Klipp E., Herwig R., Kowald A., Wierling C., Lehrach H. Systems Biology in Practice: Concepts, Implementation and Application

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Example 5-27

Assume that we can manipulate the pathway by changing the enzyme concentra-

tion in a predefined way. We would like to explore the effect of the perturbation of

the individual enzymes. For a linear pathway of the type of Eq. (5-190) consisting

of four consecutive reactions with v

= E

i–1

– k

–i

), we will calculate the flux-

control coefficients. For i = 1, …, 4, it shall hold that (1) all enzyme concentrations

= 1, (2) the rate constants k

=2,k

–i

= 1, and (3) the concentrations of the exter-

nal reactants P

= P

= 1. The resulting flux is J = 1 and the flux-control coeffi-

cients are C

= (0.533 0.267 0.133 0.067)

according to Eq. (5-180).

If we now slightly perturb the first enzyme, perhaps by performing a percentage

change of its concentration, i. e., E

+ 1%, then Eq. (5-125) implies that the

flux increases as J ?J+C

71%. In fact, the flux in the new steady state is

?1.017E

= 1.00531. Increasing E

, E

,orE

by one percent leads to flux values

of 1.00265, 1.00132, and 1.00066, respectively.

A strong perturbation would not yield similar effects. Assume that we can manip-

ulate the system at will and can double the total amount of enzyme in the system.

If we keep E

to E

constant but change E

?5E

, the resulting flux is J

?57E

1.7441. Changing instead E

?5E

yields J

?57E

= 1.0563, a lower flux increase.

Equal distribution of changes to all enzymes E

?2E

, i = 1, …, 4 results in

?27E

= 2. The maximal value for the flux J

max

= 2.2871 is obtained for the opti-

mal distribution of enzyme concentration to the individual reactions, which is

= 3.124, E

= 2.209, E

= 1.562, E

= 1.105 (see Chapter 12). Changes in en-

zyme concentrations and their effects on flux control are illustrated in Fig. 5.12

190

5 Metabolism

Fig. 5.12 Effect of enzyme concentration change

on steady-state flux and on flux-control coefficients

in an unbranched pathway consisting of four reac-

tions. In the reference state, all enzymes have the

concentration 1 (in arbitrary units), the control dis-

tribution is the same as in case (c), and the steady-

state flux is J = 1.(a) Change of E

?5E

while

keeping the other enzymeconcentrations constant

results in a remarkable drop of control of the first

enzyme. The resulting flux is J

?57E

=1.7741.

(b) The change E

?5E

corresponds to

?57E

= 1.0563. There is only slight change of

control distribution. (c) Equal enzyme concentra-

tions with E

?2E

, i = 1, …, 4 result in J

?27E

=2.

(d) Optimal distribution of enzyme concentration

= 3.124, E

= 2.209, E

=1.562,E

= 1.105 re-

sulting in the maximal steady-state flux

max

= 2.2871.

5.3.3

Extensions of Metabolic Control Analysis

5.3.3.1 Control Analysis for Variables other than Fluxes and Concentrations

Control coefficients are defined as the ratio of fractional changes in the system’s vari-

ables at steady state to fractional changes in the biochemical activity that causes the

change. The concept of control analysis can also be applied to other relevant vari-

ables of the system besides fluxes and concentrations (Schuster 1996). Such variables

can be the transition time (Section 5.2.7), free energy differences (Section 5.1.2, Wes-

terhoff and van Dam 1987), growth rate (Kacser and Beeby 1984; Dykhuizen et al.

