Modular metabolic control analysis of large responses The general case for two modules and one linking intermediate Luis Acerenza 1 and Fernando Ortega 2 1 Laboratorio de Biologı ´ a de Sistemas, Facultad de Ciencias, Universidad de la Repu ´ blica, Igua ´ , Montevideo, Uruguay 2 School of Biosciences, The University of Birmingham, UK The quantitative study of metabolic responses in intact cells is essential in research programs that require understanding of the differences in physiological and pathological cellular functioning or predicting the phenotypic consequences of genetic manipulations. To perform this type of studies, a systemic approach called metabolic control analysis (MCA) was deve- loped [1–5]. One of its central goals is to determine how the responses of system variables, quantified by control coefficients, depend on the properties of the component reactions, described by elasticity coeffi- cients. Predicting the responses of intact cellular systems to environmental and genetic changes has not been an easy task. This could explain the lack of success in many biotechnological and biomedical applications that require changing metabolic variables in a pre- established way [6,7]. Two of the major challenges to understanding metabolic responses are the structural complexity of the molecular networks sustaining cellu- lar functioning and the nonlinearity inherent in the interaction and kinetic laws involved. In the develop- ment of MCA, some strategies have been devised to deal with these difficulties. Keywords metabolic control analysis; metabolic control design; metabolic responses; modular control analysis; top-down control analysis Correspondence L. Acerenza, Laboratorio de Biologı ´ ade Sistemas, Facultad de Ciencias, Universidad de la Repu ´ blica, 4225, Montevideo 11400, Uruguay Fax: +598 2 525 8629 Tel: +598 2 525 8618–23, Ext. 139 E-mail: aceren@fcien.edu.uy Note Dedicated to the memory of Reinhart Heinrich, one of the fathers of Metabolic Control Theory (Received 5 October 2006, accepted 7 November 2006) doi:10.1111/j.1742-4658.2006.05575.x Deciphering the laws that govern metabolic responses of complex systems is essential to understand physiological functioning, pathological conditions and the outcome of experimental manipulations of intact cells. To this aim, a theoretical and experimental sensitivity analysis, called modular meta- bolic control analysis (MMCA), was proposed. This field was previously developed under the assumptions of infinitesimal changes and ⁄ or propor- tionality between parameters and rates, which are usually not fulfilled in vivo. Here we develop a general MMCA for two modules, not relying on those assumptions. Control coefficients and elasticity coefficients for large changes are defined. These are subject to constraints: summation and response theorems, and relationships that allow calculating control from elasticity coefficients. We show how to determine the coefficients from top- down experiments, measuring the rates of the isolated modules as a func- tion of the linking intermediate (there is no need to change parameters inside the modules). The novel formalism is applied to data of two experi- mental studies from the literature. In one of these, 40% increase in the activity of the supply module results in less than 4% increase in flux, while infinitesimal MMCA predicts more than 30% increase in flux. In addition, it is not possible to increase the flux by manipulating the activity of demand. The impossibility of increasing the flux by changing the activity of a single module is due to an abrupt decrease of the control of the modules when their corresponding activities are increased. In these cases, the infini- tesimal approach can give highly erroneous predictions. Abbreviations ANT, adenine nucleotide translocator; MCA, metabolic control analysis; MMCA, modular metabolic control analysis. 188 FEBS Journal 274 (2007) 188–201 ª 2006 The Authors Journal compilation ª 2006 FEBS Regarding network complexity, top-down or modu- lar strategies have been proposed [8–10]. These strat- egies abstractly divide the system into modules, lumping together irrelevant (and unknown) compo- nents and representing explicitly only the processes that we are interested in describing. The aim is to sti- mulate and measure the responses using the intact sys- tem, so that we are certain that the analysis performed and the conclusions obtained apply to this system. To deal with nonlinearity two assumptions have been made. The first is that metabolic perturbations and responses are small, so that they can be described using a first order infinitesimal treatment. The second assumption is that in vivo enzyme catalysed reaction rates are proportional to the corresponding enzyme concentrations, as is normally the case when measured in diluted in vitro conditions. It is important to note that, to our knowledge, all the developments in steady- state MCA have included at least one of these two assumptions [11,12]. However, many, if not most, of the responses exhibited by metabolic systems subject to environmental changes or genetic manipulations involve large changes in metabolic variables. Moreover, the assertion that in vivo rates are proportional to enzyme concentration is difficult to justify. The cyto- plasm of cells is far from being diluted, showing a very crowded state where the validity of the proportionality found in vitro has still not been demonstrated [13]. Attempts to extend infinitesimal control analysis to large changes in the variables have been reviewed in previous publications [5,12]. Our previous contribu- tions to extend infinitesimal modular metabolic control analysis (MMCA) consisted on the following steps. First, control coefficients for large changes were defined and summation theorems, in terms of enzyme concentrations, derived [14]. Expressions to calculate these control coefficients in terms of the elasticity coef- ficients for large changes were obtained [12,15]. How- ever, the interpretation of the results of all these previous contributions