Combining Pathway Analysis with FluxBalance Analysis for the ComprehensiveStudy of Metabolic Systems

Christophe H. Schilling,1 Jeremy S. Edwards,2 David Letscher,3

Bernhard Ø. Palsson1

1Department of Bioengineering, University of California, San Diego, 9500Gilman Drive, MC 0412, La Jolla, California 92093-0412, USA; telephone:619-534-5668; fax: 619-822-3120; e-mail: palsson@ucsd.edu2Department of Chemical Engineering, University of Delaware, Newark,Delaware, USA

3Department of Mathematics, Oklahoma State University,Stillwater, Oklahoma, USA

Received 4 March 2000; accepted 10 December 2000

Abstract: The elucidation of organism-scale metabolicnetworks necessitates the development of integrativemethods to analyze and interpret the systemic propertiesof cellular metabolism. A shift in emphasis from singlemetabolic reactions to systemically defined pathways isone consequence of such an integrative analysis of meta-bolic systems. The constraints of systemic stoichiometry,and limited thermodynamics have led to the definition ofthe flux space within the context of convex analysis.The flux space of the metabolic system, containing allallowable flux distributions, is constrained to a convexpolyhedral cone in a high-dimensional space. Frommetabolic pathway analysis, the edges of the high-dimensional flux cone are vectors that correspond to sys-temically defined “extreme pathways” spanning the ca-pabilities of the system. The addition of maximum fluxcapacities of individual metabolic reactions serves to fur-ther constrain the flux space and has led to the develop-ment of flux balance analysis using linear optimization tocalculate optimal flux distributions. Here we provide theprecise theoretical connections between pathway analy-sis and flux balance analysis allowing for their combinedapplication to study integrated metabolic function. Shiftsin metabolic behavior are calculated using linear optimi-zation and are then interpreted using the extreme path-ways to demonstrate the concept of pathway utilization.Changes to the reaction network, such as the removal ofa reaction, can lead to the generation of suboptimal phe-notypes that can be directly attributed to the loss of path-way function and capabilities. Optimal growth pheno-types are calculated as a function of environmental vari-ables, such as the availability of substrate and oxygen,leading to the definition of phenotypic phase planes. It isillustrated how optimality properties of the computedflux distributions can be interpreted in terms of the ex-treme pathways. Together these developments are ap-plied to an example network and to core metabolism ofEscherichia coli demonstrating the connections betweenthe extreme pathways, optimal flux distributions, and

phenotypic phase planes. The consequences of changingenvironmental and internal conditions of the network areexamined for growth on glucose and succinate in theface of a variety of gene deletions. The convergence ofthe calculation of optimal phenotypes through linear pro-gramming and the definition of extreme pathways estab-lishes a different perspective for the understanding ofhow a defined metabolic network is best used under dif-ferent environmental and internal conditions or, in otherwords, a pathway basis for the interpretation of themetabolic reaction norm. © 2001 John Wiley & Sons, Inc.Biotechnol Bioeng 71: 286–306, 2000/2001.Keywords: metabolic modeling; convex analysis; linearprogramming; flux balance analysis; metabolic path-ways

INTRODUCTION

Automated sequencing and high-throughput experimentaltechniques to analyze cellular functions at the genomic levelare beginning to reveal the full complexity of living sys-tems. The implication of dealing with complex systems isthat their properties are not predictable from the completedescription of their components, leading to the concept ofemergent properties of living systems (Bhalla and Iyengar,1999; Palsson, 1997; Weng et al., 1999). To understand thecomplexity inherent in cellular networks, approaches needto be implemented that focus on the systemic properties ofthe network. Subsequently, these approaches must be ap-plied to complete cellular systems. Our focus then shiftsfrom a reductionist to a holistic approach for understandingthe interdependence of gene function and the role of thegene in the context of multigenetic cellular functions orgenetic circuits.

The multigeneic nature of cellular functions presents asignificant challenge in extracting phenotypic informationfrom the genotype. Both experimental and theoreticalmethods that overcome this challenge will contribute sig-nificantly toward extracting valuable information from ge-nomic data. Nowhere is this contribution more apparent

Correspondence to:B. Ø. Palsson.Contract grant sponsors: National Institutes of Health; National Science

FoundationContract grant numbers: GM57089; 9873384; 9814092

© 2001 John Wiley & Sons, Inc.

than in cellular metabolism where it is now possible toreconstruct detailed metabolic networks from annotated ge-nome sequences, and experimentally characterize metabolicphenotypes using functional genomic technologies such aswhole-genome expression profiling and protein character-ization. Furthermore, there exists a history of studying thesystemic properties of metabolic networks. This history in-cludes approaches such as metabolic control analysis (Fell,1996; Heinrich and Rapoport, 1974; Kacser and Burns,1973), flux balance analysis (Bonarius et al., 1997; Edwardset al., 1999; Varma and Palsson, 1994b), pathway analysis(Clark, 1988; Liao et al., 1996; Mavrovouniotis et al., 1990;Schilling et al., 2000a; Schuster et al., 1999; Seressiotis andBailey, 1988), cybernetic modeling (Ramakrishna et al.,1996; Varner and Ramkrishna, 1998, 1999), biochemicalsystems theory (Savageau, 1969a, 1969b, 1970), and tem-poral decomposition (Palsson et al., 1987).

Although the ultimate goal may be the development ofdynamic models for the complete simulation of metabolicsystems, the success of such approaches is severely ham-pered by the current lack of kinetic information on the dy-namics and regulation of metabolic reactions. In the absenceof kinetic information it is still possible to accurately assessthe theoretical capabilities and operative modes of meta-bolic systems using steady-state analysis. Steady-stateanalysis is based on the stoichiometry of the metabolic re-actions, which is a structural invariant of the metabolic net-work.

In recent years, an approach known as flux balance analy-sis has been developed to describe metabolic physiology ina quantitative manner (Bonarius et al., 1997; Edwards andPalsson, 1998; Edwards et al., 1999; Sauer et al., 1998;Schilling et al., 1999a; Varma and Palsson, 1994b). Fluxbalance analysis is based on the fundamental law of massconservation and the application of optimization principlesto predict the optimal distribution of metabolic resourceswithin a network. It thus provides insightful informationabout the systemic constraints placed on metabolic function.The analysis is performed under steady-state conditions andit only requires information about the stoichiometry ofmetabolic pathways and on metabolic demands. This ap-proach is particularly applicable for postgenomic analysis,because the stoichiometric parameters can be defined fromthe annotated genome sequence of a particular organism(Edwards and Palsson, 1999, 2000; Schilling and Palsson,2000b). Flux balance analysis is distinguished from meta-bolic flux analysis (MFA) where fluxes are experimentallymeasured and used to reduce the underdetermined nature ofa metabolic system to allow for the calculation of all theunknown fluxes under steady-state conditions without in-voking optimization principles (Stephanopoulos et al.,1998).

Flux balance analysis has recently been applied to ana-lyze and predict the metabolic genotype–phenotype relationfrom a reaction-based perspective, in which flux activitiesfor individual reactions are examined (Edwards and Pals-

son, 1998, 1999, 2000; Schilling et al., 1999a). The abilityto interpret predicted metabolic phenotypes from a pathwayperspective in addition to an individual reaction-based per-spective might shed new light on the complexity of thegenotype–phenotype relationship. A pathway perspectivewill provide flux activities for systemic metabolic pathwaysthat are comprised of balanced sets of reactions. These path-ways are characterized based on their functional and struc-tural relevance within the system as opposed to their patternof historical discovery. Recent approaches have been devel-oped to perform detailed pathway decompositions of meta-bolic networks that hold promise in providing the pathwayperspective that is currently sought to complement flux bal-ance analysis and related optimization techniques.

The pathway analysis of metabolic networks has pro-gressed itself over the years from simply analyzing wallcharts of metabolic reactions to the recent development ofdetailed theories for the in-depth study of the structure ofmetabolic networks (for a recent review, see Schilling et al.[1999b]). The definition of systemic pathways in metabo-lism is essential for providing a pathway-oriented perspec-tive to overall metabolic functions and phenotypes. One ofthe more promising approaches for performing this identi-fication and analysis has capitalized on the principles ofconvex analysis. These principles include the closely relatedconcepts of elementary modes (Schuster et al., 1996, 1999)and extreme pathways (Schilling et al., 2000a), which canboth be used to define the limitations and production capa-bilities of metabolic systems. These approaches to pathwayanalysis share a common underlying mathematical frame-work with flux balance analysis. Although this connectionhas been noted by a number of investigators, a clear defi-nition and illustration of the formal relationships betweenthese two approaches is lacking.

In this article we illustrate the precise relationships thatbind metabolic pathway analysis to flux balance analysis,and illustrate their combined use to interpret shifts in meta-bolic routing that may occur in response to environmentaland internal/genetic challenges. The combination of thesetwo approaches allows for a clear interpretation of meta-bolic phenotypes and flux distributions from the traditionalreaction-based perspective and now from a pathway-oriented perspective. Many of the relations are illustratedinitially through the use of a hypothetical reaction networkto assist in developing a core understanding of both ap-proaches and their connections. We then apply the com-bined approach for the detailed study of the central meta-bolic network of Escherichia coli.The comprehensivecollection of calculations, and illustrations generated for thework presented is available on-line along with addi-tional supportive material to assist the reader (http://gcrg.ucsd.edu/supplementarydata/PathwayFBA/default.htm). The complementarities of flux balance andpathway analysis for studying metabolic systems along withthe combination of emerging genome-scale experimentaltechnologies should serve to form a useful platform fromwhich to study the complexity inherent in cellular metabo-

SCHILLING ET AL.: COMBINING METABOLIC PATHWAY ANALYSIS WITH FLUX BALANCE ANALYSIS 287

lism for both basic scientific and applied purposes of meta-bolic engineering (Stephanopoulos, 1999; Yarmush andBerthiaume, 1997) and biocommodity engineering (Lynd etal., 1999).