1987), and others. Let X be a vector of generalized state variables and Y a vector of

generalized response variables. State variables X characterize the state of the system,

which is not necessarily the steady state. Response variables Y = Y(X,p) can be calcu-

lated in terms of the state variables X and the parameters p. The reaction rates v are

also considered as functions of X and p, v = v(X, p). The matrix of non-normalized

control coefficients can be expressed as

non





1

; (5-195)

and for the matrix of normalized control coefficients, it holds that

 dgY



1



1

dgv



: (5-196)

In Eqs. (5-195) and (5-196), the total derivative dY/dp has been used since not only

the direct effect of p on Y but also the indirect effect via X must be taken into ac-

count. Schuster (1996) showed that the coefficients defined in Eq. (5-195) obey a uni-

fied summation theorem,

non

K 



1

K ; (5-197)

as well as a unified connectivity theorem,

non





1



; (5-198)

where X

is a subvector of X containing only independent variables (compare Chap-

ter 5, Section 5.2.5).

191

5.3 Metabolic Control Analysis

Example 5-28

In a metabolic system with conservation relations such as the conservation

ATP + ADP = const. considered in Example 5-14, it could be of interest to interpret

experimental results in terms of the concentration ratio X

= ATP/ADP. The vec-

tor X =(ATP/ADP, ATP + ADP)

can formally replace the concentration vector

S =(ATP ADP)

. The coefficients expressing the control of reaction k over the

concentration ratio can be expressed in terms of the concentration control coeffi-

cients

non

ATP=ADP



i ATP=ADP



ADP

non

ATP



ATP

ADP

non

ADP

: (5-199)

Since ATP + ADP = const., it holds that

non

ATP

=–

non

ADP

, and therefore,

non

ATP=ADP



ATP=ADP



ADP

non

ATP

: (5-200)

Coefficients for the control of molar free energy differences were defined by Wes-

terhoff and Van Dam (Westerhoff et al. 1987). In non-normalized form, they read

non



iDG

(5-201)

and obey the following summation and connectivity theorems:

k1

non

 J

 0 (5-202)

k1

non

 e

d

: (5-203)

The quantity e

denotes the elasticity of reaction rate v

with respect to the free

energy difference of reaction j, and d



1; if i  j

0; if i 6 j



is the Kronecker symbol.

It may also be of interest to study the control over characteristic times of the reac-

tion network. For unbranched reaction sequences like the one presented in

Eq. (5-190), the control coefficient for the transition time t

of metabolite S

(given in

Eq. (5-109)) can be defined as



: (5-204)

Their properties have been investigated by Easterby, Melendez-Hevia, Schuster,

and others (Easterby 1981, 1990; Melendez-Hevia et al. 1990; Schuster 1996). The

following simple relation holds:

192

5 Metabolism

 C

 C

: (5-205)

The summation theorem reads

i1

1 : (5-206)

This implies that the activation of all enzymes by the same fractional amount re-

duces the transition time by the same factor. The connectivity theorem for pathways

without conservation relations reads

i1

 e

d

: (5-207)

5.3.3.2 Time-dependent Control Coefficients

Time-dependent control coefficients have been introduced for the case that the

steady state is not reached within a reasonable time (Heinrich and Reder 1991).

They are given as

non

tI 

exp tN



 I



1

N (5-208)

and

non

t exp tN



 I



1

N : (5-209)

The matrices

non

(t) and

non

(t) are time-dependent operators that transform

the initial perturbation of the reaction rates into flux and concentration variations at

time t. Consideration of relaxation processes does not change the summation rela-

tionships

non



 K  K

non

tK  0 (5-210)

but affects the connectivity relationships

non

t

L 

L exp tN



non

t

L  L exp tN



 I



(5-211)

(Heinrich and Reder 1991). An extension of metabolic control analysis to non-

steady-state trajectories is presented by Ingalls and Sauro (2003).

193

5.3 Metabolic Control Analysis

5.3.3.3 Spatially Heterogeneous and Time-varying Cellular Reaction Networks

Peletier et al. (2003) took into account that during signaling processes the assump-

tion of spatial homogeneity (Eq. (5-10)) does not necessarily hold. Instead, the con-

centrations of activated proteins show temporally changing spatial gradients. These

gradients are due to protein diffusion from the receptor to the place of action. They

play an important role in the transmission of the respective signal. To tackle this

problem, protein concentrations S

are considered as functions of time and space.

