| In general, chemical reaction networks define a set of nonlinear differential equations. We present two methods for qualitatively analyzing the set of nonlinear differential equations defined by chemical reaction networks. The first method is specific to a special class of chemical reaction networks that have deficiency zero, where the deficiency is akin to the size of the kernel of the set of differential equations. The primary characteristic of deficiency zero chemical reaction networks is that the corresponding set of differential equations exhibit a unique and strictly positive steady state solution, which is indicative of linear differential equations and, in general, not nonlinear differential equations. Moreover, this steady state is asymptotically stable. We develop a bijective transformation that allows us to analytically identify the steady state manifold of the nonlinear differential equations of deficiency zero chemical reaction networks by considering a suitable set of linear differential equations. Furthermore, the steady state of the linear system is asymptotically stable so that the resulting set of linear differential equations is qualitatively equivalent to the set of nonlinear differential equations. The second method for qualitatively analyzing chemical reaction networks makes use of hypergraph theory. A hypergraph is a generalization of a graph such that an edge may be incident with more than two vertices. This generalization provides a natural mechanism for modeling the nonlinear dynamics of chemical reaction networks, wherein the nonlinear dynamics are caused by several distinct molecules interacting to produce another distinct set of molecules. It is shown that the steady state solutions to the set of differential equations correspond to the nullspace of the incidence matrix representation of the hypergraph model of the chemical reaction network. We obtain an abstraction of the nullspace and the row space of the incidence matrix by generating oriented matroids defined over the set of integers. The abstraction of these two spaces allows us to enumerate the independent steady state solutions and flux conservations, which thereby establish a rank ordering of the molecular interdependencies that define the chemical reaction network. |
