Allosteric regulation of phosphofructokinase controls theemergence of glycolytic oscillations in isolated yeast cellsAnna-Karin Gustavsson1*, David D. van Niekerk2*, Caroline B. Adiels1, Bob Kooi3, Mattias Goks€or1

and Jacky L. Snoep2,4,5,

1 Department of Physics, University of Gothenburg, Sweden

2 Molecular Cell Physiology, Department of Biochemistry, Stellenbosch University, Matieland, South Africa

3 Theoretical Biology, VU University, Amsterdam, The Netherlands

4 Molecular Cell Physiology, VU University, Amsterdam, The Netherlands

5 Manchester Centre for Integrative Systems Biology, Manchester Interdisciplinary Biocentre, The University of Manchester, UK

Keywords

cell heterogeneity; glycolysis; limit cycle

oscillation; mathematical model; microfluidic

chamber

Correspondence

J. L. Snoep, Molecular Cell Physiology,

Department of Biochemistry, Stellenbosch

University, Private Bag X1, Matieland 7602,

South Africa

Fax: +27 218085863

Tel: +27 218085844

E-mail: jls@sun.ac.za

*These authors contributed equally to this

work.

(Received 21 November 2013, revised 25

February 2014, accepted 10 April 2014)

doi:10.1111/febs.12820

Oscillations are widely distributed in nature and synchronization of oscilla-

tors has been described at the cellular level (e.g. heart cells) and at the popu-

lation level (e.g. fireflies). Yeast glycolysis is the best known oscillatory

system, although it has been studied almost exclusively at the population

level (i.e. limited to observations of average behaviour in synchronized cul-

tures). We studied individual yeast cells that were positioned with optical

tweezers in a microfluidic chamber to determine the precise conditions for

autonomous glycolytic oscillations. Hopf bifurcation points were determined

experimentally in individual cells as a function of glucose and cyanide

concentrations. The experiments were analyzed in a detailed mathematical

model and could be interpreted in terms of an oscillatory manifold in a three-

dimensional state-space; crossing the boundaries of the manifold coincides

with the onset of oscillations and positioning along the longitudinal axis of

the volume sets the period. The oscillatory manifold could be approximated

by allosteric control values of phosphofructokinase for ATP and AMP.

Database

The mathematical models described here have been submitted to the JWS Online Cellular

Systems Modelling Database and can be accessed at http://jjj.mib.ac.uk/webMathematica/

UItester.jsp?modelName=gustavsson5.

[Database section added 14 May 2014 after original online publication]

Introduction

Glycolysis is an important pathway in central carbon

metabolism of almost all organisms. Although the

pathway is mostly studied under stable steady-state

conditions, it is well known that the concentrations of

intermediates in the pathway can display oscillatory

behaviour. These oscillations were first discovered in

intact yeast cells and later in yeast extracts [1].

Although glycolytic oscillations were also shown for

various other cell types [2, 3], yeast has remained the

model organism of choice in experimental and theoret-

ical approaches.

Individual yeast cells can synchronize their oscilla-

tions in high-density cell cultures and, typically, an

average population signal is measured [1]. In such

cultures, the individual cells are coupled via extracel-

lular acetaldehyde [4, 5], although the mechanism of

Abbreviations

2D, two-dimensional; 3D, three-dimensional; ACA, acetaldehyde; CN, cyanide; DHAP, dihydroxyacetone phosphate; F6P, fructose

6-phosphate; F16bP, fructose 1,6-bisphosphate; G3PDH, glycerol-3-phosphate dehydrogenase; G6P, glucose 6-phosphate; Glc, glucose;

PFK, phosphofructokinase.

1FEBS Journal (2014) ª 2014 FEBS

synchronization has been a matter of debate. The

dependency of synchronization on population density

[6] and the apparent lack of oscillations in individual

cells [7, 8] points to the population oscillations as a

collective property [6], where loss of coherence is the

result of a loss of oscillations in all cells. However,

our recent discovery of oscillations in isolated cells [9]

and the observation of individually oscillating cells in

cultures [10] contradicts this viewpoint, and would

suggest that coherence in population oscillations is

the result of synchronization of individual oscillators

via a so-called Kuramoto transition [11, 12].

Studying oscillations in isolated cells allows for criti-

cal testing of conditions where individual cells switch

from steady-state behaviour to oscillatory behaviour

(at a so-called Hopf bifurcation point) and distinguish-

ing this from the conditions for synchronization at a

population level. We used microfluidic flow chambers

[9, 13] in combination with optical tweezers [14, 15]

for precise cell positioning and spatial and temporal

control of the extracellular conditions. Once the condi-

tions for oscillations are precisely mapped, an attempt

can be made to relate these conditions back to the

mechanism responsible for the glycolytic oscillations.

