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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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