Metabolic oscillations in budding yeast described by a ... fileconstant uptake of glucose (these...

Metabolic oscillations in budding yeast

described by a dynamical model

Wolfram Liebermeister

Charite – Universitatsmedizin Berlin

Abstract

The budding yeast Saccharomyces cerevisiae, grown under the right conditions in continuous culture,

shows spontaneous metabolic oscillations. The oscillations affect not only gas exchange rates and metabolite

concentrations, but also the expression of a large fraction of genes. I integrated time-dependent metabolite

concentrations, oxygen uptake rates, carbon dioxide and ethanol excretion rates, and thermodynamic data

into a large dynamic model of yeast metabolism. A linearised kinetic model was fitted to the data, assuming

that the oscillations in metabolism are entirely driven by the time-dependent, experimentally know oxygen

uptake. To fit a model of this size to periodic time series data, a number of tricks had to be used: (i)

the model variables were determined in a stepwise procedure, starting from a stationary flux distribution

and ending with the reaction elasticities, which describe the dynamics around this reference state; (ii) the

elasticities themselves were derived from thermodynamic forces and saturation values in order to guarantee

thermodynamically feasible reaction kinetics; (iii) and the periodic dynamics was not simulated by numerical

integration in time, but by a Fourier synthesis based on periodic response coefficients, which can be directly

computed from the reaction elasticities; (iv) the reference state was required to remain dynamically stable,

which puts strong constraints on the possible reaction elasticities. The model parameters (thermodynamic

forces and saturation values) were then fitted in such a way that their values, by construction, correspond to

thermodynamically consistent kinetic rate laws. The model predicts metabolic fluxes, concentrations of non-

measurable metabolites, and local metabolite concentrations in cell compartments (cytosol and mitochondria)

along the metabolic cycle. Computer animations of experimental data and simulation results can be found on

www.metabolic-economics.de/yeast-metabolic-oscillations/.

Keywords: Metabolic oscillation, budding yeast, dynamical model, model fit, reaction elasticity

1 Introduction

Metabolic oscillations in yeast have been studied for many years, with a focus on transcriptional and metabolic

changes [1, 2, 3], types of oscillatory dynamics [4, 5, 6], and the accompanying changes in DNA structure [7, 8, 9].

During the metabolic oscillations, cells alternate between an oxidative state with higher respiration activity and

a reductive state in which biosynthesis tends to become more active. That metabolic pathways change their

activity along the metabolic cycle is supported both by metabolomics and gene expression data [10]. However,

the pathway fluxes, which would be the ultimate criterion for “activity”, cannot be easily measured. In order to

infer the non-measurable, time-dependent metabolite concentrations and metabolic fluxes from available data, I

developed a dynamic model of metabolism in yeast that can fill these holes. The model was developed in close

collaboration with Prof. Douglas Murray (Institute for Advanced Bioscience, Keio University), who also provided

most of the experimental data. Even though some metabolite concentrations can be measured – those were used

to fit the model – a reconstruction of metabolite curves is important for two reasons: for some metabolites, no

measurements are available; and for some metabolites, it is important to determine their concentrations within

1

cell compartments (cytosol and mitochondria), a piece of information distinction that cannot be extracted from

metabolomics data alone.

A main model assumption is that metabolism responds “passively” to peridic changes in oxygen uptake, at a

constant uptake of glucose (these uptake rates agree with experimental observations). In particular, I assumed

that all enzyme levels remain constant in time. The latter assumption was made because the changes in mRNA

levels, even though they are pervasive, are not very strong in amplitude. Given the short oscillation period (a bit

less than an hour), these expression changes would translate into very small changes in protein levels. Since no

information about regulated protein modifications or protein degradation were available, constant enzyme levels

were assumed. Therefore, the only source of oscillations in the model is the predefined, periodically varying oxygen

uptake flux. From this uptake reaction, oscillations in fluxes and metabolite levels percolate into the metabolic

network like waves, where cofactors and allosteric regulation can also couple distant parts of the network.

This report starts with a brief description of the data used and of how the model was built and fitted to data.

Then, simulation results are compared to experimental data. On the website www.metabolic-economics.de/

yeast-metabolic-oscillations/, computer animations of experimental data and simulation results are shown.

There, the fluxes are also compared to time-dependent gene expression data obtained from a similar experimental

set-up [10], and to gene expression data measured in a different experimental set-up that gives rise to much slower

oscillations [2], in which periodic enzyme levels could have a much larger effect on the metabolic dynamics.

2 Model construction

The objective of this work was to construct a metabolic of yeast that reproduces the measured metabolite

profiles and exchange fluxes during metabolic oscillations in growth on glucose and to predict non-measurable

quantities from this model. Different types of data were used: (i) time-dependent exchange rates (oxygen, carbon

dioxide, ethanol) in yeast continuous cultures, provided by D. Murray; (ii) time-dependent absolute metabolite

concentrations in the same experiment, also provided by D. Murray; (iii) standard Gibbs free energies of reactions,

provided by Dr. E. Noor (computed using the component contribution method [11]). Some of the data are shown

in Figures 1 and 2.

The metabolic network (Figure 3) is a modified version of the yeast metabolic network from Jol et al. (2012)

[12], with slight modifications by my collaborators S. Hoffmann, W. Gottstein, and D. Murray. The model

in Systems Biology Markup Language (SBML) format can be downloaded from www.metabolic-economics.

de/yeast-metabolic-oscillations/. To construct and fit a dynamic model, I made the following model

assumptions:

1. Aim of model construction We aim at constructing a linearised model that is thermodynamically consistent

and that, with a given time-dependent oxygen uptake rate, reproduces the main experimental facts (CO2

and ethanol secretion rates, mean metabolite levels, time-dependent metabolite levels, heat production, and

respiratory quotient (CO2 secretion divided by O2 uptake).

2. Constant protein levels An analysis of gene expression amplitudes suggested that the changes in enzyme levels

are probably small. Therefore, all enzyme levels are assumed to be constant in the model. Allosteric regulation

of enzyme activities was still considered.

3. Dynamic response to oxygen uptake rate We assume that oxygen uptake is limited by non-metabolic

processes (possibly involving structural changes of the mitochondrial membrane. Since these processes are

outside the scope of the model, we treat the time-dependent oxygen uptake rate as an experimental fact and

impose it onto the model as a constraint.

2

(a) Gas exchange rates (b) Concentration changes betweenoxidative and reductive phase

0 2 4 6

1

2

3

4

O2 uptake[mmol/(gdw h)]

0 2 4 6

2.6

2.8

3

3.2

CO2 production [mmol/(gdw h)]

0 2 4 6

0.35

0.36

0.37

0.38

Ethanol production [mmol/(gdw h)]

0 2 4 6

8

9

10

11x 10

−3 Acetate prod. [mmol/(gdw h)]

0 2 4 60.006

0.008

0.01

0.012

0.014

Acetaldehyde production [mmol/(gdw h)]

0 2 4 6

−4

−3

−2

−1

O2 production [mmol/(gdw h)]

10−Formyltetrah..

1 3 beta D Gluc..

3−Phospho−D−gly..

1−Pyrroline−5−c..

2,3−Dihydroxy−3..


