Biology Reference
In-Depth Information
A central concept in metabolic network analysis is that of elementary flux modes [Schuster and
Hilgetag, 1994, Schuster
et al.
, 2002a]. They stand for minimal sets of enzymes that could operate at
steady state and are uniquely determined. That means that no other flux modes at steady state are proper
subsets of the elementary modes. For a better understanding of the behaviour of biochemical systems,
they can be decomposed into such simplest relevant routes. This has recently been demonstrated for
sugar cane metabolism [Rohwer and Botha, 2001] and bacterial metabolism [Van Dien and Lidstrom,
2002]. In Petri net theory, elementary modes have, as counterpart, the minimal
T
-invariants [Starke,
1990]. The concept of elementary modes is, however, more general because reversible reactions are
allowed. Colom and Silva [1990] developed an algorithm for computing minimal
P
-invariants. It
can, by transposition of the incidence matrix, be used also for computing minimal
T
-invariants. It is
based on row operations on the incidence matrix augmented with an identity matrix. In the course of
calculation, care has to be taken to eliminate non-minimal and duplicate
T
-invariants. Colom and Silva
[1990] propose two alternative tests to do so. A method for computing elementary flux modes based
on convex analysis was proposed in [Pfeiffer
et al.
, 1999; Schuster and Hilgetag, 1994; Schuster
et al.
,
2000]. Although the latter algorithm was developed with different goals (Colom and Silva [1990] did
not deal with metabolic networks) and completely independently of Petri net theory, the two algorithms
show some similarities. However, they differ in that elementary modes can involve reversible reactions.
This is taken into account by partitioning the stoichiometry matrix into “reversible” and “irreversible”
submatrices. Moreover, the test for eliminating non-minimal and duplicate
T
-invariants (flux modes) is
slightly different. For a more detailed comparison of the algorithms, see [Schuster
et al.
, 2002a].
The
T
-invariants are helpful in studying several properties of Petri nets, such as consistency. This
property means that there exists an initial marking and a corresponding firing sequence that regenerates
the initial state and contains each transition at least once. As can be seen in the system shown in Fig. 8
with either reactions 3 and 4 completely inhibited or reactions 5 and 6 completely inhibited, not every
metabolic system is consistent according to this definition. However, reactions 5 and 6 in the former case
and reactions 3 and 4 in the latter case are not covered by true
T
-invariants. If we consider only a subnet
that is covered by true
T
-invariants, such as transitions
t
1
and
t
2
, it is consistent. This is because, once the
minimal
T
-invariants (elementary flux modes) are identified, appropriate initial markings enabling these
invariants to operate can be linearly combined and a new initial marking is obtained. The system can fire
each above-mentioned
T
-invariant consecutively (the necessary resources exist due to the “construction”
of the initial marking), each transition is used at least once and the initial marking is always regenerated.
A further property studied for Petri nets, reversibility, means that for every marking
m
that can be
reached from
M
0
,
M
0
can also be reached from
M
. It holds for metabolic networks, if some constraints
are fulfilled. One constraint is that the network is covered by true
T
-invariants. The second constraint is
that all external metabolites have enough tokens to operate all true
T
-invariants. The arguments read as
follows: Let us denote the number of times the transitions ti have to fire in order to reach a marking
m
from
M
0
,by
w
i
. The numbers
w
i
are gathered in a vector
W
. Note that
W
need not fulfil Eq. (6). As
the net is covered by true
T
-invariants, we can find a vector
V
that does satisfy Eq. (6) and a sufficiently
large natural number such that
V
W
involves positive components only. This vector indicates how
many times the transitions need to be fired to reach the initial marking again.
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SIPHONS, TRAPS, DEADLOCKS AND LIVENESS
In Petri nets, special sets of places can be identified, for example, siphons, called also structural
deadlocks, and traps [Reisig, 1985]. A siphon is a set of places that - once it is unmarked - remains
