An oscillating reaction network with an exact closed form solution in the time domain

Area of Science:

• Biochemical reaction networks
• Systems biology
• Theoretical chemistry

Background:

• Oscillatory behaviors are fundamental to biological processes like cell cycles and circadian rhythms.
• Controlling oscillation characteristics (frequency, amplitude, phase) is crucial for biological functions.
• Previous oscillating reaction networks often assumed nonlinear dynamics, with no known linear examples or methods for precise OC control.

Purpose of the Study:

• To demonstrate that oscillating reaction networks can be constructed using only linear dynamics.
• To derive closed-form solutions for the frequency, amplitude, and phase of oscillations in a linear network.
• To develop an algorithm for designing oscillating reaction networks with specific, desired oscillation characteristics.

Main Methods:

• Theoretical analysis of reaction networks represented as systems of ordinary differential equations.
• Construction of a novel two species harmonic oscillator (2SHO) model.
• Derivation of closed-form mathematical expressions for oscillation parameters.

Main Results:

• Successfully constructed an oscillating reaction network (2SHO) based solely on linear dynamics.
• Obtained explicit formulas for calculating the frequency, amplitude, and phase of the 2SHO oscillations.
• Developed a parameterization algorithm enabling the design of 2SHO networks with targeted oscillation characteristics.

Conclusions:

• The 2SHO model validates the feasibility of linear dynamics for generating biological oscillations.
• Closed-form solutions allow for precise prediction and control of oscillation characteristics.
• The developed methodology facilitates the design of synthetic biological systems with desired oscillatory properties.