Efficient Stochastic Simulation of Systems with Multiple Time Scales via Statistical Abstraction

  • Luca Bortolussi
  • Dimitrios Milios
  • Guido Sanguinetti
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9308)


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.


Rate Expectation Slow Variable Fast Variable Fast Subsystem Gaussian Process Regression 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Bortolussi, L., Milios, D., Sanguinetti, G.: Smoothed model checking for uncertain continuous time Markov chains. CoRR ArXiv, 1402.1450 (2014)Google Scholar
  2. 2.
    Bortolussi, L., Paškauskas, R.: Mean-field approximation and quasi-equilibrium reduction of markov population models (2014)Google Scholar
  3. 3.
    Bruna, M., Chapman, S.J., Smith, M.J.: Model reduction for slowfast stochastic systems with metastable behaviour. J. Chem. Phys. 140(17), 174107 (2014)CrossRefGoogle Scholar
  4. 4.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cao, Y., Gillespie, D.T., Petzold, L.R.: The slow-scale stochastic simulation algorithm. J. Chem. Phys. 122(1), 14116 (2005)CrossRefGoogle Scholar
  6. 6.
    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)CrossRefGoogle Scholar
  7. 7.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Durrett, R.: Essentials of Stochastic Processes. Springer, New York (2012)CrossRefzbMATHGoogle Scholar
  9. 9.
    Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81(25), 2340–2361 (1977)CrossRefGoogle Scholar
  10. 10.
    Goutsias, J.: Quasi-equilibrium approximation of fast reaction kinetics in stochastic biochemical systems. J. Chem. Phys. 122(18), 184102 (2005)CrossRefGoogle Scholar
  11. 11.
    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)CrossRefGoogle Scholar
  12. 12.
    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)CrossRefGoogle Scholar
  13. 13.
    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)CrossRefGoogle Scholar
  14. 14.
    Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. MIT Press, Cambridge (2006)zbMATHGoogle Scholar
  15. 15.
    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)CrossRefGoogle Scholar
  16. 16.
    Zechner, C., Koeppl, H.: Uncoupled analysis of stochastic reaction networks in fluctuating environments. PLoS Comp. Bio. 10(12), e1003942 (2014)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Luca Bortolussi
    • 1
    • 2
    • 3
  • Dimitrios Milios
    • 4
  • Guido Sanguinetti
    • 4
    • 5
  1. 1.Modelling and Simulation GroupUniversity of SaarlandSaarbrückenGermany
  2. 2.Department of Mathematics and GeosciencesUniversity of TriesteTriesteItaly
  3. 3.CNR/ISTIPisaItaly
  4. 4.School of InformaticsUniversity of EdinburghEdinburghScotland, UK
  5. 5.SynthSys, Centre for Synthetic and Systems BiologyUniversity of EdinburghEdinburghScotland, UK

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