154
Chapter 5
the least square estimate for flux vector. This estimate is found by using the pseudo-inverse of
T,
- called
T/,
i.e., the fluxes are calculated from:
v
(5.27)
The pseudo-inverse o f
T2
is given as:
T2*
*
=(T2rT2)-'T2
r
(5.28)
The matrix
T2tT2
is always a square matrix. Furthermore, if
T2
has full rank then
T2tT2
is non-
singular and the matrix can be inverted. The requirement of
T2
having full rank is synonymous
with the requirement of
T2
being non-singular for the case o f a determined system, and it means
that there exists a
JxJ
sub-matrix within
T2
that is non-singular. The requirements for application
of eq. (5.28) are therefore the same as for analysis o f the determined system, i.e., that there are no
linearly dependent reaction stoichiometries and there are no metabolites with identical or linear
dependent stoichiometric coefficients (such as the co-factor couple NADH/NAD+). Furthermore,
the set of measured rates must be chosen such that the fluxes can be calculated using the matrix
equation, but this requirement is normally fulfilled for an over-determined system. Whereas it is
quite simple to consider only one of the compounds in co-factor couples it is often more difficult
to avoid linearly dependent reaction stoichiometries, and this is therefore discussed further in
Note 5.4.
Note 5.4 L inear dependency in reaction stoichiom etries.
B ecause m ost living cells are capable o f u tilizing a large variety o f com pounds as carbon, energy and
nitrogen sources, m any com plem entary pathw ays exist that w ould serve sim ilar functions if they
operated at the sam e tim e. T he inclusion o f all such pathw ays m ay give rise to problem s w hen m atrix
inversion is applied for flux analysis. T his situation usually m anifests itse lf as a m atrix singularity,
w hereby the non-observable pathw ays appear as linearly dependent reaction stoichiom etries. The fluxes
through these different pathw ays cannot be discerned b y extracellular m easurem ents alone. H ere w e w ill
consider tw o exam ples:
G lyoxylate cycle in prokaryotes
N itrogen assim ilation via the G S -G O G A T system
In prokaryotes, the T C A cycle and all anaplerotic reactions, including the glyoxylate cycle, operate in the
cytosol. O ften the glyoxylate cycle is considered as a bypass o f the T C A cycle because it shares a
num ber o f reactions w ith this cycle (see Fig. 2.5). H ow ever, the tw o p athw ays serve very different
purposes: the T C A cycle has the prim ary purpose o f oxidizing pyru v ate to carbon dioxide, w hereas the
glyoxylate cycle has the purpose o f sy n th esizin g precursor m etabolites,
e.g.,
o xaloacetate, from acetyl-
CoA . C onsidered individually the T C A cycle and the glyoxylate shunt are not linearly dependent, but if
other anaplerotic pathw ays,
e.g.
the p y ruvate carboxylase reaction, are included, a singularity arises. T his
m ay be illustrated by w riting lum ped reactions for the three pathw ays (see Fig. 2.10 for an overview o f
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