with minimal effort, one may postulate the principle of flux minimization, as follows: given the available external substrates and given a set of functionally important target fluxes requir
Trang 1The principle of flux minimization and its application to estimate stationary fluxes in metabolic networks
Hermann-Georg Holzhu¨tter
Humboldt-University Berlin, Medical School (Charite´), Institute of Biochemistry, Berlin, Germany
Cellular functions are ultimately linked to metabolic fluxes
brought about by thousands of chemical reactions and
transport processes The synthesis of the underlying enzymes
and membrane transporters causes the cell a certain effort
of energy and external resources Considering that those cells
should have had a selection advantage during natural
evo-lution that enabled them to fulfil vital functions (such as
growth, defence against toxic compounds, repair of DNA
alterations, etc.) with minimal effort, one may postulate the
principle of flux minimization, as follows: given the available
external substrates and given a set of functionally important
target fluxes required to accomplish a specific pattern
of cellular functions, the stationary metabolic fluxes have
to become a minimum To convert this principle into a
mathematical method enabling the prediction of stationary
metabolic fluxes, the total flux in the network is measured
by a weighted linear combination of all individual fluxes
whereby the thermodynamic equilibrium constants are used
as weighting factors, i.e the more the thermodynamic
equilibrium lies on the right-hand side of the reaction, the
larger the weighting factor for the backward reaction A
linear programming technique is applied to minimize the total flux at fixed values of the target fluxes and under the constraint of flux balance (¼ steady-state conditions) with respect to all metabolites The theoretical concept is applied
to two metabolic schemes: the energy and redox metabolism
of erythrocytes, and the central metabolism of Methylobac-terium extorquensAM1 The flux rates predicted by the flux-minimization method exhibit significant correlations with flux rates obtained by either kinetic modelling or direct experimental determination Larger deviations occur for segments of the network composed of redundant branches where the flux-minimization method always attributes the total flux to the thermodynamically most favourable branch Nevertheless, compared with existing methods of structural modelling, the principle of flux minimization appears to be
a promising theoretical approach to assess stationary flux rates in metabolic systems in cases where a detailed kinetic model is not yet available
Keywords: optimality principle; flux balance; kinetic model; metabolic network; systems biology
Correspondence to H.-G Holzhu¨tter, Humboldt University Berlin, Medical Faculty (Charite´), Institute of Biochemistry, Monbijoustr 2,
10117 Berlin, Germany Fax: + 49 30 450 528 942, Tel.: + 49 30 450 528 166, E-mail: hermann-georg.holzhuetter@charite.de
Abbreviations: FBA, flux-balance analysis; OAA, oxaloacetate; PHB, poly b-hydroxy butyrate.
Enzymes: hexokinase (EC 2.7.1.1); phosphohexose isomerase (EC 5.3.1.9); phosphofructokinase (EC 2.7.1.11); aldolase (EC 4.1.2.13); triose-phosphate isomerase (EC 5.3.1.1); glyceraldehyde-3-triose-phosphate dehydrogenase (EC 1.2.1.12); phosphoglycerate kinase (EC 2.7.2.3); bisphospho-glycerate mutase (EC 5.4.2.4); bisphosphobisphospho-glycerate phosphatase (EC 3.1.3.13); phosphobisphospho-glycerate mutase (EC 5.4.2.1); enolase (EC 4.2.1.11); pyruvate kinase (EC 2.7.1.40); lactate dehydrogenase (EC 1.1.1.28); adenylate kinase (EC 2.7.4.3); glucose-6-phosphate dehydrogenase (EC 1.1.1.49); phosphogluconate dehydrogenase (EC 1.1.1.44); glutathione reductase (EC 1.8.1.7); phosphoribulose epimerase (EC 5.1.3.1); ribose phosphate isomerase (EC 5.3.1.6); transketolase (EC 2.2.1.1); transaldolase (EC 2.2.1.2); phosphoribosylpyrophosphate synthetase (EC 2.7.6.1); transketolase (EC 2.2.1.1); ethanol dehydrogenase (EC 1.1.1.244); methylene H4F dehydrogenase (MtdA) (EC 1.5.1.5); methenyl H4F cyclo-hydrolase (EC 3.5.4.9); formyl H4F synthetase (EC 6.3.4.3); formate dehydrogenase (EC 1.2.1.2); formaldehyde-activating enzyme (EC unknown1); methylene H4MPT dehydrogenase (MtdB) (EC unknown); methylene H4MPT dehydrogenase (MtdA) (EC unknown); methenyl H4MPT cyclohydrolase (EC 3.5.4.27); formyl MFR:H4MPT formyltransferase (EC unknown); formyl MFR dehydrogenase (EC 1.2.99.5) serine hydroxymethyltransferase (EC 2.1.2.1); serine-glyoxylate aminotransferase (EC 2.6.1.45); hydroxypyruvate reductase (EC 1.1.1.81); glycerate kinase (EC 2.7.1.31); PEP carboxylase (EC 4.1.1.31); malate dehydrogenase (EC 1.1.1.37); malate thiokinase (EC 6.2.1.9); malyl-CoA lyase (EC 4.1.3.24); pyruvate dehydrogenase (EC 1.2.4.1); citrate synthase (EC 2.3.3.1); aconitase (EC 4.2.1.3); isocitrate dehydrogenase (EC 1.1.1.42); a-KG dehydrogenase (EC 1.2.1.52); succinyl-CoA synthetase (EC 6.2.1.4); succinyl-CoA hydrolase (EC 3.1.2.3); succinate dehydrogenase (EC 1.3.5.1); fumarase (EC 4.2.1.2); malic enzyme (EC 1.1.1.38); pyruvate carboxylase (EC 6.4.1.1); PEP carboxykinase (EC 4.1.1.32); b-ketothiolase (EC 2.3.1.16); acetoacetyl-CoA reductase (NADPH) (EC 1.1.1.36); PHB synthase (EC 2.3.1.-); PHB depolymerase (EC 3.1.1.75); b-hydroxy-butyrate dehydrogenase (EC 1.1.1.30): acetoacetate-succinyl-CoA transferase (EC 2.8.3.5); D -crotonase (EC 4.2.1.17); L -crotonase (EC 4.2.1.17); acetoacetyl-CoA reductase (NADH) (EC 1.1.1.35); crotonyl-CoA reductase (EC 1.3.1.8); propionyl-CoA carboxylase (EC 6.4.1.3); methylmalonyl-CoA mutase (EC 5.4.99.2); NADH-quinone oxidoreductase (EC 1.6.99.5); cytochrome oxidase (EC 1.10.2.2); ubiquinone oxidoreductase (EC 1.5.5.1); NDP kinase (EC 2.7.4.6); transhydrogenase (EC 1.6.1.2); 3-phosphoglycerate dehydrogenase (EC 1.1.1.95); phosphoserine transaminase (EC 2.6.1.52); phosphoserine phosphatase (EC 3.1.3.3); glutamate dehydrogenase (EC 1.4.1.4).
