Báo cáo khoa học: Metabolic control in integrated biochemical system doc

Keywords: metabolic control analysis; hierarchical control; gene expression; metabolism.. Metabolic control analysis MCA [1,2] is a framework to quantify the control of metabolic variabl

Metabolic control in integrated biochemical systems

Alberto de la Fuente1, Jacky L Snoep2, Hans V Westerhoff3,4and Pedro Mendes1

1 Virginia Bioinformatics Institute, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA, 2 Department of Biochemistry, University of Stellenbosch, Matieland, South Africa; 3 Stellenbosch Institute for Advanced Study, South Africa;

4

Departments of Molecular Cell Physiology and Mathematical Biochemistry, BioCentrum Amsterdam, Amsterdam, the Netherlands

Traditional analyses of the control and regulation of

steady-state concentrations and fluxes assume the activities

of the enzymes to be constant In living cells, a

hierar-chical control structure connects metabolic pathways to

signal-transduction and gene-expression Consequently,

enzyme activities are not generally constant This would

seem to compromise analyses of control and regulation at

the metabolic level Here, we investigate the concept of

metabolic quasi-steady state kinetics as a means of

apply-ing metabolic control analysis to hierarchical biochemical

systems We discuss four methods that enable the

experi-mental determination of metabolic control coefficients, and demonstrate these by computer simulations The best method requires extra measurement of enzyme activities, two others are simpler but are less accurate and one method is bound only to work under special conditions Our results may assist in evaluating the relative import-ance of transcriptomics and metabolomics for functional genomics

Keywords: metabolic control analysis; hierarchical control; gene expression; metabolism

Metabolic control analysis (MCA) [1,2] is a framework to

quantify the control of metabolic variables, such as a

steady-state flux or a metabolite concentration, by parameters of

the system Control is measured in terms of response

coefficients, which are defined as the ratio between the

relative change in the variable (the response of the system)

and the relative change in the parameter (imposed

exter-nally) [3] The MCA formalism is exact when response

coefficients are expressed as partial derivatives:

RY¼@Y=Y

@P=P¼@ln Y

Yis any system variable and P the perturbed parameter

Usually, one is concerned with the control of steady-state

fluxes and metabolite concentrations by the activities of the

biochemical reactions (ÔstepsÕ) in the system Then, the

discussion concerns the subset of response coefficients that

are called control coefficients and are denoted by C rather

than R:

C½Xss

v ¼@ln½Xss

CJss

v ¼@ln Jss

where [X] is the concentration of the metabolite in question,

Jthe flux and v the rate of the step Control coefficients are systemic properties that depend on all the components of the system

MCA shows that the properties of the individual enzymes (usually called ÔlocalÕ properties) that are important for control are their elasticity coefficients These measure the relative change in rate of an enzyme caused by a relative change in the concentration of any effector:

ev

x¼ @v=v

@x=x¼@ln v

Ultimately, it is the integration of all local properties of the biochemical steps that determines the pathway’s control properties, reflected in the control coefficients Various methods [4–7] exist to calculate control coefficients from elasticity coefficients

MCA, in its original form, is only concerned with the distribution of control among fixed metabolic steps In the living cell, however, metabolic pathways are part of a larger biochemical system that includes signal-transduction path-ways, transcription, translation and several post-transcrip-tional and post-translapost-transcrip-tional steps, such as mRNA splicing This ÔinterconnectionÕ of the components in the biochemical network means that the flux through a pathway is controlled by elements additionally to the metabolic enzymes

Hierarchical control analysis (HCA) [8–10] is an exten-sion to MCA that explicitly accounts for the control exerted

by subsystems not connected to the pathway by mass flow, only by kinetic effects HCA considers the enzyme activities themselves as variables of the system as they change due to translation, proteolysis, binding to other proteins, and covalent modification It is also possible to consider mRNA concentrations explicitly, which are also variables due to transcription and degradation In this setting, it has been shown [8] that transcription and translation participate in

Correspondence to P Mendes, Virginia Bioinformatics Institute,

Virginia Polytechnic Institute and State University, 1880 Pratt Drive,

Blacksburg, VA 24061-0477, USA.

Fax: + 1 540 231 2606, Tel.: + 1 540 231 7411,

E-mail: mendes@vt.edu

Abbreviations: IPTG, isopropyl thio-b- D -galactoside; MCA,

metabolic control analysis; HCA, hierarchical control analysis.

