The 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 t ransporters causes the cell a certain ÔeffortÕ 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, d efence against toxic compounds, r epair of DNA alterations, etc.) with minimal effort, one may p ostulate the principle of flux minimization, as follows: given the available external substrates and given a set of functionally important ÔtargetÕ fluxes required to accomplish a specific pattern of cellular functions, the stationary metabolic fluxes have to become a m inimum. T o convert this principle into a mathematical method enabling the prediction of stationary metabolic fluxes, the total flux in the n etwork is me asured 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 t he right-hand side o f the reaction, the larger the weighting factor for the backward reaction. A linear programming technique is applied to m inimize the total flux at fixed values of the target fluxes and under the constraint of flux balan ce (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 extorquens AM1. The flux rates predicted by the flux- minimization method exhibit significant correlations with flux rates obtained b y either k inetic modelling o r 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-phosphate dehydrogenase (EC 1.2.1.12); phosphoglycerate kinase (EC 2.7.2.3); bisphospho- glycerate mutase (EC 5.4.2.4); bisphosphoglycerate phosphatase (EC 3.1.3.13); phosphoglycerat e 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); p hospho ribosylpyro phosph ate 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); m ethenyl 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); h ydroxypyruvate 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 d ehydrogenase (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) (E C 1.1.1.35); croto nyl-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 oxidor eductase (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/holzhu tter/index.html free of charge. (Received 16 March 2004, revised 3 May 2004, accepted 12 May 2004) Eur. J. Biochem. 271, 2905–2922 (2004) Ó FEBS 2004 doi:10.1111/j.1432-1033.2004.04213.x Complex cellular functions, such as motility, growth, replication, defence against toxic compounds and repair o f molecular d amage, are ultimately linked to metabolic processes. Metabolic processes can be grossly subdivided into chemical reactions and me mbrane transport processes, most being catalysed by enzymes and facilitated by specific membrane transporters. The activity o f 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 i n computational biochemistry ( reviewed in [1]). Mathematical m odels of metabolic systems facilitate the study of systems behaviour by means of computer-based Ôin silicoÕ 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 v iew that realistic kinetic modelling of metabolic networks needs detailed r ate equations for e ach 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 r ate-vs concentra- tion relationships 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 s uccessfully undertaken only for a f ew 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 t o permit realistic mathematical modelling. To obtain at least a qualitative estimate of stationary metabolic flux rates w ithout knowledge of t he detailed kinetics of individual processes, t he so-called flux-balance analysis (FBA) has been proposed [11]. FBA makes u se of the fact that under steady-state conditions the sum of fluxes producing o r degrading any ÔinternalÕ metabolite has to be zero. Application of this m ethod is based on only t wo prerequisites, namely that (a) the topology of the metabolic network under consideration has to be known, and (b) an evaluation cr iterion is needed to identify the most likely flux distribution among all those flux distributions that are compatible with the steady-state co nditions. Th e topology of the m etabolic network i s given in terms of the so-called stoichiometric matrix, relating the time-dependent vari- ation of the metabolite concentrations 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 metab olic pathways is, meanwhile, available for numerous cell t ypes (see, for example, http://www.genome. ad.jp/kegg). In the first place, this is the result of i ntensive 