computer-aided design of metabolic networks

Computer-Aided Design of Metabolic Networks

Klaus Mauch, Stefan Buziol, Joachim Schmid and Matthias Reuss ∗

Institute of Biochemical EngineeringUniversity of Stuttgart, Germany

AbstractThis contribution presents results obtained from a model based design of metabolic networks. In the first part of thepaper, topological analysis is used for exploring the metabolic architecture. These investigations—also called pathwayanalysis or flux space analysis—are aimed at detecting the metabolic routes that lead from anyone starting point to someproducts. The technique is applied for the computation of maximal yields for amino acids and, for the first time, alsofor the analysis of metabolic networks in context with the formation of biomass. The latter study leads to an array ofmutants with different biomass yields, for which the name “Phenome” has been coined.

In the second part of the contribution, a strategy for the optimization of product formation rates is presented by meansof the ethanol formation rate in Saccharomyces cerevisiae. A dynamic model based on experimental observations atdefined anaerobic conditions serves as a starting point. Non-linear optimization of the distribution of enzyme activitiesresults in a substantial increase of ethanol formation rate. The optimum is mainly constrained by homeostasis and canbe characterized by higher activities of strongly rate limiting steps. However, some enzymes exerting almost no controlon ethanol flux (e.g. triose phosphate isomerase) are found at higher activities as well. This finding can be explained bythe enzyme’s ability of counteracting an increase of pool concentrations effectively.

KeywordsMetabolic networks, Flux analysis, Flux optimization

Introduction

One of the fascinating challenges in mastering biosystemsis to interpret living processes in a quantitative man-ner. As such, mathematical modeling is of central impor-tance. At a time of the ballooning amount of data gener-ated by the high throughput technology in genome, tran-scriptome, proteome and metabolome research, one ofthe important issues for modeling is to bridge the gap be-tween data and an integrated understanding of the com-plex functionality of biosystems. Moreover, there is anurgent need for new approaches to strengthen the modelbased design of biosystems. This activity is of increasingrelevance with respect to the optimization of yields, se-lectivities and productivities in industrial bioprocesses.Furthermore, mathematical modeling will lead to signif-icant insight through an integrative analysis of diseasesand support target identification for drug discovery.

While the potential and promise of biological systemsmodeling is substantial, also several obstacles are en-countered. Before benefits can be gathered from bio-logical systems analysis, issues like, for example, the ap-propriate balance between significance, complexity andavailability of quantitative experimental observationsneed to be addressed. Furthermore, in many cases insuf-ficient emphasis has been placed on fundamental ques-tions of purpose, intended application (Bailey, 1998),predictive power and relevance concerning significantcontributions to the solution of the aforementioned prob-lems.

This contribution aims at the design of biosystems

∗[email protected]

D N A - T e c h n o l o g y

Q u a n t i t a t i v eA n a l y s i s

D e s i g n B a s e do n Q u a n t i t a t i v e

P r e d i c t i o n s

S y n t h e s i s

Figure 1: Optimization of bioprocesses throughmetabolic engineering.

within industrial applications. Therefore, modeling ispart of a well known engineering cycle depicted in Fig-ure 1.

Central to the general modeling framework is its pre-dictive strength. A sound prediction, in turn, mustrest upon reliable experimental data. In the majorityof cases, implementation of the model-based suggestionsfor genetic reprogramming is performed with the aidof recombinant DNA-technology. Unfortunately, onlyfew models qualify for the design of metabolic networks.When extrapolation beyond the horizon of experimen-tal observations is required, the missing link mostly ispredictive strength. A critical assessment of the stateof the art and meaningful discussion of the present lim-its would require a comprehensive evaluation of severalfundamental issues of modeling biosystems. Only two ofthese issues will be addressed in this contribution: (1)Flux analysis based on topological properties and (2)Flux optimization based both on toplogical and kinetic

82

Computer-Aided Design of Metabolic Networks 83

P a t h w a yD a t a b a s e s

A n n o t a t e dG e n o m e s c D N A - A r r a y D a t a P r o t e o m i c s

C o n s t r a i n t s o n R e v e r s i b i l t i y

F l u x S p a c e A n a l y s i s

M a s s B a l a n c e a t S t e a d y S t a t e : N v = 0

M e t a b o l i c F l u x A n a l y s i s

F l u x D i s t r i b u t i o n

E x p e r i m e n t a l O b s e r v a t i o n s : - E x c h a n g e F l u x e s - L a b e l i n g D i s t r i b u t i o n

Figure 2: Quantification of flux distributions.

properties of metabolic networks.

