Modular metabolic control analysis of large responses in branched systems – application to aspartate metabolism Fernando Ortega1 and Luis Acerenza2 Computational Biology, Advanced Science and Technology Laboratory, AstraZeneca, Macclesfield, UK Systems Biology Laboratory, Faculty of Sciences, University of the Republic, Montevideo, Uruguay Keywords Asp metabolism; large metabolic responses; metabolic control analysis; metabolic modeling; modular analysis Correspondence ´ L Acerenza, Igua 4225, Montevideo 11400, Uruguay Fax: +598 5258629 Tel: +598 5258618/Ext.139 E-mail: aceren@fcien.edu.uy (Received 17 February 2011, revised 16 April 2011, accepted 17 May 2011) doi:10.1111/j.1742-4658.2011.08184.x Organisms subject to changing environmental conditions or experimental protocols show complex patterns of responses The design principles behind these patterns are still poorly understood Here, modular metabolic control analysis is developed to deal with large changes in branched pathways Modular aggregation of the system dramatically reduces the number of explicit variables and modulation sites Thus, the resulting number of control coefficients, which describe system responses, is small Three properties determine the pattern for large changes in the variables: the values of infinitesimal control coefficients, the effect of large rate changes on the control coefficients and the range of rate changes preserving feasible intermediate concentrations Importantly, this pattern gives information about the possibility of obtaining large variable changes by changing parameters inside the module, without the need to perform any parameter modulations The framework is applied to a detailed model of Asp metabolism The system is aggregated in one supply module, producing Thr from Asp (SM1), and two demand modules, incorporating Thr (DM2) and Ile (DM3) into protein Their fluxes are: J1, J2, and J3, respectively The analysis shows similar high infinitesimal control coefficients of J2 by the rates of SM1 and J2 J2 DM2 (Cv1 ¼ 0:6 and Cv2 ¼ 0:7, respectively) In addition, these coefficients present only moderate decreases when the rates of the corresponding modules are increased However, the range of feasible rate changes in SM1 is narrow Therefore, for large increases in J2 to be obtained, DM2 must be modulated Of the rich network of allosteric interactions present, only two groups of inhibitions generate the control pattern for large responses Introduction Living organisms have complex cellular machineries that have the ability to sense and adapt their metabolic states to changes in external conditions The transition between two metabolic states depends on the existence of molecular mechanisms that, most often, produce changes in many variable concentrations and fluxes The design principles behind these responses and the constraints limiting the patterns that could be achieved have been studied within the framework of metabolic control analysis (MCA) [1–5] Metabolic networks of organisms show thousands of variable concentrations and fluxes, so full application of traditional MCA to intact metabolic systems is impracticable Therefore, to overcome this problem, module-based approaches, such as modular MCA, were introduced [6–9] Modular MCA conceptually Abbreviations AdoMet, S-adenosylmethionine; AK, aspartate kinase; DHDPS, dihydrodipicolinate synthase; HSDH, homoserine dehydrogenase; MCA, metabolic control analysis; TD, threonine deaminase; TS, threonine synthase FEBS Journal 278 (2011) 2565–2578 ª 2011 The Authors Journal compilation ª 2011 FEBS 2565 Control of large changes in metabolic branches F Ortega and L Acerenza divides the system into modules, grouping together all that is irrelevant to the question of interest, including all that we ignore and knowledge of which is not required to obtain the answer On the other hand, the relevant metabolic variables, module exchange fluxes and linking intermediate concentrations remain explicit to perform a MCA on them MCA and modular MCA have been applied to analyze the control of metabolic pathways [10–14] and intact cells [15–17] One important limitation of traditional MCA and modular MCA is that they have been mainly developed for small, strictly speaking infinitesimal, changes Thus, in general, the power of MCA to forecast, for example, the flux change resulting from a change in enzyme activity is confined to cases where the enzymatic perturbation is small However, many regulatory processes in vivo and experiments that involve perturbations, as well as biotechnological process of interest, require large metabolic changes A modular MCA suitable for the analysis of large responses in complex systems has started to be developed The general theory for a system divided into two modules and one linking intermediate was obtained [18–21] Another type of sensitivity analysis for large changes is global sensitivity analysis [22–24] This studies the effect that large changes in the parameters have on the relevant outputs of the system, using random sampling of the parameter space This tool is mainly used for model characterization and validation It was not developed to analyze large responses of complex experimental systems, where detailed information of the structure and the types of rate laws governing many processes is not known For this purpose, modular approaches could be used Here, the general theory of modular MCA for large responses is developed to include the analysis of branched systems This formalism enables the analysis of systems with three modules and one explicit intermediate It may be used, for example, to predict in what region or regions of the metabolic network an effector would have to operate in order to produce a large change in a particular metabolic concentration or flux This is relevant for studying where a physiological activator or inhibitor would act to regulate a cellular process, or where the site of action of a drug would have to be to compensate for the deviation of a metabolic variable in a pathological condition The application of the new method is illustrated using a model of Asp metabolism [25] in the parameter p has on the steady-state value of a variable w (metabolite concentration, S, or flux, J) is w quantified by the response coefficient, Cp , representing the relative change in the variable divided by the relative change in the parameter (see definitions of the coefficients in Doc S1 and [21]) The number of response coefficients in cellular metabolism (number of parameters · number of variables) is very large, and measuring all of them is not practically feasible To overcome this problem, we can conceptually divide the system into a small number of modules, leaving explicit only the variable concentrations and fluxes relevant to the analysis that we want to perform [6–9] For example, in Fig 1, we represent a system divided into three modules and one linking intermediate Each module can therefore be considered