Eur J Biochem 271, 4375–4391 (2004) Ó FEBS 2004 doi:10.1111/j.1432-1033.2004.04375.x An extension to the metabolic control theory taking into account correlations between enzyme concentrations ´ ´ ´ ´ Sebastien Lion1,*, Frederic Gabriel1, Bruno Bost2, Julie Fievet1, Christine Dillmann1 and Dominique de Vienne1 UMR de Ge´ne´tique Ve´ge´tale, INRA/UPS/CNRS/INAPG, Ferme du Moulon, Gif-sur-Yvette, France; 2Institut de Ge´ne´tique et Microbiologie, CNRS UMR 8621, Universite´ Paris Sud, Orsay Cedex, France The classical metabolic control theory [Kacser, H & Burns, J.A (1973) Symp Soc Exp Biol 27, 65–104; Heinrich, R & Rapoport, T (1974) Eur J Biochem 42, 89–95.] does not take into account experimental evidence for correlations between enzyme concentrations in the cell We investigated the implications of two causes of linear correlations: competition between enzymes, which is a mere physical adaptation of the cell to the limitation of resources and space, and regulatory correlations, which result from the existence of regulatory networks These correlations generate redistribution of enzyme concentrations when the concentration of an enzyme varies; this may dramatically alter the flux and metabolite concentration curves In particular, negative correlations cause the flux to have a maximum value for a defined distribution of enzyme concentrations Redistribution coefficients of enzyme concentrations allowed us to calculate the Ôcombined response coefficientÕ that quantifies the response of flux or metabolite concentration to a perturbation of enzyme concentration The introduction of the metabolic control theory by Kacser & Burns [1] and Heinrich & Rapoport [2] was a great improvement in our understanding of the control of metabolism (for a review see [3]) Numerous extensions to the classical theory have been proposed to get rid of some restrictive hypotheses of the initial theory Extensions exist, for example, for nonproportionality of the rates of reaction to enzyme concentration [4], enzyme–enzyme interaction [5,6], time-varying systems [7,8], or supply–demand analysis [9] Nevertheless, most studies have neglected the correlations that exist between enzyme amounts in the cell Concentration is a key parameter for enzyme activity Changes in expression of enzyme genes play a central role in the physiology of the cell, and dramatic modifications of the cell proteome are consistently observed over development and differentiation, or in response to environmental changes (see http://us.expasy.org for examples in various species) In addition, genetic studies have revealed natural variability for enzyme concentration, for instance for alcohol dehydrogenase in Drosophila [10] or lactate dehydrogenase in Fundulus heteroclitus [11] Other examples can also be found [12,13] Quantitative proteomic approaches have confirmed that a majority of proteins/enzymes can display a large range of variation within species [14–19] Those physiological or genetic variations are expected to be interdependent There is evidence for cellular constraints that induce a variation of concentration of some enzymes in response to a variation of others These correlations between enzyme concentrations undoubtedly have an impact on the behaviour of metabolic systems, and hence on their evolution Two kinds of correlations will be studied in this paper The first one will be referred to as competition It is a mere physical adaptation of the cell to energetic or steric constraints The second one results from regulatory networks It will be referred to as regulation Competitive constraints on the variation of enzyme concentrations have already been pointed out Such constraints have the effect of avoiding macromolecular crowding, which can result in a modification of catalytic and/or thermodynamic properties of enzymes [20], in a limitation of solubility leading to partial protein crystallization or aggregation [21,22], or a decrease in the diffusion of essential metabolites ([23], for a review see [24]) Other arguments include the limitation of resources, the energetic cost of maintaining the cellular concentrations of enzymes [25–27], and the availability of amino acids or elements of the transcription and translation machinery, which has been shown to be a limiting factor of protein synthesis in Escherichia coli [28] and Saccharomyces cerevisiae [29] Kacser & Beeby [30] were among the first to suggest that the hyperbolic flux–activity relationship must ultimately decline, for no more profound reason than that the cell or organism must eventually reach a point at which the cost of producing excess enzyme outweighs the benefit in fitness that can be derived from possessing the excess [31,32] It is clear that such competitive constraints imply that variations of enzyme concentrations are negatively correlated: an increase in the concentration of some enzymes ´ ´ ´ ´ Correspondence to D de Vienne, UMR de Genetique Vegetale, INRA/UPS/CNRS/INAPG, Ferme du Moulon, 91190 Gif-surYvette, France Fax: +33 69 33 23 40, Tel.: +33 69 33 23 60, E-mail: devienne@moulon.inra.fr ´ ´ ´ *Present address: Laboratoire d’ecologie, Ecole normale superieure, 46, rue d’Ulm, 75005 Paris, France (Received 19 July 2004, revised 20 September 2004, accepted 22 September 2004) Keywords: biochemical modelling; cellular constraint; flux; metabolite; response coefficient Ó FEBS 2004 4376 S Lion et al (Eur J Biochem 271) causes a decrease in the concentration of other enzymes, which can lead to important metabolic perturbations, i.e to the so-called protein burden effect [33] For instance, overexpression of b-galactosidase in E coli was found to reduce the synthesis of the other proteins [34] and overexpression of glycolysis enzymes in Zymomonas mobilis has been shown to reduce glycolytic flux [35]: therefore, for large enzyme concentrations, the classical hyperbolic shape of the flux curve, as predicted by the metabolic control theory, does not describe in a satisfactory way the behaviour of the metabolic pathway Flux can be expected to decrease when enzyme concentration becomes too high, and it may be interesting to model such behaviour Regulatory correlations can be positive or negative The production and degradation of enzymes, which determines their concentration, is related to the structure of the genetic regulatory network [36] The lactose operon in E coli [37] is a well known example of a regulatory system that induces correlations between the concentrations of the enzymes involved in lactose metabolism Several experimental and theoretical studies have been devoted to the understanding of the mechanisms of regulatory networks [38–41] Metabolic engineering makes an important use of regulation of metabolic pathways to achieve overexpression of the products of interest For instance, Prati et al [42] reported a way to achieve simultaneous inhibition and activation of two glycosyltransferases of the O-glycosylation pathway in Chinese hamster ovary cells In Lactococcus lactis, several genes of glycolysis have been shown to be expressed at higher levels on glucose than on galactose [43] The authors interpreted this as a result of two different regulatory networks With the growing use of quantitative proteomics methods, we can expect to find many more examples of correlations between enzymes, even if we still lack the tools to determine whether regulatory networks actually underlie these correlations The existence of these competitive and regulatory correlations between enzymes is assumed to constrain the response of the metabolic systems Here, we present an extension of the metabolic control theory in which response coefficients allow us to quantify the change of a metabolic variable (flux or metabolite concentration) in response to a perturbation of a parameter (enzyme concentration) and