title: An algebraic-combinatorial model for the identification and mapping of biochemical pathways.
abstract: 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.
authors.affiliation: Environmental Molecular Sciences Laboratory, Theory, Modeling and Simulation Group, Pacific Northwest, National Laboratories, Richland, Washington, USA. jso@pnl.gov
authors.forename: J S
authors.initials: JS
authors.lastname: Oliveira
authors.forename: C G
authors.initials: CG
authors.lastname: Bailey
authors.forename: J B
authors.initials: JB
authors.lastname: Jones-Oliveira
authors.forename: D A
authors.initials: DA
authors.lastname: Dixon
mesh_terms: D016014:Linear Models; D008432:Mathematical Computing; D008954:Models, Biological; D008956:Models, Chemical