DESCRIBING METABOLIC SYSTEMS

A metabolic network is a collection of enzyme-catalyzedreactions and transport processes that serve to dissipate sub-strate metabolites and generate final metabolites. To de-scribe metabolic networks in a quantitative manner, dy-namic mass balances are written for each metabolite in thenetwork, which generates a system of ordinary differentialequations that describe the transient behavior of metaboliteconcentrations:

dXi

dt= (

j

Sij vj (1)

wherevj corresponds to thejth metabolic flux, [Xi] repre-sents the concentration of the metabolite, and the stoichio-metric coefficient,Sij, stands for the number of moles ofmetabolitei formed (or consumed) in reactionj. As we areoften interested in the structural characteristics or invariantproperties of the reaction network (Reder, 1988), it is rea-sonable to place the metabolic system into a steady state,thus reducing the set of differential equations to a set oflinear homogeneous equations, written in matrix nota-tion as:

S? v 4 0 (2)

Although the system is effectively closed to the passageof certain metabolites, others are allowed to enter or exit thesystem via exchange fluxes (or pseudoreactions [Clarke,1980]). These fluxes generally do not represent biochemical

conversions or transport processes like those of internalfluxes, but can be thought of as representing the inputs andoutputs to the system. For example, the demand for a me-tabolite for incorporation into cellular biomass creates anexchange flux on the intracellular metabolite from the meta-bolic network. The availability of a substrate in the extra-cellular environment creates an exchange flux on the extra-cellular metabolite that represents a source. Thus, all me-tabolites are internal to the system and a distinction is madebetween intracellular and extracellular metabolites withinthe system, closing the material balance to all metabolites,as indicated in Eq. (2).

To complete the description of the metabolic network wemust include the constraints that are placed on fluxes due toreaction thermodynamics and the systemic characteristics(input/output) of the network. To simplify matters, we candecompose all reversible internal reactions into a separateforward and reverse flux. This decomposition will create ageneral constraint that all internal fluxes must be greaterthan zero, such as in the directed graph of Figure 1a. Forexchange fluxes we can set upper and lower bounds on theconstraints that represent the corresponding metabolites’ability to enter or exit the system. (For a more detaileddiscussion see Schilling et al. [2000a].)

At this point, we introduce the reaction network shown inFigure 1a to illustrate these mathematical network descrip-tions and to provide a simplified example for many of thetheoretical concepts that will be illustrated in what follows.The system is comprised of five metabolites that are oper-ated on by seven internal fluxes. Note that fluxesv3 andv4

can be viewed as the decomposition of a reversible reactioninto a forward and a backward reaction. Also, the stoichi-ometry on fluxv7 converts 2 mol of D into 1 mol of E.Therefore, metabolite C can be directly converted into E

Figure 1. (a) Example metabolic reaction network consisting of 5 metabolites and 11 fluxes (7 internal fluxes, 4 exchange fluxes) listed along with theindividual reactions. (b) Mathematical description of the reaction network in the form of steady-state mass balances (linear equalities) and flux constraintsimposed on the network (linear inequalities). Together this system of linear equalities/inequalities defines the region of admissible flux vectors.

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throughv6; however, ifv5 andv7 are used, the efficiency iscut in half, analogous to a physiological situation in whichcarbon may be lost as CO2 due to an alternative route toproduce a metabolite. We allow metabolites C, D, and E toexit the system as products of the network (either as bio-mass precursors or final metabolic products). To representsubstrates available to the system we allow metabolites Aand C to enter the system. These potential sources and sinkson the system create four exchange fluxes operating on thesystem. Note that the exchange fluxes are defined as out-ward, thus the value of the flux is negative when the me-tabolite is entering the system and positive when it isformed as a product of the system. The steady-state massbalances are provided in Figure 1b along with the generalconstraints that are placed on both the internal and exchangefluxes. Each of the steady-state mass balances correspondsto a row of the stoichiometric matrix. This set of linearhomogeneous equations and linear inequalities describes themetabolic system under steady state.

CONVEX ANALYSIS

As a result of the inequality constraints placed on variousfluxes traditional linear algebra can no longer be used tohandle such a mathematical system of equalities/inequal-ities, forcing the use of convex analysis to study the prop-erties of the solution space for this problem. From convexanalysis the solution space for any system of linear homo-geneous equations and inequalities is a convex polyhedralcone emanating from the origin. We refer to this region ofadmissible steady-state flux vectors as the flux cone(C). Allpossible solutions, and hence flux distributions, that the sys-tem can operate in a steady state are confined to the interiorof the flux cone. To uniquely describe a convex cone wemust identify the set of extreme rays/generating vectors(pi)that span the cone (similar to the edges of a pyramid). In thecontext of metabolic systems we refer to these edges asextreme pathways, as each vector corresponds to a particu-lar pathway or active set of fluxes that satisfy Eq. (2) and theinequality constraints placed on the system. Every point(v)within the cone can be written as a nonnegative linear com-bination of these extreme pathways as:

C = Hv : v = (i = 1

k

wipi, wi $ 0 ;iJ (3)

Recently, we presented the detailed theoretical frameworkand algorithm used for the rapid identification of the set ofextreme pathways for any metabolic system (Schilling et al.,2000a). It addition to being unique for a particular systemthe set of extreme pathways is the absolute minimal set ofpathways that can be used to span the flux cone in a convexmanner. This set of pathways generating the flux cone mayalso be referred to as a convex basis, and is more properlysaid to form the conical hull of the flux cone. As an aside,for those familiar with elementary modes, the extreme path-ways are a subset of the elementary modes of a reaction

network (Schuster et al., 1999). For a more thorough expla-nation of the similarities and differences between these ap-proaches one is referred to Pfeiffer et al. (1999) and Schil-ling et al. (2000a).

The set of extreme pathways for the reaction networkdescribed in Figure 1 is provided in vector form in Table Iwith each pathway illustrated in Figure 2. There are eightextreme pathways (p1, . . . ,p8). Two of the pathways (p7,p8) display no active exchange fluxes, and thus correspondto internal cycles within the network of whichp8 is simplya result of decomposing a reversible reaction into two op-posite reactions. Using the classification scheme developedpreviously (Schilling et al., 2000a), these pathways corre-spond to type III pathways, whereas the remaining six path-ways all describe functional pathways through the network(type I) and together represent the true systemic functions orcapabilities of the network. Their combined activity can beused to describe any flux distribution of the network as inEq. (3).

The set of extreme pathways is conically or systemicallyindependent, in analogy to the concept of linear indepen-dence from linear algebra. Mathematically this means thatthe pathways cannot be formed by a positive combination ofany other vectors or pathways in the flux cone. From afunctional point of view this means that every flux distri-bution can be reached by either switching pathways off orturning them on to a certain level (this is analogous tohaving a dial controlling the activity of each extreme path-way). We will illustrate this notion in more detail in thefollowing sections.

EFFECTS OF BALANCED DEMANDS ONPATHWAY STRUCTURE

Under changing substrate/supply conditions metabolic net-works are continuously faced with a balanced set of bio-synthetic demands (i.e., production of amino acids, nucleo-tides, phospholipids, as well as energy and redox potential).This means that the network must generate a balanced ac-

Table I. The eight extreme pathway vectors for the example networkwith free outputs.a

Pathwaynumber

Internal fluxes Exchange fluxes

v1 v2 v3 v4 v5 v6 v7 b1 b2 b3 b4

p1 1 1 0 0 0 0 0 −1 1 0 0p2 1 0 1 0 0 0 0 −1 0 1 0p3 2 0 2 0 0 0 1 −2 0 0 1p4 0 0 0 0 1 0 0 0 −1 1 0p5 0 0 0 0 2 0 1 0 −2 0 1p6 0 0 0 0 0 1 0 0 −1 0 1p7 0 0 1 1 0 0 0 0 0 0 0p8 0 1 0 1 1 0 0 0 0 0 0

aThe first six pathways are type I pathways (see Fig. 2). The last twopathways are type III pathways that represent internal cycles in the network(i.e., zero weights on all exchange fluxes). For definition of the pathwayclassification scheme see Schilling et al., (2000a).

SCHILLING ET AL.: COMBINING METABOLIC PATHWAY ANALYSIS WITH FLUX BALANCE ANALYSIS 289

tivity through the exchange fluxes for the particular metabo-lites required to meet these demands. What are the impli-cations of these aggregate demands on the pathway struc-ture of a network?

To assess the systemic performance of a network in meet-ing biosynthetic demands an exchange flux is introducedinto a network representing the aggregate relative demandof the biomass precursors (analogous to an objective func-tion [Varma and Palsson, 1993a]). Additional constraintsmust also be added to the network to effectively close thematerial balances on the metabolites participating in thebiosynthetic demand (or growth) flux. The introduction of anew exchange flux and the associated restriction of existingexchange fluxes will alter some of the mass balances andlinear inequalities of the network. A new pathway structurecan be determined based on these changes to the systemicconstraints that is functionally related to the original set of

pathways, as demonstrated in what follows. To distinguishbetween the two different forms of output for a system [(1)all material balances closed with an aggregate demand/growth flux included versus (2) no aggregate demand/growth flux and material balances not closed on biosyn-thetic precursor outputs] we consider the system without abiosynthetic demand flux (as discussed in the previous sec-tion) to havefree outputs, and the system with balancednetwork demands is referred to as havinglinked outputs.

For the example system in Figure 1, we introduce anexchange flux that represents an aggregate demand on thenetwork requiring 1 mol of metabolite C and D along with2 mol of metabolite E. This flux serves as a balanced drainon the network and is written:

bz: C + D + 2E → (4)

Note that the growth flux only involves metabolites thatwere previously allowed to exit in the system with freeoutputs and subsequently to have a preexisting exchangeflux. Otherwise, the input and output characteristics of thetwo representations are no longer compatible and the path-way results will not be complementary. Furthermore, theexchange fluxes for C, D, and E are no longer allowed toserve as direct outputs for the system. This condition willforce b2 to take on a negative value while constrainingb3

andb4 to zero (see Fig. 3).The set of extreme pathways for this system is provided

in Table II (the schematic illustrations of each pathway areavailable on-line). We can see that there are ten pathways intotal (eight functional pathways of type I, and two type IIIcyclic pathways identical to those from the free output sys-tem in Table I). All functional pathways utilize either A orC, or A and C, to produce the balanced set of demandsrepresented by the growth flux.