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 i f the biochemical i dentification 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 a t the predictio n of enzyme kinetic properties and enzyme concentration profiles, ensuring optimal performance of m etabolic pathways [12–19]. T he theoretical p redictions 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 a pplication of FBA to cells with more sophisticated ambitions needs a more general criterion. Here I propose to settle this criterion on the following principle o f flux minimization: given the value of f unctionally relevant Ôta rgetÕ fluxes, i.e. those fluxes that are d irectly 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 metab olic network requires s ome ÔeffortÕ. This effort can be split into two different categorie s. First, some metabolic ef fort, in terms of ener gy and other valuable resources (e.g. essential amino acids), is required to synthesize sufficiently high amounts o f enzymes and trans- port proteins. Second, some evolutionary effort has been required t o improve the specificity, catalytic efficiency and regulatory control of an enzyme during the long-term process of n atural evolution. Whereas the metabolic effort can be measured in units of energy or mass flow, t he evolutionary effort is a measure of the probability of favourable mutational events t hat 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 ear ly phases o f 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 s tationary 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 extorquens AM1 . 2906 H G. Holzhu ¨ tter (Eur. J. Biochem. 271) Ó FEBS 2004 The mathematical m odel d escribed here has been submitted 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 v j (j ¼ 1,2,…,r), through all reactions for which at least one enzyme (or t ransport protein) can be expressed, and by the metabolites S i (i ¼ 1,2,…,n) involved in these reactions. T he stoichiometric matrix indicates how flux v j affects the concentration of meta- bolite S i :N i,j >0,N i,j molecules of m etabolite (i) are formed during a single reaction (j); N ij <0,N i,j molecules of m etabolite (i) are consumed during a single reaction (j); and N ij ¼ 0, metabolite (i) is not involved in reaction (j). For example, for the flux v 8 through the chemical reaction 2S 1 þ S 2 ! v 8 S 3 þ 3S 4 , the elements of the s toichiometric m atrix read: N 18 ¼ )2, N 28 ¼ )1, N 38 ¼ 1, and N 48 ¼ 3. In general, the fluxes v j may b e positive o r negative, i.e. the net rea ction may p roceed either in a forward or a backward direction. To deal with non-negative variables, the flux v j is decomposed into an irreversible forward flux, v ðþÞ j (the net reaction p roceeds from le ft to right), an d an irreversible backward flux, v ðÞ j (the net reaction proceeds from right to left), a s follows: v j ¼ v ðþÞ j  v ðÞ j v ðþÞ j ¼ v j Hðv j Þ; v ðÞ j ¼ v j ½Hðv j Þ1ð1Þ where Q (x) d enotes the unit-step function, i.e. by definition only one of the two components v ðþÞ j and v ðÞ j can b e 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: X r j¼1 N ij v j ¼ X r j¼1 N ij ðv ðþÞ j  v ðÞ j Þ¼0 ði ¼ 1; :::; nÞð2Þ representing the principle of conservation of mass for a homogeneous reaction system. The flux balance c onditions shown in equation system (2) constitute a homogeneous system of linear equations with respect t o t he 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. i t possesses an infinite number o f solutions. Setting target fluxes through functionally essential reactions To accomplish a particular functional state of the cell, the fluxes through a certain number of ÔtargetÕ reactions have to be maintained at nonzero values. This can be expressed by equality constraints of the form: v j ¼ L j ; L j > 0 ðj ¼ j 1 ; j 2 ; :::Þð3Þ Some of the target reactions as, for example, the production of en ergy (ATP) or the synthesis of membran e phospho- lipids, are permanently required to e nsure cell integrity. Other target reactions as, for example, the synthesis of a hormone or the detoxification of a pharmaceutical, may be only t emporarily required. The s election of target fluxes is somewhat arbitrary. For example, the demand f or a continuous synthesis o f phospholipids can be instantiated by introducing the total amount