Background: Oscillatory behavior is critical to many life sustaining processes such as cell cycles, circadian rhythms, and notch signaling. Important biological functions depend on the characteristics of these oscillations (hereafter, oscillation characteristics or OCs): frequency (e.g., event timings), amplitude (e.g., signal strength), and phase (e.g., event sequencing).
Results: We develop and conduct a theoretical study of an oscillating reaction network to quantify the relationships between OCs and the structure and parameters of the reaction network. We consider a reaction network with two species that have dynamics that can be described by a system of linear differential equations. This makes it possible to construct a two species harmonic oscillator (2SHO). We obtain exact, closed-form formulas for the OCs of the 2SHO. These formulas are used to develop the parameterizeOscillator algorithm that parameterizes the 2SHO to achieve desired oscillation characteristics.
Conclusions: The OC formulas are employed to analyze the roles of the reactions in the network, and to comment on other studies of oscillatory reaction networks. For example, others have stated that nonlinear dynamics are required to create oscillations in reaction networks. Our 2SHO is a counter example to this claim in that 2SHO is an oscillating reaction network whose dynamics are described by a system of linear differential equations. A further insight of potential interest is that our formulas show that in order to obtain oscillations in the 2SHO, the rate of the reaction that causes negative feedback must exceed the rate of the reaction that causes positive feedback.