Singular perturbation theory for open enzyme reaction networks

Area of Science:

• Biochemical Engineering
• Chemical Kinetics
• Applied Mathematics

Background:

• Enzyme reaction networks are crucial in biochemistry, but their complex dynamics can be challenging to analyze.
• The pseudo-steady-state hypothesis simplifies these networks by assuming rapid intermediate reactions reach a steady state.
• Mathematical justification for this hypothesis in general open systems is often lacking.

Purpose of the Study:

• To provide rigorous mathematical justification for the pseudo-steady-state hypothesis in general open enzyme reaction networks.
• To analyze the convergence of solutions between a complete system and its reduced form.
• To explore the asymptotic behavior of the degenerate system.

Main Methods:

• Application of singular perturbation theory on infinite time intervals.
• Derivation of a reduced system from the original complete system based on input/output function conditions.
• Analysis of the convergence between solutions of the complete and reduced systems, excluding the initial boundary layer.
• Study of the asymptotic behavior of the degenerate system, focusing on stable fixed points and periodic solutions.

Main Results:

• A condition on the input/output function is identified to derive a suitable reduced system.
• The convergence of solutions from the reduced system to the original system (outside the boundary layer) is established.
• The analysis covers cases where the degenerate system exhibits an asymptotically stable fixed point or a periodic solution.

Conclusions:

• The study provides a robust mathematical framework for applying the pseudo-steady-state hypothesis to general open enzyme reaction networks.
• The findings validate the use of simplified reduced models for analyzing complex biochemical systems.
• The results have broad applicability to various problems in chemical kinetics and systems biology.