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.