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JOURNAL REVIEWDynamic Models of Metabolism: Review of the

Cybernetic Approach

Doraiswami Ramkrishna and Hyun-Seob SongSchool of Chemical Engineering, Purdue University, West Lafayette, IN 47907

DOI 10.1002/aic.13734Published online February 2, 2012 in Wiley Online Library (wileyonlinelibrary.com).

The cybernetic approach to metabolic modeling tracing its progress from its early beginnings to its current state withregard to its relationship to other modeling approaches, applications to bioprocess modeling, metabolic engineering, andfuture prospects are described. The framework is shown to handle large metabolic networks in making dynamic predictionsfrom limited data with looming prospects of extending to genome scale networks. VVC 2012 American Institute of Chemical

Engineers AIChE J, 58: 986–997, 2012Keywords: dynamic metabolic models, the cybernetic approach, metabolic engineering, elementary modes, cellularregulation

Introduction

Cellular metabolism involves the uptake of nutrients fol-lowed by transformation through a complex network of reac-tions catalyzed by highly specific enzymes, producingnumerous intracellular components known as metabolitesand other metabolic products that are excreted into the abi-otic phase. A distinct aspect of metabolism is the presenceof regulatory processes which endow the organisms to routechemical changes by the control of the levels and activitiesof various enzymes that catalyze metabolic transformations.Consequently, the metabolic state of an organism could be afunction of its environment as much as of its genetic back-ground.

Modeling of metabolism is of engineering interest in thatits products cover a large variety of diverse applications.Quantitative understanding of metabolic processes is essen-tial not only for the optimization and control of biologicalprocesses but also toward engineering genetic changes forimproved metabolic performance. It is the objective of thisarticle to examine the status of modeling in this area and toexpound the interrelationship of various approaches in thisregard, ratiocinating ultimately in support of the frameworkof dynamic cybernetic models. The cybernetic framework,since its inception,1 has evolved progressively transcendingvarious limitations to its current stage of development,2–4

that holds unparalleled promise for a rational, quantitativedescription of metabolism. These developments will bereviewed in this article, providing a discussion of their rela-tionship to ‘‘steady state’’ methods, known as the flux bal-ance approach that has hitherto been more popular (thandynamic models) with analysis of metabolic data.

In the sequel, we first present the metabolic system as aset of chemical reactions including transport of the externalnutrients into the cell, followed by a complex network of in-ternal cellular reactions resulting in the production of variousmetabolites, and other products of metabolism. Internal orga-nization in the cell could be accommodated by partitioningthe state vector comprising the concentration levels of vari-ous chemical species into separate vectorial entities repre-senting those in different spatially distributed compartments.A theoretical framework representing continuous spatio-tem-poral distribution of the state vector within the cell is con-ceivable but difficult from a computational perspective.Thus, the compartmental model must be viewed as a ‘‘mac-roscopic’’ version obtained by averaging the continuous per-spective over each of the compartments within the cell.However, our focus, for the present, will be limited to a sin-

gle state vector, say w � w1;w2;…;wnw

h iof nw components,

whose application is most suited to prokaryotic organisms,although higher organisms have often been treated by model-ers as an approximation. We deem the vector w to comprisethe concentrations of all external species including nutrientsand excreted metabolic products, intracellular variablesreferred to the biomass that include, the various enzymesthat catalyze the different reactions. The inclusion of enzymelevels in the state vector is most fundamental to the cyber-netic models that have made them distinctive. The reactionsconstituting metabolism may be represented by

Xnwj¼1

aijWj ¼ 0; i ¼ 1; 2;…; nr (1)

where Wj denotes the species whose concentration is wj. Eq. 1represents nr reactions of metabolism which subsume bothtransport and chemical reaction with aij the stoichiometriccoefficient which, as per the usual convention, is positive,

Correspondence concerning this article should be addressed to D. Ramkrishna [email protected].

VVC 2012 American Institute of Chemical Engineers

986 AIChE JournalApril 2012 Vol. 58, No. 4

BIOENGINEERING, FOOD, AND NATURAL PRODUCTS

negative, or zero, in respective accordance with Wj being aproduct, reactant, or nonparticipant in the ith reaction. The rateassociated with the ith reaction is denoted by ri. As eachreaction is catalyzed by a specific enzyme, the level of thisenzyme and its activity would determine the rate of thereaction besides the concentrations (as dictated by the kinetics)of species participating in the reaction.

The Nature of Regulatory Processes

Metabolism in microorganisms is subject to regulation bywhich the levels of various enzymes and their activities aresubject to control. The well-known diauxic growth of bacte-ria in mixtures of sugars, discovered by Monod,5 provides aclassic example of regulation in which enzymes are preferen-tially synthesized for the utilization of one of the sugars.Similarly, the activities of enzymes are also subject to pref-erential control. Although detailed mechanisms of specificregulatory processes are known, mechanistic informationabout the phenomenon on the whole is largely unknown.From a modeling viewpoint, however, the dynamic descrip-tion of metabolic processes must comprehensively addressregulatory phenomena. This requirement, which is enforcedon dynamic models, becomes irrelevant for flux balanceapproaches which relate intracellular fluxes through stoichio-metric coupling to external fluxes that are experimentallymeasured. Consequently, such approaches cannot be whollypredictive; neither are they suitable to describe the state ofthe organism as determined by the concentration levels of allintracellular metabolites.