of phospholipids as a model v ariable a nd putting either the flux of phospholipids degradation or t he flux of phospholipids s ynthesis 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 l ow-energy, high-entropy products. The absence of a certain substrate associated with the exchange flux, v i , can be expressed by forcing the uptake component of the flux to zero, as follows: v ðuptakeÞ i ¼ 0 ð4Þ Thermodynamic evaluation of fluxes: irreversibility of reactions The direction of any flux v j is dictated by the change of Gibbs free energy: DG j ¼ DG ð0Þ j þ RT ln Q n i¼1 ½S i  N ðþÞ j Q n i¼1 ½S i  N ðÞ j 0 B B @ 1 C C A with N ðþÞ ij ¼ N ij if N ij  0; N ðÞ ij ¼N ij if N ij  0 ð5Þ where DG ð0Þ j denotes the change of Gibbs free energy under the condition that all reactants are present at un it con- centrations (¼ 1molÆL )1 ). DG ð0Þ j can be expressed through the thermodynamic equilibrium constant K equ i , as follows: DG ð0Þ j ¼RT lnðK equ j Þð6Þ where RT ¼ 2.4 8 kJÆmol )1 at room temperature (25  C). As stated above, all reactions of the network will be notated such that under s tandard conditions DG ð0Þ j £ 0, K equ i  1, and thus v j >0v ðÞ j ¼ 0. The second term in the right- hand side of Eqn (5) depends upon the actual concentra- tions of the reactants which, und er cellular conditions, may strongly deviate from unit concentrations. With accumula- ting concentrations of the reaction p roducts ( appearing i n 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 o f accomplishing the required change in the concentra tion of the reac tants. For example, the standard free energy change of the glyco- Ó FEBS 2004 Flux minimization (Eur. J. Biochem. 271) 2907 lytic reaction ( glycerol aldehyde phosphate fi dihydroxy acetone phosphate) catalyzed by the enzyme triose phos- phate isomerase amounts to DG (0) ¼ )7.94 kJ Æmol )1 K equ TIM ¼ 24.6. Nevertheless, under cellular conditions this reaction proceeds i nto a backward direction (dihydroxy acetone phosphate fi glyceraldehyde phosphate) as the reaction substrate g lycerol 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 measure F of the total flux: U ¼ X r j¼1 ðv ðþÞ j þ K equ j v ðÞ j Þð7Þ Weighting the backward flux with the thermodynamic equilibrium constant takes into account the thermodynamic effort connected with reversing the ÔnaturalÕ direction o f the flux. Below, we w ill discuss the flux-minimized steady-state of the c omplete metabolic system if the flux distribution satisfies the side c onstraints of E qns (2)–(4) a nd yield s a minimum of the flux evaluation function F defined by Eqn (7). Results Flux-minimized steady-states of the erythrocyte metabolism The method outlined above w as 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, a re depicted in Table 1. The schem e takes into a ccount two cardinal metabolic pathways of this ce ll: glycolysis, i nclu- ding the so-called 2,3-bisphosph oglycerate shunt; and the pentose phosphate cycle, comprising an oxidative a nd a nonoxidative part. The model comprises 30 reactions and 29 metabolites, whereby only 25 metabolites are independent because there are four conservation conditions: AMP + ADP + ATP ¼ const. ¼ A; NAD + NADH ¼ const. ¼ ND; NADP + NADPH ¼ const. ¼ NDP; a nd GSH + ½ GSSG ¼ const. ¼ G. Note that in the reaction scheme the orientation of the arrows corresponds to the ÔnaturalÕ 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 th e reaction scheme i n Fig. 1 , a comprehe nsive mathematical model w as used that takes into account the detailed kinetics of the participating enzymes. This m athe- 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]: v ¼ v max K m ext Glc ext  Glc K equ  1 þ Glc ext K m ext þ Glc K m in þ a Glc ext K m ext Glc K m in ð8Þ {kinetic parameters: V max ¼ 74 520 m M Æh )1 [22]; K m _ ext ¼ 1.7 m M , K m _ in ¼ 6.9 m M , a ¼ 0.54 (calculated as indicated previously [23]); K eq ¼ 1}. The mathematical model has been shown to provide reliable s imulations of tim e-dependent and s tationary 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. Reac tions, e nzymes, e quilibrium 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. 2908 H G. Holzhu ¨ tter (Eur. J. Biochem. 