to the variations of other parameters resulting from that perturbation We apply the general concept of a Ôcombined response coefficientÕ to a linear model of redistribution of enzyme concentrations in order to study the systemic consequences of enzyme correlations Control of metabolic pathways and redistribution of enzyme concentrations Control of metabolic variables Let us consider a metabolic pathway with n enzymes E1, E2, …, En catalyzing reversible reactions between substrates S1, …, Sm (m metabolites) To quantify the response of a systemic variable y, such as thefluxinthe pathwayor theconcentrationofametabolite,to an infinitesimal change in the activity (concentration) of enzyme Ek, Kacser & Burns [1] and Heinrich & Rapoport [2] introduced the control coefficient In the revised nomen- clature for metabolic control analysis, the control coefficient Cy is defined as the steady-state response in y to a change in k the local rate of step k, vk, with no reference to enzyme concentration (http://www.sun.ac.za/biochem/mcanom.html) In particular, the control coefficient of flux J with respect to step k is: CJ ¼ k vk @J J @vk and the control coefficient of metabolite concentration Si with respect to step k is: CSi ¼ k vk @Si Si @vk Summation theorems can be derived for metabolite and flux control coefficients Summing over all reactions, we have [1]: n X j¼1 CJ ¼ and j n X CSi ¼ j j¼1 These relationships show that the control of flux (or metabolite concentration) is shared among all enzymes in the pathway Control coefficients are systemic properties We can also define local properties such as the elasticity, which quantifies the effect of any parameter p that affects the local rate of an individual (isolated) step The elasticity coefficient ek for p step k is written as [1]: p @vk ek ¼ p vk @p Introducing correlations between enzyme concentrations The classical form of metabolic control theory implicitly considers that enzyme concentration can increase towards infinity, which is biologically inconsistent Competitive and regulatory constraints on enzyme concentrations exist, that can be described with a model of redistribution of enzyme concentrations We considered a system starting in a state defined by the concentrations E0 ¼ ðE01 ; E02 ; :::; E0k ; :::; E0n Þ of the n enzymes, and supposed that a variation of the concentration of a target enzyme Ek results in a variation of the concentrations of other enzymes Redistribution coefficient In order to quantify the impact of variation of enzyme Ek on enzyme Ej, we defined the redistribution coefficient (akj) as the ratio of an infinitesimal change in the concentration Ej to an infinitesimal change in the concentration Ek: @Ej ð1Þ akj ¼ @Ek In this framework, the enzyme concentrations become interdependent parameters Combined response coefficient of the flux If an effector p acts on the flux through its effect on enzyme j, the response Ó FEBS 2004 An extension to the metabolic control theory (Eur J Biochem 271) 4377 coefficient RJ is the product of the flux response coefficient p with respect to enzyme j and the elasticity of enzyme j with respect to p [1]: RJ ¼ CJ ejp j p Let us now assume that the effector p acts on more than one enzyme in a metabolic pathway We can define the overall, multisite response obtained from the n enzymes of the system as [44,45]: n X RJ ¼ CJ ejp j p j¼1 This is only true for very small changes in p because the response coefficient is defined as a first order approximation For a large change in p, we should add correction terms to account for nonlinearities Considering an effector p causing the redistribution of enzyme concentrations through the modification of concentration Ek of the enzyme Ek (e.g p is a mutation causing an increase of Ek and consequently modification of other enzyme concentrations), we can write, replacing p by Ek: n X RJ k ¼ CJ ejk j E j¼1 Assuming that the response of an isolated reaction is directly proportional to change in enzyme concentration, we have: ejk ¼ Ek akj Ej so that RJ k ¼ Ek E n X CJ j j¼1 akj Ej ð2Þ We call RJ k the combined response coefficient [46] We E will show later (in the case of a linear metabolic pathway) that the combined response coefficient can be equivalently written as: RJ k ¼ E Ek @J J @Ek Linear models of redistribution of enzyme concentrations We assumed linear redistribution, which means that akj is considered to be constant Figure shows how enzyme concentrations are redistributed due to their correlations Figure 1A corresponds to the case of independent enzyme concentrations that was studied in the founding papers of the metabolic control theory [1,2] Figure 1B–G corresponds to various constraints that result in a redistribution of enzyme concentrations over the variation of a particular enzyme Let us examine these constraints, the mathematical expressions of which are summarized in Table We focus on a linear model of redistribution of enzyme concentrations but other models are possible Let us further introduce the normalized concentration ej defined as: ej ¼ Ej Etot where where the partial derivative is taken on a set of enzyme concentrations that describes the correlations between enzymes Biologically speaking, this means that the combined response coefficient contains information about the correlations between the enzyme concentrations, hence the term ÔcombinedÕ We can see the combined response coefficient as a Ôresponse coefficient under constraintÕ We can split Eqn into two terms: X akj RJ k ẳ CJ akk ỵ Ek CJ k j E Ej j6¼k Note that akk ¼ (Eqn 1), so that X akj CJ RJ k ẳ CJ ỵ Ek k j E Ej j6¼k enzyme Ek on the flux, and (b) the effect of enzyme Ek on the others, through the redistribution rules, which is modulated by the control exerted by those enzymes on the flux Thus, even if enzyme Ek has a high control coefficient on the flux, an increase of Ek should cause a decrease of the flux if Ek is negatively correlated with the concentrations of other enzymes We can also note that the response coefficient of enzyme Ek will be higher than its control coefficient in cases where enzyme Ek is positively correlated with at least one other enzyme ofthepathway, and notcorrelated totheothers Thus, we have given a general expression for the combined response coefficient of the flux, valid for a network of any complexity, with no assumption on the rules of redistribution of enzyme concentrations In the next paragraph, we will present the theoretical framework that allowed us to describe the linear correlations between enzyme concentrations, and in the second part of the paper we will analyse in detail the particular case of a linear pathway of enzymes far from saturation, considering the response of both flux and metabolite concentrations Etot ¼ n X Ej j¼1 Competitive correlations In order to take into account the fact that enzyme concentrations are likely to be bounded, Heinrich et al [47–49], and de Vienne et al [46], have proposed to put a constraint on the total concentration Etot of the enzymes in the pathway In this paper, this is designed as competition and we limit the study to the quite rigid constraint where Etot is a constant We have: n X Ej ẳ Etot ẳ const jẳ1 3ị The effect of a variation of enzyme Ek on the flux appears then to be dependent on two factors: (a) the control exerted by Using the normalized concentration ej, the competitive constraint on the metabolic pathway reads n X ej ¼ ð4Þ j¼1 Ĩ FEBS 2004 4378 S Lion et al (Eur J Biochem 271) A B C D E F G Fig Redistribution of enzyme concentrations when the concentration of the target enzyme changes We considered a six-enzyme pathway E3 is the concentration of the target enzyme The y-axis shows the concentrations of enzymes E1, E2, E4, E5 and E6 Unless otherwise stated, the starting distribution of enzyme concentrations