They may change due to occurrence of biochemical reactions with the rates v

or due

to diffusion with the diffusion constant D

. The evolution of concentrations is given

by the balance equations (Katchalsky and Curran 1965)

 D



j1

; (5-212)

where r is the number of reactions, D 



is the Laplacian operator,

and x

,…,x

are the spatial Cartesian coordinates.

In addition to reaction and diffusion, molecules may be transported across system

boundaries (the plasma membrane or membranes of organelles) with a flux f

. The

quantity L denotes a characteristic length of the cell (e. g., the diameter) that may

change due to cell swelling or growth. For modulation of these quantities around a

reference state, dimensionless parameters a are introduced, such that

! a

 v

; D

! a

 D

; f

! a

 f

; and L ! a

 L : (5-213)

The parameters a assume the value 1 at the reference state. The control over a

steady-state variable q is expressed by the control coefficients



i ln q

i ln a

; C



i ln q

i ln a

; C



i ln q

i ln a

; and C



i ln q

i ln a

; (5-214)

i.e., by the ratio of the fractional change in q caused by the fractional change in a.

For systems with transport processes and a non-homogeneous distribution of con-

centrations, the generalized summation theorem for the steady-state flux takes the

form

j1



i1



i1

 1 ; (5-215)

i.e., the total control of 1 on any steady-state flux is shared by all diffusion, reaction,

and transport processes.

A further extension of this concept takes into account a dependence on changing

conditions in space or time. The space conditions may modify due to cell growth or

shrinkage. The dependence of the system on time may be given if initial conditions

alter with time. In an analogue way, as in Eq. (5-213), time t is rescaled according to

194

5 Metabolism

t ?a

7t. The generalization of the summation theorem for flux-control coefficients

for systems that depend on time reads

C

J t



j1

J t



i1

J t



i1

J t

 1 : (5-216)

For control over concentrations, the following summation theorem holds:

C



j1



i1



i1

 0 ; (5-217)

where S

is the concentration depending on space and time. The control exerted by

cell size is reflected in two further summation theorems, namely, for fluxes:

i1

Jt



i1

Jt

 C

J 

 1 : (5-218)

For control over concentrations, the following summation theorem holds:

i1



i1

 C

 0 : (5-219)

The theorems introduced by Peletier et al. (2003) are well suited for application in

signaling networks.

Suggested Further Reading

Schellenberger 1989; Kuby 1991; Fell 1992, 1997; Heinrich and Schuster 1996; Cor-

nish-Bowden 2004.

195

5.3 Metabolic Control Analysis

196

5 Metabolism

References

Bier, M., Teusink, B., Kholodenko, B.N. and

Westerhoff, H.V. Control analysis of glycoly-

tic oscillations (1996) Biophys. Chem. 62,

15–24.

Briggs, G.E. and Haldane, J.B.S.A note on the

kinetics of enzyme action (1925) Biochem. J.

19, 338–339.

Brown, A.J. Enzyme action (1902) J. Chem.

Soc. 81, 373–386.

Bruggeman, F.J.,Westerhoff, H.V., Hoek, J.B.

and Kholodenko, B.N. Modular response

analysis of cellular regulatory networks (2002)

J. Theor. Biol. 218, 507–20.

Cascante, M.,Torres, N.V., Franco, R., Melen-

dez-Hevia, E. and Canela, E.I. Control analy-

sis of transition times. Extension of analysis

and matrix method (1991) Mol. Cell. Biochem.

101, 83–91.

Cascante, M., Melendez-Hevia, E., Kholo-

denko, B., Sicilia, J. and Kacser, H. Control

analysis of transit time for free and enzyme-

bound metabolites: physiological and evolu-

tionary significance of metabolic response

times (1995) Biochem. J. 308 (Pt 3), 895–9.