271) Ó FEBS 2004 Table 1. Reactions of the metabolic scheme shown in Fig. 1: enzymes, thermodynamical equilibrium constants, flux dependencies and calculated flux values. Reaction Enzyme/transporter Abbr. K equ Dependency on independent fluxes In vivo value Flux minimization Kinetic model v 1 a Glc(out) fi Glc(in) Glucose transporter Glc t 1.00E+00 1.506 1.514 v 2 Glc + ATP fi Glc6P + ADP Hexokinase HK 3.90E+03 ¼ v 1 1.506 1.514 v 3 Fru6P fi Glc6P Phosphohexose isomerase GPI 2.55E+00 ¼ 5v 1 ) 3v 9 ) 14 v 26 ) 3v 16 )1.459 )1.417 v 4 Fru6P +ATPfi Fru1,6P + ADP Phosphofructokinase PFK 1.00E+05 ¼ ) v 1 +4v 26 +v 9 +v 16 1.473 1.465 v 5 DHAP + GraP fi Fru1,6P Aldolase ALD 8.77E+00 ¼ v 1 ) 4v 26 ) v 9 ) v 16 )1.473 )1.465 v 6 GraP fi DHAP Triosephosphate isomerase TPI 2.46E+01 ¼ v 1 ) 4v 26 ) v 9 ) v 16 )1.473 )1.465 v 7 1,3PG + NADH fi GraP + Pi + NAD Glyceraldehyde-3-phosphate dehydrogenase GAPDH 5.21E+03 ¼ )3v 26 ) v 9 ) v 16 )2.953 )2.953 v 8 1,3PG + ADP fi 3PG + ATP Phosphoglycerate kinase PGK 1.46E+03 ¼ 3v 26 +v 16 2.459 2.459 v 9 b 1,3PG fi 2,3PG Bisphosphoglycerate mutase DPGM 1.00E+05 0.494 0.494 v 10 2,3PG fi 3PG + Pi Bisphosphoglycerate phosphatase DPGase 1.00E+05 ¼ v 9 0.494 0.494 v 11 2PG fi 3PG Phosphoglycerate mutase PGM 6.90E+00 ¼ )3v 26 ) v 9 ) v 16 )2.953 )2.953 v 12 2PG fi PEP Enolase EN 1.70E+00 ¼ 3v 26 +v 9 +v 16 2.953 2.953 v 13 PEP + ADP fi Pyr + ATP Pyruvate kinase PK 1.38E+04 ¼ 3v 26 +v 9 +v 16 2.953 2.953 v 14 Pyr + NADH fi Lac + NAD Lactate dehydrogenase LDH 9.09E+03 ¼ 3v 26 +v 9 +v 16 2.953 2.953 v 15 Pyr + NADPH fi Lac + NADP Lactate dehydrogenase LDH(P) 1.42E+03 ¼ 12 v 1 ) 6v 9 ) 28 v 26 ) 6v 16 ) v 21 0.000 0.100 v 16 b ATP fi ADP + Pi ATPase ATPase 1.00E+05 2.382 2.382 v 17 2ADP fi ATP + AMP Adenylate kinase AK 4.00E+00 ¼ ) v 26 )0.026 )0.026 v 18 Glc6P +NADPfi 6PG + NADPH Glucose-6-phosphate dehydrogenase Glc6PD 2.00E+03 ¼ 6v 1 ) 3v 9 ) 14 v 26 ) 3v 16 0.047 0.097 v 19 6PG + NADP fi Ru5P +CO 2 + NADPH Phosphogluconate dehydrogenase 6-PGD 1.42E+02 ¼ 6v 1 ) 3v 9 ) 14 v 26 ) 3v 16 0.047 0.097 v 20 GSSG + NADPH fi 2GSH + NADP Glutathione reductase GSSGR 1.04E+00 ¼ v 21 0.093 0.093 v 21 b GSH fi GSSG Glutathione oxidation GSHox 1.00E+05 0.093 0.093 v 22 Ru5P fi X5P Phosphoribulose epimerase EP 2.70E+00 ¼ 4v 1 ) 2v 9 ) 10 v 26 ) 2v 16 0.014 0.047 v 23 Ru5P fi R5P Ribose phosphate isomerase KI 3.00E+00 ¼ 2v 1 ) 4v 26 ) v 9 ) v 16 0.033 0.049 v 24 X5P + R5P fi GraP + S7P Transketolase TK1 1.05E+00 ¼ 2v 1 ) 5v 26 ) v 9 ) v 16 0.007 0.024 v 25 S7P + GraP fi E4P + Fru6P Transaldolase TA 1.05E+00 ¼ 2v 1 ) 5v 26 ) v 9 ) v 16 0.007 0.024 v 26 b R5P + ATP fi AMP + PrPP Phosphoribosylpyro- phosphate synthetase PRPPS 1.00E+05 0.026 0.026 v 27 X5P + E4P fi GraP + Fru6P Transketolase TK2 1.20E+00 ¼ 2v 1 ) 5v 26 ) v 9 ) v 16 0.007 0.024 v 28 Pi(out) fi Pi(in) Phosphate transporter P t 1.00E+00 ¼ 3v 26 0.077 0.077 v 29 Lac(out) fi Lac(in) Lactate exchange Lact 1.00E+00 ¼ 25 v 26 +5 v 9 +5 v 16 )12 v 1 +v 21 )2.953 )3.053 v 30 Pyr(out) fi Pyr(in) Pyruvate exchange Pyr t 1.00E+00 ¼ 12 v 1 ) 28 v 26 ) 6v 9 ) 6v 16 ) v 21 0.000 0.100 a Independent flux; b given target flux. Ó FEBS 2004 Flux minimization (Eur. J. Biochem. 271) 2909 external conditions. Thus, metabolic steady states computed by means of the kinetic model can be used to assess the reliability of flux rates c omputed by means of the flux- minimization method. The target reactions consid ered in this example are (a) ATP utilization (v 16 ) which is mostly spent on the Na/K ATPase to maintain Na/K gradients across the plasma membrane, (b) glutathione (GSH) oxidation (v 21 )toprevent oxidative damage of cellular proteins and lipids, (c) formation of 2,3-bisphosphoglycerate ( v 9 ) required to modulate oxygen affinity of haemoglobin, and (d) synthesis of phosphoribosylpyrophosphate (v 26 ) required for the salvage of adenine nucleotides. The magnitude of these four target reactions depends on the specific ÔexternalÕ 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 g overning the relationship between the 25 independent metabolites and 30 reactions is given in F ig. 