metabolic oscillations in budding yeast described by a ... fileconstant uptake of glucose (these...
TRANSCRIPT
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..
2,3−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−3−hyd..
3−Carboxy−4−met..
3−Dehydroquinat..3−Dehydroshikim..
C’−(3−Indolyl)−..
3−Methyl−2−oxob..
3−Methyl−2−oxob..
3−Methyl−2−oxop..
3−Methyl−2−oxop..
3−Phospho−D−gly..
3−Phosphohydrox..
5−O−(1−Carboxyv..
4−Methyl−2−oxop..
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..
N−Acetyl−L−glut..
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..
3−Phospho−D−gly..
1−Pyrroline−5−c..
2,3−Dihydroxy−3..
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−3−hyd..
3−Carboxy−4−met..
3−Dehydroquinat..3−Dehydroshikim..
C’−(3−Indolyl)−..
3−Methyl−2−oxob..
3−Methyl−2−oxob..
3−Methyl−2−oxop..
3−Methyl−2−oxop..
3−Phospho−D−gly..
3−Phosphohydrox..
5−O−(1−Carboxyv..
4−Methyl−2−oxop..
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..
N−Acetyl−L−glut..
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..
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..
3−Phospho−D−gly..
1−Pyrroline−5−c..
2,3−Dihydroxy−3..
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−3−hyd..
3−Carboxy−4−met..
3−Dehydroquinat..3−Dehydroshikim..
C’−(3−Indolyl)−..
3−Methyl−2−oxob..
3−Methyl−2−oxob..
3−Methyl−2−oxop..
3−Methyl−2−oxop..
3−Phospho−D−gly..
3−Phosphohydrox..
5−O−(1−Carboxyv..
4−Methyl−2−oxop..
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..
N−Acetyl−L−glut..
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..
Tryptophan Tyrosine
UDP
UDPglucose
UMP
UTP
Valine
Xylulose 5−phos..
Zymosterol C27H..
10−Formyltetrah..
1 3 beta D Gluc..
3−Phospho−D−gly..
1−Pyrroline−5−c..
2,3−Dihydroxy−3..
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−3−hyd..
3−Carboxy−4−met..
3−Dehydroquinat..3−Dehydroshikim..
C’−(3−Indolyl)−..
3−Methyl−2−oxob..
3−Methyl−2−oxob..
3−Methyl−2−oxop..
3−Methyl−2−oxop..
3−Phospho−D−gly..
3−Phosphohydrox..
5−O−(1−Carboxyv..
4−Methyl−2−oxop..
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..
N−Acetyl−L−glut..
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..
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..
3−Phospho−D−gly..
1−Pyrroline−5−c..
2,3−Dihydroxy−3..
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−3−hyd..
3−Carboxy−4−met..
3−Dehydroquinat..3−Dehydroshikim..
C’−(3−Indolyl)−..
3−Methyl−2−oxob..
3−Methyl−2−oxob..
3−Methyl−2−oxop..
3−Methyl−2−oxop..
3−Phospho−D−gly..
3−Phosphohydrox..
5−O−(1−Carboxyv..
4−Methyl−2−oxop..
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..
N−Acetyl−L−glut..
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..
Tryptophan Tyrosine
UDP
UDPglucose
UMP
UTP
Valine
Xylulose 5−phos..
Zymosterol C27H..
10−Formyltetrah..
1 3 beta D Gluc..
3−Phospho−D−gly..
1−Pyrroline−5−c..
2,3−Dihydroxy−3..
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−3−hyd..
3−Carboxy−4−met..
3−Dehydroquinat..3−Dehydroshikim..
C’−(3−Indolyl)−..
3−Methyl−2−oxob..
3−Methyl−2−oxob..
3−Methyl−2−oxop..
3−Methyl−2−oxop..
3−Phospho−D−gly..
3−Phosphohydrox..
5−O−(1−Carboxyv..
4−Methyl−2−oxop..
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..
N−Acetyl−L−glut..
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..
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
D−Glucose 6−pho[..]
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
3−Phospho−D−gly[..]
Measured
Pre
dic
ted
6−Phospho−D−glu[..]
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
Alpha−D−Ribose [..]
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
D−Erythrose 4−p[..]
Measured
Pre
dic
ted
D−Fructose 1,6−[..]
Measured
Pre
dic
ted
D−Fructose 6−ph[..]
Measured
Pre
dic
ted
D−Glucose 1−pho[..]
Measured
Pre
dic
ted
D−Glucose 6−pho[..]
Measured
Pre
dic
ted
D−Glycerate 2−p[..]
Measured
Pre
dic
ted
D−Ribulose 5−ph[..]
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
N(omega)−(L−Arg[..]
Measured
Pre
dic
ted
N−Acetyl−L−glut[..]
Measured
Pre
dic
ted
N2−Acetyl−L−orn[..]
Measured
Pre
dic
ted
Nicotinamide ad[..]
Measured
Pre
dic
ted
Nicotinamide ad[..]
Measured
Pre
dic
ted
O−Phospho−L−ser[..]
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
3−Phospho−D−gly[..]
0
0.5
1
6−Phospho−D−glu[..]
0
0.5
1
ADP
0
0.5
1
ATP
0
0.5
1
Acetyl−CoA
0
0.5
1
Alpha−D−Ribose [..]
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
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 6−pho[..]
0
0.5
1
D−Glycerate 2−p[..]
0
0.5
1
D−Ribulose 5−ph[..]
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
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
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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12