Journal of Mathematical Biology

, Volume 69, Issue 3, pp 687–735

Method of conditional moments (MCM) for the Chemical Master Equation

A unified framework for the method of moments and hybrid stochastic-deterministic models
  • J. Hasenauer
  • V. Wolf
  • A. Kazeroonian
  • F. J. Theis
Article

DOI: 10.1007/s00285-013-0711-5

Cite this article as:
Hasenauer, J., Wolf, V., Kazeroonian, A. et al. J. Math. Biol. (2014) 69: 687. doi:10.1007/s00285-013-0711-5

Abstract

The time-evolution of continuous-time discrete-state biochemical processes is governed by the Chemical Master Equation (CME), which describes the probability of the molecular counts of each chemical species. As the corresponding number of discrete states is, for most processes, large, a direct numerical simulation of the CME is in general infeasible. In this paper we introduce the method of conditional moments (MCM), a novel approximation method for the solution of the CME. The MCM employs a discrete stochastic description for low-copy number species and a moment-based description for medium/high-copy number species. The moments of the medium/high-copy number species are conditioned on the state of the low abundance species, which allows us to capture complex correlation structures arising, e.g., for multi-attractor and oscillatory systems. We prove that the MCM provides a generalization of previous approximations of the CME based on hybrid modeling and moment-based methods. Furthermore, it improves upon these existing methods, as we illustrate using a model for the dynamics of stochastic single-gene expression. This application example shows that due to the more general structure, the MCM allows for the approximation of multi-modal distributions.

Keywords

Continuous-time discrete-state Markov process Chemical Master Equation Method of moments Hybrid stochastic-determinstic models Differential algebraic equations  Gene expression 

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • J. Hasenauer
    • 1
    • 2
  • V. Wolf
    • 3
  • A. Kazeroonian
    • 1
  • F. J. Theis
    • 1
    • 2
  1. 1.Institute of Computational BiologyHelmholtz Zentrum MünchenNeuherbergGermany
  2. 2.Department of MathematicsUniversity of Technology MunichGarchingGermany
  3. 3.Department of Computer ScienceSaarland UniversitySaarbrückenGermany

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