Flux Analysis

According to the source of information, balance equa-tions for the metabolic system of interest can be created(see Figure 2).

When balance equations are derived from annotatedsequence data, the resulting stoichiometric matrix re-flects the metabolic capabilities of the genotype. Whiledepending on physiological conditions, a more specificphenotype is considered when data from cDNA chips orproteome analysis are used. Generally, balance equationsof structured metabolic models can be written as

d

dtn = Nv, (1)

where matrix N (m× n) contains the stoichiometric co-efficients νi,j of the n biochemical reactions. m denotesthe number of metabolites, v the vector of reaction rates,whereas the vector of metabolites is denoted by n. Notethat in Equation 1 the transport rates across the variousmembranes of the cell as well as the dilution of metabo-lites due to growth are included in the state vector of thereaction rates v. For steady state conditions Equation 1reads:

Nv = 0. (2)

Metabolic Flux Analysis

Depending on the data available and the area of appli-cation, Equation 2 can be used for the computation offlux distributions in metabolic networks. The first routedelivers solutions for flux distributions through exper-imentally determined exchange fluxes (Stephanopouloset al., 1998; Mauch et al., 2000).

For the estimation of unknown metabolic fluxes fromexperimentally determined fluxes, we rewrite Equation 2as

[Nm|Nc][vm

vc

]= 0, (3)

where vector vm consists of q experimentally determined

fluxes. The remaining (q − n) unknown fluxes are gath-ered in vector vc. Partitioning of v into the known fluxesvm and unknown fluxes vc then yields

Nmvm + Ncvc = 0, (4)

with matrices Nm (m× q) and Nc (m× n− q) corre-sponding to vm and vc, respectively. From Equation 4we obtain

Ncvc = −Nmvm, (5)

and upon multiplying Equation 5 by the transposed ofmatrix Nc, the solution for unknown fluxes Nc is ob-tained according to

vc = −(NT

c Nc

)−1NT

c Nmvm, (6)

where the superscript -1 indicates matrix inversion. Bydefining the pseudoinverse N#

c

N#c =

(NT

c Nc

)−1NT

c , (7)

Equation 6 may be rewritten as

vc = −N#c Nmvm. (8)

Mathematically, a determined metabolic system is de-fined by

dim (vm) = n− rank (N) . (9)

That is, the amount of experimentally determinedfluxes q in vm equals the degree of freedom of themetabolic network. If no conservation relations arepresent in a determined system, the pseudoinverse N#

c

coincides with the inverse of Nc, thus(NT

c Nc

)−1NT

c = (Nc)−1

. (10)

Hence, the fluxes vc of a determined system without con-servation relations may be obtained by

vc = −N−1c Nmvm. (11)

Importantly, a solution for vc only exists for a nonsin-gular matrix Nc, that is

det (Nc) 6= 0, (12)

and for a determined metabolic system with 0 ≤ z < mconservation relations:

det(NT

c Nc

)6= 0. (13)

Provided the existence of a unique solution for vc, themetabolic system is called observable. For an under-determined metabolic network we have

dim (vm) < n− rank (N) , (14)

and since a unique solution for vc cannot be obtained,we always find

det(NT

c Nc

)= 0. (15)

84 Klaus Mauch, Stefan Buziol, Joachim Schmid and Matthias Reuss

Hence, the dimension l of the solution space for an under-determined system is given by

l = n− rank (Nc)− dim (vm) , (16)

whereas an overdetermined (i.e. redundant) metabolicsystem is characterized by

dim (vm) > n− rank (N) . (17)

Similarly to a determined system, in overdeterminedsystems a unique solution for the unknown fluxes vc

can only be derived if Equation 13 holds. Note thatan overdetermined system is not necessarily observable.Frequently we meet the situation where experimental in-formation on some fluxes is redundant while part of themetabolic network still cannot be observed. The prob-lem of non-observable fluxes can often be bypassed whenexperimental data derived from tracer experiments areavailable. Those measurements of labeled substrates areeither performed with NMR (Marx et al., 1996; Szyperskiet al., 1999) or gas chromatography/mass-spectroscopy(GC-MS) (Christensen and Nielsen, 2000; Dauner andSauer, 2000). When metabolite balance equations areextended with balance equations for the metabolite la-beling distributions, a system of non-linear equations hasto be solved to estimate the associated steady state fluxdistributions (Wiechert and de Graaf, 1997; Wiechertet al., 1997; Schmidt et al., 1997, 1999).

