The mean and noise of protein numbers in stochastic gene expression

Gene expression is the central process in cells, and is stochastic in nature. In this work, we study the mean expression level of, and the expression noise in, a population of isogenic cells, assuming that transcription is activated by two sequential exponential processes of rates kappa and lambda. We find that the mean expression level often displays oscillatory dynamics, whereas most other models suggest that it always grows monotonically. We show that, given the same average gene off duration, the asymptotic expression noise increases with |kappa - lambda|, and is thus maximized when either kappa -> a or lambda -> a, for which the two exponential processes approach to one process. It suggests that natural selection may favor two or more rate-limiting steps for gene transcription activation. Our analysis reveals that, at steady-state, the noise equals the inverse of the mean, plus the normalized covariance of the mRNA and protein copy numbers. This interesting identity partially explains a recent striking finding that the protein noises of many Escherichia coli genes were close to the inverse of the mean protein levels, and simultaneously, the protein and mRNA copy numbers within the individual cells were uncorrelated. We show further that the protein noise is close to the inverse of the mean if the gene is transcribed effectively and almost continuously, and the protein molecules are considerably more stable than the mRNAs. Such phenomenon has been observed repeatedly in the synthetic reporter genes controlled by strong promoters and tagged with fluorescent labels.