2−Aceto−2−hydro..

1−(2−Carboxyphe..

2−Dehydro−3−deo..

2−Isopropylmale..

2−Oxobutanoate

2−Oxobutanoate

2 Oxoadipate C6..

2 Oxoadipate C6..

Glycerate 2−pho..

3−(4−Hydroxyphe..

3−Carboxy−2−hyd..


3−Carboxy−4−met..

3−Dehydroquinat..3−Dehydroshikim..

C’−(3−Indolyl)−..

3−Methyl−2−oxob..


3−Methyl−2−oxop..



3−Phosphohydrox..

5−O−(1−Carboxyv..


4−Phospho−L−asp..

5−Methyltetrahy..

6−Phospho−D−glu..

6−phospho−D−glu..

L 2 Aminoadipat..

L 2 Aminoadipat..

Acetate

Acetate

Acetate

Acetaldehyde

Acetaldehyde

Acetaldehyde

Acetyl−CoA

N−Acetyl−L−glut..



O Acetyl L homo..

N2−Acetyl−L−orn..

5−Amino−1−(5−Ph..

2−Oxoglutarate

Alanine

Alanine

2−Acetolactate

Anthranilate

Adenosine 5’−ph..

Arginine

N(omega)−(L−Arg..

Asparagine

Aspartate

Aspartate 4−sem..

But 1 ene 1 2 4..

Carbamoyl phosp..

Chorismate

Citrate

Citrate

Citrulline

CMP

CO2

Coenzyme A

Cysteine

Cystathionine

DAMPDCMP

DGMP

Dihydroxyaceton..

DTMP

Erythrose 4−pho..

erythro−1−(Imid..

Ergosterol C28H..

Ethanol

Ethanol

Ethanol

Fructose 6−phos..

Fructose 1,6−bi..

Ferricytochrome..Ferrocytochrome..

Formate

Fumarate

Fumarate

Glucose 1−phosp..

Glyceraldehyde ..

Glucose 6−phosp..

GlucoseGlucose

Glutamine

Glutamate 5−pho..

Glutamate 5−sem..

Glutamate

Glutamate

Glyoxylate

Glyoxylate

Glycine

Glycine

Glycerol 3−phos..GlycerolGlycerol

Glycogen

GMP

H2O

Hydrogen sulfid..

Hydrogen sulfid..

H+

2 Hydroxybutane..

2 Hydroxybutane..

hco3

hco3

Homocysteine

hcyt[c]

Homoisocitrate ..

HistidineHistidinol phos..Histidinol

Homoserine

Isocitrate

Isocitrate

Isoleucine

3−(Imidazol−4−y..

LeucineLysine

Malate

Malate

Mannan C6H10O5

Methionine5,10−Methenylte..

5,10−Methylenet..

Ammonium

Ammonium

O2

Oxaloacetate

Oxaloacetate

Ornithine

Ornithine

Oxaloglutarate ..

Phosphatidate ..

Adenosine 3’,5’..

3’−Phosphoadeny..

Phosphatidylcho..

Phosphatidyleth..

Phosphoenolpyru..

Phenylalanine

O−Phospho−L−hom..

Phenylpyruvate

Phosphate

Prephenate

N−(5−Phospho−D−..

1−(5−Phosphorib..1−(5−Phosphorib.. 1−(5−Phosphorib..5−[(5−phospho−1..

Proline

5−Phospho−alpha..

Phosphatidylser..

O−Phospho−L−ser..Phosphatidyl 1D..

Pyruvate

Pyruvate

Pyruvate

Ubiquinone 6 C3..

Ubiquinol 6 C39..

Alpha−D−Ribose ..

Ribulose 5−phos..

Sedoheptulose 1..

Sedoheptulose 7..

L Saccharopine ..

Serine

Shikimate 5−pho..Shikimate

Sulfite

Sulfate

Sulfate

Succinate

Succinate

Succinate Succinyl−CoA

Threonine

Threonine

Alpha,alpha’−Tr..

TrehaloseTrehalose

Triglyceride y..

TryptophanTyrosine

UDP

UDPglucose

UMP

UTP

Valine

Xylulose 5−phos..

Zymosterol C27H..

−2

−1

0

1

Figure 1: Measured gas exchange rates and changes in metabolite concentrations in yeast during the metaboliccycle (experimental data provided by D. Murray). (a) Measured exchange rates. The oxygen curve (grey) isshown for comparison. (b) Measured concentration changes between oxidative and reductive phase. Relativeconcentrations are shown in colours (blue: higher in oxidative phase, pink: higher in reductive phase). Highlyconnected metabolites (e.g., cofactors) are included in the network, but not shown in the graphics. Somemetabolites exist both in cytosol and mitochondria; their data values are shown twice and refer to the overconcentration in the cell. Some reactions were omitted for clarity.

Due to the large model size and the type of data used (periodic metabolite concentrations), fitting a kinetic model

directly to the data would be difficult. Instead, to obtain a viable model, a number of tricks had to be used: the

model variables were determined in a stepwise procedure, starting from a stationary flux distribution and ending

with the reaction elasticities, which describe the dynamics around this reference state; the elasticities themselves

were derived from thermodynamic forces and saturation values in order to guarantee thermodynamically feasible

reaction kinetics; and the periodic dynamics was not simulated by numerical integration in time, but by a Fourier

synthesis based on periodic response coefficients, which can be directly computed from the reaction elasticities. To

allow for one reaction to be externally controlled (namely the oxygen uptake reaction), a new type of “restricted”

control coefficients had to be defined. (iv) the reference state was required to remain dynamically stable, which

puts strong constraints on the possible reaction elasticities. (v) Finally, to handle cell compartments with different

volumes (75 and 25 percent of the cell volume, respectively, for cytosol and mitochondria), reaction fluxes were

defined as numbers of reaction events (in moles) per time and cell volume; concentrations were not defined as

local concentrations in the cell compartments, but as amounts (in moles) in the cell compartments, but divided

by the cell volume. Therefore, the stoichiometric matrix keeps a simple form; however, the compartment volumes

enter the formulae for computing control and response coefficients.

All this resulted in a new procedure for model building. The guiding thought was to determine the values of

different types of variables step by step: first a time-averaged flux distribution; then a corresponding set of

equilibrium constants and stationary metabolite levels; and then, reaction elasticities defining the dynamics of the

model around this metabolic reference state. Altogether, the model was constructed and fitted in the following

way:

1. Compounds, reactions, and network structure The lists of metabolites and reactions, defining a stoichio-

metric matrix, were taken from a published model of yeast metabolism [13], with previous modifications by

S. Hoffmann, W. Gottstein, and D. Murray. Extracellular compounds were treated as external (i.e., with con-

centration curves being predefined, not determined by rate equations). A number of known allosteric interactions