to MMCA for large changes requires that the rates of the steps are proportional to the corresponding enzyme concentrations. In the present contribution, we develop an MMCA that applies to steady-state responses of any extent and that does not assume proportionality between reaction rates and parameters. Therefore, it applies to any parameter (enzyme concentration, external effector, etc.), irrespective of its functional relationship with the reaction rate. To achieve this, rate control coefficients (where parameters are not specified) and p-elasticity coefficients for large changes were defined. Combining these two types of newly defined coefficients, we derive response theorems, which are essential to study the response of metabolic variables to external activators or inhibitors. We also show, in the framework of large changes, that rate control coefficients verify the same constraints (summation theorems, etc.) as those satis- fied, when rates are proportional to enzyme concentra- tions, by enzyme response coefficients. Another central result is that the rate control coefficients can be used to determine the flux and intermediate changes that would be obtained by changing the rates of the isola- ted modules by large factors. These relationships are useful to analyse where to modulate the system in order to change a variable in a desirable way, or to speculate about possible sites at which cell physiology operates to modify the variables, when adapting to dif- ferent conditions. All the quantities and relationships developed here may be applied to data obtained from top-down experiments. Notably, this type of experi- ment may be performed by direct modulation of the intermediate, without changing parameters inside the modules. The way the formalism is applied and the type of conclusions that can be drawn are illustrated with two studies, taken from the literature, performed using top-down experiments: the control of glycolytic flux and biomass production in Lactococcus lactis [16] and the control of oxidative phosphorylation in isola- ted rat liver mitochondria [17]. Results The modular approach to large metabolic responses A central issue to solving many biotechnological and biomedical problems is to assess how to modulate a metabolic system in order to obtain a pre-established change in the concentration of an intermediate or a flux. Within the framework of reductionist approaches, the studies to solve this type of problem are performed on isolated component reactions, reconstructed small portions of the network or extracts. As a consequence, the results obtained may not be extrapolated with con- fidence to the in vivo system because, in the reduction process, it is more likely that relevant interactions are lost. In contrast, modular approaches study the intact system and therefore the conclusions obtained apply to this system. Let us consider a metabolic network with any num- ber of intermediates and reactions. In the modular approach, we focus on an intermediate S which divides the system into two parts or modules (Scheme 1). The system has three variables: the concentration of the linking intermediate (S), the rate at which the interme- diate is produced by the supply module (v 1 ) and the L. Acerenza and F. Ortega Metabolic control analysis of large responses FEBS Journal 274 (2007) 188–201 ª 2006 The Authors Journal compilation ª 2006 FEBS 189 rate at which it is consumed by the demand module (v 2 ) [18]. In this strategy, it is assumed that the only interactions between modules are linking intermediates [10]. A module could be an enzyme catalysed reaction, a metabolic pathway, a large portion of metabolism (i.e., carbohydrate metabolism), an organelle (e.g., mitochondria) or a cell. The rate v 1 depends on S and on all the parameters belonging to the supply module. Similarly, v 2 depends on S and on the parameters belonging to the demand module. Examples of parameters could be the concen- trations of external substrates, products or effectors and the concentrations of enzymes. The functional dependence of v 1 and v 2 on S and on the parameters could be very complex because, in the case that mod- ules are large portions of metabolism, many enzyme catalysed reactions and metabolites are involved. But, for our purposes, we only need to consider explicitly one parameter for each module: p 1 for the supply module and p 2 for the demand module. In this context, the functional dependence of the rates could be expressed as follows: v 1 ¼ v 1 (S, p 1 ) and v 2 ¼ v 2 (S, p 2 ). Note that proportionality between rates and parame- ters is not assumed in this treatment. At steady state, both rates are equal to the flux, J (v 1 ¼ v 2  J). There are three types of metabolic changes relevant to the modular control analysis that we shall develop below. In the first, p 1 (or p 2 ) is changed and the pertur- bation propagates throughout the system, with S and J settling to new steady-state values. In the second type, S is kept at a constant value by some external means so that when p 1 (or p 2 ) is changed, the perturbation will not be able to propagate to the other module, resulting in different final values of v 1 and v 2 . In the third type, one changes S without changing any parameter of the system (for example, adding an auxiliary reaction which consumes S), also resulting in different changes in the rates. These three types of metabolic changes are the basis for the definitions of response (and control), p-elasticity and e-elasticity coefficients for large chan- ges, respectively, given below. Quantification of metabolic responses The sensitivity of response of a steady-state variable, w (usually metabolite concentration, S, or flux, J)toa large change in a parameter, p i , from an initial state o to a final state f, is quantified by the mean-response coefficient (or mean-sensitivity coefficient) [14]: R w pi ¼ w f w o À 1  p f i p o i À 1 !, ð1Þ It represents the relative change in the variable divided by the relative change in the parameter that originated the variable change. This