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Solving the chemical master equation for monomolecular reaction systems analytically

Jahnke, Tobias and Huisinga, Wilhelm (2007) Solving the chemical master equation for monomolecular reaction systems analytically. Journal of Mathematical Biology. ISSN 1432-1416

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Abstract

The stochastic dynamics of a well-stirred mixture of molecular species interacting through different biochemical reactions can be accurately modelled by the chemical master equation (CME). Research in the biology and scientific computing community has concentrated mostly on the development of numerical techniques to approximate the solution of the CME via many realizations of the associated Markov jump process. The domain of exact and/or efficient methods for directly solving the CME is still widely open, which is due to its large dimension that grows exponentially with the number of molecular species involved. In this article, we present an exact solution formula of the CME for arbitrary initial conditions in the case where the underlying system is governed by monomolecular reactions. The solution can be expressed in terms of the convolution of multinomial and product Poisson distributions with time-dependent parameters evolving according to the traditional reaction-rate equations. This very structured representation allows to deduce any property of the solution. The model class includes many interesting examples and may also be used as the starting point for the design of new numerical methods for the CME of more complex reaction systems.

Item Type: Article
Keywords: Chemical master equation; Explicit solution formula; Continuous-time Markov process; Convergence to steady state; Hamilton Institute.
Subjects: Science & Engineering > Hamilton Institute
Science & Engineering > Mathematics & Statistics
Item ID: 1729
Depositing User: Hamilton Editor
Date Deposited: 10 Dec 2009 15:11
Journal or Publication Title: Journal of Mathematical Biology
Publisher: Springer
Refereed: Yes
URI:

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