3

20 40 60

0

0.02

0.04

0.06

0.08

2−Oxoglutarate

20 40 60

0

0.02

0.04

0.06

0.08

3−Phospho−D−glycerat

20 40 60

0

0.01

0.02

0.03

6−Phospho−D−gluconat

20 40 60

0

0.02

0.04

0.06

0.08AMP

20 40 60

0

0.02

0.04

ATP

20 40 60

0

20

40

Acetaldehyde

20 40 60

0

20

40

Acetate

20 40 60

0

1

2

3

x 10−3

Acetyl−CoA

20 40 60

0

0.01

0.02

0.03

Adenosine 5’−phospho

20 40 60

0

0.5

1

1.5

x 10−3

Alpha−D−Ribose 5−pho

20 40 60

0

2

4

x 10−3

Anthranilate

20 40 60

0

0.005

0.01

CMP

20 40 60

0

50

100

CO2

20 40 60

0

0.05

0.1

0.15

Citrate

20 40 60

0

2

4

x 10−3

Coenzyme A

20 40 60

0

2

4

6

8

x 10−3

D−Erythrose 4−phosph

20 40 60

0

2

4

x 10−3

D−Fructose 1,6−bisph

20 40 60

0

0.02

0.04

D−Fructose 6−phospha

20 40 60

0

0.01

0.02

D−Glucose 1−phosphat

20 40 60

0

0.05

0.1

0.15

D−Glucose 6−phosphat

20 40 60

0

0.01

0.02

D−Glycerate 2−phosph

20 40 60

0

1

2

x 10−3

D−Ribulose 5−phospha

20 40 60

0

0.5

1

x 10−3

DTMP

20 40 60

0

500

1000

Ethanol

20 40 60

0

0.02

0.04

Fumarate

20 40 60

0

2

4

6

8

x 10−3

GMP

20 40 60

0

0.2

0.4

0.6

Glycine

20 40 60

0

20

40

60

Glycogen

20 40 60

0

0.05

0.1

Hydrogen sulfide

20 40 60

0

0.02

0.04

0.06

0.08

Isocitrate

20 40 60

0

2

4

6

8L−Alanine

20 40 60

0

0.2

0.4

L−Asparagine

20 40 60

0

0.2

0.4

0.6

0.8

L−Aspartate

20 40 60

0

0.2

0.4

0.6

L−Citrulline

20 40 60

0

0.2

0.4

L−Cystathionine

20 40 60

0

2

4

6

L−Glutamate

20 40 60

0

2

4

6

L−Glutamine

20 40 60

0

0.2

0.4

0.6

L−Histidine

20 40 60

0

0.02

0.04

0.06

L−Homoserine

20 40 60

0

0.1

0.2

L−Isoleucine

20 40 60

0

0.05

0.1

L−Leucine

20 40 60

0

0.2

0.4

0.6

0.8

L−Lysine

20 40 60

0

0.1

0.2

L−Malate

20 40 60

0

0.02

0.04

0.06

0.08

L−Phenylalanine

20 40 60

0

0.1

0.2

0.3

L−Proline

20 40 60

0

0.5

1

L−Serine

20 40 60

0

0.2

0.4

0.6

0.8

L−Threonine

20 40 60

0

2

4

6

x 10−3

L−Tryptophan

20 40 60

0

0.02

0.04

L−Tyrosine

20 40 60

0

0.5

1

L−Valine

20 40 60

0

0.01

0.02

0.03

N(omega)−(L−Arginino

20 40 60

0

0.02

0.04

0.06

N−Acetyl−L−glutamate

20 40 60

0

0.02

0.04

N2−Acetyl−L−ornithin

20 40 60

0

0.02

0.04

0.06

Nicotinamide adenine

20 40 60

0

0.5

1

x 10−3

Nicotinamide adenine

20 40 60

0

2

4

6

x 10−3

O−Phospho−L−serine

20 40 60

0

200

400

O2

20 40 60

0

2

4

6

8

x 10−3

Phosphoenolpyruvate

20 40 60

0

0.02

0.04

0.06

Pyruvate

20 40 60

0

0.02

0.04

Sedoheptulose 7−phos

20 40 60

0

0.01

0.02

UDP

20 40 60

0

0.01

0.02

UMP

20 40 60

0

0.005

0.01

UTP

Figure 2: Temporal metabolite profiles (experimental data provided by D. Murray). The time axis (x-axis) showsone oscillation period. Oxygen curves (grey) are shown for comparison. Only metabolites appearing in the modelare shown.

were added to the model. The network structure is shown in Figure 3.

2. Exchange rate data: conversion and mapping Measured time series of uptake and secretion fluxes (for

oxygen, carbon dioxide, and ethanol), as well as heat production, were obtained from D. Murray (64 time points,

standardised to oscillation phase angles) and were converted into units of mM/s (referring to concentrations

inside the cell).

3. Concentration data: conversion and mapping Measured time series of metabolite concentrations were

obtained from D. Murray. The data were smoothed and interpolated, resulting in time series with 64 equally

spaced time points, roughly standardised to oscillation phase angles. The data were then converted to mM (in

the cell volume) and mapped onto the model.

4. Time-average fluxes: calculation by Flux Balance Analysis Time-averaged fluxes were computed by flux

balance analysis. The uptake and secretion rates (oxygen, carbon dioxide, ethanol) were roughly fixed to

their (time-averaged) experimental values. Fluxes were predicted in two steps, by applying a normal FBA (to

determine the maximal possible biomass production rate), followed by a minimal-flux FBA (where this biomass

production rate was used as a constraint and the sum of absolute fluxes was minimised).

5. Intracellular concentrations: adjustment by thermodynamic constraints Time-averaged intracellular con-

centrations for the model were computed based on (time-averaged) measured metabolite concentrations, on

4

10−Formyltetrah..

1 3 beta D Gluc..








2−Isopropylmale..

2−Oxobutanoate

2−Oxobutanoate

2 Oxoadipate C6..

2 Oxoadipate C6..

Glycerate 2−pho..












3−Phosphohydrox..




5−Methyltetrahy..



L 2 Aminoadipat..

L 2 Aminoadipat..

Acetate

Acetate

Acetate

Acetaldehyde

Acetaldehyde

Acetaldehyde

Acetyl−CoA




O Acetyl L homo..



2−Oxoglutarate

Alanine

Alanine

2−Acetolactate

Anthranilate

Adenosine 5’−ph..

Arginine


Asparagine

Aspartate

Aspartate 4−sem..

But 1 ene 1 2 4..

Carbamoyl phosp..

Chorismate

Citrate

Citrate

Citrulline

CMP

CO2

Coenzyme A

Cysteine

Cystathionine

DAMPDCMP

DGMP

Dihydroxyaceton..

DTMP

Erythrose 4−pho..


Ergosterol C28H..

Ethanol

Ethanol

Ethanol

Fructose 6−phos..

Fructose 1,6−bi..


Formate

Fumarate

Fumarate

Glucose 1−phosp..

Glyceraldehyde ..

Glucose 6−phosp..

GlucoseGlucose

Glutamine

Glutamate 5−pho..

Glutamate 5−sem..

Glutamate

Glutamate

Glyoxylate

Glyoxylate

Glycine

Glycine


Glycogen

GMP

H2O

Hydrogen sulfid..

Hydrogen sulfid..

H+

2 Hydroxybutane..

2 Hydroxybutane..

hco3

hco3

Homocysteine

hcyt[c]

Homoisocitrate ..


Homoserine

Isocitrate

Isocitrate

Isoleucine


LeucineLysine

Malate

Malate

Mannan C6H10O5


5,10−Methylenet..

