Stochasticity in single gene expression with both intrinsic noise and fluctuation in kinetic parameters

Stochasticity is one of the most important properties in gene expression. Noise originates from two sources: thermal fluctuation inherent in the system (intrinsic noise) and variabilities in factors external to the system that usually result to the fluctuation in the kinetic parameters (extrinsic noise). This paper studies analytically the stationary fluctuation of the number of protein molecules through a mathematical model involving both sources of noises. The results in this paper show that the two sources of noises interlock to each other to generate total fluctuation in protein numbers. In particular, the extrinsic noises effect the total fluctuation in multiple ways, including the extrinsic fluctuation, the correlation with intrinsic noise, the alternation of the time averaging of transcription and translation, and the amplification of the total fluctuation by an impact factor. The impact factor is pronounced when the fluctuations in the degradation rates of mRNA or protein are large. Moreover, the extrinsic noise to the translational rate generates large fluctuation when the translational efficiency is too low, which is contrast to the translational bursting in high translational efficiency because of intrinsic noise. These results suggest that it is important to control the mRNA and protein degradation rate as well as the translational efficiency in order to attenuate the fluctuation in gene expression in the present of both intrinsic and extrinsic noises.