Biochem. J. (2000) 345, 321–334 (Printed in Great Britain) 321
Effect of cellular interaction on glycolytic oscillations in yeast :
a theoretical investigation
Jana WOLF and Reinhart HEINRICH
1
Humboldt-Universita
$
t zu Berlin, Institut fu
$
r Biologie, Theoretische Biophysik, Invalidenstrasse 42, D-10115 Berlin, Germany
On the basis of a detailed model of yeast glycolysis, the effect of
intercellular coupling on the oscillatory dynamics is analysed
theoretically. The model includes the main steps of anaerobic
glycolysis, and the production of ethanol and glycerol. Trans-
membrane diffusion of acetaldehyde is included, since it has been
hypothesized that this substance mediates the interaction.
Depending on the kinetic parameters, the single-cell model shows
both stationary and oscillatory behaviour. This agrees with
experimental data with respect to metabolite concentrations and
phase shifts. The inclusion of intercellular coupling leads to a
variety of dynamical modes, such as synchronous oscillations,
and different kinds of asynchronous behaviour. These oscillations
can co-exist, leading to bi- and tri-rhythmicity. The corre-
sponding parameter regions have been identified by a bifurcation
analysis. The oscillatory dynamics of synchronized cell popul-
INTRODUCTION
Glycolytic oscillations are one of the best studied examples of
rhythmic behaviour on the cellular level. They occur in a broad
spectrum of cells, e.g. in yeast, muscle, heart, and in pancreatic
β-cells. The oscillations have been studied most extensively in
suspensions and extracts of yeast cells. It was observed that all
glycolytic intermediates oscillate with the same period, but with
different phases, and that this period is in the range of 1 min (for
recent reports, see [1,2]). The underlying oscillatory mechanism
was found to be strongly dependent on the kinetic properties of
the enzyme phosphofructokinase (PFK). This enzyme is highly
regulated: it is activated by its substrate, fructose 6-phosphate,
by its product, ADP and, in some cells, by its second product,
fructose 1,6-bisphosphate. ATP acts at higher concentrations as
an inhibitor, whereas AMP activates this enzyme. Theoretical
models studying the occurrence of glycolytic oscillations are
based on one or several of these regulatory effects. In addition to
core models [3–5], more detailed models [6–8] have also been
developed. Whereas the first group of models takes into account
a feedback activation by either fructose 1,6-bisphosphate or
Abbreviation used: PFK, phosphofructokinase.
System variables (concentrations of metabolites) used: S
1
, glucose ; S
2
, pool of glyceraldehyde 3-phosphate and dihydroxyacetone phosphate ; S
3
,
1,3-bisphosphoglycerate ; S
4
, pool of pyruvate and acetaldehyde in the cytosol ; S
ex
4
, coupling substance in the external solution; A
2
, ADP; A
3
, ATP; N
1
,
NAD
+
; N
2
, NADH.
Parameters used: J
0
, input flux of glucose via the cellular membrane; k
1
, rate constant of the lumped hexokinase/phosphoglucoisomerase/PFK
reaction ; k
2
, rate constant of the glyceraldehyde-3-phosphate dehydrogenase reaction ; k
3
, rate constant of the lumped phosphoglycerate
kinase/phosphoglycerate mutase/enolase/pyruvate kinase reaction ; k
4
, rate constant of the alcohol dehydrogenase reaction ; k
5
, rate constant of non-
glycolytic ATP consumption; k
6
, rate constant of the lumped reaction transforming triose phosphates into glycerol ; k, rate constant of the degradation
of the coupling substance within the extracellular medium; κ, kinetic constant of the transmembrane flux of the coupling substance ; }, ratio of the
total cellular volume to the extracellular volume; N, sum of the concentrations of NAD
+
and NADH; A, sum of the concentrations of ADP and ATP;
n, number of interacting cells ; T, oscillation period ; τ, phase shift between cells.
1
To whom correspondence should be addressed (e-mail reinhart¯heinrich!rz.hu-berlin.de).
ations are investigated by calculating the phase responses to
acetaldehyde pulses. Simulations are performed with respect
to the synchronization of two subpopulations that are oscillating
out of phase before mixing. The effect of the various processes on
synchronization is characterized quantitatively. While continu-
ous exchange of acetaldehyde might synchronize the oscillations
for appropriate sets of parameter values, the calculated synchron-
ization time is longer than that observed experimentally. It is
concluded either that, in addition to the transmembrane exchange
of acetaldehyde, other processes may contribute to intercellular
coupling, or that intracellular regulatory feedback plays a role in
the acceleration of the synchronization.
Key words: control coefficient, glycolysis, metabolic oscillations,
phase response, synchronization.
ADP, the second group includes activatory and inhibitory effects
of AMP and ATP respectively. The incorporation of adenine
nucleotide concentrations as system variables necessitates the
inclusion of ATP production in the second part of glycolysis, as
well as of non-glycolytic, ATP-consuming processes (ATPases).
It is generally thought that, in muscle cells, the generation of
oscillations is due to the regulation of PFK by fructose 1,6-
bisphosphate [8,9], whereas, in yeast, the same enzyme is mainly
regulated by the adenine nucleotides.
