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.     

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.     

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.     

Temporal analysis of metabolic systems and its application to metabolite channelling

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

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.     

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.     

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.     

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

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

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.     

Dosage, deletions and dominance: simple models of the evolution of gene expression

Dominance of the wild-type allele over spontaneous null mutations, such as deletions, can be explained in terms of the effects of changes in enzyme dose on the flux of metabolic pathways. If ever increasing levels of enzyme activity have ever decreasing effects on the flux of the biochemical pathway, then halving of dosage will always have a lesser effect on flux than half the effect of complete removal of gene activity. Furthermore, if gene expression rates are high, then halving of dose can have a negligible effect on flux and dominance will be strong. Given that strong dominance appears to be common, this leaves open the issue of why enzyme activity levels are so high that a halving of expression rates is of minimal effect. Why produce so much surplus enzyme? One explanation, suggested by Haldane, is that selection favoured high expression levels as a defence against mutation. We model this scenario formally and show that protection from mutation is an extremely weak force determining expression levels. The selective coefficients are only of the order of the mutation rate. However, if we suppose a linear mapping of flux with fitness and a monotonic cost to increased gene expression, it follows simply that here exists an optimal level of gene expression. By contrast to the mutational model, doubling of gene expression rates when the system is distant from the optimum is associated with extremely high selective coefficients (orders of magnitude higher than the mutation rate). When the cost of gene expression is slight the optimal rate of expression is such that strong dominance will follow.     

Channelling can decrease pool size.

It is widely considered that a possible advantage of metabolite channelling, in which a product of an enzyme is transferred to the next enzyme in a metabolic pathway without being released to the 'bulk' solution, is that channelling can decrease the steady-state concentrations of 'pool' intermediates. This then spares the limited solvent capacity of the cell, and reduces the loss of pathway flux due to leakage or instability of the free intermediate. Recently, however, based on simulations of a particular model of a 'dynamic' channel, Cornish-Bowden ["Failure of channelling to maintain low concentrations of metabolic intermediates" (1991) Eur. J. Biochem. 195, 103-108] has argued that this is not in fact the case; his simulations indicated that the channel was rather ineffective at decreasing the concentration of the pool intermediate, and in some cases actually increased it. However, although his simulations were restricted to very specific thermodynamic and kinetic parameters, he generalised his conclusions, arguing that "channelling has no effect on the free concentration of a channelled intermediate in a pathway". By showing that, for a number of kinetic cases, the concentration of the pool intermediate did decrease substantially with increased channelling, we demonstrate here that the conclusion of Cornish-Bowden is not correct. In particular, if the reaction catalysed by the enzymes forming the channel has an equilibrium constant K higher than 1, and if the enzyme removing the product of the channel reaction is kinetically competent, channelling in the model system studied by Cornish-Bowden (1991) can decrease the steady-state concentration of the pool by a factor of 1000, independently of the mechanism of the terminal reaction and under conditions of essentially constant overall flux. If the channel is a 'static' channel, the decrease in the pool can be to arbitrarily low levels. This conclusion also holds for a system in which other reactions may consume the pool intermediate. Thus, channelling can maintain metabolite concentrations at low levels.     

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.     

Thiol dependent oxidation of enzymes: The last chance against oxidative stress

• 1.1. A survey of known effects of oxidized thiols on enzyme activity reveals a potential concerted action on metabolic pathways determining an impairment of anabolic reduction processes and an activation of the oxidative arm of the hexose monophosphate shunt. Thus it appears that, following oxidative stress, the increase of disulphides may act in restoring a reduced state in the cell by specifically channelling the metabolic energy flux.

