Systems analysis of the tricarboxylic acid cycle in Dictyostelium discoideum. II. Control analysis.

A steady-state computer model of the tricarboxylic cycle in Dictyostelium discoideum was analyzed using metabolic control theory. The steady state had variations of less than 0.04% over the last half of the simulation for both metabolite concentrations and fluxes. Metabolite and flux control coefficients were determined by varying enzymatic activities within 2% of their initial values and simulating the responses of metabolite concentrations and fluxes to these changes. Under these conditions, summation properties were met for most metabolite and all flux control coefficients. Maximum flux control coefficients were found for succinate dehydrogenase (0.35), malic enzyme (0.24), and malate dehydrogenase (-0.18). Comparable control was found for the reaction supplying pyruvate (0.14) and for the sum of the input amino acids (0.43), which serve as an energy source for D. discoideum. The time-dependent processes by which a new steady state was established were examined after increasing malic enzyme or malate dehydrogenase activities. This provided a method for an analysis of the mechanisms by which the observed control coefficients were generated. In addition, the effects of increasing the stimuli within 5-20% of the original enzyme activity were examined. Under these conditions, more typical of experimental stimuli and measurable responses, the metabolic model failed to return to steady state, and thus summation properties were not met. Whether "true" steady states ever occur or whether metabolic control theory can be applied in vivo is discussed.     

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).     

Control in channelled pathways. A matrix method calculating the enzyme control coefficients

The usual equations expressing the enzyme control coefficients (quantitative indicators of 'global' control properties of a pathway) via the elasticity coefficients (reflecting local kinetic properties of an enzyme reaction), cannot be applied to a variety of 'non-ideal' pathways, in particular to pathways with metabolic channelling. Here we show that the relationship between the control and elasticity coefficients can be obtained by considering such a metabolic pathway as a network of elemental chemical conversions (steps). To calculate the control coefficients of enzymes one should first determine the elasticity coefficients of such elemental steps and then take their appropriate combinations. Although the method is illustrated for a channelled pathway it can be used for any non-ideal pathway including those with high enzyme concentrations where the sequestration of metabolites by enzymes cannot be neglected.     

The physiology of weak selection

Reaction rates in metabolic pathways typically exhibit a kind of diminishing returns in which small variations in the activities of the individual enzymes have very little effect on overall flux. These effects are measured by the control coefficients of the enzymes, and most systems are governed by the summation theorem stating that all control coefficients must sum to unity. One implication is that complex systems will not usually contain single rate limiting steps, but rather be controlled to a greater or lesser extent by many enzymes, each exerting relatively small control. Wright understood this principle in 1934 and used it for his physiological theory of dominance. With respect to small variations in enzyme activity, the principle implies that many small variations should have only mild effects on fitness. Analysis of nucleotide polymorphisms in the genes for glucose-6-phosphate dehydrogenase and alkaline phosphatase in Escherichia coli implies that most amino acid replacements are harmful, and that the average selection coefficient against amino acid replacements that are polymorphic in natural populations is 1 x 10(-7) to 5 x 10(-7). In experiments to determine the a priori distribution of selection coefficients among random amino acid replacements, 25 replacements in beta-galactosidase were created by genetic means, and 22 of these produced selective effects too small to be detected in chemostat competition experiments (s less than 0.004 per generation).     

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.     

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.     

The mitochondrial respiratory chain is partially organized in a supercomplex assembly: kinetic evidence using flux control analysis

The model of the respiratory chain in which the enzyme complexes are independently embedded in the lipid bilayer of the inner mitochondrial membrane and connected by randomly diffusing coenzyme Q and cytochrome c is mostly favored. However, multicomplex units can be isolated from mammalian mitochondria, suggesting a model based on direct electron channeling between complexes. Kinetic testing using metabolic flux control analysis can discriminate between the two models: the former model implies that each enzyme may be rate-controlling to a different extent, whereas in the latter, the whole metabolic pathway would behave as a single supercomplex and inhibition of any one of its components would elicit the same flux control. In particular, in the absence of other components of the oxidative phosphorylation apparatus (i.e. ATP synthase, membrane potential, carriers), the existence of a supercomplex would elicit a flux control coefficient near unity for each respiratory complex, and the sum of all coefficients would be well above unity. Using bovine heart mitochondria and submitochondrial particles devoid of substrate permeability barriers, we investigated the flux control coefficients of the complexes involved in aerobic NADH oxidation (I, III, IV) and in succinate oxidation (II, III, IV). Both Complexes I and III were found to be highly rate-controlling over NADH oxidation, a strong kinetic evidence suggesting the existence of functionally relevant association between the two complexes, whereas Complex IV appears randomly distributed. Moreover, we show that Complex II is fully rate-limiting for succinate oxidation, clearly indicating the absence of substrate channeling toward Complexes III and IV.     