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 1 , t he rate of glucose uptake into the cell. Thus, given the valu es of the four target fluxes v 9 ,v 16 , v 21 and v 26 , the values of all other stationary fluxes are fully determined by the v alue of the glucose uptake flux. Calculation of the stationary state by means of the flux- minimization method is accomplished b y 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 v 1 of glucose uptake ( cf. Fig. 4F). T his yields the value v 1 ¼ 1.51 m M Æh )1 . The las t two c olumns of Table 1 contain the flux values obtained by the flux-minimization methods and by kinetic m odelling. 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 (r 2 ¼ 0.9997) cannot hide that larger relati ve 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 be ing by far the largest target flux of the s ystem. The fluxes through the oxidative and nonoxidative pentose phosphate pathway are less strictly determined by the t arget fluxes: synthesis of phosphoribosylpyrophosphate can be brought about along either branches, and the flux through t he 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 dehydrogen ase. This accounts for the weaker Table 2. Metabolites and related fl ux ba lance c onditions for the metabolic scheme of the erythrocyte. Conserved moieties: A, sum o f a denine nucleotides (A ¼ AMP + ADP + ATP); ND, sum of pyridine nucleotides (ND ¼ NAD + N ADH 2 ); NDP, sum of P – pyridine nucleotides (NDP ¼ NADP + NADPH 2 ); and G, s um of oxidized and reduced glutathione (G ¼ GSH + GSSG/2). Detailed r ate equations, b inding equlibria and kinetic parameters of t he kinetic model have been published previously [8]. Metabolite Name Flux balance condition Glc Glucose v 1 ) v 2 ¼ 0 Glc6P Glucose-6-phosphate v 2 +v 3 ) v 18 ¼ 0 Fru6P Fructose-6-phosphate – v 3 ) v 4 +v 25 +v 27 ¼ 0 Fru(1,6)P 2 Fructose-1,6-bisphosphate v 4 +v 5 ¼ 0 GraP Glyceraldehyde-3-phosphate – v 5 ) v 6 +v 7 +v 24 ) v 25 +v 27 ¼ 0 DHAP Dihydroxyacetone phosphate – v 5 +v 6 ¼ 0 1,3(P 2 )G 1,3-Bisphospho- D -glycerate – v 7 ) v 8 ) v 9 ¼ 0 2,3(P 2 )G 2,3-Bisphospho- D -glycerate v 9 ) v 10 ¼ 0 3PG 3-Phospho- D -glycerate v 8 +v 10 +v 11 ¼ 0 2PG 2-Phospho- D -glycerate – v 11 ) v 12 ¼ 0 PEP Phosphoenolpyruvate v 12 ) v 13 ¼ 0 ATP Adenosine triphosphate – v 2 ) v 4 +v 8 +v 13 ) v 16 +v 17 ) v 26 ¼ 0 ADP Adenosine diphosphate v 2 +v 4 ) v 8 ) v 13 +v 16 )2v 17 ¼ 0 6PG Phospho- D -glucono-1,5-lactone v 18 ) v 19 ¼ 0 NADP Nicotinamide adenine dinucleotide phosphate v 15 ) v 18 ) v 19 +v 20 ¼ 0 GSH Glutathione 2 v 20 ) 2v 21 ¼ 0 Ru5P Ribulose-5-phosphate v 19 ) v 22 ) v 23 ¼ 0 X5P Xylulose-5-phosphate v 22 ) v 24 ) v 27 ¼ 0 R5P Ribose-5-phosphate v 23 ) v 24 ) v 26 ¼ 0 S7P Sedoheptulose-7-phosphate v 24 ) v 25 ¼ 0 E4P Erythrose-4-phosphate v 25 ) v 27 ¼ 0 NAD Nicotinamide adenine dinucleotide v 7 +v 14 ¼ 0 Pi Phosphate v 7 +v 10 +v 16 +v 28 ¼ 0 Lac Lactate v 14 +v 15 +v 29 ¼ 0 Pyr Pyruvate v 13 ) v 14 ) v 15 +v 30 ¼ 0 2910 H G. Holzhu ¨ tter (Eur. J. Biochem. 271) Ó FEBS 2004 performance of the fl ux-minimization method with respect to the m inor fluxes through the hexose monophosphate shunt. N evertheless, the absolute differences are still acceptable considering that the experimental uncertainty of flux measurements (e.g. by tracer methods) i s 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 m odel 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 c ompensated for by a nonvanis hing pyruvate uptake. Moreover, the flux through the oxidative pentose phosphate pathway is also higher than predicted by the flux-minimization method because a nonzero fl ux through t he 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 ÔcheaperÕ according to the flux evaluation criterion Eqn (7). However, strictly vanishing zero-fluxes can n ever be expected in any branch of the network if the substrates of the reaction a re present in finite concentrations because e nzyme activities cannot be completely switched off b y any regulatory mechanism. Therefo re, zero-fluxe s predicte d b y t he flux- minimization method have to be interpreted as ÔsmallÕ fluxes compared with other fluxes in the network. As the fluxes through the NADP-dependent LDH reaction a nd the pyruvate exchange calculated by means of the kinetic model belong to the group of small fluxes, the prediction of a zero- flux (¼ ÔsmallÕ flux) is i n qualitative agreement with predictions of the kinetic model. Remarkably, the optimal value of v 1 ¼ 1.51 m M Æh )1 , obtained by using the flux-minimization approach, is not simply dictated by intuition. Plotting the