To understand network-emergent properties such as

Hopf bifurcations, mathematical models are required

to formulate and test hypotheses. Many mathematical

models for yeast glycolysis have been constructed,

mostly focussing on either stable steady-state behaviour

[16] or limit cycle oscillations [17]. To interpret model

behaviour in terms of enzyme kinetic mechanisms, such

models must be based on experimentally verified kinetic

data for the isolated enzymes. These detailed models

are importantly different from so-called core-models

[18–22], which are typically used to test whether a

certain mechanism can display certain types of behav-

iour, whereas the detailed models can be used to test

whether a model based on known, experimentally

validated kinetic mechanisms predicts the observed

behaviour.

Two very different mechanisms for the glycolytic

oscillations have been proposed on the basis of core

model analyses: the first mechanism puts the phospho-

fructokinase (PFK) forward as the oscillophore (i.e. as

the master reaction controlling the glycolytic oscilla-

tions) [23], whereas, in the second mechanism, the

autocatalytic stoichiometry of the glycolytic pathway

was suggested as a positive feedback leading to oscilla-

tions [19]. We have criticised the oscillophore concept

[24] because the oscillations are systems properties and

no single reaction or property fully controls the oscil-

lations, although it is interesting to test the suggested

mechanisms in detailed kinetic models.

In a recent set of studies, we adjusted a previous

model [16] to display limit cycle oscillations [ 9, 25 26]

thereby obtaining a detailed oscillatory model that was

originally based on experimentally measured enzyme

kinetic parameters. This model was adapted to reflect

the microfluidic environment and to include new

experimental data (for details, see Materials and meth-

ods; Doc. S1. We analyzed the experimental data with

this detailed kinetic model for yeast glycolysis and,

despite the high dimensionality of the mathematical

model, we could interpret the onset and period of

oscillations in terms of positioning in a state-space

spanned by only three variables. These three variables

could be related back to allosteric control of the PFK

by ATP and AMP, emphasizing the importance of this

enzyme in determining the conditions for glycolytic

oscillations.

Results and Discussion

We first set out to precisely determine the conditions

under which individual cells oscillate. To be sure that

we analyzed individual cells without cell–cell interac-

tions, we positioned the cells far apart in a microflui-

dic chamber and we used relatively high flow rates to

ensure homogenous extracellular conditions with

near-zero concentrations of acetaldehyde (ACA).

ACA is the coupling agent between the cells and we

assumed that with the low extracellular concentra-

tions (< 15 nM) no cell–cell interaction occurs. For

the precise experimental set-up, see Materials and

methods.

From population studies, it is known that changes in

the concentration of extracellular glucose (Glc) and

cyanide (CN) can induce oscillations, where Glc affects

the input flux and CN acts by binding acetaldehyde

(intra- and extracellular) and dihydroxyacetone phos-

phate (DHAP) and pyruvate (intracellular), and inhib-

its respiration by binding cytochrome c oxidase [27].

To compare our individual cell results with the popula-

tion experiments, we analyzed a large number of indi-

vidual cells with respect to their dynamics and

expressed the results as percentage of oscillatory cells

times their amplitude. In this way, we analyze the aver-

age amplitude of individual cells, independent of their

phase, to distinguish between oscillatory behaviour and

synchronization. We were specifically interested in two

aspects that are currently not well-understood: first, the

presence of two Hopf bifurcations in CN titrations

and, second, the decrease in oscillatory period that is

observed upon increasing the Glc concentration. We

first present the experimental results and then discuss

the model simulations of these experiments.

2 FEBS Journal (2014) ª 2014 FEBS

Emergence of yeast glycolytic oscillations A.-K. Gustavsson et al.

We observed a single Hopf bifurcation in Glc titra-

tions (Fig. 1A) and two Hopf bifurcations in CN titra-

tion experiments (Fig. 1B). Thus, the single cell results

are qualitatively similar to those observed for popula-

tion studies, although there was a significant difference

in the concentrations at which the Hopf bifurcations

occurred. The Glc bifurcation point lies at 0.25 mM

Glc in individual cells (Fig. 1A), compared to 1.6 mM

in populations [17]. For a comparison of the CN bifur-

cation points, see Fig. 1B. In the same experiments,

the period was determined for the oscillatory cells and

the effects of Glc and CN on the oscillation period are

presented in Fig. 2.

The period of glycolytic oscillations observed in syn-

chronized yeast populations is known to have a small

negative dependence on Glc concentrations [17, 28].