Cascante, M., Boros, L.G., Comin-Anduix, B.,

de Atauri, P., Centelles, J.J. and Lee, P.W.

Metabolic control analysis in drug discovery

and disease (2002) Nat. Biotechnol. 20, 243–9.

Cornish-Bowden, A. Metabolic control therapy

and biochemical systems theory: different

objectives, different assumptions, different

results (1989) J. Theor. Biol. 136, 365–77.

Cornish-Bowden, A. Fundamentals of enzyme

kinetics, 3rd edition (2004), Portland Press,

London.

Dykhuizen, D.E., Dean, A.M. and Hartl, D.L.

Metabolic flux and fitness (1987) Genetics

115, 25–31.

Eadie, G.S. The inhibition of cholinesterase by

physostigmine and prostigmine (1942) J. Biol.

Chem. 146, 85–93.

Easterby, J.S. Coupled enzyme assays: a gen-

eral expression for the transient (1973) Bio-

chim. Biophys. Acta 293, 552–8.

Easterby, J.S. A generalized theory of the transi-

tion time for sequential enzyme reactions

(1981) Biochem. J. 199, 155–61.

Easterby, J.S. Integration of temporal analysis

and control analysis of metabolic systems

(1990) Biochem. J. 269, 255–9.

Edwards, J.S. and Palsson, B.O. The Escheri-

chia coli MG1655 in silico metabolic genotype:

its definition, characteristics, and capabilities

(2000a) Proc. Natl. Acad. Sci. USA 97,

5528–33.

Edwards, J.S. and Palsson, B.O. Metabolic flux

balance analysis and the in silico analysis of

Escherichia coli K-12 gene deletions (2000 b)

BMC Bioinformatics 1,1.

Fell, D.A. Theoretical studies of the control of

adenosine 3:5-cyclic monophosphate by the

high- and low-Km phosphodiesterases (Pro-

ceedings) (1979) Biochem. Soc. Trans. 7,

1039–40.

Fell, D.A. Metabolic control analysis: a survey

of its theoretical and experimental develop-

ment (1992) Biochem. J. 286 (Pt 2), 313–30.

Fell, D. Understanding the control of metabo-

lism (1997) 1st edition, K. Snell, ed. Portland

Press, London, Miami.

Fell, D.A. and Sauro, H.M. Metabolic control

and its analysis. Additional relationships be-

tween elasticities and control coefficients

(1985) Eur. J. Biochem. 148, 555–61.

Glansdorff, P. and Prigogine, I.Thermody-

namic theory of structure, stability and fluc-

tuations (1971) Wiley-Interscience, London.

Guldberg, C.M. and Waage, P. Über die che-

mische Affinität (1879) J. Prakt. Chem. 19, 69.

Haldane, J.B.S. Enzymes (1930) Longmans,

Green and Co., London.

Hanes, C.S. Studies on plant amylases. I. The

effect of starch concentration upon the velo-

city of hydrolysis by the amylase of germi-

nated barley (1932) Biochem. J. 26, 1406–

1421.

Haynie, D.T. Biological thermodynamics (2001)

Cambridge University Press, Cambridge.

Heinrich, R. and Rapoport,T.A. A linear

steady-state treatment of enzymatic chains.

General properties, control and effector

strength (1974) Eur. J. Biochem. 42, 89–95.

Heinrich, R. and Rapoport,T.A. Mathematical

analysis of multienzyme systems. II. Steady

state and transient control (1975) Biosystems

7, 136.

Heinrich, R. and Reder, C. Metabolic control

analysis of relaxation processes (1991) J. Theor.

Biol. 151, 343–350.

Heinrich, R. and Schuster, S. The regulation

of cellular systems (1996) Chapman & Hall,

New York.

197

References

Heinrich, R. and Schuster, S. The modelling

of metabolic systems. Structure, control and

optimality (1998) Biosystems 47, 61–77.

Heinrich, R., Rapoport, S.M. and Rapoport,

T.A. Metabolic regulation and mathematical

models (1977) Prog. Biophys. Mol. Biol. 32,

1–82.