Note: a website is available at http://www.vbi.vt.edu/mendes

(Received 24 September 2001, revised 24 June 2002,

accepted 2 July 2002)

the control of the metabolic flux DNA supercoiling [11,12]

in living Escherichia coli has recently been subjected to HCA

[13]

The control analysis of multilevel systems has been

generalized by Hofmeyr & Westerhoff [14] The

distribu-tion of control in the full system, referred to as integral

control, was shown to be expressed in terms of the control

of the modules in isolation, termed intramodular control,

and the sensitivity of the modules to each other,

intermodular response We adopt the same nomenclature

and, accordingly, refer to the quasi-steady-state of

the metabolic system as the intramodular steady-state,

and the steady-state of the full system as the global

steady-state

The issue of the diverse mechanisms through which living

cells are controlled is quite relevant in the realm of

functional genomics Whilst there has been an initial

emphasis on the transcriptome as representative for

func-tion, more recent work [15] has begun to emphasize that the

metabolome is where function resides Rather than it being

an issue of either-or, we believe that both metabolic and

gene expression regulation are important In every specific

case one should quantify each one’s contribution to

regulation The present paper is meant to optimize

opera-tional methods to do just that

M O D E L A N D M E T H O D S

To illustrate the proposed methodology with maximum

clarity, we use the simplest possible model system that

contains the essence of the problem, i.e the simplest

possible metabolic pathway that is subject to regulation by

itself through the synthesis of a new enzyme (Fig 1) It

counts only three variables: one metabolite, one enzyme

and one mRNA species Each is synthesized and

degra-ded Together, they constitute a hierarchical system of

three levels that are not connected by mass transfer

Nevertheless these levels ÔtalkÕ to each other by kinetic

effects The enzyme rate of synthesis depends on the

mRNA concentration, the rate of metabolite degradation

on enzyme concentration, and the transcription rate on

the metabolite concentration This provides a feedback

loopfor the regulation of the metabolic reaction rate,

which is implemented in the model for reaction 6 The

hierarchical regulation of reaction 5 is omitted for

simpli-city Yet, the model of Fig 1 should be sufficiently

interesting because it mimics the basic structure of

hierarchical biochemical systems including some routes

along which the hierarchical levels communicate to each

other The regulatory feedback from metabolite to mRNA

synthesis can produce homeostasis and is common in

known genetic systems The rates of all six reactions of

this model are given by Eqns (5–10):

v1¼

V1½nucleotidesK

m1

1þ½nucleotidesK

½metabolite

ð5Þ

v5¼V

f ½S

K mS5 Vr ½metabolite

K mP5

1þK½S

mS5þ½metaboliteK

mP5

ð9Þ

v6¼ ½enzymek

f cat

½metabolite

KmS6  krcatK½P

mP6

1þ½metaboliteK

mP6

ð10Þ

Here, [S] is the concentration of the pathway’s substrate, [metabolite] the concentration of the metabolite, [mRNA] the concentration of the messenger, [enzyme] the concen-tration of the enzyme, [nucleotides] the concenconcen-tration of nucleotides, V1 the limiting transcription rate, Km1 the Michaelis constant for nucleotides, Ka the activation constant of transcription by the metabolite, k2 the rate constant for mRNA degradation (and dilution due to cell growth), k3 the translation rate-constant, k4 the enzyme degradation rate-constant (and dilution due to cell growth),

V the limiting rate of reaction 5, Keq5 the equilibrium constant for reaction 5, KmS5the Michaelis constant for the substrate, KmP5the Michaelis constant of reaction 5 for the metabolite, kcatthe catalytic rate constant of the enzyme of step6, Keq6the equilibrium constant of reaction 6 and KmP6 the Michaelis constant for the pathway’s product It should

be noted that step5 is enzyme-catalyzed and its enzyme concentration is implicit in V

This system is called ÔdemocraticÕ in the terminology of HCA, as the arrows do not point only from transcription down to metabolism, but also from metabolism upto transcription This contrasts to ÔdictatorialÕ systems in which there are no arrows from metabolism upto transcription or translation; in that case the transcriptome (the collection of mRNAs in a cell) dictates everything down to the other levels

It is not clear if dictatorial systems actually exist, but it is

Fig 1 The model system Interactions between the different levels (dotted arrows) run through the dependencies oftranslation on mRNA concentration, the metabolic rate on enzyme concentration and the activation oftranscription by the metabolite Solid arrows indicate mass flow at the mRNA, protein and metabolic levels Although both reactions 5 and 6 are catalyzed by mRNA-encoded proteins, this is only shown explicitly for reaction 6 This simplifies the model without detracting from the essence of hierarchical regulation Accordingly, the model only takes into account this route for regulation through gene expression, effectively assuming that the gene encoding the enzyme of reaction 5 is expressed constitutively.

useful to refer to them, as it helps clarifying the properties of

the ubiquitous and more interesting democratic systems

In order to determine the metabolic intramodular control

coefficients, the two upper modules or the feedbacks from

metabolism to these, have to be ignored; therefore isolating

the metabolic part from the global system In this case the

enzyme and mRNA concentrations are assumed constant

When the enzyme concentration becomes constant its

product with kcat, in the numerator of Eqn (10), becomes

a parameter itself (V, known as the limiting rate)

In this model, the units of the kinetic constants and time

are arbitrary, however, their magnitudes were chosen to

meet the criterion that the rates of metabolism are much

higher than those of transcription and translation

(kt km) It was not our intention to mimic any known

system here, but rather to illustrate how the proposed

methods work

Table 1 lists the values of the rate constants used in the

simulations The values of the intramodular control

coefficients and the integral control coefficients under

these conditions are listed in Table 2 Simulations

were carried out with an Intel Pentium III 733 MHz

computer with the biochemical simulation packageGEPASI

[16–18]

