JXB Advance Access originally published online on November 15, 2004
Journal of Experimental Botany 2005 56(410):267-272; doi:10.1093/jxb/eri011
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RESEARCH PAPER |
Enzymes, metabolites and fluxes
School of Biological and Molecular Sciences, Oxford Brookes University, Headington, Oxford, OX3 0BP, UK
* Fax: +44 (0)1865 484017. E-mail: dfell{at}brookes.ac.uk
Received 20 May 2004; Accepted 30 August 2004
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Abstract |
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 Top Abstract IntroductionTheoretical backgroundResponse to modulation of...Responses to physiological...ConclusionReferences |
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Aspects of metabolic control theory and experiments from metabolic biochemistry are reviewed in order to deduce the circumstances in which experimental studies involving metabolomics have the greatest chance of success. It is concluded that metabolic changes effected mainly through a single enzyme are those most likely to lead to large changes in metabolite concentrations. Metabolic changes brought about through signal transduction mechanisms will tend to result in relatively much smaller adjustments in metabolite concentrations, whilst allowing significant changes in metabolic rates.
Key words: Control coefficient, metabolic control analysis, metabolomics, multi-site modulation, proportional activation
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Introduction |
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TopAbstractIntroductionTheoretical backgroundResponse to modulation of...Responses to physiological...ConclusionReferences |
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The experimental methodology of metabolomics is developing rapidly (Krishnan et al., 2005
), but there are still issues to be resolved about how to analyse and interpret the results. A common approach is to take samples from individuals of the same organism under different environmental conditions, or from related organisms of different genotypes in the same environmental conditions. Statistical techniques such as principal component analysis or clustering are then used to determine which metabolites carry the greatest information about the differences in the experiment. In order to assist in the design of useful metabolomics experiments, and to guide their interpretation, it is instructive to consider what is known about the impact of changes in metabolism on the concentrations of metabolites and the extent to which changes in the activity of a metabolic network will be accompanied by detectable changes in metabolite concentrations. Therefore, in this article, the effects on metabolites of making changes in the activities of enzymes in a network will be surveyed, both in terms of theoretical expectations, according to the theory of metabolic control analysis, and of the results of relevant experiments. The phrase change in activity of an enzyme is intended to be generic, covering a variety of causes that might impact on metabolism, including genetic variation manifested in allozymes with differing kinetic properties and environmental conditions that cause a change in gene expression. |
Theoretical background |
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TopAbstractIntroductionTheoretical backgroundResponse to modulation of...Responses to physiological...ConclusionReferences |
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This treatment draws on the theory of metabolic control analysis (Kacser and Burns, 1973
; Heinrich and Rapoport, 1974) which defines a number of coefficients that quantify the effects on a metabolic system of changing the activity of any one enzyme. One of these coefficients is the flux control coefficient, 
which is a measure of the effect of the ith enzyme, Ei, on a metabolic flux, J, where the flux is the rate of some part of the metabolic network. The exact definition is given elsewhere (Burns et al., 1985; Kacser et al., 1995; Fell, 1997), but for current purposes, this coefficient can be approximated as the percentage change in J for a 1% change in Ei. Similarly, for the concentration of the jth metabolite, Sj, the concentration control coefficient 
is the percentage change in Sj for a 1% change in Ei.
To examine the relative impact of a change in enzyme activity on a chosen flux and metabolite, it is useful to use the concept of the co-response coefficient, originally defined by Hofmeyr et al. (1993). For a metabolite Sj and a flux J, the co-response coefficient 
is the ratio of the percentage change in J to the percentage change in Sj caused by the same small change in Ei. It is, in fact, the ratio of the two control coefficients defined above, i.e.