Fluxes and metabolic pools as model traits for quantitative genetics. I. The L-shaped distribution of gene effects

The fluxes through metabolic pathways can be considered as model quantitative traits, whose QTL are the polymorphic loci controlling the activity or quantity of the enzymes. Relying on metabolic control theory, we investigated the relationships between the variations of enzyme activity along metabolic pathways and the variations of the flux in a population with biallelic QTL. Two kinds of variations were taken into account, the variation of the average enzyme activity across the loci, and the variation of the activity of each enzyme of the pathway among the individuals of the population. We proposed analytical approximations for the flux mean and variance in the population as well as for the additive and dominance variances of the individual QTL. Monte Carlo simulations based on these approximations showed that an L-shaped distribution of the contributions of individual QTL to the flux variance (R(2)) is consistently expected in an F(2) progeny. This result could partly account for the classically observed L-shaped distribution of QTL effects for quantitative traits. The high correlation we found between R(2) value and flux control coefficients variance suggests that such a distribution is an intrinsic property of metabolic pathways due to the summation property of control coefficients.     

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.     

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.     

Matrix method for determining steps most rate-limiting to metabolic fluxes in biotechnological processes

The metabolic control theory developed by Kacser, Burns, Heinrich, and Rapoport is briefly outlined, extended, and transformed so as optimally to address some biotechnological questions. The extensions include (i) a new theorem that relates the control of metabolite concentrations by enzyme activities to flux ratios at branches in metabolic pathways; (ii) a new theorem that does the same for the control of the distribution of the flux over two branches; (iii) a method that expresses these controls into properties (the so-called elasticity coefficients) of the enzymes in the pathway; and (iv) a theorem that relates the effects of changes in metabolite concentrations on reaction rates to the effects of changes in enzyme properties on the same rates. Matrix equations relating the flux control and concentration control coefficients to the elasticity coefficients of enzymes in simple linear and branched pathways incorporating feedback are given, together with their general solutions and a numerical example. These equations allow one to develop rigorous criteria by which to decide the optimal strategy for the improvement of a microbial process. We show how this could be used in deciding which property of which enzyme should be changed in order to obtain the maximal concentration of a metabolite or the maximal metabolic flux.     

Topological analysis of metabolic control

A topological approach is presented for the analysis of control and regulation in metabolic pathways. In this approach, the control structure of a metabolic pathway is represented by a weighted directed graph. From an inspection of the topology of the graph, the control coefficients of the enzymes are evaluated in a heuristic manner in terms of the enzyme elasticities. The major advantage of the topological approach is that it provides a visual framework for (1) calculating the control coefficients of the enzymes, (2) analyzing the cause-effect relationships of the individual enzymes, (3) assessing the relative importance of the enzymes in metabolic regulation, and (4) simplifying the structure of a given pathway, from a regulatory viewpoint. Results are obtained for (a) an unbranched pathway in the absence of feedback the feedforward regulation and (b) an unbranched pathway with feedback inhibition. Our formulation is based on the metabolic control theory of Kacser and Burns (1973) and Heinrich and Rapoport (1974).     

Instantaneous flux control analysis for biochemical systems.

Linear sensitivity analysis of steady-state control of enzymic systems has been extended to non-steady states yielding sensitivity coefficients which provide non-intuitive insights into the behavior of the system and the sites of metabolic control, and which are quantitative counterparts to traditional qualitative concepts. Because this information is provided in a readily understood format, these coefficients serve as convenient indices of metabolic control. This treatment was applied to a simple test system, consisting of two enzymes and one non-enzymatic reaction, which exhibits oscillatory behavior. The results indicate that oscillations in the concentrations of the intermediate metabolites are regulated almost exclusively by the second enzyme. Control of the flux through the pathway is apportioned equally among the three reactions during periods of low net flux, but it is due almost exclusively to the second enzyme during periods of high net flux.     

Experimental determination of group flux control coefficients in metabolic networks

Grouping of reactions around key metabolite branch points can facilitate the study of metabolic control of complex metabolic networks. This top-down Metabolic Control Analysis is exemplified through the introduction of group (flux, as well as concentration) control coefficients whose magnitudes provide a measure of the relative impact of each reaction group on the overall network flux, as well as on the overall network stability, following enzymatic amplification. In this article, we demonstrate the application of previously developed theory to the determination of group flux control coefficients. Experimental data for the changes in metabolic fluxes obtained in response to the introduction of six different environmental perturbations are used to determine the group flux control coefficients for three reaction groups formed around the phosphoenolpyruvate/pyruvate branch point. The consistency of the obtained group flux control coefficient estimates is systematically analyzed to ensure that all necessary conditions are satisfied. The magnitudes of the determined control coefficients suggest that the control of lysine production flux in Corynebacterium glutamicum cells at a growth base state resides within the lysine biosynthetic pathway that begins with the PEP/PYR carboxylation anaplorotic pathway. Copyright 1998 John Wiley & Sons, Inc.     

Competition for enzymes in metabolic pathways: implications for optimal distributions of enzyme concentrations and for the distribution of flux control

The structures of biochemical pathways are assumed to be determined by evolutionary optimization processes. In the framework of mathematical models, these structures should be explained by the formulation of optimization principles. In the present work, the principle of minimal total enzyme concentration at fixed steady state fluxes is applied to metabolic networks. According to this principle there exists a competition of the reactions for the available amount of enzymes such that all biological functions are maintained. In states which fulfil these optimization criteria the enzyme concentrations are distributed in a non-uniform manner among the reactions. This result has consequences for the distribution of flux control. It is shown that the flux control matrix c, the elasticity matrix epsilon, and the vector e of enzyme concentrations fulfil in optimal states the relations c(T)e = e and epsilon(T)e = 0. Starting from a well-balanced distribution of enzymes the minimization of total enzyme concentration leads to a lowering of the SD of the flux control coefficients.     