as a ‘super-reaction’, consisting of many enzyme-catalyzed reactions Modularization drastically reduces the number of explicit variables, but there are still a large number of response coefficients, because of the large number of parameters involved Parameter changes affect the explicit metabolite concentrations and fluxes through the effect on the rates of the modules to which the parameters belong Therefore, the effect that a parameter change has on a variable can be decomposed into two parts, namely, the effect that the parameter has on the rate of the module, and the effect that the resulting rate change Methods Fig Branched modular system The metabolic system is conceptually aggregated into one input and two output modules connected by one linking intermediate, S J1, J2 and J3 are the fluxes of modules 1, and 3, respectively Large parameter changes produce changes in the metabolic concentrations and fluxes The effect that a change 2566 J1 J2 J3 FEBS Journal 278 (2011) 2565–2578 ª 2011 The Authors Journal compilation ª 2011 FEBS F Ortega and L Acerenza has on the variable It is possible to obtain identical rate changes modifying different parameters or combinations of parameters, operating on the same rate, by different amounts Thus, we can quantify the effect that changing the rate of a module has on a variable without specifying the parameter responsible for the change For this purpose, we use the control coefficient w for large changes, Cv , representing the relative change in the variable divided by the relative change in the rate that produced the variable change In the modular representation of the system, there is a small number of explicit variables and of module rates, resulting in a small number of control coefficients This set of control coefficients quantifies the control properties of the modular representation of the system, and can be experimentally determined (see below) In the definition of flux control coefficient, J Cv ¼ ðrJ À 1Þ=ðr À 1Þ, rJ is the factor by which the flux J has changed, and r is the factor by which the rate, v, that originated the flux change was modified Note that v represents both the rate equation governing the rate of the step and the value that this rate equation takes By definition, the steady-state flux, J, is the value taken by the rate equation when embedded in a metabolic system that reaches steady state However, there is an important difference between flux change and rate change, which will be analyzed next For the sake of clarity, let us first focus on a reaction step in a metabolic system governed by a rate law where the rate is proportional to the enzyme concentration: vab = g(S a)E b g(S) is an arbitrary function of the intermediate concentration, S, and E is the enzyme concentration We will assume that when the parameter E is changed, the system goes from the reference state to a final state The superscripts a and b indicate the state at which S and E are evaluated, respectively J is the steady-state flux carried by the step Initially, the system is at the reference state, o, where the quantities involved take the values: E o, S o, v oo, and J o (with J o = v oo) The enzyme is changed to the final steady state, E f, the final values taken by the other quantities being: S f, v ff, and J f (with J f = v ff ) The flux change is: rJ = J f ⁄ Jo = v ff ⁄ voo We will call r the factor by which the enzyme concentration is changed (r = E f ⁄ E o); r is also the factor by which the rate was changed, because the rate is proportional to the enzyme concentration As we will see next, r can also be calculated from rate values The following equalities hold: v ff = g(S f)E f = g(S f )rE o = rv fo By solving this equation, r can be obtained: Control of large changes in metabolic branches r = v ff ⁄ v fo The results obtained for the flux change and the rate change remain valid if we consider a module including many reaction steps governed by a rate law that is not proportional to the parameter or group of parameters that are changed: vab = v(Sa,pb) In this general case, r is the factor by which the initial rate is effectively multiplied when one or more parameters in the module (nonproportional to the rate) are changed In general, the difference between flux change (rJ = J f ⁄ Jo = v ff ⁄ voo) and rate change (r = v ff ⁄ v fo) is that, whereas the rate change is a local change, obtained by changing one or more parameters at constant intermediate concentration, flux change is a systemic change, involving simultaneous changes in the parameters and intermediate concentration r < 1, r = and r > correspond to rate decrease, rate unchanged (or changed infinitesimally) and rate increase, respectively In the analysis and plots given below, r =1 corresponds to the values of the infinitesimal control coefficients w According to the definition of Cv , experimental determination of these coefficients requires change of parameters in the different modules in order to determine, for each module, the rates voo, v ff, and v fo This experimental approach has some drawbacks On the one hand, it is a laborious approach, because of the relatively high number of parameter modulations and measurements required On the other hand, in a large system, control is normally distributed, and most of the parameters have a relatively low effect on the fluxes, the errors involved in the determination of the coefficients being high An important result of the theory of modular w MCA for large responses is that the Cv coefficients can be calculated in an alternative way, using data obtained by modulating S and measuring the resulting rates In this approach, values of voo and v fo are sufficient to perform the calculations, measurement of v ff not being required Modulation of S may be performed by using an auxiliary reaction, in which case manipulation of parameters of the system is not necessary This alternative method, based on the theory of modular MCA for large responses, does not have the drawbacks mentioned above We will describe this alternative method in the case of a metabolic system grouped into three modules and one linking intermediate (Fig 1) Results Relationships between system responses and component responses in branched systems The responses of the rates of the isolated modules to changes in the intermediate concentration, i.e the FEBS Journal 278 (2011) 2565–2578 ª 2011 The Authors Journal compilation ª 2011 FEBS 2567 Control of large changes in metabolic branches F Ortega and L Acerenza component responses, are represented by the e-elasticity coefficients In the scheme of Fig 1, there are three e-elasticity coefficients for large changes: evi (i = 1, 2, S 3) These may be directly calculated replacing the data, module rates versus S, in the definition (see definitions of the coefficients in Doc S1 and [21]) With the values of the e-elasticity coefficients, evi , and rS = S f ⁄ S o, the S factors by which the rates are changed, ri, and the sysw