Biology Reference
In-Depth Information
Petri net (Fig. 7B) and then, analysed it using the program INA. For modelling the external metabolites,
we have chosen to introduce a self-loop for each source and each sink and to keep the same firing rule (2)
independently of the metabolite's type. If we do not impose capacities for the internal metabolites, the
net is unbounded because uridine, for example, can accumulate more and more tokens if Cdd keeps firing
while Urk1 is not. Accordingly, the reachability tree is infinite. As INA has reported, the net is strongly
connected, not pure (which is obviously due to the introduced self-loops), and not (sub-)conservative.
There is no
P
-invariant, except the external metabolites on their own. Again due to the self-loops next
to the external metabolites, the number of tokens in each of these places remains constant.
INA reports four minimal semi-positive
T
-invariants. We give them here by indicating the transitions
with positive components in the vectors representing these
T
-invariants:
1. Urk2, Kcy2
2. Cdd, Urk1, Kcy1
3. 2 Kcy1, KPR, 2 UPP, APT
4. Kcy1, KAD, KPR, UPP.
Note that some transitions (such as Kcy1 and UPP in the third invariant) have to fire twice, but not
necessarily successively. One can see that firing all the activated transitions that belong to an invariant
regenerates the initial marking. As each enzyme occurs in at least one minimal
T
-invariant, the net is
covered by these invariants. Therefore, the net is persistent and live. For simplicity's sake, although in
biological organisms the reactions KAD and KPR are reversible, we considered them irreversible. If
they are treated as reversible, care has to be taken that extra, irrelevant
T
-invariants, containing only
KAD and KAD', and KPR and KPR' respectively, result (where the primed symbols denote the reverse
reactions). They have to be discarded.
The biochemical meaning of the minimal
T
-invariants can be explained as follows: (1) production of
cytidine-diphosphate (CDP) from cytidine, (2) production of uridine-diphosphate (UDP) from cytidine,
(3, 4) two invariants producing uridine-diphosphate (UDP) from uracil in different ways. In the invariant
(3), one mole of adenine per two moles of UDP produced is formed as a by-product. This is because
ATP is used as a source of the ribose moiety, which is necessary for forming UDP from uracil. Note
that this invariant is not easy to determine by inspection. Moreover, it can be seen that the molar yield
with respect to ATP is different for the pathways (3) and (4). While invariant (3) consumes 3 moles of
ATP per mole of UDP produced, invariant (4) uses 3 moles of ATP per two moles of UDP [Schuster
et
al.
, 2002a]. All of these
T
-invariants correspond to the so-called salvage pathways, which serve to save
nucleotides from leaving the cell and redirect them to nucleotide phosphates [Stryer, 1995].
Let us now assume that ATP and ADP are internal metabolites and that the two enzymes KPR and
KAD are not expressed in a certain cell type. If we modify the network by eliminating the transitions
that stand for these enzymes (and also the arcs that connect them with their neighbouring places), we
do obtain a
P
-invariant. It can be translated in terms of a conservation relation: ATP
+
ADP
=
const.
The constant would be 2 if we define the initial token numbers of ATP and ADP to be 1 each. With
any conservation sum less than four, the four remaining transitions consuming ATP (Urk1, Urk2, Kcy1,
and Kcy2) are in mutual exclusion. Since the net does not include transitions producing ATP or (in the
second case) AMP, these substances are eventually running out, so that the places standing for ATP and
AMP are deadlocks, while in the complete net, there is neither a trap nor a deadlock. To maintain a
steady state, nucleotide metabolism requires permanent production of ATP, for example, by glycolysis.
