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Something from nothing − bridging the gap between constraint-based and kinetic modelling

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Something from nothing: bridging the gapbetween constraint-based and kinetic modelling
K. Smallbone, E. Simeonidis, D.S. Broomhead andD.B. KellOctober 2007MIMS EPrint:
2008.80
Manchester Institute for Mathematical SciencesSchool of MathematicsThe University of Manchester
Reports available from:
http://www.manchester.ac.uk/mims/eprints
And by contacting: The MIMS SecretarySchool of MathematicsThe University of ManchesterManchester, M13 9PL, UK
ISSN 1749-9097
Something from nothing
)
bridging the gap betweenconstraint-based and kinetic modelling
Kieran Smallbone
1,2
, Evangelos Simeonidis
1,3
, David S. Broomhead
1,2
and Douglas B. Kell
1,4
1 Manchester Centre for Integrative Systems Biology, The University of Manchester, UK2 School of Mathematics, The University of Manchester, UK3 School of Chemical Engineering and Analytical Science, The University of Manchester, UK4 School of Chemistry, The University of Manchester, UK
The emergent ﬁeld of systems biology involves thestudy of the interactions between the components of abiological system, and how these interactions give riseto the function and behaviour of that system (forexample, the enzymes and metabolites in a metabolicpathway). Nonlinear processes dominate such biologi-cal networks, and hence intuitive verbal reasoningapproaches are insufﬁcient to describe the resultingcomplex system dynamics [1–3]. Nor can suchapproaches keep pace with the large increases in-omics data (such as metabolomics and proteomics)and the accompanying advances in highthroughputexperiments and bioinformatics. Rather, experiencefrom other areas of science has taught us that quanti-tative methods are needed to develop comprehensivetheoretical models for interpretation, organization andintegration of this data. Once viewed with scepticism,we now realize that mathematical models, continuouslyrevised to incorporate new information, must be usedto guide experimental design and interpretation.We focus here on the development and analysis of mathematical models of cellular metabolism [4–6]. Inrecent years two major (and divergent) modellingmethodologies have been adopted to increase ourunderstanding of metabolism and its regulation. Theﬁrst is constraint-based modelling [7,8], which usesphysicochemical constraints such as mass balance,energy balance, and ﬂux limitations to describe the
Keywords
ﬂux balance analysis; linlog kinetics;
Saccharomyces cerevisiae
Correspondence
K. Smallbone, Manchester Centre forIntegrative Systems Biology, ManchesterInterdisciplinary Biocentre, 131 PrincessStreet, Manchester, M1 7 DN, UKFax: +44 161 30 65201Tel: +44 161 30 65146E-mail: kieran.smallbone@manchester.ac.ukWebsite: http://www.mcisb.org/ (Received 29 June 2007, revised 17 August2007, accepted 29 August 2007)doi:10.1111/j.1742-4658.2007.06076.x
Two divergent modelling methodologies have been adopted to increase ourunderstanding of metabolism and its regulation. Constraint-based modellinghighlights the optimal path through a stoichiometric network within certainphysicochemical constraints. Such an approach requires minimal biologicaldata to make quantitative inferences about network behaviour; however,constraint-based modelling is unable to give an insight into cellular substrateconcentrations. In contrast, kinetic modelling aims to characterize fully themechanics of each enzymatic reaction. This approach suffers becauseparameterizing mechanistic models is both costly and time-consuming. Inthis paper, we outline a method for developing a kinetic model for a meta-bolic network, based solely on the knowledge of reaction stoichiometries.Fluxes through the system, estimated by ﬂux balance analysis, are allowedto vary dynamically according to linlog kinetics. Elasticities are estimatedfrom stoichiometric considerations. When compared to a popular branchedmodel of yeast glycolysis, we observe an excellent agreement between thereal and approximate models, despite the absence of (and indeed therequirement for) experimental data for kinetic constants. Moreover, usingthis particular methodology affords us analytical forms for steady statedetermination, stability analyses and studies of dynamical behaviour.