Note: The mathematical model described here has been submitted to the Online Cellular Systems Modelling Database and can be accessed at http://jjj.biochem.sun.ac.za/database/holzhutter/index.html free of charge.
(Received 16 March 2004, revised 3 May 2004, accepted 12 May 2004)
Trang 2Complex cellular functions, such as motility, growth,
replication, defence against toxic compounds and repair of
molecular damage, are ultimately linked to metabolic
processes Metabolic processes can be grossly subdivided
into chemical reactions and membrane transport processes,
most being catalysed by enzymes and facilitated by specific
membrane transporters The activity of these proteins can
be modulated by various modes of regulation, such as
allosteric effectors, reversible phosphorylation and temporal
gene expression These regulatory mechanisms that are
operative at the molecular level have evolved during natural
evolution and enable the cell to adapt its metabolic activities
to specific functional requirements
Mathematical modelling of metabolic networks has a
long tradition in computational biochemistry (reviewed in
[1]) Mathematical models of metabolic systems facilitate
the study of systems behaviour by means of computer-based
in silico simulations This type of mathematical analysis
may provide deeper insights into the regulation and control
of the metabolic system studied [2] Moreover, kinetic
simulations of metabolic networks may partially replace
time-consuming and expensive experiments to explore
possible metabolic alterations of the cell induced by varying
external conditions (e.g pH value, concentration of
sub-strates, concentration of toxic compounds, concentration of
signalling molecules) and thus may provide a valuable
heuristics for future experimental work
It is the common view that realistic kinetic modelling of
metabolic networks needs detailed rate equations for each
of the participating metabolic processes Derivation of a
reliable rate equation requires detailed knowledge of all
physiological effectors influencing the activity of the
cata-lyzing enzyme and the determination of
rate-vs.-concentra-tion relarate-vs.-concentra-tionships for all these effectors Thus, realistic
mathematical modelling of a metabolic pathway turns out
to be a tedious, time-consuming enterprise, which, to date,
has been successfully undertaken only for a few pathways,
e.g the main metabolic pathways of erythrocytes [3–9] or
glycolysis in yeast cells [10] For most metabolic pathways,
and most cell types, the available enzyme–kinetic knowledge
is currently still insufficient to permit realistic mathematical
modelling
To obtain at least a qualitative estimate of stationary
metabolic flux rates without knowledge of the detailed
kinetics of individual processes, the so-called flux-balance
analysis (FBA) has been proposed [11] FBA makes use of
the fact that under steady-state conditions the sum of
fluxes producing or degrading any internal metabolite has
to be zero Application of this method is based on only two
prerequisites, namely that (a) the topology of the metabolic
network under consideration has to be known, and (b) an
evaluation criterion is needed to identify the most likely
flux distribution among all those flux distributions that are
compatible with the steady-state conditions The topology
of the metabolic network is given in terms of the so-called
stoichiometric matrix, relating the time-dependent
vari-ation of the metabolite concentrvari-ations to the fluxes through
all metabolic processes for which an enzyme or transport
protein is available in a given cell type The topology of
central metabolic pathways is, meanwhile, available for
numerous cell types (see, for example, http://www.genome
ad.jp/kegg) In the first place, this is the result of intensive
enzymological work carried out during the last four decades More recently, the sequencing of complete genomes and the development of biostatistical techniques
to map genes onto proteins, enable the prediction of metabolic pathways, even if the biochemical identification and characterization of the underlying enzymes is not yet available
The Darwinian interpretation of natural evolution con-siders existing biological systems as the outcome of an optimization process where the permanent change of phenotype properties, as a result of mutation and selection, leads to the optimal adaptation of an organism to given environmental conditions Based on this hypothesis, several optimization studies have been performed in the field of metabolic regulation, aimed at the prediction of enzyme kinetic properties and enzyme concentration profiles, ensuring optimal performance of metabolic pathways [12–19] The theoretical predictions obtained agree with experimental observations, at least in a qualitative manner