In individual cells, we observed a similar period depen-

dence as in populations, although this dependency

increased strongly at low Glc concentrations (Fig. 2).

At these lower Glc concentrations, no oscillations are

observed in population studies, reflecting the more

stringent conditions for synchronization compared to

oscillation. Varying the CN concentration did not have

a significant effect on the period of the oscillations

(Fig. 2).

For the analysis of the experimental data, we used a

detailed kinetic model [9] that was adapted to accu-

rately reflect the microfluidic chamber (see Materials

and methods; and for a complete model description,

see Supporting information). Experimental results indi-

cated a counter-intuitive increase in NADH concentra-

tion upon decreasing the Glc concentration, which was

not predicted either by the previous model [9] or any

other detailed kinetic model. A modification was there-

fore made to the kinetics of the glycerol branch (which

resulted in the correct model behaviour; Fig. 3) and,

A

B

Fig. 1. Amplitude dependence of glycolytic oscillations on CN and

Glc concentrations. (A) The amplitude of the oscillations in

individual cells (experimental data, with errorbars indicating the SD,

and model simulations, dashed line) as a function of Glc

concentration (amplitude calculated as NADH fluorescence

(arbitrary units) multiplied by the percentage of cells that were

oscillatory). CN concentration was kept constant at 5 mM. (B) The

amplitude of the oscillations in individual cells as a function of CN

concentration with a Glc concentrations of 20 mM. Model

simulations for the experiment are indicated by a solid line and the

grey-shaded area indicates the upper and lower boundaries of a

5% variation in activity of the Glc transporter, which has a strong

effect on the input flux of the system. The inset in (B) shows the

same experiment for a dense yeast population [17], where the

amplitude is the average of the population and dependent on

synchronization between the cells.

Fig. 2. The change in oscillation period as a function of Glc (CN

fixed at 5 mM) and CN (Glc fixed at 20 mM) concentration, with

the respective model simulations (solid and dashed lines for Glc

and CN titrations, respectively). Average frequencies were

calculated from a large number of oscillatory cells, with error bars

denoting the SD. The grey-shaded area indicates the upper and

lower boundaries of a 5% variation in activity of the Glc

transporter, which has a strong effect on the input flux of the

system.

3FEBS Journal (2014) ª 2014 FEBS

A.-K. Gustavsson et al. Emergence of yeast glycolytic oscillations

subsequently, the model was validated by its ability to

predict the correct period response upon a change in

extracellular Glc.

The calibrated model was able to semi-quantitatively

predict the CN and Glc Hopf bifurcations (Fig. 1),

and the period response to changes in Glc and CN

concentration (Fig. 2). The effect of a perturbation of

the Glc transporter (� 5%) is indicated by the grey-

shaded areas in Figs 1 and 2. We show the sensitivity

for the Glc transporter activity because this enzyme is

strongly affected during Glc starvation in the cell prep-

aration for the experiments. The conditions necessary

for oscillations in individual cells are less restrictive

than those for synchronized oscillations in population

studies, making the synchronization conditions a sub-

set of the oscillation conditions, as would be expected

in Kuramoto transitions [11, 12].

In a recent study [29], a detailed kinetic model for

yeast glycolytic oscillations was reduced to three vari-

ables on the basis of dissipative time scale differences.

Inspired by this strong model reduction, we tried to

analyze the oscillatory behaviour of our detailed,

high-dimensional model on the basis of three vari-

ables: ATP, fructose 6-phosphate (F6P) and fructose

1,6-bisphosphate (F16bP). For the 18 variable model,

we made an extensive perturbation analysis by vary-

ing: (a) the external variables Glc and CN, similarly

to the experimental perturbations; (b) multipliers (ran-

domly selected from the range 0.75–1.25) for each of

the reaction processes; and (c) by adding new reac-

tions that synthesize or consume F6P or F16bP and

varying the ATPase reaction rate, to ensure noncorre-

lated variations in the metabolites. After each pertur-

bation, the steady-state was calculated and indicated

as blue (stable steady-state) or red (unstable steady-

state) symbols, respectively, in the state-space spanned

by the three variables F6P, F16bP and ATP (Fig. 4).

The unstable steady-states indicated limit cycle oscilla-

tions, as confirmed by the presence of supercritical

Hopf bifurcations at the border between unstable

and stable steady-states, when analyzed with the

bifurcation package AUTO (for details, see Materials

and methods). The unstable steady-states were all

contained in a single, continuous volume in the

Fig. 3. Experimental (dashed line) and simulation (solid line) results

for an individual yeast cell are shown for a downshift in Glc (at

t = 10 min) from a saturating concentration (20 mM) to a

concentration close to the Hopf bifurcation (0.5 mM). The counter-

intuitive increase in NADH concentration upon decreasing the Glc

concentration was crucial for understanding the period dependency

of the oscillations with respect to the external Glc concentration.