Ammonium

Ammonium

O2

Oxaloacetate

Oxaloacetate

Ornithine

Ornithine

Oxaloglutarate ..

Phosphatidate ..



Phosphatidylcho..

Phosphatidyleth..

Phosphoenolpyru..

Phenylalanine


Phenylpyruvate

Phosphate

Prephenate



Proline


Phosphatidylser..


Pyruvate

Pyruvate

Pyruvate

Ubiquinone 6 C3..

Ubiquinol 6 C39..


Ribulose 5−phos..

Sedoheptulose 1..

Sedoheptulose 7..

L Saccharopine ..

Serine


Sulfite

Sulfate

Sulfate

Succinate

Succinate


Threonine

Threonine


TrehaloseTrehalose

Triglyceride y..

TryptophanTyrosine

UDP

UDPglucose

UMP

UTP

Valine

Xylulose 5−phos..

Zymosterol C27H..

GLYCt

dtmpSYN

TREH

EX pi e

ADSK

PRMICIi

ALDD2xALDD2y

SUCCt2r

IPPMIa

IPPMIb

EX o2 e

PRPPS

GHMT2r

ANPRT

GAPD

DHAD2m

HSERTA

IPMD

SUCOAACTrm

TKT1TKT2

CO2t

ASADi

FBA3

PGCD

G6PDH2

OCBTi

ICL

TYRTApyr

SO4tipeSYN

DHQS

FUM

HSDyi

ACGKm

OMCDC

FDH

EX acald e ACALDtex

SADT

DHQTi

OAAt2m

PFK 3

VALTA

BPNT

ampSYN

PHETA1

ORNt3m

CYOOm

IGPS

ATPS3m

PDHm

RPE

THRS

ETOHtm

P5CR

RPI

THRAACS

IPPS

dgmpSYN

CITtcm

CSc

ILETA

PGM

PGL

PGK

eleak

PGI

GLNt2m

ORNTACim

PC

SACCD1

TRPS1

ACt2r

ALATA L

ATPS

ACtm

ACOTAim

3MOPtm

pcSYN

PAPSR

FBP

ACHBSm

eleakm

ASPKi

cmpSYN

GLUDym

IGPDH

OXAGm

G3PDm

13GS

HCITS

DHAD1m

HCO3Em

FBA

SHKK

THRD Lm

SHK3D

GAAtm

triglycSYN

MALtm

IG3PS

TRE6PP

ETOHt

GLUDy

FUMm

HCITtm

HCO3E

SUCCtm

PYRt2m

ptd1inoSYN

EX glyc e

HISTDPRAMPC

ALCD2x

CSm

G5SADr

umpSYN

ASPTAm

GLXtm

PPND2

MCITDm

GLUDxi

CITtamGLUt2m

CBPSGLUSxm

PYRDC

ICDHxm

3MOBtm

FTHFL

CYOu6m

ALCD2y

paSYN

psSYN

EX pyr e

PSERT

EX h e

G5SD

GLYK

ANS

LEUTA

HSTPT

O2t

ALCD2m

TALA

HISTP

H2St

EX glc D e

EX co2 e

PSCVTi

GLU5K

2OXOADPtim

AGPRim

ACONT

HEX1

EX ac e

PPNDH

GLCP

PFK

HICITDm

dampSYN

EX h2o e

NH4t

ergstSYN

SBP

EX h2s e

AGTm

ME1m

ALDD2xm

HACNHm

ACONTm

DDPA

SUCFUMtm

ALAt2m

ACLSm

ICDHy

NADH2 u6cm

TREt2

UMPK

mannanSYN

AASAD1

ACOAHm

PSP L

HSK

GLCS2MTHFR3

EX tre e

aicarSYN

AKGDam

TPI

ARGSSr

PYRt2

NH4tm

CITttm

gmpSYN

CYSTGL

ASPTA

CHORM

CYSTS

ALDD2ym

MTHFD

MTHFC

NDPK2

ACALDtm

PPCK

G3PD1ir

EX nh4 e

ASNS1

ARGSL

THRt2m

TRE6PS

MALS

MDH

KARA1im

METS

MDHm

PYK

CHORS

AATA

PRAIi

PGMT

NADH2 u6m

ALATA Lm

ENO

PIt2r

ATPPRT

2OBUTtm

GLYt2m

KARA2im

TYRTA

dcmpSYN

SACCD2

SUCOAS1m

GLNS

EX succ e

zymstSYN

GND

PHETA1pyr

ICDHym

PRATPP

SULR

G3PT

GLCt1

EX so4 e

SUCD2 u6m

GALU

ACSm

AHSERL2

Figure 3: Model structure. Static metabolic fluxes were determined by flux balance analysis, using the time-averaged measured exchange rates as a constraint. Allosteric activation and inhibition is shown by blue and redarcs, respectively.

the previously computed flux directions, and on standard Gibbs free energies of formation. Standard reaction

Gibbs free energies of for the model were computed by E. Noor (ETH Zurich) using the component contribution

method [11]. Based on these Gibbs energies and on the measured metabolite concentrations, thermodynamic

parameter balancing [14] was used to obtain a consistent sets of Gibbs energies and concentrations, resembling

these input data and in agreement with the flux directions.

6. Reaction elasticities: structure of the elasticity matrix In the model, we first consider, as a reference state, a

steady state with the time-average fluxes and compount concentrations computed so far. To describe a dynamics

around this reference state, we define the reaction elasticities. We first assume that the structure of the elasticity

matrix follows the stoichiometric matrix (with positive substrate elasticties and negative product elasticities)

and add further entries to describe allosteric regulation. The actual elasticity values were determined in several

steps. A first version of the elasticity matrix was obtained by a simple heuristics, assuming approximately

half-saturated enzymes in all cases. These values were then refined by fitting the model to metabolite curves,

as described below. Importantly, to ensure a thermodynamically feasible model, the reactions elasticities were

not treated as independent model parameters; instead, their values were derived from thermodynamic driving

forces and saturation values, which can be independently varied without violating any constraints [15, 16], and

which were used as model parameters to be fitted.

5

7. Response to static changes and sine-wave oscillations To simulate metabolite curves (which was also

necessary to fit the model parameters), the dynamical properties of the model were analysed. The dynamic

response to oscillating inputs can be characterised by spectral response coefficients (computed from the Jacobian,

which follows from stoichiometric matrix and elasticity matrix).

8. Dynamic response to predefined time-dependent oxygen uptake To simulate the response to non-sine-

wave perturbations (most importantly, the predefined oxygen uptake time curve), we Fourier-transform the input

time series, multiply with the frequency-dependent spectral response matrices, and apply the reverse Fourier

transformation. This has several advantages over time-simulations: (i) long simulation runs due to a slow

initial relaxation are avoided. (ii) with a restriction to low-frequency components, fast noise is automatically

suppressed.

9. Reaction elasticities: parameter fit based on concentration profiles The values in the elasticity matrix

were further optimised by fitting the simulated metabolite time curves to experimental data. To do so, the

thermodynamic driving forces and saturation values, which together determine the reaction elasticities, were

iteratively optimised by a greedy optimisation method. (i) As a first fit, we considered a stationary response

to increased oxygen uptake, and fitted the resulting concentration changes to observed concentration changes

between the reductive and the oxidative phase. (ii) Then, we fitted the temporal variation profiles of metabolites

(for each metabolite, scaled by its mean concentration) to the corresponding data. Since the data refer to whole-

cell concentrations, and not to the concentrations in cytosol or mitochondria, the concentrations obtained from

the model were converted into predicted cellular concentrations.