In previous applications of FBA, the optimal production of biomass was used as an optimality criterion [19,20] Whereas the maximization of biomass production as the primary objective of the cellular metabolism makes sense for primitive cells, such as bacteria, which are born to replicate, the application of FBA to cells with more sophisticated ambitions needs a more general criterion Here I propose to settle this criterion on the following principle of flux minimization: given the value of functionally relevant
target fluxes, i.e those fluxes that are directly coupled with cellular functions, the most likely distribution of stationary fluxes within the metabolic network will be such that the weighted sum of all fluxes becomes a minimum This principle is backed up by the fact that increasing the flux through any reaction of a metabolic network requires some
effort This effort can be split into two different categories First, some metabolic effort, in terms of energy and other valuable resources (e.g essential amino acids), is required to synthesize sufficiently high amounts of enzymes and trans-port proteins Second, some evolutionary effort has been required to improve the specificity, catalytic efficiency and regulatory control of an enzyme during the long-term process of natural evolution Whereas the metabolic effort can be measured in units of energy or mass flow, the evolutionary effort is a measure of the probability of favourable mutational events that increase the fidelity of
an enzyme in the context of the metabolic network The principle of flux minimization is based on the plausible assumption that during the early phases of natural evolu-tion, the competition for limited external resources repre-sented a permanent pressure on living cells to fulfil their functions with minimal effort
Employing the principle of flux minimization for the calculation of stationary metabolic fluxes results in the solution of a constrained linear optimization problem: consider the set of all flux distributions meeting the flux balance relations dictated by the stoichiometry of the system and pick out the distribution for which the total flux becomes a minimum The first part of this report briefly outlines the mathematical basis of the method The second part presents two applications of the method to the metabolism of erythrocytes and of the microorganism, Metylobacterium extorquensAM1
Trang 3submitted to the Online Cellular Systems Modelling
Data-base and can be accessed at http://jjj.biochem.sun.ac.za/
database/holzhutter/index.html free of charge
Theory/method
We define the complete metabolic network of a specific
cell by the fluxes vj (jỬ 1,2,Ầ,r), through all reactions
for which at least one enzyme (or transport protein) can
be expressed, and by the metabolites Si (iỬ 1,2,Ầ,n)
involved in these reactions The stoichiometric matrix
indicates how flux vj affects the concentration of
meta-bolite Si: Ni,j> 0, Ni,j molecules of metabolite (i) are
formed during a single reaction (j); Nij< 0, Ni,j
molecules of metabolite (i) are consumed during a single
reaction (j); and NijỬ 0, metabolite (i) is not involved in
reaction (j) For example, for the flux v8 through the
chemical reaction 2S1ợ S2!v8
S3ợ 3S4, the elements of the stoichiometric matrix read: N18Ử)2, N28Ử)1,
N38Ử 1, and N48Ử 3
In general, the fluxes vjmay be positive or negative, i.e
the net reaction may proceed either in a forward or a
backward direction To deal with non-negative variables,
the flux vjis decomposed into an irreversible forward flux,
vđợỡj (the net reaction proceeds from left to right), and an
irreversible backward flux, vđỡj (the net reaction proceeds
from right to left), as follows:
vjỬ vđợỡj vđỡj
vđợỡj Ử vjHđvjỡ; vđỡj Ử vjơHđvjỡ 1 đ1ỡ
whereQ (x) denotes the unit-step function, i.e by definition
only one of the two components vđợỡj and vđỡj can be
different from zero The forward direction is defined as that
which would ensure a positive Gibbs free energy change
under standard conditions (where all reagents are present
at unit concentrations); at these standard conditions the
backward flux is defined to be zero
The steady-state fluxes have to obey the flux-balance
conditions:
Xr
jỬ1
NijvjỬXr
jỬ1
Nijđvđợỡj vđỡj ỡ Ử 0 đi Ử 1; :::; nỡ đ2ỡ
representing the principle of conservation of mass for a
homogeneous reaction system The flux balance conditions
shown in equation system (2) constitute a homogeneous
system of linear equations with respect to the unknown
fluxes For realistic metabolic systems, the number of fluxes
is larger than the number of metabolites, i.e r > n Thus,
equation system (2) is underdetermined, i.e it possesses an
infinite number of solutions
Setting target fluxes through functionally essential
reactions
To accomplish a particular functional state of the cell, the
fluxes through a certain number of target reactions have to
be maintained at nonzero values This can be expressed by
equality constraints of the form:
vjỬ Lj;Lj>0 đj Ử j1; 2; :::ỡ đ3ỡ Some of the target reactions as, for example, the production
of energy (ATP) or the synthesis of membrane phospho-lipids, are permanently required to ensure cell integrity Other target reactions as, for example, the synthesis of a hormone or the detoxification of a pharmaceutical, may be only temporarily required The selection of target fluxes is somewhat arbitrary For example, the demand for a continuous synthesis of phospholipids can be instantiated
by introducing the total amount of phospholipids as a model variable and putting either the flux of phospholipids degradation or the flux of phospholipids synthesis to a nonvanishing value
