Biology Reference
In-Depth Information
concepts of liveness, deadlocks, siphons and traps are connected with each other. A net satisfies the
deadlock-trap property if each non-empty siphon includes a trap and the maximal trap in each minimal
deadlock is sufficiently marked. In this case, no dead marking is reachable. So, the net is deadlock-free.
A further special class of Petri nets is made up of the free-choice nets. In these nets, each place has
at most one output transition or the input places of the output transitions of
p
consist only of
p
for any
place
p
belonging to the net. A free-choice net is live if and only if every non-empty siphon includes an
initially marked trap. This property is also known as Commoner's theorem [Commoner, 1972]. Siphon
and trap are dual notions. A siphon in a Petri net
N
is a trap of the net
N
obtained by reversing direction
of all edges of
N
. Therefore, the properties satisfied by siphons have counterparts for traps.
Liveness and deadlock-freeness are structural properties, in which the initial marking plays, however,
an essential role. An example for a net that is in deadlock is the above example A
+
B gives 2 B (Fig. 5)
if the initial token number of B is zero. The input and output transition sets of B coincide. Therefore,
{
B
}
is simultaneously a siphon and a trap.
{
B
}
=
{
A,B
}{
B
}
(where
{
B
}
denotes the input places of
the output transitions of B, that is, the co-substrates of B), but
|
A
|
=
|{
}|
=
1, so
“A
+
B gives 2 B” is a free-choice net. Due to Commoner' theorem, this net cannot be live if the token
number of A is not infinite (if A is not what we called external metabolite in Section 2) and trap
{
B
}
included in siphon
{
B
}
has not at least one token. Without tokens in B, this autocatalytic reaction then
does not start proceeding. In chemistry, such a situation is known as false equilibrium [Othmer, 1981].
A larger example is glycolysis, which requires 2 moles of ATP in its upper part and produces 4 moles
of ATP in its lower part. If no ATP is present at the beginning, the glycolytic pathway cannot proceed.
Therefore, this pathway has been said to have a turbo design [Teusink, 1998]. A test for liveness and
deadlock-freeness of the net can thus help us decide whether the metabolic system can attain a situation
where it is blocked. The detection of siphons and traps is instrumental for this purpose.
t
}|
=
1 and
|
B
|
=
|{
t
EVALUATING THE ROLE OF TPI IN TRYPANOSOMA BRUCEI METABOLISM BY
DETECTING SIPHONS AND TRAPS
T. brucei
is a unicellular, extracellular, eukaryotic parasite of the blood and tissue fluids of mammals.
It is transmitted by tsetse flies and causes sleeping sickness in humans. The infections are lethal unless
treated, but the few existing drugs have severe side-effects. Many studies [e.g. Bakker
et al.
, 1999;
Helfert
et al.
, 2001] were focused on the carbon and free energy metabolism of this organism, which
depends entirely on glycolysis. Accordingly to Scheme 1 in Helfert
et al.
[2001], glucose is imported
into the glycosome and then converted into F-1,6-P. The two consecutive enzymes (HXK and PFK) are
contracted in one step which consumes two ATP and produces two ADP. ALD converts F-1,6-P into
DHAP and GA3P. These two substances are isomerised into each other by a reversible enzyme, TPI
(triose-phosphate isomerase). DHAP is transformed into Gly3P by GPDH, with consumption of NADH
and production of NAD
+
. GAPDH1 uses NAD
+
to transform GA3P into BPGA and NADH. Gly3P
can be either converted into glycerol by GLYK with consumption of ADP and production of ATP, or
transported into the cytosol, where GPO oxidizes it to DHAP and H2O, DHAP being transported back to
the glycosome. PGK uses an ADP molecule to convert BPGA in 3-PGA and ATP. 3-PGA is transported
into the cytosol and converted, via 2-PGA, into PEP, which gives pyruvate and ATP, consuming one
ADP. A similar system was treated by Overkamp
et al.
[2002]. They maximized the yield of glycerol in
Saccharomyces cerevisiae
using metabolic engineering.
At the beginning, Helfert
et al.
[2001] supposed that glycolysis could proceed without TPI, producing
glycerol and pyruvate in the same amount. This corresponds to an elementary mode described in Schuster
