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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Oliveira, J.S., Bailey, C.G., Jones-Oliveira, J.B. et al. An algebraic-combinatorial model for the identification and mapping of biochemical pathways. Bull. Math. Biol. 63, 1163–1196 (2001). https://doi.org/10.1006/bulm.2001.0263
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DOI: https://doi.org/10.1006/bulm.2001.0263