temic responses, Cvi (w = S, Ji and i = 1, 2, 3), are calculated from the equations of Tables and 2, respectively Note that all the relationships involving system properties and component properties given in Tables and reduce to the well-known expressions of traditional MCA and modular MCA when infinitesimal changes are considered (i.e when rS = 1) Derivation of the equations given in Tables and is given in Doc S1 The formalism developed here is based on control coefficients with respect to rates, and therefore remains valid if the rates of the reaction steps are not proportional to the corresponding enzyme concentrations Concentration and flux control coefficients for large changes satisfy summation relationships For the branched metabolic network, the summation relationships are: den ¼ ev1 ỵ a ev2 ỵ a ịev3 S S S J J J Cv13 ¼ a Cv23 þ ð1 À aÞ Cv33   o o o o where: a ¼ J2 J1 and À a ¼ J3 J1 With the r-factors and control coefficients given in Tables and 2, the variable changes produced by a rate change may be calculated from the following expression: S S S Cv1 ỵ Cv2 ỵ Cv3 ẳ rS J J J Cv11 ỵ Cv12 ỵ Cv13 ẳ J J J Cv21 ỵ Cv22 ỵ Cv23 ẳ J J J Cv31 ỵ Cv32 ỵ Cv33 ¼ 1: In addition, the flux control coefficients are constrained by flux conservation relationships: J J J Cv11 ¼ a Cv21 ỵ aị Cv31 J J J Cv12 ẳ a Cv22 ỵ aị Cv32 Table r-factors versus e-elasticity coefficients Expressions used to calculate the rate changes (ri) from the component responses (evi ), the change in the intermediate concentration (rS) and the initial S  o o flux distribution (a ¼ J2 J1 )   ỵ a ev2 ỵ aị ev3 rS 1ị S S r1 ẳ þ ev1 ðrS À 1Þ S   v1 a þ eS À ð1 À aÞ ev3 ðrS À 1Þ S   r2 ẳ a ỵ ev2 rS 1ị S   aị ỵ ev1 À a ev2 ðrS À 1Þ S S   r3 ẳ aị ỵ ev3 rS 1Þ S 2568 Table Control coefficients versus e-elasticity coefficients w Expressions used to calculate the system responses (Cvi ) from the component responses (evi ), the change in the intermediate concenS  o o tration (rS), and the initial flux distribution (a ¼ J2 J1 )   S Cv1 ẳ ỵ ev1 rS 1ị den S   v2 S Cv2 ẳ a ỵ eS ðrS À 1Þ den   v3 S Cv3 ẳ a ị ỵ eS ðrS À 1Þ den    v3 J1 v2 v1 Cv1 ẳ a eS ỵ a ị eS den ỵ eS rS 1ị   v2 J1 v1 Cv2 ¼ À a eS þ eS ðrS À 1Þ den   v3 J1 v1 Cv3 ẳ a ị eS ỵ eS rS ị den   J2 v2 v1 Cv1 ẳ eS ỵ eS rS À 1Þ den    v3 J2 v1 Cv2 ẳ eS ỵ a ị eS den ỵ ev2 rS ị S   J2 v2 v3 Cv3 ¼ À ð À a ị eS ỵ eS rS ị den   J3 v3 v1 Cv1 ¼ eS þ eS ðrS À Þ den   J3 v3 v2 Cv2 ẳ a eS ỵ eS rS 1ị den    J Cv33 ẳ ev1 ỵ a ev2 den ỵ ev3 rS 1ị S S S wf w ẳ ỵ Cvi ð ri À Þ wo It is important to note that, in this general expression, w Cvi is a function of ri If, for a module i, the value of the control coefficient is close to 0, or only r-factors close to unity can be achieved, significant changes in the variable cannot be obtained by modulating this module This is a strong result, because it implies that the impossibility of changing the variable does not depend on which parameter or combination of parameters of the module is changed So, if we want to change a variable, it is necessary to change a parameter or set of parameters in a module with a control coefficient and r-factor substantially different from and 1, respectively Usefulness of module control coefficients in branched systems In the scheme of Fig 1, with one supply module (module 1) and two demand modules (modules and FEBS Journal 278 (2011) 2565–2578 ª 2011 The Authors Journal compilation ª 2011 FEBS F Ortega and L Acerenza 3), there are three concentration and nine flux control coefficients for large responses (Table 2) These quantify how the variables are affected by changes in the J2 J3 supply or demand rates For example, Cv1 and Cv1 represent how a change in supply rate affects the J2 J3 fluxes J2 and J3, producing Y and Z If Cv1 and Cv1 have similar values, those fluxes are similarly affected, in relative terms, by the supply rate change On the J2 other hand, Cv2 quantifies the effect that a change in the rate of the demand of S for Y synthesis has on the flux that produces Y A high value of this flux control coefficient, in a wide range of values of rate change r2, indicates that large flux changes could be achieved by changing the rate of the process carryJ3 ing the flux Similar considerations apply to Cv3 J2 J3 Finally, Cv3 and Cv2 quantify how one demand flux is affected by changing the rate of the competing branch J2 J3 J2 J3 J2 Normally, Cv1 , Cv1 , Cv2 and Cv3 are positive, and Cv3 J3 and Cv2 are negative, indicating increase and decrease of the flux with rate increase, respectively In the case that J2 and J3 produce molecules essential for cell functioning (e.g for protein synthesis), one would expect them to show a relatively high response to the demand rate of the corresponding modules, and thereJ2 J3 fore Cv2 and Cv3 to have relatively high values (the control exerted by supply being smaller) In addition, the change in one of these demand rates should not significantly affect the flux of the competing branch, J2 J3 the values of Cv3 and Cv2 being relatively small in absolute terms Control pattern In traditional modular MCA, the control pattern is the set of values that the infinitesimal control coefficients take at the reference state For example, in Fig the control pattern comprises the values of the 12 infinitesimal control coefficients In the framework of modular MCA for large responses, apart from the values of the infinitesimal control coefficients at the reference state, the control pattern includes two important additional properties One property is how the values of the control coefficients change when the rate is changed by a large (non-infinitesimal) amount The normal behavior of Ji flux control coefficients of the type Cvi is that their values decrease when the rate of the corresponding modJi ule increases; that is, Cvi decreases when ri increases Ji When Cvi stays approximately constant or increases, the control pattern is called sustained or paradoxical, respectively [26,27] Note that, in modular MCA for large responses, the values of the infinitesimal control Control of large changes in metabolic branches coefficients are also relevant to the control pattern Ji obtained Normally, Cvi decreases when the rate increases If the initial infinitesimal control coefficient is small, then, as the rate is increased, it will become even smaller, with little effect on the flux Therefore, to obtain a substantial increase in the flux, a relatively large infinitesimal control coefficient is usually required The other important property is the range of values of ri that can be achieved It is, in principle, possible to obtain any value of ri manipulating the parameters in module i However, some ri values would result in unrealistic values of the concentration of the linking intermediate (S), e.g either too high or too low to be compatible with the physical chemistry or the physiology of the cell The range of values of S (Smin, Smax) determines a range of values of ri (rimin, rimax) that can be achieved (Table 1) In summary, the three relevant properties characterizing the control pattern are: the value of the infinitesimal control coefficients at the reference state, the effect that non-infinitesimal rate changes have on the values of the control coefficients for large responses, and the range of rate changes that can be achieved, maintaining the concentration of the linking intermediate at feasible values Next, we will illustrate how these three properties constrain the range of values that a flux can take J In Fig 2A (inset), we represent curves of Cvi versus ri for a hypothetical system The a-curve corresponds to a starting system, and the b-curve to the system after several parameters have been changed Black circles are the values of the infinitesimal control coefficients at the reference state In this example, the value of the infinitesimal control coefficient at the reference state in the a-curve is twice the value in the b-curve However, the control coefficient in a drastically decreases when ri is increased, while the control coefficient in b remains constant (i.e shows sustained control) In Fig 2A, we show the flux as a function of ri calculated form the a-curve and b-curve appearing in the inset of the figure The b-curve, showing the lowest infinitesimal control coefficient in the reference state, results in greater increases in flux for moderate or high values of ri increase Note that, in an analysis based on the infinitesimal control coefficients at the reference state only, the opposite, erroneous conclusion, would have been reached In the previous example, we assumed, that in both systems, ri could be changed in identical ranges Let us consider another hypothetical example, represented in Fig 2B The value of the infinitesimal control coefficient at the reference state in the a-curve is also twice the value in the b-curve In this case, however, the FEBS Journal 278 (2011) 2565–2578 ª 2011 The Authors Journal compilation ª 2011 FEBS 2569 Control of large changes in metabolic branches F Ortega and L Acerenza Determination of the control coefficients from top-down experiments A 1.2 1.1 1.0 b J ri Jo a 0.9 0.8 J Cvi 0.8 a 0.0 0.7 0.7 0.8 0.9 b 0.4 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.0 1.1 1.2 1.3 1.4 ri B 1.2 1.1 1.0 b 0.9 a J ri Jo 0.8 J Cvi 0.8 a 0.4 b 0.0 0.7 0.7 0.8 0.9 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.0 1.1 1.2 1.3 1.4 ri Fig Flux control pattern and flux changes The flux control J coefficients with respect to the rate (Cvi ) and the flux relative to the reference state value (J ri ⁄ Jo) are plotted against the factor by which the rate of the module is changed (ri), for two hypothetical situations (A, B) In both cases, the infinitesimal flux control coefficient at the reference state (•) in the a-curve is twice that in the b-curve (A) The control coefficient in the a-curve decreases with module rate, whereas that in the b-curve remains constant As a consequence, higher increases in flux may be obtained in the b-system (B) The control coefficient in the a-curve is constant and that in the b-curve decreases smoothly However, the b-system can achieve larger flux changes, because the feasible range of rates in the a-curve is smaller In both cases, using only infinitesimal control coefficients to predict large flux changes leads to erroneous conclusions a-curve shows sustained control, whereas in the b-curve, the control drops smoothly as the rate is increased Importantly, in this case, the maximum ri that can be achieved in the a-curve (rimax = 1.10) is lower than that in in the b-curve (rimax = 1.40) As a consequence of this property, the b-curve would result in greater increases in flux for high values of rate increase Considering the infinitesimal control coefficients at the reference state only, once again, the opposite, erroneous, conclusion would have been obtained 2570 Here, we will show how to calculate the coefficients for large responses from top-down experiments in a branched system with three modules and one linking intermediate First, the fluxes and concentration of the linking intermediate, S, at the reference state are determined Second, S is changed by addition of an auxiliary reaction For each value of S, the rates of the three modules are measured Changing S by parameter modulation is also possible, but in this case only the rates where the parameter was not modulated are computed From the table of the rates versus S, the three e-elasticity coefficients for large changes are calculated in terms of S Introducing the e-elasticity coefficients in the equations of Tables and renders the values of the factors r1, r2 and r3 and of the control coefficients as a function of S Finally, the parametric plot of the control coefficients for large changes as a function of ri (i = 1–3) may be constructed In an experimental system, a complete modular MCA for large responses ideally requires both upmodulation and downmodulation of the linking intermediate concentration in the range of feasible values, and measurement of this concentration and the corresponding values of the rates To our knowledge, there is no set of data in the literature that allows performing a complete modular MCA for large responses in branched systems The analysis of incomplete datasets requires undesirable extrapolations to be made outside the experimental range Therefore, we decided to illustrate the new method with a mathematical model based on measured kinetic parameters, to avoid this type of extrapolation The model is manipulated in exactly the same way as an experimental system, as is described in the next section Application of modular MCA of large responses to Asp metabolism We will analyze a detailed kinetic model of Asp metabolism in Arabidopsis constructed on the basis of measured kinetic parameters [25] The structure of the metabolic system shows several branch points regulated by a relatively complex network of allosteric interactions The model was originally built for analyzing the function of these allosteric interactions in the branched pathway For this purpose, the authors performed numerical simulations and traditional MCA Here, we will apply to the model the new modular MCA framework, to study the control pattern for FEBS Journal 278 (2011) 2565–2578 ª 2011 The Authors Journal compilation ª 2011 FEBS F Ortega and L Acerenza Fig Modular aggregation of Asp metabolism The model of Asp metabolism is aggregated into one input and two output modules, Thr being the linking intermediate Modules 1, and have fluxes J1, J2, and J3, respectively, as in Fig AdoMet, Asp and Cys are external species, their concentrations remaining constant Aspartate semialdehyde (ASA), aspartyl phosphate (Asp-P), homoserine (HSer), Ile, Lys, phosphohomoserine (PHSer) and Thr are internal variable species AK, aspartate semialdehyde dehydrogenase (ASADH), cystathionine-c-synthase (CGS), DHDPS, HSDH, homoserine kinase (HSK), TD and TS represent enzyme