Flux Space Analysis

Flux space analysis (see Figure 2) summarizes strategiesfor topological analysis leading to meaningful informa-tion on the flux space in metabolic networks. This is anarea of increasing importance in integrative functionalgenomics aimed at a better understanding of the com-plex relation between genotype and phenotype (Schus-ter et al., 1999; Schilling et al., 1999, 2000; Schusteret al., 2000a; Edwards and Palsson, 1999; Palsson, 2000).This area—in terms of the “omic” revolution sometimesnamed “phenomics”—becomes urgent for developing en-gineering methods to deal with the massive amounts ofgenetic and expression data in an integrative and holisticway.

Flux space analysis (sometimes also called pathwayanalysis) aims at detecting the metabolic routes thatlead from anyone starting point to the products of inter-est steady state condtitions. The two strategies appliedfor this analysis are the null space and the convex cone.While linear algebra is used to determine the null spaceof the homogeneous system of linear equations, meth-ods of convex analysis (Vanderbei, 1998) are applied tocompute the convex cone.

The null space of matrix N is the subspace spannedby k = rank (N)− dim (v) linearly independent vectorsv satisfying Equation 2. Accordingly, every linear in-dependent base vector forms a column of the null space

matrix K (n× k), thus

NK = 0. (18)

Within the null space lie all of the flux distributionsunder which the system can operate at steady state.Thus, the null space allows to describe any flux distri-bution of a genotype (by superposing it’s base vectors).The vectors spanning the nullspace are, however, non-unique solutions of Equation 2. Moreover, the base vec-tors of the nullspace do not necessarily fulfill reversibilitycriteria of individual reactions.

A different approach rests upon convex analysis andleads to unique sets of vectors spanning the space of ad-missible fluxes, for which names like, for example, ele-mentary flux modes (Heinrich and Schuster, 1996) or ex-treme pathways (Schilling et al., 2000) have been coinedfor. In contrast to the base vectors of the nullspace, fluxvectors obtained by convex analysis always obey sign re-strictions of practically irreversible reaction steps. Theinvestigations presented in the following are based on theconcepts of the elementary flux modes.

Elementary Flux Modes

Elementary flux modes are non-decomposable flux dis-tributions admissible in steady state, including reactioncycles (Heinrich and Schuster, 1996). Normally, elemen-tary flux modes comprise a set of non-zero fluxes and aset of zero fluxes, with the latter pointing to enzymeswhich are not used to implement a specific function.An example for an elementary mode which frequentlyoccurs in cellular systems is the complete oxidation ofsubstrate in the respiratory chain. To sustain respi-ration, enzymes catalyzing anabolic reactions obviouslybecome dispensable. Another example might be an ele-mentary flux mode leading to the formation of an aminoacid where, again, larger parts of the respiratory chainare nonessential. Thus, by computing elementary fluxmodes, the metabolic capacity of a given metabolic net-work is unitized. In other words, the phenotype of acertain genotype may be characterized by the completeset of elementary flux modes.

The motivation for the study of elementary flux modesarises from various potential applications. In biotech-nology, an important objective is to increase the yield ofbiosynthetic processes where a desired product can oftenbe synthesized by various different routes. It is then ofinterest to detect and subsequently implement the routeon which the product/substrate ratio is maximum. Gen-erally, a flux pattern that uses only the optimal routecannot be obtained in practice. Nevertheless, it is help-ful to compute the upper limits for the molar yield froma given network topology.

It has turned out that the problem of maximizing theyield of a biotransformation can be solved by detectingall elementary modes in the system and choosing themode giving the best yield (Schuster et al., 1999). Al-


K

e 1

e 2

e 3

J

Figure 3: Convex polyhedral cone K spanned bythree generating vectors e1 to e3. Flux distributionJ results from a linear combination of the generatingvectors ei.

ternatively, such optimization problems were tackled bylinear programming (Fell and Small, 1986; Savinell andPalsson, 1992; van Gulik and Heijnen, 1995).

In many cases, the net direction of a reaction is known.Therefore, we decompose the flux vector into two subvec-tors, virr and vrev, corresponding to what will be calledthe irreversible and reversible reactions, respectively. Sowe have

virr ≥ 0. (19)

Equation 2 in conjunction with inequality(19) deter-mines what is called a convex polyhedral cone. The edgesof the convex cone are established by the elementary fluxmodes, and all the points on the interior of the cone canbe represented as positive combinations of these funda-mental pathways (see Figure 3). From there, the convexcone enfolds all potential stationary flux distributions ofa metabolic system.

Pursuing the goal to find basic pathways in biochemi-cal reaction networks, Schuster and Schuster (1993) haveearlier developed a method for detecting the simplest fluxvectors v fulfilling relations (2) and (19) with all reac-tions assumed to be irreversible. Generalizing the ap-proach in that both reversible and irreversible reactionsare allowed for, this has led to the concept of elementaryflux modes (Schuster and Hilgetag, 1994).

