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.