RESEARCH ARTICLE

Gene regulatory networks driven by intrinsic noise with two-time scales: a stochastic averaging approach

  • Fuke WU 1 ,
  • George YIN , 2 ,
  • Tianhai TIAN 3
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  • 1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
  • 2. Department of Mathematics, Wayne State University, Detroit, MI 48202, USA
  • 3. School of Mathematical Sciences, Monash University, Melbourne Vic 3800, Australia

Received date: 14 Apr 2014

Accepted date: 19 Jun 2014

Published date: 20 Aug 2014

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

This work focuses on gene regulatory networks driven by intrinsic noise with two-time scales. It uses a stochastic averaging approach for these systems to reduce complexity. Comparing with the traditional quasi-steady-state hypothesis (QSSH), our approach uses stochastic averaging principle to treat the intrinsic noise coming from both the fast-changing variables and the slow-changing variables, which yields a more precise description of the underlying systems. To provide further insight, this paper also investigates a prototypical two-component activator-repressor genetic circuit model as an example. If all the protein productions were linear, these two methods would yield the same reduction result. However, if one of the protein productions is nonlinear, the stochastic averaging principle leads to a different reduction result from that of the traditional QSSH.

Cite this article

Fuke WU , George YIN , Tianhai TIAN . Gene regulatory networks driven by intrinsic noise with two-time scales: a stochastic averaging approach[J]. Frontiers of Mathematics in China, 2014 , 9(4) : 947 -963 . DOI: 10.1007/s11464-014-0404-4

1
Arnold L. Stochastic Differential Equations: Theory and Applications. New York: Wiley, 1972

2
Ball C A, Torous W N. Unit roots and the estimation of interest rate dynamics. J Empirical Finance, 1996, 3: 215-238

DOI

3
Bollenbach T, Kishony R. Quiet gene circuit more fragile than its noisy peer. Cell, 2009, 139: 512-522

DOI

4
Burrage K, Tian T, Burrage P. A multi-scaled approach for simulating chemical reaction systems. Progress in Biophysics & Molecular Biology, 2004, 85: 217-234

DOI

5
Çagatay T, Turcotte M, Elowitz M B, Garcia-Ojalvo J, Gürol M S. Architecturedependent noise discriminates functionally analogous differentiation circuits. Cell, 2009, 139: 512-522

DOI

6
Cox J C, Ingersoll J E, Ross S A. A theory of the term structure of interest rates. Econometrica, 1985, 53: 385-407

DOI

7
de Jong H. Modelling and simulation of genetic regulatory systems: a literature review. J Comput Biol, 2002, 9: 67-103

DOI

8
Eldar A, Elowitz B. Functional roles for noise in genetic circuits. Nature, 2010, 467: 67-173

DOI

9
Feller W. Two singular diffusion problems. Ann Math, 1951, 54: 173-182

DOI

10
Gillespiek D T. The chemical Langevin equation. J Chem Phy, 2000, 113: 297-306

DOI

11
Hogg R V, McKean J W, Craig A T. Introduction to Mathematical Statistics. 7th ed. New Jersey: Pearson-Hall, 2012

12
Kampen N G Van. Stochastic Processes in Physics and Chemistry. Amsterdam: Elsevier B V, 2007

13
Khasminskii R Z. On an averaging principle for Itô stochastic differential equations. Kybernetika, 1968, 4: 260-279

14
Khasminskii R Z, Yin G. On transition densities of singularly perturbed diffusions with fast and slow components. SIAM J Appl Math, 1996, 56: 1794-1819

DOI

15
Khasminskii R Z, Yin G. On averaging principles: an asymptotic expansion approach, SIAM J Math Anal, 2004, 35: 1534-1560

DOI

16
Khasminskii R Z, Yin G. Limit behavior of two-time-scale diffusions revisited. J Differential Equations, 2005, 212: 85-113

DOI

17
Kushner H J. Weak convergence methods and singularly perturbed stochastic control and filtering problems. Boston: Birkh'́auser, 1990

18
Kushner H J, Yin G. Stochastic Approximation and Recursive Algorithms and Applications. 2nd ed. New York: Springer-Verlag, 2003

19
Lestas I, Vinnicombe G, Paulsson J. Fundamental limits on the suppression of molecular fluctuations. Nature, 2010, 467: 174-178

DOI

20
Lewin B. Genes IX. Burlington: Jones and Bartlett Learning, 2007

21
Mao X. Stochastic Differential Equations and Applications. Chichester: Horwood, 1997

22
Mier-y-Terán-Romero L, Silber M, Hatzimanikatis V. The origins of time-delay in template biopolymerization processes. PloS Comput Biol, 2010, 6: e1000726

DOI

23
Polynikis A, Hogan S J, di Bernardo M. Comparing different ODE modelling approaches for gene regulatory networks. J Theoretical Biology, 2009, 261: 511-530

DOI

24
Scheper T O, Klinkenberg D, Pennartz C, van Pelt J. A mathematical model for the intracellular circadian rhythm generator. J Neuroscience, 1999, 19(1): 40-47

25
Scheper T O, Klinkenberg D, van Pelt J, Pennartz C. A model of molecular circadian clocks: multiple mechanisms for phase shifting and a requirement for strong nonlinear interactions. J Biological Rhythms, 1999, 14(3): 213-220

DOI

26
Turcotte M, Garcia-Ojalvo J, Süel G M. A genetic timer through noise-induced stabilization of an unstable state. Proc Natl Acad Sci USA, 2008, 105: 15732-15737

DOI

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