Stoichiometric analysis of biochemical systems on graphs. I. Graphical rules of finding of conservation relationships

The graph-theoretical approach of finding of conservation relationships (linear integral of motion) of kinetic equations of biochemical systems is considered. The one-to-one correspondence between a connected directed bipartite graph and a stoichiometric matrix, and also a Jacobian of a complex reaction (a metabolic network), is shown. It is shown that the topological fragments and the graph images, which determine the general structure-dynamic properties of a reaction network, can be revealed by analysis of the graph created on the scheme/network of the pathway. Thus, the general structure-dynamic properties of the studied system have the topological interpretation and can be revealed and classified. The graphical rules (1) and (2) for revealing of the components, which are connected by the conservation relationships, in a complex reaction/metabolic network, are developed. The first rule is the necessary and sufficient conditions of equivalence of an isolated subgraph of the graph of reaction to the conservation relationship. The second rule is a graphic procedure of finding of unknown coefficients in the conservation relationships.