An Algebraic-combinatorial Model for the Identification and Mapping of Biochemical Pathways

• Published in: Bulletin of mathematical biology Vol. 63; no. 6; pp. 1163 - 1196
• Main Authors: Oliveira, Joseph S, Bailey, Colin G, Jones-Oliveira, Janet B, Dixon, David A
• Format: Journal Article
• Language: English
• Published: United States Elsevier Ltd 01.11.2001; Springer Nature B.V
• Subjects: Linear Models; Mathematical Computing; Models, Biological; Models, Chemical
• ISSN: 0092-8240; 1522-9602
• DOI: 10.1006/bulm.2001.0263

We develop the mathematical machinery for the construction of an algebraic-combinatorial model using Petri nets to construct an oriented matroid representation of biochemical pathways. For demonstration purposes, we use a model metabolic pathway example from the literature to derive a general biochemical reaction network model. The biomolecular networks define a connectivity matrix that identifies a linear representation of a Petri net. The sub-circuits that span a reaction network are subject to flux conservation laws. The conservation laws correspond to algebraic-combinatorial dual invariants, that are called S- (state) and T- (transition) invariants. Each invariant has an associated minimum support. We show that every minimum support of a Petri net invariant defines a unique signed sub-circuit representation. We prove that the family of signed sub-circuits has an implicit order that defines an oriented matroid. The oriented matroid is then used to identify the feasible sub-circuit pathways that span the biochemical network as the positive cycles in a hyper-digraph.