sensitivity coefficient is a sys- temic property because the effect of the parameter change propagates through all the system. Because, in Scheme 1, we have two system variables, S and J, and two parameters, p 1 and p 2 , we will consider four of these coefficients: R S p1 , R S p2 , R J p1 and R J p2 . Next, the parameter, p i , is changed, keeping S at a fixed value. For the sake of convenience, S is kept at the value S f , i.e., the value of the final state that S would reach if the parameter was changed without keeping S fixed (definition of response coefficient given above). We shall quantify the sensitivity of the rate, v i , to a large change in p i , from an initial value p o i to a final value p f i , by the mean p-elasticity coefficient: p vi pi ¼ v ff i v fo i À 1 ! p f i p o i À 1 !, ð2Þ Here we have used the compact notation: v ab i ¼ v i ðS a ; p b i Þ. Having two rates and two parameters there are four mean p-elasticity coefficient: p v1 p1 , p v1 p2 , p v2 p1 and p v2 p2 . Because v 1 is independent of p 2 and v 2 inde- pendent of p 1 it follows that: p v1 p2 ¼ p v2 p1 ¼ 0. p-Elasticity coefficients represent the sensitivities of the rates of the isolated component modules to changes in the parame- ters. Finally, we consider that the concentration, S,is changed by some external means, without changing the parameters p i . The sensitivity of the rate, v i , to a large change in S, from an initial value S o to a final value S f , is quantified by the mean e-elasticity coefficient [12]: e vi S ¼ v fo i v oo i À 1 ! S f S o À 1  ð3Þ Here we have also used the notation: v ab i ¼ v i ðS a ; p b i Þ. Having two rates and one intermediate there are two e-elasticity coefficients: e v1 S and e v2 S . These e-elasticity coefficients represent the sensitivity of the rate of the supply module to changes in the concentration of its product and the sensitivity of the rate of the demand module to changes in the concentration of its sub- strate, respectively. In the case of mean elasticity coefficients, p i and S both play the role of parameters. But note that while S v 1 v 2 supply demand Scheme 1. Metabolic system constituted by a supply module (1) and a demand module (2) linked by one intermediate S. Metabolic control analysis of large responses L. Acerenza and F. Ortega 190 FEBS Journal 274 (2007) 188–201 ª 2006 The Authors Journal compilation ª 2006 FEBS in the definition of mean p-elasticity coefficients the change in the rate with p i is performed keeping S at the final value, in the definition of mean e-elasticity coefficients the change in the rate with S is performed keeping p i at the initial value. Parameter changes affect S or J through the effects on the rates to which the parameters belong. Control coefficients can, therefore be defined in terms of rates, i.e., as relative change in the variable divided by the relative change in rate that produced the variable change [11,19,20]. More specifically, a parameter p i is changed from the initial value p o i to a final value p f i ,at fixed S, producing a change in the rate v i . As in the definition of mean p-elasticity coefficients [Eqn (2)], the rate change is evaluated at S ¼ S f . To quantify the sensitivity of response of the steady-state variable w to a large change in the rate v i we define the mean-control coefficient: C w vi ¼ w f w o À 1  v ff i v fo i À 1 !, ð4Þ Remember that: v ab i ¼ v i ðS a ; p b i Þ. The value taken by this coefficient is a system property, because the effect of the rate change propagates throughout. There are four of these coefficients: C S v1 , C S v2 , C J v1 and C J v2 . It can be easily shown, using Eqns (1), (2), and (4), that the two types of control coefficients defined above [Eqns (1) and (4)] are related by the response theorem: R w pi ¼ C w vi p vi pi ð5Þ w stands for S or J and i ¼ 1,2. This theorem states that the effect that a change in a parameter has on a metabolic variable depends on two factors: the local effect that the parameter has on the isolated rate through which it operates and the systemic effect that a change in rate has on the metabolic variable. If the parameter p i is an enzyme concentration or other inter- nal parameter its initial value, p o i , is not zero and its relative change ðp f i =p o i À 1Þ has a finite value. In this case, the coefficients R w pi and p vi pi are well defined. But, if p i is an external effector (inhibitor, activator or new enzyme activity), p o i will normally be zero and the coef- ficients would tend to infinity. This could easily be solved by replacing in the definitions of R w pi and p vi pi relative changes in p i by the corresponding absolute changes, i.e., replacing ð p f i =p o i À 1Þ by ðp f i À p o i Þ¼p f i . The rates of the supply and demand modules, v i , are non zero and therefore the coefficients C w vi are always well defined. One of the central aims of the present work is to show how the coefficients C w vi can be calculated using data obtained from top-down experiments. In this type of experiment only the rates of the modules for differ- ent values of the intermediate concentration are deter- mined, the measurement of parameter values not being necessary. However, to derive the equations that calcu- late the values of C w vi from measurements of v 1 , v 2 and S, the effect that particular changes in the parameter values would have on the variables will be analysed. These particular parameter changes and their conse- quences on the values of the variables are the subject matter below. Parameter changes We shall assume that the system starts at a reference state o, where the parameters, rates and variables take the values: p o 1 , p o 2 ; v oo 1 ; v oo 2 ; S o and J o (Table 1). We shall consider six different ways of modifying the initial state, o, which give the final states: x sp , y sp , x p , y p , x s and y s . In two of them, one parameter is changed (p 1 or p 2 ) and the variables (S and J) freely adjust to the final steady state. If p 1 is changed the final state is x sp and if p 2 is changed the final state is y sp (Table 1). The second two ways of modifying the system is to change a parameter, keeping S at a fixed value. In this case, if p 1 is changed the final state is x p , S being kept at the constant value S x , and if p 2 is changed the final state is y p , S being kept at S y (Table 1). Finally, the third two ways of modifying the system are to change S by some external means, without changing any parameter; S will be changed from S o to S x and from S o to S y , being the final states x s and y s , respectively (Table 1). We call r 1 the factor by which the rate v 1 changes when we go from state x s to state x sp , i.e., when p 1 is changed from p o 1 to p x 1 , keeping S fixed at S x . Similarly, we call r 2 the factor by which the rate v 2 changes when we go from state y s to state y sp , i.e., when p 2 is chan- ged from p o 2 to p y 2 , keeping S fixed at S y . As was men- tioned above, to develop the theory for a MMCA for large changes we need to consider particular changes in the parameters. These particular parameter changes Table 1. Different ways of modifying the reference state. Details given in text [note that v ab i ¼ v i ðS a ; p b i Þ]. p 1 p 2 v 1 v 2 SJ o p 0 1 p 0 2 v 00 1 v 00 2 S 0 J 0 x sp p x 1 p 0 2 v xx 1 v x0 2 S x J x y sp p 0 1 p y 2 v y0 1 v yy 2 S y J y x p p x 1 p 0 2 v xx 1 v x0 2 S x y p p 0 1 p y 2 v y0 1 v yy 2 S y x s p 0 1 p 0 2 v x0 1 v x0 2 S x y s p 0 1 p 0 2 v y0 1 v y0 2 S y L. Acerenza and F. Ortega Metabolic control analysis of large responses FEBS Journal 274 (2007) 188–201 ª 2006 The Authors Journal compilation ª 2006 FEBS 191 are those resulting in r 2 equal to the reciprocal of r 1 . In equations, we have (Table 1): r 1 B v xx 1 v xo 1 and r 2 B v yy 2 v yo 2 with r 2 ¼ 1 r 1 ð6Þ If p 1 and p 2 are changed so that Eqn (6) is fulfilled, the values of the variables satisfy the following rela- tionships (see Appendix for proof): S x ¼ S y and J x ¼ r 1 J y ð7Þ As a consequence of the steady state condition, and Eqns (6) and (7), eight equalities between the rates are fulfilled: J o ¼ v oo 1 ¼ v oo 2 J x ¼ v xx 1 ¼ v xo 2 ¼ v yo 2 J y ¼ v yy 2 ¼ v yo 1 ¼ v xo 1 ð8Þ Therefore, experimental determination of three rates, J o , v xo 1 and v yo 2 , allows the calculation of the 11 rates involved (Table 1). In Fig. 1, we give a graph (similar to the graph of combined rate characteristics used by Hofmeyr and Cornish-Bowden [18]) representing the effects on the rates of two sets of parameter changes, one fulfilling and the other not fulfilling the condition given in Eqn (6). Next, we will derive useful relationships involving the mean control coefficients [defined in Eqn (4)] and the mean e-elasticity coefficients [defined in Eqn (3)]. Relationships between system properties and module properties The fundamental relationships of MMCA for large changes, in the case of two modules, are the following: C J v1 ¼ e v1 S C S v1 þ e v1 S ðr S À 1Þþ1 C J v2 ¼ e v1 S C S v2 C J v1 ¼ e v2 S C S v1 C J v2 ¼ e v2 S C S v2 þ e v2 S ðr S À 1Þþ1 ð9Þ where r s ¼ S x ⁄ S o ¼ S y ⁄ S o . These four equations are the starting point to derive all the other relationships and theorems for large changes given below. Their validity can be tested using Eqns (3) (4), (6), (7) and (8), and Table 1. Equation (9) can be solved to obtain the mean control coefficients in terms of the mean e-elasticity coefficients and r s . The result is: C J v1 ¼ e v2 S ðe v1 S ðr S À 1Þþ1Þ e v2 S À e v1 S C J v2 ¼ Àe v1 S ðe v2 S ðr S À 1Þþ1Þ e v2 S À e v1 S C S v1 ¼ e v1 S ðr S À 1Þþ1 e v2 S À e v1 S C S v2 ¼ Àðe v2 S ðr S À 1Þþ1Þ e v2 S À e v1 S ð10Þ From these equations it is easily shown that mean con- trol coefficients fulfil the following summation theo- rems: C J v1 þ C J v2 ¼ 1 ð11Þ C S v1 þ C S v2 ¼ 1 À r s ð12Þ The factors r 1 and r 2 can also be calculated in terms of the mean e-elasticity coefficients and r s : r 1 ¼ 1 r 2 ¼ e v2 S ðr S À 1Þþ1 e v1 S ðr S À 1Þþ1 ð13Þ This relationship was obtained using Eqns (4), (6), (7) and (10). Fig. 1. Rates versus S. Schematic representations when condition Eqn 6 (A) is not fulfilled and (B) is fulfilled. Metabolic control analysis of large responses L. Acerenza and F. Ortega 192 FEBS Journal 274 (2007) 188–201 ª 2006 The Authors Journal compilation ª 2006 FEBS Solving Eqn (13) for (r s ) 1) and replacing the resulting expression into Eqn (10) gives: C J v1 ¼ e v2 S e v2 S À r 1 e v1 S C J v2 ¼ Àr 1 e v1 S e v2 S À r 1 e v1 S C S v1 ¼ 1 e v2 S À r 1 e v1 S C S v2 ¼ Àr 1 e v2 S À r 1 e v1 S ð14Þ These expressions constitute a different way to calcu- late the mean control coefficients in terms of the mean e-elasticity coefficients, to the one given in Eqn (10). Finally, expressions to calculate the mean e-elasticity coefficients from the mean control coefficients, i.e., the metabolic control design equations for large changes, can be readily obtained from Eqn (14). e v1 S ¼ C J v2 C S v2 e v2 S ¼ C J v1 C S v1 ð15Þ Equations (9) to (13) are valid independently of the functional relationship between the rates v 1 and v 2 , and the corresponding parameters p 1 and p 2 . They were previously derived under the restrictive assump- tion that the rates are proportional to the correspond- ing enzyme concentrations [12,14,15]. It is easy to show that when the changes of the parameters and rates are small (r 1 and r s tend to one) they reduce to the well-known relationships of traditional MCA, based on infinitesimal changes [1–5,21–23]. Up to this point, the analysis performed did not require the measurement of parameter values. In fact, to calculate C S v1 , C S v2 , C J v1 and C J v2 , only measurements of S o , S x , J o , v xo 1 and v yo 2 are needed. Nevertheless, if we want to determine R S p1 , R S p2 , R J p1 and R J p2 , the initial and final values of the parameter, p o 1 , p x 1 , p o 2 and p y 2 , and the corresponding rates have to be known, in order to calculate the mean p-elasticity coefficients (Eqn 2). With these, the mean response coefficients (Eqn 1), are obtained introducing Eqn (2) and (10) into the response theorems (Eqn 5). The relationships