Methods and Applications of Analysis

Volume 28 (2021)

Number 2

Special issue dedicated to Professor Ling Hsiao on the occasion of her 80th birthday, Part I

Guest editors: Qiangchang Ju (Institute of Applied Physics and Computational Mathematics, Beijing), Hailiang Li (Capital Normal University, Beijing), Tao Luo (City University of Hong Kong), and Zhouping Xin (Chinese University of Hong Kong)

An uncertainty quantification approach to the study of gene expression robustness

Pages: 195 – 220



Pierre Degond (Institute de Mathématiques de Toulouse, Université de Toulouse, France)

Shi Jin (School of Mathematical Sciences, Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, China)

Yuhua Zhu (Department of Mathematics, Stanford University, Stanford, California, U.S.A.)


We study a chemical kinetic system with uncertainty modeling a gene regulatory network in biology. Specifically, we consider a system of two equations for the messenger RNA and micro RNA content of a cell. Our target is to provide a simple framework for noise buffering in gene expression through micro RNA production. Here the uncertainty, modeled by random variables, enters the system through the initial data and the source term. We obtain a sharp decay rate of the solution to the steady state, which reveals that the biology system is not sensitive to the initial perturbation around the steady state. The sharp regularity estimate leads to the stability of the generalized Polynomial Chaos stochastic Galerkin (gPC‑SG) method. Based on the smoothness of the solution in the random space and the stability of the numerical method, we conclude the gPC‑SG method has spectral accuracy. Numerical experiments are conducted to verify the theoretical findings.


gene expression, generalized polynomial chaos, sensitivity analysis, spectral accuracy

2010 Mathematics Subject Classification

35Q92, 65M12, 65M70, 92C37

Received 15 October 2019

Accepted 11 June 2020

Published 10 June 2022