The Cybernetic Approach

In the face of inadequate information about mechanisticdetails of regulatory processes, the need for their incorporationinto dynamic modeling calls for an approach that is radicallydifferent from that used for modeling physical systems. Thecybernetic modeling strategy developed by Ramkrishna andcoworkers must be regarded in this light. It builds on the ideathat biological systems derive their robustness to diverse envi-ronmental changes from being able to respond by manipulat-ing their metabolism in ways perhaps with at least as much di-versity. It is generally understood that this capacity of livingsystems is an acquisition from adaptation to eons of exposureto extreme evolutionary pressures. The implication of the fore-going observation is that regulation of metabolism must beattached to a survival goal of the organism. The term ‘‘cyber-netic,’’ used to connote this goal-seeking aspect of systembehavior, was first introduced by Wiener.6 This idea of goalseeking behavior, although a long-standing feature of teleolog-ical reasoning in biology, however, has come under criticismas being antithetical to the usual scientific explanation of phe-nomena as a natural sequence of cause and effect. Further-more, natural selection through evolution is itself not consid-ered to be goal-oriented. In this connection, Mayr7 hasobserved that such goal seeking behavior could be understoodin light of how a genetic program can develop through pro-gressive learning that formulates and enforces an optimalresponse. From this point of view, Mayr has proposed the useof the word ‘‘teleonomy’’ in place of the infamous ‘‘teleol-ogy’’ for the investigation of goal oriented systems within theframework of rational science.

The speculative nature of the actual survival goal has of-ten led many researchers to be critical about the cyberneticmethodology. This criticism ignores the value of inductive

reasoning in science that leads to understanding of systemsthrough successive formulation of hypotheses and revisionby comparison with observation. It would appear that thestudy of biological systems should include cyberneticmechanisms that can provide bases for mathematical mod-els and suitable experiments for their verification. Indeedsuch practice already has come into being in quantitativeinvestigation of biological phenomena. For example, theflux balance analysis (FBA) has relied on the use of linearprogramming based on the postulate that the organism max-imizes the biomass yield.8,9 Although this approach effec-tively resolves the mismatch between the large number ofvariables (fluxes) and the considerably less number of(steady state mass balance) equations, there has been noarticulation by the practitioners of FBA on the significanceof the assumption of maximizing the biomass yield.10 Fur-ther, although the use of an optimization principle such asmaximizing the yield of biomass would seem to imply acybernetic basis, its qualification for the same is renderedsuspect as the strategy contains no element of timeliness,an attribute that would appear to be essential for survival ina dynamically varying environment. Consequently, theobjective functions in our cybernetic models would stressmaximization of the rates of one aspect of metabolic per-formance or another. We deem this point of departure ofcybernetic models from FBA to be significant not onlyfrom the point of view of applications in various ways butalso in the elegance with which model predictions relate tothose that emerge from FBA.

The means for manipulating metabolism lies in control ofthe syntheses and activities of the enzymes that catalyze thereactions in metabolism. The reactions are thus competitorsfor a limited amount of resources essential for the synthesisof enzymes. We define two vectors u � u1; u2;…; unr½ � andv � v1; v2;…; vnr½ � that we refer to as cybernetic variables.The vector u is associated with the fractional allocations ofresource for enzyme synthesis for the different reactions, sothat

Pnri¼1 ui ¼ 1. The vector v represents the activities of

the different enzymes. Thus, we must have 0 � ui, vi � 1, i¼ 1,2,…,nr. The v-variables are not required to sum to 1 asthey are not viewed as based on allocation of a commonresource. Although the possible candidates for this resourcehave been discussed in various papers,11,12 it has not becomenecessary to identify them precisely. The cybernetic varia-bles in u would be required for enzyme balances. In otherwords, if the maximum enzyme synthesis rate is rEi

forenzyme Ei, the regulated rate is uirEi

. The cybernetic varia-bles in v are required to calculate the regulated reactionrates. Thus, if ri is the rate of the ith reaction in (1) whenthe enzyme Ei is fully active, then the regulated rate of theith reaction is viri. As the reaction rates are determined bythe specification of w, u, v, we have, for the description ofsystem dynamics, the differential equation

dw

dt¼ fðw;u; vÞ (2)

We now posit that the goal of the organism is to maxi-mize (or minimize) at any instant t some functional relatedto its survival (or threat to existence) over the time interval(t, t þ s) that can be defined in terms of the state vector w.Denoting this functional by DJ � J tþ sð Þ � J tð Þ, the cyber-netic goal of the organism is given by

AIChE Journal April 2012 Vol. 58, No. 4 Published on behalf of the AIChE DOI 10.1002/aic 987

ramkrish

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ramkrish

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MaxuðtÞ;vðtÞ

DJ

such thatXnri¼1

ui ¼ 1; 0 � ui; vi � 1(3)

Notice in particular that the cybernetic variables of inter-est are only at time t although the ‘‘planning horizon’’ is thefinite interval (t, t þ s). When the solution to Eq. 3 has beenobtained to identify the cybernetic variables u and v, Eq. 2is a fully defined model except for the knowledge of kineticparameters contained in the various reaction rates. The evo-lution of cybernetic models in the last three decades hasoccurred along the following directions.

a) Formulation of objective functions.b) Solution of the optimal control problem.c) Metabolic scale, referring to the level of pathway

details.We discuss each of these directions in the following sec-

tions.

Cybernetic Variables and Their Evaluation

The formulation and solution of the optimal control prob-lem was first considered by Dhurjati et al.13 Their concernwas the growth of bacteria in a mixture of substitutable sub-strates such as glucose and xylose. Substitutable substratessatisfy the same nutritional needs of an organism.14 Nostructure was attributed to biomass other than the keyenzymes required to metabolize the two substitutable sub-strates, denoted by S1 and S2. Growth on each of the sub-strates could then be represented by

Si þ B �!Ei1þ YiÞBþ � � � ; i ¼ 1; 2ð (4)

Equation 4 is of course a gross oversimplification of themetabolism of the organism on either substrate. Thus, thestate vector w in the work of Dhurjati et al.13 was given byw � s1; s2; c; e1; e2½ �. The semicolon in brackets is used toseparate the intracellular variables (which include only theenzymes for the present case) from the extracellular nutrientsand the biomass concentration. The intracellular variablesare described as mass fractions of biomass. The objectivefunction was the maximization of the biomass produced inthe time interval (t, t þ s). Thus, J(t) ¼ c(t) and the maximi-zation was sought of DJ ¼ Dc.