Elementary flux modes are defined as follows (Heinrichand Schuster, 1996, cf.): An elementary flux mode, M,is defined as the set

M = v ∈ Rn|v = λv∗, λ > 0 , (20)

where v∗ is a n- dimensional vector (unequal to the nullvector) fulfilling the following two conditions.

(a) Steady-state condition. Nv∗ = 0.

(b) Sign restriction.

If the system involves irreversible reactions, then thecorresponding subvector virr of v∗ fulfils inequality (19).

For any couple of vectors and (unequal to the null vector)with the following properties:

• v′and v

′′obey restrictions (a) and (b),

• both v′

and v′′

contain zero elements wherever v∗

does, and they include at least one additional zerocomponent each, v∗ is not a nonnegative linear com-bination of v

′and v

′′,v∗ 6= λ1v

′+λ2v

′′, λ1, λ2 > 0.

The last condition formalizes the concept of genetic in-dependence introduced by Seressiotis and Bailey (1988).The condition says that a decomposition into two othermodes should not involve additional enzymes.

Elementary modes have been determined in a numberof biochemical networks, such as the synthesis of pre-cursors of aromatic amino acids (Liao et al., 1996), thetricarboxylic acid cycle and adjacent pathways (Schusteret al., 1999) and glycolysis and alternative pathways inbacteria (Dandekar et al., 1999).

While the promise is substantial, the value of the topo-logical analysis will not be fully utilized until algorithmsare developed capable of tackling large metabolic, sig-naling or gene networks efficiently. What are the limitsof the known algorithms in the calculation of elementaryflux modes when applied to large systems?

Most of the algorithms applied to biological systemsso far are bottom up approaches (Schuster et al., 2000b;Schilling et al., 2000). Initially, the stoichiometric matrixis augmented with the identity matrix. Next, a consec-utive computation of matrices through combination ofrows is performed until the stoichiometric matrix onlycontains zero elements. Obviously, this is a consecutiveapproach where all elementary modes are created at thefinal step. In addition, a large number of interim solu-tions are computed which finally disappear. By conse-quence, since even modest network sizes showing a largerdegree of freedom might feature thousands of elementaryflux modes, the problem easily becomes computationallyintractable. Nevertheless, (Schilling and Palsson, 2000)have recently tackled the problem of prediction of theso called extreme pathways for Haemophilus influenza,represented by a network of 461 reactions. The strategyapplied rests upon a decomposition of the network intosubsystems to get a picture of the structural informationfor the entire system.

Alternative approaches make use of top down strate-gies considering the entire system in a more direct way(Happel and Sellers, 1989). As a result, successive so-lutions containing the complete network information arecreated. Mauch (to be published) developed an algo-rithm performing combinations of the base vectors of thenull space in pairs. Figure 4 shows an example of theapplication of this algorithm for a system with 141 reac-tions in which the polymerization reactions for formationof biomass—and therefore growth—has been taken intoaccount for the prediction of the elementary flux modes.


0 . 2

0 . 4

0 . 6 Y i e l d ( g D W / g G l u c o s e )

A m i n oA c i dS y n t h e s i s

P PS h u n t

I n t r a c e l l u l a rT r a n s p o r t

T C AC y c l e

Figure 4: Elementary flux modes in S. cerevisiae.Explanation see text.

Each column shown in Figure 4 represents an elemen-tary flux mode utilizing glucose as the sole carbon andenergy source while producing biomass. Open symbolsrepresent deleted genes for various reactions in the net-work. Obviously, these different mutants—or differentphenotypes—are still able to grow albeit with varyingyield coefficients. Elementary modes given in Figure 4are ordered in descending sequence with respect to theyield of biomass on glucose. The maximal value of thebiomass yield is in agreement to what has been mea-sured for S. cerevisae growing with glucose under aero-bic conditions. For mutants with defects in the transportsystems via the mitochondrial membrane, for example,the biomass yield decreases to a value observed underanaerobic conditions. Thus, structural properties of thenetwork clearly define the upper limit of product yields.Interestingly, these findings are independent of the ki-netic properties of the network. Absolute values of fluxesas well as a dynamic response to system perturbations,however, can only be described when kinetic rate equa-tions have been assigned.

A l a A r g A s n A s p C y s G l uG l n G l y H i s I l e u L e u L y s M e t P h e P r o S e r T h r T r y T y r V a l

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

1 . 2

Yie

ld (

g A

min

o A

cid

/ g

Glu

co

se

)

Figure 5: Maximal yields of amino acids on glucosein S. cerevisiae.