Higgins, J. Dynamics and control in cellular

reactions (1965) in B. Chance, R. W. Estabrook

and J. R. Williamson, eds. Control of the

energy metabolism. Academic Press, New

York, pp 13–46.

Hill, A.V. The possible effects of the aggrega-

tion of the molecules of hemoglobin on its dis-

sociation curves (1910) J. Physiol. 40, iv-vii.

Hill, A.V. The combinations of hemoglobin

with oxygen and with carbonmonoxide (1913)

Biochem. J. 7, 471–480.

Hofer,T. and Heinrich, R. A second-order

approach to metabolic control analysis (1993)

J. Theor. Biol. 164, 85–102.

Hofmeyr, J.H. Control-pattern analysis of meta-

bolic pathways. Flux and concentration control

in linear pathways (1989) Eur. J. Biochem.

186, 343–54.

Hofmeyr, J.H. Metabolic regulation: a control

analytic perspective (1995) J. Bioenerg. Bio-

membr. 27, 479–90.

Hofmeyr, J.H. and Cornish-Bowden, A.

Co-response analysis: a new experimental

strategy for metabolic control analysis (1996)

J. Theor. Biol. 182, 371–80.

Hofmeyr, J.H. and Westerhoff, H.V. Building

the cellular puzzle: control in multi-level

reaction networks (2001) J. Theor. Biol. 208,

261–85.

Hofmeyr, J.H., Kacser, H. and van der

Merwe, K.J. Metabolic control analysis of

moiety-conserved cycles (1986) Eur. J. Bio-

chem. 155, 631–41.

Hynne, F., Dano, S. and Sorensen, P.G. Full-

scale model of glycolysis in Saccharomyces cere-

visiae (2001) Biophys. Chem. 94, 121–63.

Ingalls, B.P. and Sauro, H.M. Sensitivity ana-

lysis of stoichiometric networks: an extension

of metabolic control analysis to non-steady

state trajectories (2003) J. Theor. Biol. 222,

23–36.

Kacser, H. and Beeby, R. Evolution of catalytic

proteins or on the origin of enzyme species by

means of natural selection (1984) J. Mol. Evol.

20, 38–51.

Kacser, H. and Burns, J.A. The control of flux

(1973) Symp. Soc. Exp. Biol. 27, 65–104.

Kahn, D. and Westerhoff, H.V. Control theory

of regulatory cascades (1991) J. Theor. Biol.

153, 255–85.

Katchalsky, A. and Curran, P.F. Nonequili-

brium thermodynamics in biophysics (1965)

Harvard University Press, Cambridge, M.A.

Kholodenko, B.N. Control of mitochondrial

oxidative phosphorylation (1984) J. Theor.

Biol. 107, 179–88.

Kholodenko, B.N. and Westerhoff, H.V.

Metabolic channelling and control of the flux

(1993) FEBS Lett. 320,71–4.

Kholodenko, B.N. and Westerhoff, H.V.

Control theory of one enzyme (1994) Biochim.

Biophys. Acta 1208, 294–305.

Kholodenko, B.N., Lyubarev, A.E. and Kurga-

nov, B.I. Control of the metabolic flux in a

system with high enzyme concentrations and

moiety-conserved cycles. The sum of the flux

control coefficients can drop significantly

below unity (1992) Eur. J. Biochem. 210,

147–53.

Kholodenko, B.N., Cascante, M. and

Westerhoff, H.V. Control theory of meta-

bolic channelling (1995a) Mol. Cell. Biochem.

143, 151–68.

Kholodenko, B.N., Schuster, S., Rohwer,

J.M., Cascante, M. and Westerhoff, H.V.

Composite control of cell function: metabolic

pathways behaving as single control units

(1995b) FEBS Lett. 368, 1–4.