3 Model results

Simulation results are shown in this section. Computer animations of experimental data and simulation results

can be found on www.metabolic-economics.de/yeast-metabolic-oscillations/.

3.1 Simulated response to harmonic perturbations in oxygen level

Before fitting the model precisely to metabolite time series, I started with simpler fits and predictions in which all

Maximal

Increasing

Minimal

Decreasing

curves were approximated by phase shifted sine curves. Each such curve can be charac-

terised by a single complex-valued amplitude (or, equivalently, by a real-valued amplitude

and a phase angle). With this simplification, the goodness of fit can be computed much

faster. Probably, the simplification also helps the fitting algorithm to fit the overall shape

of many metabolite curves, instead of getting stuck in a state where some curves are fitted

very precisely, while other show no good fit at all. Results from the fitted model (fitted

and measured metabolite phase angles, and predicted phase angles of internal fluxes) are

shown in Figure 4. The colour code for phase angles is shown in the inset Figure.

6

(a) Metabolite phase angles, measured (b) Metabolite phase angles, predicted

10−Formyltetrah..

1 3 beta D Gluc..








2−Isopropylmale..

2−Oxobutanoate

2−Oxobutanoate

2 Oxoadipate C6..

2 Oxoadipate C6..

Glycerate 2−pho..












3−Phosphohydrox..




5−Methyltetrahy..



L 2 Aminoadipat..

L 2 Aminoadipat..

Acetate

Acetate

Acetate

Acetaldehyde

Acetaldehyde

Acetaldehyde

Acetyl−CoA




O Acetyl L homo..



2−Oxoglutarate

Alanine

Alanine

2−Acetolactate

Anthranilate Adenosine 5’−ph..

Arginine


Asparagine

Aspartate

Aspartate 4−sem..

But 1 ene 1 2 4..

Carbamoyl phosp..

Chorismate

Citrate

Citrate

Citrulline

CMP

CO2

Coenzyme A

Cysteine

Cystathionine

DAMPDCMP

DGMP

Dihydroxyaceton..

DTMP

Erythrose 4−pho..


Ergosterol C28H..

Ethanol

Ethanol

Ethanol

Fructose 6−phos..

Fructose 1,6−bi..


Formate

Fumarate

Fumarate

Glucose 1−phosp..

Glyceraldehyde ..

Glucose 6−phosp..

GlucoseGlucose

Glutamine

Glutamate 5−pho..

Glutamate 5−sem..

Glutamate

Glutamate

Glyoxylate

Glyoxylate

Glycine

Glycine


Glycogen

GMP

H2O

Hydrogen sulfid..

Hydrogen sulfid..

H+

2 Hydroxybutane..

2 Hydroxybutane..

hco3

hco3

Homocysteine

hcyt[c]

Homoisocitrate ..


Homoserine

Isocitrate

Isocitrate

Isoleucine


LeucineLysine

Malate

Malate

Mannan C6H10O5


5,10−Methylenet..

Ammonium

Ammonium

O2

Oxaloacetate

Oxaloacetate

Ornithine

Ornithine

Oxaloglutarate ..

Phosphatidate ..



Phosphatidylcho..

Phosphatidyleth..

Phosphoenolpyru..

Phenylalanine


Phenylpyruvate

Phosphate

Prephenate



Proline


Phosphatidylser..


Pyruvate

Pyruvate

Pyruvate

Ubiquinone 6 C3..

Ubiquinol 6 C39..


Ribulose 5−phos..

Sedoheptulose 1..

Sedoheptulose 7..

L Saccharopine ..

Serine


Sulfite

Sulfate

Sulfate

Succinate

Succinate


Threonine

Threonine


TrehaloseTrehalose

Triglyceride y..

Tryptophan Tyrosine

UDP

UDPglucose

UMP

UTP

Valine

Xylulose 5−phos..

Zymosterol C27H..

10−Formyltetrah..

1 3 beta D Gluc..








2−Isopropylmale..

2−Oxobutanoate

2−Oxobutanoate

2 Oxoadipate C6..

2 Oxoadipate C6..

Glycerate 2−pho..












3−Phosphohydrox..




5−Methyltetrahy..



L 2 Aminoadipat..

L 2 Aminoadipat..

Acetate

Acetate

Acetate

Acetaldehyde

Acetaldehyde

Acetaldehyde

Acetyl−CoA




O Acetyl L homo..



2−Oxoglutarate

Alanine

Alanine

2−Acetolactate


Arginine


Asparagine

Aspartate

Aspartate 4−sem..

But 1 ene 1 2 4..

Carbamoyl phosp..

Chorismate

Citrate

Citrate

Citrulline

CMP

CO2

Coenzyme A

Cysteine

Cystathionine

DAMPDCMP

DGMP

Dihydroxyaceton..

DTMP

Erythrose 4−pho..


Ergosterol C28H..

Ethanol

Ethanol

Ethanol

Fructose 6−phos..

Fructose 1,6−bi..


Formate

Fumarate

Fumarate

Glucose 1−phosp..

Glyceraldehyde ..

Glucose 6−phosp..

GlucoseGlucose

Glutamine

Glutamate 5−pho..

Glutamate 5−sem..

Glutamate

Glutamate

Glyoxylate

Glyoxylate

Glycine

Glycine


Glycogen

GMP

H2O

Hydrogen sulfid..

Hydrogen sulfid..

H+

2 Hydroxybutane..

2 Hydroxybutane..

hco3

hco3

Homocysteine

hcyt[c]

Homoisocitrate ..


Homoserine

Isocitrate

Isocitrate

Isoleucine


LeucineLysine

Malate

Malate

Mannan C6H10O5


5,10−Methylenet..

Ammonium

Ammonium

O2

Oxaloacetate

Oxaloacetate

Ornithine

Ornithine

Oxaloglutarate ..

Phosphatidate ..



Phosphatidylcho..

Phosphatidyleth..

Phosphoenolpyru..

Phenylalanine


Phenylpyruvate

Phosphate

Prephenate



Proline


Phosphatidylser..


Pyruvate

Pyruvate

Pyruvate

Ubiquinone 6 C3..

Ubiquinol 6 C39..


Ribulose 5−phos..

Sedoheptulose 1..

Sedoheptulose 7..

L Saccharopine ..

Serine


Sulfite

Sulfate

Sulfate

Succinate

Succinate


Threonine

Threonine


TrehaloseTrehalose

Triglyceride y..

Tryptophan Tyrosine

UDP

UDPglucose

UMP

UTP

Valine

Xylulose 5−phos..

Zymosterol C27H..

(c) Metabolite phase angles, mismatch (d) Flux phase angles, predicted

10−Formyltetrah..

1 3 beta D Gluc..








2−Isopropylmale..

2−Oxobutanoate

2−Oxobutanoate

2 Oxoadipate C6..

2 Oxoadipate C6..

Glycerate 2−pho..












