Combinatorial and topological methods in nonlinear chemical kinetics

Abstract

Combinatorial and topological techniques are developed to classify nonlinear chemical reaction networks in terms of their qualitative dynamics. A class of N coupled equations, based on a hypothesis concerning biological control by Monod and Jacob is derived. Transitions between volumes in concentration space for these equations are represented as directed edges on N cubes (hypercubes in N dimensions). A classification of the resulting state transition diagrams for N=2,3 is given. A version of a topological theorem by Poincaré and Hopf is derived which is appropriate for application to chemical systems. This theorem is used to predict the existence of critical points in continuous nonlinear equations with oscillation and bistability on the basis of their state transition diagrams. A large number of nonlinear kinetic equations proposed in previous studies by other authors are classified in terms of their state transition diagrams.