Abstract
Stiffness in chemical reaction systems is a frequently encountered computational problem, arising when different reactions in the system take place at different time-scales. Computational savings can be obtained under time-scale separation. Assuming that the system can be partitioned into slow- and fast- equilibrating subsystems, it is then possible to efficiently simulate the slow subsystem only, provided that the corresponding kinetic laws have been modified so that they reflect their dependency on the fast system. We show that the rate expectation with respect to the fast subsystem’s steady-state is a continuous function of the state of the slow system. We exploit this result to construct an analytic representation of the modified rate functions via statistical modelling, which can be used to simulate the slow system in isolation. The computational savings of our approach are demonstrated in a number of non-trivial examples of stiff systems.
L. Bortolussi is partially supported by EU-FET project QUANTICOL (nr. 600708), by FRA-UniTS, and the German Research Council (DFG) as part of the Cluster of Excellence on Multimodal Computing and Interaction at Saarland University. D.M. and G.S. are supported by the ERC under grant MLCS 306999.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This is a technically sound operation, as the fast subsystem has a unique steady state distribution, depending only on the state \({{\varvec{z}}}\) of the slow subsystem, which is reached immediately after the firing of a slow reaction.
- 2.
GP regression typically involves matrix inversion, but this can be avoided as we make no use of predictive variances.
References
Bortolussi, L., Milios, D., Sanguinetti, G.: Smoothed model checking for uncertain continuous time Markov chains. CoRR ArXiv, 1402.1450 (2014)
Bortolussi, L., Paškauskas, R.: Mean-field approximation and quasi-equilibrium reduction of markov population models (2014)
Bruna, M., Chapman, S.J., Smith, M.J.: Model reduction for slowfast stochastic systems with metastable behaviour. J. Chem. Phys. 140(17), 174107 (2014)
Cao, Y., Gillespie, D.T., Petzold, L.: Multiscale stochastic simulation algorithm with stochastic partial equilibrium assumption for chemically reacting systems. J. Comput. Phys. 206(2), 395–411 (2005)
Cao, Y., Gillespie, D.T., Petzold, L.R.: The slow-scale stochastic simulation algorithm. J. Chem. Phys. 122(1), 14116 (2005)
Cao, Y., Gillespie, D.T., Petzold, L.R.: Accelerated stochastic simulation of the stiff enzyme-substrate reaction. J. Chem. Phys. 123(14), 144917–12 (2005)
Cao, Y., Petzold, L.: Accuracy limitations and the measurement of errors in the stochastic simulation of chemically reacting systems. J. Comput. Phys. 212(1), 6–24 (2006)
Durrett, R.: Essentials of Stochastic Processes. Springer, New York (2012)
Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81(25), 2340–2361 (1977)
Goutsias, J.: Quasi-equilibrium approximation of fast reaction kinetics in stochastic biochemical systems. J. Chem. Phys. 122(18), 184102 (2005)
Haseltine, E.L., Rawlings, J.B.: Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics. J. Chem. Phys. 117(15), 6959 (2002)
Mari, L., Bertuzzo, E., Righetto, L., Casagrandi, R., Gatto, M., Rodriguez-Iturbe, I., Rinaldo, A.: Modelling cholera epidemics: the role of waterways, human mobility and sanitation. J. R. Soc. Interface 9(67), 376–388 (2011)
Rao, C.V., Arkin, A.P.: Stochastic chemical kinetics and the quasi-steady-state assumption: Application to the Gillespie algorithm. J. Chem. Phys. 118(11), 4999 (2003)
Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. MIT Press, Cambridge (2006)
Weinan, E., Liu, D., Vanden-Eijnden, E.: Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J. Chem. Phys. 123(19), 194107 (2005)
Zechner, C., Koeppl, H.: Uncoupled analysis of stochastic reaction networks in fluctuating environments. PLoS Comp. Bio. 10(12), e1003942 (2014)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Bortolussi, L., Milios, D., Sanguinetti, G. (2015). Efficient Stochastic Simulation of Systems with Multiple Time Scales via Statistical Abstraction. In: Roux, O., Bourdon, J. (eds) Computational Methods in Systems Biology. CMSB 2015. Lecture Notes in Computer Science(), vol 9308. Springer, Cham. https://doi.org/10.1007/978-3-319-23401-4_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-23401-4_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-23400-7
Online ISBN: 978-3-319-23401-4
eBook Packages: Computer ScienceComputer Science (R0)