3−Phosphohydrox..




5−Methyltetrahy..



L 2 Aminoadipat..

L 2 Aminoadipat..

Acetate

Acetate

Acetate

Acetaldehyde

Acetaldehyde

Acetaldehyde

Acetyl−CoA




O Acetyl L homo..



2−Oxoglutarate

Alanine

Alanine

2−Acetolactate


Arginine


Asparagine

Aspartate

Aspartate 4−sem..

But 1 ene 1 2 4..

Carbamoyl phosp..

Chorismate

Citrate

Citrate

Citrulline

CMP

CO2

Coenzyme A

Cysteine

Cystathionine

DAMPDCMP

DGMP

Dihydroxyaceton..

DTMP

Erythrose 4−pho..


Ergosterol C28H..

Ethanol

Ethanol

Ethanol

Fructose 6−phos..

Fructose 1,6−bi..


Formate

Fumarate

Fumarate

Glucose 1−phosp..

Glyceraldehyde ..

Glucose 6−phosp..

GlucoseGlucose

Glutamine

Glutamate 5−pho..

Glutamate 5−sem..

Glutamate

Glutamate

Glyoxylate

Glyoxylate

Glycine

Glycine


Glycogen

GMP

H2O

Hydrogen sulfid..

Hydrogen sulfid..

H+

2 Hydroxybutane..

2 Hydroxybutane..

hco3

hco3

Homocysteine

hcyt[c]

Homoisocitrate ..


Homoserine

Isocitrate

Isocitrate

Isoleucine


LeucineLysine

Malate

Malate

Mannan C6H10O5


5,10−Methylenet..

Ammonium

Ammonium

O2

Oxaloacetate

Oxaloacetate

Ornithine

Ornithine

Oxaloglutarate ..

Phosphatidate ..



Phosphatidylcho..

Phosphatidyleth..

Phosphoenolpyru..

Phenylalanine


Phenylpyruvate

Phosphate

Prephenate



Proline


Phosphatidylser..


Pyruvate

Pyruvate

Pyruvate

Ubiquinone 6 C3..

Ubiquinol 6 C39..


Ribulose 5−phos..

Sedoheptulose 1..

Sedoheptulose 7..

L Saccharopine ..

Serine


Sulfite

Sulfate

Sulfate

Succinate

Succinate


Threonine

Threonine


TrehaloseTrehalose

Triglyceride y..

Tryptophan Tyrosine

UDP

UDPglucose

UMP

UTP

Valine

Xylulose 5−phos..

Zymosterol C27H..

10−Formyltetrah..

1 3 beta D Gluc..








2−Isopropylmale..

2−Oxobutanoate

2−Oxobutanoate

2 Oxoadipate C6..

2 Oxoadipate C6..

Glycerate 2−pho..












3−Phosphohydrox..




5−Methyltetrahy..



L 2 Aminoadipat..

L 2 Aminoadipat..

Acetate

Acetate

Acetate

Acetaldehyde

Acetaldehyde

Acetaldehyde

Acetyl−CoA




O Acetyl L homo..



2−Oxoglutarate

Alanine

Alanine

2−Acetolactate


Arginine


Asparagine

Aspartate

Aspartate 4−sem..

But 1 ene 1 2 4..

Carbamoyl phosp..

Chorismate

Citrate

Citrate

Citrulline

CMP

CO2

Coenzyme A

Cysteine

Cystathionine

DAMPDCMP

DGMP

Dihydroxyaceton..

DTMP

Erythrose 4−pho..


Ergosterol C28H..

Ethanol

Ethanol

Ethanol

Fructose 6−phos..

Fructose 1,6−bi..


Formate

Fumarate

Fumarate

Glucose 1−phosp..

Glyceraldehyde ..

Glucose 6−phosp..

GlucoseGlucose

Glutamine

Glutamate 5−pho..

Glutamate 5−sem..

Glutamate

Glutamate

Glyoxylate

Glyoxylate

Glycine

Glycine


Glycogen

GMP

H2O

Hydrogen sulfid..

Hydrogen sulfid..

H+

2 Hydroxybutane..

2 Hydroxybutane..

hco3

hco3

Homocysteine

hcyt[c]

Homoisocitrate ..


Homoserine

Isocitrate

Isocitrate

Isoleucine


LeucineLysine

Malate

Malate

Mannan C6H10O5


5,10−Methylenet..

Ammonium

Ammonium

O2

Oxaloacetate

Oxaloacetate

Ornithine

Ornithine

Oxaloglutarate ..

Phosphatidate ..



Phosphatidylcho..

Phosphatidyleth..

Phosphoenolpyru..

Phenylalanine


Phenylpyruvate

Phosphate

Prephenate



Proline


Phosphatidylser..


Pyruvate

Pyruvate

Pyruvate

Ubiquinone 6 C3..

Ubiquinol 6 C39..


Ribulose 5−phos..

Sedoheptulose 1..

Sedoheptulose 7..

L Saccharopine ..

Serine


Sulfite

Sulfate

Sulfate

Succinate

Succinate


Threonine

Threonine


TrehaloseTrehalose

Triglyceride y..

Tryptophan Tyrosine

UDP

UDPglucose

UMP

UTP

Valine

Xylulose 5−phos..

Zymosterol C27H..

Figure 4: Periodic metabolite concentrations and fluxes caused by a harmonically varying perturbation of oxygenuptake. Phase angles are colour-coded (inset Figure in section 3.1). (a) Measured metabolite phase angles. (b)Predicted metabolite phase angles. (c) Differences between predicted and measured metabolite phase angles. (d)Predicted flux phase angles.

7

3.2 Metabolic dynamics in response to temporal oxygen uptake

Figure 5 shows simulated metabolic concentrations, assuming a predefined periodic oxygen uptake rate (experi-

mentally measured oxygen uptake curve). The model parameters (reaction elasticities) were fitted to the measured

metabolite curves. Measured and fitted metabolite curves are compared in Figure 6. The phase angles of many

metabolites were fitted relatively well. However, some phase angles in glycolysis were wrongly predicted.

0

0.5

1

2−Oxoglutarate

0

0.5

1

3−Phospho−D−gly[..]

0

0.5

1

6−Phospho−D−glu[..]

0

0.5

1

AMP

0

0.5

1

ATP

0

0.5

1

Acetaldehyde

0

0.5

1

Acetate

0

0.5

1

Acetyl−CoA

0

0.5

1

Adenosine 5’−ph[..]

0

0.5

1

Alpha−D−Ribose [..]

0

0.5

1

Anthranilate

0

0.5

1

CMP

0

0.5

1

CO2

0

0.5

1

Citrate

0

0.5

1

Coenzyme A

0

0.5

1

D−Erythrose 4−p[..]

0

0.5

1

D−Fructose 1,6−[..]

0

0.5

1

D−Fructose 6−ph[..]

0

0.5

1

D−Glucose 1−pho[..]

0

0.5

1


0

0.5

1

D−Glycerate 2−p[..]

0

0.5

1

D−Ribulose 5−ph[..]

0

0.5

1

DTMP

0

0.5

1

Ethanol

0

0.5

1

Fumarate

0

0.5

1

GMP

0

0.5

1

Glycine

0

0.5

1

Glycogen

0

0.5

1

Hydrogen sulfid[..]

