Examples include: cell cycles, circadian rhythms, notch signaling in the development of the nervous system, tissue development, gene transcription, and efficient signaling. Oscillatory behavior is critical to many life sustaining processes. Finally, our sensitivity analysis suggests an approach to constructing a 2SHO for a biochemical system. With these formulas, we can design 2SHO reaction networks to have desired oscillation characteristics. Because it is a linear system, we can derive closed form expressions for the frequency, amplitude, and phase of oscillations, something that has not been done for other published reaction networks. Our 2SHO demonstrates the feasibility of creating an oscillating reaction network whose dynamics are described by a system of linear differential equations. Our contributions are: (a) construction of an oscillating, two species reaction network that has no nonlinearity (b) obtaining closed form formulas that calculate frequency, amplitude, and phase in terms of the parameters of the 2SHO reaction network, something that has not been done for any published oscillating reaction network and (c) development of an algorithm that parameterizes the 2SHO to achieve desired oscillation, a capability that has not been produced for any published oscillating reaction network. This is a theoretical study that analyzes reaction networks in terms of their representation as systems of ordinary differential equations. Finally, no one has published an algorithm for constructing oscillating reaction networks with desired OCs. Further, no one has obtained closed form solutions for the frequency, amplitude and phase of any oscillating reaction network. No one has shown that oscillations can be produced for a reaction network with linear dynamics. Some investigators claim that oscillations in reaction networks require nonlinear dynamics in that at least one rate law is a nonlinear function of species concentrations. Numerous oscillating reaction networks have been documented or proposed. 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). Oscillatory behavior is critical to many life sustaining processes such as cell cycles, circadian rhythms, and notch signaling.
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