The intramodular control coefficients, to be indicated by

lower case ÔcÕ, can be expressed as a function of the elasticity

coefficients:

cJss

v 5 ¼ e

v6

½X

ev6

½X ev5

½X

ð11Þ

cJ ss

v6 ¼ e

v5

½X

c½Xss

ev6

½X ev5

½X

ð13Þ

c½Xss

ev5

½X ev6

½X

ð14Þ

Where [X] stands for the metabolite concentration, and J for metabolic flux When considering the whole system, the integral control coefficients (indicated by capital ÔCÕ) can be derived similarly:

CJ ss

v5 ¼ e

v6

½X

ev6

½X ev5

½X

CJss

v6 ¼ e

v 5

½X

ev5

½X ev6

½X

where:

C½Xss

ev6

½X ev5

½X



v1

½X ev3½Nev6½E

ev2½Nev1½N



ev4½Eev3½E



ev5½Xev6½X



0

@

1 A ð18Þ

C½Xss

v6 ¼ C½Xss

Nstands for mRNA and E for enzyme

R E S U L T S

The control exerted by an enzyme of a metabolic pathway

on a metabolite concentration is defined in terms of the effect that a modulation of the former has on the steady-state magnitude of the latter This is called metabolic intramodular control if the enzyme activities remain constant If these are also subject to changes, a more global control reigns, leading to a different magnitude of the quantifier of control, i.e the integral control coefficient Comparison of Eqn (13), for the intramodular control of enzyme 5 on the metabolite, to Eqn (18), for the integral

Table 1 Parameter values used in the simulations ofthe model systems

described in Fig 1 and Eqns (5–10).

k r

Table 2 The values for the control coefficients according to Eqns (11– 19), obtained using the elasticity coefficients calculated numerically by Gepasi at the standard parameter set in Table 1.

Type of control Control coefficient Value Intra-modular c J ss

c J ss

c½X ss

c½X ss

Integral C J ss

C J ss

C½X ss

C½X ss

v1

½Xev5

½Xev3

½Nev6

½E

ev6

½X ev5

½X



ev2

½N ev1

½N



ev4

½E ev3

½E



ev5

½X ev6

½X



 ev1

½Xev3

½Nev6

½E

control of enzyme 5 on the metabolite, reveals the

differ-ence Compared to the intramodular control, the integral

control is attenuated by a rather complex factor involving

interlevel elasticity coefficients As most actual systems have

connections between regulatory levels, the question is if and

how metabolic intramodular control can be measured

There are two ways of measuring the metabolic

compo-nent of control One relies on the metabolic response being

faster than the gene-expression response, and analyzes the

system when the former has settled, while the latter is hardly

changed The second adds an inhibitor of transcription, so

as to eliminate the nonmetabolic response Less obvious

methods include one in which various modulations of the

system are performed and global control is measured, after

which intralevel control can be calculated; and another in

which one measures and then corrects for the adjusting

enzyme activity Each of these methods is now illustrated in

detail using the model of Fig 1

Method 1: based on metabolite time-courses

This method requires one to follow the time evolution of

the metabolite concentration after a perturbation has been

introduced The motivation comes from an anticipated

wide difference in time-scale between the metabolic

reactions, on the one hand, and the reactions of mRNA

and protein levels, on the other [14,19] After a

perturba-tion of the limiting rate V, the concentraperturba-tion of the

metabolite should first evolve to a metabolic

quasi-steady-state This apparent steady-state should be close to the

one that the metabolic system would approach if

decou-pled from gene expression Only subsequently should the

system evolve, slower, towards the global steady-state

(Fig 2, at the lower rate constants for transcription)

When transcription, translation and metabolism operate

at similar time-scales, the concentration of the metabolite

and its flux both move to the global steady-state without

exhibiting a metabolic quasi-steady-state (Fig 2, at high

rate constants for transcription)

In order to determine the values of the metabolic

intramodular control coefficient, we simulated the

model system for several values of the rate constants of

transcription, mRNA degradation, translation, and protein

degradation The parameters were varied to obtain ratios of about 500, 50, 5 and 0.5 between the characteristic times of metabolism and the other levels Simulations were per-formed such that the state concentrations, steady-state fluxes, and global control coefficients were equal, so as

to allow for meaningful comparisons The parameter values corresponding to these operations are given in Table 1 and the legend of Fig 2 The metabolic intramodular control coefficients were calculated using the time series, taking the highest point in metabolite concentration as the new metabolic intramodular steady-state after the perturbation:

cYss

v5 ¼ YssðnewÞ YssðinitialÞ

V5ðperturbedÞ V5ðinitialÞ

V5ðinitialÞ

YssðinitialÞ ð20Þ where Y represents any system variable, e.g the flux through the pathway [as in Eqn (1)] The modulation of v5 was kept small, i.e 1% If the value of the final (global) steady-state is used in Eqn (20), then the global control coefficient is obtained