that we have derived show that the control coefficients for large changes are subject to constraints, which condition the responses of the meta- bolic variables to parameter changes. As a conse- quence, an important issue in MCA is to determine how a variable (w) would respond if a parameter or a rate of the system is modulated with a large change. The mean control coefficients can be used to perform this calculation, employing the following equation, derived from Eqn (4). w f w o ¼ 1 þ C w vi ðr i À 1Þ with i ¼ 1; 2 ð16Þ where w 0 and w f are the initial and final values of the variable (intermediate or flux), respectively, C w vi is the mean control coefficient (Eqn 10), and r i is the factor by which the rate of the isolated module i has been changed (Eqn 13). If C w vi and (r i – 1) have the same sign the variable increases and if they have opposite signs the variable decreases. Rate changes are pro- duced by parameter changes. The change in the vari- able that results from the change in a particular parameter, p i , can be calculated with an analogous equation to Eqn (16): w f =w o ¼ 1 þ C w vi p vi pi p f i =p o i À 1  with i ¼ 1; 2: Calculation of systemic responses from top-down experiments Next, we shall show how the mean control coefficients may be calculated from top-down experiments using the relationships derived in the previous section. Adding to Scheme 1 an auxiliary reaction, it is poss- ible to modulate the concentration of the intermediate, S, and measure the rates of the supply and demand modules, v 1 and v 2 . Applying fitting procedures to the table of experimental values v 1 , v 2 and S, continuous functions, represented by v 1 (S) and v 2 (S), can be obtained. These two functions are the basis for all the calculations. In the reference state, o, the auxiliary rate is zero: S ¼ S o , v 1 ¼ v oo 1 ¼ v 1 ðS o Þ and v 2 ¼ v oo 2 ¼ v 2 ðS o Þ. When the auxiliary rate is gradually changed, the values taken by intermediate and rates are: S ¼ S x ¼ S y , v 1 ¼ v xo 1 ¼ v 1 ðSÞ and v 2 ¼ v yo 1 ¼ v 2 ðSÞ. The mean e-elas- ticity coefficients (Eqn 3), expressed in terms of the fitting functions, are given by: e v1 S ¼ v 1 ðSÞ v 1 ðS o Þ À 1  S S o À 1  e v2 S ¼ v 2 ðSÞ v 2 ðS o Þ À 1  S S o À 1  ð17Þ Introducing these functions and r s ¼ S ⁄ S o into Eqns (10) and (13) we obtain C S v1 , C S v2 , C J v1 , C J v2 , r 1 and r 2 as a function of S. With these functions several plots can be built. We can represent C S v1 , C S v2 , C J v1 , C J v2 , L. Acerenza and F. Ortega Metabolic control analysis of large responses FEBS Journal 274 (2007) 188–201 ª 2006 The Authors Journal compilation ª 2006 FEBS 193 C S v1 þ C S v2 and C J v1 þ C J v2 as a function of S ⁄ S o . These plots show how the overall control, given by the sum- mation theorems (Eqns 11 and 12), is distributed among the blocks. On the other hand, we can repre- sent C S v1 and C J v1 as a function of r 1 , and C S v2 and C J v2 as a function of r 2 . These are useful to analyse how the control of each module varies as its activity chan- ges. In the case of the flux the control normally drops when the activity is increased. The procedure of analysis that we have described does not require the measurement of parameter values. But, as was mentioned above, to calculate the mean p-elasticity coefficient, p v1 p1 and p v2 p2 , and the mean response coefficients, R S p1 , R S p2 , R J p1 and R J p2 , the param- eter values, p o 1 , p x 1 , p o 2 and p y 2 , and the rates for these parameter values must be measured. The calculations for the case of parameters acting, say, on the rate v 1 are performed as follows. The increase in the param- eter from p o 1 to p x 1 , results in a new steady state in the intermediate, S x . p v1 p1 , C S v1 and C J v1 are evaluated at S x . Introducing these values in the response theorems (Eqn 5), R S p1 and R J p1 are obtained. An analogous pro- cedure can be followed to calculate R S p2 and R J p2 . Finally, using Eqn (16), the mean control coefficients can be used to calculate the change in the system vari- able (w ¼ J or S) that could be obtained with a large change in the rate of the isolated module by a factor r. For this purpose, the ratios J f ⁄ J o and S f ⁄ S o are plotted as a function of r, for each one of the modules. These plots show where and in what extent the system has to be modulated in order to obtain a desirable change in a variable. Below, we will apply this analysis to data deter- mined with top-down experiments obtained from the literature. Analysis of experimental cases Here, we shall apply the formalism developed in two studies, performed using top-down experiments. The first analyses the control of glycolytic flux and biomass production of L. lactis [16] and the other studies the control of oxidative phosphorylation in isolated rat liver mitochondria [17]. The choice of these cases was not based on the particular interest of the systems studied, but on the appropriateness of the examples to illustrate the application of the analysis developed in this work. In the study of Koebmann and colleagues [16], energy metabolism of L. lactis was split into a supply module, that produces ATP (glycolytic module or module 1), and a demand module, that consumes ATP (biomass production module or module 2). The inter- mediate is the ratio of concentrations ATP ⁄ ADP (Scheme 1 with S ¼ ATP ⁄ ADP). Top-down experi- ments consisted of varying the ATP ⁄ ADP ratio and measuring the supply and demand rates independently. The decrease in the ATP ⁄ ADP ratio was achieved by overexpressing the hydrolytic part of the F1 domain of the (F 1 F 2 )H + -ATPase, that increases ATP consump- tion. To perform an infinitesimal top-down control analysis at the reference state, the authors obtained fit- ting functions for the experimental values of v 1 and v 2 versus S. These functions are adequate for their pur- pose, but they are not sufficiently good for points away from the reference state, which should be consi- dered when performing a top-down control analysis for large changes. Here, the