Frontiers of Mathematics in China >
Gene regulatory networks driven by intrinsic noise with two-time scales: a stochastic averaging approach
Received date: 14 Apr 2014
Accepted date: 19 Jun 2014
Published date: 20 Aug 2014
Copyright
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.
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
|
3 |
Bollenbach T, Kishony R. Quiet gene circuit more fragile than its noisy peer. Cell, 2009, 139: 512-522
|
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
|
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
|
6 |
Cox J C, Ingersoll J E, Ross S A. A theory of the term structure of interest rates. Econometrica, 1985, 53: 385-407
|
7 |
de Jong H. Modelling and simulation of genetic regulatory systems: a literature review. J Comput Biol, 2002, 9: 67-103
|
8 |
Eldar A, Elowitz B. Functional roles for noise in genetic circuits. Nature, 2010, 467: 67-173
|
9 |
Feller W. Two singular diffusion problems. Ann Math, 1951, 54: 173-182
|
10 |
Gillespiek D T. The chemical Langevin equation. J Chem Phy, 2000, 113: 297-306
|
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
|
15 |
Khasminskii R Z, Yin G. On averaging principles: an asymptotic expansion approach, SIAM J Math Anal, 2004, 35: 1534-1560
|
16 |
Khasminskii R Z, Yin G. Limit behavior of two-time-scale diffusions revisited. J Differential Equations, 2005, 212: 85-113
|
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
|
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
|
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
|
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
|
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
|
/
〈 | 〉 |