When the integral control exceeds the intramodular control, as is the case for the flux-control of reaction 5, another method needs to be applied, because the traject-ory fails to exhibit an extremum (the global steady-state would be an extremum, but it was not reached in the interval of the measurements) In Fig 2B, the transient flux rapidly increased towards the intramodular state and then increased further towards the global steady-state A transient quasi-steady-state has the characteristic that the first derivative of the time-course is zero Therefore, in order to locate the intramodular steady-state, first derivatives of the time-course were estimated; the point at which the derivative was closest to zero was taken to be the quasi-steady-state This value was used as the new steady-state flux in Eqn (20) to calculate the intramodular flux-control coefficient It may be noted that this method differs from that of Liao & Delgado [20], which uses the time-course to estimate the control coefficients directly Here the trajectory is only used to locate the metabolic quasi-steady-state achieved after the perturbation It also differs form the method used by Sorribas et al [21] who determined kinetic orders from the time series and then used a matrix method to determine

Fig 2 Time simulations ofthe model system at different magnitudes oftranscription and mRNA degradation rates Parameter values are indicated as

in Table 1, except that the rate constants on the translational level were k 3 ¼ 10 and k 4 ¼ 1, and the rates on the level of transcription were: squares:

k 1 ¼ k 2 ¼ 0.001, triangles 0.01, diamonds 0.1 and circles 1 Rate v 5 was perturbed by increasing V (Eqn 8) by one percent The asterisks show the value of the intramodular steady-state after the perturbation (A) Metabolite concentration (B) Metabolic flux.

the logarithmic gains (which correspond to control

coefficients)

Figure 3 shows the concentration- and flux-control

coefficients, calculated using this method for different

combinations of parameter values on the levels of

tran-scription and translation Only the smaller rate constants of

the level of transcription or the smaller rate constants of

translation, were the control coefficients estimated at an

accuracy exceeding 95%

Method 2: based on inhibition of transcription

Inhibition of transcription or translation destroys the

feedback loops from metabolism to gene expression If the

mRNA or protein degradation rates are much smaller than

the metabolic rates, metabolism will behave as if isolated on

a short time-scale At this time-scale, one can measure

intramodular control coefficients

In practice, global transcription can be inhibited by

adding rifampicine to the medium, while global translation

by adding chloramphenicol Here, we mimicked the action

of a strong transcription inhibitor by setting the rate of

transcription to 10)25 in the simulations Transcription should be inhibited at the same time as the metabolic perturbation is made The effect of inhibiting transcrip-tion, together with the perturbation of rate v5, on the metabolite concentration and flux is shown in Fig 4 When transcription is abolished, this system cannot reach

a finite global steady-state as the concentrations of mRNA and protein decay to zero and the metabolic pathway reaches chemical equilibrium (no metabolic flux) As with method 1, we studied how this would work at several values for the rate constants of transcript-degradation and translation/enzyme degradation differing over three orders

of magnitude Under conditions that lead to time separ-ation, i.e metabolic rates much higher than those of transcription and translation, the metabolite concentration first increased to the metabolic steady-state and then slowly evolved to the global equilibrium The flux first moved to the metabolic steady-state and then decreased to zero Without this separation in time-scales, no quasi steady-state could be detected

To calculate the intramodular concentration-control coefficient in this example, we used the same procedure as

Fig 4 Intra-modular control coefficients as a function of mRNA-degradation rate and translation/protein degradation rate using method 2 for the case ofonly one variable enzyme (Fig 1) Measured control coefficients are scaled to the theoretical value of the intramodular control coefficient (1 indicates a perfect determination) Rates on the level of translation differed as follows: squares: 10 · k 4 ¼ k 3 ¼ 0.01, triangles 0.1, diamonds 1 and circles 10 Asterisks indicate the analytical value for the integral control coefficient (A) Concentration-control coefficients (B) Flux-control coefficients.

Fig 3 The metabolic intramodular control coefficients as a function of transcription/mRNA degradation rate and translation/protein degradation rate using method 1 Measured control coefficients are scaled to the value of the theoretical value for the intramodular control coefficient A value of 1 indicates a perfect determination of the intramodular control coefficient Rates on the level of translation were varied k 3 ¼ 10 · k 4 and for squares

k 4 ¼ 0.01, for triangles k 4 ¼ 0.1, for diamonds k 4 ¼ 1 and for circles k 4 ¼ 10 The asterisks indicate the analytical value for the integral control coefficient (A) Concentration-control coefficients (B) Flux-control coefficients.

