Hybrid method for the chemical master equation

Hellander, Andreas; Lötstedt, Per

doi:10.1016/j.jcp.2007.07.020

Cited by 72 publications

(89 citation statements)

References 37 publications

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“…This approach for reducing the CME has been proposed by Andreas Hellander and Per Lötstedt in [20] using a different derivation. Although similar models are known in other application areas (cf.…”

Section: Hellander-lötstedt Model the Number Of Degrees Of Freedom Cmentioning

confidence: 99%

“…not via stochastic simulation), and to use hybrid approaches as a model reduction technique which decreases the huge number of degrees of freedom down to a small fraction. Such approaches have been proposed, e.g., in [18,19,20,4,21]. The reduction of the problem, however, comes at the cost of a lower accuracy because in addition to the approximation error caused by solving the differential equations numerically, there is now a modeling error due to the fact that the CME is partly replaced by coarser descriptions.…”

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confidence: 99%

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On Reduced Models for the Chemical Master Equation

Jahnke¹

2011

Multiscale Model. Simul.

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Abstract. The chemical master equation plays a fundamental role for the understanding of gene regulatory networks and other discrete stochastic reaction systems. Solving this equation numerically, however, is usually extremely expensive or even impossible due to the huge size of the state space. Thus, the chemical master equation must often be replaced by a reduced model which operates with a considerably smaller number of degrees of freedom but hopefully still provides the essential information about the dynamics of the full system. We prove error bounds for two reduced models which have previously been proposed in the literature. Based on the error analysis, an alternative model reduction approach for the chemical master equation is introduced and discussed, and its advantage is illustrated by numerical examples.Key words. Chemical master equation, model reduction, hybrid models, error bounds AMS subject classifications. 34K60, 65C20, 60J27, 92D251. Introduction. Many processes in nature can be considered as reaction systems in which d ∈ N different species interact via r ∈ N reaction channels. The time evolution of such a system is usually modeled by the traditional reaction-rate equations, a set of d coupled ordinary differential equations which indicate how the concentrations of the d species change in time. This approach is simple and computationally cheap, but fails in situations where the influence of inherent stochastic noise cannot be ignored, and where certain species have to be described in terms of integer particle numbers instead of real-valued, continuous concentrations. This is the case in gene regulatory networks, viral kinetics with few infectious individuals, and many other biological systems.The chemical master equation (CME) respects the discreteness and randomness of the problem and thus provides a more accurate model. The system is considered as a random variable Z(t) which evolves according to a Markov jump process on N

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Section: Hellander-lötstedt Model the Number Of Degrees Of Freedom Cmentioning

confidence: 99%

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confidence: 99%

On Reduced Models for the Chemical Master Equation

Jahnke¹

2011

Multiscale Model. Simul.

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“…This is avoided in the equivalent formulation (17). The price to pay, however, is the fact that the linear system (17) has twice as many unknowns as (18). The doubling of the linear system can be avoided if the 2-stage Gauss-Runge-Kutta method is replaced by a singly diagonally implicit Runge-Kutta method (cf.…”

Section: Remarksmentioning

confidence: 99%

“…R 7 and R 8 model the decay of the respective species, while the expression of proteins is described by reactions R 5 and R 6 . The parameters for the reaction channels are with the initial distribution being a "discrete Gaussian" with a small variance, centered at µ = (20,18,22,5), which closely resembles a delta peak located at µ. The number of degrees of freedom in this example is 2 20 , the state space being 32×32×32×32 (d = 4).…”

Section: Extended Toggle Switchmentioning

confidence: 99%

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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“…We note that simulations of these systems are impossible even when employing massively parallel computer architectures. In order to overcome this difficulty, novel multiscale methods have been proposed, [11][12][13] which combine stochastic, microscopic, deterministic, and coarsegrained descriptions.…”

Section: Introductionmentioning

confidence: 99%

Multiresolution stochastic simulations of reaction–diffusion processes

Bayati

Chatelain

Koumoutsakos

2008

Phys. Chem. Chem. Phys.

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Stochastic simulations of reaction-diffusion processes are used extensively for the modeling of complex systems in areas ranging from biology and social sciences to ecosystems and materials processing. These processes often exhibit disparate scales that render their simulation prohibitive even for massive computational resources. The problem is resolved by introducing a novel stochastic multiresolution method that enables the efficient simulation of reaction-diffusion processes as modeled by many-particle systems. The proposed method quantifies and efficiently handles the associated stiffness in simulating the system dynamics and its computational efficiency and accuracy are demonstrated in simulations of a model problem described by the Fisher-Kolmogorov equation. The method is general and can be applied to other many-particle models of physical processes. Please check this proof carefully. Our staff will not read it in detail after you have returned it. Translation errors between word-processor files and typesetting systems can occur so the whole proof needs to be read. Please pay particular attention to: tabulated material; equations; numerical data; figures and graphics; and references. If you have not already indicated the corresponding author(s) please mark their name(s) with an asterisk. Please e-mail a list of corrections or the PDF with electronic notes attached --do not change the text within the PDF file or send a revised manuscript.Please bear in mind that minor layout improvements, e.g. in line breaking, table widths and graphic placement, are routinely applied to the final version.Please note that, in the typefaces we use, an italic vee looks like this: n, and a Greek nu looks like this: n.We will publish articles on the web as soon as possible after receiving your corrections; no late corrections will be made.Please return your final corrections, where possible within 48 hours of receipt, by e-mail to: proofs@rsc.org Reprints-Electronic (PDF) reprints will be provided free of charge to the corresponding author. Enquiries about purchasing paper reprints should be addressed via: http://www.rsc.org/Publishing/ReSourCe/PaperReprints/. Costs for reprints are below: Stochastic simulations of reaction-diffusion processes are used extensively for the modeling of complex systems in areas ranging from biology and social sciences to ecosystems and materials processing. These processes often exhibit disparate scales that render their simulation prohibitive even for massive computational resources. The problem is resolved by introducing a novel stochastic multiresolution method that enables the efficient simulation of reaction-diffusion processes as modeled by many-particle systems. The proposed method quantifies and efficiently handles the associated stiffness in simulating the system dynamics and its computational efficiency and accuracy are demonstrated in simulations of a model problem described by the Fisher-Kolmogorov equation. The method is general and can be applied to other many-particle mod...

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Hybrid method for the chemical master equation

Cited by 72 publications

References 37 publications

On Reduced Models for the Chemical Master Equation

On Reduced Models for the Chemical Master Equation

Solving Chemical Master Equations by an Adaptive Wavelet Method

Multiresolution stochastic simulations of reaction–diffusion processes

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