Reachability, persistence, and constructive chemical reaction networks (part III): a mathematical formalism for binary enzymatic networks and application to persistence

摘要

Chemical Reaction Network Theory is concerned with understanding the properties of systems of reactions from their structure. Enzymatic networks receive significant attention in the field because they are crucial in biochemistry and often illustrate the network features that are studied. In this paper we propose a formalism for binary enzymatic networks which can be used to research their mathematical properties. The networks are binary in that every enzyme-substrate complex consists of one enzyme and one substrate. Many connected concepts, e.g. futile enzymatic cycles and enzymatic cascades, are defined rigorously and so as to reflect the corresponding biochemical phenomena. We prove that binary enzymatic networks that are futile and cascaded are vacuously persistent: no species will tend to extinction if all species are implicitly present at initial time. This result extends prior work of Angeli, De Leenheer and Sontag in which a theorem was applied to show that certain particular enzymatic networks are persistent. This paper completes a series of three articles. It applies both the first paper which studies vacuous persistence and the second paper which describes a formalism for species composition.