In view of the linear dependence of the reaction rate on theenzyme levels, the function f(w, u, v) has a linear dependenceon the cybernetic variables making the optimality problem asingular control problem.13 The solution of the singular controlproblem was accomplished by these authors to predict diauxicgrowth in a batch culture of bacteria with utilization of thefaster growth supporting substrate, S1 (glucose) first and theslower growth supporting substrate next. The computationaloverhead for solution of the singular control problem, however,appeared unsuitable for more general applications. There werevarious other conceptual issues1 in the foregoing that foundfavor with a different approach as recounted below.

The matching and proportional laws

If the planning horizon is reduced to only the instant t, apolicy that Young2 referred to as ‘‘greedy,’’ then the goal

here becomes lims!0

Dcs � biomass production rate at t. Thus, the

maximization in this case became one of allocating resourcesfor enzyme syntheses for S1 and S2 such that the combinedgrowth rate on the two substrates is maximized. The maximi-zation is performed under the assumption that progressiveinvestment on any substrate leads to diminishing returns asmeasured by the growth rate. This problem has been analyzedby Kompala et al.11 to arrive at the matching law result

ui ¼ riP2j¼1

rj

; i ¼ 1; 2 (5)

Expression (5) is a remarkably simple heuristic to describethe regulatory process of catabolite repression. The controlof enzyme activity was determined by Kompala et al.11 tobe given by the proportional law

vi ¼ rimax r1; r2ð Þ ; i ¼ 1; 2 (6)

which was derived based on providing the maximum activityfor the enzyme supporting the fastest reaction while activatingthe other enzymes proportionally to the reaction rates fromthem. As the reaction rates depend on the levels of enzymesthat catalyze reactions, dynamic enzyme balances are calledfor. The enzyme balance is written in terms of the massfraction of enzymes in cell mass as follows.

deidt

¼ ai þ uirEi� bi þ rGð Þei (7)

The foregoing balance accounts for a constitutive rate ai,an inductive rate rEi

which is regulated, a degeneration ratebi, and a dilution term due to growth rate rG.

15 The differen-tial equations for biomass and substrates are readily writtenas in Kompala et al.11

Diauxic growth

Kompala et al.11 have presented the successful applicationof the cybernetic laws for diauxic growth in a batch reactoron several substrate pairs based on parameters obtained fromexperiments on each of the substrates singly. The differentialequations for batch growth are readily identified and may befound in the above publication. Some fine tuning of the sin-gle substrate parameters improves the model simulations formixed substrate growth while not noticeably affecting singlesubstrate behavior. Their simulations alongside experimentaldata for glucose and xylose are shown in Figure 1.

Growth in low carbon substrate environments

The growth of microorganisms in a medium of very low(carbon) substrate concentration, however, presents a differ-ent regulatory scenario even with single substrates than thatpredicted by the model of Kompala et al.11 Turner et al.16,17

discuss the issue of maintenance processes and their model-ing. They postulated the appearance of nongrowth associatedmaintenance processes for growth rates below their maxi-mum value. They further proposed that the primary objectivewas the maximization of the growth rate while the secondaryobjective was maximizing the total substrate uptake forgrowth and maintenance. The resulting model is shown todescribe the behavior of slowly growing batch cultures ofKlebsiella oxytoca perturbed by the addition of pulses of

988 DOI 10.1002/aic Published on behalf of the AIChE April 2012 Vol. 58, No. 4 AIChE Journal

glucose. The behavior in mixed cultures was investigated byTurner et al.16 Retaining the same enzymes for growth andmaintenance with each substrate, and the same objective func-tions, they were able to accurately describe growth on glucoseperturbed by pulse additions of arabinose, fructose, and xylose.However, Baloo and Ramkrishna18,19 noted that transientbehavior in the form of overshoots and undershoots of biomassin continuous cultures following step-ups and step-downs indilution rate were not described adequately by the models ofTurner et al.16 At severely lower levels of substrate concentra-tions encountered in continuous cultures, Baloo and Ramk-rishna envisaged growth and maintenance processes to be incompetition for cellular resources. In other words, the survivalstrategy in very lean environments was viewed to favor main-tenance of viability in preference to engage in active growth.The models were able to describe complex transient behaviorfound in experiments with step-ups and step-downs in dilutionrates. An example is shown in Figure 2.

Simultaneous uptake patterns

Narang et al.20 performed experiments with Escherichiacoli on substrate mixtures of glucose and organic acids suchas fumarate and pyruvate and found that uptake of substratescould also take place simultaneously. With glucose pyruvatemixtures, the uptake pattern depended on preculturing; se-quential utilization (with glucose first) when precultured onglucose and simultaneous utilization when precultured on py-ruvate. The cybernetic model of Ramakrishna et al.,21 whichwas able to describe these uptake patterns successfully, wasbuilt by expanding the gross pathway (4) to include an ana-bolic component involving synthesis of biomass from twodifferent kinds of growth precursors produced from thebreakdown of the two substrates. Thus

Si þ B �!EiMi; i ¼ 1; 2

M1 ÐE3

E4

M2

M1 þM2 ! B

(8)

The above model derives its success from predicting si-multaneous consumption when the precursor pools are filled

more expeditiously by consuming the two substrates thanwhen consuming alone. Conversely, diauxic growth on aspecific substrate occurs when that substrate is more effec-tive on filling the precursor pools. The dependence on theuptake pattern on preculturing is represented by initialenzyme levels that can tip the balance in favor of one or theother. In regard to the foregoing model, it is well to recall aslight correction in the growth rate by Namjoshi and Ramk-rishna,22 which helps to ensure mass conservation.

