Graph Transformations, Semigroups, and Isotopic Labeling

The Double Pushout (DPO) approach for graph transformation naturally allows an abstraction level of biochemical systems in which individual atoms of molecules can be traced automatically within chemical reaction networks. Aiming at a mathematical rigorous approach for isotopic labeling design we convert chemical reaction networks (represented as directed hypergraphs) into transformation semigroups. Symmetries within chemical compounds correspond to permutations whereas (not necessarily invertible) chemical reactions define the transformations of the semigroup. An approach for the automatic inference of informative labeling of atoms is presented, which allows to distinguish the activity of different pathway alternatives within reaction networks. To illustrate our approaches, we apply them to the reaction network of glycolysis, which is an important and well understood process that allows for different alternatives to convert glucose into pyruvate.