A detailed analysis of the stability and flipping dynamics of a delayed exclusive toggle switch is performed. We use forward flux sampling method combined with delayed stochastic simulation algorithm to get the stationary distribution function, the switching rate, and path- ways, as well as the transition state ensemble. Interestingly, under the influence of time delay, the stationary distribution corresponding to the stable states become narrower and the population in the transition region is significantly enhanced. In addition, the flipping rate increases monotonically with delay. Such findings demonstrate that time delay could reduce the stability of the bistable genetic switch dramatically. Furthermore, the transition pathways, characterized by the difference in the protein numbers and the state of operator, show larger discrepancy between the forward and backward switching process with increas- ing delay, indicating that transcriptional and translational delay can remarkably affect the flipping dynamics. Specifically, for the transition state, the difference in the probability of finding the operator site bound by the two different protein dimers is enlarged by delay, which further illustrates the crucial role of time delay on the stability and switching dynamics of genetic toggle switches.
We investigate the impact of coupling on the reliability of the logic system as well as the logical stochastic resonance （LSR） phenomenon in the coupled logic gates system. It is found that compared with single logic gate, the coupled system could yield reliable logic outputs in a much wider noise region, which means coupling can obviously improve the reliability of the logic system and thus enhance the LSR effect. Moreover, we find that the enhancement is larger for larger system size, whereas for large enough size the enhancement seems to be saturated. Finally, we also examine the effect of coupling strength, it can be observed that the noise region where reliable logic outputs can be obtained evolves non-monotonically as the coupling strength increases, displaying a resonance-like effect.