Fig. 4. Interpretation of the oscillatory behaviour as a 3D manifold.

The simulation results of an extensive parameter perturbation of

the detailed model are plotted in three dimensions. For each

perturbation we calculated the steady-state concentrations of

F16bP, F6P and ATP and we plotted their values (respectively as x,

y and z values) for the unstable solutions, shown as red dots in the

central plot. These points define the region where limit cycle

oscillations are observed. Independent of the perturbation, a

unique, continuous 3D manifold contains all oscillatory solutions.

The inset shows a magnification of the region where the Glc and

CN titrations occur and indicates the change in period, decreasing

from red to turquoise, as the manifold is traversed (note that the

blue region indicates stable steady-states). Here, two trajectories

are shown: Glc varying from 1 to 50 mM (CN fixed at 5 mM)

indicated by the solid line, and CN varying from 0 to 30 mM (Glc

fixed at 20 mM) indicated by the dashed line, corresponding to the

experiments shown in Fig. 2. The closed circles on the trajectories

indicate the Hopf bifurcation points, where the transition from

steady-state (blue points) to oscillatory behaviour occurs (one Hopf

bifurcation point for the Glc titration, two for the CN titration). The

inset shows cross-sections of the 3D manifold where blue

indicates the stable non-oscillatory steady-states surrounding the

oscillatory volume.

4 FEBS Journal (2014) ª 2014 FEBS

Emergence of yeast glycolytic oscillations A.-K. Gustavsson et al.

three-variable state-space, in agreement with the previ-

ous study [29]. However, the border of the oscillatory

manifold was not exact. When analyzed in AUTO, we

saw small differences in the exact positioning of the

Hopf bifurcation points. In the perturbation method,

we noted that, with large perturbations in the multi-

pliers for the reaction processes, there was some over-

lap between the red and blue areas, although this was

only the case for very small ranges in the F6P, F16bP

and ATP concentrations. This overlap indicates that

some of the other variables of the system had a small

contribution to the Hopfbifurcation points.

The period dependency of the glycolytic oscillations

could be interpreted as a function of the positioning in

the three-dimensional (3D) oscillatory volume, show-

ing a strong dependency upon longitudinal movement

and very little changes in period upon transversal

movement (Fig. 4). Simulating the CN titration in the

model shows that the experiment cuts transversal

through the 3D oscillatory manifold, leading to the

two Hopf bifurcations and no significant change in

period, whereas the Glc titration moves longitudinally

into the manifold, with a much stronger effect on the

period of the oscillations. These results indicate that

the contributions of these three variables to the eigen-

values of the Jacobian matrix, both to the real part

(positioning of the Hopf bifurcation points) as well as

to the imaginary part (correlated to the period

response) of the complex conjugate pair, dominate the

contributions of the other variables.

The three variables that span the 3D oscillatory

state-space are all effectors of the PFK, an enzyme

that has been suggested as the oscillophore [23] for

glycolytic oscillations on the basis of its kinetic mecha-

nism [18, 20], where allosteric regulation by ATP and

AMP lead to instabilities causing the oscillations. We

tested this hypothesis by analysing the elasticity [30] of

PFK for ATP and AMP in the 3D volume. Strikingly,

the 3D manifold (red region in Fig. 5C,D) can be

approximated within specific allosteric control values

emPFKATP (blue and purple regions in Fig. 5A), and emPFKAMP

(green and cyan regions in Fig. 5B); oscillations only

occur at moderately positive response values for AMP

(0:29\emPFKAMP\1:09) and moderate to strong negative

response values for ATP (�6:46\ emPFKATP \ � 1:47);

indicating that oscillatory behaviour is determined by

allosteric control of PFK.

We also tested whether a single closed volume of

oscillatory states can be obtained with other combina-

tions of three variables (Figs S1 and S2). Figures S1

and S2 show cross-sections of the different bifurcation

manifolds obtained when substituting F6P with

A B

C D

Fig. 5. Allosteric control of PFK determines the oscillatory behaviour. We tested whether the 3D volume that contains all the oscillatory

states can be bounded by the allosteric control strength of ATP and AMP on the phosphofructokinase. (A) Plotting specific ranges of the

elasticity coefficient [30] of the PFK for ATP that approximate the 3D Hopf manifold (emPFKATP [ –1.47, blue; emPFKATP \ –6.46, purple). (B) Plotting

specific ranges of the elasticity coefficient of the PFK for AMP that approximate the 3D Hopf manifold (emPFKAMP [ 1.09, cyan; emPFKAMP\ 0.29,

green). (C, D) The combined constraints on allosteric control values of AMP and ATP approximate the volume in which glycolytic oscillations

are observed. Indicated in red are the same data points as those shown in Fig. 4.

