Open Access

Stochastic Oscillations in Genetic Regulatory Networks: Application to Microarray Experiments

EURASIP Journal on Bioinformatics and Systems Biology20062006:59526

DOI: 10.1155/BSB/2006/59526

Received: 19 January 2006

Accepted: 27 June 2006

Published: 3 October 2006


We analyze the stochastic dynamics of genetic regulatory networks using a system of nonlinear differential equations. The system of -functions is applied to capture the role of RNA polymerase in the transcription-translation mechanism. Using probabilistic properties of chemical rate equations, we derive a system of stochastic differential equations which are analytically tractable despite the high dimension of the regulatory network. Using stationary solutions of these equations, we explain the apparently paradoxical results of some recent time-course microarray experiments where mRNA transcription levels are found to only weakly correlate with the corresponding transcription rates. Combining analytical and simulation approaches, we determine the set of relationships between the size of the regulatory network, its structural complexity, chemical variability, and spectrum of oscillations. In particular, we show that temporal variability of chemical constituents may decrease while complexity of the network is increasing. This finding provides an insight into the nature of "functional determinism" of such an inherently stochastic system as genetic regulatory network.

[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38]

Authors’ Affiliations

Division of Cancer Prevention, Biometry Research Group, National Cancer Institute


  1. Bower JM, Bolouri H (Eds): Computational Modeling of Genetic and Biochemical Networks. MIT Press, Cambridge, Mass, USA; 2001.Google Scholar
  2. Boxler P: A stochastic version of center manifold theory. Probability Theory and Related Fields 1989,83(4):509-545. 10.1007/BF01845701MATHMathSciNetView ArticleGoogle Scholar
  3. Bradley R: Basic properties of strong mixing conditions. A survey and some open questions. Probability Surveys 2005, 2: 107-144.MATHMathSciNetView ArticleGoogle Scholar
  4. Bressan A: Tutorial on the Center Manifold Theory. 2003. SISSA, Trieste, Italy,Google Scholar
  5. Cai L, Friedman N, Xie XS: Stochastic protein expression in individual cells at the single molecule level. Nature 2006,440(7082):358-362. 10.1038/nature04599View ArticleGoogle Scholar
  6. Carr J: Applications of Center Manifold Theory. Springer, New York, NY, USA; 1981.View ArticleGoogle Scholar
  7. Chen F: Introduction to Plasma Physics and Controlled Fusion. Plenum Press, New York, NY, USA; 1984.View ArticleGoogle Scholar
  8. Chen T, He HL, Church GM: Modeling gene expression with differential equations. Pacific Symposium on Biocomputing (PSB '99), Mauna Lani, Hawaii, USA, January 1999 29-40.Google Scholar
  9. De Jong H: Modeling and simulation of genetic regulatory systems: a literature review. Journal of Computational Biology 2002,9(1):67-103. 10.1089/10665270252833208View ArticleGoogle Scholar
  10. Elf J, Ehrenberg M: Fast evaluation of fluctuations in biochemical networks with the linear noise approximation. Genome Research 2003,13(11):2475-2484. 10.1101/gr.1196503View ArticleGoogle Scholar
  11. García-Martínez J, Aranda A, Pérez-Ortín JE: Genomic run-on evaluates transcription rates for all yeast genes and identifies gene regulatory mechanisms. Molecular Cell 2004,15(2):303-313. 10.1016/j.molcel.2004.06.004View ArticleGoogle Scholar
  12. Gardiner CW: Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer, New York, NY, USA; 1983.MATHView ArticleGoogle Scholar
  13. Gillespie D: Exact stochastic simulation of coupled chemical reactions. The Journal of Physical Chemistry 1977,81(25):2340-2361. 10.1021/j100540a008View ArticleGoogle Scholar
  14. Kauffman S, Peterson C, Samuelsson B, Troein C: Random Boolean network models and the yeast transcriptional network. Proceedings of the National Academy of Sciences of the United States of America 2003,100(25):14796-14799. 10.1073/pnas.2036429100View ArticleGoogle Scholar
  15. Kim JT, Martinetz T, Polani D: Bioinformatic principles underlying the information content of transcription factor binding sites. Journal of Theoretical Biology 2003,220(4):529-544. 10.1006/jtbi.2003.3153MathSciNetView ArticleGoogle Scholar