Kholodenko, B.N., Schuster, S., Garcia, J.,

Westerhoff, H.V. and Cascante, M. Control

analysis of metabolic systems involving quasi-

equilibrium reactions (1998) Biochim.

Biophys. Acta 1379, 337–52.

Kholodenko, B.N., Brown, G.C. and

Hoek, J.B. Diffusion control of protein phos-

phorylation in signal transduction pathways

(2000) Biochem. J. 350 Pt 3, 901–7.

Kohen, E., Kohen, C., Hirschberg, J.G.,

Wouters, A.W.,Thorell, B.,Westerhoff,

H.V. and Charyulu, K.K. Metabolic control

and compartmentation in single living cells

(1983) Cell. Biochem. Funct. 1, 3–16.

Koshland, D.E., Jr., Nemethy, G. and

Filmer, D. Comparison of experimental bind-

ing data and theoretical models in proteins

containing subunits (1966) Biochemistry 5,

365–85.

Kuby, S.A. A study of enzymes. I. Enzyme cata-

lysis, kinetics, and substrate binding (1991)

CRC Press, Boca Raton, FL.

198

5 Metabolism

Lehninger, A.L. Biochemistry, 2nd edition,

(1975), New York,Worth, p 397.

Liebermeister,W., Klipp, E., Schuster, S. and

Heinrich, R. A theory of optimal differential

gene expression (2004) Biosystems, in press.

Lineweaver, H. and Burk, D. The determina-

tion of enzyme dissociation constants (1934) J.

Am. Chem. Soc. 56, 658–660.

Llorens, M., Nuno, J.C., Rodriguez,Y.,

Melendez-Hevia, E. and Montero, F.

Generalization of the theory of transition

times in metabolic pathways: a geometrical

approach (1999) Biophys. J. 77, 23–36.

Melendez-Hevia, E., Torres, N.V., Sicilia, J.

and Kacser, H. Control analysis of transition

times in metabolic systems (1990) Biochem. J.

265 195–202.

Michaelis, L. and Menten, M.L. Kinetik der

Invertinwirkung (1913) Biochem. Z. 49,

333–369.

Michal, G. Biochemical pathways (1999) Spek-

trum Akademischer Verlag, Heidelberg.

Monod, J.,Wyman, J. and Changeux, J.P. On

the nature of allosteric transitions: A plausible

model (1965) J. Mol. Biol. 12, 88–118.

Peletier, M.A.,Westerhoff, H.V. and Kholo-

denko, B.N. Control of spatially heteroge-

neous and time-varying cellular reaction net-

works: a new summation law (2003) J. Theor.

Biol. 225, 477–87.

Pfeiffer, T., Sanchez-Valdenebro, I.,

Nuno, J.C., Montero, F. and Schuster, S.

METATOOL: for studying metabolic networks

(1999) Bioinformatics 15, 251–7.

Ramakrishna, R., Edwards,J.S.,McCulloch,A.

and Palsson, B.O. Flux-balance analysis of

mitochondrial energy metabolism: conse-

quences of systemic stoichiometric constraints

(2001) Am. J. Physiol. Regul. Integr. Comp.

Physiol. 280, R695–704.

Reder, C. Metabolic control theory: a structural

approach (1988) J. Theor. Biol. 135, 175–201.

Reich, J.G. and Sel’kov, E.E. Time hierarchy,

equilibrium and non-equilibrium in metabolic

systems (1975) Biosystems 7, 39–50.

Sauro, H.M., Small, J.R. and Fell, D.A. Meta-

bolic control and its analysis. Extensions to

the theory and matrix method (1987) Eur. J.

Biochem. 165, 215–21.

Schellenberger, A. (ed.) Enzymkatalyse

(1989) VEB Gustav Fischer Verlag, Jena.

Schilling, C.H. and Palsson, B.O. Assess-

ment of the metabolic capabilities of Haemo-

philus influenzae Rd through a genome-scale

pathway analysis (2000) J. Theor. Biol. 203,

249–83.