5FEBS Journal (2014) ª 2014 FEBS

A.-K. Gustavsson et al. Emergence of yeast glycolytic oscillations

various other metabolites and fixing the concentration

of F16bP. Here, the lack of a clearly defined boundary

between the red oscillatory and blue steady-state

regions can be seen when a metabolite different from

F6P [and glucose 6-phosphate (G6P)] is used. That

both G6P and F6P can be used as the third variable

together with F16bP and ATP is a result of the fast

phosphoglucose isomerase reaction that links the two

metabolites, essentially keeping the reaction in equilib-

rium, leading to a strict correlation between the two

variables. We show that, for the chosen three vari-

ables, the best defined boundary exists between oscilla-

tory and non-oscillatory states.

To further check the uniqueness of the PFK in con-

trolling the oscillatory states of the cells, we have also

conducted similar analyses for the other three kinases

in the glycolytic pathway: phosphoglycerate kinase,

pyruvate kinase and glucokinase. Figure S3a,b shows

the 3D manifold for the choice of variables around the

phosphoglycerate kinase enzyme. This manifold was

obtained by perturbing Glc, CN, K-ATPase, bisphos-

phoglycerate and glycerate 3-phosphate. Figure S4a,b

shows similar results for perturbations around pyru-

vate kinase and Fig. S5a,b shows the results for per-

turbations around glucokinase. Together, Figs S1 to

S5 serve as motivation for our choice of the variables

F6P, F16bP and ATP for the 3D state-space analysis,

indicating the lack of well-defined boundaries between

oscillatory and stable steady-state regions in state-

space for kinases different from PFK.

The PFK has been suggested to act as an oscillo-

phore [23]. Our observations that oscillatory behaviour

can always be related to the three variables ATP, F6P

and F16bP, which are all effectors of the PFK, and,

additionally, that this 3D volume can be related to

allosteric regulation of the enzyme, are in agreement

with the suggested role of PFK as an oscillophore.

The original observations that, in yeast cell extracts,

oscillations can be observed with Glc, G6P or F6P as

substrates, but not when F16bP is used, also pointed

at PFK as the oscillophore without which no oscilla-

tions can be observed [31]. However, it could be

argued that bypassing both of the ’sparking’ reactions,

hexokinase and PFK, by using F16bP as substrate in

the cell free extract studies, disabled the oscillatory

mechanism suggested by Sel’kov [19]. An oscillophore

as a classic ’master’ enzyme has been defined by Hess

as an enzyme, ’which might be a primary source of the

oscillation of the whole process, in contrast to other

enzymes with kinetic properties insufficient to maintain

an autonomous oscillatory state’, [23]. PFK has desta-

bilizing kinetic properties [substrate (ATP) inhibition

and product (AMP) activation] [23], which can fulfil

the necessary condition to function as an oscillophore.

Clearly, the PFK does not fully control the oscilla-

tions; for example the input flux of Glc has a large

effect on the glycolytic oscillations, which are only

observed in a narrow range of input fluxes. For this

reason, the classic definition of the oscillophore as a

master reaction has been criticised [24, 32] because the

oscillatory characteristics of a system are never fully

controlled by a single reaction step.

In a distributed control approach, it was suggested

that two quantitative measures should be used based

on metabolic control analysis to determine to what

extent a process is oscillophoric [i.e. control (sensitiv-

ity) coefficients of both the real and the complex part

of the smallest eigenvalue of the Jacobian matrix] [24].

This approach takes into account that whether or not

a system oscillates is determined by multiple, if not all,

components in the network. Thus, for a given set of

conditions, the contribution of each of the components

to the oscillatory behaviour can be quantified with this

method. Dependent on the conditions, these control

coefficients can vary greatly in value; for example for

the three experimental Hopf bifurcations (i.e. in the

CN and Glc titrations), the control distributions are

quite different, with most control residing in the top

part of glycolysis for the Glc and the low CN bifurca-

tion points, whereas at the high CN bifurcation point,

most control resides in the CN binding reactions. Inde-

pendent of the conditions, the Glc transport reaction

always had the highest destabilizing control coefficient

and the highest period control coefficient. The PFK

reaction did not have a high control coefficient for any

of the experimental Hopf bifurcation points.

