FLuX CONTROL ANALYSIS: BASIC PRINCIPLES

Kacser’s MCA theory facilitates the assessment of not only how perturbation of a particular enzy-mic activity affects metabolic flux, but also by how much. The response to changes in the concen-tration of a particular enzyme on flux varies over a wide range. For example, the response could be immediate, with a strong correlation between the increase in flux and the increase in enzymic activity, as is the case for adenylate kinase en route to ATP synthesis. In this case, one might justifi-ably describe such an enzyme as rate limiting or a pacemaker. However, most enzymes do not enjoy such a high profile and as such an increase in flux may or may not be brought about by increasing the activity of a single enzyme.

Furthermore, the degree of control exerted by a particular enzyme on the overall flux of a given pathway is not purely dependent on the numerical value of its intracellular concentration but rather on whether the enzyme has the capacity for higher throughput, which can only be ascertained from the value of the enzyme’s “flux control coefficient,” the first pillar of the theory of MCA.

7.2.1 Flux Control CoeFFICIent

The flux control coefficient is a parameter that describes in quantitative terms the relative contribu-tion of a particular enzyme to flux control in a given pathway. It is not an intrinsic property of the enzyme per se but rather a system property and so is subject to change as the environment changes.

It is generally expressed as the fractional change in flux in response to a fractional change in the concentration of the enzyme in question, and its value ranges between 0 and 1.0.

The measurement of the flux control coefficient of a particular enzyme allows an accurate pre-diction of how flux through a given pathway might fluctuate in response to a specific change in the enzyme’s catalytic activity or concentration. Although a change in the concentration can be brought about by cloning and subsequent overexpression of the structural gene encoding the enzyme in question, changes in the catalytic activity of the enzyme without changing its concentration can be brought about through site-directed mutagenesis and protein engineering techniques. For example, consider pyruvate dehydrogenase (PDH) with the view of assessing its effect or influence on flux to acetate excretion in Escherichia coli (Figure 7.1). The influence of PDH in that direction can be assessed from the enzyme’s flux control coefficient, which can be calculated from the tangent to the curve of a log-log plot of flux (J) as a function of enzymic activity or concentration (E). Assuming that a small increase in the concentration of PDH (dEpdh) was accompanied by a small increase in the steady-state flux (J) of the enzyme acetate kinase (dJak), it follows that if we were to change the concentration of PDH very slightly, then the ratio dJak/dEpdh becomes equal to the slope of the tangent to the curve of Jak against Epdh as depicted in Figure 7.2. However, analyzing the data in this way is somewhat imperfect because the numerical value of enzyme concentration and units of enzymic activity will be different from one enzyme to another. This problem could be overcome if we were to relate the fractional changes in flux through acetate kinase to the fractional increase in the concentration of PDH (i.e., dJak/Jak and dEpdh/Epdh), and as such the flux control coefficient will assume a value between 0 and 1.0, which can then be expressed in terms of a percentage.

However, it is possible for an enzyme to have a flux control coefficient with a negative value, as is the case at branch points where one metabolite has to be partitioned between two enzymes. In such a case the increase in flux through one branch is generally at the expense of the other, as exemplified

in the case study for the partition of carbon flux at the junction of isocitrate (see Section 7.3). At this junction, any increase in the concentration of isocitrate dehydrogenase (ICDH) is concomitant with a decrease in flux through the competing enzyme, namely isocitrate lyase (ICL). It is possible, there-fore, to describe ICDH as having a negative flux control coefficient on flux through ICL. Although any increase in the concentration of ICDH is accompanied by a decrease in flux through ICL, the opposite is not true for reasons that will become apparent later on; for further details, see El-Mansi et al. (1994).

7.2.2 summatIon theorem

The summation theorem, the second pillar of the MCA theory, states that the total sum of flux control coefficients of all enzymes in a given pathway adds up to 1.0. The summation theorem also shows that the flux control coefficient of an enzyme is a system property because any increase in

NAD⁺ NADH.H⁺

Pyruvate

E1 E2 E3

J_pdh J_pta J_ak

Intermediates

Reservoir

Key: pdh (E₁), pyruvate dehydrogenase pta (E₂), phosphotransacetylase ak (E₃), acetate kinase

Sink pta

pdh AC.CoA AC. Phosphate ak Acetate

HS-CoA ADP ATP

Pi

FIGuRE 7.1 The enzymes and metabolites en route to acetate excretion.

j

log10 flux through acetate kinase (E3)

log10 concentration of pyruvate dehydrogenase (Ee 1) dlog₁₀ j_ak

dlog₁₀ E_pdh = C_EpdhJ_ak

FIGuRE 7.2 Graphical determination of PDH flux control coefficient with respect to acetate excretion. The graph depicts a typical pattern of variation in flux to acetate; measured as acetate kinase (Jak), in response to changes in the concentration of pyruvate dehydrogenase (Epdh).

the concentration of a particular enzyme is accompanied by a decrease in its flux control coefficient.

