Unbounded solutions of models for glycolysis

Area of Science:

• Biochemistry
• Mathematical Biology
• Dynamical Systems

Background:

• The Selkov oscillator is a simplified model of glycolysis using ordinary differential equations and mass action kinetics.
• Previous research established fundamental properties of the Selkov oscillator's solutions.

Purpose of the Study:

• To prove the existence of solutions diverging to infinity in an oscillatory manner for the Selkov oscillator.
• To analyze the behavior of a related system with Michaelis-Menten kinetics.

Main Methods:

• Poincaré compactification was applied to both the mass action and Michaelis-Menten systems.
• A shooting argument was employed to demonstrate oscillatory divergence.

Main Results:

• Solutions of the Selkov oscillator (mass action kinetics) diverge to infinity in an oscillatory manner.
• The Michaelis-Menten kinetics system also exhibits solutions diverging to infinity, but monotonically.
• The Michaelis-Menten system admits subcritical Hopf bifurcations, leading to unstable periodic solutions.

Conclusions:

• The existence of unbounded solutions raises questions about the biological relevance of the Selkov oscillator.
• Comparison with other models for glycolysis is discussed in light of these findings.