Metabolic flux control analysis of branch points: an improved approach to obtain flux control coefficients from large perturbation data

An overview of published approaches for the metabolic flux control analysis of branch points revealed that often not all fundamental constraints on the flux control coefficients have been taken into account. This has led to contradictory statements in literature on the minimum number of large perturbation experiments required to estimate the complete set of flux control coefficients C(J) for a metabolic branch point. An improved calculation procedure, based on approximate Lin-log reaction kinetics, is proposed, providing explicit analytical solutions of steady state fluxes and metabolite concentrations as a function of large changes in enzyme levels. The obtained solutions allow direct calculation of elasticity ratios from experimental data and subsequently all C(J)-values from the unique relation between elasticity ratio's and flux control coefficients. This procedure ensures that the obtained C(J)-values satisfy all fundamental constraints. From these it follows that for a three enzyme branch point only one characterised or two uncharacterised large flux perturbations are sufficient to obtain all C(J)- values. The improved calculation procedure is illustrated with four experimental cases.     

Steady-state analysis of metabolic pathways: comparing the double modulation method and the lin-log approach

The increasing interest in studying enzyme kinetics under in vivo conditions requires practical methods to estimate control parameters from experimental data. In contrast to currently established approaches of dynamic modelling, this paper addresses the steady-state analysis of metabolic pathways. Within the framework of metabolic control analysis (MCA), elasticity coefficients are used to describe the control properties of a local enzyme reaction. The double modulation method is one of the first experimental approaches to estimate elasticity coefficients from measurements of steady-state flux rates and metabolite concentrations. We propose a generalized form of the double modulation method and compare it to the recently developed linear-logarithmic approach.     

The identification of enzyme targets for the optimization of a valine producing Corynebacterium glutamicum strain using a kinetic model

The enzyme targets for the rational optimization of a Corynebacterium glutamicum strain constructed for valine production are identified by analyzing the control of flux in the valine/leucine pathway. The control analysis is based on measurements of the intracellular metabolite concentrations and on a kinetic model of the reactions in the investigated pathway. Data‐driven and model‐based methods are used and evaluated against each other. The approach taken gives a quantitative evaluation of the flux control and it is demonstrated how the understanding of flux control is used to reach specific recommendations for strain optimization. The flux control coefficients (FCCs) with respect to the valine excretion rate were calculated, and it was found that the control is distributed mainly between the acetohydroxyacid synthase enzyme (FCC = 0.32), the branched chain amino acid transaminase (FCC = 0.27), and the exporting translocase (FCC = 0.43). The availability of the precursor pyruvate has substantial influence on the valine flux, whereas the cometabolites are less important as demonstrated by the calculation of the respective response coefficients. The model is further used to make in‐silico predictions of the change in valine flux following a change in enzyme level. A doubling of the enzyme level of valine translocase will result in an increase in valine flux of 31%. By optimizing the enzyme levels with respect to valine flux it was found that the valine flux can be increased by a factor 2.5 when the optimal enzyme levels are implemented. © 2009 American Institute of Chemical Engineers Biotechnol. Prog., 2009     

Defining control coefficients in non-ideal metabolic pathways

The extent to which an enzyme controls a flux has been defined as the effect on that flux of a small modulation of the activity of that enzyme divided by the magnitude of the modulation. We here show that in pathways with metabolic channelling or high enzyme concentrations and conserved moieties involving both enzymic and non-enzymic species, this definition is ambiguous; the magnitude of the corresponding flux control coefficient depends on how the enzyme activity is modulated. This is illustrated with two models of biochemically relevant pathways, one in which dynamic metabolite channelling plays a role, and one with a moiety-conserved cycle. To avoid such ambiguity, we view biochemical pathways in a more detailed manner, i.e., as a network of elemental steps. We define 'elemental control coefficients' in terms of the effect on a flux of an equal modulation of the forward and reverse rate constant of any such elemental step (which may correspond to transitions between enzyme states). This elemental control coefficient is independent of the method of modulation. We show how metabolic control analysis can proceed when formulated in terms of the elemental control coefficients and how the traditional control coefficients are related to these elemental control coefficients. An 'impact' control coefficient is defined which quantifies the effect of an activation of all elemental processes in which an enzyme is involved. It equals the sum of the corresponding elemental control coefficients. In ideal metabolic pathways this impact control coefficient reduces to the traditional flux control coefficient. Differences between the traditional control coefficients are indicative of non-ideality of a metabolic pathway, i.e. of channelling or high enzyme concentrations.     