Flux constraints arising from the availability
of external metabolites The nonequilibrium state of biochemical reaction systems is maintained by a steady uptake of energy-rich, low-entropy substrates and the release of low-energy, high-entropy products The absence of a certain substrate associated with the exchange flux, vi, can be expressed by forcing the uptake component of the flux to zero, as follows:
vđuptakeỡi Ử 0 đ4ỡ
Thermodynamic evaluation of fluxes: irreversibility
of reactions The direction of any flux vj is dictated by the change of Gibbs free energy:
DGjỬ DGđ0ỡj ợ RT ln
Qn iỬ1
ơSiNđợỡj
Qn iỬ1
ơSiNđỡj
0 B
@
1 C
A with Nđợỡij Ử Nij
if Nij 0; Nđỡij Ử Nij if Nij 0 đ5ỡ where DGđ0ỡj denotes the change of Gibbs free energy under the condition that all reactants are present at unit con-centrations (Ử 1 molẳL)1) DGđ0ỡj can be expressed through the thermodynamic equilibrium constant Kequi , as follows:
DGđ0ỡj Ử RT lnđKequj ỡ đ6ỡ where RTỬ 2.48 kJẳmol)1 at room temperature (25C)
As stated above, all reactions of the network will be notated such that under standard conditions DGđ0ỡj ặ 0, Kequi 1, and thus vj> 0 vđỡj Ử 0 The second term in the right-hand side of Eqn (5) depends upon the actual concentra-tions of the reactants which, under cellular condiconcentra-tions, may strongly deviate from unit concentrations With accumula-ting concentrations of the reaction products (appearing in the nominator) and/or vanishing concentrations of the reaction substrates (appearing in the denominator), the concentration-dependent term in Eqn (5) may assume arbitrarily large negative values, i.e in principle the direction
of a chemical reaction can always be reversed provided that other reactions in the system are capable of accomplishing the required change in the concentration of the reactants For example, the standard free energy change of the
Trang 4glyco-lytic reaction (glycerol aldehyde phosphate fi dihydroxy
acetone phosphate) catalyzed by the enzyme triose
phos-phate isomerase amounts to DG(0)Ử)7.94 kJẳmol)1
KequTIMỬ 24.6 Nevertheless, under cellular conditions this
reaction proceeds into a backward direction (dihydroxy
acetone phosphate fi glyceraldehyde phosphate) as the
reaction substrate glycerol aldehyde phosphate is rapidly
converted into 1,3-bisphosphoglycerate along the glycolytic
pathway This example shows that a sharp classification
into reversible and irreversible reactions on the sole basis of
DG(0)can be problematic Instead, we will use the value of
the equilibrium constant as a weighting factor for the
measureF of the total flux:
UỬXr
jỬ1
đvđợỡj ợ Kequj vđỡj ỡ đ7ỡ
Weighting the backward flux with the thermodynamic
equilibrium constant takes into account the thermodynamic
effort connected with reversing the natural direction of the
flux Below, we will discuss the flux-minimized steady-state
of the complete metabolic system if the flux distribution
satisfies the side constraints of Eqns (2)Ờ(4) and yields
a minimum of the flux evaluation functionF defined by
Eqn (7)
Results
Flux-minimized steady-states of the erythrocyte
metabolism
The method outlined above was applied to the metabolic
scheme for erythrocytes depicted in Fig 1 The meaning of
the abbreviations used in the scheme, and the numerical
values of the equilibrium constants of the reactions, are depicted in Table 1 The scheme takes into account two cardinal metabolic pathways of this cell: glycolysis, inclu-ding the so-called 2,3-bisphosphoglycerate shunt; and the pentose phosphate cycle, comprising an oxidative and a nonoxidative part The model comprises 30 reactions and 29 metabolites, whereby only 25 metabolites are independent because there are four conservation conditions:
const.Ử ND; NADP + NADPH Ử const Ử NDP; and GSH + ơ GSSGỬ const Ử G Note that in the reaction scheme the orientation of the arrows corresponds to the
natural direction of the reactions which, as declared above,
is defined as that direction which would ensure a positive Gibbs free energy change under standard conditions For the calculation of stationary and time-dependent states of the reaction scheme in Fig 1, a comprehensive mathematical model was used that takes into account the detailed kinetics of the participating enzymes This mathe-matical model comprises the rate equations outlined previ-ously [8] and, additionally, a rate equation for the transport
of glucose between the cytoplasm and the external space [21]:
vỬ
vmax
K m extGlcext GlcKequ
1ợ Glcext
K m extợ GlcK
m inợ a Glcext
K m ext
Glc
Km in
đ8ỡ
{kinetic parameters: VmaxỬ 74 520 mMẳh)1[22]; Km_extỬ 1.7 mM, Km_inỬ 6.9 mM, aỬ 0.54 (calculated as indicated previously [23]); KeqỬ 1}
The mathematical model has been shown to provide reliable simulations of time-dependent and stationary metabolic states of the erythrocyte under a variety of
Fig 1 Metabolic scheme depicting parts of the erythrocyte metabolism analysed by using the flux-minimization method Note that the reaction arrows point in the direction of the net reaction under standard conditions for which reactions 3, 5, 6, 7, 11 and 29 differ from the direction under in vivo conditions Reactions, enzymes, equilibrium constants and metabolites are as explained in Tables 1 and 2 Target reactions with fixed flux values are indicated by red arrows, exchange fluxes with the external medium are symbolized by blue arrows Reaction numbers (Table 1) are given in green.
