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et al.
[2000]. However, TPI knockout mutants turned out to be inviable. Thus, they built a kinetic model
to explain the unexpected result that all system fluxes (PYR, GLYCEROL) decrease. We now give a
structural explanation, thus ignoring the kinetics. In Fig. 9, the Petri net model corresponding to Scheme
1 in Helfert
et al.
[2001] is given. It should be noted that this network is not a free-choice net because,
for example,
|
GLY3P
|
=
|{
t2, GLYK
}|
=2 and
{
GLY3P
}
=
{
GLY3P, ADP
}{
GLY3P
}
. The first property
is known in Petri net theory as conflict, because two transitions
{
t2, GLYK
}
compete for the same
resources (the tokens from place B). But in metabolic networks, the token number is large enough that
the transitions in competition will simply “agree” on the tokens distribution depending on their reaction
rate. Taking this aspect into account, we do not need to know the reaction rates, but we only assume that
the flux through transition
T
2
in Fig. 9 is always greater than zero. Another important knowledge that
we use is that ALD is reversible and therefore inhibited by its products [Stryer, 1995].
Let us consider the case when TPI is knocked out.
t
1
=
{
NADH, NAD
+
}
forms a siphon and a trap
at the same time. Its input transitions set
{
GPDH, GAPDH
}
coincides with its output transitions. This
means that once this set of places is sufficiently marked, it keeps its tokens. Moreover,
{
GPDH, GAPDH
}
forms also a
P
-invariant, their tokens sum remaining constant during the whole process. Another trap
(
T
2) consists of
{
DHAPc, DHAPg, GLY3Pc, GLY3Pg, Gly
}
because its input transitions set
{
ALD,
GPDH, GPO, GLYK,
t
1
,
t
2
}
. If the flux
through
t
2
were equal to zero, Gly would accumulate, but because the flux through
t
2
is greater than
zero, GLY3P is partially transformed back into DHAPg. Let us start proceeding with the marking m0.
Following the firing sequence
{
}
includes its output transitions set
{
GPDH, GPO, GLYK,
t
1
,
t
2
e
3
, ALD, GPDH, GAPDH,
t
2
, GPO,
t
1
, GPDH,
t
2
, GPO,
t
1
, PGK,
t
3
,
e
1
,
e
2
}
as Table 2 illustrates, the network reaches a dead marking m1, because DHAP accumulates - no
NADH being available for further converting it. This is because GPO is draining the flux, consuming
NADH faster than GAPDH can produce it. This continues until the product inhibition of ALD is so
strong that ALD ceases operating. Therefore, the whole system is dead, no more Gly and Pyr being
produced. For deriving this result, it is informative that after deleting TPI, the three transitions
t
1
,
t
2
and
GPO are not involved in any elementary mode (minimal
T
-invariants) anymore.
The importance of TPI can be seen if its corresponding transitions are added in the model. TPI
being a reversible enzyme, two transitions, one acting forwards and one - backwards, have to be added.
Of course, the
T
-invariant
{
TPI1, TPI2
}
has to be ignored, having no biological significance. In this
new context,
T
2
is not any more a trap. Another minimal trap occurs in
t
3
=
{
DHAPC, DHAPg,
GLY3Pc, GLY3Pg, GLY, GA3P, BPGA, 3PGK, 2PGK, PEP, PYR
}
corresponding to the accumulation
of GLY and PYR. Whenever DHAPg tends to accumulate, due to NADH lack, TPI1 converts a part of
DHAPg tokens into GA3P. Due to the sufficient amount of NAD
+
, GAPDH fires with production of
the necessary NADH, which gives GPDH the possibility to fire. Also if NAD
+
is deficient, but GA3P
is sufficient, TPI2 converts GA3P into DHAPg, GPDH fires and produces the required NAD
+
. Taking
into account almost only structural properties of the given network, especially the presence of traps, we
could evaluate the role of TPI in glycolysis and glycerol production. In the next section an example
taken from nucleotide metabolism will be used to illustrate the notions presented above. The program
ILLUSTRATION OF PETRI NET PROPERTIES ON A SYSTEM EXTRACTED FROM
NUCLEOTIDE METABOLISM
Let us now consider the biochemical system depicted in Fig. 7A. It represents part of nucleotide
metabolism, as it occurs, for example, in human liver [Stryer, 1995]. We have translated it in terms of a