activities The five groups of allosteric interactions, four inhibitions (G-I to G-IV) and one activation (G-V), are indicated by dashed lines For additional information, see Doc S1 and [25] large responses of Thr and Ile incorporation into protein To this end, a modular aggregation of the model in three modules and one linking intermediate, Thr, was made (Fig 3) Module is the ‘supply’ module producing Thr from Asp Module is a ‘demand’ module consuming Thr for protein synthesis Module is another ‘demand’ module, producing Ile from Thr, used for protein synthesis It is important to emphasize, before beginning our analysis, that there are two key differences between the Control of large changes in metabolic branches MCA applied in Curien et al [25] and the modular MCA that we will perform next The first is that, whereas the MCA used by Curien et al applies to small (strictly speaking, infinitesimal) changes around the reference state only, our modular MCA is also valid for large changes To study the effect of large modulations, Curien et al used numerical simulation to obtain the effects that large changes in particular parameters have on selected variables The second difference is that Curien et al determined the control coefficients of all the variable metabolite concentrations and fluxes with respect to the rate of all the steps, and we will use a modular aggregation of the model, applying modular MCA to one supply and two demand modules connected by Thr Therefore, our conclusions will not be referred to the control by individual steps, but to the control by regions of the network relevant to the particular metabolic processes that we aim to understand Our analysis will consist of two stages First, we will calculate the control pattern of large responses of the modular system, as was described in ‘Control pattern’ The analysis of this pattern will show, for example, how the system responds to large changes in supply of and demand for Thr for protein synthesis In this first stage, the modules will be treated as ‘black boxes’ The perturbations and determination of the responses in the model will follow the same steps described in ‘Determination of control coefficients from top-down experiments’ It is important to emphasize that none of the conclusions obtained at this stage require knowledge about the processes taking place inside the modules In the second stage, we will look inside the modules and study how the control pattern, determined in the first stage, is affected by eliminating the allosteric interactions operating in the system This study will allow investigation of which allosteric interactions are relevant for establishing the control pattern of Thr and Ile incorporation into protein and which are not The rate equations and parameter values used are given in Doc S1 To start the analysis, the model was manipulated following the same general procedure that would be performed on an experimental system The Thr concentration was changed up and down, and the corresponding rates of the three modules were computed With these quantities, all of the coefficients and factors were calculated The elasticity coefficients for large responses ( ev1 , ev2 S S and ev3 ) were obtained in a range of Thr concentration S between 60 and 3000 lm, i.e approximately between ⁄ and 10 times the reference steady-state value, Thro = 303 lm (see Discussion below) This range of FEBS Journal 278 (2011) 2565–2578 ª 2011 The Authors Journal compilation ª 2011 FEBS 2571 Control of large changes in metabolic branches F Ortega and L Acerenza 10 10 Thr ri Thr Thr Cvi o –5 1 –10 2 3 4 5 ri Fig Concentration control pattern The concentration control Thr coefficients with respect to the rates (Cvi ) and Thr concentration relative to the reference state value (Thr ri ⁄ Thr o) are plotted against the factor by which the rate of module i (ri, i = 1, 2, 3) is changed, for the system in Fig Starting at the reference state (•), moderate increases in r1 or decreases in r2 and r3 result in relatively high changes in the control coefficients (r1min, r1max) = (0.36, 1.69), (r2min, r2max) = (0.23, 5.2) and (r3min, r3max) = (0.14, 4.0) are the ranges of ri values that could be achieved without producing unfeasible Thr concentrations values of Thr concentration is the one used in all of the calculations, and will determine the ranges of values of ri that could be achieved The product elasticity coefficient, ev1 , is negative and the substrate elasticity coeffiS cients, ev2 and ev3 , are positive in of all the range of S S values of Thr concentration When the concentration of Thr increases, the three coefficients decrease, in absolute terms These decreases correspond to increases in saturation of the processes (Fig S1) In Fig 4, the concentration control coefficients for Thr Thr Thr large responses (Cv1 , Cv2 and Cv3 ) are represented as a function of the factors by which the rates of the modules were changed (r1, r2, and r3) The signs of these control coefficients are those normally expected: Thr Cv1 , quantifying control with respect to ‘supply’, is Thr Thr positive, and Cv2 and Cv3 , quantifying control with respect to ‘demand’, are negative The absolute values taken at the reference state are close to the minimum values attained in all of the range of plausible rates Moderate increases in r1 or decreases in r2 and r3 result in relatively high changes in the concentration control coefficients The ranges of ri values, (rimin, rimax) i = 1–3, that could be achieved without producing unfeasible concentrations of Thr, are: (r1min, r1max) = (0.36, 1.69), (r2min, r2max) = (0.23, 5.2), and (r3min, r3max) = (0.14, 4.0) Importantly, these are the ranges of rate changes that can be achieved when the fluxes adapt to changing external conditions or when the system is manipulated for biotechnological purposes 2572 In Fig 5, we represent the flux control coefficients J1 J2 J3 J1 J2 J3 J1 J2 for large responses, Cv1 , Cv1 , Cv1 , Cv2 , Cv2 , Cv2 , Cv3 , Cv3 J3 and Cv3 , and the steady-state fluxes, J1, J2 and J3, as a function of the corresponding factors r1, r2, and r3 The flux control coefficients with respect to v1 are fairly constant, in most of the range of r1, and show reasonably high values However, the range of r1 values, maintaining the concentration of the linking intermediate at plausible values, is relatively narrow: (r1min, r1max) = (0.36, 1.69) As a consequence, the maximum increases in the output fluxes, J2 and J3, that can be achieved by increasing the rate of the supply module are modest: 29% and 16%, respectively (Fig 5A) On the other hand, the ranges of rate changes of the demand modules are much wider: (r2min, r2max) = (0.23, 5.2) and (r3min, r3max) = (0.14, 4.0) In addition, if we look in the insets of Fig 5B,C, J2 J3 the flux control coefficients Cv2 and Cv3 at the reference state are high, and show only moderate decreases when rate is increased This is why the fluxes J2 and J3 can achieve increases of 