described for method 1, determining the quasi-steady-state

point from estimates of the first derivatives The metabolic

flux-control coefficients were then calculated in the same

way as described for method 1: the maximum in the time

series was taken to be the quasi steady-state value and was

used in Eqn (20) Figure 4 shows the results of simulations

for various values of the rate constants of transcription and

translation Again, the intramodular control coefficients

were only estimated accurately when the transcription or

translation rate constants were small

In our model, only one of the enzymes was variable in

time This assumes that the rate of degradation of the

second enzyme (or its mRNA) is infinitely slower than the

degradation rate of the first (or its mRNA) We did

simulations of a model system that is similar to the one

described in Fig 1, Eqns (5–10) and Table 1, but where the

transcription and translation of the gene coding for the

enzyme producing the metabolite are explicit Degradation

and translation kinetics are identical to that of the gene for

step6 The transcription kinetics is assumed to be

insensi-tive to the metabolite, and therefore its rate is constant, and

set to 10)25 (as for step6) to mimic the effect of the

transcription inhibitor We performed simulations with this

system, analyzed the data as described above, and found

accurate estimates of control coefficients In this case both

proteins decay to zero at the same rate so that both the

production and consumption rates of the metabolite

decrease in the same proportion, decoupling metabolism

from gene expression (agreeing with the summation

the-orem for concentration control) Only when the time-scales

of metabolism and gene expression are close were the

estimates of concentration control coefficient poor

(Fig 5A) The accuracy of the measured flux control

coefficients was still low (Fig 5B), similar to the results of

Fig 4B

Proteins can have degradation rates varying over several

orders of magnitudes Therefore, the systems we studied

here are special cases, illustrating the extremes of behavior

that can be observed We expect that the results that can be

obtained using this method will be somewhere between the

results of these two extremes

Method 3: based on external gene induction This method is based on replacing the gene promoter by another whose activity does not depend on the metabolite concentration A popular method is the replacement of the original promoter by the IPTG inducible lac-type promoter, described in the context of metabolic control analysis by Jensen et al [22] By this substitution of promoters, one transforms the system to one of dictatorial control, where transcription is insensitive to the other levels In our model, this substitution of promoters is represented by introducing a new parameter, i.e the concentration of an external transcription activator, which

is now the modifier in Eqn (5), instead of the pathway metabolite This implies that there is no significant transcription without the presence of this external tran-scription activator, which is used to adjust the transcrip-tion rate independently from metabolism (just as IPTG has been used by Jensen et al [22]) The activation constant of the external transcription activator was set to

100, and its concentration adjusted such that the steady-state would have the same concentrations of metabolite and enzyme as originally Without the feedback loop, the response of metabolism to a perturbation in v5 is purely intramodular (i.e at the level of metabolism alone) In simulations, we found that the response is identical to the response that the metabolic pathway would have if considered in isolation The rates of transcription were varied following the same methodology as in the previous two methods An estimate of the intramodular control coefficient was obtained by inserting the values of the new steady-state variables in Eqn (20) In this case, the ability

to measure the intramodular control coefficients was independent of the separation of time-scales between metabolism and gene expression

Method 4: measuring and correcting for the altered enzyme activity

In this method, we made use of the fact that the rate equation for a metabolic stepcan be expressed by the

Fig 5 Intra-modular control coefficients as a function of mRNA-degradation rate and translation/protein degradation rate using method 2 when both enzymes are variable and have equal degradation rates Measured control coefficients are scaled to the value of the theoretical value for the intramodular control coefficient A value of 1 indicates a perfect determination of the intramodular control coefficient Translation rates differed as follows: squares: 10 · k 4 ¼ k 3 ¼ 0.01, triangles 0.1, diamonds 1 and circles 10 Rate constants for the expression of mRNA and protein for the metabolic step5 are taken to vary identically to mRNA and Enzyme for step6 (A) Concentration-control coefficients (B) Flux-control coefficients.

product of two factors, one dependent only on the enzyme

activity (ei) and another representing the kinetic mechanism

(ui) [23]:

The kinetic part can be perturbed independently of the

activity using non-tight-binding inhibitors The

concentra-tion of the enzyme will change due to the change in

metabolite through the regulatory feedback loop All newly

synthesized enzyme molecules will be inhibited to the same

proportion as those originally present Consequently, u stays

constant during the whole measurement To obtain the

global control coefficient one measures the change in flux or

metabolite concentration and differentiates the logarithm of

that change towards the logarithm of the perturbation [see

Supplementary material for derivation of Eqns (24, 25, 27,

and 28)]

CJss

v 6 ¼dln Jss

C½Xss

v 6 ¼dln½Xss

dln u6

ð23Þ For the intramodular control coefficients these

expres-sions have to be corrected for the change in enzyme

concentration (which could be seen as an additional

perturbation to the metabolic level) The logarithm of the

change in flux (or concentration) should then be

differen-tiated towards the logarithm of the whole rate equation:

cJss

v 6 ¼ dln Jss

dln u6þ d ln e6

J ss

v 6

1þ d ln e6=dln u6

ð24Þ

c½Xss

v6 ¼ dln½Xss

dln u6þ d ln e6

X ss

v6

1þ d ln e6=dln u6

ð25Þ

In order to calculate the intramodular control coefficient

using this method, one needs to measure the enzyme

concentration additionally to the fluxes and metabolite

concentrations Results of this method on the model system

of Fig 1 are given in the toprows of Table 3 It is seen that

this method is rather accurate

Eqns (24) and (25) are only valid when the enzyme of the

stepunder consideration was the only enzyme that changed

concentration To remove such a restriction, we extended

the model by explicitly taking account of the mRNA and

enzyme concentrations of the metabolic step5 Degradation and translation kinetics are identical to that of the gene for step6 Transcription of this gene is affected by the meta-bolite through a mechanism of competitive inhibition:

v7¼

V7 ½nucleotides

Km7

1þ½nucleotidesK

m7 þ½metaboliteK

I7

ð26Þ

V7is the limiting transcription rate for this gene, Km7the Michaelis constant for the nucleotides and KI7the inhibition constant of the metabolite Parameter values are