values o f repre- sentative fluxes vs. values of v 1 (Fig. 4A–E), the only obvious restriction for v 1 arises below the threshold value v 1 ¼ 1.50 m M Æh )1 . Glucose up take below t his threshold value w ould i mply a thermodynamically unfavourable regime where the flux thro ugh the oxidative p entose phosphate pathway had to be reversed to maintain the target fluxes. Then, the NADPH needed to drive the reactions of the o xidative pathway into a backwards direction and to form hexose phosphates from ribose phosphates by CO 2 fixation must be delivered by the NADP-dependent LDH. However, there does not exist an obvious upper threshold restricting v 1 to v alues close to 1.51 m M Æh )1 .Uptothevalueofv 1 ¼ 2.98 m M Æh )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 g lucose exceeding t he ATP- controlled demand of glycolysis can be c ompensated by a Fig. 2. Stoichiometric matrix of the reactions constituting the metabolic scheme for the erythrocyte shown in Fig. 1. Ó FEBS 2004 Flux minimization (Eur. J. Biochem. 271) 2911 correspondingly high flux through the hexose monophos- phate shunt. In t hat c ase, 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 h ypothetical g lucose-wasting and p yruvate- 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 s ystem 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 c ombinations o f tar get 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 h igh. 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, w hich is mostly a result of the fact that the flux-minimization method puts the flux through t he NADP-dependent lactate dehydrogenase to zero, whereas the value calculated by means of the kinetic model is  0.1 m M Æh )1 for all 81 cases. To balance the NADPH utilized by the LDH(P) reaction, the flux through the oxidative pentose phosphate pathway is a ctually higher than the flux through the NADPH-consuming g lutathione 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 i s still sufficiently high t o satisfy the supply of t he 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 t he n onoxidative pentose phosphate pathway will also be predicted b y the kinetic model (data not shown). Flux-minimized steady-states of the central metabolism in Methylobacterium extorquens AM1 As a second example, the flux-minimization method was applied t o t he central metabolism of M. extorquens AM1. This bacterium is c apable of growth using C 1 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 by 13 C- label t racing and mass spectroscopy [24], thus allowing assessment of the reliability of the results obtained by t he flux-minimization method. The underlying metabolic scheme is shown in Fig. 6 . In b rief, formaldehyde i s produced from methanol by the methanol dehydrogenase complex. The formaldehyde may react with two pools of folate compounds: tetrahydrofolate (H 4 F) and tetrahydro- methanopterin (H 4 MPT). Each of the methylene ad ducts is involved in further reactions. The scheme in Fig. 6 compri- ses the following subsystems: formaldehyde metabolism, glycolysis and gluconeogen esis, 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 e xchanged with the external medium by free or facilitated diffusion: methanol, CO 2 , 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 t o unity. If available, the values of the equilibrium constants were as published previously [34], otherwise they were fixed to the standard values 1 ðDG ð0Þ j ¼ 0Þ and 100.0000 ðDG ð0Þ j ¼ 28:6kJmol 1 Þ for reactions known to proceed near or very far from equilibrium, respectively. The stoichiometric matrix relating the 77 m etabolites 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 s o-called biomass of the bacterium, or are formed during biomass synthesis. Utilization or prod uction 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 pane l: re actions w ith fl ux 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). 2912 H G. Holzhu ¨ tter (Eur. J. Biochem. 