Abstract
The Jacobian matrix of a dynamic system and its principal minors play a prominent role in the study of qualitative dynamics and bifurcation analysis. When interpreting the Jacobian as an adjacency matrix of an interaction graph, its principal minors reate to sets of disjoint cycles in this graph and conditions for qualitative dynamic behaviors can be inferred from its cycle structure. The Jacobian of chemical reaction systems decomposes into the product of two matrices, which allows more fine-grained analyses by studying a corresponding bipartite species-reaction graph. Several different bipartite graphs have been proposed and results on injectivity, multistationarity, and bifurcations have been derived. Here, we present a new definition of the species-reaction graph that directly connects the cycle structure with determinant expansion terms, principal minors, and the coefficients of the characteristic polynomial. It encompasses previous graph constructions as special cases. This graph has a direct relation to the interaction graph, and properties of cycles and sub-graphs can be translated in both directions. A simple equivalence relation enables simplified decomposition of determinant expansions and allows simpler and more direct proofs of previous results.
Highlights
The analysis of chemical reaction systems is hampered by the fact that parameters such as kinetic rate constants are inherently difficult to obtain from experimental data, and that in-vitro parameters might not translate to in-vivo experiments
When interpreting the Jacobian as an adjacency matrix of an interaction graph, its principal minors reate to sets of disjoint cycles in this graph and conditions for qualitative dynamic behaviors can be inferred from its cycle structure
We develop and emphasize the direct relation between determinant expansions, principal minors, the Jacobian matrix and its interaction graph, and our directed species-reaction (DSR)-graph, making use of long-known relations [12]
Summary
The analysis of chemical reaction systems is hampered by the fact that parameters such as kinetic rate constants are inherently difficult to obtain from experimental data, and that in-vitro parameters might not translate to in-vivo experiments. Several approaches have been proposed that exploit these constraints and allow to determine—without knowledge of parameter values and with minimal conditions on the rate laws—if a reaction network is capable of specific qualitative dynamics such as oscillations and multiple equilibria, and to establish stability of equilibria While methods such as Chemical Reaction Network Theory [14,10,9], Stoichiometric Network Analysis [4], and Biochemical Systems Theory [17,18,19] exploit the particular algebraic structure of reaction systems, more recent methods focus on graphical representations of the Jacobian matrix of the dynamical system and its properties. We present refined criteria for sign-definiteness of the determinant of the Jacobian and its principal minors based on a simple equivalence relation of subgraphs of our DSR-graph and show the equivalence of two criteria developed independently in [16] and [6]
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