  16. Lemon B, Tjian R: Orchestrated response: a symphony of transcription factors for gene control. Genes & Development 2000,14(20):2551-2569. 10.1101/gad.831000View ArticleGoogle Scholar
  17. Lewin B: Genes VIII. Prentice-Hall, Upper Saddle River, NJ, USA; 2004.Google Scholar
  18. Lewis D: A qualitative analysis of S-systems: Hopf bifurcation. In Canonical Nonlinear Modeling. S-System Approach to Understanding Complexity. Edited by: Voit E. Van Nostrand Reinhold, New York, NY, USA; 1991:304-344.Google Scholar
  19. Loeve M: Probability Theory, The University Series in Higher Mathematics. Van Nostrand, New York, NY, USA; 1963.Google Scholar
  20. Lorenz EN: Deterministic nonperiodic flow. Journal of the Atmospheric Sciences 1963,20(2):130-141. 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2View ArticleGoogle Scholar
  21. Lotka AJ: Elements of Physical Biology. Williams and Wilkins, Baltimore, Md, USA; 1925.MATHGoogle Scholar
  22. Maquat LE: Nonsense-mediated mRNA decay in mammals. Journal of Cell Science 2005,118(9):1773-1776. 10.1242/jcs.01701View ArticleGoogle Scholar
  23. McAdams HH, Arkin A: Stochastic mechanisms in gene expression. Proceedings of the National Academy of Sciences of the United States of America 1997,94(3):814-819. 10.1073/pnas.94.3.814View ArticleGoogle Scholar
  24. McAdams HH, Arkin A: It's a noisy business! Genetic regulation at the nanomolar scale. Trends in Genetics 1999,15(2):65-69. 10.1016/S0168-9525(98)01659-XView ArticleGoogle Scholar
  25. Newman M: The structure and function of complex networks. SIAM Review 2003,45(2):167-256. 10.1137/S003614450342480MATHMathSciNetView ArticleGoogle Scholar
  26. Parr RG, Yang W: Density Functional Theory of Atoms and Molecules. Oxford University Press, New York, NY, USA; 1989.Google Scholar
  27. Perko L: Differential Equations and Dynamical Systems. 3rd edition. Springer, New York, NY, USA; 2001.MATHView ArticleGoogle Scholar
  28. Peytavi R, Raymond FR, Gagné D, et al.:Microfluidic device for rapid ( min) automated microarray hybridization. Clinical Chemistry 2005,51(10):1836-1844. 10.1373/clinchem.2005.052845View ArticleGoogle Scholar
  29. Ptashne M: Regulated recruitment and cooperativity in the design of biological regulatory systems. Philosophical Transactions of the Royal Society A 2003,361(1807):1223-1234. 10.1098/rsta.2003.1195View ArticleGoogle Scholar
  30. Rosenfeld N, Young JW, Alon U, Swain PS, Elowitz MB: Gene regulation at the single-cell level. Science 2005,307(5717):1962-1965. 10.1126/science.1106914View ArticleGoogle Scholar
  31. Savageau M, Voit E: Recasting nonlinear differential equations as S-systems: a canonical nonlinear form. Mathematical Biosciences 1987, 87: 83-115. 10.1016/0025-5564(87)90035-6MATHMathSciNetView ArticleGoogle Scholar
  32. Sorribas A, Savageau MA: Strategies for representing metabolic pathways within biochemical systems theory: reversible pathways. Mathematical Biosciences 1989,94(2):239-269. 10.1016/0025-5564(89)90066-7MATHMathSciNetView ArticleGoogle Scholar
  33. Voit E (Ed): Canonical Nonlinear Modeling. S-System Approach to Understanding Complexity. Van Norstand Reinhold, New York, NY, USA; 1991.MATHGoogle Scholar
  34. Wang W, Cherry JM, Botstein D, Li H: A systematic approach to reconstructing transcription networks in Saccharomyces cerevisiae. Proceedings of the National Academy of Sciences of the United States of America 2002,99(26):16893-16898. 10.1073/pnas.252638199View ArticleGoogle Scholar
  35. Wang R, Jing Z, Chen L: Modelling periodic oscillation in gene regulatory networks by cyclic feedback systems. Bulletin of Mathematical Biology 2005,67(2):339-367. 10.1016/j.bulm.2004.07.005MathSciNetView ArticleGoogle Scholar
  36. Wuensche A: Genomic regulation modeled as a network with basins of attraction. Pacific Symposium on Biocomputing (PSB '98), Maui, Hawaii, USA, January 1998 3: 89-102.Google Scholar
  37. Zhang D, Gyorgyi L, Peltier WR: Deterministic chaos in the Belousov-Zhabotinsky reaction: experiments and simulations. Chaos 1993,3(4):723-745. 10.1063/1.165933View ArticleGoogle Scholar
  38. Zumdahl S: Chemical Principles. Houghton Mifflin, New York, NY, USA; 2005.Google Scholar


© Simon Rosenfeld. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.