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.     

Control analysis of time-dependent metabolic systems

Metabolic Control Analysis is extended to time dependent systems. It is assumed that the time derivative of the metabolite concentrations can be written as a linear combination of rate laws, each one of first order with respect to the corresponding enzyme concentration. The definitions of the control and elasticity coefficients are extended, and a new type of coefficient ("time coefficient", "T") is defined. First, we prove that simultaneous changes in all enzyme concentrations by the same arbitrary factor, is equivalent to a change in the time scale. When infinitesimal changes are considered, these arguments lead to the derivation of general summation theorems that link control and time coefficients. The comparison of two systems with identical rates, that only differ in one metabolite concentration, leads to a method for the construction of general connectivity theorems, that relate control and elasticity coefficients. A mathematical proof in matrix form, of the summation and connectivity relationships, for time dependent systems is given. Those relationships allow one to express the control coefficients in terms of the elasticity and time coefficients for the case of unbranched pathway.     

Control analysis of biochemical pathways: a novel procedure for calculating control coefficients, and an additional theorem for branched pathways

A novel method for calculating control coefficients of individual enzymes on fluxes and concentrations in metabolic pathways is presented. This method is derived by applying the theorem on implicit functions to the equations defining the steady state metabolite concentrations; it allows verification of the existing summation theorems and connectivity relations, and leads to a novel theorem for flux control coefficients in branched pathways. The method and the novel theorem are illustrated by several examples.     

Control analysis of metabolic networks. 2. Total differentials and general formulation of the connectivity relations

The mathematical background of the connectivity relations of metabolic control theory is analysed. The connectivity relations are shown to reflect general properties of total differentials of reaction rate vi, flux J, and metabolite concentration Xj. Connectivity relations hold for any metabolic network in which all vi are homogeneous functions of enzyme concentration Ei. This notion allows established algebraic methods to be used for the formulation of connectivity relations for metabolic systems in which numerous constraints are imposed on metabolite concentrations. A general procedure to derive connectivity relations for such metabolic systems is given. To encourage a broader audience to apply control theory to physiological systems, an easy-to-use graphical procedure is derived for formulating connectivity relations for biochemical systems in which no metabolite is involved in more than one constraint.     

Dynamic simulation and metabolic re-design of a branched pathway using linlog kinetics

This paper presents a new mathematical framework for modeling of in vivo dynamics and for metabolic re-design: the linlog approach. This approach is an extension of metabolic control analysis (MCA), valid for large changes of enzyme and metabolite levels. Furthermore, the presented framework combines MCA with kinetic modeling, thereby also combining the merits of both approaches. The linlog framework includes general expressions giving the steady-state fluxes and metabolite concentrations as a function of enzyme levels and extracellular concentrations, and a metabolic design equation that allows direct calculation of required enzyme levels for a desired steady state when control and response coefficients are available. Expressions giving control coefficients as a function of the enzyme levels are also derived. The validity of the linlog approximation in metabolic modeling is demonstrated by application of linlog kinetics to a branched pathway with moiety conservation, reversible reactions and allosteric interactions. Results show that the linlog approximation is able to describe the non-linear dynamics of this pathway very well for concentration changes up to a factor 20. Also the metabolic design equation was tested successfully.     

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.     

Quantitative assessment of regulation in metabolic systems

We show how metabolic regulation as commonly understood in biochemistry can be described in terms of metabolic control analysis. The steady-state values of the variables of metabolic systems (fluxes and concentrations) are determined by a set of parameters. Some of these parameters are concentrations that are set by the environment of the system; they can act as external regulators by communicating changes in the environment to the metabolic system. How effectively a system is regulated depends both on the degree to which the activity of the regulatory enzyme with which a regulator interacts directly can be altered by the regulator (its regulability) and on the ability of the regulatory enzyme to transmit the changes to the rest of the system (its regulatory capacity). The regulatory response of a system also depends on its internal organisation around key variable metabolites that act as internal regulators. The regulatory performance of the system can be judged in terms of how sensitivity the fluxes respond to the external stimulus and to what degree homeostasis in the concentrations of the internal regulators is maintained. We show how, on the level of both external and internal regulation, regulability can be quantified in terms of an elasticity coefficient and regulatory capacity in terms of a control coefficient. Metabolic regulation can therefore be described in terms of metabolic control analysis. The combined response relationship of control analysis relates regulability and regulatory capacity and allows quantification of the regulatory importance of the various interactions of regulators with enzymes in the system. On this basis we propose a quantitative terminology and analysis of metabolic regulation that shows what we should measure experimentally and how we should interpret the results. Analysis and numerical simulation of a simple model system serves to demonstrate our treatment.     

Optimization-based metabolic control analysis

In this work, a novel optimization-based metabolic control analysis (OMCA) method is introduced for reducing data requirement for metabolic control analysis (MCA). It is postulated that using the optimal control approach, the fluxes in a metabolic network are correlated to metabolite concentrations and enzyme activities as a state-feedback control system that is optimal with respect to a homeostasis objective. It is then shown that the optimal feedback gains are directly related to the elasticity coefficients (ECs) of MCA. This approach requires determination of the relative "importance" of metabolites and fluxes for the system, which is possible with significantly reduced experimental data, as compared with typical MCA requirements. The OMCA approach is applied to a top-down control model of glycolysis in hepatocytes. It is statistically demonstrated that the OMCA model is capable of predicting the ECs observed experimentally with few exceptions. Further, an OMCA-based model reconciliation study shows that the modification of four assumed stoichiometric coefficients in the model can explain most of the discrepancies, with the exception of elasticities with respect to the NADH/NAD ratio.     