Figure 2. Routing patterns for the six type I extreme pathways calculatedfor the example reaction network. Detailed pathway vectors indicating theindividual flux levels in each pathway are provided in Table I.

Figure 3. (a) The example network presented in Figure 1 with the addition of an aggregate demand flux that requires 1 mol of metabolite C and D forevery 2 mol of metabolite E. (b) Mathematical description of the network with the changes from Figure 1b indicated in gray type. All changes are a resultof the inclusion of the demand exchange flux (a linked output) and the associated alterations to the exchange flux constraints of the participating metabolites.

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If only the growth flux is added to the free output system,and no additional constraints are placed on the exchangefluxes, the set of extreme pathways would include all of theextreme pathways listed in Tables I and II (16 total). Byadding one additional flux the dimension in which the fluxcone resides has increased by one (fromR11 to R12), and thecone has gained six additional edges. If we project the fluxcone down onto the original dimensions inR11 it is seen thatthe six additional pathways all reside within the interior ofthe flux cone described by the pathway vectors in Table I.Importantly, these new pathways can be interpreted as posi-tive combinations of the extreme pathways from the freeoutput representation that generates the cone as described inEq. (3).

To determine the appropriate combination of extremepathways we have to redistribute the values for the growthflux back to the original exchange fluxes of the variousmetabolites. This redistribution can be achieved by multi-plying the growth flux value by the stoichiometric coeffi-cient of each of the metabolites in the reaction and thenadding this value to the appropriate exchange fluxes. Thevalues for the internal fluxes remain unchanged. This op-eration is purely mathematical and does not affect the func-tionality of the system. Thus, for pathwayp85 the value forthe exchange fluxes would shift from [−6,0,0,0,1] forb1, b2,b3, b4, andbz to [−6,1,1,2,0]. From this we can determinethat pathway (p85) for the linked output system is equivalentto the following combination of pathways for the free outputsystem:p1 + p2 + 2 * p3. The equivalencies between all ofthe free and linked output pathways are provided in Table II.

Outlining the pathway equivalencies offers the ability tointerpret flux distributions in terms of the pathways for thelinked outputs that represent global metabolic routing pat-terns used to meet balanced sets of demands, and then fur-ther decomposes these patterns into the individual pathwaysthat comprise such a distribution pattern. These comple-mentary perspectives for interpreting metabolic functionwill be illustrated in the following two sections where wefocus on the interpretation of optimal flux distribution cal-culated using flux balance analysis.

CALCULATING OPTIMAL FLUX DISTRIBUTIONS

The performance capabilities of any metabolic network re-side in the flux cone that we have now seen described by theset of extreme pathways. In fact, the answer to any questionrelated to the general structure and fitness of the networklies within this cone. The pathways offer a convenient wayof interpreting metabolic function, but how do we best ex-plore the capabilities and functioning of a metabolic net-work?

One approach that has recently been used to explore therelationship between the metabolic genotype and phenotypefor a number of organisms is flux balance analysis (Edwardsand Palsson, 1998, 1999, 2000; Schilling et al., 1999a,2000). This approach uses linear optimization techniques todetermine the optimal flux distributions within a network soas to maximize/minimize a particular objective function.The standard form of a linear programming problem is asfollows, where a linear objective function is maximized orminimized subject to a series of linear equalities and in-equality constraints:

Maximize/minimize Z = (j

cj vj

subject to (i,j

Sij vj = bi, aj # vj # bj (5)

Note that Eq. (5) is analogous to the system of linear equali-ties/inequalities that forms the general description of themetabolic network in Figure 1, where all the components ofthe b vector are zero. In fact, this system of equations ismerely a special case of the general system defined by themass balances and physicochemical constraints (as shownin Fig. 1b). Linear programming will determine the optimalvalue for an objective that lies within the flux cone spannedby the extreme pathways. Furthermore, the foundations oflinear programming are built on the principles of convexanalysis, which can be used to link the pathway analysis ofmetabolic networks outlined earlier with flux balance analy-sis.

For metabolic applications, the linear objective function

Table II. The ten extreme pathway vectors for the example network in Figure 1 with a linked output.a

Pathwaynumber

Internal fluxes Exchange fluxesPathway equivalences

linked ∼ freev1 v2 v3 v4 v5 v6 v7 b1 b2 b3 b4 bz

p18 5 0 5 0 0 0 2 −5 −1 0 0 1 p18 ∼ p2 + 2p3

p28 1 0 1 0 0 2 0 −1 −3 0 0 1 p28 ∼ p2 + 2p6

p38 0 0 0 0 5 0 2 0 −6 0 0 1 p38 ∼ p4 + 2p5

p48 0 0 0 0 1 2 0 0 −4 0 0 1 p48 ∼ p4 + 2p6

p58 6 1 5 0 0 0 2 −6 0 0 0 1 p58 ∼ p1 + p2 + 2p3

p68 4 3 1 0 0 2 0 −4 0 0 0 1 p68 ∼ 3p1 + p2 + 2p6

p78 6 6 0 0 5 0 2 −6 0 0 0 1 p78 ∼ 6p1 + p4 + 2p5

p88 4 4 0 0 1 2 0 −4 0 0 0 1 p88 ∼ 4p1 + p4 + 2p6

p98 0 0 1 1 0 0 0 0 0 0 0 0 p98 ∼ p7

p108 0 1 0 1 1 0 0 0 0 0 0 0 p108 ∼ p8

aThe first eight pathways correspond to type I pathways, whereas the last two pathways are type III. Pathway equivalencies between the free and linkedoutput representations are provided for each pathway.

SCHILLING ET AL.: COMBINING METABOLIC PATHWAY ANALYSIS WITH FLUX BALANCE ANALYSIS 291

(Z) to be maximized or minimized can correspond to anumber of diverse objectives ranging from particular meta-bolic engineering design objectives (energy and metaboliteproduction) to the maximization of cellular growth (in amanner similar to that seen in the previous section). Re-gardless of the objective function chosen, the optimal solu-tions will lie within the flux cone that is defined by the massbalance and physicochemical constraints (i.e., linear in-equalities) placed on the system. Otherwise, the solution tothe optimization problem will be infeasible. This solutionspace is virtually the exact same flux cone that we have justseen described in the previous section in terms of the set ofextreme pathways for the linked output system. The onlydifference is that the flux cone must become bounded (abounded polytope) to calculate an optimal solution.

We consider the growth exchange flux introduced in Eq.(4) as the objective function. To determine the optimal uti-lization of the metabolic network (so as to maximize theobjective function), additional physicochemical constraintsmust be considered. Specifically, in this example system,the upper bound on the exchange fluxes for metabolites Aand C must be set to limit the amount of metabolites A andC available for intake by the network. Physiologically, theseconstraints may correspond to limited substrate availabilityor maximal uptake rates or even maximum rates ofdiffusion-mediated transport. These maximum capacityconstraints serve to place certain bounds on the flux conesuch that it is no longer an unbounded polyhedral cone.Regions of the flux cone become bounded by the hyper-planes defined from these constraints. Without these con-straints the maximum value of any flux in the network couldbe infinite, and the linear programming problem is un-bounded.

Mathematically, the solution space is the sum of a convexpolytope (closed surface) and a new polyhedral cone (un-bounded) formed by a subset of the original extremal rays(see Appendix for details). The conical hull of the polyhe-dral cone now corresponds to the pathways that are unaf-fected by the added constraints, and the convex hull of thepolytope is essentially generated from the remaining path-ways that have now been constrained. Importantly, all of thevertices of the polytope correspond to bounded feasible so-lutions of the linear programming problem, whereas theedges of the flux cone correspond to the unbounded feasiblesolutions. Both the polytope and the polyhedral cone arecontained within the original flux cone, which allows for theinterpretation of any point in the solution space using theoriginal set of extreme pathways from the linked outputsystem as well as the free output system through theirequivalencies, as we will now illustrate.

LINKING FLUX BALANCE ANALYSIS WITHPATHWAY ANALYSIS

Consider three cases that represent different environmentaland internal challenges that the example network may facein meeting its demands: case 1, only metabolite A is avail-

able to the network to meet the demands of the growth fluxin Eq. (4); case 2, only metabolite A is available, but theinternal flux,v6, is not functional and is thus constrained toequal zero; and case 3, only metabolite C is available to thenetwork to meet the demands of the growth flux. In eachcase, we will arbitrarily constrain the input of the availablesubstrate metabolite to a value of 1 (to obtain substratenormalized distributions). We can consider the shift fromcase 1 to case 2 as a sudden challenge to the internal struc-ture of the network due to a loss of function, and the shiftfrom case 1 to case 3 represents an environmental challenge,namely a change in substrate availability.

Using a commercially available linear programmingpackage (Lindo Systems Inc., Chicago, IL) the optimal fluxdistributions for these three cases were calculated and arepresented in Figure 4. We see that the loss in function ofv6

in case 2 decreases the maximum value of the objectivefunction by 33% from 0.25 to 0.17, essentially forcing thenetwork to perform suboptimally. There is a subsequentshift in the utilization of the metabolic pathways that occursbetween case 1 and case 2, and to a more severe extent inthe shift from case 1 to case 3 due to the change in substrate.It is these shifts in metabolic routing that we seek to inter-pret from a pathway perspective using the set of extremepathways and equivalencies provided in Table II.

