Numerical method for coupling the macro and meso scales in stochastic chemical kinetics

Ferm, Lars; Lötstedt, Per

doi:10.1007/s10543-007-0150-z

Cited by 7 publications

(6 citation statements)

References 27 publications

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“…For δ r = 0.08 where the reaction rate equations reach a steady state, see Fig. 6, while the solution computed with the hybrid solver continues to produce reliable oscillations in agreement with the numerical experiments in [5,7] and the conclusions in [34]. However, with the splitting used in the above example, small time steps are needed in order to ensure oscillations.…”

Section: Circadian Rhythm Modelsupporting

confidence: 57%

Hybrid method for the chemical master equation

Hellander

Lötstedt

2007

Journal of Computational Physics

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The chemical master equation is solved by a hybrid method coupling a macroscopic, deterministic description with a mesoscopic, stochastic model. The molecular species are divided into one subset where the expected values of the number of molecules are computed and one subset with species with a stochastic variation in the number of molecules. The macroscopic equations resemble the reaction rate equations and the probability distribution for the stochastic variables satisfy a master equation. The probability distribution is obtained by the Stochastic Simulation Algorithm due to Gillespie. The equations are coupled via a summation over the mesoscale variables. This summation is approximated by Monte Carlo and Quasi Monte Carlo methods. The error in the approximations is analyzed. The hybrid method is applied to three chemical systems from molecular cell biology.

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Section: Circadian Rhythm Modelsupporting

confidence: 57%

Hybrid method for the chemical master equation

Hellander

Lötstedt

2007

Journal of Computational Physics

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“…We remark, however, that model reduction based on conditional moments is not a new idea. In fact, our approach (5.8)-(5.9) is closely related to a corresponding model reduction approach proposed in [18,19] for the Fokker-Planck equation, and to similar techniques in the context of polymerization kinetics (cf. [31]).…”

Section: Model Reduction Based On Conditional Expectations (Mrce) Thmentioning

confidence: 89%

“…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%

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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“…In terms of probabilities, the chemical reaction i with a stoichiometric vector z i increases the probability of state x + z i by some amount and decreases the probability of state x by the same amount. Chemical reactions thus define fluxes of probabilities j i and one gets the following differential equations for a system with r reactions [4]:…”

Section: The Chemical Master Equationmentioning

confidence: 99%

Approximating the solution of the chemical master equation by aggregation

Hegland¹

2008

ANZIAMJ

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The chemical master equation is a continuous time discrete space Markov model of chemical reactions. The chemical master equation is derived mathematically and it is shown that the corresponding initial value problem has a unique solution. Conditions are given under which this solution is a probability distribution. We present finite state and aggregation-disaggregation approximations and provide error bounds for the case of piecewise constant disaggregation. The aggregation-disaggregation approximation allows the solution of the chemical master equation for larger state spaces and is also an important tool for the solution of multidimensional problems.

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Numerical method for coupling the macro and meso scales in stochastic chemical kinetics

Cited by 7 publications

References 27 publications

Hybrid method for the chemical master equation

Hybrid method for the chemical master equation

On Reduced Models for the Chemical Master Equation

Approximating the solution of the chemical master equation by aggregation

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