Realizations of kinetic differential equations

Area of Science:

• Chemical Kinetics
• Systems Biology
• Computational Chemistry

Background:

• Reaction networks with mass action kinetics generate polynomial differential equations.
• Determining if a given system of differential equations can arise from a reaction network is a key challenge.

Purpose of the Study:

• To determine if a kinetic realization exists for a given system of polynomial differential equations.
• To identify if such realizations possess chemically relevant properties like detailed balancing or mass conservation.

Main Methods:

• The study employs constructive mathematical approaches to address the existence and properties of kinetic realizations.
• Analysis involves exploring the relationship between differential equations and underlying reaction network structures.

Main Results:

• Provides constructive answers to the problem of finding kinetic realizations for systems of polynomial differential equations.
• Demonstrates the possibility of finding realizations with specific chemically relevant properties.
• Highlights applications in fitting differential equations to data and analyzing dynamic behaviors.

Conclusions:

• The existence and properties of kinetic realizations are systematically investigated.
• Results offer practical utility in systems biology and computational chemistry for model building and analysis.
• The findings have implications for solving related mathematical problems, such as the existence of positive solutions to algebraic equations.