In each case, the flux distribution corresponds to an exactscalar multiple of one of the extreme pathways for thelinked output system in Table II, implying that the solutionsare on the extreme edges of the flux cone. As the environ-mental/genetic conditions change, the corresponding opti-mal pathway utilizations shift accordingly across faces ofthe flux cone. If a reaction is eliminated or constrained tozero, all of the pathways that utilize this reaction will beeliminated. From basic principles of vector addition it be-comes apparent that the utilization or weight (wi) on anextreme pathway in comprising a flux distribution mustequal zero if the pathway utilizes the internal flux/reactionthat has been eliminated. Therefore, every extreme pathwayin the system that has a positive flux value for the elimi-nated reaction will havewi 4 0 in Eq. (3). This is thesituation in case 2 where all of the pathways that can pro-duce metabolite E optimally have been eliminated by theloss of v6, leading to the utilization of other pathways tomeet the demands of the system.

The changes in pathway utilization allow for the inter-pretation of metabolic shifts in terms of the underlying path-ways of the network. These example cases offer an illus-tration of how pathways may be switched on and off toachieve different flux distributions in a network. Using theequivalencies in Table II we can generate pathway signa-tures or profiles based on the utilization of pathways for thefree output representation as shown in the lower portion ofFigure 4. These signatures offer the visual interpretation ofpathways being “switched” on and off in order to establishthe optimal flux distributions. Taken together, this illus-trates how flux distributions and associated metabolic phe-notypes can be interpreted from a pathway-based perspec-

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tive. To return to a reaction-based description, it is onlynecessary to perform a coordinate transformation by multi-plying the weights of the pathways by the pathway vectors.

From pathway analysis we arrive at a fundamentally dif-ferent perspective from which to view metabolic behaviorsuch as that predicted through the use of flux balance analy-sis. In addition to viewing the activity of individual reac-tions we can effectively interpret metabolic function interms of the activity of individual pathways. The aforemen-tioned example is useful in illustrating many of the theoret-ical concepts for a pathway interpretation of metabolic func-tion and, in what follows, we extend these concepts to ana-lyze complex metabolic networks. Although linearoptimization is used to calculate flux distributions in all ofthe subsequent examples, the concepts of pathway repre-sentations of flux distributions are general to any flux dis-tribution regardless of whether it is derived experimentallyor theoretically.

APPLICATION TO E. COLICENTRAL METABOLISM

The reconstruction of metabolic networks from annotatedgenome sequence information is currently an active area ofresearch. From these reconstructions it is possible to studythe system characteristics and capabilities of metabolic net-

works and explore the metabolic genotype–phenotype rela-tionship in fully sequenced organisms using both flux bal-ance analysis and genome-scale pathway analysis. Usingflux balance analysis, in silico studies on the systemic prop-erties of theHaemophilus influenzae(Edwards and Palsson,1999) andE. coli (Edwards and Palsson, 2000) metabolicnetworks have recently been completed. In addition, path-way analysis has now been successfully applied to assessthe production capabilities and general fitness of a recon-structed metabolic network forH. influenzae(Schilling andPalsson, 2000b). In what follows, we will consider a sub-system of theE. coli metabolic genotype comprised of thecentral metabolic pathways. Flux balance analysis will beused to explore the metabolic capabilities and predictedfunctions of this network under various substrate condi-tions. Then flux balance analysis will be combined withpathway analysis to provide a complete view of the capa-bilities and steady-state behavior of the metabolic networkfrom a pathway-oriented perspective.

The schematic illustration of the reactions comprising thecentral metabolic network ofE. coli examined in what fol-lows is shown in Figure 5 (the complete table of reactions isprovided on-line). Briefly, the network is comprised of thecomplete set of glycolytic reactions, the pentose phosphateshunt, and the tricarboxylic acid (TCA) cycle without theglyoxylate shunt, along with the necessary transport reac-

Figure 4. Flux distributions and pathway signatures for the three cases considered. (a) For case 1, only metabolite A is allowed to enter the system leadingto an optimal yield of 0.25 with the complete primal solution to the optimization provided. Flux levels throughbz are redistributed back to the originalexchange fluxes and the flux distribution is illustrated. The extreme pathways for the free output system utilized to achieve the flux distribution are overlaidonto the reaction network with their relative magnitudes provided in the pathway signature charts. (b) In case 2, the internal flux,v6, is restricted to zerowith metabolite A remaining as the only substrate. (c) For case 3, the activity ofv6 is restored while the substrate availability is switched from metaboliteA to C.

SCHILLING ET AL.: COMBINING METABOLIC PATHWAY ANALYSIS WITH FLUX BALANCE ANALYSIS 293

tions required to import and export metabolites from theextracellular space. The necessary reactions of the electrontransport chain are included with the P/O ratio set to 4/3.The system boundary is drawn around all of the reactions inthe network including the transport reactions. Exchangefluxes have been included to allow for the supply of glucoseand succinate as carbon sources available to the system,while ethanol and acetate are allowed byproducts of thesystem. Oxygen, carbon dioxide, and inorganic phosphateare free to enter and exit the system as needed. (Note thatthis network does not completely describe central metabo-lism in E. coli. We have chosen this representation as acompromise between successful representation of the basicaspects of central metabolism and providing a useful ex-ample of the salient features of this combined approach forstudying metabolic systems.)

Pathway Analysis for the Free Output System

To study the pathway structure of the system with freeoutputs it is necessary to include exchange fluxes for all ofthe necessary precursors and cofactors that are utilized togenerate biomass. In this example we define the metabolicdemands as those provided in Table III, which were studiedin an earlier model ofE. coli central metabolism (Varmaand Palsson, 1993a). The metabolic demands considered foreach metabolite are used to generate the growth flux usedwhen considering the linked output system. We focus firston the free output system without the aggregate demandflux.

For the case of free outputs we introduce an exchangeflux for each of the biosynthetic precursors that is con-strained only to allow the metabolite to exit the system.

Figure 5. Central metabolic network ofE. coliconsidered in the pathway and flux balance analysis. The system is comprised of 53 metabolites, 78 internalfluxes, and 8 exchange fluxes. A complete list of the net reaction capable of occurring in the network is provided on-line with complete reaction names.

294 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 71, NO. 4, 2000/2001

Unconstrained exchange fluxes are added for currency me-tabolites such as ATP, ADP, AMP, NADH, and NADPH.Therefore, we expect to generate extreme pathways thatlead directly from a substrate to a particular precursor ormetabolic byproduct (type I pathways) along with a numberof pathways and cycles that interconvert only cofactors andcurrency metabolites (type II pathways). In the case whereglucose is considered as the only available substrate thetotal number of extreme pathways is 1093 (1053 type I, 10type II, and 30 type III). For the case with succinate as theonly available substrate, the number of pathways reduces to195 extreme pathways (155 type I, 10 type II, 30 type III).These sets of pathways contain all of the optimal as well assuboptimal conversion ratios of each substrate to all of theprecursors and can be used to interpret the specific capa-bilities of the network and other systemic properties, aspreviously demonstrated forH. influenzae(Schilling andPalsson, 2000b). However, our interest is in studying thephenotype or the utilization of the network to meet thebalanced set of precursor demands required to generate bio-mass, rather than focusing on the production of particularprecursors of interest. Therefore, we now examine thelinked output system using pathway analysis initially, andthen combine this with flux balance analysis.

Pathway Analysis for the Linked Output System

For the case of linked outputs a growth flux is introducedthat represents the metabolic demands presented in Table IIIfor generating 1 g of dry weight (DW) biomass. We alsoremove the exchange fluxes for the precursors and currencymetabolites that were introduced in the free output system,thus closing the mass balances on all of these metabolites.The system now has eight exchange fluxes (extracellular

glucose, succinate, ethanol, acetate, carbon dioxide, oxygen,inorganic phosphate, and the intracellular biomass). In allfurther calculations, glucose and succinate are constrainedto only enter the system, whereas ethanol, acetate, and bio-mass are constrained to only exit the system. The remainingexchange fluxes are unconstrained (oxygen, carbon dioxide,and inorganic phosphate). With the decomposition of re-versible internal fluxes into a forward and reverse reactionthe system is comprised of 78 internal fluxes and 8 ex-change fluxes. The extreme pathways for the linked outputsystem will involve flux distribution patterns that the systemcan use to generate biomass, completely oxidize the carbonsource, or convert it into one of the byproducts.

First, we consider glucose to be the only substrate avail-able to the network. In this case, there are a total of 115extreme pathways (85 type I, 30 type III). The type IIIpathways are all identical to those of the free output systemand no type II pathways remain as all of the cofactors arenow forced to balance. There are four futile cycles in thesystem (sets of reactions that can utilize and convert ATP toADP and inorganic phosphate), those sets being (pfkA/fbp),(pckA, ppc), (pykF, ppsA, adk), and the ATP drain flux,which is used to represent non–growth-associated mainte-nance.

After careful inspection of the 85 type I pathways it isapparent that many of them are in sets of four pathways thatare identical, except for the utilization of a different futilecycle for the dissipation of ATP so as to balance the netproduction and consumption of ATP. All of these pathwaysin a set exhibit the same values for the exchange fluxes,illustrating their functional equivalence. To generate a re-duced set of pathways that represents the full capabilities ofthe network we retain only pathways from these sets thatutilized the ATP drain flux instead of one of the three futilecycles listed earlier. The type III pathways, which aremainly a consequence of the decomposition of reversiblereactions into a forward and a reverse reaction, are alsoremoved from consideration, as they show no activity in theexchange fluxes. Following these reductions we are leftwith 30 pathways. Table IV provides the details for theexchange fluxes of these pathways normalized to the inputvalue for the substrate glucose.

The pathway analysis was also performed with succinateas the sole carbon source for the system, generating thecomplete set of 66 extreme pathways (36 type I, 30 type III).Following the same simplification procedure just described,a reduced set of 12 pathways is generated from the completeset (Table V). All of the detailed pathway vectors for thecomplete and reduced sets in both cases are provided on-line.