Trang 5v1
v2
v1
v3
v26
v4
v1
v5
v1
v9
v16
v6
v1
v9
v16
v7
v9
v16
v8
v9
v10
v9
v11
v9
v16
v12
v13
v14
v15
v1
v26
v21
v16
v17
v26
v18
v26
v19
O2
v26
v20
v21
v21
v22
v26
v23
v9
v16
v24
v9
v16
v25
v9
v16
v26
v27
v9
v16
v28
Pt
v29
v26
v9
v16
v1
v30
v1
v26
v21
b given
Trang 6external conditions Thus, metabolic steady states computed
by means of the kinetic model can be used to assess the
reliability of flux rates computed by means of the
flux-minimization method
The target reactions considered in this example are (a)
ATP utilization (v16) which is mostly spent on the Na/K
ATPase to maintain Na/K gradients across the plasma
membrane, (b) glutathione (GSH) oxidation (v21) to prevent
oxidative damage of cellular proteins and lipids, (c)
formation of 2,3-bisphosphoglycerate (v9) required to
modulate oxygen affinity of haemoglobin, and (d) synthesis
of phosphoribosylpyrophosphate (v26) required for the
salvage of adenine nucleotides The magnitude of these
four target reactions depends on the specific external
conditions of the cell, such as osmolarity of the blood (or
preservation medium), oxidative stress caused by reactive
oxygen species or lowering of the oxygen tension during
hypoxia
The equilibrium constants of the reactions are depicted in
Table 1 The flux-balance conditions for the metabolites are
listed in Table 2 The stoichiometric matrix governing the
relationship between the 25 independent metabolites and 30
reactions is given in Fig 2 Owing to the linear flux
dependencies imposed by the 25 flux-balance conditions,
there exist only five independent fluxes through which the
remaining 25 fluxes can be expressed as linear combinations
(see column six of Table 1) Four of these five independent
fluxes are the target fluxes; the fifth independent (nontarget)
flux is chosen to be v, the rate of glucose uptake into the
cell Thus, given the values of the four target fluxes v9, v16,
v21and v26, the values of all other stationary fluxes are fully determined by the value of the glucose uptake flux Calculation of the stationary state by means of the flux-minimization method is accomplished by expressing all fluxes through the linear combinations given in column six
of Table 1 and determining the minimum of the flux evaluation function Eqn (7) with respect to the flux v1of glucose uptake (cf Fig 4F) This yields the value v1¼ 1.51 mMÆh)1 The last two columns of Table 1 contain the flux values obtained by the flux-minimization methods and
by kinetic modelling The correlation between these inde-pendent sets of flux values is shown in Fig 3 For better visualization, fluxes possessing low and high values are shown in two different panels The excellent overall correlation (r2¼ 0.9997) cannot hide that larger relative differences remain for the minor fluxes, mostly pertaining to the hexose monophosphate shunt This is plausible consid-ering that under normal in vivo conditions the glycolytic flux
is well determined by the demand of ATP utilization being
by far the largest target flux of the system The fluxes through the oxidative and nonoxidative pentose phosphate pathway are less strictly determined by the target fluxes: synthesis of phosphoribosylpyrophosphate can be brought about along either branches, and the flux through the oxidative pentose phosphate pathway is not only deter-mined by the NADPH consumption of the glutathione reductase but also by the flux through the NADP-depend-ent lactate dehydrogenase This accounts for the weaker
Table 2 Metabolites and related flux balance conditions for the metabolic scheme of the erythrocyte Conserved moieties: A, sum of adenine nucleotides (A ¼ AMP + ADP + ATP); ND, sum of pyridine nucleotides (ND ¼ NAD + NADH 2 ); NDP, sum of P – pyridine nucleotides (NDP ¼ NADP + NADPH 2 ); and G, sum of oxidized and reduced glutathione (G ¼ GSH + GSSG/2) Detailed rate equations, binding equlibria and kinetic parameters of the kinetic model have been published previously [8].
