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Error Bound for Hybrid Models of Two-Scaled Stochastic Reaction Systems

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Extraction of Quantifiable Information from Complex Systems

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 102))

Abstract

Biochemical reaction systems are often modeled by a Markov jump process in order to account for the discreteness of the populations and the stochastic nature of their evolution. The associated time-dependent probability distribution is the solution of the Chemical Master Equation (CME), but solving the CME numerically is a considerable challenge due to the high dimension of the state space. In many applications, however, species with rather small population numbers interact with abundant species, and only the former group exhibits stochastic behavior. This has motivated the derivation of hybrid models where a low-dimensional CME is coupled to a set of ordinary differential equations representing the abundant species. Using such a hybrid model decreases the number of unknowns significantly but – in addition to the numerical error – causes a modeling error. We investigate the accuracy of the MRCE (= model reduction based on conditional expectations) approach with respect to a particular scaling of the reaction system and prove that the error is proportional to the scaling parameter.

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Notes

  1. 1.

    We do not make this dependency explicit in the notation in order to keep the equations as simple as possible.

  2. 2.

    The remainder term R j is not to be mixed up with the reaction channel R j in (15.1).

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Authors and Affiliations

  1. Karlsruhe Institute of Technology, Kaiserstr. 89-93, 76133, Karlsruhe, Germany

    Tobias Jahnke & Vikram Sunkara

Authors
  1. Tobias Jahnke
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  2. Vikram Sunkara
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Corresponding author

Correspondence to Tobias Jahnke .

Editor information

Editors and Affiliations

  1. FB 12 Mathematik und Informatik, Philipps-Universität Marburg, Marburg, Germany

    Stephan Dahlke

  2. Inst. Geometrie & Praktische Mathemathik, RWTH Aachen University, Aachen, Germany

    Wolfgang Dahmen

  3. Universität Bonn Institut für Numerische Simulation, Bonn, Germany

    Michael Griebel

  4. Max-Planck-Inst. f. Mathem. in d. Naturwissenschaften, Leipzig, Sachsen, Germany

    Wolfgang Hackbusch

  5. Fachbereich Mathematik, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Klaus Ritter

  6. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Reinhold Schneider

  7. Seminar für Angewandte Mathematik, ETH Zürich, Zürich, Switzerland

    Christoph Schwab

  8. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Harry Yserentant

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Jahnke, T., Sunkara, V. (2014). Error Bound for Hybrid Models of Two-Scaled Stochastic Reaction Systems. In: Dahlke, S., et al. Extraction of Quantifiable Information from Complex Systems. Lecture Notes in Computational Science and Engineering, vol 102. Springer, Cham. https://doi.org/10.1007/978-3-319-08159-5_15

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