V7¼ 0.001, Km7¼ 1 and KI7 ¼ 100; [nucleotides] as in Table 1

Corrections due to the changes in enzyme concentration need to be taken in account, too For the intramodular flux control coefficient, we obtained

cJ ss

v6 ¼ dln Jss d ln e5

dln u6þ d ln e6 d ln e5

J ss

v6  d ln e5=dln u6

1þ d ln e6=dln u6 d ln e5=dln u6

ð27Þ and for the intramodular concentration control coefficient

c½Xss

dln u6þ d ln e6 d ln e5

½X ss

v 6

1þ d ln e6=dln u6 d ln e5=dln u6

ð28Þ Results of applying this method are given in the bottom rows of Table 3 Again, this proved to be an accurate method

D I S C U S S I O N

We proposed four alternative strategies to measure meta-bolic (or intramodular) control in hierarchical biochemical systems The proposed methods were illustrated using a kinetic model and its parameters were chosen to obtain a high ratio between the intramodular and the integral control coefficients (also called A-coefficient; [14]) In real bio-chemical systems the values of two different types of control coefficients might be either closer or further apart When the values of the two coefficients are closer, it will be more difficult to distinguish the two

Table 3 Values ofglobal and intramodular control coefficients calculated using method 4, Eqns (24,25) for the system with one variable enzyme and Eqns (27,28) for the system with two variable enzymes.

System with one variable enzyme

C ½X  ss

c½X ss

System with two variable enzymes

C½X ss

The first two methods are motivated by the time-scale

separation that might exist between the dynamics of

intermediary metabolism and gene expression Such time

separation is mainly determined by the difference in the

degradation rates of the different levels [14,19] When this

difference is sufficiently large, the initial behavior of the

system is determined by the intramodular (metabolic)

control, and the final behavior by the integral control In

our models a difference of two to three orders of magnitude

proved sufficient to observe this effect Smaller differences in

time-scales reduced the accuracy at which the intramodular

control coefficients could be measured Estimating for

major metabolic pathways of E coli metabolite turnover

times (concentration divided by flux) of a few seconds,

whereas most enzymes last many cell cycles of longer than

30 min, the characteristic times may indeed be more than a

factor of 600 apart That is of the order of the required

factor of 100–500 Similar estimates apply to yeast

glyco-lysis However, glucose transporters can be downregulated

by internalization at time-scales of a few minutes,

compro-mising the distinction between metabolic and hierarchical

regulation In various anabolic routes, both the flux and the

concentrations are often lower by a factor of 100, leading to

the same metabolic turnover times and the same gap

between metabolic and gene-expression regulation times

scales In cases of metabolite channeling, metabolic response

times will even be faster

In EGF-induced signal transduction in mammalian cells,

there is a first fast phase that is at a time-scale close to

metabolic time-scales [25] Yet, much of the final effect

happens at the much slower, gene-expression regulation

time-scale of hours and perhaps days It is not yet clear the

significance of the early fast dynamics of this system, if not

to turn on a switch [26] The ability to discriminate between

fast and slow control, as elaborated in the present

manu-script, may help understand the function of signal

trans-duction networks, which often have more than one

characteristic time constant

For our model system, we note that should the

feedback interaction of the metabolite to transcription

be stronger (lower Ka), the time-scales would come closer

(as measured by eigenvalues of the Jacobian [27] or by

transient times [28]) With decreasing Ka, the fast

time-scale decreased towards the slow time-time-scale (results not

shown) One should thus be cautious when reasoning

solely on the basis of rate constants of transcription and

metabolism, without knowledge of the strength of

inter-actions between these levels

In our demonstration of methods 1 and 2, the sampling

frequency of the measurements was rather high because

both methods require one to locate a minimum of the

first-order derivative of the curve In practice, it might be difficult

to make measurements at this frequency and thus the

quasi-steady-state could be missed or misplaced, resulting in larger

error It is advisable to fit the time-course to a function first

and then locate the quasi-steady-state from the zero of the

derivative of this function

Method 2 did not prove any better than method 1 This is

because the method itself perturbs the steady-state at

about the same amount, as the relaxation phase sets in

that separates intralevel from global control The rate of

change of mRNA should be inhibited in order to keep

the concentration of mRNA constant By inhibiting the

transcription rate alone, one makes an additional change in the concentration of mRNA As the mRNA continues to be degraded, while its synthesis is being stopped, its steady-state balance is perturbed An alternative method would inhibit both transcription and mRNA degradation, such that the level of mRNA would remain constant This is difficult to achieve experimentally and thus was not considered here

Methods 3 and 4 are similar in that both remove the feedback loopfrom metabolism to gene expression, either physically (by replacing the promoter) or mathematically The problem with method 3 is that it only works when there is a single feedback loop(or two in case of method 4) In living cells there are certainly more feedback loops from metabolism to gene expression, so one still measures the global control of the system, but without that particular feedback loop [11] In order to measure intramodular control, one would have to replace all promoters Method 4 is applicable to systems consisting of many variable enzymes, provided that one measures all those enzymes and the control coefficients of all but one

of them, severely limiting its application to real systems In the case of a system with two enzymes the summation theorems can be used to express the control coefficients of one stepin terms of the control coefficients of the other When considering more enzymes one would need to measure the global control for all but one of them, and the concentration change for all of them, to be able to solve a set of equations, like Eqn (5), for the intramodular control coefficients