is the vector E0 ¼ (0.04,0.02,0.04,0.37,0.44,0.09), indicated with dots on the figures The concentration of the target enzyme varies either between and Etot ¼ 1, or between E3,min and E3,max, depending on the constraints imposed on the system (A) Independence between enzyme concentrations (B) Pure regulation with positive correlations The redistribution coefficients are a3 ¼ b3 ¼ (0.99,0.63,1,0.94,0.43,0.29) (C) Pure regulation with one enzyme being negatively correlated a34 ¼ )0.94, the other redistribution coefficients being the same as in (B) (D) Competition when the starting distribution of enzyme concentrations is the optimal one, which maximizes the flux (E) Competition when the starting distribution of enzyme concentrations is E0 (F) Regulation with competition The starting distribution of enzyme concentrations is E1 ¼ (0.13,0.13,0.31,0.04,0.02,0.37) and the redistribution coefficients are a3 ¼ (0.05,0.05,1,0.5,0.5,-2.1) (G) Regulation with competition when the starting distribution of enzyme concentrations is E0 and coregulation coefficients b (Eqn 6) are as in (B) In systems with only competitive constraints, the concentrations of enzymes Ej ("j „ k) decrease when the concentration of enzyme Ek increases and the proportions of enzymes Ej remain constant So we can define the competition coefficient ckj between enzymes Ek and Ej (i.e the constant proportion between these enzymes) as Ó FEBS 2004 An extension to the metabolic control theory (Eur J Biochem 271) 4379 Table Mathematical expressions of the redistribution coefficients of enzyme concentrations akj when introducing competitive and/or regulatory constraints akj is the ratio of a change in the concentration Ej to a change in the concentration Ek Note that values of akj are only true for j „ k because akk is always equal to unity The subscript k refers in this table to the enzyme whose concentration we want to vary, for instance through experimental or genetic means (see Appendices A to D for more details) akj ¼ bkj À ej Bk À ek B k ð7Þ where Bk ¼ n X bkj : j¼1 No competition Competition (Etot is not constant) (Etot is constant) No regulation 8j 6¼ k akj ¼ akk ¼ exists between the redistribution coefficient akj and the coregulation coefficients bkj (Appendix A): ej 8j 6¼ k akj ¼ ckj ¼ À À ek akk ¼ ckk ¼ n X ckj ¼ This relationship does not involve explicitly the competition coefficient ckj But when there is no coregulation in the system, we have "j „ k bkj ¼ and Bk ¼ bkk ¼ 1, so that: ej ¼ ckj akj ¼ À À ek j¼1 Regulation 8j 6¼ k akj ¼ bkj bkk ¼ 8j 6¼ k akj ¼ bkj À ej Bk À ek Bk Application: the case of a linear pathway of enzymes akk ¼ Xn a ¼0 j¼1 kj 8j ¼ k ckj ¼ À We applied our model of redistribution of enzyme concentrations to the linear pathway of enzymes far from saturation studied by Kacser & Burns [1] ej À ek ð5Þ Thus partial derivation of Eqn with respect to ek leads to @ej ¼ ckj @ek Flux and metabolite concentrations in a linear pathway Let us consider a linear metabolic pathway, with n enzymes E1, E2, …, En converting a substrate X0 into a final product Xn by a series of unimolecular reversible reactions: E1 E2 E3 EnÀ2 @e and we have ckk ¼ As by definition akj ¼ @ekj , we have for pure competitive systems, a simple relationship between competition and redistribution coefficients (Table 1, Appendix A): ej 8j ¼ k akj ¼ ckj ¼ À À ek En The enzymes are supposed to be Michaelian and far from saturation The steady-state flux through the pathway is [1,2]: J¼ X n P akk ¼ ckk ¼ which can also be derived from summation of Eqn Regulatory correlations When redistribution of enzyme concentrations is only due to regulatory mechanisms, total enzyme content has no upper limit, but enzyme concentrations are correlated Variation of the concentration of 00 enzyme Ek from Ek to Ek drives the system to a new state 00 00 00 E1 ; :::; Ej ; :::; En , where ð6Þ where Ej is the concentration of enzyme Ej before the variation of enzyme Ek, and bkj is the coregulation coefficient between enzymes Ej and Ek The coefficients can be positive, negative or null, but at least one is different from It is worth noting that bkk ¼ In systems with only regulatory constraints, the coregulation coefficient corresponds to the redistribution coefficient, i.e akj ¼ bkj, as shown in Appendix A (also Table 1) Redistribution coefficients in competitive-regulatory pathways When both competition and regulation are present in a pathway, it is interesting to note that a simple relationship 8ị Vj jẳ1 M j 8j E00 ẳ Ej ỵ bkj E00 Ek ị j k EnÀ1 X0 ¢ S1 ¢ S2 ¢ ¢ SnÀ2 ¢ SnÀ1 ¢ Xn K0;jÀ1 and the steady-state concentration of metabolite Si is X J Xn X A ỵ Si ẳ K0;i @X0 Vj V X K0;jÀ1 K0;n j i j K0;jÀ1 j>i Mj ð9Þ Mj where X0 and Xn are the concentrations of substrate X0 and product Xn, respectively, and X ¼ X0 ) Xn/K0,n X0 and Xn are considered as fixed parameters of the systems, while the intermediate metabolite concentrations Si (1 £ i £ n ) 1) are variables Vk is the maximum velocityQ enzyme Ek, Mk of is its Michaelis constant, and K0;k ¼ k KjÀ1;j is the j¼1 product of the equilibrium constants of reactions 1, 2,…, k To make apparent the concentration of enzymes, Ek, in Eqns and 9, we used the relationship: Vk ¼ kcat;k Ek where kcat,k is the turnover number of enzyme Ek We can then define the activity parameter Ak of enzyme Ek by: kcat;k K0;kÀ1 Ak ¼ Mk with K0,0 ¼ by convention Ó FEBS 2004 4380 S Lion et al (Eur J Biochem 271) The steady-state flux through the pathway is thus Jẳ X n P jẳ1 10ị Aj E j and the steady-state concentration of metabolite Si is ! X J Xn X 11ị ỵ Si ¼ K0;i X0 X Aj Ej K0;n j i Aj Ej j>i Below, we will consider the catalytic component Ak is constant and only consider variations of enzyme concentrations, in order to study how biological constraints on these concentrations can modify the behaviour of metabolic pathways Variation of flux and metabolite concentrations in unconstrained pathways Aj E j The concentration of a metabolite located downstream of the variable enzyme increases until it reaches a plateau (Appendix C): K0;i B X Xn X C Sdown ¼ P @X0 ỵ A i Aj Ej K0;n j i Aj Ej j>i Aj E j j6¼k j6¼k The concentration of a metabolite located upstream of the variable enzyme decreases until it reaches a minimum value: X X C K0;i B Xn Sup ¼ P @X0 ỵ A i Aj Ej K0;n j i Aj Ej j>i Aj E j j6¼k j6¼k It is clear that the level of the asymptotes depends on the concentrations of the other enzymes Figure describes the change in flux that results from different types of correlations and corresponds to various constraints that result in a redistribution of enzyme concentrations over the variation of a particular enzyme Flux 0.002 0.001 0.000 0.2 0.4 0.6 Concentration of enzyme 0.8 1.0 0.0 0.2 0.4 0.6 Concentration of enzyme 0.8 1.0 1.0 Metabolite concentration 1.2 1.4 1.6 1.8 2.0 0.0 B j6¼k Consequences of enzyme redistribution on the flux 0.003 A 0.004 When enzyme concentrations are not correlated, i.e when there are no competitive or regulatory constraints, both flux and metabolite concentrations reach a plateau when the concentration of a particular enzyme Ek increases (Fig 2) Considering the concentrations of the other enzymes as constants, the maximum flux value is (Eqn 10 and a general theoretical background in Appendix B): X Jmax ¼ P Fig Variation of metabolic variables in a linear unconstrained pathway with respect to concentration of one enzyme Unless stated otherwise, all plots describe a six-enzyme pathway with activities A1 ¼ 0.32, A2 ¼ 0.83, A3 ¼ 0.72, A4 ¼ 0.04, A5 ¼ 0.40, A6 ¼ 0.16 The x-axis is the concentration of enzyme The concentrations of enzyme are (0.04,0.02,E3,0.37,0.44,0.09) Furthermore, we choose X0 ¼ and Xn/K0,n ¼ (therefore, X ¼ 1) (A) Variation of flux in a linear unconstrained pathway with respect to concentration of one enzyme The figure shows