The enhanced capacity of the model by Ramakrishna et al.21

is a consequence of additional details on the growth process.

Growth on complementary substrates

Alexander and Ramkrishna23 developed a cybernetic modelfor the production of siderophores (iron-chelating agents)associated with iron-limiting conditions. Their data showedthat, while final biomass levels were governed by exhaustionof the carbon source, iron limitations also contributed todetermination of the biomass yield. They developed a rela-tively detailed structured model including an iron-limitedenergy resource production, which was able to quantitativelyaccount for this apparent dual-substrate limitation over a widerange of batch and continuous operating conditions. The ex-perimental data featured large variations in iron uptake overthe wide range of operating conditions and iron levels investi-gated. The model, which included a low and high (sidero-phore-mediated) affinity iron transport, and siderophore pro-duction, however, closely simulated results that were in goodquantitative agreement with the siderophore, medium and celliron levels, in both batch and steady-state continuous culturesfor a wide range of operating conditions. Alexander’s mod-els24 involved more pathway details than those of his prede-cessors. However, a more systematic attempt to look at meta-bolic pathways in terms of basic units was due to Straight.25

Pathway Considerations

Straight,25 in his doctoral dissertation, introduced interest-ing ideas in the extension of cybernetic models to moredetailed consideration of pathway structure. This exten-sion26,27 recognized that metabolic networks contain basicunits such as linear segments, converging and divergingunits, and cyclic pathways. The matching and proportionallaws of Kompala et al.11 were extended to local objectives

Figure 2. Simulations of Baloo and Ramkrishna19 forcontinuous culture with mixed feed of glucoseand xylose, showing overshoot of biomasstransients during shift-down of dilution rate.

Reprinted with permission from John Wiley and Sons.

Figure 1. Data of Kompala et al.11 on diauxic growthand model simulations: transient concentra-tions of biomass and substrates.

Cell growth profile and data reprinted with permission

from John Wiley and Sons.


for each of the units. For example, a diverging unit leadingto multiple products could use a local objective of maximiz-ing either the products’ sum or product, the latter yieldingmore aggressive control than the former. Similar strategieswere used for cyclic pathways. The common underlying reg-ulatory feature was the optimal distribution of resources witha fixed allotment for each unit toward accomplishing thelocal goal. Straight and Ramkrishna26 applied these ideassuccessfully to bacterial growth on carbon and nitrogen sub-strates under dual limiting conditions. To describe growth oncomplementary substrates under interactive and noninterac-tive situations, they introduced the concept of multiple bio-synthetic intermediates all of which were necessary for bio-mass synthesis but some of which could arise from either ofthe complementary substrates. Their figure is reproducedalongside simply to show the type of network structure usedto produce interactive behavior (Figure 3). The diverse pre-dictive potential of such a minimal model can be seen fromStraight and Ramkrishna.26 They also investigated thegrowth of E. coli in continuous cultures fed with carbon andnitrogen substrates under various limiting conditions such as(i) carbon-limited, (ii) nitrogen-limited, and (iii) dual carbonand nitrogen-limited.27 The simplified network considered bythem is presented in Figure 4. The network can be seen tofeature the basic metabolic units considered by these authorsfor cybernetic formulations by assigning local objectives asmentioned earlier. Figure 5 presents their steady state pro-files under carbon limitation on the left and under nitrogenlimitation on the right. The biomass trends are remarkablydifferent at lower dilution rates in the two cases because ofoverflow metabolism and accumulation of storage com-pounds. Similarly, data are compared of steady state yieldsof biomass as a function of dilution rate under different lim-iting conditions. In addition, the studies also included tran-sient effects following shift-ups and shift-downs of dilutionrate under different limiting conditions. Although, for a fulldiscussion of the quality of performance of the models, thereader is referred to Straight and Ramkrishna,27 their modelhas had considerable success in describing data notablydiverse in nature and difficult to describe with kinetic mod-els devoid of regulatory features. Indeed we are evenunaware of kinetic models that have addressed such diversetransient patterns.

The concepts developed in Straight’s work25 were remark-able in extending the success of cybernetic ideas to describ-

ing complex interactive behavior in environments of comple-mentary substrates with relatively simple networks. How-ever, the recognition of basic metabolic units toward themission of modeling larger metabolic networks lacked anactual manner of synthesis. The underlying conundrum wasto conjure a collection of smaller subnetworks with an allo-cation strategy for resources toward synthesis of enzymesthat would support a global objective of the organism. Fur-ther, the attractiveness of the idea of distributing a resourceamong many metabolic subunits was confounded by thecombinatorial complexity of their choice, and the hazard ofthrottling fluxes through indiscriminate application of thelocal objectives for subunits.

Varner and Ramkrishna28,29 addressed detailed metabolicnetworks in which transcriptional details were included forthe synthesis of enzymes. They introduced cybernetic varia-bles to regulate the synthesis of m-RNA’s for different pro-teins. Subsequently, Varner30 presented a model of centralcarbon metabolism of E coli, describing batch aerobicgrowth on glucose, in which transcription, translation, andactivity of the gene products of 45 genes were included. Themodel featured 122 species including metabolites, enzymes,mRNA pools, and biomass with 46 reactions. An attractivefeature of Varner’s model was in its capacity to evaluate theeffect of various gene knockouts. For example, his modelaccurately captured the metabolic reprogramming resultsfrom the deletion of pyruvate kinase, a fact that went unno-ticed in the subsequent work of Segre et al.31 Varner’sexpansion of the cybernetic framework with the potential toincorporate proteomic and microarray data could be impor-tant in the future.