values of v 1 and v 2 versus S were fitted to the following functions: v 1 (S) ¼ 82.14 S 0.4 ⁄ (0.8574 + 0.2107 S 0.75 ) and v 2 (S) ¼ 2.325 S 3.5 ⁄ (2.253 + 0.02219 S 3.5 ) (Scheme 1). As mentioned above, these two functions are the basis for all our cal- culations. The parameters of the fitting functions do not have units, because S (i.e., ATP⁄ ADP) is dimen- sionless and the values of the rates are expressed as a percentage of the rate at the reference state. The refer- ence state is S o ¼ 9.7 and the ratios S ⁄ S o , studied experimentally, are in the interval (0.49, 1). The mean e-elasticity coefficients, e v1 S and e v2 S , are calculated replacing the fitting curves given above, v 1 (S) and v 2 (S), in Eqn (17). e v2 S is always positive. This is the sign normally expected because a substrate is an acti- vator of the reaction rate, its increase normally result- ing in an increase in rate. e v1 S , a product elasticity, exhibits the normal (negative) sign around the refer- ence state (S o ¼ 9.7). However, at approximately S ¼ 6.24 the elasticity vanishes, taking a positive sign under this value. This behaviour represents ‘product activa- tion’ of S on the rate of module 1. Finally, C S v1 , C S v2 , C J v1 , C J v2 , r 1 and r 2 are obtained, introducing the expres- sions for the mean e-elasticity coefficients and r s ¼ S ⁄ S o into Eqn (10) and (13). At the reference state (when S tends to S o ), these expressions give the values of the infinitesimal control coefficients: C J v1 ¼ 0:80, C J v2 ¼ 0:20, C S v1 ¼ 6:55 and C S v2 ¼À6:55. In Fig. 2 we represent C S v1 , C S v2 , C J v1 , C J v2 , C S v1 þ C S v2 and C J v1 þ C J v2 as a function of S ⁄ S o . C J v1 þ C J v2 is always one, according to what it states in the flux summation relationship for large changes (Eqn 11). In the region of S⁄ S o values between 0.49 and 0.65, C J v1 > 1 and C J v2 < 0. This is due to the posit- ive sign of the product elasticity, e v1 S , in this region. In addition, the values C J v1 and C J v2 are quantitatively very different from those obtained with infinitesimal chan- ges (Fig. 2A). The concentration summation relation- ship (Eqn 12) states that in the case of large changes C S v1 þ C S v2 is not equal to zero. Because in all the Metabolic control analysis of large responses L. Acerenza and F. Ortega 194 FEBS Journal 274 (2007) 188–201 ª 2006 The Authors Journal compilation ª 2006 FEBS experimental range S £ S o , the sum of the coefficients is positive. In this case, C S v2 is negative and slightly smaller in absolute value than C S v1 , which is positive. Only at the reference state, both coefficients take the same absolute value, i.e., when the changes are infini- tesimal (Fig. 2B). Next, we represent C S v1 and C J v1 as a function of r 1 , and C S v2 and C J v2 as a function of r 2 in two parametric plots: C J v1 and C J v2 in Fig. 3A and C S v1 and C S v2 in Fig. 3B. These are useful plots to analyse how the flux and concentration control of each module changes as the activity of the corresponding module is increased. For the flux control, we obtain the normal behaviour, i.e., the control of both modules diminishes as their activity is increased (Fig. 3A). In addition, C J v1 is greater than C J v2 in all the range of r factors studied (0.72, 1.38), but they both fall dramatically in this rather small range. C J v1 decreases from 1.04 to 0.09 and C J v2 from 0.91 to )0.04. For the concentration control, the control of the supply module increases and the control of the demand module decreases, in absolute terms, when the corresponding activity is increased (Fig. 3B). In the range studied (0.72, 1.38), C S v1 increa- ses from 1.9 to 15.9 and À C S v2 decreases from 22.0 to 1.4. At r ¼ 0.72, ÀC S v2 is more than 11 times greater than C S v1 and, at r ¼ 1.38, C S v1 is more than 11 times greater than À C S v2 .Atr ¼ 1, where the mean coeffi- cients coincide with the infinitesimal coefficients, C S v1 and ÀC S v2 are equal. Finally, we determine the changes in the flux and intermediate that could be obtained by changing the rates of the modules. This calculation is performed using Eqn (16) and is represented in Fig. 4. Figure 4A shows that it is not possible to increase the flux signifi- cantly, which is due to the abrupt decrease in C J v1 and C J v2 with r 1 and r 2 , respectively. In this respect, a 40% increase in the activity of the supply module (mod- ule 1) results in less that 4% increase in flux and, in practice, increasing the activity of the demand module Fig. 2. Mean control coefficients versus S ⁄ S o in L. lactis. (A) Flux mean control coefficients and their sum and (B) intermediate mean control coefficients and their sum. The reference state is indicated by d at S ⁄ S o ¼ 1. The range of S ⁄ S o represented corresponds to the experimental range reported in [16]. Fig. 3. Mean control coefficients versus module activity, r,in L. lactis. (A) Flux mean control coefficients and (B) intermediate mean control coefficients. Solid lines represent values in the experi- mental range and dashed lines give values extrapolated outside this range. The reference state is indicated by d at S ⁄ S o ¼ 1. L. Acerenza and F. Ortega Metabolic control analysis of large responses FEBS Journal 274 (2007) 188–201 ª 2006 The Authors Journal compilation ª 2006 FEBS 195 (module 2) the flux decreases (there is a very small increase in the flux when increasing the activity of module 2 for r between 1 and 1.1, which is, for all practical purposes, irrelevant). In contrast, decreasing both rates, independently, produces significant and similar decreases of the flux. In this region, flux and rate are approximately proportional for both modules, the decrease in rate of module 1 producing a slightly bigger decrease of the flux. Note that, in this example, there is no way to obtain significant increases in the flux by changing the activity of a single module. Regarding the intermediate, Fig. 4B shows that decreasing the supply rate or increasing the demand rate produces moderate decreases (less than 50%), while