Biology Reference
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Fig. 1. Basically, Petri nets are constructed using three kinds of symbols for places, transitions, and arcs. In Cell Illustrator, both
sets of places and transitions are classified into discrete and continuous types, and places and transitions can be replaced with
pictures reflecting the biological images. This replacement makes the HFPN model of a biological pathway more comprehensible
for biologists.
HFPN was defined by Matsuno
et al.
[Matsuno
et al.
, 2003b] as an extension of a hybrid Petri net
[Alla and David, 1998]. HFPN has two kinds of places, namely, discrete and continuous (depicted as a
double circle) and two kinds of transitions, discrete and continuous (depicted as an unfilled rectangle).
The concepts of discrete place and discrete transition are the same as those in the traditional discrete
Petri net.
1
A continuous place can hold a real number as its content. A continuous transition fires
continuously at the speed of the parameter assigned to the continuous transition. The traditional symbols
of these places and transitions are shown in Fig. 1.
Three types of arcs are used in HFPN. A specific value is assigned to each arc as a weight. When a
normal arc (a solid arc in Fig. 1) with weight
w
is attached to a discrete/continuous transition, a certain
number of tokens are transferred through the normal arc only if the content of the place at the source of
the normal arc exceeds the weight
w
. The firing rule of a test arc is the same as that of a normal arc in
terms of the weight, but the content of the place at the source of the test arc is not consumed by firing. A
test arc (a dashed line arc in Fig. 1) can be used to represent enzyme activity since the enzyme itself is
not consumed. An inhibitory arc (a line terminated with the small bar in Fig. 1) with weight
w
enables
the transition to fire only if the content of the place at the source of the arc is less than or equal to
w
.For
example, an inhibitory arc can be used to represent repressive activity in gene regulation (inhibitory arcs
are not used in this paper).
HFPN model construction based on the literature
Figure 2 shows an HFPN model which has been constructed by compiling and interpreting the
information of p53-MDM2 interactions in the literature [Barak
et al.
, 1993; Miyashita and Reed, 1995;
Honda
et al.
, 1997; Kamijo
et al.
, 1998; Pomerantz
et al.
, 1998; Zhang
et al.
, 1998; Tao and Levine,
1999; Zhang and Xiong, 1999]. We changed the symbols of “place” and “transition” to biological
images. Although these changes have no effect on mathematical meaning, it is helpful for biologists to
1
A discrete place and a discrete transition are represented by symbols of a single circle and a filled rectangle, respectively.
These symbols are not used in the HFPN of Fig. 2.
