model, one can determine the Markovian transition rates. The theory is illustrated with a two-ion one-well channel. The stationary probability of states is compared with that from both Brownian dynamics simulation and the hierarchical Fokker-Planck equations. The conductivity of the channel is also studied, and the optimal geometry maximizing ion flux is computed. © 2014 Society for Industrial and Applied Mathematics.

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From Brownian Dynamics to Markov Chain: An Ion Channel Example

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