Stochastic Oscillations in Genetic Regulatory Networks: Application to Microarray Experiments

Abstract

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.

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References

  1. 1.

    Bower JM, Bolouri H (Eds): Computational Modeling of Genetic and Biochemical Networks. MIT Press, Cambridge, Mass, USA; 2001.

    Google Scholar 

  2. 2.

    Boxler P: A stochastic version of center manifold theory. Probability Theory and Related Fields 1989,83(4):509-545. 10.1007/BF01845701

    MATH  MathSciNet  Article  Google Scholar 

  3. 3.

    Bradley R: Basic properties of strong mixing conditions. A survey and some open questions. Probability Surveys 2005, 2: 107-144.

    MATH  MathSciNet  Article  Google Scholar 

  4. 4.

    Bressan A: Tutorial on the Center Manifold Theory. 2003.http://www.math.psu.edu/bressan/PSPDF SISSA, Trieste, Italy,

    Google Scholar 

  5. 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/nature04599

    Article  Google Scholar 

  6. 6.

    Carr J: Applications of Center Manifold Theory. Springer, New York, NY, USA; 1981.

    Google Scholar 

  7. 7.

    Chen F: Introduction to Plasma Physics and Controlled Fusion. Plenum Press, New York, NY, USA; 1984.

    Google Scholar 

  8. 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. 9.

    De Jong H: Modeling and simulation of genetic regulatory systems: a literature review. Journal of Computational Biology 2002,9(1):67-103. 10.1089/10665270252833208

    Article  Google Scholar 

  10. 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.1196503

    Article  Google Scholar 

  11. 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.2004.06.004

    Article  Google Scholar 

  12. 12.

    Gardiner CW: Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer, New York, NY, USA; 1983.

    Google Scholar 

  13. 13.

    Gillespie D: Exact stochastic simulation of coupled chemical reactions. The Journal of Physical Chemistry 1977,81(25):2340-2361. 10.1021/j100540a008

    Article  Google Scholar 

  14. 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.2036429100

    Article  Google Scholar 

  15. 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.2003.3153

    MathSciNet  Article  Google Scholar 

  16. 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.831000

    Article  Google Scholar 

  17. 17.

    Lewin B: Genes VIII. Prentice-Hall, Upper Saddle River, NJ, USA; 2004.

    Google Scholar 

  18. 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. 19.

    Loeve M: Probability Theory, The University Series in Higher Mathematics. Van Nostrand, New York, NY, USA; 1963.

    Google Scholar 

  20. 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;2

    Article  Google Scholar 

  21. 21.

    Lotka AJ: Elements of Physical Biology. Williams and Wilkins, Baltimore, Md, USA; 1925.

    Google Scholar 

  22. 22.

    Maquat LE: Nonsense-mediated mRNA decay in mammals. Journal of Cell Science 2005,118(9):1773-1776. 10.1242/jcs.01701

    Article  Google Scholar 

  23. 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.94.3.814

    Article  Google Scholar 

  24. 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-X

    Article  Google Scholar 

  25. 25.

    Newman M: The structure and function of complex networks. SIAM Review 2003,45(2):167-256. 10.1137/S003614450342480

    MATH  MathSciNet  Article  Google Scholar 

  26. 26.

    Parr RG, Yang W: Density Functional Theory of Atoms and Molecules. Oxford University Press, New York, NY, USA; 1989.

    Google Scholar 

  27. 27.

    Perko L: Differential Equations and Dynamical Systems. 3rd edition. Springer, New York, NY, USA; 2001.

    Google Scholar 

  28. 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.2005.052845

    Article  Google Scholar 

  29. 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.2003.1195

    Article  Google Scholar 

  30. 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.1106914

    Article  Google Scholar 

  31. 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)90035-6

    MATH  MathSciNet  Article  Google Scholar 

  32. 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)90066-7

    MATH  MathSciNet  Article  Google Scholar 

  33. 33.

    Voit E (Ed): Canonical Nonlinear Modeling. S-System Approach to Understanding Complexity. Van Norstand Reinhold, New York, NY, USA; 1991.

    Google Scholar 

  34. 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.252638199

    Article  Google Scholar 

  35. 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.2004.07.005

    MathSciNet  Article  Google Scholar 

  36. 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. 37.

    Zhang D, Gyorgyi L, Peltier WR: Deterministic chaos in the Belousov-Zhabotinsky reaction: experiments and simulations. Chaos 1993,3(4):723-745. 10.1063/1.165933

    Article  Google Scholar 

  38. 38.

    Zumdahl S: Chemical Principles. Houghton Mifflin, New York, NY, USA; 2005.

    Google Scholar 

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Correspondence to Simon Rosenfeld.

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Rosenfeld, S. Stochastic Oscillations in Genetic Regulatory Networks: Application to Microarray Experiments. J Bioinform Sys Biology 2006, 59526 (2006). https://doi.org/10.1155/BSB/2006/59526

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Keywords

  • Regulatory Network
  • System Biology
  • Microarray Experiment
  • Genetic Regulatory
  • Genetic Regulatory Network