Schilling, C.H., Schuster, S., Palsson, B.O.

and Heinrich, R. Metabolic pathway analy-

sis: Basic concepts and scientific applications

in the post-genomic era (1999) Biotechnol.

Prog. 15, 296–303.

Schilling, C.H., Letscher, D. and Palsson,

B.O. Theory for the systemic definition of

metabolic pathways and their use in inter-

preting metabolic function from a pathway-

oriented perspective (2000) J. Theor. Biol. 203,

229–48.

Schuster, S. Control analysis in terms of gener-

alized variables characterizing metabolic sys-

tems (1996) J. Theor. Biol. 182, 259–68.

Schuster, S. and Westerhoff, H.V. Modular

control analysis of slipping enzymes (1999)

Biosystems 49, 1–15.

Schuster, S., Dandekar,T. and Fell, D.A.

Detection of elementary flux modes in bio-

chemical networks: a promising tool for path-

way analysis and metabolic engineering

(1999) Trends Biotechnol. 17, 53–60.

Schuster, S., Fell, D.A. and Dandekar, T.

A general definition of metabolic pathways

useful for systematic organization and analy-

sis of complex metabolic networks (2000) Nat.

Biotechnol. 18, 326–32.

Schuster, S., Hilgetag, C.,Woods, J.H. and

Fell, D.A. Reaction routes in biochemical re-

action systems: algebraic properties, validated

calculation procedure and example from

nucleotide metabolism (2002) J. Math. Biol.

45, 153–81.

Small, J.R. and Fell, D.A. Metabolic control

analysis. Sensitivity of control coefficients to

elasticities (1990) Eur. J. Biochem. 191,

413–20.

Snoep, J.L. and Olivier, B.G. Java Web Simula-

tion (JWS); a web based database of kinetic

models (2002) Mol. Biol. Rep. 29, 259–63.

Snoep, J.L., Jensen, P.R., Groeneveld, P.,

Molenaar, D., Kholodenko, B.N. and

Westerhoff, H.V. How to determine control

of growth rate in a chemostat. Using meta-

bolic control analysis to resolve the paradox

(1994) Biochem. Mol. Biol. Int. 33, 1023–32.

Thomas, S. and Fell, D.A. Design of metabolic

control for large flux changes (1996) J Theor

Biol 182, 285–98.

Van Dam, K.,Wiechmann, A.H.,Westerhoff,

H.V. and Hellingwerf, K.J. Proton gradients

199

References

across energy-transducing membranes (1977)

Biochem. Soc. Trans. 5,28–9.

Varma, A. and Palsson, B.O. Metabolic flux

balancing: basic concepts, scientific and prac-

tical use. (1994a) Biotechnology 12, 994–998.

Varma, A. and Palsson, B.O. Stoichiometric

flux balance models quantitatively predict

growth and metabolic by-product secretion in

wild-type Escherichia coli W3110 (1994 b) Appl.

Environ. Microbiol. 60, 3724–31.

Wegscheider, R. Über simultane Gleichge-

wichte und die Beziehungen zwischenTher-

modynamik und Reaktionskinetik homogener

Systeme (1902) Z. Phys. Chem. 39, 257–303.

Westerhoff, H.V. and van Dam, K.Thermo-

dynamics and control of biological free-energy

transduction. (1987) Elsevier, Amsterdam.

Westerhoff, H.V., Plomp, P.J., Groen, A.K.

and Wanders, R.J. Thermodynamics of the

control of metabolism (1987) Cell. Biophys.

11, 239–67.

Westerhoff, H.V., Getz,W.M., Bruggeman, F.,

Hofmeyr, J.H., Rohwer, J.M. and Snoep, J.L.

ECA: control in ecosystems (2002) Mol. Biol.

Rep. 29, 113–7.

Wiback, S.J. and Palsson, B.O. Extreme path-

way analysis of human red blood cell metabo-

lism (2002) Biophys. J. 83, 808–18.