222
Chapter 6
pathway the number of intermediates becomes
1-2
when the total number of enzymes is /, and the
E matrix in Eq. (6.42) is therefore not quadratic. But Eq. (6.45) represents an additional constraint
(extra row), which allows calculation of all the control coefficients from the elasticity coefficients,
for branched pathways also. In branched pathways negative flux-control coefficients can appear
since an increase of the activity of an enzyme in one branch could well have a negative influence on
the flow in the other branch.
MCA is a powerful tool for qualitative studies of metabolic pathways. A serious drawback of the
method is, however, the requirement that either the elasticity coefficients or the control coefficients
have to be measured. This is not an easy task, and practical applications are as yet rarely seen in the
literature. In some cases one may learn something of the enzyme kinetics from
in vitro
experiments,
and this may be used to calculate the elasticity coefficients as illustrated in Example 6.4 and in an
appealingly simple study by Delgado
et al.
(1993) for two glycolytic enzymes. It is, however,
unlikely that the
in vivo
enzyme kinetics is the same as that determined from
in vitro
experiments,
and conclusions based on
in vitro
experiments must therefore be regarded as tentative only.
The most direct way of obtaining the control coefficients is to examine the effect of variations of
the enzyme activities on the pathway flux. The enzyme level may be varied genetically, e.g., by
insertion of stronger promoters or additional copies of the genes, as illustrated by Flint
et al.
(1981),
who examined the arginine pathway of the filamentous fungus
Neurospora crassa.
From
measurements of fluxes in different mutants expressing different levels of the individual enzymes,
the control coefficients could be found directly. This approach is, however, very time-consuming
and it is not generally applicable since different mutants of the applied strain are not normally
available.
Another limitation of MCA is that it is based on a steady state of the pathway, and strictly speaking
the analysis is valid only close to a given operating point
(s,p).
However, the purpose of metabolic
engineering is to design a strain with a completely different flux distribution than the parental
strain, and the analysis of the parental strain based on MCA is therefore not likely to hold for the
new strain. Despite these drawbacks MCA is still a useful tool for examination of metabolic
pathways, but it should be used in conjunction with other methods. Both Stephanopolous et al.
(1998) and Fell (1997) give detailed accounts of the theoretical as well as the experimental
foundation of MCA. One of the original contributors to the development of MCA has proposed
(Small and Kacser, 1993) that as long as the pathway flux
.1
is a downwards convex function of the
activity of the ;th enzyme then the control coefficient can be obtained experimentally also by
measuring the change in flux mediated by an integral change in e, This important result will be
proved and used in Example 6.5.
The drawbacks of the steady-state assumptions applied in MCA are circumvented in a method
described by Liao and Lightfoot (1988). It is a generalization of a classical analysis by Wei and
Prater (1962) of
characteristic reaction paths
to identify rate-controlling steps using
time-scaling
separation.
In an extension of these techniques, Delgado and Liao (1991) illustrated the use of
measurements of the pathway intermediates in a transient state after a perturbation is applied to the
cells. The method is based on a linearization of all the reaction rates in the pathway, i.e.,
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