Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The finest decompositions' architecture of a reaction network

Bryan S. Hernandez, Juan Paolo C. Santos, Patrick Vincent N. Lubenia, Eduardo R. Mendoza

Abstract

Biochemical and environmental modeling typically relies on reaction networks to represent complex transformations. While the Linkage Class Decomposition (LCD) partitions networks based on visual standard connectivity, it often misaligns with the algebraic properties governing long-term dynamics. This work establishes the Finest Decompositions' Architecture (FDA) framework by analyzing hierarchical relationships between the LCD and two algebraic structures: the Finest Independent Decomposition (FID) and the Finest Incidence-Independent Decomposition (FIID). These algebraic decompositions serve as the respective building blocks for characterizing general equilibria and complex-balanced equilibria of a reaction network. Under the partial order of "coarsens to," we categorize reaction networks into six architectures, distinguishing three subclasses of Independent Linkage Classes (ILC) from three subclasses of Dependent Linkage Classes (DLC). To facilitate the classification, we introduce the Deficiency Difference (Delta), measuring the discrepancy between total and subnetwork deficiencies, and the Common Complexes Cardinality CC of the FID. Results show that Delta uniquely identifies all the ILC classes and one DLC subclass, while CC distinguishes the remaining DLC subclasses. A number of results on mass action systems such as the Deficiency One Theorem as well as on power law systems essentially rely on the ILC property of the underlying networks. These suggest that the FDA classification of ILC and DLC networks signify a certain alignment of both structural and kinetic attributes. This work opens up direction for the study of the structure and equilibria analysis of reaction networks across diverse decomposition architectures.

The finest decompositions' architecture of a reaction network

Abstract

Biochemical and environmental modeling typically relies on reaction networks to represent complex transformations. While the Linkage Class Decomposition (LCD) partitions networks based on visual standard connectivity, it often misaligns with the algebraic properties governing long-term dynamics. This work establishes the Finest Decompositions' Architecture (FDA) framework by analyzing hierarchical relationships between the LCD and two algebraic structures: the Finest Independent Decomposition (FID) and the Finest Incidence-Independent Decomposition (FIID). These algebraic decompositions serve as the respective building blocks for characterizing general equilibria and complex-balanced equilibria of a reaction network. Under the partial order of "coarsens to," we categorize reaction networks into six architectures, distinguishing three subclasses of Independent Linkage Classes (ILC) from three subclasses of Dependent Linkage Classes (DLC). To facilitate the classification, we introduce the Deficiency Difference (Delta), measuring the discrepancy between total and subnetwork deficiencies, and the Common Complexes Cardinality CC of the FID. Results show that Delta uniquely identifies all the ILC classes and one DLC subclass, while CC distinguishes the remaining DLC subclasses. A number of results on mass action systems such as the Deficiency One Theorem as well as on power law systems essentially rely on the ILC property of the underlying networks. These suggest that the FDA classification of ILC and DLC networks signify a certain alignment of both structural and kinetic attributes. This work opens up direction for the study of the structure and equilibria analysis of reaction networks across diverse decomposition architectures.

Paper Structure

This paper contains 45 sections, 29 theorems, 56 equations, 1 figure, 7 tables.

Table of Contents

  1. Introduction
  2. Fundamentals of reaction networks and kinetic systems
  3. Chemical reaction networks
  4. Kinetic systems
  5. Decomposition theory of reaction networks
  6. Overview of the finest decompositions' architecture (FDA)
  7. The three finest decompositions of a CRN
  8. Review of FID and FIID concepts and key properties
  9. Discussion of the LCD as the coarsest $\mathscr{C}$-decomposition
  10. The definition of the FDA of a CRN
  11. Algebraic aspects of CRNs
  12. The finest decompositions' architecture classes of independent linkage class (ILC) networks
  13. Characterization and examples of ILC-1 networks
  14. The subclass of FDE networks
  15. Non-FDE ILC-1 networks
  16. ...and 30 more sections

Key Result

Lemma 2.15

Hernandez:WRCF For any network decomposition $\mathcal{N}=\mathcal{N}_1\cup\mathcal{N}_2\cup \ldots \cup\mathcal{N}_k$, the following statements are equivalent:

Figures (1)

  • Figure 1: The finest decompositions' architecture of a chemical reaction network. The FDA hierarchy is established by the partial order 'coarsens to'. While ILC networks are characterized by deficiency additivity over their linkage classes, DLC networks lack this alignment. The solid arrow $\rightarrow$ indicates that the decomposition at the tail is a refinement of (or "coarsens to") the decomposition at the head. The inequality sign $\ne$ marks where a coarsening relationship exists but the two decompositions are not equal.

Theorems & Definitions (94)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 84 more