# Hydrobates

A mathematician thinks aloud

## Albert Goldbeter and glycolytic oscillations

This Christmas, at my own suggestion, I was given the book ‘La Vie Oscillatoire’ by Albert Goldbeter as a present. This book is concerned with oscillatory phenomena in biological systems and how they can be explained and modelled mathematically. After the introduction the second chapter is concerned with glycolytic oscillations. I had a vague acquaintance with this subject but the book has given me a much better picture. The chapter treats both the theoretical and experimental aspects of this subject.

If yeast cells are fed with glucose they convert it into alcohol. Those of us who appreciate alcoholic beverages can be grateful to them for that. In the presence of a supply of glucose with a small constant rate alcohol is produced at a constant rate. When the supply rate is increased something more interesting happens. The output starts to undergo periodic oscillations although the input is constant. It is not that the yeast cells are using some kind of complicated machine to produce these. If the cells are broken down to make yeast extract the effect persists. In fact for yeast extract the oscillations go away again for very high concentrations of glucose, an effect not seen for intact cells. This difference is not important for the basic mechanism of production of oscillations. The breakdown of sugar in living organisms takes place via a process called glycolysis consisting of a sequence of chemical reactions. By replacing the input of glucose by an input of each of the intermediate products it was possible to track down the place where the oscillations are generated. The enzyme responsible is phosphofructokinase (PFK), which converts fructose-6-phosphate into fructose-1,6-bisphosphate while converting ATP to ADP to obtain energy. Now ADP itself increases the activity of PFK, thus giving a positive feedback loop. This is what leads to the oscillations. The process can be modelled by a two-dimensional dynamical system called the Higgins-Selkov oscillator. Let $S$ and $P$ denote the concentrations of substrate and product respectively. The substrate concentration satisfies an equation of the form $\dot S=k_0-k_1SP^2$. The substrate is supplied at a constant rate and used up at a rate which increases with the concentration of the product. (Here we are thinking of ADP as the product and ignoring other possible effects.) The product concentration correspondingly satisfies $\dot P=k_1 SP^2-k_2 P$.

The Higgins-Selkov oscillator gives rise to a limit cycle by means of a Hopf bifurcation. The ODE system is similar to the Brusselator. There are two clear differences. The substance which is being supplied from ouside occurs linearly in the nonlinear term in the Higgins-Selkov system and quadratically in the Brusselator. In the Higgins-Selkov system the nonlinear term occurs with a negative sign in the evolution equation for the substance being supplied from outside while in the Brusselator it occurs with a positive sign. In the book of Goldbeter the Higgins-Selkov oscillator seems to play the role of a basic example to illustrate the nature of biological oscillations.
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