Biochemical Reaction Networks
169
-1005’,
= ~iV2 +Vi)A
(3)
-
100
S
2
= ~{V2 + Vl)Ai
(4)
where
St
indicates the enrichment of the first carbon atom in the substrate and 5) indicates the enrichment
of the second carbon atom in the substrate (similarly for
A,
and
A2).
The balances simply state that
enriched carbon entering the first position in the metabolite pool
A
balance the enriched carbon in the
first position leaving this metabolite pool. Similar to (3)-(4) balances can be written for all metabolites,
and in matrix formulation we get:
M
V2
+ Vi)
0
0
0
0
0
r
v
'
- 1 0 0 5 ,
'
0
- (V2
+ V3 )
0
0
0
0
a 2
-
10052
V3
0
~ v 6
0
V4
0
5 ,
0
0
V3
0
~ v 6
0
v 4
B 2
0
0
V2
0
0
“ (V4 + Vs )
0
C,
0
l
V2
0
0
0
0
" ( V4 + V5 ),
\^~2 )
,
0
,
Equation (5) can be used to calculate the fluxes, but they cannot be obtained directly from the matrix
equation which is non-linear due to the occurrence of products of fluxes and labeling. However, the
matrix has full rank, and if the fluxes are given the L,C enrichment can be calculated by a simple matrix
inversion. This is the basis for the iterative method of Figure 5.10._________________________________
As discussed in Example 5.10 estimation of the fluxes is not straightforward and it requires a
robust estimation routine. One approach to estimate the fluxes is illustrated in Fig. 5.10, but other
approaches have been described in the literature (see Wiechert (2001) for a recent review).
Instead of using the fractional labeling of the metabolites one may also directly calculate the
NMR spectra that would arise for a given isotopomer distribution and then compare the
calculated NMR spectra with the experimentally determined spectra. Schmidt
et al.
(1999) used
this approach for estimation of the fluxes in
E. coll
based on experimental NMR data. A
requirement for any estimation procedure is a mathematical model describing the carbon
transitions and including all relevant biochemical reactions in the network. The carbon
transitions
for most biochemical
reactions
are
described
in biochemical
textbooks,
and
procedures to implement these carbon transitions in a formalized way have been developed
(Zupke and Stephanopoulos (1995); Schmidt
et al.
(1997); Wiechert
et al.
(1997)). Although
there are good procedures for estimation of the fluxes, it is still important to specify the correct
model structure, and in many cases it is necessary to evaluate different model structures in order
to identify a proper network that fits the experimental data. In these cases the network
identification and flux quantification goes hand in hand.
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