Solving the chemical master equation for monomolecular reaction systems analytically

Jahnke, Tobias; Huisinga, Wilhelm

doi:10.1007/s00285-006-0034-x

Cited by 287 publications

(426 citation statements)

References 19 publications

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“…However, since the stable attractor is unique, we seek exact solutions that are of smallest possible dimension, rather than fully general solutions (for arbitrary initial data) as found in [4].…”

Section: Examples Of Markov Processes With Invariant Manifoldsmentioning

confidence: 99%

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Invariant manifold reductions for Markovian ion channel dynamics

Keener

2008

J. Math. Biol.

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We show that many Markov models of ion channel kinetics have globally attracting stable invariant manifolds, even when the Markov process is time dependent. The primary implication of this is that, since the dimension of the invariant manifold is often substantially smaller than the full master equation system, simulations of ion channel kinetics can be substantially simplified, with no approximation. We show that this applies to certain models of potassium channels, sodium channels, ryanodine receptors and IP 3 receptors. We also use this to show that the original Hodgkin-Huxley formulations of potassium channel conductance and sodium channel conductance are the exact solutions of full Markov models for these channels.

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Section: Examples Of Markov Processes With Invariant Manifoldsmentioning

confidence: 99%

“…Recently several more exact solutions have been found [4]. These include the multinomial distribution, the generalized Poisson distribution and the negative binomial distribution.…”

Section: Examples Of Markov Processes With Invariant Manifoldsmentioning

confidence: 99%

Invariant manifold reductions for Markovian ion channel dynamics

Keener

2008

J. Math. Biol.

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“…The steps 2 and 3 in the algorithm are equivalent to applying the 2-stage Gauss method to the projected CME (22). The local error of the Gauss method is bounded by…”

Section: Theorem 1 Let P N Be the Approximation Computed In The N-thmentioning

confidence: 99%

“…The purpose of this very simple example is to check the behavior of the error with respect to the tolerance selected by the user. This is made possible because the exact solution of the corresponding CME is known: all reactions are of monomolecular type, and for such systems an explicit formula has been derived in [22]. For any x ∈ N 2 , N ∈ N and any r = (r 1 , r 2 ) with r 1 , r 2 ∈ [0, 1] and r 1 + r 2 ≤ 1, the multinomial distribution M(x, N, r) is defined by…”

Section: Merging Modesmentioning

confidence: 99%

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Solving Chemical Master Equations by an Adaptive Wavelet Method

Jahnke¹,

Galan²

2008

AIP Conference Proceedings

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Solving chemical master equations numerically on a large state space is known to be a difficult problem because the huge number of unknowns is far beyond the capacity of traditional methods. We present an adaptive method which compresses the problem very efficiently by representing the solution in a sparse wavelet basis that is updated in each step. The step-size is chosen adaptively according to estimates of the temporal and spatial approximation errors. Numerical examples demonstrate the reliability of the error estimation and show that the method can solve large problems with bimodal solution profiles.

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Stochastic modeling analysis of sequential first-order degradation reactions and non-Fickian transport in steady state plumes

2014

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Stochastic analyses were performed to examine sequential first-order monomolecular reactions at the microscopic scale and both Fickian and non-Fickian plume reactive transport at the macroscopic scale. An analytical solution was derived for the chemical master equation (CME) for a closed system of irreversible first-order monomolecular reactions. Taking a Lagrangian reference frame of particles migrating from a source, analyses show that the relative concentration of each species in the deterministic analytical solution for 1-D steady state plug flow with first-order sequential degradation is mathematically equivalent to the mean of a multinomial distribution of plume particles moving at constant velocity with sequential transformations described by transition probabilities of a discrete state, continuous-time Markov chain. In order to examine the coupling of reaction and transport terms in subdiffusive-reactive transport equations, a closed-form multispecies analytical solution also was derived for steady state advection, dispersion, and sequential first-order reaction. Using a 1-D continuous-time random walk (CTRW) embedded in Markov chains, computationally efficient Monte Carlo simulations of particle movement were performed to more fully examine effects of subdiffusive-reactive transport with an application to steady state, sequentially degrading multispecies plumes at a site in Palm, Bay, FL. The simulation results indicated that non-Fickian steady state plumes can resemble Fickian plumes because linear reactions truncate the waiting time between particle jumps, which removes lower velocity particles from the broad spectrum of velocities in highly heterogeneous media. Results show that fitting of Fickian models to plume concentration data can lead to inaccurate estimates of rate constants because of the wide distribution of travel times in highly heterogeneous media.

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Solving the chemical master equation for monomolecular reaction systems analytically

Cited by 287 publications

References 19 publications

Invariant manifold reductions for Markovian ion channel dynamics

Invariant manifold reductions for Markovian ion channel dynamics

Solving Chemical Master Equations by an Adaptive Wavelet Method

Stochastic modeling analysis of sequential first-order degradation reactions and non-Fickian transport in steady state plumes

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