the classical hyperbolic flux curve (B) Variation of metabolite concentration in a linear unconstrained pathway with respect to concentration of one enzyme The figure shows the variation of concentration of a metabolite upstream (solid line) and a metabolite downstream (dashed line) the variable enzyme Competitive-regulatory pathways Introducing both competitive and regulatory correlations in the system will alter the flux curve with respect to enzyme concentration These constraints result in a limited range of variation for enzyme concentrations, and in the variation of concentration of an enzyme being limited by that of the others If the concentration of a given enzyme becomes high, the concentration of another is likely to vanish, therefore bringing the value of the flux to zero Therefore, in this model, each enzyme has a range of variation [emin,emax] in which the flux is positive Over the range of variation of the concentration of an enzyme, the flux increases to a maximal value, then decreases when the concentration is lower or higher This is due to the fact that at least one redistribution coefficient must be negative when competition is introduced Moreover, everything else being equal, each set of redistribution coefficients results in a particular flux curve As will be mentioned later, all the possible curves are restricted by an envelope curve Relationship between redistribution and combined response coefficients In order to analyse the response of the flux to the variation of enzyme concentration in constrained pathways, we used the combined response coefficient we have defined previously All the results in this section depend on the assumption that the pathway is linear with massaction kinetics, which ensures analytical tractability J Replacing Cj by its expression in Eqn 2, we easily find an analytical expression for the combined response coefficient: Ó FEBS 2004 An extension to the metabolic control theory (Eur J Biochem 271) 4381 A B C D eà k ¼ p1 ffiffiffiffi Ak n P pffiffiffiffi Aj j¼1 50 In Appendix D we present another derivation using the @J fact that RJk ¼ eJk @ek e Unlike the control coefficient, the flux combined response coefficient can become negative, as a consequence of competition and/or negative coregulation (Fig 4) Limits of flux combined response coefficient are +1 for the minimal value of ek and –1 for the maximal value (Appendix D) Flux reaches an absolute maximal value for a vector of enzyme concentrations (e1,…, en) defined by [47,50] such that –50 ð12Þ Flux combined response coefficient n X akj J ek XEtot j¼1 Aj e2 j –100 RJk ¼ e 100 Fig Relationship between flux and enzyme concentration for different models of enzyme redistribution We consider a six-enzyme linear pathway with activities as in Fig The other parameters are the same as in Fig The vertical dashed line indicates the point corresponding to reference distribution E0 (A) Pure regulation with positive or null correlations Solid line: all the ai’s are positive Dashed line: a36 ¼ Dotted line: a36 ¼ and a35 ¼ Dashed-dotted line: all the ai’s are zero (B) Pure regulation with one negative correlation Dashed line: with one enzyme being negatively correlated Solid line: pure regulation with positive correlations [compare with (A)] (C) Competition Solid line corresponds to the redistribution coefficients in Fig 1D, and dashed line to those in Fig 1E (D) Regulation with competition Solid line is competition as shown in (C); dashed and dotted lines describe regulation with competition (dotted line corresponds to the redistribution coefficients in Fig 1F, dashed line to those in Fig 1G) 0.2 0.3 0.4 Concentration of enzyme 0.5 0.6 ð13Þ and the flux combined response coefficient is null when ek ¼ e à k The envelope curve For each value of ek, we can determine a maximum value for the flux (Appendix E) and consider the curve that passes through all these points This curve will be called the envelope curve Does this envelope curve correspond to a peculiar redistribution system or to a mere mathematical construction? In Appendix E, we used the optimization method proposed by Heinrich et al [47,50] to show that the envelope Fig Variation of flux combined response coefficient with respect to the concentration of one enzyme, under the constraint of Eqn (competition) Flux combined response coefficient is positive for e < e*, null for e ¼ e* and negative for e > e*, where e* is the concentration leading to the optimal value of flux, as defined by [47] (Appendix E) curve corresponds to a pure competitive model It passes through the absolute maximum of flux under the constraint of Eqn 4, which is reached for a vector of concentrations (eà ; :::; eà ) as defined in Eqn 13 For all values of ek, we have: n rffiffiffiffiffi ei Aj ¼ 8i; j ¼ k ej Ai Ó FEBS 2004 4382 S Lion et al (Eur J Biochem 271) Moreover, we show that the redistribution coefficients of the envelope curve are given by: eà j akj ¼ À À eà k Therefore, whatever the redistribution rules, for a given set of fixed activities, all flux curves will be bounded by the envelope curve we have defined Pure regulatory pathways When only regulatory constraints are present, two subcases of interest should be mentioned: positive and negative correlations Positive correlation is presented in Fig 3A When all coregulation coefficients are positive, the flux asymptotically tends towards a straight line, as the concentration of enzyme Ek increases The coefficients of this line are given in Appendix B If, starting from the situation where all the enzymes are positively coregulated, we choose to set p coregulation coefficients to zero, the flux curve will reach a plateau, which is characterized by the following ux value (Appendix B): X Jpị ẳ P max j2Ip Aj Ej0 with i Ip if aki ¼ (i.e enzymes Ei are independent from enzyme Ek) The higher the number of coregulation coefficients (i.e the less the enzymes are coregulated), the lower the plateau, and when no enzymes are coregulated with enzyme Ek, the ðnÀ1Þ maximum flux value Jmax is the one found by Kacser & Burns [1]: X Jn1ị ẳ P max j6ẳk Aj Ej0 Negative correlation is presented in Fig 3B When at least one redistribution coefficient is negative, the flux curve reaches a maximum beyond which it declines towards zero (Appendix B) Pure competitive pathways Here, we consider that the total concentration of the enzymes is constant (Eqn 4) and that only proportional redistribution occurs between the ej enzymes of the pathway (akj ¼ ckj ¼ À1Àek ) This means that an increase in the concentration of a given enzyme Ek causes a decrease in the concentrations of the others, in proportions remaining constant Mathematically speaking, this is equivalent to setting "j „ k, bkj ¼ in the expression of akj (Eqn 7) Therefore, in this particular model the flux response coefficient reads (Appendix D):   ek J RJk ¼ À1 e À ek XEtot Ak e2 k for the enzyme Ek that causes the redistribution of the other concentrations, and: 8j ¼ k RJj ¼ À e for the other enzymes J XEtot Ak e2 k Thus it must be stressed that because of the absence of coregulation, the limit of the combined response coefficients for ek ¼ is not +1 anymore, but is now (Appendix D) As in the competitive-regulatory model, the shape of the flux-activity curve is altered and points out that enzymes should possess an optimal concentration beyond which the flux decreases (Fig 3C), as it had been predicted previously [30,49,51] When concentration of enzyme Ek varies from to Etot, we only need to know the proportional redistribution coefficients in order to determine the concentrations of the other enzymes Therefore, we can draw several flux curves, each one determined by a set of proportional redistribution rules When ek ¼ or ek ¼ 1, the flux is null The maximum value of the flux curve depends on the redistribution