Decomposable and Essentially Univariate Mass-Action Systems: Extensions of the Deficiency One Theorem.

Area of Science:

• Chemical kinetics
• Systems biology
• Computational chemistry

Background:

• Feinberg's deficiency one theorems are foundational for analyzing reaction networks with mass-action kinetics.
• Existing theorems require specific conditions like independent linkage classes and a single absorbing strong component.
• These limitations restrict the analysis of complex reaction systems.

Purpose of the Study:

• To extend the applicability of deficiency theorems to more complex reaction networks.
• To introduce and utilize the concept of monomial dependency.
• To develop a generalized deficiency theorem for mass-action systems.

Main Methods:

• Development of a dependency one theorem for parametrized systems of polynomial equations.
• Application of this theorem to mass-action systems, relaxing previous structural constraints.
• Derivation of the extended deficiency one theorem as a specific case.

Main Results:

• A novel dependency one theorem for essentially univariate and decomposable polynomial systems.
• A generalized deficiency one theorem for mass-action systems accommodating arbitrary subnetwork deficiencies and multiple absorbing strong components.
• Demonstration that the extended deficiency one theorem is a special case of the new framework.

Conclusions:

• The monomial dependency framework provides a more general approach to analyzing reaction networks.
• This generalization expands the scope of systems amenable to rigorous equilibrium analysis.
• The findings offer new tools for understanding complex chemical kinetics and systems biology models.