Biochemical Reaction Networks
the pathw ays). In all cases w e use pyruvate as the starting point:
TCA cycle: - pyruvate + 3CO, + 4 N A D H + FA D H : + GTP = 0
Glyoxylate shunt: - 2pyruvate + 2C O : + oxaloacetate + 4N A D H + FA D H 3 = 0
Pyruvate carboxylase: - pyruvate - A TP - C 0 2 + oxaloacetate = 0
If ATP and G TP are pooled to gether (w hich is often done in the analysis o f cellu lar reactions), it is quite
obvious that the glyoxylate shunt is a linear com bination o f the tw o o ther pathw ays, and all three
pathw ays cannot be d eterm ined independently by flux analysis. It m ay be a d ifficult task to decide w hich
pathway should be elim inated. Fortunately, these p athw ays rarely operate at the sam e tim e as their
enzymes are induced differently. Inform ation about induction and regulation o f the corresponding
enzym atic activities is critical in m aking a decision as to the exact pathw ay to be considered at a given
set o f environm ental conditions. For exam ple, expression o f isocitrate lyase (the first enzym e o f the
glyoxylate shunt) is repressed by glucose in m any m icroorganism s, and co nsequently the glyoxylate
shunt is inactive for grow th on glucose. In eukaryotes, the p resence o f the glyoxylate shunt does not give
rise to a linear dependency due to co m partm entation o f the different reactions,
the TC A cycle
operates in the m itochondria and the glyoxylate shunt eith er in the cytosol or in m icrobodies. In practice
there are m any other reactions in the netw ork that involve interm ediates o f the T C A cycle and the
glyoxylate shunt, and these reactions m ay lead to a rem oval o f the lin ear dependency betw een these tw o
pathways (see also E xam ple 5.6). E ven in cases w here the tw o pathw ays are not linearly dependent the
inclusion o f both pathw ays in the m odel m ay lead to an ill-co n d itio n ed system , i.e., the condition num ber
may be high (see N ote 5.4).
Another exam ple o f linearly dependent reactions is the tw o am m onia assim ilation routes: the glutam ate
dehydrogenase catalyzed reaction and the G S -G O G A T system (see Section 2.4.1). The stoichiom etries o f
these tw o routes are
GDH: - a -ketoglutarate - N H 3 - N A D PH + glutam ate = 0
GS-GOGAT: - a -ketoglutarate - N H 3 - NA D PH - ATP + glutam ate = 0
Thus, the only differen ce is that A TP is used in the G S -G O G A T route (w hich is a high-affinity system )
but not in the G D H reaction. T he p roblem here is that an A T P balance is not easy to utilize due to lack o f
sufficient inform ation about all A T P -consum ing reactions. In the absence o f an A T P balance to
differentiate b etw een them , the tw o n itrogen assim ilation reactions are linearly dependent and, as such,
non-observable. B ecause the only d ifference betw een the tw o routes is the consum ption o f A TP in the
GS-GOGAT system , distinction betw een the tw o routes m ay not be im portant, and they are therefore
often lum ped into a single reaction in stoichiom etric m odels.
If a singularity arises in the stoichiom etric m atrix, one has the follow ing tw o options:
(1) Rem ove the linearly dependent reaction(s) from the m odel, invoking (or postulating) inform ation
about specific enzym e regulation and induction.
(2) Introduce additional inform ation such as the relative flux o f the tw o pathw ays. Such inform ation m ay
be derived from m easurem ents o f enzym e activities,
the relative activity o f key enzym es in the
two routes. H ow ever, this approach is h am pered by the fact that
in vitro
enzym e activity
m easurem ents o ften bear little relationship to actual
in vivo
flux distributions. A m ore pow erful
technique is the use o f labeled substrates,
’’C -enriched glucose, follow ed by m easurem ents o f
the labeling p attern o f in tracellular m etabolites as discussed in Section 5.4.2._________________________
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