Calculation errors of time-varying flux control coefficients obtained from elasticity coefficients by means of summation and connectivity theorems in metabolic control analysis

This paper investigates the accuracy of a matrix method proposed by other researchers to calculate time-varying flux control coefficients (dynamic FCCs) from elasticity coefficients by means of summation and connectivity theorems in the framework of metabolic control analysis. A mathematical model for the fed-batch penicillin V fermentation process is used as a case example for discussion. Calculated results reveal that this method produces significant calculation errors because the theorems are essentially valid only in steady state, although it may provide rough time-transient behaviors of FCCs. Strictly, therefore, dynamic FCCs should be directly calculated from the differential equations for metabolite concentrations and sensitivities.     

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.     

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.     

Sensitivity and control analysis of periodically forced reaction networks using the Green's function method

A general sensitivity and control analysis of periodically forced reaction networks with respect to small perturbations in arbitrary network parameters is presented. A well-known property of sensitivity coefficients for periodic processes in dynamical systems is that the coefficients generally become unbounded as time tends to infinity. To circumvent this conceptual obstacle, a relative time or phase variable is introduced so that the periodic sensitivity coefficients can be calculated. By employing the Green's function method, the sensitivity coefficients can be defined using integral control operators that relate small perturbations in the network's parameters and forcing frequency to variations in the metabolite concentrations and reaction fluxes. The properties of such operators do not depend on a particular parameter perturbation and are described by the summation and connectivity relationships within a control-matrix operator equation. The aim of this paper is to derive such a general control-matrix operator equation for periodically forced reaction networks, including metabolic pathways. To illustrate the general method, the two limiting cases of high and low forcing frequency are considered. We also discuss a practically important case where enzyme activities and forcing frequency are modulated simultaneously. We demonstrate the developed framework by calculating the sensitivity and control coefficients for a simple two reaction pathway where enzyme activities enter reaction rates linearly and specifically.     

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.     

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).     

A new framework for the estimation of control parameters in metabolic pathways using lin-log kinetics.

The control properties of biochemical pathways can be described by control coefficients and elasticities, as defined in the framework of metabolic control analysis. The determination of these parameters using the traditional metabolic control analysis relationships is, however, limited by experimental difficulties (e.g. realizing and measuring small changes in biological systems) and lack of appropriate mathematical procedures (e.g. when the more practical large changes are made). In this paper, the recently developed lin-log approach is proposed to avoid the above-mentioned problems and is applied to estimate control parameters from measurements obtained in steady state experiments. The lin-log approach employs approximative linear-logarithmic kinetics parameterized by elasticities and provides analytical solutions for fluxes and metabolite concentrations when large changes are made. Published flux and metabolite concentration data are used, obtained from a reconstructed section of glycolysis converting 3-phosphoglycerate to pyruvate [Giersch, C. (1995) Eur. J. Biochem. 227, 194-201]. With the lin-log approach, all data from different experiments can be combined to give realistic elasticity and flux control coefficient estimates by linear regression. Despite the large changes, a good agreement of fluxes and metabolite concentrations is obtained between the measured and calculated values according to the lin-log model. Furthermore, it is shown that the lin-log approach allows a rigorous statistical evaluation to identify the optimal reference state and the optimal model structure assumption. In conclusion, the lin-log approach addresses practical problems encountered in the traditional metabolic control analysis-based methods by introducing suitable nonlinear kinetics, thus providing a novel framework with improved procedures for the estimation of elasticities and control parameters from large perturbation experiments.     

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: software for the analysis of the control of metabolic networks

The program CONTROL is based on metabolic control theory anduses the method developed by Reder (1988). In this theory, twosets of parameters are defined in the vicinity of a steady-state:the elasticity coefficients which describe the local behaviourof the isolated enzymes, and the control coefficients whichexpress the response of the whole metabolic network to perturbationsat a given step. The theory shows that relationships exist betweenthe control coefficients (summation relationships or structuralrelationships) and also between the two types of coefficients(control and elasticity coefficients: connectivity relationships).The program CONTROL is divided into two parts (sub-menus). Thefirst one calculates all the control coefficients (flux andconcentrations) of a metabolic network from the elasticity coefficients.Using the second menu, the symbolic relationships are obtainedbetween the control coefficients (summation relationships) andbetween the control coefficients and the elasticity coefficients(connectivity relationships). These two sub-menus can be appliedindependently to any metabolic network (to date limited to 19steps and 19 metabolites).     

Response to temporal parameter fluctuations in biochemical networks

Metabolic response coefficients describe how variables in metabolic systems, like steady state concentrations, respond to small changes of kinetic parameters. To extend this concept to temporal parameter fluctuations, we define spectral response coefficients that relate Fourier components of concentrations and fluxes to Fourier components of the underlying parameters. It is also straightforward to generalize other concepts from metabolic control theory, such as control coefficients with their summation and connectivity theorems. The first-order response coefficients describe forced oscillations caused by small harmonic oscillations of single parameters: they depend on the driving frequency and comprise the phases and amplitudes of the concentrations and fluxes. Close to a Hopf bifurcation, resonance can occur: as an example, we study the spectral densities of concentration fluctuations arising from the stochastic nature of chemical reactions. Second-order response coefficients describe how perturbations of different frequencies interact by mode coupling, yielding higher harmonics in the metabolic response. The temporal response to small parameter fluctuations can be computed by Fourier synthesis. For a model of glycolysis, this approximation remains fairly accurate even for large relative fluctuations of the parameters.