It remains to be seen if there are real biological systems

in which gene expression and metabolism operate on similar time-scales In that case methods 1 and 2 would be hard to apply It is also an open research topic whether gene expression and metabolism are tightly coupled by feedback, although the technology to determine this is becoming available We suspect that, as usual, diversity will prove to be abundant and each system will have its own characteristics The concept of intramodular control and the methods introduced for its measurement will be much more relevant in situations where there is only loose coupling When the coupling between metabolism and gene expression is strong the concept of hierarchies is less useful Even though they would continue to carry significance conceptually, one should then perhaps treat all levels together as a single system HCA is therefore beneficial when compared to MCA, merely because it simplifies the mathematics

In many cases, there is a considerable time-scale separation between metabolism and the mRNA and protein levels For these cases, the relevance of HCA and the present method is that they are able to distinguish between the control exerted all within one level (e.g between the metabolic reactions) from the control of one level over another HCA allows one

to describe these two types of control and has exact laws to relate them The methods we have proposed here allow their experimental implementation

The distinction between metabolic and global control is crucial for the understanding of the regulation of cell physiology An example is catabolite repression by glucose, which is very common in biology This works via metabolic effects, signal transduction and gene-expression The impli-cations of the three types of mechanism differ greatly for the

dynamics and persistence of the regulation A persistent

catabolite repression mechanism would make baker’s yeast

useless for the baker, who uses mostly maltose For humans,

gene-expression regulation of glucose uptake after a rich

meal should result in a subsequent undershoot in glucose

levels, unless compensated by additional insulin-dependent

regulation On the other hand, gene expression-mediated

regulation is the one that permits the best homeostasis of

intracellular metabolites, and may hence lead to the most

optimal state

Our approach is fundamentally different from the work

of Acerenza et al [29] and Heinrich & Reder [30], who

studied the time-dependent control analysis (i.e quantifying

control of reactions on the relaxation processes) Although

based on observation of time-courses, our methods do not

extend MCA to the time domain Simply, we describe a way

of locating a quasi-steady-state on the time-course, followed

by analysis with the traditional MCA approach, as if it was

a true steady-state This has led to an emphasis on small

changes (perhaps smaller than may be experimentally

feasible), steady-states, control, and regulation Aspects of

spatial heterogeneity, and experimental errors [21] deserve

scrutiny in future work We note that the present results are

essentially the same when we applied 10% rather than 1%

perturbations (data not shown)

The enhanced ability to distinguish between metabolic

and hierarchical regulation will greatly increase our

understanding of living organisms This becomes acute

with the greatly enhanced abilities to measure gene

expression (transcriptome [31] and proteome [32]) and the

metabolome [15] in parallel and quantitatively As cell

function depends on both, and in many interconnected

ways [13,23], progress may well depend on our ability to

dissect metabolic from hierarchical regulation The four

methods developed in this paper would therefore be

relevant to further studies

A C K N O W L E D G E M E N T S

ALF and PM thank the Commonwealth of Virginia and the National

Science Foundation (grant Bes-0120306) for financial support.

R E F E R E N C E S

1 Kacser, H & Burns, J.A (1973) The control of flux Symp Soc.

Exp Biol 27, 65–104.

2 Heinrich, R & Rapoport, T.A (1974) A linear steady-state

treatment of enzymatic chains General properties, control and

effector strength Eur J Biochem 42, 89–95.

3 Burns, J.A., Cornish-Bowden, A., Groen, A.K., Heinrich, R.,

Kacser, H., Porteous, J.W., Rapoport, S.M., Rapoport, T.A &

Tager, J.M (1985) Can we agree on metabolic control? Trends

Biochem Sci 10, 152.

4 Fell, D.A & Sauro, H.M (1985) Metabolic control and its

ana-lysis Additional relationships between elasticities and control

coefficients Eur J Biochem 148, 555–561.

5 Sauro, H.M., Small, J.R & Fell, D.A (1987) Metabolic control

and its analysis Extensions to the theory and matrix method Eur.

J Biochem 165, 215–221.

6 Westerhoff, H.V & Kell, D.B (1987) Matrix method for

determining steps most rate-limiting to metabolic fluxes in

biotechnological processes Biotechnol Bioeng 30, 101–107.

7 Reder, C (1988) Metabolic control theory A structural approach.

J Theoret Biol 135, 175–201.

8 Westerhoff, H.V & van Workum, M (1990) Control of DNA structure and gene expression Biomed Biochim Acta 49, 839– 853.

9 Kahn, D & Westerhoff, H.V (1991) Control theory of regulatory cascades J Theor Biol 153, 255–285.

10 Westerhoff, H.V & Kahn, D (1993) Control involving metabo-lism and gene expression: the square-matrix method for modular decomposition Acta Biotheor 41, 75–83.

11 Jensen, P.R., Van Der Weijden, C.C., Jensen, L.B., Westerhoff, H.V & Snoep, J.L (1999) Extensive regulation com-promises the extent to which DNA gyrase controls DNA super-coiling and growth rate of Escherichia coli Eur J Biochem 266, 865–877.

