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

Metabolic control in integrated biochemical 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 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: 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 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. 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 control Control 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 in integrated biochemical 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 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 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 in integrated biochemical 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) 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 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 control Control 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 in integrated biochemical 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. 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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