Asymptotic behavior of gene expression with complete memory and two-time scales based on the chemical Langevin equations

Yun Li; Fuke Wu; George Yin

doi:10.3934/dcdsb.2019125

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2019, Volume 24, Issue 8: 4417-4443. Doi: 10.3934/dcdsb.2019125

This issue Previous Article The hipster effect: When anti-conformists all look the same Next Article Some monotone properties for solutions to a reaction-diffusion model

Asymptotic behavior of gene expression with complete memory and two-time scales based on the chemical Langevin equations

1.
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China
2.
Department of Mathematics, Wayne State University, Detroit, MI 48202 USA

^* Corresponding author: Fuke Wu
^* Corresponding author: Fuke Wu

Received: June 30, 2018

Revised: October 31, 2018

Early access: June 2019

Published: July 2019

Yun Li is supported by in part by the National Natural Science Foundations of China (Grant Nos. 61873320 and 11761130072). Fuke Wu is supported by the National Natural Science Foundations of China (Grant Nos. 61873320 and 11761130072) and the Royal Society-Newton Advanced Fellowship. George Yin is supported by the National Science Foundation under grant DMS-1710827

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Abstract

Gene regulatory networks, which are complex high-dimensional stochastic dynamical systems, are often subject to evident intrinsic fluctuations. It is deemed reasonable to model the systems by the chemical Langevin equations. Since the mRNA dynamics are faster than the protein dynamics, we have a two-time scales system. In general, the process of protein degradation involves time delays. In this paper, we take the system memory into consideration in which we consider a model with a complete memory represented by an integral delay from $ 0 $ to $ t $. Based on the averaging principle and perturbed test function method, this work examines the weak convergence of the slow-varying process. By treating the fast-varying process as a random noise, under appropriate conditions, it is shown that the slow-varying process converges weakly to the solution of a stochastic differential delay equation whose coefficients are the average of those of the original slow-varying process with respect to the invariant measure of the fast-varying process.

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Mathematics Subject Classification: Primary: 60H20, 65C30, 60H30, 34K50; Secondary: 62L20, 92C45, 93C70, 34E10.

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