12 Snoep, J.L., van de Weijden, C.C., Andersen, H.W., Westerhoff, H.V & Jensen, P.R (2000) Hierarchical control of DNA super-coiling in Escherichia coli: how to study homeostatically controlled sytems using control analysis BTK2000: Animating the Cellular Map (Hofmeyr, J.-H.S., Rohwer, J.H & Snoep, J.L., eds), Stel-lenbosch University Press, StelStel-lenbosch.

13 Snoep, J.L., Van der Weijden, C.C., Andersen, H.W., Westerhoff, H.V & Jensen, P.R (2002) DNA supercoiling in Escherichia coli is under tight and subtle homeostatic control, involving gene-expression and metabolic regulation of both topoisomerase I and DNA gyrase Eur J Biochem 269, 1662–1669.

14 Hofmeyr, J.H & Westerhoff, H.V (2001) Building the cellular puzzle: control in multi-level reaction networks J Theoret Biol.

208, 261–285.

15 Raamsdonk, L.M., Teusink, B., Broadhurst, D., Zhang, N., Hayes, A., Walsh, M.C., Berden, J.A., Brindle, K.M., Kell, D.B., Rowland, J.J., Westerhoff, H.V., van Dam, K & Oliver, S.G (2001) A functional genomics strategy that uses metabolome data

to reveal the phenotype of silent mutations Nat Biotechnol 19, 45–50.

16 Mendes, P (1993) GEPASI: a software package for modelling the dynamics, steady-states and control of biochemical and other systems Comput Appl Biosci 9, 563–571.

17 Mendes, P (1997) Biochemistry by numbers: simulation of bio-chemical pathways with Gepasi 3 Trends Biochem Sci 22, 361– 363.

18 Mendes, P & Kell, D.B (1998) Non-linear optimization of bio-chemical pathways: applications to metabolic engineering and parameter estimation Bioinformatics 14, 869–883.

19 de la Fuente van Bentem, A., Mendes, P., Westerhoff, H.V & Snoep, J.L (2000) Can metabolic control analysis be applied to hierarchical regulated metabolism? MCA versis HCA BTK2000: Animating the Cellular Map (Hofmeyr, J.-H.S., Rohwer, J.H & Snoep, J.L., eds), pp 191–198 Stellenbosch University Press, Stellenbosch.

20 Liao, J.C & Delgado, J (1992) Dynamic metabolic control theory – a methodology for investigating metabolic regula-tion using transient metabolic data Ann NY Acad Sci 165, 27–38.

21 Sorribas, A., Samitier, S., Canela, E.I & Cascante, M (1993) Metabolic pathway characterization from transient response data obtained in-situ Parameter estimation in S-system models.

J Theoret Biol 162, 81–102.

22 Jensen, P.R., Westerhoff, H.V & Michelsen, O (1993) The use of lac-type promoters in control analysis Eur J Biochem 211, 181– 191.

23 ter Kuile, B.H & Westerhoff, H.V (2001) Transcriptome meets metabolome: hierarchical and metabolic regulation of the glyco-lytic pathway FEBS Lett 500, 169–171.

24 Westerhoff, H.V., Jensen, P.R., Egger, L., Van Heeswijk, W.C., Van Spanning, R.J.M., Kholodenko, B.N & Snoep, J.L (1998) Hierarchical control of electron-transfer paper presented at the NATO/ESF Workshop Bioelectron-Transfer Chains NATO/ESF, Tomar, Portugal.

25 Kholodenko, B.N., Demin, O.V., Moehren, G & Hoek, J.B.

(1999) Quantification of short term signaling by the epidermal

growth factor receptor J Biol Chem 274, 30169–30181.

26 Kholodenko, B.N., Hoek, J.B & Westerhoff, H.V (2000) Why

cytoplasmic signalling proteins should be recruited to cell

mem-branes Trends Cell Biol 10, 173–178.

27 Stucki, J.W (1978) Stability analysis of biochemical systems A

practical guide Progr Biophys Mol Biol 33, 99–187.

28 Easterby, J.S (1981) A generalized theory of the transition time

for sequential enzyme reactions Biochem J 199, 155–161.

29 Acerenza, L., Sauro, H.M & Kacser, H (1989) Control analysis of

time dependent metabolic systems J Theoret Biol 137, 423–444.

30 Heinrich, R & Reder, C (1991) Metabolic control analysis of

relaxation processes J Theoret Biol 151, 343–350.

31 Schena, M., Shalon, D., Davis, R.W & Brown, P.O (1995) Quantitative monitoring of gene expression patterns with a com-plementary DNA microarray Science 270, 467–470.

32 Mann, M., Hendrickson, R.C & Pandey, A (2001) Analysis of proteins and proteomes by mass spectrometry Ann Rev Biochem.

70, 437–473.

S U P P L E M E N T A R Y M A T E R I A L

The following material is available from http://www blackwell-science.com/products/journals/suppmat/EJB/EJB 3088/EJB3088sm.htm

The derivation of Eqns (25), (27) and (28)