By contrast to this, independent of the conditions or

perturbations that we made, we could always relate

the emergence of oscillations to PFK kinetics, and spe-

cifically to the regulatory control by ATP and AMP.

Apparently, the complete system sets the environment

around the PFK, and thereby controls whether or not

oscillations will occur. This lead us to look for an

extension of the condition-dependent definition of the

oscillophoric control with a condition-independent

component. If the system is considered as defined

merely by its network topology (i.e. with its parame-

ters still undefined), it is no longer possible to ask

which parameter (e.g. the activity of which enzyme)

determines the oscillations most. In that case, it is

preferable to ask which aspect of the topology (such

as the presence of a metabolite, or the presence, sign

or nature of a regulatory loop or kinetic mechanism)

is essential for oscillations to occur at all (i.e. indepen-

dent of the enzyme expression levels or external condi-

tions). This extension is useful in that it defines those

6 FEBS Journal (2014) ª 2014 FEBS

Emergence of yeast glycolytic oscillations A.-K. Gustavsson et al.

components of the system essential for oscillations

(necessary condition) and, together with the rest of the

system, can lead to a sufficient condition. In this sense,

the oscillophore need not have a large control on oscil-

latory properties such as amplitude and period; it is an

enabler of the oscillations but not necessarily a con-

troller.

For yeast glycolysis, both the PFK and the auto-

catalytic stoichiometry [19] have been proposed as os-

cillophores and, indeed, in mathematical models,

oscillations have been obtained in the absence of the

special properties of PFK or of the autocatalytic stoi-

chiometry. To test whether a reaction or a kinetic

property is essential for observing oscillations, it

needs to be perturbed or removed completely before

testing whether oscillations are still possible. To test

the existence of oscillations in a possibly enormous

parameter space, we would propose not to make

changes in the network structure or in kinetic equa-

tions but rather to test whether changes in expression

levels of the systems components can induce oscilla-

tions. We tested this for the current model and found

that making small changes in the allosteric binding

constants of the PFK for either AMP or ATP elimi-

nated oscillations in the system, which could not be

rescued by perturbations in any of the other reac-

tions. Reducing the autocatalytic stoichiometry of the

network for ATP would also remove the oscillations,

although these could be rescued by a simple reduc-

tion in activity of the ATPase. This suggests that

allosteric regulation of PFK activity is the key kinetic

component without which no oscillations can be

observed, independent of the other enzyme activities.

This makes the PFK the oscillophore in terms of the

necessary kinetic component, namely an oscillation e-

nabler.

The enzyme mechanistic interpretation of the precise

transition of individual cells from stationary to oscilla-

tory behaviour at the Hopf bifurcation point, and the

prediction of the period response in oscillatory cells,

demonstrates an unsurpassed level of understanding of

these emergent system properties. Key points in the

study were: (a) working with individual cells in micro-

fluidic chambers, which enabled us to precisely deter-

mine the oscillatory conditions (separate from

synchronization issues, which interfere in population

studies) and (b) analyzing the experimental data with a

detailed mechanistic model. The most striking result of

the present study is not so much that PFK can act as

the oscillophore (necessary conditions) but rather that

the sufficiency condition of the high-dimensional

model can be related to allosteric control of PFK by

ATP and AMP.

Materials and methods

Experimental procedures

Yeast cells (Saccharomyces cerevisiae X2180) were grown in

Glc rich YNB medium (pH 5.0), at 30 ∘C until Glc was

exhausted as described [1]. After harvesting, the cells were

starved on a rotary shaker at 30 ∘C for 3 h, washed and stored

at 4 ∘C until use. Washing and starvation was carried out in

100 mM potassium phosphate buffer at pH 6.8. After this

preparation, the cells were introduced into the microfluidic

flow chamber and positioned in arrays on the bottom using

optical tweezers as described previously [9]. To investigate sin-

gle cells in isolation, the inter-cell distance was set to 10 lm.

The extracellular environment was controlled by adjusting the

flow rates in the different inlet channels and the flow rates and

position of the array were determined by simulations in COM-

SOL MULTIPHYSICS, as described previously [9] (Fig. 6). Images

of the NADH autofluorescence from the individual cells were

acquired every other second.

In all experiments, a microfluidic flow chamber with three

inlet channels was used (Fig. 6A). The cells were introduced

in channel C and the Glc/KCN solutions with varying con-

centrations were introduced in channel A. Here, all channels

contained 100 mM potassium phosphate buffer at pH 6.8.