coefficients for this curve These different curves correspond to unoptimized distributions of the  concentrations, i.e distributions where akj ẳ e e ị, j k eà being defined by Eqn 13 (the optimum distribution) As k the optimum distribution corresponds to the envelope curve (see above), all the curves corresponding to unoptimized distributions have flux values less than those of the envelope curve Consequences of enzyme redistribution on metabolite concentrations When the metabolite concentrations are considered as systemic variables, similar treatment applies because the results on redistribution, competition and coregulation coefficients between enzyme concentrations are still valid Metabolite concentrations are always bounded For a linear pathway with a positive flux, the metabolite concentration will have a lower limit corresponding to the weighted concentration of the input substrate of the system (X0K0,i) and an upper limit corresponding to XnK0,i/K0,n (Eqn 11) The range of variation of the concentration of metabolite Si is therefore equal to K0,iG, where G is the equilibrium ratio X0K0,n/Xn This means that the further the system is from equilibrium, the more the metabolites are free to vary Interestingly, this also implies that the variation of metabolite concentrations is constrained by environmental parameters, independently of the catalytic properties of the enzymes It is important to note that this feature only results from the particular expression of Si in a linear pathway and not from the introduction of redistribution rules As for the flux, we summarize in Fig the change in metabolite concentrations (actually Si/K0,i and not Si) as a result of various correlations between enzyme concentrations Relationship between redistribution and metabolite combined response coefficients As in the case of the flux, we can define a combined response coefficient for the concentration of metabolite Si, with respect to enzyme Ek: RSki ¼ e n X j¼1 CSi ejek ¼ k n X j¼1 CSi akj k ek ej We can show that it is equivalent to calculate: Ó FEBS 2004 An extension to the metabolic control theory (Eur J Biochem 271) 4383 A B C D Fig Relationship between metabolite concentrations and enzyme concentration for different models of enzyme redistribution The parameters are the same as in Fig Note that plots are Si/K0,i and not Si alone (A) Independent enzymes (B) Pure regulation with positive correlations (C) Competition (D) Regulation with competition The two upper curves represent metabolites upstream of the variable enzyme; the three other are downstream metabolites RSki ¼ e ek @Si Si @ek on a suitable set of enzyme concentrations describing the constraints In competitive-regulatory systems, we have the following relationship between the metabolite combined response coefficient and the redistribution coefficients (Appendix F): ! X akj X ek J2 K0;i X akj X Si Rek ¼ À Si XE2 Aj e2 j>i Aj ej Aj e2 j i Aj ej tot j j j i j 1 > > k i RSi ¼ J K0;i > ek > < A j ej A k ek À ek XE2 Si tot j>i ! X > 1 > k>i RSi ¼ À J K0;i > > ek > A j ej A k ek À e k XE2 Si : tot j i Competitive-regulatory pathways The competitive and regulatory constraints also change the pattern of variation of metabolites concentrations when enzyme concentration becomes too high The global behaviour of metabolite concentration with respect to enzyme concentration is dramatically altered for large enzyme concentrations As we take into account competition in this section, there is at least one negative redistribution coefficient, which means that the concentration of at least one enzyme ‘ vanishes for ek ¼ emax Therefore, metabolite concentration will be minimal when ek ¼ emax if ‘ > k and maximal if ‘ £ k (Appendix C) The behaviour of the system is fully determined by the sign and the magnitude of the redistribution coefficients between enzymes Therefore, three kinds of behaviour can be distinguished in this system (Fig 5D): (a) a ÔU-shapedÕ variation whereby the upstream metabolite decreases from XnK0,i/K0,n to a minimum value and then increases until the maximal concentration is reached again; (b) a monotonous variation allowing the metabolite concentration to describe the whole range of variation; (c) a Ôhump-shapedÕ variation whereby the downstream metabolite concentration increases from X0K0,i to a maximum value, and then decreases towards X0K0,i Hence, the model can account for a variety of behaviours of the metabolite concentrations We can say that the behaviour of the system depends on the position in the pathway of the enzyme whose concentration becomes when the target enzyme reaches its maximal value The key point is to know whether the enzyme is located upstream or downstream the metabolite (Appendix C) Thus, an increase in the concentration of an enzyme can induce either an increase or a decrease in the concentration of a metabolite This can be related to what is observed in many human metabolic diseases, which can be caused either by an excess 4384 S Lion et al (Eur J Biochem 271) or by a lack in a given metabolite By extending our model to nonlinear correlations and pathways, we can expect to observe similar patterns with nonbounded metabolite concentrations Pure regulatory pathways In this case, the total enzyme concentration is not constant but at least one regulation coefficient is non-zero As for flux, two subcases of interest should be mentioned If there are positive correlations (Fig 5B), i.e all the redistribution coefficients are positive, the metabolite concentration curve reaches a plateau whatever the position of the metabolite in the pathway with respect to the enzyme: downstream or upstream The level of the plateau is different from that of the unconstrained case and is given in Appendix C With negative correlations, i.e when at least one redistribution coefficient is negative, the concentration of metabolite Si is seen to increase or decrease towards the upper or the lower limits of metabolite concentrations (Appendix C) This is due to the fact that, when two enzymes are negatively correlated, an increase in the first one ultimately causes the second one to vanish Pure competitive pathways In a pure competitive pathway (Fig 5C), the behaviour of metabolite concentrations is not affected by the introduction of proportional redistribution and the general behaviour of the system is the same as the one predicted by the classical metabolic control theory (Fig 5A) Upstream metabolites are found to decrease until a plateau is reached (when all enzyme concentrations are null except the varying enzyme), whereas downstream metabolites are found to increase until a plateau is reached The values of the plateau for both an upstream and a downstream enzyme are different from those of an unconstrained pathway (Appendix C) The pure competitive model shows therefore that taking into account proportional redistribution between enzymes can dramatically modify the flux through the pathway without altering the qualitative behaviour of metabolite concentrations Discussion We developed an extension to the metabolic control theory that takes into account the existence of correlations between enzyme concentrations in the cell In our model, enzyme concentrations are linked by so-called redistribution coefficients, which account for the effect of the variation of one enzyme concentration onto the concentration of other enzymes We have distinguished two kinds of correlations: competition and regulation This distinction is not a mere artifice In the literature, there are multiple examples of correlations due to regulatory mechanisms, at the transcriptional and/or (post) translational levels Competition is less popular, but is also documented For instance Snoep et al [35] showed experimentally that overexpression of plasmid-encoded protein in Z mobilis