Dynamic Models with Local and GlobalCybernetic Variables

Elementary Modes

We will now turn our attention to the new optimality con-siderations that Young2 (see also Young and Ramkrishna4)introduced. Young revisited the optimality problem (Eqs. 2and 3) with the proposition that the planning horizon for theorganization at any instant is a time interval s is small but

Figure 3. Simplified metabolic network of Straight andRamkrishna26 accommodating both interac-tive and noninteractive behavior of comple-mentary substrates.


Figure 4. Simplified pathway of Straight and Ramk-rishna27 growth of E. coli under carbon andnitrogen limitations.



distinct from zero. The smallness of s permits linearizationof the function f(w, u,v) with respect to w, while introducinga quadratic penalty function on the control (cybernetic) vari-ables to account for diminishing returns from each invest-ment. This perspective was all that was required for optimalcontrol theory to yield the most elegant solution to representthis framework. The cybernetic variables could be deter-mined analytically for arbitrary objective functions. A dis-tinctive quality of Young’s development2 is that the optimal-ity perspective at any instant over a finite (albeit short) timeinterval lent an anticipatory aspect to the cell’s planning pro-cess without it being unreasonably futuristic. Young appro-priately termed this policy as temperate as against the greedimplied in the matching law development of Kompalaet al.11 that was based on a zero planning horizon. As aresult, the framework based on Young’s work will display asense of robustness that has for long been held to be a prop-erty of biological systems without a concomitant effort toincorporate it in models.

With the optimality framework firmly secured through theapplication of mathematical control theory, Young’s doctoraldissertation2 also addressed the problem of developing aview of the metabolic network for identification of appropri-ate objective functions. In accomplishing this, the decompo-sition of a metabolic network into so-called elementarymodes, which represent metabolic pathway options for theorganism, became a key aspect of the development. Thisdecomposition has its roots in the steady state view of me-tabolism in which all intracellular metabolites (with someexceptions such as those involved in overflow metabolismthat accumulate within the cell that have slower dynamics)are at steady state with respect to the uptake of nutrients bythe cell. We will indulge in some detail on introducing ele-mentary modes in this review because it provides us with aneffective way of comparing our dynamic cybernetic modelsto the flux balance approach. The steady state mass balancefor the metabolites, neglecting the dilution term due togrowth, leads to the homogeneous equation

Smr ¼ 0; r � 0 (9)

where Sm is the stoichiometric matrix associated withintracellular participants in all metabolic processes and r isthe metabolic flux vector. Eq. 9 reveals the metabolic fluxvector to be in the null space of Sm; however, r, in view of its

components being intrinsic reaction rates that are non-negative, must lie within and on the boundary of a ‘‘fluxcone,’’ a convex set in the non-negative orthant of <nr . Thisflux cone clearly has its vertex at the origin. Each solutionvector espouses reactions that form a sub-pathway of theoverall network. The cone edges are special solutions that arein fact elementary modes mentioned earlier (it is customary inthe metabolic literature to distinguish between elementarymodes, extreme pathways, and generating modes, the distinc-tion among which comes about when there are reversiblereactions. This set of elementary modes defines a convex basisset of solution vectors in the sense that every solution of Eq. 9is a convex combination of these vectors. For a full discussionof elementary modes, the reader is referred to Schusteret al.,32,33 Klamt and Stelling,34 and Wagner and Urbanczik.35

See also Trinh et al.36 for the review of recent developmentand applications of elementary modes). It is also convenient torepresent the potential metabolic states in a yield vector spacein which the yields of all extracellular products arerepresented.37 The flux cone transforms to a convex hull inthis space. The edges of the flux cone collapse into the verticesof the foregoing convex hull. The steady state metabolicperformance will clearly be a point in this hull obtained by aconvex combination of its vertices. Each elementary mode isviewed as a metabolic option of the organism for substrateuptake. All fluxes in the mode are calculable from aspecification of this uptake rate. Young’s cybernetic perspec-tive4 proposed that the organism invests its total resourcesavailable for enzyme synthesis among elementary modeschosen to realize a global objective of the organism such asmaximizing biomass or uptake of carbon during the timeinterval t to t þ s (in other words the average rate ofmetabolism during this interval). Thus, elementary modes areviewed to compete for resources to drive metabolismeffectively to meet the foregoing global objective. Individualreactions in each mode, however, compete for resourcesallocated to that mode through the realization of a localobjective, which is to maximize the harmonic mean of all thefluxes in that mode. No reaction in the mode has therefore thepotential to thwart the flux through that mode because ofdeficiency in the amount or activity of the enzyme catalyzingthe reaction in question, a feature that also further contributesto the property of robustness. The joint realization of theglobal and local objectives of the organism will thereforeensure the survival effort of the organism. The incorporation

Figure 5. Model simulations (continuous, dashed and dotted lines) of Straight and Ramkrishna27 alongside dataobtained for E. coli in continuous cultures with glucose and ammonia substrates.

The figure to the left is under carbon-limiting conditions while that to the right is under nitrogen-limiting conditions. Reprinted

with permission from John Wiley and Sons.


of elementary modes by Young2 resolved the basic difficultyalluded to earlier associated with finding subunits and anappropriate resource allocation strategy that would allow acoordinated flow of fluxes.

The application of Young’s approach to E. coli is presentedby Young et al.3 They show that parameters can be identifiedfor multiple strains of the organism with identical values foroverlapping parameters so that their metabolic performancecan be evaluated which includes the rates of growth and sub-strate uptake. This predictive element of the cybernetic modelis one that distinguishes it from other approaches to modelingmetabolism. As no quasi-steady state assumption is involvedfor intracellular variables in the model of Young et al.,3 meta-bolic transients will appear as a curve in yield space terminat-ing at a point when steady state is attained for all intracellularvariables. We note that this steady state point could be an in-terior point of the convex hull while, FBA, in view of its lin-ear programming strategy to maximize the biomass yield,would require it to be at some vertex.38 This would be thecase even with the dynamic version of FBA for describingmetabolic transients. We will again return to this issue in ourdiscussion of the so-called hybrid cybernetic models (HCM).

