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MCA |
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Back to the MCA homepage.
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Although very often used as examples, linear pathways (as in
figure 5) are not the only structures to be found
in metabolic maps (see figure 4). The latter are
populated with branches and cycles. Also, the restriction that the rate of
reactions should be linearly proportional to the enzyme concentration is not
needed to apply MCA. A very brief description of how MCA deals
with such cases is presented here.
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Branched Pathways |
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In a previous section it was mentioned that one
can calculate the values of all control coefficients of a pathway given the
knowledge of the values of all enzyme elasticities using the summation theorem
and the connectivity relations. When a pathway contains one or more branches
(see figure 6), the number of equations is smaller
than the number of steps (enzymes), and so one needs more equations for a
complete solution. Those extra equations relate the enzyme elasticities and the
control coefficients at the branch points and are known as the branch-point
theorems. Including them here would make these web pages too heavy and so the
interested reader is pointed to the following papers: Kacser
(1983), Fell & Sauro
(1985), Westerhoff & Kell
(1987) and Small & Fell
(1989).
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Moiety-Conserved Cycles |
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Reich & Sel'kov (1981)
introduced the term moiety-conserved cycle to name metabolic cycles in
which there is conservation of a certain chemical group (moiety) after a
full turn of the cycle. These cycles (see figure 7)
are quite common in metabolism, an example of which is the NADH-NAD+ cycle. One
important characteristic of these cycles is that the the entry and exit of
other moieties into and out of the cycle is done via bimolecular reactions and
there cannot be any branches from the cycle (see figure 7).
It is easy to mistake other metabolic structures for
moiety-conserved cycles when they are not. A notable example is that of
the TCA cycle, in which there is no conservation of moieties. In fact the TCA
cycle is a branched pathway (that we can draw as a circle...)
Because there is mass-conservation in moiety-conserved cycles,
the concentration of one of the metabolites of the cycle is a linear
combination of the concentrations of the other metabolites of the cycle, and
therefore the former is not an independent variable. Furthermore, the sum of
the concentrations of the metabolites of the cycle is constant, this sum is
thus an external parameter of the patway (the total [NADH]+[NAD+] is an
example). Because of this, one can define response coefficients for the total
concentration of the moiety. The existence of one or more of these cycles in a
pathway also implies that some connectivity relations and the summation theorem
for concentration-control coefficients will be different from those of a linear
pathway . Hofmeyr et al. (1986)
gives a detailed study of MCA and moiety-conserved cycles and is
best consulted for further details.
Reder (1988) introduced a general methodology
to calculate control coefficients from elasticities that takes into account
moiety-conservation (see also the next section).
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Substrate Cycles |
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Substrate cycles, also known as futile cycles, are
metabolic structures in which there are two independent enzymes operating in
opposite directions interconverting two metabolites. In some substrate cycles a
full turn of the cycle is accompanied by the breakdown of ATP (and this is the
origin of the term futile, meaning that the cycle is just "wasting"
energy). Figure 8 depicts a substrate cycle.
Inspection of figure 8 reveals that in a
substrate cycle there is no conservation of any moiety, rather the cycle is
formed by two sequential branches. The application of metabolic control
analysis to substrate cycles is based on that of metabolic branches and is well
covered in Fell & Sauro (1985).
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Enzyme-Enzyme Interactions |
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MCA as developed by Kacser & Burns ( 1973)
and Heinrich & Rapoport (1974)could
not deal with metabolic systems in which the concentration of one enzyme affected
reactions other than the one it catalysed. Furthermore, although the text in
Kacser & Burns (1973) is general enough to
handle systems in which the rate of reaction is not linearly proportional to the
enzyme concentration, the concept of control coefficients became equated with the
changes in enzyme concentration rather than rate of reaction
(see definition of control coefficient in
Kacser & Burns 1973 and
above).
This would result that the summation theorems
would be invalid whenever there are enzyme-enzyme interactions.
There are many cases of enzyme-enzyme interactions in metabolism. A well
known case is that of the mitochondrial electron-transfer chain in free-energy
transducing membranes, or more generally, group transfer chains. In these
structures, each enzyme directly transfers a metabolite to the next enzyme (be
it electrons or molecules). Metabolic channelling (see
srere 1987,
Ovádi 1991,
Mendes et al. 1995) can also be
seen as a form of group transfer. Click here to
read something more about metabolic channelling.
Several papers have addressed enzyme-enzyme interactions in the context
of MCA and there are a few methods that can be used to describe
such systems using MCA. See
Westerhoff & Kell (1988),
Kell & Westerhoff (1990),
Kacser et al. (1990),
Sauro & Kacser (1990),
Kholodenko & Westerhoff (1993),
and van Dam et al. (1993).
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Interconvertible Enzyme Cascades |
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Interconvertible enzyme cascades are sequences of enzymes catalysing the
formation of covalent bonds between other enzymes and chemical moieties. This
is a special case of enzyme-enzyme interactions in which enzymes are substrates
of other enzymes, forming cycles of active-inactive forms (more and less active
forms, to be more precise). A good example of interconvertible enzyme cascades
is that of the interconversion of glycogen phosphorylase a and glycogen
phosphorylase b catalysed by phosphorylase-b kinase and
phosphorylase-a phosphatase (see for example
Chock & Stadman 1980).
Figure 9 depicts an interconvertible enzyme
cascade.
The application of MCA to enzyme cascades was first done on a
mechanistic level by
Cárdenas & Cornish-Bowden (1989),
later followed by
Small & Fell ( 1990) and
Schuster et al. ( 1993)
with mechanism-independent studies. The latter work is also applicable
to other modular systems such as protein synthesis.
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