could lead to the dilution of other enzymes and therefore cause a reduction in the glycolytic flux This protein burden effect is likely to be more critical in organisms like Z mobilis, where 50% of the cytoplasmic proteins are Ó FEBS 2004 reserved for glycolytic enzymes [52], than in E coli where these enzymes are present at low concentration In the same line, Parsch et al [53] showed in Drosophila that the deletion of a conserved regulatory element in the Adh gene resulted in increased ADH overexpression and activity, but delayed development As enzyme concentrations are not independent, the control of flux or metabolite concentrations cannot be quantified with the classical control coefficient anymore Using the concept of response coefficients, we have showed how correlations between enzyme concentrations can affect flux and metabolite concentrations in a pathway, and how this effect can be quantitatively measured For the flux, we gave a general expression (Eqn 3) showing how the interplay of the redistribution of enzyme concentrations and of the control of enzymes on the flux determine the response of any metabolic pathway to a variation of enzyme concentration A similar treatment can be applied to metabolite concentrations The combined response coefficient can take any positive or negative value, while the control coefficient varies between and (at least in simple linear pathways) As a major and general conclusion of the model, we showed that, if the concentration of an enzyme is negatively correlated with the concentrations of other enzymes, increasing the concentration of that enzyme will cause the flux to have a maximum, even if the control of this enzyme on the flux is strong The influence of the redistribution coefficients on the combined response coefficient means that the correlations between enzyme concentrations modify the control distribution pattern within the pathway However the combined response coefficients not exhibit a simple summation property, unlike the classical control coefficients Thus, it would be hazardous to use control coefficient summation property in top-down control analysis to estimate the control of steps that have not been studied through modulation of enzyme efficiency, especially in cases where competition and/or regulation are likely to be present Another approach to study the distribution of control has been developed by Westerhoff’s group [54,55], which applies to multilevel networks These networks are divided into modules where reactions are linked by mass transfer, whereas modules can interact with each other only by regulatory effects This approach allows the determination of the role of enzyme level in metabolic control, by considering that nonmetabolic modules can have a share of the metabolic control For the sake of analytical tractability, we chose to study in detail a simple linear metabolic pathway We showed that introducing correlations between enzyme concentrations alters the shape of the flux and metabolite curves For the flux, there is indeed a maximum value for any redistribution rule, provided that at least one coefficient is negative (due to competition or negative regulation): the enzymes have an optimal concentration beyond which the flux decreases, as already predicted or demonstrated [30,47,50,51] in the case of competition alone The only case where there is no maximum flux value is when all enzymes are positively correlated In metabolic engineering, the only way to have high fluxes is to increase all enzyme concentrations simultaneously As it can be technically difficult, it should be more practicable to optimize the distribution of enzyme amounts in the system Ó FEBS 2004 An extension to the metabolic control theory (Eur J Biochem 271) 4385 with fine regulation of gene expression, with tools such as synthetic promoter libraries in microorganisms Hartl et al [51] have underlined that in most cases, the maximum of the flux curve is expected to be broad, which is consistent with the plateau typically observed when plotting flux against enzyme concentration However, in particular cases, the maximum could be sharply defined Koehn [31] has emphasized that the ÔbreadthÕ of the maximum depends on the enzyme, as a function of the turnover cost Variation of the concentrations of enzymes allowed us to obtain flux curves with sharp or broad maximum Furthermore, it is clear that a less rigid constraint on total enzyme amount would lead to a broader maximum, as an enzyme amount should increase in a given range without limiting the others The extension of our model to the metabolite concentrations allowed us to clarify how constraints at the enzyme amount level can interfere with metabolite pools In systems with competition alone, the behaviour of metabolite concentrations is qualitatively the same as in systems without any constraint Upstream metabolites decrease when the variable enzyme amount increases, whereas downstream metabolites increase Changes are observed for the asymptotic values of the metabolite concentrations when the variable enzyme amount increases to Etot Thus metabolite pool sizes are not very affected by competition as compared to the flux With only positive regulations, the metabolite behaviour can be modified, with upstream metabolites increasing when the variable enzyme amount increases, which is impossible in systems with no constraint Introducing negative regulations leads to a great variety in metabolite behaviours, with different types of curves ranging from ÔU-shapedÕ to Ôhump-shapedÕ, with also monotonous variations The behaviour of a metabolite depends both on its position relative to the variable enzyme (upstream or downstream) and on the sign of the correlations in the system Metabolites displaying ÔU-shapedÕ and Ôhump-shapedÕ curves have a restricted range of variation as compared to other metabolites Our linear approximation of enzyme correlations is certainly too naive It might be more realistic to consider correlation coefficients that are null for small values of enzyme concentration and non-null for large concentrations, or to consider coefficients that are a growing function of enzyme concentration Experimentally, the parameters of the constrained model could be estimated, for example using overexpression libraries in bacteria or yeast [56], or exploiting natural variability of protein concentration It is expected that non-null redistribution parameters would be observed only for highly expressed proteins Furthermore we chose a quite rigid constraint on total enzyme concentration, because we considered Etot as a constant It would be more appropriate to consider that the quantity of protein the cell allocates to a particular metabolic pathway is between a lower and an upper bound Of course, it would be interesting to investigate the predictions of the model for more complex pathways (branched, with feedback inhibition, etc.) However, we think the results we obtained from this simplified example already shed light on some interesting consequences of correlations between enzymes, or at least on some questions that should be tackled in future work An interesting extension of the competition model could be to consider competition between metabolic pathways or networks For a given pathway, it is likely that the distribution of total enzyme amount between the enzymes should be optimized by evolution with respect to catalytic properties of the enzymes But at the cellular level, there would be competition between different pathways for the allocation of the total resources in protein In this case, enzyme concentrations within a given pathway will be positively correlated (as in Fig 1B), whereas the total amount of the pathway will be negatively correlated with total amounts of other pathways The redistribution between