Approximating the solution of the chemical master equation by aggregation

Hegland, Markus

doi:10.21914/anziamj.v50i0.1426

Cited by 6 publications

(3 citation statements)

References 10 publications

(12 reference statements)

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“…[27]. We emphasize that the semigroup operators e tA are not defined via the exponential series; since A is an unbounded operator, its exponential series does not converge.…”

Section: The Chemical Master Equation As An Abstract Cauchy Problemmentioning

confidence: 99%

Efficient simulation of discrete stochastic reaction systems with a splitting method

Jahnke

Altintan

2010

Bit Numer Math

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Stochastic reaction systems with discrete particle numbers are usually described by a continuous-time Markov process. Realizations of this process can be generated with the stochastic simulation algorithm, but simulating highly reactive systems is computationally costly because the computational work scales with the number of reaction events. We present a new approach which avoids this drawback and increases the efficiency considerably at the cost of a small approximation error. The approach is based on the fact that the time-dependent probability distribution associated to the Markov process is explicitly known for monomolecular, autocatalytic and certain catalytic reaction channels. More complicated reaction systems can often be decomposed into several parts some of which can be treated analytically. These subsystems are propagated in an alternating fashion similar to a splitting method for ordinary differential equations. We illustrate this approach by numerical examples and prove an error bound for the splitting error.Keywords Stochastic simulation algorithm · discrete stochastic reaction systems · splitting methods · analytic solution formulas · error bounds · chemical master equation Mathematics Subject Classification (2000)

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“…[27]. We emphasize that the semigroup operators e tA are not defined via the exponential series; since A is an unbounded operator, its exponential series does not converge.…”

Section: The Chemical Master Equation As An Abstract Cauchy Problemmentioning

confidence: 99%

Efficient simulation of discrete stochastic reaction systems with a splitting method

Jahnke

Altintan

2010

Bit Numer Math

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“…There is an ongoing effort to tackle the chemical master equation numerically, the major challenge being its high dimensionality: for a system of d interacting species the chemical master equation is a differential equation with state space N d 0 , N 0 the set of nonnegative integers. Various numerical methods have been proposed, for instance Galerkin methods [1], spectral methods [3], sparse grid methods [7,9], wavelet methods [15,24], tensor methods [2,8,14,16], and hybrid methods [9,10,13,19]. In the papers on numerical methods known to us, no attention is paid to the regularity (or the decay) of the solutions to the chemical master equation, although such properties are inherently related to their approximability and are particularly relevant in high dimensions.…”

Section: Introductionmentioning

confidence: 99%

Regularity and approximability of the solutions to the chemical master equation

Gauckler

Yserentant

2014

ESAIM: M2AN

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The chemical master equation is a fundamental equation in chemical kinetics. It underlies the classical reaction-rate equations and takes stochastic effects into account. In this paper we give a simple argument showing that the solutions of a large class of chemical master equations are bounded in weighted 1-spaces and possess high-order moments. This class includes all equations in which no reactions between two or more already present molecules and further external reactants occur that add mass to the system. As an illustration for the implications of this kind of regularity, we analyze the effect of truncating the state space. This leads to an error analysis for the finite state projections of the chemical master equation, an approximation that forms the basis of many numerical methods.

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“…Various methods for approximate stochastic analysis have been suggested: Monte Carlo sampling of probability density functions of species' counts over time, known as Gillespie's algorithm [11]; approximations of this sampling [4,19]; explicit treatment of fluctuations with stochastic differential equations [12]; consideration of subspace of system states with highest probability mass [6,24]; aggregation of states [17]. Moment closure (MC) is a promising method for approximate analysis of the behavior of stochastic systems.…”

Section: Introductionmentioning

confidence: 99%

Symmetry-Based Model Reduction for Approximate Stochastic Analysis

Batmanov

Kuttler

Lemaire

et al. 2012

Computational Methods in Systems Biology

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Abstract. For models of cell-to-cell communication, with many reactions and species per cell, the computational cost of stochastic simulation soon becomes intractable. Deterministic methods, while computationally more efficient, may fail to contribute reliable approximations for those models. In this paper, we suggest a reduction for models of cell-to-cell communication, based on symmetries of the underlying reaction network.To carry out a stochastic analysis that otherwise comes at an excessive computational cost, we apply a moment closure (MC) approach. We illustrate with a community effect, that allows synchronization of a group of cells in animal development. Comparing the results of stochastic simulation with deterministic and MC approximation, we show the benefits of our approach. The reduction presented here is potentially applicable to a broad range of highly regular systems.

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Approximating the solution of the chemical master equation by aggregation

Cited by 6 publications

References 10 publications

Efficient simulation of discrete stochastic reaction systems with a splitting method

Efficient simulation of discrete stochastic reaction systems with a splitting method

Regularity and approximability of the solutions to the chemical master equation

Symmetry-Based Model Reduction for Approximate Stochastic Analysis

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