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.     

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.     

Control involving metabolism and gene expression

Control of DNA supercoiling by the free-energy of hydrolysis of ATP that involves gene expression is analyzed in terms of three levels of unconnected metabolic pathways. These are synthesis and breakdown of topoisomerase mRNAs, synthesis and breakdown of topoisomerase proteins and supercoiling and relaxation of DNA. The so-called square-matrix method previously developed for the control of metabolic pathways, is extended to deal with this hierarchical control system. It turns out that also in this case, the matrix of control coefficients is equal to the inverse of the so-called elasticity matrix, which contains all relevant elasticity coefficients as well as information about the structure and connectedness of the pathways involved. For a simpler case of a hierarchy of two systems, we demonstrate that the explicit matrix inversion method may be replaced by an implicit method in which the regulatory effects that run through the other level are described by an additional elasticy coefficient which may then be treated as if local.     

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.     

Metabolic control analysis of Aspergillus niger L-arabinose catabolism

A mathematical model of the L-arabinose/D-xylose catabolic pathway of Aspergillus niger was constructed based on the kinetic properties of the enzymes. For this purpose L-arabinose reductase, L-arabitol dehydrogenase and D-xylose reductase were purified using dye-affinity chromatography, and their kinetic properties were characterized. For the other enzymes of the pathway the kinetic data were available from the literature. The metabolic model was used to analyze flux and metabolite concentration control of the L-arabinose catabolic pathway. The model demonstrated that flux control does not reside at the enzyme following the intermediate with the highest concentration, L-arabitol, but is distributed over the first three steps in the pathway, preceding and following L-arabitol. Flux control appeared to be strongly dependent on the intracellular L-arabinose concentration. At 5 mM intracellular L-arabinose, a level that resulted in realistic intermediate concentrations in the model, flux control coefficients for L-arabinose reductase, L-arabitol dehydrogenase and L-xylulose reductase were 0.68, 0.17 and 0.14, respectively. The analysis can be used as a guide to identify targets for metabolic engineering aiming at either flux or metabolite level optimization of the L-arabinose catabolic pathway of A. niger. Faster L-arabinose utilization may enhance utilization of readily available organic waste containing hemicelluloses to be converted into industrially interesting metabolites or valuable enzymes or proteins.     

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.     

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.     

Enzymology in vivo using NMR and molecular genetics

Models of metabolic flux regulation are frequently based on an extrapolation of the kinetic properties of enzymes measured in vitro to the intact cell. Such an extrapolation assumes a detailed knowledge of the intracellular environment of these enzymes in terms of their free substates and effectors concentrations and possible interaction with other cellular macromolecules, which may modify their kinetic properties. These is a considerable incentive, therefore, to study the properties of enzymes directly in vivo. We have been using non-invasive NMR techniques, in conjunction with molecular genetic manipulation of enzyme levels, to study the kinetic properties of individual enzymes in vivo. We have also developed a novel strategy which has allowed us to monitor, by NMR, the ligand binding properties and mobilities of enzymes in the intact cell. This technique may also allow us to measured the diffusion coefficients of these proteins in the cell. These studies should give new insight into the properties of enzymes in vivo     

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).     

Control of gluconeogenesis in rat liver cells. Flux control coefficients of the enzymes in the gluconeogenic pathway in the absence and presence of glucagon.

We have used control analysis to quantify the distribution of control in the gluconeogenic pathway in liver cells from starved rats. Lactate and pyruvate were used as gluconeogenic substrates. The flux control coefficients of the various enzymes in the gluconeogenic pathway were calculated from the elasticity coefficients of the enzymes towards their substrates and products and the fluxes through the different branches in the pathway. The elasticity coefficients were either calculated from gamma/Keq. ratios (where gamma is the mass-action ratio and Keq. is the equilibrium constant) and enzyme-kinetic data or measured experimentally. It is concluded that the gluconeogenic enzyme pyruvate carboxylase and the glycolytic enzyme pyruvate kinase play a central role in control of gluconeogenesis. If pyruvate kinase is inactive, gluconeogenic flux from lactate is largely controlled by pyruvate carboxylase. The low elasticity coefficient of pyruvate carboxylase towards its product oxaloacetate minimizes control by steps in the gluconeogenic pathway located after pyruvate carboxylase. This situation occurs when maximal gluconeogenic flux is required, i.e. in the presence of glucagon. In the absence of the hormone, when pyruvate kinase is active, control of gluconeogenesis is distributed among many steps, including pyruvate carboxylase, pyruvate kinase, fructose-1,6-bisphosphatase and also steps outside the classic gluconeogenic pathway such as the adenine-nucleotide translocator.     

Total cell protein concentration as an evolutionary constraint on the metabolic control distribution in cells.