pathways should be influenced by environmental changes (adaptation) or by specific induction When a given pathway is activated by an increase of total enzyme amount, the flux responds by a linear increase only if all enzymes are positively correlated If at least one enzyme is not correlated with the others, a plateau will limit the response of the flux to activation The existence of operons coding for enzymes of a specific pathway (such as amino acid biosynthesis in bacteria) makes sense in this context A last, but important point we need to discuss is the assumption we make that enzyme concentrations are the only genetically variable parameters Indeed, genetic variability also affects the kinetic parameters M and/or kcat ([10,57–59] give classical examples Introducing variable Ai parameters in our models is of course possible and would not modify the theoretical framework However it is not clear for us whether constraints may exist on the variations of the Ai’s, and, if any, how they could result in correlations between parameters of different enzymes In any case, theoretical studies and experimental data suggest that enzyme concentrations are more likely to vary than their catalytic properties Pettersson [60] showed for example that enzyme catalytic efficiency has an upper limit depending on the diffusion rate of the molecules in the cell In addition, molecular polymorphism of enzymes shows conservation of sequences required for enzyme functionality [61], while there is large natural polymorphism in regulatory sequences [12,62,63] In this connection, studies in quantitative proteomics have reported high levels of genetic variability in protein amounts [14–19] Therefore, variations in enzyme concentrations are expected to be more frequent and larger than variations in catalytic properties The classical metabolic control theory has been used for addressing theoretical questions in evolutionary genetics [64,65], and to study the effect of natural [51,66,67] and artificial [68,69] selection on the flux considered as model quantitative traits The biologically relevant constraints on enzyme variations we introduced in the metabolic control theory have noticeable consequences on the behaviour of flux and metabolites This opens interesting questions about the selective pressures on enzyme correlations in the evolution of metabolic pathways These features will be investigated in more detail in other publications Acknowledgements ´ We are very grateful to Marı´ a Luz Cardenas-Cerda for helpful discussions and carefully reading the manuscript We also want to ´ thank some anonymous referees for very useful comments J Fievet ` was supported by a Ph.D grant from the French Ministere de la ´ Jeunesse, de l’Education nationale et de la Recherche 4386 S Lion et al (Eur J Biochem 271) References Kacser, H & Burns, J.A (1973) The control of flux Symp Soc Exp Biol 27, 65–104 Heinrich, R & Rapoport, T (1974) A linear steady-state treatment of enzymatic chains General properties, control and effector strength Eur J Biochem 42, 89–95 Fell, D.A (1992) Metabolic control analysis: a survey of its theoretical and experimental background Biochem J 286, 313–330 Melendez-Hevia, E., Torres, N & Sicilia, J (1990) A generalization of metabolic control analysis to conditions of no 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(1988) Stabilizing selection and metabolism Heredity 61, 433–438 68 Ward, P.J (1990) The inheritance of metabolic flux – expressions for the within-sibship mean and variance given the parental genotypes Genetics 125, 655–667 69 Keightley, P.D (1996) Metabolic models of selection response J.Theor Biol 182, 311–316 Appendix A Decomposition of redistribution coefficients Let us consider that the P initial enzyme concentrations are E1, …, Ej, …, En with n Ej ¼ Etot Variation of the j¼1 00 concentration of enzyme Ek from Ek to Ek drives the 00 00 00 system to a new state E1 ; :::; Ej ; :::; En , due to redistribution between enzymes In a first step, we consider that the variations of concentrations are only due to coregulations 00 Therefore Ej ẳ Etot ỵ E, where E can be positive or negative As defined in Eqn we have bkj ¼ E00 À Ej j E00 À Ek k and n X bkj ¼ Bk ¼ j¼1 E00 k E À Ek Introducing proportional redistribution in the system, due to competition, affects all the concentrations and drives the system to the final state E01 ; :::; E0j ; :::; E0n with Pn the constraint j¼1 Ej ¼ Etot , as a consequence of competition-regulation driven redistribution Following Eqn 1, we have akj ¼ E0j À Ej E0k À Ek or, using normalized concentrations, akj ¼ e0j À ej e0k À ek Due to proportional redistribution, the relationship between E00 and E0j is j E00 E0j j ẳ Etot ỵ E Etot or e00 ¼ e0j j Therefore, and e0k À ek ¼ E00 Ek k Etot ỵ E Etot As we have E00 ẳ Ej ỵ bkj E00 Ek j k summing for j ¼ to n, we get Etot ỵ E ẳ Etot ỵ Bk E00 À Ek k After some rearrangements, we get Ó FEBS 2004 4388 S Lion et al (Eur J Biochem 271) akj ẳ pị is independent The value Jmax of the plateau when p enzymes are independent is given by: bkj À ej Bk À ek B k With only competition in the system, we have "j „ k, bkj ¼ and Bk ¼ bkk ¼ 1, and thus we come back to the relationship ej ¼ Àckj akj ¼ À À ek With only coregulation in the system, 8j; E0j ¼ E00 and j thus we have a correspondance between the redistribution and coregulation coefficients: E00 À Ej j akj ¼ 00 ¼ bkj Ek À Ek Appendix B Asymptotic behaviour of flux curves in pure regulatory models Let us consider a pure regulatory model, i.e a model where variation of the concentration of enzyme k from to Ek causes the enzyme to vary from Ej0 (concentration when Ek ¼ 0) to Ej with a coefcient bkj: 8j ẳ k Ej ẳ Ej0 ỵ bkj Ek n X Ej ¼ Etot j¼1 We have J¼ X XAk Ek ¼P n P  Ak  ỵ jẳ1 X Jpị ẳ P max j2Ip ðpÞ It is clear that Jmax gets smaller when p increases and that ðnÀ1Þ Jmax corresponds to the maximum value of the flux found by Kacser & Burns [1] We have: X Jn1ị ẳ P max j6ẳk Appendix C Asymptotic behaviour of metabolite concentration curves We still consider a pure regulatory model (Appendix B), with p independent enzymes The coregulation coefficients are first assumed to be positive We note i Pp (resp i Dp) if bki ¼ and k £ i (resp bki ¼ and k > i) We can write the concentration of metabolite i ‡ k as follows: Si ¼ K0;i k Let us further assume that all coefficients bkj are strictly positive Taking the limit of the precedent expression, we get: Ek ỵ IndEk ị ỵ CorrEk ị P Ak =Aj P Ak Ek ỵ Ej0 ỵb Ek ỵ Aj Ej0 j= Pp [Dp IndEk ị ẳ X0 It is easy to calculate that J asymptotically tends towards a straight line aEk + b where: XAk P k ỵ AAb j b ẳ XAk CorrEk ị ẳ X0 1ỵ j6ẳk !2 Ak Aj bkj Let us assume that p enzymes are independent We note i Ip if bki ¼ We can write J as follows: J¼ XAk P B  Ek @ j6¼k k   Ak Aj Xn X Ak Aj þ Ej0 þ bkj Ek K0;n j= P Ej0 þ bkj Ek p j6¼k Taking the limit of the expression of Si when Ek fi 1, we see that the metabolite concentration reaches a plateau as soon as at least one enzyme is independent of the others ðpÞ The value Si of the plateau when p enzymes are independent is given by: P Xn P X0 Aj Ej0 ỵ K0;n Aj Ej0 j2Dp P Ak Ak C ỵ A Aj Ej0 ỵ Ek Ej0 j= Ip Aj bkj ỵ E X j= Dp Aj b kj P X Ak Xn X Ak ỵ AE K0;n j2Pp Aj Ej0 j2Dp j j0 and P Aj Ej0 j6¼k j2Pp[Dp j6¼k kj j6¼k kj where lim J ẳ ỵ1 Ek !1 aẳ Aj Ej0 If we now assume that at least one coefficient bk‘ is negative, it is clear that there exists a value of Ek where E‘ ¼ This value is Emax ) E‘0/bk‘ Therefore, we want to calculate the limit of J when Ek fi Emax It is easy to see that this limit is zero This means that one negative correlation is enough for the flux curve to reach a maximum Ej0 j6ẳk Aj bkj ỵ E Aj E j Aj Ej0 pị Si ẳ K0;i j2Pp[Dp j2Ip Taking the limit of this expression when Ek fi 1, we see that the flux reaches a plateau as soon