Metabolic design based on a coupled gene expression-metabolic network model of tryptophan production in Escherichia coli

The presumably high potential of a holistic design approach for complex biochemical reaction networks is exemplified here for the network of tryptophan biosynthesis from glucose, a system whose components have been investigated thoroughly before. A dynamic model that combines the behavior of the trp operon gene expression with the metabolic network of central carbon metabolism and tryptophan biosynthesis is investigated. This model is analyzed in terms of metabolic fluxes, metabolic control, and nonlinear optimization. We compare two models for a wild-type strain and another model for a tryptophan producer. An integrated optimization of the whole network leads to a significant increase in tryptophan production rate for all systems under study. This enhancement is well above the increase that can be achieved by an optimization of subsystems. A constant ratio of control coefficients on tryptophan synthesis rate has been identified for the models regarding or disregarding trp operon expression. Although we found some examples where flux control coefficients even contradict the trends of enzyme activity changes in an optimized profile, flux control can be used as an indication for enzymes that have to be taken into account in optimization.     

Summation theorems for flux and concentration control coefficients of dynamic systems

Metabolic control analysis (MCA) was developed to quantify how system variables are affected by parameter variations in a system. In addition, MCA can express the global properties of a system in terms of the individual catalytic steps, using connectivity and summation theorems to link the control coefficients to the elasticity coefficients. MCA was originally developed for steady-state analysis and not all summation theorems have been derived for dynamic systems. A method to determine time-dependent flux and concentration control coefficients for dynamic systems by expressing the time domain as a function of percentage progression through any arbitrary fixed interval of time is reported. Time-dependent flux and concentration control coefficients of dynamic systems, provided that they are evaluated in this novel way, obey the same summation theorems as steady-state flux and concentration control coefficients, respectively.     

Subtleties in control by metabolic channelling and enzyme organization

Because of its importance to cell function, the free-energy metabolism of the living cell is subtly and homeostatically controlled. Metabolic control analysis enables a quantitative determination of what controls the relevant fluxes. However, the original metabolic control analysis was developed for idealized metabolic systems, which were assumed to lack enzyme-enzyme association and direct metabolite transfer between enzymes (channelling). We here review the recently developed molecular control analysis, which makes it possible to study non-ideal (channelled, organized) systems quantitatively in terms of what controls the fluxes, concentrations, and transit times. We show that in real, non-ideal pathways, the central control laws, such as the summation theorem for flux control, are richer than in ideal systems: the sum of the control of the enzymes participating in a non-ideal pathway may well exceed one (the number expected in the ideal pathways), but may also drop to values below one. Precise expressions indicate how total control is determined by non-ideal phenomena such as ternary complex formation (two enzymes, one metabolite), and enzyme sequestration. The bacterial phosphotransferase system (PTS), which catalyses the uptake and concomitant phosphorylation of glucose (and also regulates catabolite repression) is analyzed as an experimental example of a non-ideal pathway. Here, the phosphoryl group is channelled between enzymes, which could increase the sum of the enzyme control coefficients to two, whereas the formation of ternary complexes could decrease the sum of the enzyme control coefficients to below one. Experimental studies have recently confirmed this identification, as well as theoretically predicted values for the total control. Macromolecular crowding was shown to be a major candidate for the factor that modulates the non-ideal behaviour of the PTS pathway and the sum of the enzyme control coefficients.     