GraP Glyceraldehyde-3-phosphate – v 5 ) v 6 + v 7 + v 24 ) v 25 + v 27 ¼ 0
ATP Adenosine triphosphate – v 2 ) v 4 + v 8 + v 13 ) v 16 + v 17 ) v 26 ¼ 0
NADP Nicotinamide adenine dinucleotide phosphate v 15 ) v 18 ) v 19 + v 20 ¼ 0
Trang 7performance of the flux-minimization method with respect
to the minor fluxes through the hexose monophosphate
shunt Nevertheless, the absolute differences are still
acceptable considering that the experimental uncertainty
of flux measurements (e.g by tracer methods) is at least
of the same order of magnitude The most striking
discrepancies occur with respect to the flux rate through
the NADP-dependent lactate dehydrogenase reaction and,
as a consequence of that, the pyruvate uptake The
flux-minimization method predicts a vanishing flux through the
lactate dehydrogenase [LDH(P)] reaction so that the release
of lactate equals exactly the glycolytic flux In contrast, the
kinetic model yields a nonvanishing flux through the
LDH(P) reaction, having approximately the same
magni-tude as the fluxes in the oxidative pentose phosphate
pathway The additional consumption of pyruvate by the
LDH(P) has to be compensated for by a nonvanishing
pyruvate uptake Moreover, the flux through the oxidative
pentose phosphate pathway is also higher than predicted by
the flux-minimization method because a nonzero flux
through the LDH(P) reaction is associated with an
additional consumption of NADPH required for the
reduction of glutathione reductase (GSSG) This
discrep-ancy results from the fact that the flux-minimization method
will force some of the fluxes to zero if alternative reactions
or pathways exist in the network that are cheaper
according to the flux evaluation criterion Eqn (7) However,
strictly vanishing zero-fluxes can never be expected in any
branch of the network if the substrates of the reaction are
present in finite concentrations because enzyme activities
cannot be completely switched off by any regulatory mechanism Therefore, zero-fluxes predicted by the flux-minimization method have to be interpreted as small fluxes compared with other fluxes in the network As the fluxes through the NADP-dependent LDH reaction and the pyruvate exchange calculated by means of the kinetic model belong to the group of small fluxes, the prediction of a zero-flux (¼ small zero-flux) is in qualitative agreement with predictions of the kinetic model
Remarkably, the optimal value of v1¼ 1.51 mMÆh)1, obtained by using the flux-minimization approach, is not simply dictated by intuition Plotting the values of repre-sentative fluxes vs values of v1 (Fig 4A–E), the only obvious restriction for v1 arises below the threshold value
v1¼ 1.50 mMÆh)1 Glucose uptake below this threshold value would imply a thermodynamically unfavourable regime where the flux through the oxidative pentose phosphate pathway had to be reversed to maintain the target fluxes Then, the NADPH needed to drive the reactions of the oxidative pathway into a backwards direction and to form hexose phosphates from ribose phosphates by CO2 fixation must be delivered by the NADP-dependent LDH However, there does not exist an obvious upper threshold restricting v1 to values close to 1.51 mMÆh)1 Up to the value of v1¼ 2.98 mMÆh)1, all strongly exergonic reactions [hexokinase (HK), phospho-fructokinase (PFK), pyruvate kinase (PK), glucose-6-phos-phate dehydrogenase (Glc6PD)] proceed into a forward direction and the uptake of glucose exceeding the ATP-controlled demand of glycolysis can be compensated by a Fig 2 Stoichiometric matrix of the reactions constituting the metabolic scheme for the erythrocyte shown in Fig 1.
Trang 8correspondingly high flux through the hexose
monophos-phate shunt In that case, the surplus of NADPH not
required for reductive processes can be utilized by the
LDH(P), converting pyruvate into lactate There is no
thermodynamic or kinetic principle excluding the existence
of such a hypothetical glucose-wasting and
pyruvate-utilizing metabolic regime However, the flux-minimization
principle does!
