Delays-based protein switches in a stochastic single-gene network

•We study the protein switch in a single-gene network with time delays and noises.•Noises and delays can induce the protein switch between the ON state and OFF one.•It is shown that noises and delays can enhance stability of the OFF state in a single-gene network.

In this paper, the protein switch in a single-gene network with time delays is investigated, where the gene expression is assumed to be disturbed by multiplicative and additive noises. The impacts of time delays τdτd and τsτs in degradation and synthesis processes, time delay τgτg in global process and cross-correlation between two noises (λiλi, and i=d,s,gi=d,s,g) on the probability distribution and switch time (ST) from low protein level (OFF state) to high one (ON state) are discussed, respectively. Our results show that (1) the increase of the cross-correlation between two noises (λiλi) can induce protein switches from ON state to OFF one; (2) for λi⩾0λi⩾0, the increase of τdτd can induce protein switches from OFF state to ON one, while τsτs (or τgτg) can induce protein switches from the ON state to the OFF one, but for λi<0λi<0, the τdτd (or τsτs) can induce protein switches from the OFF state to the ON one, while τgτg can induce protein switches from the ON state to the OFF one; (3) the ST as functions of the noise intensities exhibits a maximum, which is the signature of the noise enhanced stability (NES) of the OFF state, while the increase of τdτd can cause the NES phenomenon to disappear; and (4) τdτd and τsτs play opposing roles in the ST, i.e., the impacts of the time delays τdτd and τsτs on ST can be canceled each other out.