Metaboliccontrolinintegratedbiochemical systems
Alberto de la Fuente
1
, Jacky L. Snoep
2
, Hans V. Westerhoff
3,4
and Pedro Mendes
1
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 metaboliccontrol analysis to hierarchical biochemical
systems. We discuss four methods that enable the experi-
mental determination of metaboliccontrol 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: metaboliccontrol 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:
R
Y
P
¼
@Y=Y
@P=P
¼
@ ln Y
@ ln P
ð1Þ
Y is 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:
C
½X
ss
v
¼
@ ln½X
ss
@ ln v
ð2Þ
C
J
ss
v
¼
@ ln J
ss
@ ln v
ð3Þ
where [X] is the concentration of the metabolite in question,
J the 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:
e
v
x
¼
@v=v
@x=x
¼
@ ln v
@ ln x
ð4Þ
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-translational 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)
Eur. J. Biochem. 269, 4399–4408 (2002) Ó FEBS 2002 doi:10.1046/j.1432-1033.2002.03088.x
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 controlin 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 metabolicsystem 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.
MODEL AND METHODS
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
loop for 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):
v
1
¼
V
1
½nucleotides
K
m
1
1 þ
½nucleotides
K
m
1
þ
K
a
½metabolite
ð5Þ
v
2
¼ k
2
[mRNA] ð6Þ
v
3
¼ k
3
[mRNA] ð7Þ
v
4
¼ k
4
[enzyme] ð8Þ
v
5
¼
V
f
½S
K
mS5
À V
r
½metabolite
K
mP5
1 þ
½S
K
mS5
þ
½metabolite
K
mP5
ð9Þ
v
6
¼½enzyme
k
f
cat
½metabolite
K
mS6
À k
r
cat
½P
K
mP6
1 þ
½metabolite
K
mS6
þ
½P
K
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 concentration of
nucleotides, V
1
the limiting transcription rate, K
m1
the
Michaelis constant for nucleotides, K
a
the activation
constant of transcription by the metabolite, k
2
the rate
constant for mRNA degradation (and dilution due to cell
growth), k
3
the translation rate-constant, k
4
the enzyme
degradation rate-constant (and dilution due to cell growth),
V the limiting rate of reaction 5, K
eq5
the equilibrium
constant for reaction 5, K
mS5
the Michaelis constant for the
substrate, K
mP5
the Michaelis constant of reaction 5 for the
metabolite, k
cat
the catalytic rate constant of the enzyme of
step 6, K
eq6
the equilibrium constant of reaction 6 and K
mP6
the Michaelis constant for the pathway’s product. It should
be noted that step 5 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 up to
transcription. This contrasts to ÔdictatorialÕ systems in which
there are no arrows from metabolism up to 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 of translation on mRNA
concentration, the metabolic rate on enzyme concentration and the
activation of transcription 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.
4400 A. de la Fuente et al. (Eur. J. Biochem. 269) Ó FEBS 2002
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 k
cat
, 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
(k
t
( k
m
). 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 package
GEPASI
[16–18].
The intramodular control coefficients, to be indicated by
lower case ÔcÕ, can be expressed as a function of the elasticity
coefficients:
c
J
ss
v
5
¼
e
v
6
½X
e
v
6
½X
À e
v
5
½X
ð11Þ
c
J
ss
v
6
¼
e
v
5
½X
e
v
5
½X
À e
v
6
½X
ð12Þ
c
½X
ss
v
5
¼
1
e
v
6
½X
À e
v
5
½X
ð13Þ
c
½X
ss
v
6
¼
1
e
v
5
½X
À e
v
6
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)canbe
derived similarly:
C
J
ss
v
5
¼
e
v
6
½X
e
v
6
½X
À e
v
5
½X
þ T ð15Þ
C
J
ss
v
6
¼
e
v
5
½X
e
v
5
½X
À e
v
6
½X
À T ð16Þ
where:
C
½X
ss
v
5
¼
1
e
v
6
½X
À e
v
5
½X
1 À
e
v
1
½X
e
v
3
½N
e
v
6
½E
e
v
2
½N
Àe
v
1
½N
e
v
4
½E
Àe
v
3
½E
e
v
5
½X
Àe
v
6
½X
0
@
1
A
ð18Þ
C
½X
ss
v
6
¼ÀC
½X
ss
v
5
ð19Þ
N stands for mRNA and E for enzyme.
RESULTS
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 of the model systems
described in Fig. 1 and Eqns (5–10).
Rate Parameter Value
v
1
V
1
0.01
[nucleotides] 1
K
m1
10
K
a
100
v
2
k
2
0.01
v
3
k
3
0.1
v
4
k
4
0.01
v
5
V
f
100
V
r
1
[S] 1
K
mS5
1
K
mP5
10
v
6
k
f
cat
100
k
r
cat
1
[P] 0.1
K
mS6
10
K
mP6
1
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 controlControl coefficient Value
Intra-modular c
J
ss
v
5
0.28
c
J
ss
v
6
0.72
c
½X
ss
v
5
1.10
c
½X
ss
v
6
)1.10
Integral C
J
ss
v
5
0.60
C
J
ss
v
6
0.40
C
½X
ss
v
5
0.61
C
½X
ss
v
6
)0.61
T ¼
e
v
1
½X
e
v
5
½X
e
v
3
½N
e
v
6
½E
e
v
6
½X
À e
v
5
½X
e
v
2
½N
À e
v
1
½N
e
v
4
½E
À e
v
3
½E
e
v
5
½X
À e
v
6
½X
À e
v
1
½X
e
v
3
½N
e
v
6
½E
ð17Þ
Ó FEBS 2002 MCA inintegratedbiochemical systems (Eur. J. Biochem. 269) 4401
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 concentration of the
metabolite should first evolve to a metabolic quasi-steady-
state. This apparent steady-state should be close to the
one that the metabolicsystem 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 steady-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:
c
Y
ss
v
5
¼
Y
ssðnewÞ
À Y
ssðinitialÞ
V
5ðperturbedÞ
À V
5ðinitialÞ
Á
V
5ðinitialÞ
Y
ssðinitialÞ
ð20Þ
where Y represents any system variable, e.g. the flux
through the pathway [as in Eqn (1)]. The modulation of v
5
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 steady-
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 of the model system at different magnitudes of transcription 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.