 | (1) |
). Thus the arguments that follow apply equally to metabolic changes induced by these other modes of altering enzyme activity. |
Response to modulation of a single enzyme |
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TopAbstractIntroductionTheoretical backgroundResponse to modulation of...Responses to physiological...ConclusionReferences |
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Theory
For small changes in the activity of an enzyme, the effects on flux and metabolite concentrations are predicted by the enzyme's control coefficients. Therefore, some conclusions about the likely size of the effects can be derived through the theory of metabolic control analysis. First, the flux control coefficients are constrained by the flux summation theorem (Kacser and Burns, 1973
):
 | (2) |
In a linear pathway, where there can be no negative flux control if the enzyme kinetics are conventional, this means that the maximum value for any enzyme flux control coefficient is 1, and then only if all the other enzymes have zero flux control coefficients. Metabolic networks with branches and cycles are more complicated because not only is there then more than one flux value that can be altered by the change in an enzyme activity, but also some of these control coefficients can be negative (implying that an increase in the activity of an enzyme in some part of the network causes a decrease in the flux elsewhere). Nevertheless, the flux summation theorem, backed up by the algebraic analysis of the relationship between flux control coefficients and the kinetics of all the enzymes of the network (Kacser and Burns, 1973; Fell and Sauro, 1985; Fell, 1997), led to the expectation that flux control will normally be shared, in variable proportions, between the enzymes and that flux control coefficients of magnitude 1 or greater would be extreme situations.
By contrast, the summation theorem for metabolite concentration control coefficients (Westerhoff and Chen, 1984) is, for each metabolite in the network:
 | (3) |
It follows that there are necessarily both positive and negative control coefficients on any metabolite. This is as expected, since an increase in activity of an enzyme in a linear section of a metabolic network would normally lower the concentration of metabolites on its upstream or substrate side, and increase them on its downstream or product side. Further, even in a linear pathway, there are no bounds on the value of concentration control coefficients.
Again, the theory of control analysis allows relationships to be made between the kinetics of the enzymes and the values of the concentration control coefficients (Sauro et al., 1987). From there, the next step is to determine the relative size of the flux and metabolite control coefficients for a particular enzyme, and as stated above, the metabolite-flux co-response coefficient provides the measure for this comparison. The supply–demand analysis initiated by Hofmeyr and Cornish-Bowden illustrates how this can be done, and how the co-response coefficients can be related to the kinetics of the pathway enzymes (Hofmeyr and Cornish-Bowden, 1991; Hofmeyr et al., 1993). If this analysis is pursued, it emerges (Hofmeyr, 1995; Thomas and Fell, 1996) that an enzyme can have concentration control coefficients that are significantly larger in magnitude than its flux control coefficients, but, conversely, it is difficult to envisage circumstances where a flux control coefficient can be substantially larger than an enzyme's concentration control coefficients. In essence, this reflects the sub-sensitive response of enzyme rates to substrates, exemplified by rectangular hyperbolic kinetics (see Fell, 1997, for details): a change in substrate concentration generally delivers a smaller relative change in enzyme rate. Only where there is an enhanced response of rate to metabolite concentrations, as with co-operativity, or kinetics of near-equilibrium enzymes, can the concentration change required for a given change in rate be diminished.
Taken together, the theory suggests that a change in the activity of a single enzyme will most likely produce a larger signal in the metabolite concentrations than in the metabolic fluxes, so a metabolomics study is a good option for detecting the effects. Although it is not an invariable rule, enzymes catalysing near-equilibrium reactions are unlikely to produce large effects on either fluxes or concentrations and will therefore be less visible.
Practice
Experimental illustration of the above principles can be obtained where the expression level of an enzyme has been changed, either by expression of a transgene or anti-sense RNA. An example is the effects on potato tuber metabolism of substantial over-expression of phosphofructokinase (Burrell et al., 1994). Over 35-fold over-expression caused no measurable change in glycolytic flux, consistent with calculations on a theoretical model of potato tuber glycolysis that gave a low value (about 0.14) for the flux control coefficient of the enzyme (Thomas et al., 1997). On the other hand, there were up to 10-fold increases in downstream glycolytic metabolites (Fig. 1) that showed an enzyme dose–response relationship. The model of potato tuber glycolysis also predicted values for the concentration control coefficients (Thomas et al., 1997). The three largest for phosphofructokinase on the metabolites measured were those on phosphoenolpyruvate (0.6), fructose 1,6-bisphosphate (0.37), and 3-phosphoglycerate (0.28), and these can be seen generally to have the largest responses in the experiment (Fig. 1). However, phosphofructokinase did not have the largest concentration control coefficients: phosphoglycerate mutase is predicted to have a coefficient of 8.7 on fructose 1,6-bisphosphate, followed by enolase with a value of –7.7 on the same metabolite.