According to the summation theorem, such a decrease will have to be balanced by increasing the flux control coefficient of another enzyme—or more than one enzyme—within the same pathway so that the sum of all flux control coefficients remains constant (i.e., 1.0). For example, in a linear pathway consisting of enzymes with usual kinetic properties (i.e., where substrates stimulate and products inhibit reaction rate), the flux control coefficients for every enzyme must be 0 or higher with a total sum of 1.0. If an enzyme were to show a flux control coefficient of 1.0 with all other enzymes showing flux control coefficients of 0, such an enzyme could justifiably be described as rate limiting. The summation theorem also shows that this is not necessarily the case because it is also possible for some or all of the enzymes to have values greater than 0 providing that the total does not exceed 1.0. In practice, we would expect a pathway flux to be influenced mainly by enzymes in that pathway, and to a much lesser extent by closely related pathways, and that distantly connected enzymes would have negligible influence or none at all. In other words, the flux control coefficients of hundreds or even thousands of enzymes that are not directly related or connected to the pathway in question will be zero although flux control is shared among all enzymes.

Another consequence of the highly branched and intricate nature of cellular metabolism is that the central pathways provide biosynthetic precursors and energy for other pathways. So, as biosyn-thetic precursors are made, some are fed directly into the biosynbiosyn-thetic routes, which in turn dimin-ish flux through the central metabolic pathways. Therefore, It follows that biosynthetic enzymes are likely to have negative flux control coefficients with respect to flux through the central metabolic pathways. According to the summation theorem, if one or more enzymes possess a negative value of flux control coefficient, then it is possible to see some other enzymes displaying a flux control coef-ficient higher than the numerical value of 1.0. This is because if there are negative flux control coefficients, one or more flux control coefficients would have to be greater than 1.0 so that the total sum adds up to 1.0. This shows that the flux control coefficient is not an intrinsic property of the enzyme itself but rather a property of the whole system.

7.2.3 elastICIty CoeFFICIent

The flux control coefficient of an enzyme is influenced by the enzyme’s ability to respond to changes in the concentration of its immediate substrate, as well as its ability to influence the concentrations of other metabolites in the pathway, a linkage that was first demonstrated by Heinrich and Rapaport (1974). The elasticity coefficient, the third pillar of the MCA theory, was therefore introduced to describe how flux is influenced by changes in the concentration of a given metabolite. In other words, elasticity is a parameter that describes, in quantitative terms, the sensitivity and responsive-ness of an enzyme to a metabolite.

Unlike the flux control coefficient, elasticity is a property of individual enzymes and not of the pathway. The elasticity of an enzyme to a metabolite is defined by the slope of the curve of enzyme units (reaction rate) plotted as a function of metabolite concentrations, with the measure-ments taken at the metabolite concentration found in vivo. By analogy with the flux control coeffi-cient (Figure 7.2), the value of the elasticity coefficoeffi-cient, which can be calculated from the slope, will depend upon the units used for the measurement of enzymic activities, which may vary from one enzyme to another. This can be avoided, as described earlier for the flux control coefficient, by directly calculating the elasticity coefficient from a log-log plot of catalytic activity versus metabolite concentration to give the fractional change in enzymic activity as a function of the fractional change in the concentration of the substrate. As highlighted in the case study presented in Section  7.3, elasticities have positive values for metabolites that stimulate enzymic activity (substrates, activators) and negative values for those that decrease reaction rate, such as products and inhibitors. Therefore, elasticity is a parameter that describes, in quantitative terms, the sen-sitivity and responsiveness of an enzyme to a particular metabolite that could be a substrate, a product, or an effector.

7.2.4 ConneCtIvIty t^heorem

This theorem, the fourth pillar of the MCA theory, addresses the question of how the flux control coefficient of a given enzyme can be related to its kinetic properties. Such an inter-relationship is governed by the connectivity theorem, which states that the sum of all connectivity values in a given pathway is zero. The connectivity value for any given enzyme can be calculated by multiplying its flux control coefficient by its elasticity with respect to the metabolite in question. Enzymes not affected by the metabolite in question will naturally have an elasticity of zero and as such will make no contribution toward the final sum obtained. Further analysis of connectivity values has revealed that large elasticities are associated with small flux control coefficients and vice versa. The math-ematical equations relating the connectivity theorem to linear pathways, branch points, and cycles have been extensively described and dealt with elsewhere (Fell 1997).

7.2.5 response CoeFFICIents

Induction and repression of enzyme synthesis in response to internal or external environmental stimuli are widely distributed in nature and are very effective in “turning on” and “switching off”

transcription. Covalent modification through reversible phosphorylation is another mechanism that regulates the activity of existing enzymes by rendering them active or inactive, as is the case for ICDH in E. coli during adaptation to acetate (Koshland 1987; Cozzone 1988). In addition to degra-dation of mRNA and proteins, enzymes may also be the subject of allosteric control mechanisms, which change the enzyme’s affinity toward its substrate and/or cofactor(s).

The response coefficient, the fifth pillar of the MCA theory, reflects the effectiveness of a par-ticular effector on flux through a given pathway and is dependent on two factors: the flux control coefficient of the target enzyme and the strength of the effector, which Is given by its elasticity coef-ficient. For an effector to have a significant effect on flux, each of the above parameters with respect to the target enzyme clearly has to be of a value higher than zero.

Under circumstances in which a particular effector may activate or inactivate more than one enzyme in a given pathway, the total response will be the sum of the individual responses from each enzyme affected. However, this is only true when the changes in the concentration of the effector are very small because of the nonlinear relationship of the kinetics in metabolic systems (Hofmeyr and Cornish-Bowden 1991).

Now, let us consider how carbon flux is partitioned at the junction of isocitrate during growth of E. coli on acetate.

7.3 CONTROL OF CARBON FLuX AT THE juNCTION OF