increasing the supply rate or decreasing the demand rate produces increases by a large factor (up to more than seven times). Let us now analyse the second experimental case, concerning the control of oxidative phosphorylation in isolated rat liver mitochondria [17]. Oxidative phos- phorylation was divided into two modules linked by the fraction of mitochondrial matrix ATP [S ¼ ATP ⁄ (ADP + ATP)]. The demand module (ATP-consuming module or module 2) is the adenine nucleotide translocator (ANT) and the supply module (ATP-producing module or module 1) is the rest of mitochondrial oxidative phosphorylation, including respiratory chain, ATP synthesis and the associated transport processes. Membrane potential (Dw)isan intermediate included inside module 1. In the following analysis, we shall assume that the direct effect of this intermediate on module 2 can be neglected, existing only an indirect effect through S. Experimental evi- dence for this assumption was reported by Ciapaite et al. [24]. Under these conditions, the analysis remains valid even if large changes in Dw take place when the system is modulated with effectors. One of these effec- tors is palmitoyl-CoA, an inhibitor of module 2 (ANT) that has no direct effect on module 1. To apply the top-down control analysis developed in the present work to this case, we fitted the experimental points reported in Fig. 5 of [17] to continuous functions. Fig. 4. Fluxes (A) and intermediate concentrations (B) produced by independent modulations in the activity of the supply or demand in L. lactis. Solid lines represent values in the experimental range and dashed lines give values extrapolated outside this range. The refer- ence state is indicated by d at S ⁄ S o ¼ 1. Fig. 5. Mean control coefficients versus S ⁄ S o in isolated rat liver mitochondria. (A) Flux mean control coefficients and their sum and (B) intermediate mean control coefficients and their sum. The refer- ence state is indicated by d at S ⁄ S o ¼ 1. The range of S ⁄ S o repre- sented corresponds to the experimental range reported in [17]. Metabolic control analysis of large responses L. Acerenza and F. Ortega 196 FEBS Journal 274 (2007) 188–201 ª 2006 The Authors Journal compilation ª 2006 FEBS The rates of module 1 and 2 are given by: v 1 (S) ¼ 14.04 ⁄ (0.03625 + S 10.19 ) and v 2 (S) ¼ 1259S ⁄ (1.136 + S) (Scheme 1). When 5 lmolÆL )1 of palmi- toyl-CoA (I ¼ 5) was added, the rate v 1 was described by the same function [v 1 (S, I ¼ 5) ¼ v 1 (S)] and the rate v 2 changed, being described by: v 2 (S, I ¼ 5) ¼ 378.3S ⁄ (0.4796 + S). The reference states, without and with 5 lmolÆL )1 of palmitoyl-CoA, were S o ¼ 0.49 and S o I ¼ 0:70, respectively. The mean e-elasticity coeffi- cients, e v1 S and e v2 S , are calculated replacing the fitting curves, v 1 (S) and v 2 (S), into Eqn (17). For the entire range of S studied, e v2 S is positive and e v1 S is negative as would normally be expected. Introducing e v1 S , e v2 S and r s ¼ S ⁄ S o into Eqn (10) and (13) C S v1 , C S v2 , C J v1 , C J v2 , r 1 and r 2 are obtained. Note that, v 1 and v 2 were meas- ured for different ranges of values of S (see Fig. 5 of [17]). As a consequence, all values of mean-control coefficients calculated by this analysis involve values of mean-elasticity coefficients extrapolated outside the experimental range. Accordingly, in the figures that we will present next, no distinction between experimental and extrapolated range will be made (in contrast to Figs 2–4). In Fig. 5, we plot C S v1 , C S v2 , C J v1 , C J v2 , C S v1 þ C S v2 and C J v1 þ C J v2 as a function of S ⁄ S o . C J v1 þ C J v2 is always one (Eqn 11) and, in this case, 0 < C J v1 < 1 and 0 < C J v2 < 1 because e v1 S and e v2 S show normal signs (Fig. 5A). C S v1 > 0 and C S v2 < 0 in all the range of S ⁄ S o values (Fig. 5B). For S ⁄ S o <1, C S v1 > C S v2       and C S v1 þ C S v2 > 0 (total concentration con- trol dominated by supply), while for S ⁄ S o >1, C S v1 < C S v2       and C S v1 þ C S v2 < 0 (total concentration con- trol dominated by demand). C S v1 þ C S v2 ¼ 0 at the refer- ence state only (Eqn 12). Finally, we have quantified the effect of palmytoil- CoA (I, specific inhibitor of module 2) on the interme- diate, S, and the flux, J, using the corresponding mean response coefficients, R S I and R J I . Here, definitions involving absolute changes in I are used because the initial value of I is zero [ R S I ¼ðS f =S o À 1Þ=ðI f À I o Þ and R J I ¼ðJ f =J o À 1Þ=ðI f À I o Þ]. These coefficients are calcu- lated using the response theorems for large changes (Eqn 5), i.e., R S I ¼ C S v2 p v2 I and R J I ¼ C J v2 p v2 I , where p v2 I is the mean p-elasticity coefficient, defined in terms of absolute changes in I ½ p v2 I ¼ðv ff 2 =v fo 2 À 1Þ=ðI f À I o Þ¼ v 2 ðS; I ¼ 5Þ=v 2 ðSÞÀ1=ð5 À 0Þ. In Fig. 6, we represent R S I , R J I and p v2 I as a function of S=S o I . In the range of values analysed, p v2 I varies between, approximately, )0.07 and )0.1. Therefore, its effect, in the response theorem is, roughly speaking, to lower by a tenth the absolute value of the mean control coefficients, C S v2 and C J v2 , and to change their sign (compare Figs 5 and 6). Another interesting representation would be to plot R S I , R J I and p v2 I as a function of the concentration of inhibitor I. This was not possible for this example because the data available was determined at one inhibitor concentration only. Discussion In MCA, elasticity analysis is the procedure that allows calculation of the control coefficients in terms of elasticity coefficients. In this contribution, we develop a completely general modular elasticity analy- sis of large metabolic responses, for the case of two modules and one intermediate, which also constitutes an extension of the infinitesimal supply demand analy- sis developed by Hofmeyr and Cornish-Bowden [18] to large changes. The stages to achieving this goal were the following: In the first elasticity analysis of large metabolic responses that we previously developed [12], the equations obtained were valid for