A positive and a negative feedback loop can induce bistability and oscillation, respectively, in biological networks. Nevertheless, they are frequently interlinked to perform more elaborate functions in many gene regulatory networks. Coupled positive and negative feedback loops may exhibit either oscillation or bistability depending on the intensity of the stimulus in some particular networks. It is less understood how the transition between the two dynamic modes is modulated by the positive and negative feedback loops. We developed an abstract model of such systems, largely based on the core p53 pathway, to explore the mechanism for the transformation of dynamic behaviors. Our results show that enhancing the positive feedback may promote or suppress oscillations depending on the strength of both feedback loops. We found that the system oscillates with low amplitudes in response to a moderate stimulus and switches to the on state upon a strong stimulus. When the positive feedback is activated much later than the negative one in response to a strong stimulus, the system exhibits long-term oscillations before switching to the on state. We explain this intriguing phenomenon using quasistatic approximation. Moreover, early switching to the on state may occur when the system starts from a steady state in the absence of stimuli. The interplay between the positive and negative feedback plays a key role in the transitions between oscillation and bistability. Of note, our conclusions should be applicable only to some specific gene regulatory networks, especially the p53 network, in which both oscillation and bistability exist in response to a certain type of stimulus. Our work also underscores the significance of transient dynamics in determining cellular outcome.

• Received 4 June 2017
• Revised 25 January 2018
• Corrected 8 January 2020

DOI:https://doi.org/10.1103/PhysRevE.97.042412