In order to check whether the flux-minimization method
is capable of providing reasonable estimates of stationary
fluxes within a physiologically reliable range of the target
fluxes, steady-state flux distributions of the system were
calculated at different combinations of target fluxes where
the values of each of the target fluxes was normal, increased
by a factor of 2 or decreased by a factor of 0.5 For these 81
different combinations of target fluxes, the values of three
representative flux rates obtained by flux minimization and
by kinetic modelling are plotted against each other in Fig 5
The correlation between these values is very high Both
methods provide almost identical flux rates of glucose
uptake However, the flux rates through the two branches of
the hexose monophosphate shunt exhibit a constant shift
against each other, which is mostly a result of the fact that the flux-minimization method puts the flux through the NADP-dependent lactate dehydrogenase to zero, whereas the value calculated by means of the kinetic model is
0.1 mMÆh)1 for all 81 cases To balance the NADPH utilized by the LDH(P) reaction, the flux through the oxidative pentose phosphate pathway is actually higher than the flux through the NADPH-consuming glutathione reductase reaction This causes an extra supply of ribose phosphates for the synthesis of phosphoribosylpyrophos-phate Thus, the flux through the oxidative pentose phosphate pathway is still sufficiently high to satisfy the supply of the phosphoribosylpyrophosphate synthetase with ribose phosphates where the flux minimization method already predicts negative fluxes through the nonoxidative pentose phosphate pathway By increasing the flux through the phosphoribosylpyrophosphate synthetase by more than twofold, negative flux rates through the nonoxidative pentose phosphate pathway will also be predicted by the kinetic model (data not shown)
Flux-minimized steady-states of the central metabolism
inMethylobacterium extorquens AM1
As a second example, the flux-minimization method was applied to the central metabolism of M extorquens AM1 This bacterium is capable of growth using C1 compounds such as methanol as the only carbon and energy source Flux rates through the major pathways of the central metabolism of this bacterium have been determined by13 C-label tracing and mass spectroscopy [24], thus allowing assessment of the reliability of the results obtained by the flux-minimization method The underlying metabolic scheme is shown in Fig 6 In brief, formaldehyde is produced from methanol by the methanol dehydrogenase complex The formaldehyde may react with two pools of folate compounds: tetrahydrofolate (H4F) and tetrahydro-methanopterin (H4MPT) Each of the methylene adducts is involved in further reactions The scheme in Fig 6 compri-ses the following subsystems: formaldehyde metabolism, glycolysis and gluconeogenesis, the tricarboxylic acid (TCA) cycle, pentose phosphate shunt, serine cycle, poly b-hydroxy butyrate synthesis, respiration and oxidative phosphoryla-tion The following metabolites can be exchanged with the external medium by free or facilitated diffusion: methanol,
CO2, formate, glycine, serine, succinate, inorganic phosphate and formaldehyde All reactions and corres-ponding enzymes are given in Table 3 As in the first example, the reactions are notated such that they proceed from left to right under standard conditions, i.e all equilibrium constants are larger than or equal to unity
If available, the values of the equilibrium constants were
as published previously [34], otherwise they were fixed to the standard values 1 ðDGð0Þj ¼ 0Þ and 100.0000 ðDGð0Þj ¼ 28:6 kJmol1Þ for reactions known to proceed near or very far from equilibrium, respectively The stoichiometric matrix relating the 77 metabolites to the 78 reactions of the metabolic scheme in Fig 6 is given in Fig 7 Several metabolites of the central metabolism serve as precursors of the so-called biomass of the bacterium, or are formed during biomass synthesis Utilization or production
of a metabolite associated with biomass production is
Fig 3 Comparison of fluxes obtained by the flux-minimization method
and by kinetic modelling [8] In vivo values of the target fluxes:
v 9 ¼ 0.49 m M Æh)1, v 16 ¼ 2.38 m M Æh)1, v 21 ¼ 0.093 m M Æh)1, v 26 ¼
0.026 m M Æh)1 Upper panel: reactions with flux values lower than
0.2 m M Æh)1 Lower panel: reactions with flux values higher than
0.2 m M Æh)1 Significant differences between the two types of flux values
occur for the reaction of LDH(P) and the influx of pyruvate (indicated
by a red point).
Trang 9indicated by the red arrows in Fig 6 The biomass of this
bacterium consists mainly of proteins, poly b-hydroxy
butyrate and higher carbohydrates [33] Reactions
descri-bing the incorporation of precursor metabolites into the
biomass are considered as the target reactions of the system
As the stoichiometric proportions with which the precursor
metabolites are consumed or produced during biomass
production have been determined experimentally [24], all
fluxes connecting the precursor metabolites with the
biomass can be expressed through a single flux, the flux of
biomass production (v78), multiplied by the corresponding
stoichiometric coefficient (see reaction 78 in Table 3)
Using the flux-minimization method, the steady state of
the central metabolism of M extorquens was calculated for
a chemostat-grown culture of bacteria where methanol is
the only carbon source, i.e the uptake fluxes v69–v76 of
exchangeable carbon compounds, except v75(exchange of
methanol), were constrained to zero The obtained flux
values (given relative to a basis of 10 mol of C1 units
entering the system through reaction 1) are given in the last
column of Table 3 Intriguingly, 22 (!) fluxes are predicted to
be zero in the flux-minimized state, i.e they are dispensable
provided that biomass production is the only function to be
accomplished by the central metabolism of the bacterium
The reduced reaction scheme referring to the flux-minimized