4402 A. de la Fuente et al. (Eur. J. Biochem. 269) Ó FEBS 2002
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 v
5
,onthe
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
movedtothemetabolicsteady-stateandthendecreasedto
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
of only 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.
Ó FEBS 2002 MCA inintegratedbiochemical systems (Eur. J. Biochem. 269) 4403
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
step 6. 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 step 6) 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 v
5
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 step can 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 step 5 are taken to vary identically to mRNA and Enzyme for step 6. (A) Concentration-control coefficients. (B) Flux-control coefficients.
4404 A. de la Fuente et al. (Eur. J. Biochem. 269) Ó FEBS 2002
product of two factors, one dependent only on the enzyme
activity (e
i
) and another representing the kinetic mechanism
(u
i
) [23]:
v
6
¼ u
6
 e
6
ð21Þ
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)].
C
J
ss
v
6
¼
d ln J
ss
d ln u
6
ð22Þ
C
½X
ss
v
6
¼
d ln½X
ss
d ln u
6
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:
c
J
ss
v
6
¼
d ln J
ss
d ln u
6
þ d ln e
6
¼
C
J
ss
v
6
1 þ d ln e
6
=d ln u
6
ð24Þ
c
½X
ss
v
6
¼
d ln½X
ss
d ln u
6
þ d ln e
6
¼
C
X
ss
v
6
1 þ d ln e
6
=d ln u
6
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 top rows of Table 3. It is seen that
this method is rather accurate.
Eqns (24) and (25) are only valid when the enzyme of the
step under 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 step 5. Degradation
and translation kinetics are identical to that of the gene for
step 6. Transcription of this gene is affected by the meta-
bolite through a mechanism of competitive inhibition:
v
7
¼
V
7
½nucleotides
K
m
7
1 þ
½nucleotides
K
m
7
þ
½metabolite
K
I
7
ð26Þ
V
7
is the limiting transcription rate for this gene, K
m7
the
Michaelis constant for the nucleotides and K
I7
the inhibition
constant of the metabolite. Parameter values are
V
7
¼ 0.001, K
m7
¼ 1andK
I7
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
c
J
ss
v
6
¼
d ln J
ss
À d ln e
5
d ln u
6
þ d ln e
6
À d ln e
5
¼
C
J
ss
v
6
À d ln e
5
=d ln u
6
1 þ d ln e
6
=d ln u
6
À d ln e
5
=d ln u
6
ð27Þ
and for the intramodular concentration control coefficient
c
½X
ss
v
6
¼
d ln½X
ss
d ln u
6
þ d ln e
6
À d ln e
5
¼
C
½X
ss
v
6
1 þ d ln e
6
=d ln u
6
À d ln e
5
=d ln u
6
ð28Þ
Results of applying this method are given in the bottom
rows of Table 3. Again, this proved to be an accurate
method.
DISCUSSION
We proposed four alternative strategies to measure meta-
bolic (or intramodular) controlin 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 of global 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.
Type of controlControl coefficient Real value Calculated
System with one variable enzyme
Integral C
J
ss
v
6
0.4 0.4
C
½X
ss
v
6
)0.61 )0.62
Intra-modular c
J
ss
v
6
0.72 0.73
c
½X
ss
v
6
)1.1 )1.12
System with two variable enzymes
Integral C
J
ss
v
6
0.43 0.43
C
½X
ss
v
6
)0.54 )0.54
Intra-modular c
J
ss
v
6
0.68 0.69
c
½X
ss
v
6
)1.12 )1.14
Ó FEBS 2002 MCA inintegratedbiochemical systems (Eur. J. Biochem. 269) 4405
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 K
a
), the time-scales would come closer
(as measured by eigenvalues of the Jacobian [27] or by
transient times [28]). With decreasing K
a
, the fast time-
scale decreased towards the slow 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 loop from 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
thecaseofasystemwithtwoenzymesthesummation
theorems can be used to express the control coefficients of
one step in 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
4406 A. de la Fuente et al. (Eur. J. Biochem. 269) Ó FEBS 2002
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.
ACKNOWLEDGEMENTS
ALF and PM thank the Commonwealth of Virginia and the National
Science Foundation (grant Bes-0120306) for financial support.
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SUPPLEMENTARY MATERIAL
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).
4408 A. de la Fuente et al. (Eur. J. Biochem. 269) Ó FEBS 2002
. helps clarifying the properties of
the ubiquitous and more interesting democratic systems.
In order to determine the metabolic intramodular control
coefficients,. 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