Advertisement

Approximation of Event Probabilities in Noisy Cellular Processes

  • Frédéric Didier
  • Thomas A. Henzinger
  • Maria Mateescu
  • Verena Wolf
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5688)

Abstract

Molecular noise, which arises from the randomness of the discrete events in the cell, significantly influences fundamental biological processes. Discrete -state continuous-time stochastic models (CTMC) can be used to describe such effects, but the calculation of the probabilities of certain events is computationally expensive.

We present a comparison of two analysis approaches for CTMC. On one hand, we estimate the probabilities of interest using repeated Gillespie simulation and determine the statistical accuracy that we obtain. On the other hand, we apply a numerical reachability analysis that approximates the probability distributions of the system at several time instances. We use examples of cellular processes to demonstrate the superiority of the reachability analysis if accurate results are required.

Keywords

Markov Chain State Space Event Probability Transition Class Precision Level 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arkin, A., Ross, J., McAdams, H.H.: Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-infected E. coli cells. Genetics 149, 1633–1648 (1998)PubMedPubMedCentralGoogle Scholar
  2. 2.
    Barkai, N., Leibler, S.: Biological rhythms: Circadian clocks limited by noise. Nature 403, 267–268 (2000)PubMedGoogle Scholar
  3. 3.
    Blake, W.J., Kaern, M., Cantor, C.R., Collins, J.J.: Noise in eukaryotic gene expression. Nature 422, 633–637 (2003)CrossRefPubMedGoogle Scholar
  4. 4.
    Bremaud, P.: Markov Chains. Springer, Heidelberg (1998)Google Scholar
  5. 5.
    Burrage, K., Hegland, M., Macnamara, F., Sidje, R.: A Krylov-based finite state projection algorithm for solving the chemical master equation arising in the discrete modelling of biological systems. In: Proc. of the Markov 150th Anniversary Conference, Boson Books, pp. 21–38 (2006)Google Scholar
  6. 6.
    Elowitz, M.B., Levine, M.J., Siggia, E.D., Swain, P.S.: Stochastic gene expression in a single cell. Science 297, 1183–1186 (2002)CrossRefPubMedGoogle Scholar
  7. 7.
    Fedoroff, N., Fontana, W.: Small numbers of big molecules. Science 297, 1129–1131 (2002)CrossRefPubMedGoogle Scholar
  8. 8.
    Fox, B.L., Glynn, P.W.: Computing Poisson probabilities. Communications of the ACM 31(4), 440–445 (1988)CrossRefGoogle 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.
    Gillespie, D.T.: Markov Processes. Academic Press, New York (1992)Google Scholar
  11. 11.
    Gonze, D., Halloy, J., Goldbeter, A.: Robustness of circadian rhythms with respect to molecular noise. PNAS, USA 99(2), 673–678 (2002)CrossRefGoogle Scholar
  12. 12.
    Gonze, D., Halloy, J., Goldbeter, A.: Stochastic models for circadian oscillations: Emergence of a biological rhythm. Quantum Chemistry 98, 228–238 (2004)CrossRefGoogle Scholar
  13. 13.
    Goutsias, J.: Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems. J. Chem. Phys. 122(18), 184102 (2005)CrossRefPubMedGoogle Scholar
  14. 14.
    Hasty, J., Pradines, J., Dolnik, M., Collins, J.J.: Noise-based switches and amplifiers for gene expression. PNAS, USA 97, 2075 (2000)CrossRefGoogle Scholar
  15. 15.
    Hellander, A.: Efficient computation of transient solutions of the chemical master equation based on uniformization and quasi-Monte carlo. J. Chem. Phys. 128(15), 154109 (2008)CrossRefPubMedGoogle Scholar
  16. 16.
    Henderson, D.A., Boys, R.J., Proctor, C.J., Wilkinson, D.J.: Linking systems biology models to data: a stochastic kinetic model of p53 oscillations. In: O’Hagan, A., West, M. (eds.) Handbook of Applied Bayesian Analysis. Oxford University Press, Oxford (2009)Google Scholar
  17. 17.
    Henzinger, T., Mateescu, M., Wolf, V.: Sliding window abstraction for infinite Markov chains. In: Proc. CAV. LNCS. Springer, Heidelberg (to appear, 2009)Google Scholar
  18. 18.
    van Kampen, N.G.: Stochastic Processes in Physics and Chemistry, 3rd edn. Elsevier, Amsterdam (2007)Google Scholar
  19. 19.
    Kierzek, A., Zaim, J., Zielenkiewicz, P.: The effect of transcription and translation initiation frequencies on the stochastic fluctuations in prokaryotic gene expression. Journal of Biological Chemistry 276(11), 8165–8172 (2001)CrossRefPubMedGoogle Scholar
  20. 20.
    Law, A., Kelton, D.: Simulation Modelling and Analysis. McGraw-Hill Education, New York (2000)Google Scholar