160% and 182%, changing the J3 rates of the corresponding modules (Fig 5B,C) Cv2 shows, in all the range of rates, low values in absolute terms (Fig 5B) As a consequence, J3 suffers only minor perturbations if the rate of the competing modJ2 ule is changed Cv3 shows higher absolute values than J3 Cv2 , and J2 decreases to a greater extent than J3, when the rate of the competing module is increased (Fig 5C), although the effect is not dramatic In summary, with the control pattern found, the fluxes J2 and J3 show a large response to the demand of the corresponding modules, the effect of changing supply being much smaller In addition, the change in one of the demand rates does not severely affect the flux of the competing branch These properties are those to be expected when the products of the demand modules are essential, simultaneously, for cell functioning, as is the case in the system under study If a modular analysis based on infinitesimal control coefficients only were performed, some of these conclusions would have been different For instance, as the infinitesimal J2 J2 module control coefficients Cv1 and Cv2 take similar J2 J2 values (Cv1 ¼ 0:6 and Cv2 ¼ 0:7), this information alone would suggest that the increases in the flux J2 that could be achieved by changing, independently, the rate of supply and the rate of demand are quantitatively similar As we have seen, the analysis for large responses shows that this conclusion based on infinitesimal module control coefficients only is erroneous In the work of Curien et al [25], where MCA was applied for infinitesimal changes and without module aggregation, the authors found a rather high level of control of protein-forming fluxes in the reaction FEBS Journal 278 (2011) 2565–2578 ª 2011 The Authors Journal compilation ª 2011 FEBS F Ortega and L Acerenza Control of large changes in metabolic branches A 1.2 1.0 Jir1 Jio 0.8 0.8 Ji Cv1 0.6 0.0 0.4 0.4 0.6 0.8 0.4 0.4 0.6 1.0 0.8 1.0 1.2 1.2 1.4 1.4 1.6 1.6 r1 B 3.5 0.8 3.0 0.4 Ji Cv2 0.0 2.5 Jir2 –0.4 2.0 o Ji 1.5 1.0 0.5 r2 C 4.0 3.0 0.8 3.5 0.4 Ji Cv3 0.0 Jir3 Jio 2.5 2.0 2 3 –0.4 1.5 1.0 0.5 r3 Ji Ji Fig Flux control pattern The flux control coefficients (Cv , Cv Ji and Cv ) and the flux values, relative to the reference state value    (Jir1 Jio ,Jir Jio and Jir Jio ) are plotted against the factor by which the rate of the module is changed (r1, r2 and r3, respectively), for the system in Fig The three numbered curves in each plot correspond to the three fluxes Ji (i = 1, 2, 3) We can see that there is a relatively high infinitesimal control of the output fluxes, J2 J3 J2 and J3, by the rate of the supply module [i.e Cv1 and Cv , represented in the inset of (A), are relatively high in all the feasible range of r1], but, owing to the narrow range of feasible supply rates, substantial increases of these fluxes require increases in  o r2 the rates of demand modules, r2 and r3 [see J2 J2 in (B) and  o r3 J3 J3 in (C)] catalyzed by isoform AK1 of aspartate kinase (AK), located in the supply region, suggesting that important increases in the fluxes could be achieved by modulating supply However, if the maximum velocity of AK1 at the reference state (AK1 = 0.25) is multiplied by a factor 2.26, r1 reaches its maximum feasible value (1.69) (Fig 5A) As was discussed above, the parameter increase could produce, at most, a 29% in J2 and a 16% increase in J3 Therefore, increases in supply rate may produce only moderate increases in protein-forming fluxes Note that the upper bounds to output fluxes increases are independent of which parameter or combination of parameters of the supply module are changed and to what extent, as was discussed above The modular MCA performed for large responses and the conclusions obtained up to now did not require knowledge of details from inside the modules Now we will look inside the modules to analyze the effect of eliminating the allosteric interactions on the control pattern for large responses (Fig 3) In module 1, there are several groups of allosteric interactions: inhibition of both activities of bifunctional AK-HSDH (two isoforms: AKI-HSDHI and AKII-HSDHII) by Thr (G-II), inhibition of the isoforms of monofunctional AK (AK1 and AK2) by Lys (G-III), inhibition of the isoforms of DHDPS (DHDPS1 and DHDPS2) by Lys (G-IV), and activation of TS (TS1) by AdoMet (G-V) In module 3, there is only one protein subject to allosteric regulation, namely, TD, which is inhibited by Ile (G-I) and, in module 2, there is no allosteric interaction (for a full description of the allosteric regulations in the model, see [25] and references therein) Next, we will study the control pattern after elimination of the five groups of interactions (G-I to G-V), one at a time, to assess the relative importance that these groups have in determining the type of control pattern for large responses exhibited by the system It is important to note that this type of modification will also produce the undesirable effect of affecting the reference values of the variable metabolite concentrations and fluxes, i.e the reference steady state of the system To avoid this simultaneous effect on state and control, we have modified the maximal rates in the rate equations where the allosteric interaction is eliminated in such a way that the starting values of the concentrations and fluxes remain unaltered In Fig 6, we represent the effect of eliminating the feedback inhibition of TD by Ile (G-I in Fig 3) The main differences from the original control pattern are as follows The range of r1 increases, the modified system showing large increases in control by supply of J3 FEBS Journal 278 (2011) 2565–2578 ª 2011 The Authors Journal compilation ª 2011 FEBS 2573 Control of large changes in metabolic branches A 12 1.2 10 0.8 Jir1 Jio F Ortega and L Acerenza 0.4 J Cv1i 0.0 2 123 r1 B 0.8 0.4 –0.4 Jir2 J Cv2i 0.0 –0.8 Jio 1 3 0 r2 C 1 Jir3 0.8 0.4 0.0 –0.4 J Cv3i 10 12 14 3 Jio 2 0 10 12 14 r3 Fig Effect of eliminating G-I allosteric interactions The flux control coefficients and the flux values, relative to the reference state value, are plotted against the factor by which the rate of the module is changed, for the system in Fig (solid lines) and the system modified by eliminating G-I allosteric interactions (dashed lines) The main effects on the control pattern of eliminating G-I are as follows: (A) the feasible range of supply rates and the control of J3 by supply increase dramatically; (B) a large increase in J2 can still be achieved by increasing r2, but there is a concomitant large decrease in the flux of the competing branch, J3; and (C) by increasing r3, the same maximum value of J3 can be achieved, but requires much higher increases in rate Therefore, eliminating G-I interactions produces undesirable effects on the control pattern 2574 (the flux of the module where the feedback was elimiJ3 nated) but not J2 (Fig 6A) Cv2 increases in absolute terms, producing an undesirably large change in