For both glucose and succinate as the sole carbon sourcethere are pathways that generate biomass and either produceCO2, CO2 and acetate, or CO2 and ethanol as byproducts.Similar pathways also exist for generating these byproductswithout generating biomass. It is important to note that thenetwork does not allow for the generation of formate orsuccinate as metabolic byproducts, due to the increased

Table III. Metabolic demands of precursors and energy/redox cofactorsrequired for the generation of 1-g biomass yield as used in Varma andPalsson (1993a) and estimated from cellular composition data (Ingraham etal., 1983).a

Metabolite Demand (mmol)

ATP 41.3NAD+ 3.5NADPH 18.2G6P 0.2F6P 0.1R5P 0.9E4P 0.4GA3P 0.13PG 1.5PEP 0.5PYR 2.8ACCOA 3.7OXA 1.8AKG 1.1SUCCOA (Trace)

aThese demands do not include energy requirements for non–growth-associated maintenance. The growth flux is a summation of the drains ofeach metabolite multiplied by the demand weights given in the table.

SCHILLING ET AL.: COMBINING METABOLIC PATHWAY ANALYSIS WITH FLUX BALANCE ANALYSIS 295

Tab

leIV

.F

unct

iona

lcha

ract

eris

tics

ofth

ere

duce

dse

tof

30ex

trem

epa

thw

ays

calc

ulat

edfo

rgl

ucos

eas

the

sole

carb

onfo

rth

eE

.co

lim

etab

olic

netw

ork.a

Pat

hway

num

ber

Exc

hang

eflu

xesb

Net

path

way

reac

tion

bala

nce

GLC

xt/G

LCxt

ET

Hxt

/GLC

xtA

Cxt

/GLC

xtG

RO

/GLC

xtP

Ixt/G

LCxt

CO

2xt/G

LCxt

O2x

t/GLC

xt

2−

1.00

00

00.

105

−0.

390

1.49

8−

1.37

6G

LCxt

+0.

39P

Ixt+

1.37

6O

2xt

⇒0.

105

GR

O+

1.49

8C

O2x

t24

−1.

000

00

0.10

5−

0.38

81.

522

−1.

402

GLC

xt+

0.38

8P

Ixt+

1.40

2O

2xt

⇒0.

105

GR

O+

1.52

2C

O2x

t21

−1.

000

00

0.10

4−

0.38

31.

575

−1.

455

GLC

xt+

0.38

3P

Ixt+

1.45

5O

2xt

⇒0.

104

GR

O+

1.57

5C

O2x

t23

−1.

000

00.

125

0.10

2−

0.37

61.

410

−1.

293

GLC

xt+

0.37

6P

Ixt+

1.29

3O

2xt

⇒0.

102

GR

O+

1.41

CO

2xt+

0.12

5A

Cxt

38−

1.00

00

0.17

80.

101

−0.

373

1.34

2−

1.22

6G

LCxt

+0.

373

PIx

t+1.

226

O2x

t⇒

0.10

1G

RO

+1.

342

CO

2xt+

0.17

8A

Cxt

4−

1.00

00

00.

099

−0.

367

1.76

0−

1.64

6G

LCxt

+0.

367

PIx

t+1.

646

O2x

t⇒

0.09

9G

RO

+1.

76C

O2x

t42

−1.

000

0.19

90

0.09

6−

0.35

71.

482

−1.

172

GLC

xt+

0.35

7P

Ixt+

1.17

2O

2xt

⇒0.

096

GR

O+

1.48

2C

O2x

t+0.

199

ET

Hxt

39−

1.00

00

0.27

50.

095

−0.

353

1.37

7−

1.26

7G

LCxt

+0.

353

PIx

t+1.

267

O2x

t⇒

0.09

5G

RO

+1.

377

CO

2xt+

0.27

5A

Cxt

44−

1.00

00.

151

0.12

40.

095

−0.

353

1.37

7−

1.11

6G

LCxt

+0.

35P

Ixt+

1.12

O2x

t⇒

0.09

5G

RO

+1.

38C

O2x

t+0.

15E

TH

xt+

0.12

AC

22−

1.00

00.

255

00.

093

−0.

344

1.51

6−

1.15

4G

LCxt

+0.

344

PIx

t+1.

154

O2x

t⇒

0.09

3G

RO

+1.

516

CO

2xt+

0.25

5E

TH

xt43

−1.

000

0.31

00

0.09

1−

0.33

61.

503

−1.

089

GLC

xt+

0.33

6P

Ixt+

1.08

9O

2xt

⇒0.

091

GR

O+

1.50

3C

O2x

t+0.

31E

TH

xt29

−1.

000

0.33

00

0.08

8−

0.32

61.

577

−1.

145

GLC

xt+

0.32

6P

Ixt+

1.14

5O

2xt

⇒0.

088

GR

O+

1.57

7C

O2x

t+0.

33E

TH

xt32

−1.

000

00.

330

0.08

8−

0.32

61.

577

−1.

476

GLC

xt+

0.32

6P

Ixt+

1.47

6O

2xt

⇒0.

088

GR

O+

1.57

7C

O2x

t+0.

33A

Cxt

35−

1.00

00

00.

088

−0.

326

2.23

8−

2.13

7G

LCxt

+0.

326

PIx

t+2.

137

O2x

t⇒

0.08

8G

RO

+2.

238

CO

2xt

45−

1.00

00.

595

00.

078

−0.

287

1.49

3−

0.80

8G

LCxt

+0.

287

PIx

t+0.

808

O2x

t⇒

0.07

8G

RO

+1.

493

CO

2xt+

0.59

5E

TH

xt26

−1.

000

00

0.07

0−

0.26

12.

993

−2.

912

GLC

xt+

0.26

1P

Ixt+

2.91

2O

2xt

⇒0.

07G

RO

+2.

993

CO

2xt

3−

1.00

00

00.

068

−0.

251

3.10

5−

3.02

7G

LCxt

+0.

251

PIx

t+3.

027

O2x

t⇒

0.06

8G

RO

+3.

105

CO

2xt

18−

1.00

00

00.

066

−0.

243

3.20

0−

3.12

5G

LCxt

+0.

243

PIx

t+3.

125

O2x

t⇒

0.06

6G

RO

+3.

2C

O2x

t10

−1.

000

00

0.06

3−

0.23

53.

289

−3.

216

GLC

xt+

0.23

5P

Ixt+

3.21

6O

2xt

⇒0.

063

GR

O+

3.28

9C

O2x

t14

−1.

000

00

0.06

2−

0.22

93.

359

−3.

288

GLC

xt+

0.22

9P

Ixt+

3.28

8O

2xt

⇒0.

062

GR

O+

3.35

9C

O2x

t46

−1.

000

1.00

00

00

4.00

0−

3.00

0G

LCxt

+3.

0O

2xt

⇒4.

0C

O2x

t+1.

0E

TH

xt49

−1.

000

01.

000

00

4.00

0−

4.00

0G

LCxt

+4.

0O

2xt

⇒4.

0C

O2x

t+1.

0A

Cxt

52−

1.00

00

00

06.

000

−6.

000

GLC

xt+

6.0

O2x

t⇒

6.0

CO

2xt

57−

1.00

02.

000

00

02.

000

0G

LCxt⇒

2.0

CO

2xt+

2.0

ET

Hxt

61−

1.00

00

2.00

00

02.

000

−2.

000

GLC

xt+

2.0

O2x

t⇒

2.0

CO

2xt+

2.0

AC

xt65

−1.

000

00

00

6.00

0−

6.00

0G

LCxt

+6.

0O

2xt

⇒6.

0C

O2x

t69

−1.

000

00

00

6.00

0−

6.00

0G

LCxt

+6.

0O

2xt

⇒6.

0C

O2x

t72

−1.

000

01.

667

00

2.66

7−

2.66

7G

LCxt

+2.

667

O2x

t⇒

2.66

7C

O2x

t+1.

667

AC

xt75

−1.

000

1.66

70

00

2.66

7−

1.00

0G

LCxt

+1.

0O

2xt

⇒2.

667

CO

2xt+

1.66

7E

TH

xt84

−1.

000

00

00

6.00

0−

6.00

0G

LCxt

+6.

0O

2xt

⇒6.

0C

O2x

t

a Pat

hway

sar

eor

dere

dba

sed

onac

tivity

ofth

egr

owth

flux

norm

aliz

edto

the

gluc

ose

upta

ke.P

athw

aynu

mbe

rsco

inci

dew

ithth

eor

igin

alnu

mbe

rsof

the

pat

hway

vect

ors

reta

ined

from

the

com

plet

ese

t(av

aila

ble

on-li

ne).

All

exch

ange

flux

valu

esar

eno

rmal

ized

toth

egl

ucos

ein

take

leve

l(ne

gativ

eva

lues

are

rela

tive

upta

kera

tios,

posi

tive

valu

esar

epr

odu

ctio

nra

tios)

.bG

LC,

gluc

ose;

ET

H,

etha

nol;

AC

,ac

etat

e;P

I,in

orga

nic

phos

phat

e;C

O2,

carb

ondi

oxid

e;O

2,ox

ygen

;G

RO

,bi

omas

s/gr

owth

flux

(see

Tab

leIII

).

296 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 71, NO. 4, 2000/2001

number of pathways that would be generated based on theseconditions. This increase in pathway number is sufficientenough to distract from the main theoretical issues beingillustrated herein, and was therefore not considered. Forglucose we see that the maximal yield of biomass in anypathway is 0.105 g DW/mmol glucose (0.583 g DW/g glu-cose), whereas the maximal yield of succinate is 0.051 gDW/mmol (0.432 g DW/g succinate). These yields cannotbe exceeded, as the combination of any set of extreme path-ways according to Eq. (3) will not surpass the ratio of bio-mass/substrate of the optimal pathway. In each case, theminimum biomass yield is zero, as pathways do exist that donot produce biomass. For glucose, the range of oxygen con-sumption to substrate uptake varies from 0.0 to 6.0 molO2/mol glucose, whereas, for succinate, the range narrowsfrom 0.5 to 3.5 mol O2/mol succinate. Any oxygen/substrateconsumption ratios outside of these ranges are infeasible.Other similar systemic constraints can be defined from thesepathways that will set boundaries on the functional capa-bilities of the network. Recall that these sets of extremepathways define the edges of the flux cone that contain allof the possible metabolic phenotypes possibly displayed bythe network. Therefore, it is possible to bracket the behaviorof the network from the pathway analysis. Next, we illus-trate how the pathways can be used to gain a deeper under-standing of shifting metabolic behaviors as determined us-ing flux balance analysis.