  21. 21.
    Little, J.W., Shepley, D.P., Wert, D.W.: Robustness of a gene regulatory circuit. The EMBO Journal 18(15), 4299–4307 (1999)CrossRefPubMedPubMedCentralGoogle Scholar
  22. 22.
    Losick, R., Desplan, C.: Stochasticity and Cell Fate. Science 320(5872), 65–68 (2008)CrossRefPubMedPubMedCentralGoogle Scholar
  23. 23.
    Maamar, H., Raj, A., Dubnau, D.: Noise in gene expression determines cell fate in Bacillus subtilis. Science 317(5837), 526–529 (2007)CrossRefPubMedGoogle Scholar
  24. 24.
    McAdams, H.H., Arkin, A.: Stochastic mechanisms in gene expression. PNAS, USA 94, 814–819 (1997)CrossRefGoogle Scholar
  25. 25.
    McAdams, H.H., Arkin, A.: It’s a noisy business! Trends in Genetics 15(2), 65–69 (1999)CrossRefPubMedGoogle Scholar
  26. 26.
    Munsky, B., Khammash, M.: The finite state projection algorithm for the solution of the chemical master equation. J. Chem. Phys. 124, 044144 (2006)CrossRefGoogle Scholar
  27. 27.
    Ozbudak, E.M., Thattai, M., Kurtser, I., Grossman, A.D., van Oudenaarden, A.: Regulation of noise in the expression of a single gene. Nature Genetics 31(1), 69–73 (2002)CrossRefPubMedGoogle Scholar
  28. 28.
    Patel, P., Arcangioli, B., Baker, S., Bensimon, A., Rhind, N.: DNA replication origins fire stochastically in fission yeast. Mol. Biol. Cell 17, 308–316 (2006)CrossRefPubMedPubMedCentralGoogle Scholar
  29. 29.
    Paulsson, J.: Summing up the noise in gene networks. Nature 427(6973), 415–418 (2004)CrossRefPubMedGoogle Scholar
  30. 30.
    Rao, C., Wolf, D., Arkin, A.: Control, exploitation and tolerance of intracellular noise. Nature 420(6912), 231–237 (2002)CrossRefPubMedGoogle Scholar
  31. 31.
    Sandmann, W.: Stochastic simulation of biochemical systems via discrete-time conversion. In: Proceedings of the 2nd Conference on Foundations of Systems Biology in Engineering, pp. 267–272. Fraunhofer IRB Verlag (2007)Google Scholar
  32. 32.
    Sandmann, W., Maier, C.: On the statistical accuracy of stochastic simulation algorithms implemented in Dizzy. In: Proc. WCSB, pp. 153–156 (2008)Google Scholar
  33. 33.
    Sandmann, W., Wolf, V.: A computational stochastic modeling formalism for biological networks. Enformatika Transactions on Engineering, Computing and Technology 14, 132–137 (2006)Google Scholar
  34. 34.
    Sandmann, W., Wolf, V.: Computational probability for systems biology. In: Fisher, J. (ed.) FMSB 2008. LNCS (LNBI), vol. 5054, pp. 33–47. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  35. 35.
    Sidje, R., Burrage, K., MacNamara, S.: Inexact uniformization method for computing transient distributions of Markov chains. SIAM J. Sci. Comput. 29(6), 2562–2580 (2007)CrossRefGoogle Scholar
  36. 36.
    Srivastava, R., You, L., Summers, J., Yin, J.: Stochastic vs. deterministic modeling of intracellular viral kinetics. Journal of Theoretical Biology 218, 309–321 (2002)CrossRefPubMedGoogle Scholar
  37. 37.
    Stewart, W.J.: Introduction to the Numerical Solution of Markov Chains. Princeton University Press, Princeton (1995)Google Scholar
  38. 38.
    Swain, P.S., Elowitz, M.B., Siggia, E.D.: Intrinsic and extrinsic contributions to stochasticity in gene expression. PNAS, USA 99(20), 12795–12800 (2002)CrossRefGoogle Scholar
  39. 39.
    Thattai, M., van Oudenaarden, A.: Intrinsic noise in gene regulatory networks. PNAS, USA 98(15), 8614–8619 (2001)CrossRefGoogle Scholar
  40. 40.
    Turner, T.E., Schnell, S., Burrage, K.: Stochastic approaches for modelling in vivo reactions. Computational Biology and Chemistry 28, 165–178 (2004)CrossRefPubMedGoogle Scholar
  41. 41.
    van Moorsel, A., Sanders, W.: Adaptive uniformization. ORSA Communications in Statistics: Stochastic Models 10(3), 619–648 (1994)Google Scholar
  42. 42.
    Warmflash, A., Dinner, A.: Signatures of combinatorial regulation in intrinsic biological noise. PNAS 105(45), 17262–17267 (2008)CrossRefPubMedPubMedCentralGoogle Scholar
  43. 43.
    Wilkinson, D.J.: Stochastic Modelling for Systems Biology. Chapman & Hall, Boca Raton (2006)Google Scholar
  44. 44.
    Zhang, J., Watson, L.T., Cao, Y.: A modified uniformization method for the solution of the chemical master equation. TR-07-31, Computer Science, Virginia Tech. (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Frédéric Didier
    • 1
  • Thomas A. Henzinger
    • 1
  • Maria Mateescu
    • 1
  • Verena Wolf
    • 1
    • 2
  1. 1.EPFLSwitzerland
  2. 2.Saarland UniversityGermany

Personalised recommendations