Metabolic control and its analysis. Additional relationships between elasticities and control coefficients

Existing theorems from the analysis of metabolic control have been taken and embedded in a simple matrix algebra procedure for calculating the flux control coefficients of enzymes (formerly known as sensitivities) in a metabolic pathway from their kinetic properties (their elasticities). New theorems governing the flux control coefficients of branched pathways and substrate cycles have been derived to allow the procedure to be applied to complex pathway configurations. Modifications to the elasticity terms used in the equations have been theoretically justified so that the method remains valid for pathways with conserved metabolites (for example, the adenine nucleotide pool or the intermediates of a catalytic cycle such as the tricarboxylic acid cycle) or with pools of metabolites kept very near to equilibrium by very rapid reactions. The matrix equations generated using these theorems and relationships may be solved algebraically or numerically. Algebraic solutions have been used to determine the factors responsible for the degree of amplification of flux control coefficients by substrate cycles and to show that it is possible to derive expressions for the elasticities of a group of enzymes.     

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.     

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.     

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.     

Experimental determination of flux control distribution in biochemical systems: in vitro model to analyze transient metabolite concentrations

Determination of the control coefficients allows the identification of rate-controlling steps in a reaction system. However, the measurement of the flux control coefficients in a biochemical system is not a trivial task, except for some special cases. We have developed a theoretical basis for the direct determination of these coefficients from dynamic responses. In order to show the validity of this methodology experimentally, the dynamic approach is applied to an in vitro reconstituted partial glycolytic pathway to determine the flux control coefficients of hexokinase and phosphofructokinase. It is shown that the dynamic approach gives consistent results, which agree well with values obtained by the direct enzyme titration method. The detailed procedure and potential applications to other systems, such as immobilized enzyme or cell reactors, are discussed. (c) 1993 Wiley & Sons, Inc.     

In vivo modular control analysis of energy metabolism in contracting skeletal muscle

We used (31)P MRS (magnetic resonance spectroscopy) measurements of energetic intermediates [ATP, P(i) and PCr (phosphocreatine)] in combination with the analytical tools of metabolic control analysis to study in vivo energy metabolism in the contracting skeletal muscle of anaesthetized rats over a broad range of workload. According to our recent MoCA (modular control analysis) used to describe regulatory mechanisms in beating heart, we defined the energetic system of muscle contraction as two modules (PCr-Producer and PCr-Consumer) connected by the energetic intermediates. Hypoxia and electrical stimulation were used in this in vivo study as reasonably selective modulations of Producer and Consumer respectively. As quantified by elasticity coefficients, the sensitivities of each module to PCr determine the control of steady-state contractile activity and metabolite concentrations. The magnitude of the elasticity of the producer was high (4.3+/-0.6) at low workloads and decreased 5-fold (to 0.9+/-0.2) at high workloads. By contrast, the elasticity of the consumer remained low (0.5-1.2) over the range of metabolic rates studied. The control exerted by each module over contraction was calculated from these elasticities. The control of contraction was found on the consumer at low workloads and then swung to the producer, due to the workload-dependent decrease in the elasticity of producer. The workload-dependent elasticity and control pattern of energy production in muscle is a major difference from heart. Since module rate and elasticity depend on the concentrations of substrates and products, the absence of homoeostasis of the energetic intermediates in muscle, by contrast with heart, is probably the origin of the workload-dependent elasticity of the producer module.     

A 'top-down' approach to the determination of control coefficients in metabolic control theory

A new approach to the determination of flux and concentration control coefficients in metabolic pathways is outlined. Linear pathways are conceptually divided in two around an intermediate metabolite (or group or metabolites) and the control coefficients of the two parts are derived from the elasticity coefficients of the two parts to the intermediate. Branched pathways are treated similarly, the control coefficients of the branches being derived either from the elasticities of the branches to their common intermediate or from the relative flux changes of the branches. Repeating this analysis around other intermediates in the pathway allows the control coefficients of smaller and smaller groups of enzymes to be determined. In complex systems this approach to describing control may have several advantages over determining the control coefficients of individual enzymes and is a potentially useful complementary approach.     

Sensitivity analysis of stoichiometric networks: an extension of metabolic control analysis to non-steady state trajectories

A sensitivity analysis of general stoichiometric networks is considered. The results are presented as a generalization of Metabolic Control Analysis, which has been concerned primarily with system sensitivities at steady state. An expression for time-varying sensitivity coefficients is given and the Summation and Connectivity Theorems are generalized. The results are compared to previous treatments. The analysis is accompanied by a discussion of the computation of the sensitivity coefficients and an application to a model of phototransduction.     