Abbreviations
BPG, 1,3-bisphosphoglycerate; ETOH, ethanol; FBA, ﬂux balance analysis; PFK, phosphofructokinase.
5576
FEBS Journal
274
(2007) 5576–5585
ª
2007 The Authors Journal compilation
ª
2007 FEBS
potential behaviour of an organism. The biochemicalstructure of (at least the central) metabolic pathways ismore or less well-known, and hence the stoichiometriesof such a network may be deduced. In addition, theﬂux of each reaction through the system may be con-strained through, for example, knowledge of its V
max
,or irreversibility considerations. From the steady statesolution space of all possible ﬂuxes, a number of tech-niques have been proposed to deduce network behav-iour, including ﬂux balance and extreme pathway orelementary mode analysis. In particular, ﬂux balanceanalysis (FBA) [9] highlights the most effective andefﬁcient paths through the network in order to achievea particular objective function, such as the maximiza-tion of biomass or ATP production.The key beneﬁt of FBA lies in the minimal amount of biological knowledge and data required to make quanti-tative inferences about network behaviour. However,this apparent free lunch comes at a price – constraint-based modelling is concerned only with ﬂuxes throughthe system and does not make any inferences nor anypredictions about cellular metabolite concentrations. Bycontrast, kinetic modelling aims to characterize fully themechanics of each enzymatic reaction, in terms of howchanges in metabolite concentrations affect local reac-tion rates. However, a considerable amount of data isrequired to parameterize a mechanistic model; if com-plex reactions like phosphofructokinase are involved,an enzyme kinetic formula may have 10 or more kineticparameters [6]. The determination of such parameters iscostly and time-consuming, and moreover many may bedifﬁcult or impossible to determine experimentally. The
in vivo
molecular kinetics of some important processeslike oxidative phosphorylation and many transportmechanisms are almost completely unknown, so thatmodelling assumptions about these metabolic processesare necessarily highly speculative.In this paper, we deﬁne a novel method for the gen-eration of kinetic models of cellular metabolism. Likeconstraint-based approaches, the modelling frameworkrequires little experimental data regarding variablesand no knowledge of the underlying mechanisms foreach enzyme; nonetheless it allows inference of thedynamics of cellular metabolite concentrations. Theﬂuxes found through FBA are allowed to vary dynam-ically according to linlog kinetics [10–12]. Linlog kinet-ics, which draws ideas from thermodynamics andmetabolic control analysis, is known to be moreappropriate for approximating hyperbolic enzymekinetics than are other phenomenological relationssuch as power laws [13]. Indeed, when a version usinglinlog kinetics is compared with the srcinal and mech-anistic branched yeast glycolysis model of Teusink
et al
. [14], we observe an excellent agreement betweenthe real and approximate models. Moreover, we showthat a model framed within the linlog format affordsanalytical forms for steady state determination, stabil-ity analyses and studies of dynamical behaviour. Assuch, it does not suffer from the usual [15] computa-tional scalability problems, and could therefore beapplied to existing genome scale models of metabolism[8,16–18]. Such a model has powerful predictive powerin determining cellular responses to environmentalchanges, and may be considered a stepping-stone to afull kinetic model of cell metabolism: a ‘virtual cell’.
Results
The linlog approximation [10–12] is a method for sim-plifying reaction rate laws in metabolic networks.Drawing ideas from metabolic control analysis, itdescribes the effect of metabolite levels on ﬂux as a lin-ear sum of logarithmic terms (Eqn 2). By deﬁnition, itwill provide a good approximation near a chosen refer-ence state. Moreover, thermodynamic considerationsshow that we can expect a logarithmic response tochanges in metabolite concentrations [10,13], andhence that the approximation may be valid some dis-tance from the reference state. Indeed, linlog kineticsare known to be appropriate for approximating hyper-bolic enzyme kinetics, and, in this case, are superior toother phenomenological relations such as power laws(including generalized mass action and S-systems) [13].To illustrate this graphically, we compare in Fig. 1
10
−1
10
0
10
1
00.511.522.5Substrate concentration x / x
0
R e a c t i o n r a t e v / v
0
Fig. 1.
