Hypergraphic Oriented Matroid Relational Dependency Flow Models of Chemical Reaction Networks

In this paper we derive and present an application of hypergraphic oriented matroids for the purpose of enumerating the variable interdependencies that define the chemical complexes associated with the kinetics of non-linear dynamical system representations of chemical kinetic reaction flow networks. The derivation of a hypergraphic oriented matroid is obtained by defining a closure operator on families of n-subsets of signed multi-sets from which a "Z-module" is obtained. It has been observed that every instantiation of the closure operator on the signed multiset families define a matroid structure. It is then demonstrated that these structures generate a pair of dual matroids corresponding respectively to hyperspanning trees and hypercycles obtained from the corresponding directed hypergraphs. These structures are next systematically evaluated to obtain solution sets that satisfy systems of non-linear chemical kinetic reaction flow networks in the MAP Kinase cascade cell-signaling pathway.