Chemical interpretation of oscillatory modes at a Hopf point

Abstract

We present two complementary methods for studying the oscillatory mechanisms in a chemical reaction network in the neighbourhood of a supercritical Hopf bifurcation. The first method is a modification of metabolic control analysis (a form of sensitivity analysis), and focuses on the reactions rather than the chemical species. By rephrasing metabolic control analysis in terms of the amplitude equation of the Hopf bifurcation, we show that control of amplitude and frequency of the oscillations should be considered separately, and that the amplitude control is directly related to the control of the stability of the stationary state. Generally, the frequency of the oscillations is controlled by more reactions than the amplitude is, and those reactions controlling amplitude will generally also exert control of the frequency. The second method focuses on the role of the chemical species. By considering their relative phases and amplitudes, the method reveals to what extent a simple activator–inhibitor interpretation of the amplitude equation associated with the Hopf bifurcation corresponds to an equally simple chemical interpretation. If applicable, the method identifies the activating and inhibiting modes chemically. Prior knowledge of the underlying reaction network is not needed, only phase and amplitude measurements are used in the analysis. Hence, this method is a top-down approach well suited for systems biology. Both methods are exemplified by calculations on the Oregonator model for the Belousov–Zhabotinsky reaction.