1- and 2-topology of reaction networks

Abstract

The kinds of reaction networks introduced earlier by this writer are capable of diverse applications in chemical physics, biochemistry, chemical engineering, economics, ecological, and other dynamics. All possible mechanisms or pathways as a function of the numbers of reaction steps ρ or species σ are generated with them. They also give the rate laws, multiplicity of steady states, the nature of their dynamic instabilities, oscillations, etc. These properties are related to a large extent on the 1- and 2-topology of the networks, {𝒩}. The {𝒩} are graphs of two kinds of lines and two kinds of vertices. They can be planar or nonplanar. The genus t and thickness t of any 𝒩 are related to ρ, σ and the numbers of catalytic and autocatalytic cycles. The Betti numbers Bi(p) of 1- and 2-complexes constituted by 𝒩 and other topological invariants of the networks under two kinds of homeomorphisms are given. A number of theorems are stated and proved. The above reaction networks are interesting mathematical objects in that they help classify coupled nonlinear differential equations.