271) Ó FEBS 2004 indicated b y the red arrows in Fig. 6 . T he biomass of this bacterium consists mainly of proteins, poly b-hydroxy butyrate and higher carbohydrates [33]. Reactions descri- bing the incorporation o f 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 p roduction (v 78 ), m ultiplied by the corresponding stoichiometric coefficient (see reaction 78 in Table 3). Using the flux-minimization method, the s teady state of the central metabolism of M. extorquens was calculated for a c hemostat-grown culture of bacteria where methanol is the only carbon source, i.e. the uptake fluxes v 69 –v 76 of exchangeable carbon compounds, except v 75 (exchange of methanol), were constrained to zero. The obtained flux values (given relative to a b asis of 10 mol of C 1 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 a re dispensable provided that biomass production is the only function to be accomplished b y t he centr al metabolis m o f t he bacter ium. The reduced reaction scheme referring to the flux-minimized solution is shown in Fig. 8, w here all reactions with predicted zero fluxes are indicated by using light-grey arrows. O ne 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 w hich are not essential for m aintaining n onzero fluxes in those branches of t he complete network that are relevant for biomass production. An example of such a dis pensable subsystem is the acetyl- CoA conversion pathway c omprising 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). I ntriguingly, the two oxidative decarboxylation reactions catalyzed by pyruvate dehydro- genase (reaction 22) and a-ketoglutarate dehydrogenase (reaction 26), c ommonly regarded to play a c entral role in the intermediary m etabolism, 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 suffic iently good ( r 2 ¼ 0.68). Striking discrepancies Fig. 4. Hypothetical fluxes through represen- tative reac tions of the erythrocyte metabolism (A–E) and flux evaluation (F) at varying flux of glucose uptake. Thegraphsshownin(A–E) correspond to the linear dependencies dictated by the steady-state conditions (Table 1, c ol- 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 fl ux min imization, 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. Ó FEBS 2004 Flux minimization (Eur. J. Biochem. 271) 2913 remain with respect to the reactions connecting phos- phoenolpyruvate with m alate. The flux-minimized solution predicts the c onversion of phosphoenolpyruvate to malate to proceed mainly along the branch catalyzed by p yruvate 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. 1 0). 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 a lternative routes consists of the consumption o f CO 2 and N ADH a nd the formation of ATP (GTP). H owever, the two reactions 42 and 43, constituting the route favoured by flux-minimization, pro- ceed both in the ÔnaturalÕ direction, whereas t he direction of the GTP-d elivering p yruvate carboxykinase reaction (v 45 ) has to be reversed. The flux through reaction 45 will be weighted (¼ punished), with weight K 45 ¼ 12, by the flux- minimization method. On the o ther hand, avoiding this thermodynamically unfavourable reactio n and i nstead 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 OA A fi malate, predicted by the flux-minimi- zation method as a dominant flux route, seems indeed to be the more r easonable one. The discrepancies between predicted and observed fluxes thus may have kinetic or genetic reasons. Apparently, t he 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 lim itations of th e 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, m etabolites) can be used to build up m athematical models of the g ene-regulatory, signal-transducing a nd metabolic networks of a cell and to integrate them i nto whole-cell Ôin silicoÕ models. The predictive capacity of such models w ill 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 knowled ge avail- able to build up realistic k inetic models. As the number of enzymological studies has dramatically decreased since 1998 (according to statistics b ased on entries of enzymological papers into the d atabase http://www.brenda.uni-koeln.de), there is little hope that this situation will improve in the near future. Structural modelling a pproaches 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 s toi- 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. Atotal 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. 2914 H G. Holzhu ¨ tter (Eur. J. Biochem. 271) Ó FEBS 2004 [...]