By a closer inspection of the solution shown in Fig-ure 4, it is possible to separate more flexible regions ofthe network from rigid parts. For instance, we cannotdelete any reaction in the pathways leading to aminoacids. This is a reasonable finding for growth in thepresence of synthetic media. In contrast, greater flexi-bility is observed in the citric acid cycle (TCA), pentosephosphate (PP) shunt and intracellular transport. It isinteresting to recognize that these variations also havebeen identified as key modulations during the process ofevolution.

The design aspect of these predictions may be bet-ter elaborated by examples of biotechnological relevantproduct formation. A desired product can often be syn-thesized by various different routes or pathways withinthe network. It is then of interest to detect the routeon which the product/substrate ratio is maximum. Fig-ure 5 illustrates an example showing the maximal yieldof the individual 20 amino acids on substrate glucose inS. cerevisiae.

The results shown in Figure 5 have been obtained bydetecting all elementary modes in the system and thenchoosing the mode giving the best yield. Sometimes,however, implementation of an elementary mode leadingto a slightly non-optimal yield might be easier from apractical point of view. In contrast to linear optimiza-tion, the complete set of elementary flux modes immedi-ately provides the complete spectrum of alternative im-plementations.

Flux Optimization

The driving force for selection of an optimal pathwayis the maximization of the yield of the product. How-ever, economic considerations also require optimizationof the product formation rate (productivity). This prob-lem leads to the question of an optimal modulation ofenzyme activities in metabolic networks.

At first glance, the example chosen for thisdiscussion—ethanol production with the yeast S.


cerevisiae—appears to be rather boring because it hasbeen tackled so many times by yeast geneticists. Theprimary aim of these empirical attempts to modulate(mostly amplify) the key enzymes within glycolysisis the maximization of ethanol production rate whichcorrelates with carbon dioxide evolution and, in turn,with the baking power of yeast. While addressing thisproblem it should be emphasized that the interest inanswering the question of an optimal redistribution ofenzyme activities is much broader. Generally, knowledgeof the rate controlling steps in the central metabolism(glycolysis and PP shunt) is of central importance forcell cultures used for producing proteins as well as forthe analysis of potential targets in cancer cells. Anotherfield of interest is the identification of potential targetsfor antitrypanosomal drugs, important for treatment ofthe african sleeping sickness (Bakker et al., 1999). Thelast named authors concluded that “Despite the greatinterest, it is not yet known completely for any organismhow the control of the glycolytic flux is distributed”.

Dynamic Model

Similar to the strategy of Bakker et al. (1999), the prob-lem can be approached from the basis of experimentallydetermined kinetic properties of the key enzymes whichare then aggregated to a dynamic model. Individual rateexpressions including their kinetic parameter have beenidentified in vivo by a stimulus response methodology(Theobald et al., 1997; Rizzi et al., 1997; Vaseghi et al.,1999): A pulse of glucose or alternative stimuli are in-troduced into a continuous culture operating at steadystate and the transient response of several intracellularmetabolite and cometabolite pools is experimentally de-termined in time spans of seconds or, recently, also mil-liseconds (Buziol et al., to be published). Within theserelatively short time spans, enzyme concentrations areconsidered to be in a “frozen” state. Figure 6 summa-rizes examples for some of the experimental observationsof metabolites and cometabolites from the yeast S. cere-visiae growing under anaerobic conditions.

The metabolome’s response due to dynamic systemexcitation has been used to identify the dynamic systembehavior by a stepwise internalization of metabolites sim-ilar to the method proposed by Rizzi et al. (1997). Todescribe the dynamic system behavior, deterministic ki-netic rate equations for the pathways for the reactionshave been formulated.

The general form of this rate equations can be writtenas

ri = rmax,i f (c,p) , (21)

where the maximal rate (capacity) rmax,i is obtainedfrom the vector of model parameters p, the vector com-prising metabolite, cometabolite and effector concentra-tions c and the flux distribution J at the systems’s steady

G L C

P Y R

R I B U 5 P

X Y L 5 P R I B 5 P

E 4 P F 6 P

G 6 P

F 6 P

G A P

A L D

P E P

E T O H

1 8 1

1 7 5

1 2

3

578 2

C y t o s o l

F B P

D H A P

6

S E D 7 P G A P

6 P G

G L Y P

6

P Y R

G L C

G L Y C

1 0 0 1 2

8 9

8 9

8 9

< 0 . 1

1 0 0

< 0 . 1

1 8 1

1 7 5< 0 . 1

A T P + H 2 O A D P + P i6 4 A M P + A T P 2 A D P8

O A A , A C C O A

G 1 P

4

4

Figure 7: Flux distribution within central metabolicpathways of S. cerevisiae under anaerobic conditionsat a specific growth rate of µ = 0.1 h−1. All fluxesare related to the influx of glucose.

