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

Oliveira, Joseph S.; Bailey, Colin G.; Jones-Oliveira, Janet B.; Dixon, David A.

In: Bulletin of Mathematical Biology 63 (6), pages 1163-1196. November 2001.

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.

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