as at least one enzyme j2Pp P n1ị j6ẳk Aj Ej0 It is clear that Si corresponds to the level of the plateau found by Kacser & Burns [1] In the case where k > i, we have: Ó FEBS 2004 An extension to the metabolic control theory (Eur J Biochem 271) 4389 X0 P Aj Ej0 j2Dp pị Si j6ẳk ẳ K0;i Xn ỵ K0;n P j2Pp[Dp P j2Pp Aj Ej0 ỵ X0 K0;i j>i Slim ẳ K0;i i ẳ 1ỵ P Xn ỵ K0;n P j6ẳk K0;i j i j6¼k j i P Aj akj j6¼k akj ¼ À if k>i P X0 Slim i ¼ K0;i þ j>i Aj Ej0 Xn þ K0;n P X0 Slim i K0;i A‘ ek0 ¼ because P j>i j6¼k ¼ Aj Ej0 when ‘ i A similar calculus leads to: P j i Aj Ej0 j6ẳk if Aj Ej0 Xn ỵ K0;n P j6ẳk Taking the limit, we get: P j i k i Aj Ej0 if k>i Aj Ej0 Note that the values for the plateaus are different from those in the unconstrained pathway, because the values of the Ej0 are different in both cases Appendix D lim Si ¼ K0;i X0 Ek !Emax when ‘>i We see that the values of Si when one enzyme concentration is null can be K0,iX0 or K0,iXn/K0,n depending on the position of the enzyme with respect to the metabolite In particular, this consideration allows us to derive the value of Si when Ek ¼ If we now consider a pure competitive linear pathway, we have: ej ¼ akj À ek P j6¼k j6¼k Aj E j Xn lim Si ¼ K0;i Ek !Emax K0;n ej0 ¼ Àej0 À ek0 Replacing akj by this value in the expressions of Silim , we see that the levels of the plateaus are given by: j i j6¼k À Aj akj j>i Slim ¼ K0;i i Aj bkj Aj bkj j>i E‘ Aj akj Finally, we have for k > i: P Xn P X0 Aj1 kj ỵ K0;n Aj1 kj a a B P C Xn P E‘ @X0 Aj1 j0 ỵ K0;n Aj1 j0 A ỵ AXn0;n E E K ¼ j i j6¼k P j6¼k If we now assume that at least one coefficient bk‘ is negative, it is clear that there exists a value of Ek where E‘ ¼ This value is Emax ) E‘0/bk‘ Therefore, we want to calculate the limit of when Ek fi Emax Let us write Si (‘ £ i) as follows: ð0Þ Si P By definition, we have: Aj bkj j6ẳk Xn ỵ K0;n Aj akj kj j6ẳk 0ị Si P j>i Aj Ej0 If we suppose that all enzymes are linked by positive correlations (p ¼ 0), the limits for Si are given by: P Xn P ỵ X0 Aj1 ỵ K0;n Aj1 bkj bkj j>i j i 0ị Si j6ẳk P ẳ if k i K0;i ỵ Aj1 b P X0 and n X ej ¼ Relationship between redistribution and flux combined response coefficients Competitive-regulatory pathways Analytical expression of the flux reads: J¼ j¼1 j6¼k Aj ej Using the fact that: j¼1 We can write the metabolite concentration as follows, if we assume ‘ £ i: P Xn P k Xn X0 Aj1 kj ỵ K0;n Aj1 kj ỵ 1ek K0;n a a Ak e j>i j i Si j6¼k ẳ P 1ek K0;i Aj akj ỵ Ak ek XEtot n P @ej ¼ akj @ek we get: n P @J ¼ XEtot @ek j¼1 n P j¼1 Taking the limit of this expression when ek fi 1, we see that the Si reaches a plateau The value of the plateau is: akj Aj e2 j !2 Aj ej Multiplying this expression by ek/J, we get after some rearrangement: Ó FEBS 2004 4390 S Lion et al (Eur J Biochem 271) RJk ¼ e n X akj J ek XEtot j¼1 Aj e2 j Appendix E Characterization of flux envelope curve We can now derive the right and left limits of the flux combined response coefficient First, let us observe that these limits are reached when the concentration of a particular enzyme E‘ becomes zero We have therefore: P a e2 Ajkj2 ỵ ak A ek j6¼‘ ej P RJk ¼ e e‘ e‘ Aj ej ỵ A j6ẳ The limit of the second fraction when e‘ becomes zero is akj because the sums are finite Therefore, we have: ek RJk / ak‘ e e‘ For a nonpure competitive model, the limit on the right is unchanged (–1) but the limit on the left is generally not Indeed, we have k „ ‘ and ak‘>0, so the limit is +1 Pure competitive pathways Replacing akj by its expression in Eqn 12, we have: ! X J ej J À Rek ¼ ek XEtot Ak e2 j6¼k Aj e2 À ek j k eà j ¼ pffiffiffiffi Aj n P k¼1 p1 ffiffiffiffi Ak Let us now consider that the value of ek is fixed (ek ¼ e) We define for each value of e a new constraint that reads: X ej ¼ À e j6¼k Using the method of Lagrange multipliers, we calculate the maximal value of the flux when ek ¼ e, under the new constraint The optimal vector of concentrations e° is solution of the following set of equations: !! X @ JÀk ej À þ e ¼0 8j ¼ k @ej j6¼k This can be simplified as: This reads: RJk ¼ ek e Heinrich et al [47,50] have used optimization principles to study the evolution of metabolic systems For instance, they showed that under the constraint of Eqn 4, flux has an optimal value on the range [0, Etot] This value is reached for a unique vector of concentrations (eà ; :::; eà ) n defined as [47]: 1 X À À ek j6¼k Aj ej A k ek J XEtot !  ! J 1 XEtot À À XEtot Ak e2 À ek A k ek J k   J XEtot þ À ¼ ek XEtot Ak e2 Jð1 À ek Þ Ak ek ð1 À ek Þ k   J 1 XEtot À ¼ ek XEtot Ak e2 ð1 À ek Þ À ek J k 8j ¼ k Using the expression of J, we get: ¼ ek Finally we get: RJj ¼ e   ek J À1 À ek XEtot Ak e2 k For the other enzymes j „ k, the redistribution coefficients are: ej i ¼ k aij ¼ ei ajk ¼ À À ek ej J XEtot Ak e2 k In this case, lim RJk ¼ 1, whereas lim RJk ¼ À1 e e ek !0 8j ¼ k J2 ¼k XEtot Aj e j Taking two arbitrary indices i and j and dividing the two equations, we obtain: sffiffiffiffiffi e Ai j 8i; j ¼ k ¼ e Aj i Therefore, we have: X pffiffiffiffiffiffiffiffiffi X pffiffiffiffiffi ¼ À e e ¼ Ai e 8i ¼ k i j Aj j6¼k j6¼k We deduce the expression of the vector e : pffiffiffiffiffi Ai 8i ¼ k e ¼ eị P ẳ e eị i i pffiffiffiffi j6¼k Replacing these values in the expression of combined response coefficient leads to: RJj ¼ À e @J ¼k @ej ek !1 Aj e ¼ e ¼ e ðeÞ k k We define the function J° as J°(e) ¼ J(e°(e)) J° is clearly an envelope function Moreover, we can easily show that: pffiffiffiffiffi Ai @e eà i ¼ÀP ¼À i à 8i ¼ k pffiffiffiffi @e À ek j6¼k Aj Ĩ FEBS 2004 An extension to the metabolic control theory (Eur J Biochem 271) 4391 @e k ¼1 @e Finally, after some rearrangement we get: where eà is the optimal value of ei found by Heinrich et al k [47] Therefore, J° is a pure competitive flux curve, defined by a particular set of redistribution of coefficients: j¼k eà j akj ¼ À À eà k Appendix F Relationship between redistribution and metabolite concentration combined response coefficients X akj X @Si J2 K0;i X akj X ¼ À 2 @ek XEtot j i Aj ej j>i Aj ej j>i Aj e2 j i Aj ej j ! Pure competitive pathways As in appendix D, we replace akj by its expression in Eqn 14 We have, if k £ i: X akj X X akj X À Aj ej j>i Aj ej j>i Aj e2 j i Aj ej j j i X 1 CX B ¼@ À A Aj ej À ek j>i Aj ej A k ek j i j6¼k Competitive-regulatory pathways The concentration of metabolite Si is ! X J Xn X K0;i X0 þ Si ¼ XEtot Aj ej K0;n j i Aj ej j>i Using the fact that @ P j i j6ẳk  X 1  ek 1ỵ ẳ A j ej A k e2 À ek k j>i ! Aj ej ¼À @ek X akj A j e2 j j i Finally, we get for metabolites Si that are downstream of enzyme Ek: ! J2 K0;i X 1 Si k i Rek ¼ S Aj ej Ak ek À ek XEtot i j>i we get: " @Si K0;i @J ¼ @ek XEtot @ek X Xn X X0 ỵ Aj ej K0;n j i Aj ej j>i !# X akj Xn X akj ỵ J X0 Aj e2 K0;n j i Aj e2 j j j>i ! Similarly, we have for metabolites Si that are upstream of enzyme Ek: ! J2 K0;i X 1 Si : k>i Rek ¼ À A j ej A k ek À ek XE2 Si tot j i and n @J J2 X akj ¼ @ek XEtot j¼1 Aj e2 j Therefore, we have: X ej X ỵ Aj e2 À ek j i Aj ej j j>i 13 X X X C7 B ¼ À À @ A5 A j ej A k e2 À e k j i A j ej A j ej k j>i j i P Xn K0;n Aj ej P ! Aj ej n P akj Aj e2 j ỵ X0 7 j>i j i j¼1 @Si J2 K0;i 6 ! ¼ 2 n P akj P @ek X Etot Xn P akj X0 ỵ K0;n Aj e2 Aj ej Aj e2 j>i j j i j j¼1 ... existence of these competitive and regulatory correlations between enzymes is assumed to constrain the response of the metabolic systems Here, we present an extension of the metabolic control theory. .. proportion between these enzymes) as Ó FEBS 2004 An extension to the metabolic control theory (Eur J Biochem 271) 4379 Table Mathematical expressions of the redistribution coefficients of enzyme concentrations. .. induces correlations between the concentrations of the enzymes involved in lactose metabolism Several experimental and theoretical studies have been devoted to the understanding of the mechanisms