Control of molecular transformations in polyenzyme systems: quantitative theory of the regulation of metabolism

An attempt of a comprehensive treatment of the theory of metabolic control is presented. The introductory section giving an outline of the early development of the theory, is followed by definitions quantifying the control in the metabolic system. By means of the perturbation method the complete system of equations is obtained which allows one to express all the enzyme control coefficients ("global" coefficients) through the elasticity coefficients characterizing kinetic properties of individual enzymes ("local" coefficients) and through the steady-state values of metabolic fluxes and concentrations. It is shown how connectivity relations between global and local coefficients should be modified when conserved sums of intermediates are present in the system. A new theorem is derived, it allows one to express the global response of the system to any change in the external parameter (such as external effector concentration, or temperature, pH, ionic strength, ets.) through the control coefficients and local responses of individual reaction steps. Explicit formulas are derived for response coefficients of the fluxes and concentrations to changes in the conserved sums of intermediates, which express the values of these global coefficients through the control and elasticity coefficients of enzymes and steady-state pools. The results obtained comprise as a special case all the results published so far in the literature.     

Hexokinase and not glycogen synthase controls the flux through the glycogen synthesis pathway in frog oocytes

Here we set out to evaluate the role of hexokinase and glycogen synthase in the control of glycogen synthesis in vivo. We used metabolic control analysis (MCA) to determine the flux control coefficient for each of the enzymes involved in the pathway. Acute microinjection experiments in frog oocytes were specifically designed to change the endogenous activities of the enzymes, either by directly injecting increasing amounts of a given enzyme (HK, PGM and UGPase) or by microinjection of a positive allosteric effector (glc-6P for GS). Values of 0.61 ± 0.07, 0.19 ± 0.03, 0.13 ± 0.03, and −0.06 ± 0.08 were obtained for the flux control coefficients of hexokinase EC 2.7.1.1 (HK), phosphoglucomutase EC 5.4.2.1 (PGM), UDPglucose pyrophosphorylase EC 2.7.7.9 (UGPase) and glycogen synthase EC 2.4.1.11 (GS), respectively. These values satisfy the summation theorem since the sum of the control coefficients for all the enzymes of the pathway is 0.87. The results show that, in frog oocytes, glycogen synthesis through the direct pathway is under the control of hexokinase. Phosphoglucomutase and UDPG-pyrophosphorylase have a modest influence, while the control exerted by glycogen synthase is null.     

Theory of metabolism regulation: a complete system of equations for regulation coefficients

Basic quantitative parameters of control in a metabolic system are considered: control coefficients of enzymes with respect to metabolic fluxes and concentrations, and in the case when there are conservation laws, the response coefficients of metabolic fluxes and concentrations to changes in the conserved sums of metabolite concentrations (e. g. conserved moieties). Relationships are obtained which generalize the well known connectivity relations for the case of metabolites binding by conservation laws. Additional relationships are obtained which complement the set of connectivity relations up to the complete system of equations for determining all the control coefficients. The control coefficients are expressed through the enzyme elasticity coefficients, steady state metabolic fluxes and concentrations. Formulas are derived which express response coefficients of flux and concentrations through the enzyme control and elasticity coefficients and metabolite concentrations.     

Determining and understanding the control of flux. An illustration in submitochondrial particles of how to validate schemes of metabolic control.

Two complementary methods were used to determine how the rate of respiration and that of ATP hydrolysis were controlled in rat liver submitochondrial particles. In the first, 'direct control analysis' method, respiration was titrated with malonate, antimycin or cyanide at 20, 30 and 37 degrees C, to determine the flux control exerted by succinate dehydrogenase, cytochrome bc1 complex and cytochrome c oxidase, respectively. Together, the three respiratory complexes only controlled the flux by about 50%, leaving the other 50% of flux control to the H+ leak. In the second, 'elasticity based' method, the elasticity coefficients of the respiratory chain or the H+-ATPase and the H+ leak towards the H+ gradient were determined. Then, the flux control coefficients were calculated using the connectivity and summation laws of metabolic control theory. The correspondence between the flux control coefficients determined in the two ways validated the two methods. This allowed us to use the second method to analyse what was the kinetic origin of the observed distribution of control. Control of ATP hydrolysis by the ATPase decreased with increasing ATPase activity; hence, the control exerted by the H+ leak increased with increasing ATPase activity, due to a diminishing elasticity towards the H+ gradient. Reverse electron transport was mainly controlled by the ATPase; the sum of flux control coefficients of succinate dehydrogenase, NADH-CoQ oxidoreductase, and H+-ATPase yielded a value greater than one, indicating that the H+ leak exerted a significant negative control on this pathway.     

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.