A comparison of Michaelis–Menten kinetics
v
(
x
)
¼
V x
⁄
(
x
+
K
m
) (o) to its linlog approximation
u
(
x
)
¼
v
* (1 +
e
log(
x
⁄
x
*))(solid line) and its power law approximation
x
(x)
¼
v
*(
x
⁄
x
*)
e
(dashed line), where
v
*
¼
v
(
x
*) and
e
¼
K
m
⁄
(
x
* +
K
m
). Parametervalues used are
x
*
¼
V
¼
K
m
¼
1.K. Smallbone
et al.
Constraint-based meets kinetic modelling
FEBS Journal
274
(2007) 5576–5585
ª
2007 The Authors Journal compilation
ª
2007 FEBS
5577
typical irreversible Michaelis–Menten kinetics (o) withits linlog (solid line) counterpart. Notice that theabscissa is logarithmic, and Michaelis-Menten kineticsappears to be close to linear in this plotting regime.Thus linlog serves as an excellent approximation. Evenan order of magnitude away from the reference state,the functions have comparable values. Also shown isthe power law approximation (dashed line). We seethat linlog provides a better approximation than powerlaw for substrate concentrations greater than the refer-ence state, whilst the two approximations are equallyvalid for concentrations less than the reference state.Having described the validity of linlog kinetics atthe single reaction level, we move on to apply theapproximation to a full network: the branched modelof yeast glycolysis of Teusink
et al
. [14], available inSBML format from JWS Online [19]. Taking the mod-el’s steady state as our reference state, elasticities maybe calculated analytically from the kinetic equationsusing Eqn (3). Eqn (10) may then be used to predictchanges in internal metabolite concentrations withexternal metabolite changes.In the Teusink
et al
. model, there are three externaleffectors: ethanol, glucose and glycerol; in Fig. 2 weshow, as an example, internal changes in response tochanges in ethanol (ETOH). We see that both thezeroth and ﬁrst derivatives of the linlog kineticsare correct around the reference state [ETOH]
¼
[ETOH]
0
, and hence the approximation is good in aregion near this point. Moreover, we see that inmany cases the approximation remains valid whenthe ethanol concentration is changed by an order of magnitude.Linlog provides a good approximation to enzymekinetics, and moreover (as we show in Eqns 10–13)affords analytical forms for steady state determination,stability analyses and temporal dynamics. However,the good ﬁt in Fig. 2 was obtained through our exactknowledge of the underlying kinetic formulae. Phe-nomenological relations such as linlog are unlikely tobe of such interest when all enzymatic mechanisms andcorresponding parameters are known; rather the inter-est lies in their applicability when such information isnot available and we require a best guess model of the
01234
R e l a t i v e c h a n g e i n B P G
0.511.522.5
R e l a t i v e c h a n g e i n G L C i
0.511.52
R e l a t i v e c h a n g e i n P 2 G
10
–1
10
0
10
1
00.511.522.53
R e l a t i v e c h a n g e i n P E P
Relative change in ETOH10
–1
10
0
10
1
10
–1
10
0
10
1
10
–1
10
0
10
1
Relative change in ETOH
Fig. 2.
From Eqn (10). Elected variations in steady state intracellular metabolite concentrations with changes in ethanol (ETOH) concentra-tion in the branched model of yeast glycolysis of Teusink
et al
. [14]. Shown are the real model solutions (o) and the predictions of the linlogapproximation (solid line). BPG, 1,3-bisphosphoglycerate; GLCi, glucose in cytosol; P2G, 2-phosphoglycerate; PEP, phosphoenolpyruvate.
Constraint-based meets kinetic modelling
K. Smallbone
et al.