FLUX BALANCE ANALYSIS AND PHENOTYPICPHASE PLANES

Flux balance analysis can be used to examine quantitativelythe linked output system detailed earlier. Geometrically, theconstraints that are imposed on the input values of the ex-change fluxes for the substrates will bound the flux conedefined by the extreme pathways; thus resulting in abounded polyhedron. Optimal solutions within this spacewill then lie on a vertex of the polyhedron. These are thebounded feasible solutions of the linear programming prob-lem. First, we calculate the optimal flux distributions forgrowth on glucose and succinate normalized to 1 mmol ofsubstrate. Illustrations of the optimal flux distributions ascalculated for both conditions are provided on-line. Theoptimal biomass yields determined with flux balance analy-sis are 0.105 g DW/mmol glucose and 0.051 g DW/mmolsuccinate, which are identical to the optimal yield calculatedfrom the pathways (pathway #2 for glucose, and pathway #1for succinate). This result reveals that the optimal solutionlies directly on the vertex of the polyhedron that is definedby the extreme pathway containing the optimal ratio of bio-mass/substrate. In each case, the inclusion of any other path-way into the solution will decrease the yield from the maxi-mum ratios indicated earlier. (For comparison to unpub-lished data, the experimental optimal yield that we haveachieved for growth ofE. coli K-12 in batch culture onglucose is 0.096 g DW/mmol [n4 6, SD 4 0.010]. Forgrowth on succinate the optimal yield is 0.050 g DW/mmol[n 4 13, SD4 0.006].)

All metabolic phenotypes (metabolic flux distributions)attainable from a defined metabolic system are mathemati-cally confined to the flux cone. Flux balance analysis is usedto search through the flux cone for a solution that maxi-mizes/minimizes a given objective function. It has beenshown that, under nutritionally rich growth conditions (i.e.,cell is not starved for phosphate or nitrogen),E. coli growsin a stoichiometrically optimal manner (Varma and Palsson,1994a); thus, setting the growth flux as the objective func-tion produces physiologically meaningful results. The opti-mal flux distribution is only meaningful when interpreted interms of the specific environmental conditions. Therefore,phenotype phase planes (PhPPs) have been calculated todefine the range of optimal phenotypic behavior in the con-text of a number of environmental parameters (Edwards andPalsson, 1999; Varma et al., 1993b). These phase planesillustrate how optimal phenotypic behavior is dependent onthe environmental conditions, and can be used to delineatefundamentally different regions of metabolic behavior in amultiparameter space. In what follows, we show that thedemarcations on the PhPP that are generated by flux balanceanalysis (FBA) and linear programming are identical to theprojections of the extreme pathways onto the relevant sub-space.

The methodology for defining PhPPs has been describedby Edwards et al. (currently in review by B & B). Herein,we briefly describe the construction of PhPPs in the contextof the metabolic pathway analysis. The exchange flux ornutrient uptake rates for two metabolites (such as the carbonsubstrate and oxygen) can form two axes on an (x,y)-plane(these exchange fluxes were two unit vectors inRn). Theoptimal metabolic flux distribution is calculated for allpoints in this plane using linear programming. Each of thetwo exchange fluxes for the two metabolites considered inthe phase plane is set to equal a precise value based on thecoordinates in the plane, and optimal solutions are calcu-lated. In other words, the maximum value of the objectivefunction is found as the position of the hyperplanes thatbound the flux cone in the respective directions is moved.

Through an examination of the shadow prices (linear pro-gramming dual variable), it is determined that there is afinite number of fundamentally different optimal metabolicflux distributions present in such a plane (Varma et al.,1993b). The shadow prices are measurements associatedwith every metabolite in the network, and indicate the in-trinsic value of the metabolite. They are calculated from thedual problem of a linear optimization and are defined asfollows:

gi = −dZ

dbi(6)

The shadow prices (gi) essentially define the sensitivity ofthe objective function (Z) to changes in the availability ofeach metabolite/substrate, whereasbi defines the violationof the mass balance constraint and is equivalent to an uptakereaction. The shadow price can be negative, zero, or posi-

SCHILLING ET AL.: COMBINING METABOLIC PATHWAY ANALYSIS WITH FLUX BALANCE ANALYSIS 297

tive, depending on the value of the metabolite to the objec-tive function. Through observations of the shadow prices itcan be determined when a new basic feasible solution hasbeen introduced into the optimal solution. Because the ex-treme pathways of the linked output system are equivalentto the basic feasible solutions, this situation indicates a shiftin the metabolic behavior corresponding to a new extremepathway being utilized. Thus, the weight (w) on a particularextreme pathway increases from zero to a positive value indescribing the flux distribution, as in Eq. (3). This procedureleads to the demarcation of distinct regions, or “phases,” inthe plane where the same pathways are always being uti-lized to varying extents. In each phase, the optimal use ofthe metabolic network is fundamentally different, corre-sponding to different optimal phenotypes.

In terms of the pathway analysis, the PhPP is basicallygenerated by the projection of selected edges of the fluxcone into a two-dimensional or, in some cases, a three-dimensional space spanned by the fluxes used to generatethe particular phase plane. The boundaries of the phaseswithin the PhPP correspond to particular extreme pathways.The entire region corresponding to a particular phase issubsequently the positive combination of the two extremepathways that bound the phase. Thus, as we traversethrough the phase plane we are essentially moving along thefaces of the flux cone. In addition, the internal fluxes thatwill define the distinct phenotypes in each phase can beidentified directly from the extreme pathways. The differ-ence between the FBA phenotype phase plane and the ex-treme pathway projection is that the phase plane analysisonly identifies the pathways that are used under optimalconditions, whereas the extreme pathway analysis maps allthe pathways onto the same two-dimensional space. Theremaining pathways that do not generate the boundaries ofthe PhPP represent suboptimal solutions or alternate optimalsolutions. We now illustrate these concepts for growth onsuccinate using the succinate–oxygen PhPP.

SUCCINATE–OXYGEN PHENOTYPICPHASE PLANE

The succinate–oxygen PhPP contains four distinct meta-bolic phenotypes (Fig. 6). The dark lines define the bound-aries of each region in the phase plane as calculated usinglinear programming, and the lighter lines are the additionalextreme pathways projected onto the two- and three-dimensional spaces. There are several distinguishing char-acteristics of the succinate–oxygen PhPP. First, the PhPPillustrates that the metabolic network is unable to utilizesuccinate as the sole carbon source for growth in an anaero-bic environment, as evidenced by the PhPP only covering aregion of the positive quadrant and the fact that all extremepathways that generate biomass contain input exchangefluxes for succinate and oxygen. Second, the PhPP has threefutile regions. (These are defined as regions where increaseduptake of one of the substrates has a negative effect on theability of the metabolic network to support growth [Ed-

wards et al., currently in review by B & B]: one whereoxygen is in excess [region 1]; and two where the carbonsource is in excess [regions 3 and 4].) The futile regions canbe identified as regions in which the slope of the isoclines ispositive, where the isocline is defined as the locus of pointsfor which the combination of the two exchange fluxes willprovide for the same optimal value of the objective function.The succinate–oxygen PhPP also contains one dual-substrate limitation region (region 2).

Region 1 (Fig. 6a) is an oxygen-excess-futile region. Ad-ditional carbon is consumed to eliminate the oxygen that isprovided in excess. Therefore, this region is wasteful, andless carbon is available for biomass production because it isoxidized to eliminate the excess oxygen. The ability of themetabolic network to produce high-energy phosphate bondsis not limiting growth in the futile region. The pathways inTable V are projected onto the phase planes based on theratio of oxygen uptake to substrate uptake (Fig. 6b). It isseen that pathways #23 and #35, which project onto thesame line, bound region 1 at a maximum oxygen/succinateuptake ratio of 6. These pathways do not generate biomassand, subsequently, the biomass yield decreases as the oxy-gen/succinate ratio increases moving the position in region1 closer to the upper boundary than the lower boundary.

Separating region 1 from region 2 is a line defined as theline of optimality. This line represents the optimal relationbetween the two exchange fluxes. The biomass yield on theline of optimality is 0.051 g DW/mmol succinate. The lineof optimality corresponds to the extreme pathway #1 inTable V. The optimal utilization of the metabolic pathwaysinvolves the cyclic operation of the TCA cycle, no flux inthe oxidative branch of the pentose phosphate pathway(PPP), the utilization of the malic enzyme to produce py-ruvate and NADPH, and CO2 production as the only meta-bolic byproduct. The metabolic flux distribution for a pointon this line is available on-line.

In region 2, the availability of succinate and oxygen islimiting the biomass production. Thus, this region is con-sidered a dual-substrate-limited region, and the dual limita-tion is immediately apparent by the negative slope of theisoclines in region 2. The optimal metabolic flux distribu-tion in this region leads to acetate production, and thus anextreme pathway that includes the acetate exchange flux isturned on. The activated pathway is the extreme pathwayseparating region 2 from region 3 and corresponds to path-way #20 in Table V, which produces acetate and biomasswith a yield of 0.047 g DW/mmol succinate. Thus, everypoint in region 2 is a positive combination of pathway #1and pathway #20, which results in a biomass yield rangingfrom 0.047 to 0.051 g DW/mmol succinate, and an acetateproduction level ranging from 0 to 0.158 mmol acetate/mmol succinate.