Comparisons of flux control exerted by mitochondrial outer-membrane carnitine palmitoyltransferase over ketogenesis in hepatocytes and mitochondria isolated from suckling or adult rats.

The primary aim of this paper was to calculate and report flux control coefficients for mitochondrial outer-membrane carnitine palmitoyltransferase (CPT I) over hepatic ketogenesis because its role in controlling this pathway during the neonatal period is of academic importance and immediate clinical relevance. Using hepatocytes isolated from suckling rats as our model system, we measured CPT I activity and carbon flux from palmitate to ketone bodies and to CO2 in the absence and presence of a range of concentrations of etomoxir. (This is converted in situ to etomoxir-CoA which is a specific inhibitor of the enzyme.) From these data we calculated the individual flux control coefficients for CPT I over ketogenesis, CO2 production and total carbon flux (0.51 +/- 0.03; -1.30 +/- 0.26; 0.55 +/- 0.07, respectively) and compared them with equivalent coefficients calculated by similar analyses [Drynan, L., Quant, P.A. & Zammit, V.A. (1996) Biochem. J. 317, 791-795] in hepatocytes isolated from adult rats (0.85 +/- 0.20; 0.23 +/- 0.06; 1.06 +/- 0.29). CPT I exerts significantly less control over ketogenesis in hepatocytes isolated from suckling rats than those from adult rats. In the suckling systems the flux control coefficients for CPT I over ketogenesis specifically and over total carbon flux (< 0.6) are not consistent with the enzyme being rate-limiting. Broadly similar results were obtained and conclusions drawn by reanalysis of previous data {from experiments in mitochondria isolated from suckling or adult rats [Krauss, S., Lascelles, C.V., Zammit, V.A. & Quant, P.A. (1996) Biochem. J. 319, 427-433]} using a different approach of control analysis, although it is not strictly valid to compare flux control coefficients from different systems. Our overall conclusion is that flux control coefficients for CPT I over oxidative fluxes from palmitate (or palmitoyl-CoA) differ markedly according to (a) the metabolic state, (b) the stage of development, (c) the specific pathway studied and (d) the model system.     

A simple matrix method for determining flux control coefficients in complex pathways

Flux control coefficients express in quantitative terms the extent to which the steady state flux through a metabolic pathway is controlled by a particular parameter. Enzyme flux control coefficients can be calculated using matrix algebra methods which express the control coefficients in terms of parameters which can be determined experimentally (enzyme elasticities, flux ratios, metabolite ratios). This paper describes an algorithm based on a 'constraint' matrix which enables expressions for enzyme control coefficients to be written for pathways of any complexity.     

Optimal re-design of primary metabolism in Escherichia coli using linlog kinetics

This paper examines the validity of the linlog approach, which was recently developed in our laboratory, by comparison of two different kinetic models for the metabolic network of Escherichia coli. The first model is a complete mechanistic model; the second is an approximative model in which linlog kinetics are applied. The parameters of the linlog model (elasticities) are derived from the mechanistic model. Three different optimization cases are examined. In all cases, the objective is to calculate the enzyme levels that maximize a certain flux while keeping the total amount of enzyme constant and preventing large changes of metabolite concentrations. For an average variation of metabolite levels of 10% and individual changes of a factor 2, the predicted enzyme levels, metabolite concentrations and fluxes of both models are highly similar. This similarity holds for changes in enzyme level of a factor 4-6 and for changes in fluxes up to a factor 6. In all three cases, the predicted optimal enzyme levels could neither have been found by intuition-based approaches, nor on basis of flux control coefficients. This demonstrates that kinetic models are essential tools in Metabolic Engineering. In this respect, the linlog approach is a valuable extension of MCA, since it allows construction of kinetic models, based on MCA parameters, that can be used for constrained optimization problems and are valid for large changes of metabolite and enzyme levels.     