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Finite change analysis
The example above supports the arguments developed from metabolic control theory, but a potential limitation is that control coefficients are defined for small changes in enzyme activity, and the control coefficients themselves change as the enzyme activity changes. This can be seen in Fig. 1, where the response of the concentrations is much larger near the starting, wild-type level of phosphofructokinase (1.0–10.0 on the scale) than at the highest levels of over-expression. Drawing general conclusions for large changes in enzyme activity is difficult because the precise relationships between the metabolic variables and enzyme activities are complex, non-linear functions that are usually unknown and are different in every case. An intermediate approach developed by Small and Kacser does, however, allow extension of the conclusions to large changes in enzyme activity, subject to the limitation that the effects do not result in significant changes in the degrees of saturation of the enzymes with their substrates, products, and effectors. In this case, the response of flux to change in a single enzyme activity follows a rectangular hyperbolic curve (Kacser and Burns, 1973
; Small and Kacser, 1993a). On this basis, Small and Kacser (1993a) derived the following approximate equation for the fold-change, f, in pathway flux produced by an r-fold change in the activity of an enzyme with flux control coefficient 
 | (4) |
.) It has previously been illustrated (Fell and Thomas, 1995, 1999) how this equation shows that even very large changes in a single enzyme activity can only have a limited impact on flux unless the flux control coefficient is very close to one. Note that the equation above implies that the limit of the flux change, when r»1 is
 | (5) |
Thus for a flux control coefficient of 0.5, the maximum flux change that can ever be achieved is a factor of 2. The corresponding equation for the impact of enzyme activity changes on metabolite concentrations turns out to be more complicated (Small and Kacser, 1993b) since it depends on both the concentration and flux control coefficient of the enzyme. Furthermore, in addition to the limitations above, no account is taken of restrictions on the concentration changes placed by the equilibrium constants. The equation is, with the same definitions as above:
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 | (7) |
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Thus, extending the theoretical arguments to cover the case of large changes in enzyme activity, it is still likely to be the metabolite concentrations that demonstrate the largest effects.
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Responses to physiological shifts in metabolism |
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The observations
The previous section argued that modulation of the activity of a single enzyme would be likely to produce a larger change in metabolite concentrations than fluxes, so that a metabolite–flux co-response coefficient would be greater than one. With the exception of top-down control analysis experiments (Brown et al., 1990
), there have been few attempts to measure these co-response coefficients experimentally. There are cases, however, where both concentration and flux changes have been measured in response to a physiological or environmental stimulus, which, it is supposed, is mediated by a factor X. In these cases, the changes are often large, and cannot be attributed a priori to the change in a single enzyme, and therefore cannot be represented as measures of the co-response coefficient. It has been argued (Fell and Thomas, 2000) that an approximate finite change co-response can be calculated from experimental data as:
 | (8) |
) for physiological changes in metabolism in a variety of animals, and in most cases the finite change co-response coefficient was very small, and in no case exceeded 1. These included cases where there were substantial flux changes; in the extreme, glycolytic flux in insect muscle increased several hundred-fold. The surprising constancy of metabolite concentrations in vertebrate muscle in the rest-exercise transition has been extensively documented and analysed by Hochachka (1994). More recently, a large dataset was analysed for the increase in glycolytic flux in pig heart during ischaemia (Vogt et al., 2002), and in nearly all the cases where the coefficient could be estimated with statistical significance, its value was relatively small, most particularly where the heart had previously been exposed to ischaemia. (The relevant values are the reciprocals of the b values in Table II of Vogt et al., 2002.) The reason for the difficulty in determining some of the co-response coefficients was precisely because some of the metabolite changes were so small, even though the flux change was easily measurable. Similarly, in a recent analysis of the effects of insulin on muscle glycogen synthesis, the glucose 6-phosphate to glycogen synthesis flux co-response coefficient was computed as between 0.15 and 0.28 (reported as 
in Schafer et al., 2004), corresponding to approximately 3-fold changes in the metabolite for a 50-fold change in flux.
