An algebraic-combinatorial model for the identification and mapping of biochemical pathways

We develop the mathematical machinery for the construction of an algebraic- combinatorial model using Petri nets to construct an oriented matroid repre sentation of biochemical pathways. For demonstration purposes, we use a mod el metabolic pathway example from the literature to derive a general bioche mical reaction network model. The biomolecular networks define a connectivi ty matrix that identifies a linear representation of a Petri net. The sub-c ircuits that span a reaction network are subject to flux conservation laws. The conservation laws correspond to algebraic-combinatorial dual invariant s, that are called S- (state) and T- (transition) invariants. Each invarian t has an associated minimum support. We show that every minimum support of a Petri net invariant defines a unique signed sub-circuit representation. W e prove that the family of signed sub-circuits has an implicit order that d efines an oriented matroid. The oriented matroid is then used to identify t he feasible sub-circuit pathways that span the biochemical network as the p ositive cycles in a hyper-digraph. (C) 2001 Society for Mathematical Biolog y.