Cell arrays with 494 cells were used and during cell position-

ing, the flow rates were set to 40–85–85 nL�min�1 in channels

A–C, respectively. At time zero of the experiment, the flow

rates were changed to 1000-0-0 nL�min�1, exposing the cells

to the Glc/KCN solution for the remainder of the experi-

ment.

COMSOL simulations were made to characterize the flow in

the microfluidic chamber to confirm that all yeast cells in the

cell array area (white squares in Fig. 6A) experienced the

same medium in the different flow regimes. The simulations

were performed 0.1 lm and 13.5 lm from the bottom of the

system for the concentration distributions and the velocity

profiles respectively. The top inlet channel is 210 lm wide

and the bottom two are 110 lm wide. The chamber is 27 lmhigh and the outlet channel 410 lm wide. Figure 6A shows

the results of the flow simulations in the microfluidic envi-

ronments used to characterize the concentration gradients

and flow properties of different chemical species. Also shown

is the brightfield image of a typical cell array used in the

experiments.

Data analysis

Images acquired were analyzed as described previously [9].

The NADH fluorescence intensity data from each cell was,

after background subtraction, analyzed using the software

MATLAB (MathWorks, Inc., Natick, MA, USA). A mini-

mum of 10 cells was set as a requirement for a concentra-

tion to be analyzed, otherwise the data set was discarded.

The data sets were then Fourier transformed in the time

7FEBS Journal (2014) ª 2014 FEBS

A.-K. Gustavsson et al. Emergence of yeast glycolytic oscillations

interval 10–15 min, after subtracting the mean of each cell

signal in the interval to reduce the DC component in the

spectrum. Only cells having a peak in the single-sided

amplitude spectrum larger than 20 arbitary units for period

times shorter than 120 s were regarded as oscillatory. The

amplitudes of the oscillations of cells with a qualified peak

in the amplitude spectrum were then measured in the time

intervals 7–12 min, 10–15 min and 13–18 min, by subtract-

ing the mean value of the valleys of the oscillations from

the mean value of the peaks in the three different intervals,

respectively, and dividing the difference by 2. The ampli-

tudes were then normalized by division by the mean value

of the signal in the analyzed time interval. To discard tran-

sient oscillations, the ratio between the amplitudes in time

intervals 13–18 min and 7–12 min, Aratio, had to be within

0.5 < Aratio < 2, otherwise the cell was regarded as non-

oscillatory. For cells fulfilling these criteria for sustained

oscillations, the amplitude and period time in the time

interval 10–15 min were calculated. The mean � SD (where

applicable) was then used for each concentration.

Model adaptations

Our current model is an adaptation of the Teusink model

[16], which was changed to display limit cycle oscillations

as described previously [ 25, 26]. Although we had to go

through a number of optimization iterations, we did not

change the intracellular reaction network, nor the kinetic

rate equations. We tried to make minimal changes to

enzyme kinetic parameters and only changed seven parame-

ters (out of 108) with more than a factor 2 (five of these

parameters were related to the PFK). The resulting model

was applied for the simulation of individual yeast cells [9].

The model presented here was adapted from the gustavs-

son1 model [9] to reflect the observed NADH upshift upon

a downshift in Glc concentration if a cell is in the oscilla-

tory regime of parameters. The previous model [9] shows a

consistent increase (decrease) in NADH upon an upshift

(downshift) in Glc. Model behavior that was qualitatively

similar to experimental results was obtained by modifying

the glycerol branch (which consumes NADH) to increase

the elasticity of the glycerol-3-phosphate dehydrogenase

(G3PDH) enzyme for DHAP [i.e. modifying the kinetics of

this enzyme to reflect cooperativity for DHAP (and G3P)].

Accordingly, the kinetic rate equation for this enzyme was

changed from a simple reversible Michaelis–Menten to a

reversible Hill equation with a Hill coefficient of 4 (to

reflect the 4 subunits of G3PDH that might be involved in

catalysis). The parameters KmDHAP, KmG3P and

VmG3PDH were then obtained by fitting the new model to

the same G3PDH flux as the original model at the bifurca-

tion point. This calibrated model was then validated in its

qualitatively correct prediction of the experimentally

observed period response upon a shift in Glc. In addition,

A

B C

Fig. 6. Experimental set-up. (A) Simulation results of the concentration distributions and velocity profiles (arrows) for the indicated flow

velocities in the three-channel microfluidic chamber. The flow rates are given in nL�min�1, for channels A–C, respectively. Cells were

positioned in arrays at the bottom of the chamber in the areas marked with white rectangles. (A) Brighfield image of a typical 4 9 4 cell

array used in the experiments. (B) Simulation results of the flow velocity field for a typical (3 9 3) cell array in the microfluidic chamber with

blue indicating low flow (0 dm�min�1) and red high flow (0.5 dm�min�1), respectively. (C) Simulation results of the extra- and intracellular

acetaldehyde concentrations with blue indicating low concentrations (0 lM) and red high concentrations (5 lM).