     

Modular decomposition of metabolic systems via null-space analysis

We describe a method by which the reactions in a metabolic system may be grouped hierarchically into sets of modules to form a metabolic reaction tree. In contrast to previous approaches, the method described here takes into account the fact that, in a viable network, reactions must be capable of sustaining a steady-state flux. In order to achieve this decomposition we introduce a new concept--the reaction correlation coefficient, phi, and show that this is a logical extension of the concept of enzyme (or reaction) subsets. In addition to their application to modular decomposition, reaction correlation coefficients have a number of other interesting properties, including a convenient means for identifying disconnected subnetworks in a system and potential applications to metabolic engineering. The method computes reaction correlation coefficients from an orthonormal basis of the null-space of the stoichiometry matrix. We show that reaction correlation coefficients are uniquely defined, even though the basis of the null-space is not. Once a complete set of reaction correlation coefficients is calculated, a metabolic reaction tree can be determined through the application of standard programming techniques. Computation of the reaction correlation coefficients, and the subsequent construction of the metabolic reaction tree is readily achievable for genome-scale models using a commodity desk-top PC.     

Increasing the flux in metabolic pathways: A metabolic control analysis perspective

The problems of engineering increased flux in metabolic pathways are analyzed in terms of the understanding provided by metabolic control analysis. Over-expression of a single enzyme is unlikely to be effective unless it is known to have a high flux control coefficient, which can be used as an approximate predictive tool. This is likely to rule out enzymes subject to feedback inhibition, because it transfers control downstream from the inhibited enzyme to the enzymes utilizing the feedback metabolite. Although abolishing feedback inhibition can restore flux control to an enzyme, it is also likely to cause large increases in the concentrations of metabolic intermediates. Simultaneous and coordinated over-expression of most of the enzymes in a pathway can, in principle, produce substantial flux increases without changes in metabolite levels, though technically it may be difficult to achieve. It is, however, closer to the method used by cells to change flux levels, where coordinated changes in the level of activity of pathway enzymes are the norm. Another option is to increase the demand for the pathway product, perhaps by increasing its rate of excretion or removal. Copyright 1998 John Wiley & Sons, Inc.     

Experimental determination of control of glycolysis in Lactococcus lactis

The understanding of control of metabolic processes requires quantitative studies of the importance of the different enzymatic steps for the magnitude of metabolic fluxes and metabolite concentrations. An important element in such studies is the modulation of enzyme activities in small steps above and below the wild-type level. We review a genetic approach that is well suited for both Metabolic Optimization and Metabolic Control Analysis and studies on the importance of a number of glycolytic enzymes for metabolic fluxes in Lactococcus lactis. The glycolytic enzymes phosphofructokinase (PFK), glyceraldehyde-3-phosphate dehydrogenase (GAPDH), pyruvate kinase (PYK) and lactate dehydrogenase (LDH) are shown to have no significant control on the glycolytic flux in exponentially growing cells of L. lactis MG1363. Introduction of an uncoupled ATPase activity results in uncoupling of glycolysis from biomass production. With MG1363 growing in defined medium supplemented with glucose, the ATP demanding processes do not have a significant control on the glycolytic flux; it appears that glycolysis is running at maximal rate. It is likely that the flux control is distributed over many enzymes in L. lactis, but it cannot yet be excluded that one of the remaining glycolytic steps is a rate-limiting step for the glycolytic flux.     

Flux control analysis for biphenyl metabolism by Rhodococcus pyridinovorans R04

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

Dominance is not inevitable

In a diploid organism, a mutant gene that results in elimination of an enzyme activity in the homozygote is almost universally found to be recessive, so that the heterozygote phenotype is virtually indistinguishable from the wild type. It has been argued (H. Kacser & J. A. Burns, Genetics 97, 639-666 (1981)) that there is no need to look to evolution for an explanation of this phenomenon, as it is an inevitable consequence of the low control coefficients for metabolic flux possessed by nearly all enzymes. However, it is possible to envisage pathways in which every enzyme is more than half-saturated, so that moderate changes in the concentration of any enzyme result in substantial changes in metabolic flux. Such behaviour can occur, for example, if the limiting rates of the enzymes decrease as one proceeds along the pathway and the precursor concentration is large compared with the Michaelis constants of all the enzymes. Consequently one does require an explanation in terms of natural selection of why such pathways are apparently not observed in nature.