0

0.5

1

Isocitrate

0

0.5

1

L−Alanine

0

0.5

1

L−Asparagine

0

0.5

1

L−Aspartate

0

0.5

1

L−Citrulline

0

0.5

1

L−Cystathionine

0

0.5

1

L−Glutamate

0

0.5

1

L−Glutamine

0

0.5

1

L−Histidine

0

0.5

1

L−Homoserine

0

0.5

1

L−Isoleucine

0

0.5

1

L−Leucine

0

0.5

1

L−Lysine

0

0.5

1

L−Malate

0

0.5

1

L−Phenylalanine

0

0.5

1

L−Proline

0

0.5

1

L−Serine

0

0.5

1

L−Threonine

0

0.5

1

L−Tryptophan

0

0.5

1

L−Tyrosine

0

0.5

1

L−Valine

0

0.5

1

N(omega)−(L−Arg[..]

0

0.5

1

N−Acetyl−L−glut[..]

0

0.5

1

N2−Acetyl−L−orn[..]

0

0.5

1

Nicotinamide ad[..]

0

0.5

1

Nicotinamide ad[..]

0

0.5

1

O−Phospho−L−ser[..]

0

0.5

1

O2

0

0.5

1

Phosphoenolpyru[..]

0

0.5

1

Pyruvate

0

0.5

1

Sedoheptulose 7[..]

0

0.5

1

UDP

0

0.5

1

UMP

0

0.5

1

UTP

Figure 5: Metabolite concentration curves, measured (red) and simulated (blue). The simulated curves representmodel fits, not independent predictions. All curves were shifted and scaled to span the (y-axis) range from 0 to1.

8

(a) Concentration curves, measured vs fitted (b) Metabolite phase angles, measured vs fitted

Measured

Pre

dic

ted

2−Oxoglutarate

Measured

Pre

dic

ted


Measured

Pre

dic

ted


Measured

Pre

dic

ted

AMP

Measured

Pre

dic

ted

ATP

Measured

Pre

dic

ted

Acetaldehyde

Measured

Pre

dic

ted

Acetate

Measured

Pre

dic

ted

Acetyl−CoA

Measured

Pre

dic

ted

Adenosine 5’−ph[..]

Measured

Pre

dic

ted


Measured

Pre

dic

ted

Anthranilate

Measured

Pre

dic

ted

CMP

Measured

Pre

dic

ted

CO2

Measured

Pre

dic

ted

Citrate

MeasuredP

redic

ted

Coenzyme A

Measured

Pre

dic

ted


Measured

Pre

dic

ted


Measured

Pre

dic

ted


Measured

Pre

dic

ted


Measured

Pre

dic

ted


Measured

Pre

dic

ted


Measured

Pre

dic

ted


Measured

Pre

dic

ted

DTMP

Measured

Pre

dic

ted

Ethanol

MeasuredP

redic

ted

Fumarate

Measured

Pre

dic

ted

GMP

Measured

Pre

dic

ted

Glycine

Measured

Pre

dic

ted

Glycogen

Measured

Pre

dic

ted

Hydrogen sulfid[..]

Measured

Pre

dic

ted

Isocitrate

Measured

Pre

dic

ted

L−Alanine

Measured

Pre

dic

ted

L−Asparagine

Measured

Pre

dic

ted

L−Aspartate

Measured

Pre

dic

ted

L−Citrulline

MeasuredP

redic

ted

L−Cystathionine

Measured

Pre

dic

ted

L−Glutamate

Measured

Pre

dic

ted

L−Glutamine

Measured

Pre

dic

ted

L−Histidine

Measured

Pre

dic

ted

L−Homoserine

Measured

Pre

dic

ted

L−Isoleucine

Measured

Pre

dic

ted

L−Leucine

Measured

Pre

dic

ted

L−Lysine

Measured

Pre

dic

ted

L−Malate

Measured

Pre

dic

ted

L−Phenylalanine

MeasuredP

redic

ted

L−Proline

Measured

Pre

dic

ted

L−Serine

Measured

Pre

dic

ted

L−Threonine

Measured

Pre

dic

ted

L−Tryptophan

Measured

Pre

dic

ted

L−Tyrosine

Measured

Pre

dic

ted

L−Valine

Measured

Pre

dic

ted


Measured

Pre

dic

ted


Measured

Pre

dic

ted


Measured

Pre

dic

ted

Nicotinamide ad[..]

Measured

Pre

dic

ted

Nicotinamide ad[..]

Measured

Pre

dic

ted


Measured

Pre

dic

ted

O2

Measured

Pre

dic

ted

Phosphoenolpyru[..]

Measured

Pre

dic

ted

Pyruvate

Measured

Pre

dic

ted

Sedoheptulose 7[..]

Measured

Pre

dic

ted

UDP

Measured

Pre

dic

ted

UMP

Measured

Pre

dic

ted

UTP

−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

2−Oxoglutarate

3−Phospho−D−glycerate

6−Phospho−D−gluconate

AMP

ATP

Acetaldehyde

Acetate

Acetyl−CoA

Adenosine 5’−phosphosulfate

Alpha−D−Ribose 5−phosphate

Anthranilate

CMP

CO2

Citrate

Coenzyme A

D−Erythrose 4−phosphate

D−Fructose 1,6−bisphosphateD−Fructose 6−phosphate

D−Glucose 1−phosphateD−Glucose 6−phosphate

D−Glycerate 2−phosphate

D−Ribulose 5−phosphateDTMPEthanol

Fumarate

GMP

Glycine

Glycogen

Hydrogen sulfide

Isocitrate

L−AlanineL−Asparagine

L−Aspartate

L−CitrullineL−Cystathionine

L−Glutamate

L−Glutamine

L−Histidine

L−Homoserine

L−IsoleucineL−Leucine

L−Lysine

L−Malate

L−Phenylalanine

L−Proline

L−Serine

L−Threonine

L−Tryptophan

L−Tyrosine

L−Valine

N(omega)−(L−Arginino)succinate

N−Acetyl−L−glutamate

N2−Acetyl−L−ornithine

Nicotinamide adenine dinucleotide phosphate

Nicotinamide adenine dinucleotide phosphate − reducedO−Phospho−L−serine

O2

PhosphoenolpyruvatePyruvate

Sedoheptulose 7−phosphate

UDP

UMP

UTP

Metabolite phase angles (data)

Me

tab

olit

e p

ha

se

an

gle

s (

pre

dic

ted

)

Phases scatter plot

(c) Metabolite phase differences on network (d) Metabolite phase difference, histogram

