Bifurcation from the potential field analog of some chemical systems

Abstract

Multistep enzyme-catalyzed reaction sequences with feedback chemical regulation are common features of cellular reaction networks. The stability properties of these reaction and control systems may influence enzyme and metabolite production in fermentation processes. The dynamic equations that describe an enzymatic reaction system with feedback regulation are shown to be reducible to a two-dimensional dynamic system which is analogous to the description of particle motion in a potential field. The mechanical analog of the chemical system provides useful intuitive insights and is used as a guide to study the chemical system dynamics. The approach is valid for any three-dimensional dynamic system of constant negative divergence that is reducible to a special normal form. Certain types of bifurcations of the chemical system are seen to correspond to a change in structure of the potential function. Hopf bifurcation is shown to correspond to loss of “friction” at a minimum of the potential function. A formula is obtained to determine the stability of the Hopf-induced limit cycles, and conclusions about the possible types of global dynamic behaviour are obtained.