In yeast cell suspensions the oscillations of glycolysis have
been followed experimentally on the population level via moni-
toring mean concentrations, i.e. of NADH. In this way, only
limited information has been obtained about the dynamics of a
single cell within such a population [10]. Obviously, the existence
of sustained oscillations on the population level implies that the
individual cells also oscillate, and that the single cells are
synchronized, at least to a certain extent. The existence of
synchronizing interactions was also demonstrated directly by the
mixing of two subpopulations that oscillate out of phase. In
the resulting population, the oscillations are first damped, but
reoccur after several cycles [11–13].
# 2000 Biochemical Society

322 J. Wolf and R. Heinrich
The models of glycolytic oscillations mentioned above are
restricted to the case of single cells. The mathematical description
of the effects of cellular coupling is still an open problem; in
particular, for complex metabolic processes. A number of experi-
ments, studying the nature of coupling, have suggested that cells
interact by the exchange of a metabolic intermediate via the
external medium. Ethanol, pyruvate and acetaldehyde have been
discussed as candidates for the coupling metabolites [10,11,14].
Recent studies have yielded strong evidence that acetaldehyde
mediates the coupling [13]. In particular, it was demonstrated
that acetaldehyde is secreted by the cells, and that the extracellular
concentration of this compound oscillates. Moreover, the cells
respond to acetaldehyde pulses. These observations imply that
the dynamics of a population are determined not only by the
intracellular kinetics, but also by the exchange of substances
between cells. The consequences of this hypothesis are addressed
in the present theoretical study.
There are some indications that populations of cells do not
always synchronize, but might also lose their synchrony [10].
This corresponds to theoretical results obtained for a two-
component model, in which oscillations originate from a
feedback-activation mechanism, and in which the effects of
interactions within a population of identical cells were investi-
gated in detail [15]. It was found that the transmembrane diffusion
of a metabolite might lead to synchronized, as well as to
desynchronized, states, depending on the values of the system
parameters. In this basic model, two different types of asyn-
chronous limit cycles may arise, corresponding (i) to regular
asynchronous behaviour, where the amplitudes of the oscillating
variables are the same in all cells and where the phase shifts
between the cells are multiples of the reciprocal value of the
number of cells, and (2) to non-regular asynchronous oscillations
with non-identical oscillations in different cells and non-regular
phase shifts. Simulations show that this dynamical behaviour
might result in ‘hidden oscillations ’, where large variations in the
cellular concentrations are not represented by the corresponding
mean values.
In the present study, we extend our analysis of intercellular
communication to the case of glycolytic oscillations. To meet this
aim, we have developed a detailed model for anaerobic energy
metabolism in yeast cells. Compared with our previous investi-
gations [15], we have arrived at a model comprising many
variables for a single cell. A consideration of the coupling
of these cells leads to a further increase of complexity, which can
be analysed only by numerical methods. The identification
of parameter regions with different dynamic behaviour was
performed by bifurcation analysis using the software package
AUTO [16].
THE MODEL
The reaction scheme
The starting point for our model is energy metabolism in yeast
cells under anaerobic conditions, with the respiratory chain in a
completely inhibited state and only alcoholic fermentation taking
place. The reaction network of a single cell is represented by
Scheme 1. It contains the main reactions of glycolysis, and
adjacent reactions producing ethanol and glycerol.
The individual processes in Scheme 1 are shown to represent
the following: flux 0 (J
!
), input of glucose via the cellular
membrane; reaction 1 (where the reactions are shown in the
Scheme by Š
"
, Š
#
, etc.), lumped reactions of hexokinase, phospho-
glucoisomerase and PFK; reaction 2, glyceraldehyde-3-phos-
phate dehydrogenase reaction; reaction 3, lumped reactions of
phosphoglycerate kinase, phosphoglycerate mutase, enolase and
Scheme 1 Reaction scheme for a single cell
The scheme shows the main reactions of anaerobic glycolysis in yeast, in addition to
transmembrane transport of glucose and the coupling substance (pyruvate and/or acetaldehyde).
For further details and explanation of the symbols used, see the text.
pyruvate kinase ; reaction 4, alcohol dehydrogenase reaction;
reaction 5, non-glycolytic ATP consumption; reaction 6,
formation of glycerol from triose phosphates ; and reaction 7,
degradation of the coupling substance in the extracellular me-
dium. Furthermore, the model includes the membrane transport
of the coupling substance, characterized by the flux, J.
A
$
and A
#
, and N
"
and N
#
denote the concentrations of ATP
and ADP, and NAD
+
and NADH, respectively. Owing to the
fact that several glycolytic reactions are omitted and that other
reactions are lumped, the model variables denote, in some cases,
the concentrations of pools of intermediates, rather than concen-
trations of individual compounds. This concerns the pool of the
triose phosphates, glyceraldehyde 3-phosphate and dihydroxy-
acetone phosphate (variable S
#
), and the pool of pyruvate and
acetaldehyde (variable S
%
). The concentration of glucose is
represented by the variable S
"
, and that of 1,3-bisphospho-
glycerate by S
$
. Furthermore, the lumping process implies that
the concentrations of some compounds do not appear as separate
model variables. Such a reduction in model complexity may be
justified by quasi-steady-state approximations for the concen-
tration of metabolites located between the lumped reactions, as
shown in detail previously [17].
# 2000 Biochemical Society