1 . 2 1

0 . 1 2 30 . 0 1

0 . 0 0 10 . 0 0 9 0 . 0 0 6

0 . 0 0 0 10 . 0 0 7

0 . 0 0 0 1

0 . 5

1 . 0

1 . 5J E t h a n o l

E iC

p e r m h k p g i p f k a l d o t i sg a p d h

m u t

e n o l

p k p d c a d h

0 . 0 0 4

Figure 8: Distribution of flux control coefficients onethanol formation route. Enzymes involved: Hexosetransporter (perm), hexokinase(hk), phosphoglucoseisomerase (pgi), phosphofructokinase (pfk), aldolase(aldo), triose phosphate isomerase (tis), gap dehydro-genase (gapdh), phospoglyceromutase (mut), enolase(enol), pyruvate kinase (pk), pyruvatedecarboxylase(pdc) and alcohol dehydrogenase (adh).

state (e.g. µ = 0.1 h−1); accordingly

rmax,i = 1/rsteady statei f

(csteady state,p

). (22)

Intracellular flux distribution has been estimated by ex-perimentally determined uptake and excretion rates ofglucose, carbon dioxide, ethanol, glycerol and biomass.The results are documented in Figure 7.


0

1

2

3

4

0 . 3

0 . 6

0 . 9

1 . 2

1 . 5

1 . 8

0

1

2

3

4

5

0

1

2

3

4

5

6

7

01

2

3

4

5

6

7

0 . 0 6

0 . 1 2

0 . 1 8

0 . 2 4

0 . 3 0

- 4 0 - 2 0 0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0

0 . 0 6

0 . 1 2

0 . 1 8

0 . 2 4

0 . 3 0

A T P

A D P

A M P

N A D H

N A D

N A D P H

N A D P

- 4 0 - 2 0 0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 04

6

8

1 0

1 2

1 4

1 6

1 8

2 0

P i

( m

mol

/ l )

t i m e t ( s e c )

0

1

2

3

4

5

6

0

1

2

3

- 4 0 - 2 0 0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 00

1 0

2 0

3 0

4 0

5 0

6 0

7 0

0 . 3

0 . 4

0 . 5

0 . 6

0 . 7

0 . 8

0 . 9

1 . 0

0 . 2

0 . 3

0 . 4

0 . 5

0 . 6

0 . 3

0 . 4

0 . 5

0 . 6

0 . 7

0 . 8

0 . 9

1 . 0

1 . 1

- 4 0 - 2 0 0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0

8 4 0

9 0 0

9 6 0

1 0 2 0

1 0 8 0

G L C e x

F D P

G A P

P E P

P Y R

E T O H e x

F 6 P

G 6 P

0 . 0 0

0 . 0 4

0 . 0 8

0 . 1 2

0 . 1 6

0 . 2 0

( m

mol

/ l )

M e t a b o l i t e s C o m e t a b o l i t e s

t i m e t ( s e c )t i m e t ( s e c )t i m e t ( s e c )

S t o p F e e d

G l u c o s e P u l s e

Figure 6: Comparison between model simulation and measured concentrations of glycolytic metabolites and cometabolitesafter dynamic system excitation. Data shown left from the broken lines represent steady state values at a growth rate ofµ = 0.1 h−1.

Sensitivity Analysis

The next step towards solving the envisaged design prob-lem is to calculate the so called flux control coefficients,or—in the terminology of engineering—sensitivity coeffi-cients. The flux control coefficient CJ

Eihas been defined

as the fractional change of the network flux J caused bya fractional change in the level of enzyme activity Ei.Thus,

Flux Control Coefficient =dJ (Ei)

dEi. (23)

or normalized

CJEi

=dJ/J

dEi/Ei=

d lnJ (Ei)d lnEi

. (24)

Figure 8 depicts the results of these calculations forthose enzymes involved on the path from glucose toethanol.