Young’s models obviously retain the property of their pre-cursors to represent regulatory phenomena without the aid ofparameters in addition to those that must of course be usedfor reaction kinetics. However, the parameter identificationissue for the foregoing models is a formidable one especiallyfor large metabolic networks as it would require the fullblown power of metabolomic measurements with accuracy.That identification can, however, be accomplished for rela-tively smaller networks with dynamic measurements ofextracellular variables as shown by Young et al.3 and repro-duced in Figure 6 here.

HCM

The viewpoint of elementary modes being metabolicoptions for an organism leads naturally to an attractivemodel framework, which, in its application to larger meta-bolic networks, could retain the cybernetic features intro-duced earlier by Kompala et al.11 This framework which weshall refer to as the HCM accomplishes an interesting syn-thesis of the early cybernetic approach with the quasi-steadystate assumption for intracellular variables used in FBA. Anexposition of this approach can be found in Kim et al.39 Thequasi-steady state assumption relegates consideration of reg-ulatory processes associated with the reactions in each ele-mentary mode to that of substrate uptake through that mode.In other words, a single enzyme is assigned to each modethe control of whose synthesis and activity will represent theregulation of all reactions in the elementary mode. Thus,regulation is viewed to distribute resources among differentmodes so that a global objective of the organism is satisfied.Note in particular that, in realizing the above objective,which involves the maximization of a metabolic rate, a con-vex combination of many modes is involved. It is well toreflect at this stage the point of departure of HCM from thatof FBA, as the latter, in focusing on metabolic yield, seeksout a single (except when the linear programming has multi-ple solutions) mode (vertex in the convex hull) as the operat-ing metabolic state in yield space. The differences betweenthe two approaches are, however, deeper. FBA relegates reg-ulatory effects to the measurement of uptake rates so thatthe consequence of regulation falls outside the scope of pre-

diction. Attempts to correct for this through the developmentof kinetic expressions for uptake rates would in fact elimi-nate the input of regulation through experiment.

The intracellular fluxes in each mode are determined oncethe uptake rate through the mode is known. The onlydynamic variables of the HCM are the concentrations ofextracellular variables such as substrates, fermentation prod-ucts, and cell mass. The model parameters are the kineticconstants associated with uptake of substrate through eachmode, which must be determined by comparing the dynamicpredictions of extracellular variables with suitably spacedmeasurements. As large metabolic networks lead to a largenumber of elementary modes, the consequent imbalancebetween the number of parameters and the number of avail-able measurements could greatly complicate the identifica-tion of such models. To contain such an imbalance, itbecomes necessary to explore a process of mode reductionand/or expand the measurements of extracellular variables toinclude as many of the intracellular fluxes as possiblethrough, for example, C13 labeling.40–42

Song and Ramkrishna37 have developed strategies formode reduction (see also Provost et al.43 for otherapproaches) based on what they refer to as metabolic yieldanalysis, which involves exploring metabolic data on theyield vector space. As observed earlier, the modes appear asvertices of a convex hull in yield vector space and the point,that represents the metabolic function in terms of the yieldsof all extracellular products, is a convex combination of allthe vertices (Figure 7). It is now possible to select a minimalset of modes whose convex combination can yield the givenpoint. If, it turns out that the point lies outside the convexhull either due to experimental measurement errors or per-haps due to missing modes, a minimal set of the existingones is selected based on their proximity to the point inquestion. In fact Song and Ramkrishna37 have a well laid

Figure 6. Comparison of Young’s model3 simulationswith data on two different strains of E. coli.



out strategy for substantial mode reduction that has beenapplied in various cases.44,45

The mode reduction strategy is inspired by the belief thatan organism’s explosive repertoire of elementary modes is aconsequence of evolutionary experience not all of whichmay be essential to respond to a controlled laboratory orindustrial reactor environment. Thus, highly reduced lowerorder models could be envisaged for control applications tobioprocesses. Conversely, an exuberant effort to eliminatemodes runs the risk of limiting metabolic engineering direc-tives that could accrue from models. Thus, some balance iscalled for in condensing the number of modes. The reader isreferred to Song and Ramkrishna37 for a more detailedaccount of the mode reduction process.

The hybrid model equations are given by differentialequations for biomass, substrates, fermentation products, andenzyme levels for uptake of substrates together with steadystate balances for intracellular variables. The intracellularfluxes relate to the uptake rates through all the elementarymodes. The growth rate is expressed as a known linear com-bination of intracellular fluxes. Kim et al.39 produced tworeduced order models of E coli, one with nine elementary

modes and the other containing 20. Their model simulationsare shown along with those of FBA and Young’s model forcomparison with experimental data obtained by Kim et al.39

(Figure 8). Of particular significance is the fact that the for-mation of succinate in experiments is upheld by both HCMand Young’s model while linear programming, which maxi-mizes the biomass yield in FBA, selects out a mode that pro-duces no succinate.

Song et al.44 have successfully applied HCM to the co-utili-zation of glucose and xylose by a recombinant strain of Sac-charomyces. Figure 9 shows the HCM simulations (continu-ous lines) alongside data obtained by Krishnan. Dashed linesare simulations of what has been termed as macroscopic bio-reaction model (MBM) due to Provost et al. The MBM servesas a useful comparison of regulatory effects considered byHCM in the optimal distribution of substrate uptake amongthe different elementary modes of metabolism as against afixed distribution in MBM. Harsher scenarios can be con-ceived for comparing the two models, however. In particular,it will be of interest to examine how HCM can predict eventsin a chemostat, an issue to be addressed in a subsequent sec-tion (see Nonlinear Behavior of Cybernetic Models).

