Impact of time delays on oscillatory dynamics of interlinked positive and negative feedback loops

Interlinking a positive feedback loop (PFL) with a negative feedback loop (NFL) constitutes a typical motif in genetic networks, performing various functions in cell signaling. How time delay in feedback regulation affects the dynamics of such systems still remains unclear. Here, we investigate three systems of interlinked PFL and NFL with time delays: a synthetic genetic oscillator, a three-node circuit, and a simplified single-node model. The stability of steady states and the routes to oscillation in the single-node model are analyzed in detail. The amplitude and period of oscillations vary with a pointwise periodicity over a range of time delay. Larger-amplitude oscillations can be induced when the PFL has an appropriately long delay, in comparison with the PFL with no delay or short delay; this conclusion holds true for all the three systems. We unravel the underlying mechanism for the above effects via analytical derivation under a limiting condition. We also develop a stochastic algorithm for simulating a single reaction with two delays and show that robust oscillations can be maintained by the PFL with a properly long delay in the single-node system. This work presents an effective method for constructing robust large-amplitude oscillators and interprets why similar circuit architectures are engaged in timekeeping systems such as circadian clocks.