Oscillations in three-reaction quadratic mass-action systems

Area of Science:

• Chemical Kinetics
• Systems Biology
• Biochemical Networks

Background:

• Rank-two bimolecular mass-action systems are known not to admit limit cycles.
• Understanding oscillation in small mass-action systems requires exploring networks beyond simple bimolecular reactions.

Purpose of the Study:

• To investigate rank-two mass-action systems with bimolecular sources and higher molecularity target complexes.
• To identify minimal-sized oscillatory networks, focusing on three-reaction systems.
• To characterize networks admitting periodic orbits and Andronov-Hopf bifurcations.

Main Methods:

• Analysis of mass-action systems with varying reaction stoichiometries.
• Characterization of networks based on the presence of periodic orbits and bifurcations.
• Focus on three-reaction, two-species systems with bimolecular sources.

Main Results:

• Isolated periodic orbits do not occur in three-reaction, trimolecular, mass-action systems with bimolecular sources.
• Identified a new network, besides Lotka and Ivanova reactions, admitting a center and a vertical Andronov-Hopf bifurcation.
• Characterized two families of two-species, three-reaction, bimolecular-sourced networks admitting a supercritical Andronov-Hopf bifurcation, leading to stable limit cycles. These require target complexes of molecularity at least four.

Conclusions:

• The study expands the understanding of oscillatory behavior in mass-action systems by considering higher molecularity target complexes.
• Specific network structures capable of generating stable limit cycles through supercritical Andronov-Hopf bifurcations have been identified.
• These findings are crucial for designing synthetic biological circuits and understanding complex biochemical dynamics.