Hyperdigraph-theoretic analysis of the EGFR signaling network: initial steps leading to GTP:Ras complex formation

We construct an algebraic-combinatorial model of the SOS compartment of the EGFR biochemical network. A Petri net is used to construct an initial representation of the biochemical decision making network, which in turn defines a hyperdigraph. We observe that the linear algebraic structure of each hyperdigraph admits a canonical set of algebraic-combinatorial invariants that correspond to the information flow conservation laws governing a molecular kinetic reaction network. The linear algebraic structure of the hyperdigraph and its sets of invariants can be generalized to define a discrete algebraic-geometric structure, which is referred to as an oriented matroid. Oriented matroids define a polyhedral optimization geometry that is used to determine optimal subpaths that span the nullspace of a set of kinetic chemical reaction equations. Sets of constrained submodular path optimizations on the hyperdigraph are objectively obtained as a spanning tree of minimum cycle paths. This complete set of subcircuits is used to identify the network pinch points and invariant flow subpaths. We demonstrate that this family of minimal circuits also characteristically identifies additional significant biochemical reaction pattern features. We use the SOS Compartment A of the EGFR biochemical pathway to develop and demonstrate the application of our algebraic-combinatorial mathematical modeling methodology.