Biology Reference
In-Depth Information
system of “discrete” and “continuous” dynamics [Matsuno
et al.,
2000] by employing a hybrid Petri net
(HPN) architecture [Alla and David, 1998; Drath, 1998]. It has also been observed [Ghosh and Tomlin,
2001] that biological pathways can be handled as hybrid systems. For example, protein concentration
dynamics, which behave continuously, being coupled with discrete switches. Another example is protein
production that is switched on or off depending on the expression of other genes, i.e. the presence or
absence of other proteins in sufficient concentrations.
Recently, by extending the notion of HPN, Matsuno
et al.,
2003b, introduced the hybrid functional
Petri net (HFPN) in order to give a more intuitive and natural modeling method for biological pathways
than the existing Petri nets. Although the paper demonstrates the effectiveness of an HFPN with two
examples, gene regulation mechanism for circadian rhythms and apoptosis signaling pathway, it only
gives the constructed HFPN models for these examples. The purpose of this paper is to demonstrate
how biological pathways can be created with an HFPN describing step-by-step the process to construct
a model of the
lac
operon gene regulatory mechanism and glycolytic pathway. The constructed HFPN
model is verified by simulations of five mutants of
lac
operon on the Genomic Object Net (GON) software
package (
http://www.GenomicObject.Net/
)
, [Matsuno
et al.,
2003b]. This software was developed based
on the notion of the HFPN together with a GUI specially designed for biological pathway modeling.
MODELING BIOLOGICAL PATHWAY WITH HYBRID FUNCTIONAL PETRI NET
In this chapter, first we give a brief definition of the hybrid functional Petri net and give a summary
for the modeling method of biological pathways with the hybrid Petri net.
Hybrid functional Petri net: An extended hybrid Petri net for modeling biological reactions
Petri net is a network consisting of place, transition, arc, and token. A place can hold tokens as its
content. A transition has arcs coming from places and arcs going out from the transition to some places.
A transition with these arcs defines a firing rule in terms of the contents of the places where the arcs are
attached.
Hybrid Petri net (HPN) [Alla and David, 1998] has two kinds of places
discrete place
and
continuous
place
and two kinds of transitions
discrete transition
and
continuous transition
. A discrete place and a
discrete transition are the same notions as used in the traditional discrete Petri net [Reddy
et al.,
1993].
A continuous place can hold a nonnegative real number as its content. A continuous transition fires
continuously at the speed of a parameter assigned at the continuous transition. The graphical notations
of a discrete transition, a discrete place, a continuous transition, and a continuous place are shown in
Fig. 1, together with three types of arcs. A specific value is assigned to each arc as a weight. When a
normal arc is attached to a discrete/continuous transition,
w
tokens are transferred through the normal
arc, in either of normal arcs coming from places or going out to places. An
inhibitory arc
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. A
test arc does not consume any content of the place at the source of the arc by firing. For example, a test
arc can be used to represent enzyme activity, since the enzyme itself is not consumed.
Hybrid dynamic net
(HDN) [Drath, 1998] has a similar structure to the HPN, using the same kinds
of places and transitions as the HPN. The main difference between HPN and HDN is the firing rule
of continuous transition. As we can know from the description about HPN above, for a continuous
transition of HPN, the different amounts of tokens can be flowed through the two types of arcs, coming