As pointed out before, the focus of FBA on maximizingbiomass yield results in a daring selection of a single ele-mentary mode to represent metabolism under some stipu-lated growth or uptake rate. Of course variability occurs inthe mode selected with change in the metabolic rate but withthe proviso of having to specify the latter from experimentalobservation. This is a constraint that strips the methodologyof the essential need of a modeling framework to contributeto experiment. In other words, a quest for the conditionsunder which specific metabolic states can manifest is withoutdirection. We shall have more to say about this issue in asubsequent section of this article.

Lumped HCM

The size of the network which HCM can effectively han-dle would depend on the tradeoff between the number ofavailable extracellular measurements and intracellular fluxes(as by using isotopic carbon labeling) and the number of ele-mentary modes that can be accommodated. The attractive-ness of FBA lies in its ability to address the estimation of

Figure 8. Comparison of simulation of E. coli metabo-lism by Kim et al.39 with data.


[Color figure can be viewed in the online issue, which is

available at wileyonlinelibrary.com.]

Figure 9. HCM simulations of Song et al.44 for recombi-nant Saccharomyces producing bioethanolversus experimental data of Krishnan et al.46

in glucose-xylose environments.


Figure 7. Convex hull in yield vector space from Songand Ramkrishna37 showing elementarymodes as vertices.



metabolic fluxes with the minimum of data as, for example,the substrate uptake rate. It is of interest to see to whatextent cybernetic models, with their dynamic features intact,can be exploited to obtain information on metabolic per-formance of large metabolic networks with minimal data. Inthis connection, we discuss next what we have termedLumped HCM (L-HCM), which provide dynamic models forvery large networks that can be identified with limited data.Further, L-HCM can be used to interpret the cybernetic mod-els of the older generation. Song and Ramkrishna47,48 haveshown how it may be possible to recognize ‘‘families’’ of el-ementary modes each containing several elementary modeswith one or more unifying characteristics with respect tometabolic function. Thus, key enzymes are envisaged foreach of the EM families, their syntheses receiving resourceswhich are then shared among the various modes within eachfamily. This sharing is of course related to their contributionto satisfying the organism’s goal that was chosen to be themaximization of carbon uptake rate. A notable additionalfeature of L-HCM is a simplification, enabled by analysis ofthe enzyme synthesis equations, which relates the cyberneticvariables for jth EM in a given family to a quantity termed‘‘structural return on investment’’ denoted as gj. This quan-tity, which has a purely stoichiometric origin, represents the‘‘quality’’ of the EM in its contribution to the organism’sgoal. Furthermore, ‘‘tuning parameters’’ were introduced intothe formulation that would use dynamic experimental data todistill out the significant EMs in a family that were obscuredby lumping. For details, the reader is referred to the publica-tions of Song and Ramkrishna.47,48 Here, we take particularnote of a change in lumping rule between the two cited pub-lications with the later one showing improvement in predic-tive quality. Their results are included in Figure 10 belowwhich shows how L-HCM, with parameters identified bysome measured variables (represented in solid circles in Fig-ure 10), is able to predict the dynamics of other variablesshown in open circles. It follows from Figure 10 that L-HCM can make dynamic predictions with considerably less

data because of the reduction in parameters. However, thequality of prediction depends upon its sensitivity to the datachosen for model identification. Song and Ramkrishna48

have presented several more successful dynamic predictions.It is useful to reflect on the old cybernetic models such as

those of Kompala et al.11 and Baloo and Ramkrishna18,19 inlight of current developments. As they were designed onlyto deal with biomass and extracellular variables (aside fromkey enzymes), and were free of EMs, it will be convenientto refer to them as lumped cybernetic models (LCM). Figure11 shows a conceptual comparison of LCM shown in (a)and HCM in (b). Part (c) shows L-HCM with two families,one consuming S1 and the other consuming S2, while (d)shows the LCM of Baloo and Ramkrishna.18 It shouldbecome clear that HCM and L-HCM provide avenues forlinking extracellular variables to intracellular metabolism.

Nonlinear Behavior of Cybernetic Models

A discussion on nonlinear behavior is spurred by funda-mental as well as practical issues. At the fundamental level,it behooves one to inquire into how the inherent nonlinearityof the cybernetic model leads to dynamic variations in themanner in which pathway options are negotiated by the or-ganism toward satisfying its dynamic goal (e.g., maximizecarbon uptake rate). At the practical level, it is of interest todetermine the transient and steady state behavior of continu-ous bioreactors.

Although nonlinearity arises in the kinetics of reactions inmetabolism, the primary source of nonlinearity in the cyber-netic model may be attributed to the cybernetic variables. Asthe cybernetic variables are functions of the reaction rates,they also inherit the nonlinearity of the kinetic expressions.The underlying feature of nonlinear behavior of cyberneticmodels is the dynamic manipulation of pathway options bythe organism to survive in a dynamic environment. Thus,metabolic processes may change in the extent to which spe-cific reactions are commissioned by the organism as survivalmeasures. Steady state multiplicity could therefore arise

Figure 10. L-HCM predictions (continuous lines) of Song and Ramkrishna48 for dynamic fermentation data shownin open circles, based on measurements shown in solid circles.

Predictions (a) based on measurements of glucose alone leads to good predictions of biomass, formate, ethanol but not of lactate

and succinate. When measurements of glucose and lactate are both used for model identification (b), all predictions (shown in

continuous lines) are considerably improved. Reprinted with permission from John Wiley and Sons. [Color figure can be viewed

in the online issue, which is available at wileyonlinelibrary.com.]


because of the availability of multiple metabolic choices thatensure survival criteria. Oscillatory dynamic behavior mayalso come into play because the interaction of the organismwith its environment may call for dynamically periodicswitches in metabolism.