10fthf c

13BDglcn c

13dpg c

1pyr5c c

23dhmb m

23dhmp m

2ahbut m

2cpr5p c

2dda7p c

2ippm c

2obut c

2obut m

2oxoadp c2oxoadp m

2pg c

34hpp c

3c2hmp c

3c3hmp c

3c4mop c

3dhq c3dhsk c

3ig3p c

3mob c

3mob m

3mop c

3mop m

3pg c3php c

3psme c

4mop c

4pasp c

5mthf c

6pgc c6pgl c

L2aadp6sa c

L2aadp c

ac c

ac e

ac m

acald cacald e

acald m

accoa m

acg5p m

acg5sa m

acglu m

achms c

acorn m

aicar c

akg m

ala L c

ala L m

alac S m

anth c aps c

arg L c

argsuc c

asn L c

asp L m

aspsa c

b124tc mcbp c

chor c

cit c

cit m

citr L c

cmp c

co2 e

coa m

cys L ccyst L c

damp cdcmp c

dgmp c

dhap c

dtmp c

e4p c

eig3p c

ergst c

etoh c

etoh e

etoh m

f6p c

fdp c

ficytc mfocytc m

for c

fum c

fum m

g1p c

g3p c

g6p c

glc D cglc D e

gln L m

glu5p c

glu5sa c

glu L c

glu L m

glx cglx m

gly c

gly m

glyc3p cglyc cglyc e

glycogen cgmp c

h2o e

h2s c

h2s e

h e

hcit c

hcit m

hco3 c

hco3 m

hcys L c

hcyt c

hicit m

his L chisp chistd c

hom L c

icit c

icit m

ile L c

imacp c

leu L clys L c

mal L c

mal L m

mannan c

met L cmethf c

mlthf c

nh4 e

nh4 m

o2 e

oaa c

oaa m

orn corn m

oxag m

pa SC c

pap c

paps c

pc SC cpe SC c

pep c

phe L c

phom c

phpyr c

pi e

pphn c

pran c

prbamp cprbatp c prfp c prlp c

pro L c

prpp c

ps SC c

pser L c ptd1ino SC c

pyr c

pyr e

pyr m

q6 m

q6h2 m

r5p c

ru5p D c

s17bp c

s7p c

saccrp L c

ser L c

skm5p cskm c

so3 c

so4 c

so4 e

succ c

succ e

succ m succoa m

thr L c

thr L m

tre6p c

tre ctre e

triglyc SC c

trp L c tyr L c

udp c

udpg c

ump c

utp c

val L c

xu5p D c

zymst c

−180

−90

0

90

180

−180 −90 0 90 1800

2

4

6

8

10

12Phase differences (positive: predicted curve peaks too late)

Figure 6: Comparison between measured and simulated metabolite curves. (a) Measured (x-axis) versus simulated(y-axis) metabolite time series. Dots close to a rising diagonal indicate good predictions, dots close to a fallingdiagonal indicate poor predictions. (b) Measured and predicted phase angles of oscillating metabolite levels. Dotsin the grey zone indicate good predictions, dots in the white zone indicate poor predictions. Phase angles weredetermined from concentration curves by taking the first Fourier component. (c) Mismatch between measuredand predicted phase angles. Phase mismatches are colour-coded (inset Figure in section 3.1). (d) Histogram ofphase mismatches. The peak around the phase angle 0 represents the large number of dots in the grey zone in(b).

9

3.3 Simulated drop in glucose level

As an independent test, the model was used to predict the metabolic dynamics in another experiment, i.e., data

that had not been used during model construction. The simulation concerns a part of the experimental time series

in which the cells that underwent a sudden drop in available external glucose. The metabolomics data show how

metabolite levels glycolysis are dropping fast, while some of them recover after a while. In the simulations, the

model starts in a steady state with standard external glucose concentration. At the beginning of the simulation,

this concentration is set to a lower value, causing a drop in internal metabolite levels as seen in the experimental

data (Figure 7). The experimentally observed recovery of some metabolite concentrations may be caused by a

shift to ethanol consumption, which is not appropriately captured by the model. The oxygen level remained fixed

in this simulation.

0

0.5

1

2−Oxoglutarate

0

0.5

1


0

0.5

1


0

0.5

1

ADP

0

0.5

1

ATP

0

0.5

1

Acetyl−CoA

0

0.5

1


0

0.5

1

Anthranilate

0

0.5

1

CMP

0

0.5

1

Citrate

0

0.5

1

Coenzyme A

0

0.5

1


0

0.5

1


0

0.5

1


0

0.5

1


0

0.5

1


0

0.5

1


0

0.5

1

DCMP

0

0.5

1

Dihydroxyaceton[..]

0

0.5

1

Fumarate

0

0.5

1

GMP

0

0.5

1

Isocitrate

0

0.5

1

L−Arginine

0

0.5

1

L−Asparagine

0

0.5

1

L−Citrulline

0

0.5

1

L−Cystathionine

0

0.5

1

L−Glutamate

0

0.5

1

L−Histidine

0

0.5

1

L−Histidinol

0

0.5

1

L−Homocysteine

0

0.5

1

L−Homoserine

0

0.5

1

L−Isoleucine

0

0.5

1

L−Leucine

0

0.5

1

L−Lysine

0

0.5

1

L−Malate

0

0.5

1

L−Phenylalanine

0

0.5

1

L−Proline

0

0.5

1

L−Serine

0

0.5

1

L−Threonine

0

0.5

1

L−Tyrosine

0

0.5

1

L−Valine

0

0.5

1


0

0.5

1


0

0.5

1


0

0.5

1

Nicotinamide ad[..]

0

0.5

1

Nicotinamide ad[..]

0

0.5

1


0

0.5

1

Ornithine

0

0.5

1

Phosphoenolpyru[..]

0

0.5

1

Sedoheptulose 7[..]

0

0.5

1

Shikimate

0

0.5

1

UDP

0

0.5

1

UMP

0

0.5

1

UTP

Figure 7: Concentration curves for glucose drop (scaled per metabolite), smoothed measured (red) and simulated(blue). All curves are shifted and scaled to cover the range from 0 to 1.

4 Discussion

It is important to remember that the current model only describes a subsystem of the cell – the metabolic network.

The model is not meant to explain the origin of the observed oscillations, but only what shape these oscillations

assume in different parts of the network. An explanation of the oscillations themselves would require a whole-cell

model that couples metabolism to other processes, for example, to the physiological changes in mitochondrial

10

structure that may be responsible for the periodic breakdown of respiration and for its subsequent recovery. Also

other questions remain untouched: How do cells synchronize each other? Do single cells oscillate also in other

experimental conditions, where oscillations are not observed on a cell population level? What are the possible

fitness advantages or disadvantages provided by the oscillations? Other types of models would be required to

answer such questions.

The results presented here are still preliminary. Some details of the model still need to be improved, and the

numerical optimisation used for model fitting may need some improvements as well. At the current stage, however,

there are already good matches for many metabolite curves. Despite the large number of model parameters

(the number of reaction elasticities is given by the number of reactions, multiplied with the average number of

substrates, products, and allosteric regulators per reaction), it is highly unlikely that that this match is a mere

result of overfitting because the only dynamical input to the model was the (experimentally determined) time-

dependent oxygen uptake rate, the model’s network structure restricts the possible shapes and phase shifts of

metabolite levels quite strongly, and the fitted reaction elasticities were constrained to yield a dynamically stable

state. Whether overfitting occurs, or to what extent, will have to be checked by crossvalidiation.

Acknowledgments

I thank Douglas Murray for his great support and hospitality during the project. I also thank Willi Gottstein,

Rainer Machne, and Sabrina Hoffmann, on whose work I’m building, for plenty of help and ideas they contributed.

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