Skip to main content

Quasi Product Form Approximation for Markov Models of Reaction Networks

  • Conference paper

Part of the Lecture Notes in Computer Science book series (TCSB,volume 7625)

Abstract

In cell processes, such as gene regulation or cell differentiation, stochasticity often plays a crucial role. Quantitative analysis of stochastic models of the underlying chemical reaction network can be obstructed by the size of the state space which grows exponentially with the number of considered species. In a recent paper [1] we showed that the space complexity of the analysis can be drastically decreased by assuming that the transient probabilities of the model are in product form. This assumption, however, leads to approximations that are satisfactory only for a limited range of models. In this paper we relax the product form assumption by introducing the quasi product form assumption. This leads to an algorithm whose memory complexity is still reasonably low and provides a good approximation of the transient probabilities for a wide range of models. We discuss the characteristics of this algorithm and illustrate its application on several reaction networks.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Angius, A., Horváth, A.: Product form approximation of transient probabilities in stochastic reaction networks. Electronic Notes on Theoretical Computer Science 277, 3–14 (2011)

    CrossRef  Google Scholar 

  2. Arkin, A., Ross, J., McAdams, H.H.: Stochastic kinetic analysis of the developmental pathway bifurcation in phage lambda-infected escehrichia coli cells. Genetics 149(4), 1633–1648 (1998)

    Google Scholar 

  3. Bennett, D.C.: Differentiation in mouse melanoma cells: initial reversibility and an in-off stochastic model. Cell 34(2), 445–453 (1983)

    CrossRef  Google Scholar 

  4. Boucherie, R.J., Taylor, P.: Transient product form distributions in queueing networks. Discrete Event Dynamic Systems: Theory and Applications 3, 375–396 (1993)

    CrossRef  MATH  Google Scholar 

  5. Cao, Y., Gillespie, D.T., Petzold, L.R.: The slow-scale stochastic simulation algorithm. J. Chem. Phys. 122(1) (2005)

    Google Scholar 

  6. Chandy, K.M., Herzog, U., Woo, L.S.: Parametric analysis of queueing networks. IBM Journal of R. & D. 19(1), 36–42 (1975)

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Ciocchetta, F., Degasperi, A., Hillston, J., Calder, M.: Some investigations concerning the CTMC and the ode model derived from bio-pepa. Electron. Notes Theor. Comput. Sci. 229, 145–163 (2009)

    CrossRef  MathSciNet  Google Scholar 

  8. Cook, D.L., Gerber, A.N., Tapscott, S.J.: Modeling stochastic gene expression: implications for haploinsufficiency. Proc. Natl. Acad. Sci. USA 95(26), 15641–15646 (1998)

    CrossRef  Google Scholar 

  9. Cordero, F., Horváth, A., Manini, D., Napione, L., Pierro, M.D., Pavan, S., Picco, A., Veglio, A., Sereno, M., Bussolino, F., Balbo, G.: Simplification of a complex signal transduction model using invariants and flow equivalent servers. Theor. Comput. Sci. 412(43), 6036–6057 (2011)

    CrossRef  MATH  Google Scholar 

  10. Dayar, T., Mikeev, L., Wolf, V.: On the numerical analysis of stochastic Lotka-Volterra models. In: Proc. of the Workshop on Computer Aspects of Numerical Algorithms (CANA 2010), pp. 289–296 (2010)

    Google Scholar 

  11. Engblom, S.: Computing the moments of high dimensional solutions of the master equation. Appl. Math. Comput. 180, 498–515 (2006)

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81(25), 2340–2361 (1977)

    CrossRef  Google Scholar 

  13. Gillespie, D.T.: A rigorous derivation of the chemical master equation. Physica A 188(1), 404–425 (1992)

    CrossRef  MathSciNet  Google Scholar 

  14. Gillespie, D.T.: Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys. 115, 1716–1733 (2001)

    CrossRef  Google Scholar 

  15. Hasty, J., McMillen, D., Isaacs, F., Collins, J.J.: Computational studies of gene regulatory networks: in numero molecular biology. Nature Reviews Genetics 2(4), 268–279 (2001)

    CrossRef  Google Scholar 

  16. Henzinger, T.A., Mikeev, L., Mateescu, M., Wolf, V.: Hybrid numerical solution of the chemical master equation. In: CMSB, pp. 55–65 (2010)

    Google Scholar 

  17. Jensen, A.: Markoff chains as an aid in the study of Markoff processes. Skandinavisk Aktuarietidskrift 36, 87–91 (1953)

    Google Scholar 

  18. Kurtz, T.G.: Solutions of ordinary differential equations as limits of pure jump Markov processes. Journal of Applied Probability 1(7), 49–58 (1970)

    CrossRef  MathSciNet  Google Scholar 

  19. Kurtz, T.G.: The Relationship between Stochastic and Deterministic Models for Chemical Reactions. J. Chem. Phys. 57(7), 2976–2978 (1972)

    CrossRef  Google Scholar 

  20. Loinger, A., Lipshtat, A., Balaban, N.Q., Biham, O.: Stochastic simulations of genetic switch systems. Phys. Rev. E 75, 021904 (2007), http://link.aps.org/doi/10.1103/PhysRevE.75.021904

    CrossRef  Google Scholar 

  21. Mateescu, M., Wolf, V., Didier, F., Henzinger, T.A.: Fast adaptive uniformisation of the chemical master equation. IET Systems Biology 4(6), 441–452 (2010)

    CrossRef  Google Scholar 

  22. Moler, C., Loan, C.V.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Review 45(1), 3–49 (2003)

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. Rathinam, M., Petzold, L.R., Cao, Y., Gillespie, D.T.: Stiffness in stochastic chemically reacting systems: The implicit tau-leaping method. J. Chem. Phys. 119(24), 12784–12794 (2003)

    CrossRef  Google Scholar 

  24. Singh, A., Hespanha, J.P.: Moment closure techniques for stochastic models in population biology. In: American Control Conference, pp. 4730–4735 (2006)

    Google Scholar 

  25. Stewart, W.J.: Introduction to the Numerical Solution of Markov Chains. Princeton University Press (1995)

    Google Scholar 

  26. Zhang, J., Watson, L.T., Cao, Y.: Adaptive aggregation method for the chemical master equation. Int. J. of Computational Biology and Drug Design 2(2), 134–148 (2009)

    CrossRef  Google Scholar 

  27. Zhang, J., Watson, L.T., Cao, Y.: A modified uniformization method for the solution of the chemical master equation. Computers & Mathematics with Applications 59(1), 573–584 (2010)

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. Zhou, J.X., Brusch, L., Huang, S.: Predicting pancreas cell fate decisions and reprogramming with a hierarchical multi-attractor model. PLoS ONE 6(3) 6(3), 16 (2011), http://dx.plos.org/10.1371/journal.pone.0014752

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Angius, A., Horváth, A., Wolf, V. (2012). Quasi Product Form Approximation for Markov Models of Reaction Networks. In: Priami, C., Petre, I., de Vink, E. (eds) Transactions on Computational Systems Biology XIV. Lecture Notes in Computer Science(), vol 7625. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35524-0_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-35524-0_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-35523-3

  • Online ISBN: 978-3-642-35524-0

  • eBook Packages: Computer ScienceComputer Science (R0)