Biology Reference
In-Depth Information
Fig. 3. Simple example of capacity limitation in a metabolic system.
columns are transposed. The entries of these matrices have a nonzero value (equal to the weight of the
arc), if an arc exists, and zero otherwise.
Further, the topological structure of a Petri net can be represented by an integer matrix,
C
, called an
incidence or flow matrix.
C
is an
nm
matrix whose
m
columns correspond to the transitions and
n
rows
correspond to the places of the net. The following relation holds true:
C
=
post
T
-
pre
. The mappings
pre
and
post
can be reconstructed from the matrix
C
in the following simple way: post(
t
j
,
p
i
)
= max
{
C
ij
,0
}
,
pre(
p
i
,
t
j
)
= min
{
C
ij
,0
}
.
It is worth finding whether another state can be reached from a given state. This is related with the
property of reachability. In metabolic networks, we can search all possible subsequent states, knowing
the initial state of resources. Another interesting problem is to deduce all appropriate initial states from
a desired later state.
Places can hold an arbitrary number of tokens or can be restricted by a given number - capacitated
places. In Fig. 3, the unnamed places are considered external (with inexhaustible numbers of tokens).
Transitions
T
1
and
T
2
are activated if there is at least one token in the place ATP (the currency of
metabolic energy), respectively ADP. If the initial marking is [c, c, c, c, 1, 0], transition
T
1
can fire and
produce the marking [c, c, c, c, 0, 1]. Now, transition
T
2
is enabled. It fires and we obtain again the initial
marking. In this example, ATP
+
ADP
=
1, independent of the system state. This is a conservation
relation, which leads to boundedness of the capacity of all the internal places. Usually the places of
biological systems are not considered to be limited because the limitation due to the finite size of living
cells is not critical to most biochemical processes. There are only cases where the limitation comes from
a conservation relation such as in the above case.
Another situation important in biology is the presence of inhibitors. The corresponding Petri net model
can be extended by a special element, called inhibitory arc (Fig. 1i). The inhibitor is represented by a
place. If there is a token at that place, the transition is not enabled, so it does not fire.
Note that the incidence matrix corresponds to the stoichiometric matrix [ Erdi and T oth, 1989; Heinrich
and Schuster, 1996] for metabolic networks if the Petri nets are pure. That means that the networks
do not involve self-loops (Fig. 4), because self-loops cannot be represented in the incidence matrix: a
coefficient of 1 and a coefficient of
−
1 cancel each other to yield zero in the matrix, thus losing track
of the self-loop. Thus, we should identify the situations that produce self-loops and the way to treat
them without losing the biological meaning. One such situation occurs when enzymes are considered
as normal substrates. There are algorithms for deleting/eliminating self-loops: First, one can treat the