Why are metabolite concentration changes so small?
If organisms' typical responses to physiological and environmental signals are to make large changes in metabolic flux with relatively smaller changes in metabolite concentrations, the inescapable conclusion is that these flux changes are not brought about by action on a single enzyme, since the experimental and theoretical analyses of the effects of activation of a single enzyme reviewed above show that it does not have the same characteristics. What mechanisms could be contributing to this stability of metabolite concentrations? Hofmeyr and Cornish-Bowden (1991) made a persuasive case that co-operativity in allosteric enzymes exhibiting feedback inhibition is a mechanism that improves homeostasis in metabolites rather than enhancing the flux-controlling potential of the enzyme. Whilst the force of this case and the contribution the mechanism makes to homeostasis can be accepted, the theoretical analysis of Thomas and Fell (1996) suggests that it is not enough.
The alternative that has been proposed (Fell and Thomas, 1995) is multi-site modulation: large changes in metabolic flux must be brought about by parallel changes in activity of enzymes along the path of the flux change. This seems a common element in the way signal transduction mechanisms act on metabolism, both through short-term adjustments achieved by covalent modification, and longer-term adjustments by changes in enzyme levels. One example is the way the rubisco activase plus thioredoxin systems act together to cause light-activated increases in enzyme activity at multiple points in the Calvin cycle (Fell, 1997). According to metabolic control theory (Kacser and Burns, 1973; Kacser and Acerenza, 1993), the effect of simultaneously changing the activity of all the enzymes in a pathway by the same factor is to change the flux by that factor whilst the concentrations of metabolic intermediates remain unchanged, even for large factors. These consequences are, in effect, equivalent statements to the flux summation and concentration summation theorems above (equations 2 and 3).
At the same time as multi-site modulation was proposed as a general mechanism, Korzeniewski and colleagues proposed the related concept of proportional activation: invariance of a metabolic intermediate through a flux change implies near-equal relative changes in the activities of the up-stream (supply) and down-stream (demand) pathways (Korzeniewski et al., 1995) for the reasons given above. They also showed how to determine, where metabolite homeostasis was not perfect, the ratio of the relative activity changes in the supply and demand sides. The initial application to activation of mitochondrial oxidative phosphorylation has been followed by analysis of the rest-exercise transition in mammalian muscle (Korzeniewski, 2003). In the case of insulin stimulation of mammalian muscle glycogen synthesis, glucose 6-phosphate concentrations change much less than the flux because the activation of glycogen synthase is about 40–80% of the activation of the glucose transport/phosphorylation system, even though the latter possesses all the flux control (Schafer et al., 2004). That is, the signal transduction mechanism activating glycogen synthase functions principally to ensure metabolite homeostasis, not to control flux.
Responses to physiological stimuli, nutrition and environment also cause longer-term adjustments in enzyme levels. A large set of examples has been assembled from many organisms that shows how all the enzymes in the affected pathways are up- or down-regulated to very similar degrees (Fell, 2000). (There are slight differences between the changes in level of different enzymes in a pathway, but this is a minor variation overlaid on the main effect.) Although few of the experiments assembled for this analysis specifically addressed the point, the expectation from theory is that these co-ordinated changes in enzyme levels will allow metabolic fluxes to change without any large differences occurring in metabolite concentrations.
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Conclusion |
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TopAbstractIntroductionTheoretical backgroundResponse to modulation of...Responses to physiological...ConclusionReferences |
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The above analysis suggests that certain types of experimental investigations will derive more benefit from metabolomics approaches than others. Where there is a change in a single enzyme, then there is more likely to be a measurable impact on metabolite concentrations than on fluxes in the metabolic network. This is the reason for proposing metabolomics as a strategy for identifying the site of phenotypically silent mutations in yeast (Raamsdonk et al., 2001
). Where the metabolic response is a natural one to normal physiological or environmental stimuli, the mechanisms that have evolved seem to ensure a relative stability in the concentrations of intermediates. This will demand much higher levels of precision in the measurements in order to demonstrate significant changes. |
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