Solving the chemical master equation for monomolecular reaction systems analytically
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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 easily many properties of the solution. The model class includes many interesting examples. For more complex reaction systems, our results can be seen as a first step towards the construction of new numerical integrators, because solutions to the monomolecular case provide promising ansatz functions for Galerkin-type methods.
KeywordsChemical master equation Explicit solution formula Continuous-time Markov process Convergence to steady state
Mathematics Subject Classification (2000)92C45 60J25 34A05
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- 1.Alfonsi A., Cancès E., Turinici G., Ventura B.D., Huisinga W. (2005) Adaptive simulation of hybrid stochastic and deterministic models for biochemical systems. ESAIM Proc. 14, 1–13Google Scholar
- 2.Anderson D.H. (1983) Compartmental Modeling and Tracer Kinetics. Number 50 in Lecture Notes in Biomathematics. Springer, Berlin Heidelberg New YorkGoogle Scholar
- 3.Burrage, K., Tian, T.: Poisson Runge–Kutta methods for chemical reaction systems. In: Proceedings of the Hong Kong Conference on Scientific Computing, 2003 (in press).Google Scholar
- 7.Fall C.P., Marland E.S., Wagner J.M., Tyson J.J. (2002) Computational Cell Biology, volume 20 of Interdisciplinary Applied Mathematics. Springer, Berlin Heidelberg New YorkGoogle Scholar
- 9.Gardiner C.W. (1985) Handbook of Stochastic Methods. Springer, Berlin Heidelberg New York, , 2nd enlarged editionGoogle Scholar
- 13.Gillespie D.T. (2002) The chemical Langevin and Fokker–Planck equations for the reversible isomerization reaction. J. Phys. Chem. 106, 5063–5071Google Scholar
- 17.Kotz, S. Johnson, N.L., Read, C.B.: editors. Encyclopedia of Statistical sciences, vol. 5. Wiley, New York, Chichester, Brisbane, Toronto, Singapore (1985)Google Scholar
- 18.Lang S. (1987) Linear algebra. Undergraduate Texts in Mathematics, 3rd edn. Springer, Berlin Heidelberg New YorkGoogle Scholar
- 20.Salis H., Kaznessis Y. (2005) Accurate hybrid simulation of a system of coupled chemical or biochemical reactions. J. Chem. Phys. 122 Google Scholar