Use of implicit methods from general sensitivity theory to develop a systematic approach to metabolic control. I. Unbranched pathways

It is shown that metabolic control theory (MCT), is its present form, is a particular case of general sensitivity theory, which studies the effects of parameter variations on the behavior of dynamic systems. It has been shown that metabolic control theory is obtained from this more general theory for the particular case of steady-state and linear relationships between velocities and enzyme concentrations. In such conditions the relationships between elasticities and flux control coefficients are easily obtained. These relationships are in the form of a matrix product constructed in a priori form. Relationships between combined response coefficients and concentration control coefficients are presented. The use of implicit methodology from general sensitivity theory provides a generalization of MCT, which is applied to unbranched pathways. For this particular case, provided the matrices have been properly constructed, the matrix of global properties (flux and concentration control coefficients) can be obtained by inversion of the matrix of local properties (elasticities). The theorems of MCT (concentration summation, flux summation, flux connectivity, and concentration connectivity) applicable for unbranched pathways are directly obtained by inspection of the matrix product. With these results, the present theoretical basis of MCT is extended with a more structured framework that allows a wider range of application. The results make clearer the relatedness of MCT to the more general approach provided by biochemical systems theory (BST).     

In silico analysis of lactic acid secretion metabolism through the top-down approach: Effect of grouping in enzyme kinetics

A top-down approach is known to be a useful and effective technique for the design and analysis of metabolic systems. In this study, we have constructed a grouped metabolic network forLactococcus lactis under aerobic conditions using grouped enzyme kinetics. To test the usefulness of grouping work, a non-grouped system and grouped systems were compared quantitatively with each other. Here, grouped systems were designed as two groups according to the extent of grouping. The overall simulated flux values in grouped and non-grouped models had pretty similar distribution trends, but the details on flux ratio at the pyruvate branch point showed a little difference. This result indicates that our grouping technique can be used as a good model for complicated metabolic networks, however, for detailed analysis of metabolic network, a more robust mechanism should be considered. In addition to the data for the pyruvate branch point analysis, some major flux control coefficients were obtained in this research.     

Flux control analysis for biphenyl metabolism by Rhodococcus pyridinovorans R04

The flux control coefficients of the four enzymes involved in the upper pathway of biphenyl degradation were determined from transient metabolite concentrations. The first enzyme was indicated as the major rate-limiting step of the pathway with a control coefficient of 0.48. The flux control coefficients of the other three enzymes were 0.03, 0.23 and 0.27, respectively. This is the first experimental evidence of the control step in the pathway of biphenyl degradation using metabolic control analysis.     

In vitro control analysis of an enzyme system: Experimental and analytical developments

In this paper we describe a flow-through system for reconstituting parts of metabolism from purified enzymes. This involves pumping continuously into a reaction chamber, fresh enzymes and reagents so that metabolic reactions occur in the chamber. The waste products leave the chamber via the outflow so that a steady state can be setup. The system we chose consisted of a single enzyme, lactate dehydrogenase. This enzyme was chosen because it consumes NADH in the chamber which could be monitored spectrophotometrically. The aim of the work was to investigate whether a steady state could be achieved in the flow system and whether a metabolic control analysis could be done. We measured two control coefficients, C_LDH and C_pump for the enzyme flux and NADH concentration and confirmed that the summation theorem applied to this system. The advantage of a flow-through system is that the titrations necessary to estimate the control coefficients can be easily and precisely controlled; this means that accurate estimates for the control coefficients can be obtained. In the paper, we discuss some statistical aspects of the data analysis and some possible applications of the technique, including a method to determine the presence of metabolic channelling between two different enzymes.     