... biomass production at a given total flux is obviously equivalent to maintaining a given rate of biomass production at a minimum of the total flux Insofar, the principle of maximal biomass production is a special case of the more general principle of flux minimization It has to be noted, furthermore, that minimization of fluxes in a metabolic system is closely linked to minimization of enzyme levels, because... lumping together redundant routes to pseudo-reactions In view of all the results obtained, the flux -minimization method should be considered as a serious alternative to currently existing structure-based concepts to assess stationary flux distributions in metabolic networks if detailed kinetic information is lacking References Fig 10 Flux values for the reactions involved in the conversion of pyruvate to. .. the principle of flux minimization This principle captures the obvious fact that gaining functional fitness with minimal expense of external resources and along the shortest route in the evolutionary landscape must have been a decisive selection factor during the natural evolution of cellular systems For the special case that the functionality of a cell is reducible to rapid selfreproduction, gaining... equilibrium constant of the reaction increases Although this way of weighting the backward fluxes is purely empirical and lacks straightforward physical or chemical reasoning, it has the advantage of avoiding any a priori assumptions on the irreversibility of reactions By applying the flux -minimization method to cellular metabolic networks, one has to identify the so-called target Ó FEBS 2004 Flux minimization. .. erythrocyte) and the substrates fuelling the reaction are both present On the other hand, despite some systematic differences to the results of kinetic modelling, the flux -minimization method correctly describes the flux changes induced by changes of the target fluxes This property could render the flux -minimization method a valuable tool for predicting metabolic changes to external perturbations Application of. .. and function of the underlying proteins (enzymes, transporters) It thus has to be doubted that a single evolutionary principle alone may account for the sophisticated regulation of metabolic systems of currently existing cells Resting the computational prediction of system properties on a single optimization criterion a priori holds a considerable degree of arbitrariness This principal objection, of. .. M.E (2002) Stoichiometric model for evaluating the metabolic capabilities of the facultative methylotroph Methylobacterium extorquens AM1, with application to reconstruction of C3 and C4 metabolism Biotechnol Bioeng 78, 296–312 25 Schuster, S., Dandekar, T & Fell, D.A (1999) Detection of elementary flux modes in biochemical networks: a promising tool for pathway analysis and metabolic engineering Trends... type, and the metabolic prerequisites to enable these functions to take place, will be necessary to arrive at a reasonable selection of target fluxes The reliability of stationary fluxes predicted by the fluxminimization method was assessed for two metabolic schemes of different complexity: the energy and redox metabolism of erythrocytes, and the central carbon metabolism of M extorquens For the metabolic. .. adenine nucleotide metabolism of human erythrocytes I Reaction-kinetic statements, analysis of in vivo state and determination of starting conditions for in vitro experiments] Acta Biol Med Ger 40, 1659–1682 5 Schauer, M., Heinrich, R & Rapoport, S.M (1981) [Mathematical modelling of glycolysis and of adenine nucleotide metabolism of human erythrocytes II Simulation of adenine nucleotide Ó FEBS 2004 2922... & Heinrich, R (1999) Competition for enzymes in metabolic pathways: implications for optimal distributions of enzyme concentrations and for the distribution of flux control Biosystems 54, 1–14 Varner, J & Ramkrishna, D (1998) Application of cybernetic models to metabolic engineering: investigation of storage pathways Biotechnol Bioeng 58, 282–291 Varner, J & Ramkrishna, D (1999) Metabolic engineering . The principle of flux minimization and its application to estimate stationary fluxes in metabolic networks Hermann-Georg Holzhu¨. a given total flux is obviously equivalent to maintaining a given rate of biomass production at a minimum of the total flux. Insofar, the principle of maximal