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Adaptive Discrete Galerkin Methods Applied to the Chemical Master Equation

Please always quote using this URN: urn:nbn:de:0297-zib-9471
  • \begin{abstract} In systems biology, the stochastic description of biochemical reaction kinetics is increasingly being employed to model gene regulatory networks and signalling pathways. Mathematically speaking, such models require the numerical solution of the underlying evolution equat ion, also known as the chemical master equation (CME). Up to now, the CME has almost exclusively been treated by Monte-Carlo techniques, the most prominent of which is the simulation algorithm suggest ed by Gillespie in 1976. Since this algorithm requires an update for each single reaction event, realizations can be computationally very costly. As an alternative, we here propose a novel approach, which focuses on the discrete partial differential equation (PDE) structure of the CME and thus allows to adopt ideas from adaptive discrete Galerkin methods (as designed by two of the present authors in 1989), which have proven to be highly efficient in the mathematical modelling of polyreaction kinetics. Among the two different options of discretizing the CME as a discrete PDE, the method of lines approach (first space, then time) and the Rothe method (first time, then space), we select the latter one for clear theoretical and algorithmic reasons. First numeric al experiments at a challenging model problem illustrate the promising features of the proposed method and, at the same time, indicate lines of necessary further research. \end{abstract}

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Author:Peter Deuflhard, Wilhelm Huisinga, Tobias Jahnke, Michael Wulkow
Document Type:ZIB-Report
Tag:adaptive Rothe method; adaptive discrete Galerkin methods; chemical master equation; discrete Chebyshev polynomials; stochastic reactions kinetics
Date of first Publication:2007/02/15
Series (Serial Number):ZIB-Report (07-04)
Published in:A rev. version appeared in: SIAM J. Sci. Comput. 30, pp. 2990-3011 (2008)
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