Temporal analysis of metabolic systems and its application to metabolite channelling

When a metabolic system undergoes a transition between steady states, the lag or transition time of the system is determined by the aggregated lifetimes of the metabolite pools. This allows the transition time, and hence the temporal responsiveness of the system, to be estimated from a knowledge of the starting and finishing steady states and obviates the need for dynamic measurements. The analysis of temporal response in metabolic systems may be integrated with the general field of metabolic control analysis by the definition of a temporal control coefficient (C) in terms of flux and concentration control coefficients. The temporal control coefficient exhibits summation and other properties analogous to the flux and concentration control coefficients. For systems in which static metabolite channels exits, the major kinetic advantage of channelling is a reduction in pool sizes and, as a result, a more rapid system response reflected in a reduced transition time. The extent of the channelling advantage may therefore be assessed from a knowledge of the system transition time. This reveals that no channelling advantage is achieved at high enzyme concentration (i.e., comparable to K_m) or, in the case of ‘leaky’ channels, where rapid equilibrium kinetic mechanisms obtain. In the case of a perfect channel with no leakage and direct transfer of metabolite between adjacent enzyme active sites, the transition time is minimized and equal to the lifetime of the enzyme–substrate complex.     

The matrix method of metabolic control analysis: its validity for complex pathway structures

The sensitivities of the variables of a metabolic system (such as fluxes and concentrations) to variations in enzyme concentration are expressed in metabolic control analysis as control coefficients. The matrix method is a system of writing matrix equations that generate expressions for the control coefficients in terms of the characteristics of the components (principally the enzymes). Previously, the matrix method has been considered in terms of simple pathway structures; here we justify its applicability to complex pathways, such as those with multiple branches. It is shown that this requires modification of the branch point relationship to take account of changes of flux along the limbs of the branch and of stoichiometric factors. The method of deriving the flux control coefficients with respect to different fluxes in the system is extended to cope with these circumstances.     

Toward predicting metabolic fluxes in metabolically engineered strains

Predicting metabolic fluxes of a genetically engineered organism is an important step toward rational pathway design. However, because of various regulatory mechanisms, which are complex, often ill-characterized, and sometimes undiscovered, predicting metabolic fluxes using kinetic simulation is difficult. We propose to incorporate regulatory constraints in flux calculation to allow prediction of the steady-state fluxes without complete kinetics. The regulatory constraint, in its linear form, is derived from the dynamic metabolic control theory and involves the flux control coefficients. It is shown that with these constraints, the responses to metabolic perturbation can be predicted. Conversely, the regulatory constraints and the control coefficients can be determined by comparing the experimental data with the prediction. Therefore, this approach may offer a practical direction toward prediction of fluxes for metabolically engineered organisms.     

Kinetic properties required for sustained or paradoxical control of metabolic fluxes under large changes in enzyme activities

The effect that an increase in the activity of an enzyme has on its flux normally decreases with activity increase. To achieve a large increase in flux by manipulating a single step would therefore require a high initial effect that maintains or increases when the activity is increased, what has been called sustained or paradoxical control. Using metabolic control analysis for large responses, we derive conditions for sustained or paradoxical control in terms of elasticity coefficients. These are used to characterise types of rate laws contributing to this behaviour. The result that simple pathways, with normal kinetics, subject to large activity changes can lead to paradoxical control behaviour suggests that this type of pattern may be much more ubiquitous than could have, in principle, been suspected.     

Metabolic control analysis of plant metabolism

Metabolic control analysis and its major coefficients are introduced. The importance of measuring both elasticity and concentration-control coefficients as well as flux-control coefficients is stressed. The conditions that need to be met before control analysis can be applied experimentally are emphasized. It is argued that successful application of this approach requires methods for the measurement of flux, maximum catalytic activities of enzymes and substrate contents. The measurement of flux by consumption of substrate, production of product, and the distribution of isotope after metabolism of labelled substrates is discussed. The need to ensure that measurements of enzymes and substrates are reliable and authenticated is stressed. Particular emphasis is placed on the ease with which such measurements can be invalidated in plant tissues, and ways for countering these difficulties are discussed.