Since only a subset of the flux control coefficients withrespect to ethanol formation are shown in Figure 8 (ex-cluding, for example, the coefficients of enzymes involvedin the PPP shunt), flux coefficients of this subset do notnecessarily sum up to one. From the hierarchy of sensi-tivities it can be concluded that the enzyme responsiblefor the transport of glucose via the cell membrane (per-mease) shows the overwhelming control strength uponthe ethanol production rate. Consequently, amplifica-

Y E p H X T 5

7 7 9 5 b p s

1 0 0 0

2 0 0 0

3 0 0 0

4 0 0 0

5 0 0 0

6 0 0 0

7 0 0 0

A s p ,

A s pA s p ,

P v u I I,

B a m H I,

E c o R V ,

P s t I,

B a m H I,S m a I,A s pS a c I,E c o R I,

P v u I I,

H p a I,

IC l a,

IS t u,IS c a,

R VE c o,

IN d e,

IS c a,

I IP v u,d I I IH i n,

IS p h,

H X T 5

H A - t a g

H X T 7 T e r m

U R A 3

A m p R

H X T 7 P r o m

Figure 9: HXT5 multi copy plasmid.

tion of this enzyme should lead to an increased flux fromglucose to ethanol.

Synthesis—Amplification of Hexose Transporters

Experimental verification of the above-named designproposal has been performed in a collaboration projectwith Institute of Microbiology from the University of


q S

0 . 2

0 . 4

( g E t h a n o l / g D W h - 1 )

0 . 0 5

0 . 1 0

0 . 1 5

0 . 2 0

( 1 ) W i l d T y p e

0 . 4 2 0 . 4 80 . 3 9

0 . 1 7 0 . 1 6 0 . 1 9

( g G l u c o s e / g D W h - 1 )

( 2 ) H X T 5S i n g l e C o p y

( 3 ) H X T 5M u l t i C o p y

q P

Figure 10: Specific substrate uptake rate qs andspecific product formation rate qp during anaerobicchemostat cultivation. (1) Wild strain. (2) HXT5with a single copy of the gene integrated the chro-mosome and constitutively expressed. (3) Multi copyplasmid with HXT5.

Dusseldorf (Dr. Boles). Within this project, the kinet-ics of the three most important hexose transporters outof the 17 transporters identified from the yeast genomeproject are investigated (Boles et al., 1997; Buziol et al.,to be published). One of these s—HXT5—has been alsoexpressed with a multi copy plasmid, shown in Figure 9(Boles, to be published).

The transporter gene in HXT5 is flanked by an en-hanced HXT7 promoter and HXT7 terminator, respec-tively. HXT7 is a high affinity transporter expressedat low glucose concentrations. The construct shown inFigure 9 ensures expression of HXT5 at target growthconditions. The same transporter has been integrated assingle copy in the chromosome of a yeast strain in whichall the genes of the other transporters has been knockedout (Wieczorke et al., 1999). As a result of these geneticconstructions, it is possible to compare the flux of glu-cose through the glycolysis between two strains differingonly in the amount of the hexose transporter. Accord-ing to the hierarchy of flux control coefficients discussedin context with Figure 8, one would expect a noticeableincrease of ethanol flux.

Experiments with three different strains were per-formed in continuous cultures at a dilution rate of D =0.07 h−1 (Buziol et al., to be published). The results ofthese experiments are summarized in Figure 10.

Compared to the wild type, ethanol excretion rate andsubstrate uptake rate of the strain with higher trans-

porter activity have found to be 10% and 25% higher,respectively. The discussion of the relevance of these re-sults from industrial point of view is beyond the horizonof this paper. However, the rather modest effect promptsthe question: Are there any other alternative design pos-sibilities resulting in a substantial increase of the ethanolproduction rate?

Objective Function

The apparent failure to produce significant increase ofthe glycolytic flux points to the fundamental questionif the underlying assumptions leading to the design sug-gestion are adequate. Keeping in mind that the pathwayof interest is part of a whole—the living cell—, the ideathat one need to amplify a single or multiple enzymes ac-cording to the hierarchy of flux control coefficients maynot correspond to physiological reality. There are two as-pects that should attract attention. First, an increasedexpression of enzymes is linked with energetically expen-sive protein synthesis. Glycolytic enzymes in yeast areknown to contribute in the order of 30% to the totalamount of cellular proteins. Thus, it seems to be likelythat overexpression of these enzymes result in a stress sit-uation with an unforeseeable impact on cell physiology.Instead of a single or simultaneous elevation of enzymeactivities, a more robust strategy should try to keep thetotal concentration of proteins at a constant value andredistribute the activities according to the required ob-jective. The optimization problem may then be statedas

Maximize J(rPath

max

)(25)

subject to1w

w∑i=1

rmaxi

rmaxi,reference

≤ Ω. (26)

In writing Equation 26 we assume that the maximalrate rmax

i is proportional to the enzyme amount. There-fore, Equation 26 specifies a fixed level for the total en-zyme activity Ω.