Quality control measures for cervical cytology laboratories

The results of three quality control measures for evaluating a cytopathology laboratory's performance in the diagnosis of cervical abnormalities are presented. The sensitivities of cervical cytology were estimated to be 95.5% or 93.1% (using two different methods of analysis) for the detection of histologically diagnosed invasive squamous cell carcinoma of the cervix and 60% for the detection of adenocarcinoma and adenosquamous carcinoma of the cervix in 1983. The positive predictive values for a histologic diagnosis of neoplasia after cytologic reports of CIN III and invasive carcinoma were 92.5% and 99%, respectively. Repeatability of a negative cytologic result exceeded 98%. These results indicate that accurate cervical cytologic reporting can be achieved. Regular monitoring of the type described, which is both practical and reasonably comprehensive, is recommended for all laboratories.     

Modular metabolic control analysis of large responses in branched systems--application to aspartate metabolism

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: J(1), J(2), and J(3), respectively. The analysis shows similar high infinitesimal control coefficients of J(2) by the rates of SM1 and DM2 (C(v1)(J2) = 0.6 and C(v2)(J2) = 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 J(2) 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.     

Control of the metabolic flux in a system with high enzyme concentrations and moiety-conserved cycles. The sum of the flux control coefficients can drop significantly below unity.

In a number of metabolic pathways enzyme concentrations are comparable to those of substrates. Recently it has been shown that many statements of the 'classical' metabolic control theory are violated if such a system contains a moiety-conserved cycle. For arbitrary pathways we have found: (a) the equation connecting coefficients CEiJ (obtained by varying the Ei concentration) and CviJ (obtained by varying the kicat), and (b) modified summation equations. The sum of the enzyme control coefficients (equal to unity under the 'classical' theory) appears always to be below unity in the systems considered. The relationships revealed were illustrated by a numerical example where the sum of coefficients CEiJ reached negative values. A method for experimental measurements of the above coefficients is proposed.     

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).     

Metabolic control analysis. The effects of high enzyme concentrations

Differing views have been given in the literature as to whether the presence in a pathway of an enzyme at a concentration comparable to that of its substrate affects the values of control coefficients and the theorems of metabolic control analysis. Here we argue in favour of one of those views: that there is no effect unless the enzyme sequesters a substrate that contains a conserved moiety. In this particular case, we derive both a general criterion for estimating whether such an effect will be of a significant magnitude, and equations for determining the changes in the flux control coefficients. The nature of the phenomenom and the application of the equations are illustrated with a numerical simulation.     

Analysis of the control of respiration rate, phosphorylation rate, proton leak rate and protonmotive force in isolated mitochondria using the 'top-down' approach of metabolic control theory

The rate of respiration of isolated mitochondria was set at different values by addition of either oligomycin or an ADP-regenerating system (glucose and different amounts of hexokinase). We measured the relationship between respiration rate and membrane potential as respiration was titrated by the addition of malonate under each condition. We used the flux control summation and connectivity theorems and the branching theorem of metabolic control theory to calculate the control over respiration rate exerted by the respiratory chain (and associated reactions), phosphorylating system (and associated reactions) and proton leak at each respiration rate. The analysis also yielded the flux control coefficients of these three reactions over phosphorylation rate and proton leak rate and their concentration control coefficients over protonmotive force. We found that respiration rate was controlled largely by the proton leak under non-phosphorylating conditions, by the phosphorylating system at intermediate rates and by both the phosphorylating system and the respiratory chain in state 3. The rate of phosphorylation was controlled largely by the phosphorylating system itself in state 4 and at intermediate rates, while state 3 control was shared between the phosphorylating system and the respiratory chain; the proton leak had insignificant control. In all states the phosphorylating system had large negative control over the proton leak; the chain and the proton leak both had large positive control coefficients. The protonmotive force was controlled by the chain and by the phosphorylating system; the proton leak had little control.     

Adipocyte plasma-membrane Gi and Gs in insulinopenic diabetic patients.

The matrix method for calculating the overall sensitivities (including control coefficients) of a metabolic system, described by Crabtree & Newsholme [Biochem. J. 247, 113-129 (1987)], is simplified by a preliminary partitioning of the initial matrix equation. This reduces the size of the matrix to be inverted and thereby removes a major drawback with the original method. The resulting procedure is simpler and more systematic than the alternative methods currently available, especially when the system is extensively branched.     

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.     

Dependence of process variables on fermentation parameters during start-up of a continuous flow reactor with recombinant microorganisms

A mathematical model is proposed to describe how the parametric dependences (sensitivities) of key process variables in a continuous flow fermenter using recombinant microorganisms vary with time. Solution of the model, with typical values for the parameters, indicates that most sensitivities increase several fold in about 20 hours. But the sensitivities of the product concentration either vanish or remain constant at low values. The presence of some plasmid-free cells when the fermentation begins does not seem to affect the sensitivities.