variable elasticity coefficients and could be applied to analyse model si- mulations involving this type of coefficient. However, they could not be applied to analyse top-down experi- ments that result in variable elasticity coefficients, because the relationship between the factor r and the elasticity coefficients had not been deduced. In this context, we applied the analysis to an experimental case where the elasticity coefficients were reported to be approximately constant [12]. In a recent contribu- tion [15], the relationship between r and the elasticity coefficients was established and applied to an experi- mental case with variable elasticity coefficients. These two preceding formalisms still relied, for the interpret- ation of the results obtained, on the assumption that all the reaction rates are proportional to the corres- ponding enzyme concentrations. In addition, they Fig. 6. Mean response coefficients versus S=S o I in isolated rat liver mitochondria. The mean p-elasticity coefficient of the demand block with respect to the inhibitor (I, palmitoyl-CoA), p v2 I , is also represen- ted. The reference state (with 5 lmolÆ L )1 of palmitoyl-CoA) is indi- cated by d at S=S o I ¼ 1. L. Acerenza and F. Ortega Metabolic control analysis of large responses FEBS Journal 274 (2007) 188–201 ª 2006 The Authors Journal compilation ª 2006 FEBS 197 [...]... develop a completely general modular elasticity analysis of large metabolic responses, for the case of two modules and one intermediate, which also constitutes an extension of the infinitesimal supply demand analysis developed by Hofmeyr and Cornish-Bowden [18] to large changes The stages to achieving this goal were the following: In the first elasticity analysis of large metabolic responses that we previously... measurements of the concentrations of intermediate and r, the change in the activity of the module (Eqn 13), requires an elaborate calculation The plots as a function of S ⁄ So can be used to study how control is distributed and how this distribution is modified with the extent of the change On the other hand, the plots as a function of r describe how the control changes with respect to the variation in the. .. inside the supply module and measure the changes in the intermediate and the demand rate and, second, to perturb the demand module and measure the changes in the intermediate and the supply rate This is the way that the experiments were designed by Ciapaite et al [17], to obtain the data related to the control of oxidative phosphorylation in isolated rat liver mitochondria that we analysed in the second... by changing the activities of the modules Note that, a 40% increase in the activity of the supply module, results in less than 4% increase in flux This is a rather unexpected result from the point of view of the infinitesimal Metabolic control analysis of large responses treatment, taking into account that, at the reference state, the supply module has 80% of the control J (Cv1 ¼ 0:80) Using the infinitesimal... analysis determines the response coefficients, from the control coefficients, the p-elasticity coefficients and the response theorems Only in this last part, the parameters appear explicitly in the analysis There are two ways to perform the modulations of the system in top-down experiments, in order to measure the effect that changes in the intermediate that links the modules has on their rates One is to add... described above, there are two ways to modulate the intermediate: with an auxiliary branching reaction or perturbing the modules If we use an auxiliary branch that consumes the intermediate, S decreases (S < So) and the effect of this decrease on the rates of the modules is measured With this data and the theory here developed, the mean control coefficients as a function of S ⁄ So can be calculated and 198 plotted... absolute value of the mean control coefficients, Cv2 J and Cv2 , and to change their sign (compare Figs 5 and 6) Another interesting representation would be to plot RS , RJ and pv2 as a function of the concentration of I I I Metabolic control analysis of large responses Fig 6 Mean response coefficients versus S=SIo in isolated rat liver mitochondria The mean p-elasticity coefficient of the demand block with... consequence, response theorems were not obtained The modular elasticity analysis of large changes, developed in the present contribution, can be applied to modules of any structure, size and kinetic properties The first part of the analysis determines the control coefficients as a function of the elasticity coefficients, being valid irrespective of the parameter that has produced the rate change The parameter... view, a completely general analysis for the case of two modules and one linking intermediate Its limitations in the application to intact systems are similar to those of infinitesimal treatments [10,27,28] One important limitation is that the modules defined do not interact significantly through intermediates different to the linking one This may be checked by changing different sets of parameters that... produces the intermediate (S > So) On the other hand, if we want to cover experimentally all the range of r-values with the alternative way of modulating the intermediate, i.e., perturbing the modules, we would have to do four separate experiments, using one specific inhibitor and one specific activator of each module When the full set of experimental results required is obtained, by either way of modulating . Modular metabolic control analysis of large responses The general case for two modules and one linking intermediate Luis Acerenza 1 and Fernando Ortega 2 1 Laboratorio. measure the changes in the intermediate and the demand rate and, second, to perturb the demand module and measure the changes in the intermediate and the supply rate. This is the way that the experiments. of large metabolic responses, for the case of two modules and one intermediate, which also constitutes an extension of the infinitesimal supply demand analy- sis developed by Hofmeyr and Cornish-Bowden