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[ (ADP,
→
1'D), (ATP,
→
1'D) ], [ (NADP
+
,
→
1'D), (NADPH,
→
1'D) ],
[ (GSSG,
→
2'D), (GSH,
→
1'D) ], and [ (NAD
+
,
→
1'D), (NADH,
→
1'D) ].
Doing this yields the null defect in all cases, which means that the S-vectors above constitute S-
invariants.
3
Clearly, it is of much greater importance to deal with the
full
set of all primary (non-ubiquitous)
substances. In steady state, there should exist an S-invariant comprising that
full primary set
. To find out
the weight factors of a
full
S-vector, i.e., covering this set, we proceed step by step. First we observe that
each molecule of FBP is transformed into two GAP-molecules by the reactions
l
4 and
l
5. We conclude
that in any S-vector, to finally become an invariant, the place markings (number of molecules) of the
glycolysis pathway
GP
from Gluc unto FBP must get a weight factor twice as high as GAP and the
following places down to Lac. Trying to adopt the same principle to the pentose phosphate pathway
PPP
, however, leads to a non-null defect.
To be more specific, the simulation of
P
∗
and also its T-invariants computed in the next subsection
show that, starting with 3 Gluc molecules, the
GP
produces 6, and the
PPP
5 Lac molecules. Because
these alternative paths share the metabolites G6P, F6P, FBP, GAP, and BPS, it is not possible to find
integer weight factors for these substances to make the full S-vector an invariant. We conclude that it is
impossible to find a full S-invariant, in model
P
rev
, with 'standard' means like low-level Petri nets or
METATOOL [Pfeiffer
et al.
, 1999]. Hence, neither in Reddy
et al.
, 1996, nor in Schuster
et al.
, 1996, a
full S-invariant has been reported.
The use of high-level nets with individual tokens offers the possibility to distinguish the mentioned
metabolites according to the paths along which they are produced and consumed. Constructing the
desired S-invariant (not shown), we have to choose the weight 2 for all metabolites from Gluc unto
FBP and E4P, irrespectively of the path on which they occur. For GAP, a threefold distinction has to be
made: GAP-molecules produced by
r
3or
r
4' (
x
=
G) or by
r
5(
x
=
H) get the weight 2, whereas those
produced by
l
4or
l
5(
x
=
D) get the weight 1. And this distinction is kept also for BPS and Lac.
The result then is an S-invariant which, however, lacks a sensible biochemical interpretation because
it is impossible to distinguish molecules of the same substance in organisms. Anyhow, this invariant lets
us conclude that an essential product must be missed in the model of the
PPP
. Inspecting the
PPP
more
carefully, shows that this product is the CO
2
-molecule produced by the reaction complex
m
1:
G6P
+2
NADP
+
+
H
2
O
→
Ru5P
+2
NADPH
+2
H
+
+
CO
2
.
This means that the model
P
(and that of Reddy
et al.
, 1996) have to be revised for our purposes.
Introducing both CO
2
and H
2
O into the model means to add an input place H
2
O and an output place
CO
2
to
m
1 and an output place H
2
Oto
l
8. The latter one originates in the H
2
O produced by the reaction
DPG
→
PEP
+
H
2
O. Of course, also places for H
+
could be added to the model, but we decided to
refrain from this in order to keep the readability of the model.
3
Adopting the definitions at the beginning of section “Steady state pathways, elementary modes” and the conventions of
the package SY in a simplified version, an S-vector is written as a list of pairs (place si, distribution of si), where the second
element - in our case - degenerates to a mapping of a color variable (or the don't-care symbol “
” in case of a constant) to a
linear combination over the standard colorset CS (cf. footnote 1).
The defect of an S-vector
σ
, computed symbolically by the function DEFECT, is given as a list of members t: lico(CS), in
which lico(CS) denotes a linear combination of tokens that has to be added to an input or output place of transition t to make
σ
an S-invariant.
Dealing with the syntactic details of SY is far beyond the scope of this paper.