J3 when the rate of the competing branch is increased (Fig 6B) The range of r3 increases But, since the absolute values of the control coefficients with respect to v3 are reduced, the maximum effects on the fluxes with r3 remain approximately unchanged (Fig 6C) However, the system with the feedback inhibition has the advantage of requiring a smaller r3 to achieve the same J3 In Fig 7, we represent the effect of eliminating the feedback inhibition of bifunctional AKI-HSDHI and AKII-HSDHII by Thr (G-II in Fig 3) In contrast to what was observed for the inhibition of TD, the ranges of values of r1, r2 and r3 decrease After removal of the inhibition, the maximum values of the fluxes that can be obtained by changing r1 are almost the same (Fig 7A), but the changes in r1 required are smaller, which strengthens the control by supply In addition, the maximum effects on J2 of changing r2 and on J3 of changing r3 are drastically reduced (Fig 7B,C), impinging on the potential of the system to control the output fluxes by demand Finally, the maximum reduction of J2 by r3 and of J3 by r2 resulting from branch competition remains unchanged after elimination of the inhibition, but is achieved with smaller changes in rate, what is another disadvantage for the independent regulation of the branches Eliminating the feedback inhibition of AK1 and AK2 by Lys (G-III in Fig 3) has minor effects on the control pattern, the flux changes that can be achieved being similar to those of the original system (Fig S2) The inhibition of DHDPS1 and DHDPS2 by Lys (G-IV in Fig 3) also has minor effects on the control pattern Finally, because, in the model, AdoMet is treated as a parameter, eliminating the activation of TS1 by AdoMet (G-V in Fig 3) and compensating by changing the maximal rate has no effect on the control pattern (data not shown) In summary, only elimination of G-I and elimination of G-II (Fig 3) produce important changes in the control pattern Moreover, the resulting changes in the control pattern impair the regulatory responses of the system: weakening the control by demand, which is needed, strengthening the unwanted control by supply, reducing the desirable independence between competing branches, or a combination of these Therefore, these two groups of allosteric inhibitions appear to be essential for establishing the adequate control pattern A natural question is whether the main factor responsible for generating the control pattern in the inhibition of AKI-HSDHI and AKII-HSDHII by Thr FEBS Journal 278 (2011) 2565–2578 ª 2011 The Authors Journal compilation ª 2011 FEBS F Ortega and L Acerenza Control of large changes in metabolic branches A 1.2 1.0 Jir1 0.8 Jio 0.6 Ji Cv1 0.8 0.4 0.0 0.4 0.4 1.2 0.6 0.8 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.0 1.2 1.4 1.6 r1 B 3.5 0.8 2 3.0 0.4 1 2.5 Jir2 Jio Ji Cv2 0.0 –0.4 2.0 1.5 1.0 0.5 3 is the AK activity, the HSDH activity, or both activities To answer this question, we studied the effect of removing the inhibition of AKI and AKII, and the inhibition of HSDHI and HSDHII, separately In both procedures, the corresponding maximal rates were adjusted such that the starting values of the concentrations and fluxes remain unaltered (as previously) Inhibition of the AKI and AKII activities makes the main contribution to the generation of the control pattern (Fig S3) There is one interesting feature regarding the four AK isoenzymes (AK1, AK2, AKI, and AKII) Eightyeight per cent of the steady-state input flux of the system at the reference state is carried by the AK1 and AK2 activities However, as we have seen, these isoenzymes have only minor effects in determining the control pattern of the system (Fig S2) Therefore, AK1 and AK2 function as ‘flux-generating isoenzymes’ On the other hand, AKI and AKII carry only 12% of the input flux However, feedback inhibition of these isoenzymes (together with TD) is mainly responsible for the control pattern AKI and AKII operate as ‘control pattern-generating isoenzymes’ Briefly speaking, AK1 and AK2 determine the values of the fluxes at the reference state, and AKI and AKII determine the responses of the fluxes r2 C Discussion 4.0 0.8 3.5 0.4 Ji 0.0 Cv3 3.0 Jir3 2.5 –0.4 2.0 Jio –0.8 1.5 1.0 0.5 2 r3 Fig Effect of eliminating G-II allosteric interactions The flux control coefficients and the flux values, relative to the reference state value, are plotted against the factor by which the rate of the module is changed, for the system in Fig (solid lines) and the system modified by eliminating G-II allosteric interactions (dashed lines) The main effects on the control pattern of eliminating G-II are as follows: (A) the maximum fluxes achieved by changing r1 remain unchanged, but smaller changes in r1 are required; (B) the maximum effect on J2 of increasing r2 is drastically reduced; and (C) the maximum effect on J3 of increasing r3 is drastically reduced These consequences represent undesirable effects on the control pattern Traditional MCA and modular MCA use the values of infinitesimal control coefficients to predict changes in metabolite concentrations and fluxes produced by small changes in the rates of reaction steps or pathways This consists of extrapolating the value of the infinitesimal control coefficient to a small region around the reference state When rate changes are large, the value of the infinitesimal control coefficient is not sufficient to make this type of prediction As we have shown, in modular MCA for large responses in branched systems, three properties determine the control pattern for large changes in the variables: (a) the value of the infinitesimal control coefficient at the reference state (b) the effect that non-infinitesimal rate changes have on the value of the control coefficient and (c) the range of rate changes that can be achieved, consistent with keeping the concentration of the linking intermediate at feasible values As has been shown, using only values of infinitesimal control coefficients to predict large variable changes can lead to erroneous conclusions A central result of the theory here developed is that the changes in the variables may be obtained using the  w w equation wf wo ẳ ỵ Cvi ri À Þ , where ri and Cvi FEBS Journal 278 (2011) 2565–2578 ª 2011 The Authors Journal compilation ª 2011 FEBS 2575 Control of large changes in metabolic branches F Ortega and L Acerenza are calculated by introducing the values of the e-elasticity coefficients (which are directly determined from experimental data) in the expressions of Tables and It is important to emphasize that this analysis is based on the control coefficients with respect to rates, the conclusions obtained being valid independently of the parameter changes that produced the rate change For w instance, if the value of the control coefficient, Cvi , is close to 0, or the range of rate changes, ri, is very narrow, significant changes in the variable could not be obtained This is a strong result, because it