Cell protein occupies 15-35% of cell volume. This level is argued to be the maximum compatible with cell function. Because of this constraint, selection pressure during evolution is likely to have maximized pathway fluxes for minimum total protein level. Pathways optimized in this way are shown to have the following characteristics: (1) the "simple" flux control coefficients of all enzymes are equal, (2) the normal flux control coefficients depend on the relative kinetic constants of the enzymes, such that enzymes with low specific activity are present at relatively high levels and have high flux control, (3) the normal flux control coefficients are proportional to enzyme levels. A single rate limiting step located at the first step in a pathway is likely to be inefficient in terms of protein levels, and the major metabolic pathways are therefore expected to have control distributed throughout the pathway. This has important implications for metabolic control.     

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.     

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.     

A new theorem in the theory of metabolism regulation: how external parameters control the flow and concentration of metabolites

A simple theorem is derived concerning "global" and "local" coefficients of metabolic network. The coefficients of the flux and concentration response to changes in the external parameters (temperature, ionic strength, pH, external effectors) are considered. These coefficients are expressed through the sum of the control coefficients multiplied by the elasticity coefficients for all the enzymes of the metabolic pathway.     

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.     

Metabolic regulation: A control analytic perspective

A possible basis for a quantitative theory of metabolic regulation is outlined. Regulation is defined here as the alteration of reaction properties to augment or counteract the mass-action trend in a network reactions. In living systems the enzymes that catalyze these reactions are the handles through which such alteration is effected. It is shown how the elasticity coefficients of an enzyme-catalyzed reaction with respect to substrates and products are the sum of a massaction term and a regulatory kinetic term; these coefficients therefore distinguish between massaction effects and regulatory effects and are recognized as the key to quantifying regulation. As elasticity coefficients are also basic ingredients of metabolic control analysis, it is possible to relate regulation to such concepts as control, signalling, stability, and homeostasis. The need for care in the choice of relative or absolute changes when considering questions of metabolic regulation is stressed. Although the concepts are illustrated in terms of a simple coupled reaction system, they apply equally to more complex systems. When such systems are divided into reaction blocks, co-response coefficients can be used to measure the elasticities of these blocks.I dedicate this paper to Henrik Kacser, co-founder of and guiding light in the field of metabolic control analysis. His recent death leaves us bereft of a fount of wisdom and kindness, but his work remains as a monument along the path of our search for an understanding of metabolic behavior.     

In vivo interpretation of model predicted inhibition in acrylate pathway engineered Lactococcus lactis

To maximize the productivity of engineered metabolic pathway, in silico model is an established means to provide features of enzyme reaction dynamics. In our previous study, Escherichia coli engineered with acrylate pathway yielded low propionic acid titer. To understand the bottleneck behind this low productivity, a kinetic model was developed that incorporates the enzymatic reactions of the acrylate pathway. The resulting model was capable of simulating the fluxes reported under in vitro studies with good agreement, suggesting repression of propionyl-CoA transferase (Pct) by carboxylate metabolites as the main limiting factor for propionate production. Furthermore, the predicted flux control coefficients of the pathway enzymes under steady state conditions revealed that the control of flux is shared between Pct and lactoyl-CoA dehydratase. Increase in lactate concentration showed gradual decrease in flux control coefficients of Pct that in turn confirmed the control exerted by the carboxylate substrate. To interpret these in silico predictions under in vivo system, an organized study was conducted with a lactic acid bacteria strain engineered with acrylate pathway. Analysis reported a decreased product formation rate on attainment of inhibitory titer by suspected metabolites and supported the model.     

Characterization of UDP-N-acetylglucosamine:glycoprotein N-acetylglucosamine-1-phosphotransferase from Acanthamoeba castellanii.

The kinetic properties of UDP-N-acetylglucosamine:glycoprotein N-acetylglucosamine-1-phosphotransferase (GlcNAc-phosphotransferase) partially purified from the soil amoeba Acanthamoeba castellanii have been studied. The transferase phosphorylated the lysosomal enzymes uteroferrin and cathepsin D 3-90-fold better than nonlysosomal glycoproteins and 16-83-fold better than a Man9GlcNAc oligosaccharide. Deglycosylated uteroferrin was a potent competitive inhibitor of the phosphorylation of intact uteroferrin (Ki of 48 microM) but did not inhibit the phosphorylation of RNase B or the simple sugar alpha-methylmannoside. Deglycosylated RNase (RNase A) did not inhibit the phosphorylation of RNase B or uteroferrin. These results indicate that purified amoeba GlcNAc-phosphotransferase recognizes a protein domain present on lysosomal enzymes but absent in most nonlysosomal glycoproteins. The transferase also exhibited a marked preference for oligosaccharides containing mannose alpha 1,2-mannose sequences, but this cannot account for the high affinity binding to lysosomal enzymes. A. castellanii extracts do not contain detectable levels of N-acetylglucosamine-1-phosphodiester alpha-N-acetylglucosaminidase, the second enzyme in the biosynthetic pathway for the mannose 6-phosphate recognition marker. We conclude that A. castellanii does not utilize the phosphomannosyl sorting pathway despite expression of very high levels of GlcNAc-phosphotransferase.