Biology Reference
In-Depth Information
For this augmented model, a stepwise construction leads to the following full S-invariant (in a drastically
simplified notation, just writing i instead of
→
i'D):
σ
c
=
[ (Gluc, 6), (G6P, 6), (F6P, 6), (FBP, 6), (CO2, 1), (Ru5P, 5), (R5P, 5), (Xu5P, 5), (S7P, 7), (E4P,
4), (DHAP, 3), (GAP, 3), (BPS, 3), (Lac, 3)].
An inspection of
σ
C
reveals that the integer weight factor of any substance is equal to the number
of C-atoms bound in it. Thus, the S-invariant
σ
c
expresses the conservation rule that the sum of C-
atoms bound by all involved substrates is constant. And this clearly represents a sensible biochemical
interpretation.
Next we compute an S-invariant concerning the number of all O-atoms.
In this case we obviously
have to include also H
2
O which, of course, did not appear in
σ
C
. We get
σ
o
=
[(Gluc, 6), (G6P, 6), (F6P, 6), (FBP, 6), (CO2, 2), (H2O, 1), (Ru5P, 5), (R5P, 5), (Xu5P, 5), (S7P,
7), (E4P, 4), (DHAP, 3), (GAP, 3), (P
i
, 1), (BPS, 4), (Lac, 3) ].
Interestingly, the weights of the substances reflect only the number of O-atoms
outside
the P-groups
PO
2
3
,ifany. AsP
i
=
HO-PO
2
3
in our case, it contributes one O-atom to the total number and,
consequently, appears with the factor 1 in the vector above. Hence,
σ
o
represents the conservation rule
that the number of all O-atoms (outside the P-groups) in the pathway is constant.
If looking for the P-atoms (or P-groups) we cannot expect to get a
full
S-invariant, as some of the
primary substances do not contain a P. The S-vector
σ
p
=
[(ADP, 2), (ATP, 3), (G6P, 1), (F6P, 1), (FBP, 2), (Ru5P, 1), (R5P, 1), (Xu5P, 1), (S7P, 1), (E4P, 1),
(DHAP, 1), (GAP, 1), (P
i
, 1), (BPS, 2) ] is a
partial
S-invariant saying that the total number of P-atoms is
constant. Moreover,
σ
P
is already reported in Reddy
et al.
, 1996. This is due to the fact that it contains
neither CO
2
nor H
2
O which are missing in their net model, as we know.
Finally, it should be mentioned that we also computed a full S-invariant concerning the sum of H-atoms,
using a model that additionally includes all H
+
-ions (not shown).
Effects and T-invariants
T-vectors and T-invariants of a Petri net describe processes. This means that we have to take into
account that reversible reactions may run in the backward direction.
As mentioned in section “Steady state pathways, elementary modes”, in case of a reversible transition
t
, we add its complementary transition
t
to the net. We start with the sample net model
P
(Fig. 2),
augmented by the places for CO
2
and H
2
O. The reactions
l
1,
l
3, and
m
1 can be treated as irreversible,
because we want to consider the
GP
and
PPP
, but not the gluconeogenesis. The linear path from BPS to
Lac, replaced by the substitution transition
l
8, contains the irreversible reaction from PEP to Pyr. Thus,
l
8 is also treated as irreversible. Hence, we introduce the new complementary transitions
l
2',
l
4',
l
5',
l
7', and
r
1' to
r
5' to the augmented net
P
and thus obtain the model
P
rev
in Fig. 3.
4
rev
may entail additional critical conflicts which have
to be resolved when dealing with T-vectors and simulation. We observe:
-
l
4',
l
5',
l
7', and
r
1' to
r
5' merely create uncritical loops and can be deleted,
-
l
2' must not appear in the steady state
GP
.
The introduction of the reverse transitions in
P
4
Note 1. Transition l6 in the model
P
is identical to l5' in
P
rev
.
Note 2. Transitions m2 and m3 are treated as irreversible as they merely restore the consumed NADP
+
-molecules.
Note 3. The S-invariants of
P
rev
are identical to those computed in subsection “Defects and S-invariants” for
P
.