8 FEBS Journal (2014) ª 2014 FEBS

Emergence of yeast glycolytic oscillations A.-K. Gustavsson et al.

CN binding to internal metabolites pyruvate, ACA and

DHAP was added as described previously [25], which lead

to the correct qualitative prediction of two bifurcations as

a function of CN. Subsequently, ACA diffusion was

decreased by a factor of 4 and the binding affinities were

adjusted slightly to obtain quantitatively similar CN bifur-

cation points. Finally, time scaling was adjusted by multi-

plying all rate equations by a factor 0.487. A complete

model description is provided in Doc. S1.

Simulations and analyses

The finite element analysis software COMSOL MULTIPHYSICS

was used to investigate spatiotemporal cell dynamics. For

this purpose, we constructed a detailed model consisting of

modules for geometry, laminar flow, transport of diluted

species (for diffusion and convection of extracellular acetal-

dehyde and ethanol) and ordinary differential equations

(for the detailed intracellular kinetic model).

Investigation of the Hopf bifurcations and state space

of the detailed model was conducted in MATHEMATICA

using an ordinary differential equation boundary-layer

model [9], which showed good agreement with the

COMSOL results for intracellular metabolites. Standard

numerical routines were used to integrate the differential

equations to obtain time traces for metabolites (leading

to oscillation amplitudes and frequencies) and bifurcation

points were determined by searching for complex eigen-

values (with positive real parts) of the Jacobian matrix.

Characterization of the spaces tangent to the 3D mani-

fold was achieved by subtracting the manifold from the

background of phase points for which the elasticities

were calculated (with the adenylate kinase reaction in

equilibrium) using the standard definition [30].

To verify the Hopf bifurcation analysis in MATHEMATICA

and to test whether the Hopf bifurcation manifold can be

described exactly within a 3D analysis, we used bifurcation

theory [33] as a tool to analyze the long-term dynamics

with continuation techniques implemented in AUTO [34],

which can be used to perform numerical bifurcation analy-

ses and the results presented in bifurcation diagrams. Of

particular interest are Hopf bifurcations indicating the bor-

ders of regions in the parameter space where the dynamics

of population dynamical systems is oscillatory. AUTO [34]

was used to locate and to continue the Hopf bifurcation

point. As starting points, reference steady-state points on

the Hopf bifurcation manifold were used and several con-

tinuations of these bifurcation points were made in two

dimensional (2D) parameter space.

Acknowledgements

We thank Albert Goldbeter, Steven Strogatz, Hans

Westerhoff and Bernhard Mehlig for critically reading

the manuscript and providing valuable advice for

improvement. We acknowledge the financial support

provided by the Swedish Research Council to M.G.;

the UK Biotechnology and Biological Sciences

Research Council (SysMO-DB) to J.L.S.; and the

National Research Foundation in South Africa to

D.D.vN. and J.L.S. (SARChI of the DST and NRF of

South Africa).

Author contributions

A.K.G. and C.B.A. designed the experiments. A.K.G.

conducted the experiments and analyzed the data.

J.L.S., A.K.G. and D.D.vN. wrote the paper. D.D.vN.

and J.L.S. adapted the model and performed the simula-

tions and model analyses. B.K. performed the AUTO sim-

ulations, M.G., C.B.A. and J.L.S. supervised the project

and provided guidance throughout. All authors dis-

cussed the results and commented on the manuscript.

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Supporting information

Additional supporting information may be found in

the online version of this article at the publisher’s web

site:Doc. S1. Complete model description.

Fig. S1. Cross-sections of the state space for different

metabolites and ATP.

Fig. S2. Cross-sections of the state space for different

metabolites and ATP.

Fig. S3. 3D plot of the state space (a) and 2D cross-

section (b) for substrates and products of the PGK

enzyme.

Fig. S4. 3D plot of the state space (a) and 2D cross-

section (b) for substrates and products of the PYK

enzyme.

Fig. S5. 3D plot of the state space (a) and 2D cross-

section (b) for substrates and products of the GLK

enzyme.

10 FEBS Journal (2014) ª 2014 FEBS

Emergence of yeast glycolytic oscillations A.-K. Gustavsson et al.