Another relevant issue concerns the pool concentra-tion of the metabolites within the cell. Attempts toincrease metabolic fluxes by changes in individual en-zyme concentrations may lead to substantial changes inmetabolite concentrations. Again, a substantial changein metabolite concentrations either proves to be cyto-toxic or at least leads to an undesired flux diversion(Kell and Mendes, 2000). Therefore, preservation of themetabolite concentrations close to the steady-state val-ues of the wild strain is at any rate desirable to meet thebasic property of well established metabolic systems forwhich the metaphor homeostasis has been coined (Reichand Selkov, 1981). It is also known that part of an op-timal performance of cells with respect to the use of,for example, energy and carbon sources is a process ofadaptation leading to a change of structure to controlhomeostasis. Interestingly, such structural changes may


0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0 2 0 0 2 2 0

2 0

4 0

6 0

8 0

1 0 0

1 2 0

1 4 0

1 6 0

D e v i a t i o n f r o m p o o l c o n c e n t r a t i o n Q ( % )

x

Inc

rea

se o

f E

tha

no

lF

orm

ati

on

Ra

te (

%)

Figure 11: Increase of the Ethanol formation rate asa function of the maximal deviation from initial poolconcentrations.

include a change of enzyme concentrations via regulationof enzyme synthesis. Hence, homeostasis puts a furtherconstraint on the metabolic redesign. Mathematically,this could take the form

1m

m∑i=1

∣∣∣csteady statei,optimum − csteady state

i,reference

∣∣∣csteady statei,reference

≤ Θ. (27)

Finally, the optimum must be constraint to stablesteady states, leading to condition (28)

d

dtc = 0 and Re (λi,optimum) ≤ 0, (28)

with Re (λi) denoting the real part of the system’s eigen-values.

Optimal Solutions

Retaining the total activity at the system’s initial stateat Ω = 1, non-linear optimization of enzyme activitiesyielded significantly higher ethanol formation rates (seeFigure 11.

As shown in Figure 11, amplification of ethanol for-mation largely depends on the allowed deviation fromthe initial pool concentrations Θ. Since the ratios ofenzyme levels on the elementary flux mode glucose—ethanol are subject to modifications, no amplificationcan be expected for Θ = 0% while the maximal pos-sible amplification of ethanol formation is found to beas much as 144% at Θ = 210%. No stable steady stateshave been detected above this value.

Figure 12 shows the optimal distribution of enzymeactivities for the selected example, 40% increase of poolconcentrations resulting in an amplification of ethanolformation of 63%.

The optimized modulation of the enzyme activities re-sults in an unexpected and interesting redistribution ofenzyme activities. Due to its large share in the control of

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

1 . 2

1 . 4

1 . 6

Rel

ativ

e E

nzy

me

Act

ivit

y (-

)

p e r m h k p g i p f k a l d o t i sg a p d h

m u t

e n o l

p k p d c a d h

Figure 12: Optimal distribution of enzyme activities.All activities are related to their initial values.

ethanol formation rate, the activity of the hexose trans-porter (perm) is found to be significantly higher in theoptimized metabolic system. In contrast, even thoughthe control coefficients of tis, gapdh, mut, enol and adhare relatively small, activities of those enzymes are alsofound to be larger compared to the non-optimized sys-tem. This outcome can be explained by the enzymesability of counteracting an increase of pool concentra-tions provoked by a risen glucose influx effectively. Ob-viously, this is the result of the superposition of the threeobjectives: maximization of flux at more or less home-ostatic conditions and unchanged total amount of en-zymes.

Concluding Remarks

This paper has presented typical examples of computeraided design problems in Metabolic Engineering. Theexamples shown refer to the two different characteristicsof biological systems: (1) Topological properties and (2)Kinetic properties of the individual reactions.

Topological analysis of metabolic networks turns outto be of immense value for relating genotypes and pheno-types. The further application of the illustrated conceptof elementary flux modes critically depends on the de-velopment of new and more effective algorithms to treatlarger networks. In case of biotechnological productionprocesses, the most important application concerns theprediction of optimal topological properties for maximiz-ing the product yield.

The second example illustrates model based design ofan enhanced product formation rate. In addition to in-formation on the network topology, the suggested so-lution of this important task in Metabolic Engineeringrequires detailed knowledge on kinetic rate expressions.Armed with a dynamic model for the most importantpart of the system, it is then possible to apply non-linear


optimization methods. The advantages of this approachare twofold: (1) Effects on large changes in enzyme con-centrations can be easily studied and (2) Constraintssuch as limits on the total amount of enzymes and devi-ations from steady state metabolite pool concentrationscan be taken into account in parallel.

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