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Modeling Biological Oscillations: Integration of Short Reaction Pauses into a Stationary Model of a Negative Feedback Loop Generates Sustained Long Oscillations

Sustained oscillations are frequently observed in biological systems consisting of a negative feedback loop, but a mathematical model with two ordinary differential equations (ODE) that has a negative feedback loop structure fails to produce sustained oscillations. Only when a time delay is introduced into the system by expanding to a three-ODE model, transforming to a two-delay differential equations (DDE) model, or introducing a bistable trigger do stable oscillations present themselves. In this study, we propose another mechanism for producing sustained oscillations based on periodic reaction pauses of chemical reactions in a negative feedback system. We model the oscillatory system behavior by allowing the coefficients in the two-ODE model to be periodic functions of time-called pulsate functions-to account for reactions with go-stop pulses. We find that replacing coefficients in the two-ODE system with pulsate functions with microscale (several seconds) pauses can produce stable system-wide oscillations that have periods of approximately 1 to several hours long. We also compare our two-ODE and three-ODE models with the two-DDE, three-ODE, and three-DDE models without the pulsate functions. Our numerical experiments suggest that sustained long oscillations in biological systems with a negative feedback loop may be an intrinsic property arising from the slow diffusion-based pulsate behavior of biochemical reactions.