Using Virtual Reaction Scheme in the Multiple Time Scales System for Stochastic Modelling

Chin Foon Khoo


A virtual reaction scheme and the notion of quasi-independent is proposed in this paper to assist in model partitioning for the biochemical reactions with multiple time scales. Subsequently, the singular perturbation method is applied to the stochastic model, in particular the chemical master equation of the multiple time scales system to reduce the model dimension. As a result, a lower dimensional approximation for the chemical master equation is derived through this approach. Therefore, the high dimensional chemical master equation can be solved with a lower computational cost.


Multiple time scales system; Singular perturbation method; Stochastic modelling; Virtual reaction scheme.

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E. L. Haseltine and J. B. Rawlings, Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics, Journal of Chemical Physics, 117, 2002, 6959-6969.

C. Rao and A. Arkin, Stochastic chemical kinetics and the quasi-steady-state assumption: application to the Gillespie algorithm, Journal of Chemical Physics, 118, 2003, 4999-5010.

Y. Cao, D. T. Gillespie and L. R. Petzold, The slow-scale stochastic simulation algorithm, Journal of Chemical Physics, 122, 2005, 014116-1-18.

J. Goutsias, Quasi equilibrium approximation of fast reaction kinetics in stochastic biochemical systems, Journal of Chemical Physics, 122, 2005, 184102-1-15.

C. H. Lee and R. Lui, A reduction method for multiple time scale stochastic reaction networks, Journal of Mathematical Chemistry, 46, 2009, 1292–132.

X. Kan, C. H. Lee and H. G. Othmer, A multi-time-scale analysis of chemical reaction networks: II. Stochastic systems, Journal of Mathematical Biology, 73, 2016, 1081–1129.

P. Lecca, F. Bagagiolo and M. Scarpa, Hybrid deterministic/stochastic simulation of complex biochemical systems, Molecular Biosystems, 3(12), 2017, 2672-2686.

J. K. Kim and E. D. Sontag, Reduction of multiscale stochastic biochemical reaction networks using exact moment derivation, PLoS Computational Biology, 13(6), 2017, S1-S7.

C. F. Khoo and M. Hegland, The total quasi-steady state assumption: its justification and its application to the chemical master equation, ANZIAM Journal (CTAC2008), 50, 2008, C429-C443.

A. E. Mastny, E. L. Haseltine and J. L. Rawlings, Two classes of quasi-steady-state model reductions for stochastic kinetics, Journal of Chemical Physics, 127, 2007, 094106-1-16.

M. Hegland, C. Burden, L. Santoso, S. MacNamara and H. Booth, A solver for the stochastic master equation applied to gene regulatory network, Journal of Computational and Applied Mathematics, 205, 2006, 1-17.

J. Garcke, Sparse grids in a nutshell, in Sparse Grids and Applications, J. Garcke and M. Griebel (eds), Berlin: Springer, 88, 2012, 57-80

B. Munsky and M. Khammash, The finite state projection algorithm for the solution of chemical master equation, Journal of Chemical Physics, 124, 2006, 044104-1-13.


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