means that it is not possible to change the variable independently of the parameter or combination of parameters that are changed The essence of the power of this approach is that we can obtain valuable information about the effects that changes in parameters have on the variables without having to modulate these parameters Moreover, as we will discuss next, in the experiments to obtain the data to apply the theory, it is not necessary to change parameters of the system According to the theory of modular MCA for branched system, the control pattern for large responses can be determined from data obtained by changing the concentration of the linking intermediate and measuring the corresponding rates of the modules (i.e from module rates versus S data) Changes in the intermediate can be achieved by incorporating auxiliary reactions that produce or consume it Therefore, in contrast to what could be concluded from the definitions of control coefficients in terms of rates, modulating parameters of the system is not necessary to determine the three fundamental properties of the control pattern This procedure has several practical advantages, as was discussed above Changing the intermediate concentration by parameter modulation is also possible, but in this case only the rates of the modules where the parameter was not modulated may be used to calculate the elasticity coefficients for large responses Several studies investigating the effects of mutations on amino acid metabolism in Arabidopsis have measured intermediate concentrations in the wild type and in the mutant [28,29] However, the relevant rates were normally not measured, preventing calculation of the elasticity coefficients On the other hand, we find in the literature experiments designed to perform MCA for infinitesimal changes in branched modular aggregations, where data were obtained over a wide range of values [30,31] However, these only included modulations of the linking intermediate concentration in one direction (up or down), which is not sufficient to perform an MCA for large responses Application of modular MCA for large responses to branched systems starts by conceptual aggregation of 2576 the system in three modules and one linking intermediate Ideal aggregations fulfill two conditions: metabolites in different modules are not linked by conservation relationships, and molecules belonging to one module are not effectors of processes in another module [8] The aggregation used to analyze the model of Asp metabolism (Fig 3) fulfils these two conditions To determine the control pattern, the feasible range of concentrations must be estimated from experimental information In the model of Asp metabolism analyzed above, the range of values of Thr concentration used, between 60 and 3000 lm (i.e approximately between ⁄ and 10 times the reference state value), is a tentative range estimated from information available for Arabidopsis mutants In a mutant of Arabidopsis in which inhibition of AK by Lys was abolished, a six-fold increase in the Thr concentration was found [28] This is a lower bound to the maximum concentration of Thr that can be achieved On the other hand, a mutation in the enzyme methionine S-methyltransferase was reported to produce a two-fold decrease in the concentration of Thr [32], which is an upper bound to the minimum feasible Thr concentration On the basis of these experimental lower and upper bounds, we defined the tentative range of feasible Thr concentrations Refinement to obtain more precise bounds would require additional experimental work Another factor that may restrict the range of rates that can be achieved when rate increases are obtained by increasing the expression of enzymes is the limitation in the capacity to accommodate protein molecules in the cell [33] The control pattern of large responses for the model of Asp metabolism shows several characteristic features Fluxes incorporating Thr and Ile into protein are mainly controlled by demand rate, supply rate making only a minor contribution A change in one demand rate does not produce a major effect in the flux of the competing branch The control pattern found is the one expected when the products of the output branches are essential for cell functioning, as is the case in the system under study Note that, in other branch points where the output limbs enter into operation under different external conditions, changing the rate of one of the output processes may produce dramatic shifts between the pathways; this is called the ‘branch point effect’ [34] There is an asymmetry regarding the way in which the feasible range of concentrations constrains the effect of supply and demand rates Increases in supply rate normally increase the intermediate concentration Therefore, supply rate increases are limited by the maximum feasible concentration In contrast, increases in demand rate most often decrease the intermediate FEBS Journal 278 (2011) 2565–2578 ª 2011 The Authors Journal compilation ª 2011 FEBS F Ortega and L Acerenza concentration, these rate increases being limited by the minimum feasible concentration In the model of Asp metabolism [25] and the experimental system that it represents [29], the Thr concentration shows large increases when supply is increased The consequence of this high sensitivity is that, with moderate increases in supply rate, the maximum feasible Thr concentration is reached This severely limits the increases in the output fluxes that can be obtained modulating supply On the other hand, higher increases in demand rate may be performed before the lower feasible Thr concentration is reached, allowing higher increases in the output fluxes The expressions previously derived (Tables and 2) are useful for the analysis of branched metabolic systems They tell us what the response to a large change (quantified by the control coefficients) will be if the component steps or modules show particular kinetic properties (quantified by the e-elasticity coefficients) On the other hand, if we wanted to design a system with a particular pattern of values of control coefficients, expressions to calculate the e-elasticity coefficients from the control coefficients for large changes would be needed One set of this design expressions is: J J J S S S ev1 ¼ Cv12 Cv2 , ev2 ¼ Cv21 Cv1 and ev3 ¼ Cv31 Cv1 S S S (see Doc S1) High-throughput techniques reveal an extraordinary complexity in the changes taking place when organisms respond to perturbations For instance, DNA microarray studies show that the switch from anaerobic to aerobic growth upon depletion of glucose in Saccharomyces cerevisiae is correlated with increases or decreases in the expression of 30% of the approximately 6400 genes by factors of at least [35,36] Modular MCA for large responses could contribute to our understanding of the logic behind the way in which this type of genome scale changes act upon the metabolic 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