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Finest decomposition coarsening of reaction networks of biochemical systems

Biochemical reaction networks are typically modeled by $\dfrac{dx}{dt}=N\cdot K(x)=Y\cdot I_a\cdot K(x)$, with $x$ and $K(x)$ as the concentration and rate vectors, respectively, and $N$, $Y$, and $I_a$ as the stoichiometric, molecularity, and incidence matrices, respectively. Steady states, which describe their long-term behaviors, are determined by solving $N\cdot K(x)=0$, while complex balanced steady states are found by solving $I_a \cdot K(x)=0$. To investigate these complex networks, decomposition techniques are important, in particular, for computing steady states. Previously, we identified a widespread property across many networks: the existence of independent and incidence-independent decompositions, characterized by the ability to directly sum the stoichiometric and incidence matrices of the subnetworks, respectively, to match those of the entire network. Here, we discover the ubiquitous property that we call the Finest Decomposition Coarsening (FDC), where the finest independent decomposition (FID) is a coarsening of the finest incidence-independent decomposition (FIID). To support the analysis of this property, we introduce a MATLAB package designed to compute both these decompositions. We then characterize the FDC property and its relationship to structural factors such as the invertibility of the molecularity matrix. We also introduce and characterize the Finest Decompositions Equality (FDE) property, where FIID equals FID. Notably, we show that all deficiency zero networks exhibit the FDE property. Furthermore, we establish important relationships of the FID and FIID with decomposition of the network into its connected components. Our results highlight the prevalence of the coarsening property in reaction networks and deepens the understanding of the algebraic structure and dynamics of biochemical networks.

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