In regions 3 and 4, the uptake of additional succinate hasa negative effect on the objective function. The succinateutilized by the metabolic network beyond the demarcation isnot utilized to increase the objective function. Rather, cel-lular resources are required to eliminate the excess succi-

298 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 71, NO. 4, 2000/2001

nate. Region 3 is bounded by pathway #20 and pathway#19, which produces ethanol as a byproduct. Therefore, anycombination of these two pathways will lead to the com-bined production of acetate (from pathway #20) and ethanol(from pathway #19). Starting in region 2 and traversing intoregion 3 the metabolic network will turn off pathway #1 andturn on pathway #19. From the complete pathway vectorsthat provide the flux values for the internal fluxes (providedon-line), it is seen that thesucA and sucC reactions arepredicted to shut off and theadhE reaction is turned on,while a number of other sets of reactions will carry increas-ing or decreasing flux levels. Traversing into region 4, themetabolic network will turn off pathway #20 and turn onpathway #31, which generates ethanol and CO2 with nobiomass production, thus serving to decrease the objectivefunction. When the oxygen/succinate uptake ratios de-creases to 2.00 the network will utilize pathway #31 only,and biomass production will reach zero. An oxygen/succinate uptake ratio below 2.00 is considered infeasiblefor network processing (these points will lie outside of theflux cone).

By determining the extreme pathways that demarcate theregions of the PhPP we are able to determine the precisetheoretical range of biomass yield, oxygen/substrate uptakeratios, and the exact combination of pathways that charac-terize the predicted optimal metabolic phenotypes in eachregion. This pathway perspective may assist in the identi-fication of pathways and associated reaction sets that arepredicted to be switched on/off during a metabolic transitioninto an alternative functional modality.

GLUCOSE–OXYGEN PHENOTYPIC PHASE PLANE

The glucose–oxygen PhPP contains five regions (Fig. 7a)also bounded by sets of extreme pathways (Fig. 7b). Region1 is an oxygen-excess-futile region, similar to region 1 forsuccinate. It is unlikely that the network would ever operatein this region, as it is futile. Based on the positive slope ofthe isoclines in this region we would expect the network todecrease the oxygen uptake rate, which will increase thebiomass yield and bring the behavior closer to the line of

Figure 6. Succinate–oxygen phenotypic phase plane. (a) Four regions of metabolic behavior are shown on the two- and three-dimensional phase portraits.The isocline shading indicates regions in which the objective function (biomass yield) remains constant. (b) Projection of the 12 pathways in Table Vontothe phase planes. Pathways are plotted based on the slope of the line as determined from the ratio of the oxygen exchange flux versus the succinate exchangeflux. Bold lines and the corresponding bold pathway numbers indicate the pathways corresponding to the boundaries of the phases. Thin lines andcorresponding pathway numbers reveal the additional pathways that project onto either the interior of the cone or a face of the cone as evidenced in thethree-dimensional phase plane.

SCHILLING ET AL.: COMBINING METABOLIC PATHWAY ANALYSIS WITH FLUX BALANCE ANALYSIS 299

optimality. The line of optimality corresponds to pathway#2 from Table IV yielding 0.105 g DW/mmol glucose. Theoptimal flux distribution is provided on-line.

An oxygen consumption decreases to below 1.376 molO2/mol glucose (the slope of pathway #2 in Fig. 7b), thenetwork moves into region 2, where acetate begins to besecreted. Acetate secretion occurs due to the activation ofpathway #38 that bounds the right side of region 2. Tra-versing through region 2 yields only a modest decrease inthe biomass yield from 0.105 to 0.101 g DW/mmol glucose.As the oxygen consumption continues to decrease to below1.226 mol O2/mol glucose, the phenotype moves from re-gion 2 into region 3 and the network begins to secrete etha-nol in addition to acetate, and the biomass yield decreasesfurther. In terms of the pathways, region 3 is bound to theleft by pathway #38 and to the right by pathway #44 (Fig.7b). Therefore, moving from region 2 into region 3 shiftsthe pathway utilization by inactivating pathway 2 com-pletely while switching pathway #44 on to combine withpathway #38. From the pathways it is seen that the neces-sary reactions for ethanol production and transport are allactivated, whereas many of the reactions in the TCA cycle(sucA, sucC, sdhA1, fumA, mdh, sdhA2) are inactivated asthe cycle is no longer used to generate energy and reductivepotential because the decreased oxygen availability has lim-ited the use of the electron transport chain to generate ATP.

Region 4 is bracketed by pathway #44 and #45. As everypoint in this region is a positive combination of these twopathways the biomass yield will range from 0.078 to 0.095g DW/mol glucose for an oxygen consumption range of0.808 to 1.116 mol O2/mole glucose. ThepykFreaction willbe activated while theppsA reaction will be inactivated.Moving below an oxygen consumption rate of 0.808 molO2/mol glucose places the network in region 5 where it iscompletely oxygen limited, and the biomass yield dropssharply as the oxygen availability decreases. Region 5 isbounded by pathway #45 and #57 in Table IV. Note thatwhen the oxygen consumption rate is zero the network isunable to produce biomass, and therefore the network pre-dicts that completely anaerobic growth is unachievable.This result is due to the fact that no oxygen is available toconsume the redox equivalents generated to produce thebiomass (as part of the objective function). If we include thepyruvate formate lyase reaction in the network and allowformate to be an allowed byproduct the network can supportanaerobic growth. In this case, the complete set of extremepathways would increase from 85 to 192 simply from theaddition of this one reaction. To avoid excessive numbers ofpathways in this introductory and conceptual investigationwe chose not to include this reaction. Therefore, the pre-dicted inability to grow anaerobically is attributed to themodel construction that was simplified so as to demonstratemany of the concepts just described without drowning thereader in excessive information and excessive numbers ofpathways.

As in the previous section it is again seen that the extremepathways that span the flux cone demark regions of optimalT

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300 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 71, NO. 4, 2000/2001

metabolic utilization as calculated using linear program-ming. Using phenotypic phase planes, it is realized howpathways can be interpreted as being switched on and off asenvironmental parameters change in time. This interpreta-tion made use of the extreme pathways for the linked sys-tem, which correspond to the basic feasible solutions of thelinear programming problem created through the introduc-tion of maximum capacity constraints. As demonstrated ear-lier, the extreme pathways for the linked output system thatare determined to comprise a flux distribution may be de-composed into pathways calculated for free output descrip-tion, which provides a more detailed look at pathway utili-zation.

ALTERATIONS TO INTERNAL STRUCTURE OFTHE NETWORK

In the previous two sections, environmental challenges tothe metabolic network were considered through examina-tion of the optimal network performance across a completeparameter space for exchange fluxes. In this section thefocus shifts to assessing the effects of internal challenges to

the network on overall performance and pathway utiliza-tion. All pathways in Tables IV and V that do not form theboundary of a region in the PhPP for either glucose orsuccinate (as shown in Figs. 6 and 7) are suboptimal path-ways and will not be used under optimal conditions. Thesesuboptimal extreme pathways will be utilized when alter-ations are made to the reaction network/metabolic genotypethat lead to the removal of one or more of the optimalextreme pathways. Removal of internal reactions througheither gene deletions or a loss in enzymatic function has theeffect of completely eliminating any extreme pathway thatutilizes the reaction being removed.

To illustrate the correlation between removal of a reac-tion and the corresponding impact on pathway structure andoptimal performance of the network, each reaction in Figure5 was individually eliminated from the network and optimalflux distributions were calculated using flux balance analy-sis. This was performed for both substrate growth condi-tions, and optimal biomass yields are provided in Table VI.To determine the effects on the pathway structure, all ex-treme pathways generating biomass that utilize the disabled

Figure 7. Glucose–oxygen phenotypic phase plane. (a) Five regions of metabolic behavior are shown on the two- and three-dimensional phase portraits.The isocline shading indicates regions in which the objective function (biomass yield) remains constant. (b) Projection of the 30 pathways in Table IV ontothe phase planes. Pathways are plotted based on the slope of the line as determined from the ratio of the oxygen exchange flux versus the glucose exchangeflux. Bold lines and the corresponding bold pathway numbers indicate the pathways corresponding to the boundaries of the phases. Thin-dashed linescorrespond to additional pathways that project onto either the interior of the cone or a face of the cone as evidenced in the three-dimensional phase plane.

SCHILLING ET AL.: COMBINING METABOLIC PATHWAY ANALYSIS WITH FLUX BALANCE ANALYSIS 301

reaction are indicated in Table VI. These pathways can nolonger be used by the system. If there are no extreme path-ways remaining that can be utilized to generate biomassthen the network is completely disabled. However, in manycases, there are viable pathways that remain intact despitethe loss of a reaction. From the remaining pathways, the onewith the highest biomass yield can be determined from theinformation in Tables IV and V, and often it will be sub-optimal with respect to a pathway that has been eliminated.In comparing the optimal yields as calculated using linearprogramming versus the extreme pathways available to thesystem in the face of the internal challenges, it is seen thatthe predicted optimal growth yield and flux distribution cor-responds precisely with one of the remaining extreme

pathways that has the highest ratio of biomass to substrateuptake. This gain provides a clear representation of the con-nection between pathway analysis and flux balance analy-sis. For a number of reactions the effects of removal areidentical due to the fact that a number of reactions belong tothe same reaction/enzyme subset (Pfeiffer et al., 1999;Schilling and Palsson, 2000b). These subsets correspond toreactions that always occur together in the same ratio in allpathways in which they are active (i.e., they are dependenton each other). Table VII lists the enzyme subsets for theE.coli central metabolic network considered in both cases.

Internal challenges will also have a profound effect onhow the network responds in the face of environmentalchallenges. To examine these combined challenges, the

Table VI. Pathway and flux balance analysis results associated with removal of individual reactions or genes.a

aPathways that utilized the removed reaction are indicated in dark gray boxes, with the X indicating that they are no longer available due to the removalof the reaction. Light gray boxes are pathways still available that generate a biomass yield. The pathway with the highest yield in each case is indicatedby the double-bordered boxes. Optimal biomass yields as calculated using linear programming are indicated for each deletion on both substrates in unitsof grams dry weight per millimole of substrate.