solution is shown in Fig 8, where all reactions with
predicted zero fluxes are indicated by using light-grey arrows One group of reactions with zero fluxes comprises the exchange fluxes that are directly linked with compounds that are not present in the external medium or not produced
in excess (reactions 70, 71, 73, 74 and 76) A second group
of reactions predicted to possess zero fluxes in the flux-minimized state belong to metabolic subsystems that are not linked with biomass production and which are not essential for maintaining nonzero fluxes in those branches of the complete network that are relevant for biomass production
An example of such a dispensable subsystem is the acetyl-CoA conversion pathway comprising reactions 49–52 Although the reaction chain composed of reactions 49–51 allows production of the biomass precursor poly b-hydroxy butyrate from acetoacetyl-CoA, the flux-minimization method favours a shorter path comprising only two reactions (46 and 48) Intriguingly, the two oxidative decarboxylation reactions catalyzed by pyruvate dehydro-genase (reaction 22) and a-ketoglutarate dehydrodehydro-genase (reaction 26), commonly regarded to play a central role in the intermediary metabolism, also belong to the predicted group of dispensable reactions
Figure 9 compares the flux rates calculated by means
of the flux-minimization method with experimental data available for 16 reactions (out of 78) The overall correlation
is sufficiently good (r2¼ 0.68) Striking discrepancies
Fig 4 Hypothetical fluxes through
represen-tative reactions of the erythrocyte metabolism
(A–E) and flux evaluation (F) at varying flux of
glucose uptake The graphs shown in (A–E)
correspond to the linear dependencies dictated
by the steady-state conditions (Table 1,
col-umn six) The values of the four target fluxes
are the same as in Fig 2 The value of v 1 ¼
1.51 m M Æh)1, obtained by flux minimization, is
indicated by the dotted vertical line Below
v 1 ¼ 1.50 m M Æh)1, the reaction of the
glucose-6-phosphate dehydrogenase (Glc6PD) has to
proceed in a backwards direction Up to v 1 ¼
2.98 m M Æh)1, all strongly exergonic reactions
[hexokinase (HK), phosphofructokinase
(PFK), pyruvate kinase (PK), Glc6PD]
pro-ceed in a forward reaction.
Trang 10remain with respect to the reactions connecting
phos-phoenolpyruvate with malate The flux-minimized solution
predicts the conversion of phosphoenolpyruvate to malate
to proceed mainly along the branch catalyzed by pyruvate
kinase and the malic enzyme (reactions 43 and 42), whereas the isotope experiment indicates the main flux to proceed along an alternative branch, having oxalacetate as an intermediate (Fig 10) Although the relative flux contribu-tion of the two alternative branches was not correctly predicted by the flux-minimization method, the predicted flux of the overall reaction phosphoenolpyruvate fi malate
is close to the experimental value Interestingly, the overall reaction along both alternative routes consists of the consumption of CO2 and NADH and the formation of ATP (GTP) However, the two reactions 42 and 43, constituting the route favoured by flux-minimization, pro-ceed both in the natural direction, whereas the direction of the GTP-delivering pyruvate carboxykinase reaction (v45) has to be reversed The flux through reaction 45 will be weighted (¼ punished), with weight K45¼ 12, by the flux-minimization method On the other hand, avoiding this thermodynamically unfavourable reaction and instead achieving the flux to oxaloacetate (OAA) through reaction
18 (phosphoenolpyruvate carboxylase, reaction 18), no GTP is formed, which, compared with the ATP-producing pyruvate kinase reaction, is an disadvantage from the energetic point of view Hence, from the thermodynamic and energetic viewpoint, the route phosphoenolpyru-vate fi OAA fi malate, predicted by the flux-minimi-zation method as a dominant flux route, seems indeed to be the more reasonable one The discrepancies between predicted and observed fluxes thus may have kinetic or genetic reasons Apparently, the activity of the enzymes catalyzing the predicted reaction route phosphoenolpyru-vate fi OAA fi malate is reduced in vivo owing to a low expression level or to kinetic regulation This example highlights certain limitations of the flux-minimization method, despite its obvious capacity to provide valuable information about flux distributions in metabolic networks
Discussion
Biology is now facing the era of systems biology Different types of biological information (DNA, RNA, protein, protein interactions, enzymes, metabolites) can be used to build up mathematical models of the gene-regulatory, signal-transducing and metabolic networks of a cell and
to integrate them into whole-cell in silico models The predictive capacity of such models will increase as more details of the underlying elementary processes become incorporated With respect to metabolic networks, the current situation is such that only for a few pathways and a few cell types is sufficient enzyme-kinetic knowledge avail-able to build up realistic kinetic models As the number of enzymological studies has dramatically decreased since 1998 (according to statistics based on entries of enzymological papers into the database http://www.brenda.uni-koeln.de), there is little hope that this situation will improve in the near future
Structural modelling approaches have been proposed as alternatives to mechanism-based kinetic modelling to better understand the architecture and regulation of metabolic networks These approaches have in common that they work without enzyme-kinetic information Only the stoi-chiometry of the system and, if available, some plausible side conditions constraining the external fluxes, are used as
Fig 5 Comparison of fluxes obtained by the flux-minimization method
and by kinetic modelling at various combinations of target fluxes A total
of 3 4
¼ 81 combinations of the four target fluxes was generated by the
stationary solutions of the kinetic model, setting the maximal activities
to 100%, 50% and 200% of the original value.