Analysis of metabolic systems with complex slow and fast dynamics

Area of Science:

• Mathematical modeling
• Biophysical chemistry
• Systems biology

Background:

• Analysis of systems with time hierarchy is complex.
• The Tikhonov theorem is a classical tool for such analyses.
• Limitations exist for systems with complex fast dynamics.

Purpose of the Study:

• To present a new theorem for rigorous analysis of systems with complex fast dynamics.
• To discuss the applicability and comparison of this new theorem with existing methods.
• To rigorously study a model enzymatic reaction with product activation and slow enzyme turnover.

Main Methods:

• Review of existing literature on time hierarchy systems and the Tikhonov theorem.
• Statement and discussion of a newly proved theorem for complex fast dynamics.
• Rigorous mathematical analysis of a model enzymatic reaction system.

Main Results:

• The new theorem allows for rigorous analysis of systems previously intractable with the Tikhonov theorem.
• A model enzymatic reaction demonstrated parameter regions where fast variables oscillate, rendering the Tikhonov theorem inapplicable.
• The newly presented theorem and numerical solutions were compared for efficacy.

Conclusions:

• The classical Tikhonov theorem has limitations in analyzing systems with complex fast dynamics.
• A newly developed theorem provides a rigorous method for analyzing such systems.
• Alternative methods, including the new theorem and numerical solutions, are necessary for specific dynamic systems.