Namjoshi and Ramkrishna22 show how the cyberneticmodels of Kompala et al.11 and Ramakrishna et al.21 for bac-teria in a chemostat (continuous reactor) fed with a mixtureof carbon sugars such as glucose and xylose may lead tomultiple steady states for a range of dilution rates and feedcompositions. This multiplicity arises from the model beingable to meet a specific growth rate by metabolizing the mix-ture of glucose and xylose with differing preferences, one inwhich more glucose is used and the other in which morexylose is used. Their results are shown in Figure 12. Themechanism for multiplicity can be clearly seen from the fig-ure as changing uptake patterns with different preferencesfor the carbon sources as represented by the cybernetic vari-able u1. For example, at low concentrations of glucose in thefeed and high dilution rates the ‘‘upper’’ steady state showspreference for glucose while the ‘‘lower’’ one shows a pref-erence for xylose because of very low residual concentrationof glucose. A similar situation has been observed by Kim49

for E. coli both experimentally and using HCM. Kim’smodel displays as many as five steady states (three stableand two unstable) with different uptake patterns for certainranges of dilution rate and fraction glucose in a glucose–py-ruvate mixture as feed (unpublished work). Such operatingranges, which can only be diagnosed by the use of dynamicmodels, are difficult to discover by any form of experimentaldesign. It was this issue to which we referred earlier inregard to the importance of dynamic models.

Multiple steady states have been observed by Hu and co-workers50 and independently by Stephanopoulos and co-workers51 in mammalian cell cultures. A cybernetic modelformulated by Namjoshi et al.52 provides a close interpreta-tion of the observed multiplicity as shown in Figure 13.Jones and Kompala53 showed that oscillations observed incultures of Saccharamomyces cerevisiae could be explainedby a cybernetic model. Although oscillatory phenomena

have been known in yeast cultures, instances of such oscilla-tions in bacterial cultures are evident. Kim’s analysis49

shows that oscillations could be expected in continuous cul-tures of E. coli at very low dilution rates with glucose feed.Experimental confirmation is yet to be made of such phe-nomena. Straight and Ramkrishna54 found fairly bizarredynamic behavior in the simulated growth of K. oxytoca onlactose because of metabolic switches. It must be said, how-ever, that the extent to which nonlinear phenomena of meta-bolic systems have been investigated hardly probes the sur-face of what could well be a treasure-trove of pathologicalbehaviors. In this connection, one is tempted to speculatethat detailed determination of gene expression profiles couldserve as fine diagnostic tools for such nonlinear behavior.

Process Applications

We are concerned in this section about modeling the useof microbes for bioprocesses in which the focus is onincreasing productivity of the bioproduct by (i) varying pro-cess conditions and (ii) metabolic engineering of strainsthrough genetic manipulation. In addressing (i), the dynamicnature of cybernetic models is a crucial asset that can beused not only to optimize bioprocesses for maximum pro-ductivity but also institute continuous control.55 Controlbased on cybernetic models has the advantage of accountingfor regulatory responses of the organism to variation of itsenvironment. Song et al.56 have recently shown how bioetha-nol productivity can be notably improved by varying processconfigurations using HCM on fermentation of a mixture ofglucose and xylose.

Metabolic engineering

Metabolic engineering, initiated by Bailey,57 is an excitingarea of application of metabolic models. The special claimsof cybernetic models to being an effective tool in metabolicengineering are derived from its dynamic nature which ena-bles focus on productivity of the metabolite of interest.Steady state approaches are constrained to address theimprovement of yield often beset by a drop in productivitydue to a drop in growth rate caused by ‘‘metabolic burden.’’Cybernetic models, which have a mechanism to address met-abolic burden, therefore represent an attractive tool in this

Figure 12. Steady state multiplicity in bioreactor fedwith glucose and xylose shown as a func-tion of dilution rate D and fraction glucosein the feed c (from Namjoshi and Ramk-rishna22).

Figure 11. Comparative illustration of cybernetic mod-eling concepts (from Song and Ramk-rishna47): (a) LCM (e.g., Kompala et al.11), (b)HCM, (c) L-HCM, and (d) LCM (e.g., Balooand Ramkrishna18).


[Color figure can be viewed in the online issue, which

is available at wileyonlinelibrary.com.]


regard. Young’s models are specially suited for metabolicengineering as they address intracellular reactions withoutthe assumption of steady state. However, their applicabilityis constrained by the current limitation in the size of themetabolic network that can be used. A similar limitation innetwork size applies to HCM. Although HCM assumessteady state for intracellular metabolites, the uptake rateassociated with an EM could be varied to examine its effecton the productivity of any metabolite. Thus, the metabolicengineering quest would first focus on determining the EMsthat would most contribute to increase the productivity of themetabolite, followed by a search for intracellular reactions thatwould most effectively accomplish the required change inuptake rates of the EMs.58 The L-HCM, because of its abilityto handle large metabolic networks represents a perhaps morepromising direction for metabolic engineering. Recent esti-mates by Song and Ramkrishna (Metabol Eng., submitted)shows impressive dynamic predictions of several KO strains ofE. coli based on limited data on the wild-type strain. Consider-able further work remains in this area, however.

Future Prospects

The future of the cybernetic approach is enhanced notonly by the progress to date in handling increasingly largersized metabolic networks but also in the upcoming prospectsof being able to handle genome scale networks. FBA derivesits capability for genome scale networks because of the facil-ity for solving large scale linear programming problems but,in view of its focus on maximizing biomass yield, a greatcollection of pathway options is discarded. The L-HCMapproach of Song and Ramkrishna47,48 recruits many path-way options that could be superior to those that maximizebiomass yield because of higher uptake rates. Thus, adynamic genome scale L-HCM represents a distinct possibil-ity in the metabolic modeling horizon.

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Manuscript received July 19, 2011, revision received Nov. 19, 2011, and finalrevision received Jan. 10, 2012.


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