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Delay and noise induced regime shift and enhanced stability in gene expression dynamics

A quantitative model of autoregulatory gene expression involving a single gene with time delays and cross-correlated noise sources is investigated. The probability density and mean first passage time (MFPT) of the protein concentration are obtained. The impacts of multiplicative (σM) and additive (σA) noise intensities, cross-correlation intensity λ between two noises, time delays τ in the degradation process and θ in the synthesis process and time delay β in both processes on the probability density and MFPT of the regime shifts between high and low protein concentration states are discussed, respectively. These results indicate that (i) the regime shifts from a high (or low) protein concentration state to a low (or high) one can be induced by σM, λ and θ (or σA and β); (ii) the MFPT as a function of the noise intensity σM or σA exhibits one maximum value in the case of λ > 0 or θ > 0, this maximum is a signature of the noise's enhanced stability phenomenon for high protein concentration state; and (iii) τ and β can weaken the stability of high protein concentration state but, in contrast, λ and θ can enhance it in the gene expression dynamics.