Based on the kinetic model of genetic regulation system proposed by Smolen et al. [Am. J. Physiol. 274, c531 (1998)], the effects of fluctuations in the degradation reaction rate and the synthesis reaction rate of the transcription factor have been investigated through numerical computation and analysis theory. In the case of uncorrelated noises, it is shown that only the fluctuation of degradation reaction rate can induce a switch process, and the mean first passage time (MFPT) from the high

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... ncentration state to the low concentration one is decreased when the noise intensity of degradation reaction rate is increased. In the case of correlations between noises, a switch process can also be induced by the cross-correlation intensity between noises and by the fluctuation of the synthesis reaction rate in the genetic regulatory system. It is found that, under large cross-correlation intensity, a successive switch process (i.e., "on"→"off"→"on," which we call the reentrance transition or twice switch ) occurs with an increase of noise intensities, and a critical noise intensity exists at which the MFPT of the switch process is the largest. While the system is initially in the high concentration state with an increase of the cross correlation, the stationary probability distribution (SPD) of the transcription factor activator monomer concentration at the low concentration state is increased, yet the MFPT is increased due to the decreasing of the SPD of the transient states between the two steady stable states.