Chaos In Oscillating Chemical Reactions: The Peroxidase-Oxidase Reaction

The peroxidase-oxidase (PO) reaction is an important example of how oscillating reactions arise in living organisms. The reaction is important mathematically because exhibits many characteristics of chaos. Of these characteristics, sensitive dependence on initial conditions illustrates how predicting the future state of the PO reaction is next to impossible. For this reason, chaotic behavior in living organisms presents many obstacles to chemists and biologists trying to predict how systems will react to perturbations. This paper explores the chaotic behaviors of the PO reaction, using a system of four differential equations as a model. The topics analyzed are timeseries data, chaotic attractors, bifurcations, tests for sensitive dependence, Lyapunov exponents, and onedimensional time delay embedding. Oscillating Reactions Less than a century ago, oscillating chemical reactions were thought to be nothing more than erroneous results caused by chemical impurities. Few believed that a reaction, in proceeding to equilibrium, could keep oscillating back and forth between two or more colors. The LotkaVolterra Oscillator and the Belousov-Zhabotinsky reactions were two of the first oscillating reactions to be studied extensively and universally accepted to have oscillating concentrations of reactants [1]. Finding chaos in oscillating reactions is a much more recent undertaking, gaining popularity with the rise of computers. In fact there is still debate as to whether oscillating chemical reactions truly exhibit chaos, or if they are merely oscillatory with very large period [2]. However since the discovery and eventual acceptance of oscillatory reactions, many more such reactions have been designed and discovered. One of the main areas of discovery of oscillating reactions is in living systems. The peroxidaseoxidase (PO) reaction is a prime example of such a reaction. It was one of the first reactions outside of the BZ reactions to be classified as oscillating and chaotic, and is an extensively studied example of an in vivo oscillating reaction (the reaction can be carried out both in vivo and in vitro). Molecular oscillating reactions are an important area of study, as they are essential to understanding the more complex oscillating systems of organism (eg. a heartbeat)[1]. The Peroxidase-Oxidase Reaction The peroxidase-oxidase reaction is an enzyme catalyzed redox reaction. Nicotinamide adenine dinucleotide (NADH) is oxidized and molecular oxygen acts as an electron receiver. The net reaction is 2NADH + O2 + 2H → 2NAD + 2H2O O2 and NADH are continuously added to and products are continuously removed from the experimental system via the use of a continuousflow, stirred tank reactor (CSTR). Experimentation has shown that the PO reaction exhibits oscillatory behavior and a period-doubling1 route to chaos [3]. These characteristics have been effectively modeled using a simplified eight-step 1Here I have adopted the terminology of [3]. This does not mean period-doubling in the traditional sense. From here on out, period-doubling refers to the maxima in the timeseries data. When the number of maxima in a period in the timeseries data doubles, we call it period-doubling.