5578
FEBS Journal
274
(2007) 5576–5585
ª
2007 The Authors Journal compilation
ª
2007 FEBS
underlying kinetics. Returning to Eqn (10), we see thatto predict steady state behaviour in response tochanges in external effectors, estimates are required forthe reference system ﬂux and the elasticities.The ﬁrst point, estimation of system ﬂuxes with lim-ited information, may be addressed through appealingto ﬂux balance analysis [9]. This method allows us toidentify the optimal path through the network toachieve a particular objective, such as biomass yield orATP production. Biologically, this kind of objectivefunction assumes that an organism has evolved overtime to lie close to its maximal metabolic efﬁciency,within its underlying physicochemical, topobiological,environmental and regulatory constraints [8].FBA (Eqn 15) is applied to the model of Teusink
et al.
deﬁning the objective function as cellular ATPproduction; the results are presented in Table 1. Wesee that FBA does provide a good estimate to the realﬂuxes through the system as predicted by Teusink
et al
. The discrepancy between the real and FBA solu-tion is due to FBA disregarding the branches of thepathway not involved in ATP production, namely glyc-erol, glycogen, succinate and trehalose synthesis. It isinteresting to note that, in the full model, the ﬂuxesthrough these branches are relatively small; the major-ity of ﬂux is used to generate ATP as assumed byFBA.It remains to estimate the elasticities. Of course theseshould ideally be measured explicitly using traditionalenzyme assays, for example. In the absence of suchinformation, assuming knowledge only of reactionstoichiometries, we follow the tendency modellingapproach of Visser
et al
. [20] (see Materials and meth-ods). The results of elasticity estimation when appliedto Teusink
et al
.’s model are presented in Table 2. Wesee that this is a reasonable method for a ﬁrst estima-tion of elasticities; in most cases the estimate fallswithin an order of magnitude of the true elasticity. Itis interesting to observe that in one case, the estimatehas the incorrect sign – the phosphofructokinase(PFK) reaction with respect to high energy phosphates.Whilst ATP is a substrate of PFK, at the referencestate an increase in ATP leads to a decrease in reactionrate. Such a result is counter-intuitive and could not
Table 1.
Results from Eqn (15). A comparison between ﬂuxes inTeusink
et al
. [14] and those predicted by FBA with ATP productionmaximization. For reaction abbreviation deﬁnitions, see supplemen-tary Table S2.ReactionFlux (m
M
Æ
min
)
1
)Teusink FBAADH 129 176ALD 77.3 88.1ATP 84.5 176ENO 136 176G3PDH 18.1 0GAPDH 136 176GLK 88.1 88.1GLYCO 6 0PDC 136 176PFK 77.3 88.1PGI 77.3 88.1PGK 136 176PGM 136 176PYK 136 176SUC 3.64 0Treha 2.4 0
Table 2.
A comparison between elasticities in Teusink
et al
. andthose estimated through stoichiometric considerations. For reactionand metabolite abbreviation deﬁnitions, see supplementaryTables S1–S2.Reaction MetaboliteElasticityTeusink EstimateADH ACE 3.20 1ETOH
)
2.95
)
1NAD
)
3.04
)
1NADH 3.20 1ALD F16P 1.89 1TRIO
)
3.08
)
2ATP P 1.80 1ENO P2G 0.826 1PEP
)
0.384
)
1GAPDH BPG
)
8.00
·
10
)
2
)
1NAD 0.144 1NADH
)
9.14
·
10
)
2
)
1TRIO 0.919 1GLK G6P
)
1.65
·
10
)
2
1GLCi 0.458
)
1P 1.02 1GLT GLCi
)
7.20
·
10
)
2
)
1GLC
O
2.54
·
10
)
2
1PDC ACE 0
)
1PYR 0.423 1PFK
F
16P
)
0.402
)
1F6P 0.936 1P
)
3.21 1PGI
F
6P
)
0.709
)
1G6P 1.18 1PGK BPG 2.81 1P
)
9.47
)
1P3G
)
2.24
)
1PGM P2G
)
2.36
)
1P3G 2.94 1PYK P
)
1.82
)
1PEP 0.765 1PYR
)
0.243
)
1K. Smallbone
et al.
Constraint-based meets kinetic modelling
FEBS Journal
274
(2007) 5576–5585
ª
2007 The Authors Journal compilation
ª
2007 FEBS
5579

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