A mathematical approach to emergent properties of metabolic networks: partial coupling relations, hyperarcs and flux ratios.

Emergent properties in systems biology are those which arise only when the biological system passes a certain level of complexity. In this study, we introduce some of the emergent properties which appear in the constraint-based analysis of metabolic networks. These properties generally appear as a result of existence of hfdeyperarcs and irreversible reactions in networks. Here, we present examples of metabolic networks in which there exist at least two reactions whose fluxes cannot be written as products and/or ratios of the stoichiometric coefficients of the network. We show that any such network contains at least one hyperarc. Additionally, we prove that partial coupling cannot appear in consistent metabolic networks with less than four reactions, or with less than three irreversible reactions, or without hyperarc(s).