Interacting Hopf Oscillators

Anders Bisgaard

Department of Chemistry, Lab. 3, Universitetsparken 5, DK-2100 Copenhagen, Denmark

Abstract: We investigate the phenomenon of oscillations in the concentrations of chemical species in a population of living yeast cells. The biological conditions for the transient oscillations of the NADH concentration in yeast cells were discovered by Chance et. al. in 1964. Chance showed that the oscillations had to do with the glycolysis in each of the individual yeast cells. Thus the individual yeast cell functions as an oscillator oand the ensemble of cells works as a conglomerate of independent oscillators. The oscillations can be observed in experiments (Sune Danoe et. al.) collectively or macroscopically only when all yeast cells are oscillating in phase or close synchrony. For the cells to oscillate in a coherent fashion requires that each cell can communicate with the surrounding cells signalling the state of oscillation, that is amplitude and phase. The cell communication is brought about by the exchange of chemical species between the cell and the surrounding extracellular medium. The modelling of this complex yeast cell system is based upon an accurate and rather complete full-scale model of differential equations describing the biochemical reaction network of the glycolysis taking place in the individual yeast cell (Hynne et. al.). The full-scale model has, on the basis of substantial experimental data, been developed and optimized such as to comply with experiments in a quantitative manner. The objective is to apply the concepts from nonlinear dynamics especially the method of amplitude equations to derive a mathematical model, based on the full-scale model for each cell, that captures the key properties of the entire yeast cell ensemble. In this study we will analyse the model and develope synchronization conditions.