302 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 71, NO. 4, 2000/2001

PhPP studies examined in the previous section can be re-peated for each of the network deficiencies examined inTable VI. In Figure 8, the 12 extreme pathways for growthon succinate are projected into the oxygen–succinate plane

and the regions of the phase plane are calculated using fluxbalance analysis to determine the qualitatively different re-gions of metabolic function. In comparing the wild-typesituation (Fig. 8a) to the deletion of any one of the genescomprising the oxidative branch of the pentose phosphateshunt (Fig. 8b), there is only one pathway that remainscapable of generating biomass, and this pathway forms theboundary between the two regions of the phase plane. Theother two pathways that generate the boundaries do notproduce biomass. Figure 8c represents the loss of either thesucAor sucC reaction. In this case, many of the originalpathways remain intact with the exception of the optimalpathway utilized in the standard wild-type network. Onlyone of the boundaries to the phase plane shifts, indicatingthe utilization of a different pathway that is now optimal dueto the loss of the original optimal pathway. Taken together,this illustrates how an internal challenge to a network forcesthe utilization of a different set of pathways under changingenvironmental conditions.

Table VII. Enzyme subsets calculated for theE. coli central metabolicnetwork considering both glucose succinate as possible substrates.

Reaction/enzyme subsets

(AC trx, ackA, pta)(ETH trx, adhE)(O2 trx, cyoA)(fba, tpiA)(gapA, pgk)(gpmA, eno)(ppsA, adk)(zwf, pgl, gnd)(tktA1, talA)(gltA, acnA, icdA)(sucA, sucC)(sdhA1, fumA, mdh, sdhA2)

Figure 8. Pathway projections and phase plane outlines for mutant strains with succinate as the substrate. For (a) wild-type, (b)zwf, pgl,or gnddeletion,(c) sucAor sucCdeletion, each of the pathways available to the network are indicated by the solid black lines, whereas those no longer available due tothe deletion are indicated by the solid gray lines. Points outlining the different regions of the phase plane for each case are indicated by the filled circlesas calculated using linear programming optimizing for the biomass yield.

SCHILLING ET AL.: COMBINING METABOLIC PATHWAY ANALYSIS WITH FLUX BALANCE ANALYSIS 303

DISCUSSION

Using a simple example network and a detailed network ofE. coli central metabolism, we have illustrated how meta-bolic phenotypes (flux distributions) and shifts in metabolicbehavior can be interpreted from a pathway-based perspec-tive. From the physical principles of mass conservation andthe constraints of reaction thermodynamics, the mathemati-cal concepts of convex analysis are used to define sets ofextreme pathways that span the entire flux space/cone forcomplex metabolic networks. These pathways representbalanced sets of metabolic reactions whose functional char-acteristics together represent the capabilities of the meta-bolic network. From a pathway-oriented perspective, sys-tems can be studied in terms of their ability to generateparticular metabolic precursors (free outputs) or examinedfor their ability to generate a balanced set of these biosyn-thetic demands (linked outputs). For both approaches, net-work flux distributions, such as those generated using op-timization techniques (e.g., flux balance analysis), can beinterpreted as nonnegative combinations of the extremepathways as indicated mathematically in Eq. (3). These dif-ferent approaches allow for two different pathway perspec-tives (free versus linked outputs), which are related to eachother by precise pathway equivalencies (not to be confusedwith equivalencies among reactions).

Using flux balance analysis, the complete range of opti-mal metabolic phenotypes can be examined under definedenvironmental and genetic conditions through the use ofphenotypic phase planes (PhPPs). For theE. coli network,the PhPPs were generated for growth on succinate and glu-cose spanning particular regions and dimensions of the fluxcone. Projecting the set of extreme pathways onto the PhPPreveals the precise pathway utilizations that are occurring ineach region of metabolic behavior. The pathways serve tobracket the behavior in each region, which creates ranges ofbiomass yield along with oxygen and substrate uptake limi-tations. In the face of internal challenges to the structure ofthe network the set of viable extreme pathways is reduced,in turn limiting the manner in which a network operatesunder changing environmental conditions, leading tochanges in the phase plane. The flux cone becomes smallerdue to the loss of extreme pathways, which imposes furtherconstraints on its functional capabilities.

Using the pathways it is possible to translate back to areaction-based interpretation of the predicted metabolicphenotypes. This reveals sets of reactions that are com-pletely inactivated or activated as well as subtle changes ineither the increased or decreased utilization of various re-actions. Combining flux balance analysis and pathwayanalysis provides the ability to examine metabolic capabili-ties and interpret optimal metabolic phenotypes from bothreaction- and pathway-oriented perspectives. Together theseapproaches can be used to examine quantitatively the con-straints placed on metabolic activity due to the structure andconnectivity of the reaction network and the physicochem-ical principles that govern biological systems.

Ideally, we would like to study the pathway structure and

utilization for an entire organism with this combined ap-proach. Flux balance analysis is easily capable of handlinglarge networks as linear programming has been designedover the years to handle problems with millions of variables.The same cannot be said for convex analysis algorithmsassociated with enumerating pathways in networks in anyfield. For small networks, such as the one in Figure 1,computing the set of extreme pathways is quite trivial. How-ever, as the size of the network increases in a linear fashion,the time to calculate the extreme pathways as well as thenumber of pathways increases in an exponential fashion. Inthe traditional language of theoretical computer science theproblem of enumerating the vertices of a polyhedral cone isreferred to as an NP-hard problem, such as the “travelingsalesman” problem (Papadimitriou and Steiglitz, 1998). Al-though it is feasible to construct comprehensive flux bal-ance models for completely reconstructed metabolic geno-types, it is not feasible to enumerate all pathways present insuch a network for interpreting metabolic function. A veryconservative estimate would place the number of pathwaysin E. coli well over 106. Although it is important to calculatethese complete sets for the identification of more detailedsystems properties related to the capabilities of the network,the shear number of pathways may complicate the com-bined pathway/flux balance analysis approach to an extentthat makes the interpretation overwhelming. This problemof excessive pathways can be partially dealt with by eitherfocusing on particular situations (both environmental andgenetic) that would reduce the system as we have done inthis article, or to properly subdivide the metabolic system tosolve for pathways in each subsystem that may be piecedtogether to interpret metabolic capabilities (Schilling andPalsson, 2000b).

Neither flux balance analysis nor pathway analysis incor-porates information on reaction kinetics and regulation, lim-iting their insight into dynamic responses. From these ap-proaches it is possible to realize some of the fundamentalconstraints that metabolic systems are faced with and definethe flux cone that contains all admissible steady-state fluxvectors. As detailed information on in vivo reaction dynam-ics is accumulated, the ability to analyze, interpret, andperhaps predict dynamic responses of metabolic networks toenvironmental and genetic perturbations using dynamicmodeling approaches may become more feasible. For theoptimal phenotypes calculated in a wild-type strain to berealized in vivo a regulatory network that achieves this statemust be present. This would then assume that a regulatoryscheme has evolved to optimize the same parameters opti-mized in the linear programming (i.e., growth). Althoughthis may or may not be the case, in general, regulatoryschemes and reaction dynamics will serve to further con-strain metabolic behavior to operate in confined subspacesof the flux cone. Identifying these regions from both thetheoretical and experimental side represents a challenge forthe future.

A number of experimental technologies have now madethe holistic study of biological systems feasible beyond thetheoretical realm. The ability to assimilate DNA chip-based

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and protein expression technologies providing genome-scale information with computational methods for meta-bolic network analysis will become important in advancingthe study of metabolic physiology and the practice of meta-bolic engineering. Currently, the interpretation of high-throughput experimental information on systemic behavioris limited by a lack of analysis capabilities. Can systems-based quantitative in silico approaches, such as flux balanceand pathway analysis, be used to assist in understanding thisflood of data? This is a question that will need to be an-swered as interest builds in the genomics community forquantitative systems analysis.

The authors thank the Whitaker Foundation for their supportthrough graduate fellowships in bioengineering.

APPENDIX

A convex polyhedron is a region inRn determined by linearequalities and inequalities (Fig. A1). One of the propertiesof a polyhedron is convexity: Any line segment connectingtwo points in the polyhedron is completely contained in thepolyhedron. If a polyhedron is bounded then it is called apolytope. It is called a polyhedral cone if every ray throughthe origin and any point in the polyhedron are completelycontained in the polyhedron.

A point →v in a polyhedron is called a vertex if it is not theconvex combination of two other points in the polyhedron.[A convex combination of the points→p and→q has the formt→p + (1 − t)→q ,where 0 <t < 1.] A ray→r is called an extremalray if no point on the ray is a positive linear combination ofpoints in the polytope but not on the ray. Note that a cone

only has extremal rays and a polytope only has vertices (Fig.A2).

The set of vertices and extremal rays is unique for anypolyhedron. In fact, any polyhedron (P) can be writtenuniquely in the form:

P = Conv(→v 1, . . . ,→v k) + Cone(→r 1, . . . ,→r l) (A-1)

where the convex hull and the conical hull are defined as:

Conv~vW1,…, vWk! = $t1vW1 + … + tkvWk|ti $ 0, t1 + … + tk = 1%Cone~rW1,…, rWl! = $t1rW1 + … + tlrWl|ti $ 0% (A−2)

Note that {→v 1, . . . ,→v k} are the vertices, and {→r 1, . . . ,→r l}are the extremal rays ofP. An alternate interpretation is thatvery polyhedron is the sum of a polytope and a cone (Fig.A3). Both vertices and rays can be found efficiently andhave significance in a variety of situations extending wellbeyond biochemical reaction networks. In linear program-ming, the vertices and rays are known as basic feasiblesolutions. They have the property that any linear function ona polyhedron achieves its maximum value at a vertex or it isunbounded and lies on an extremal ray.

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