Biology Reference
In-Depth Information
Fig. 2. Model for reaction decomposing dimers to monomers. (a) Reaction decomposing dimers to monomers. (b) Hybrid Petri
net. In continuous places
P
d
and
P
m
, concentrations of the dimer and the monomer are stored, respectively. At continuous
transitions
T
1
and
T
2
, same firing speeds are assigned as a reaction speed. Integers “1” represent weights of arcs. (c)
hybrid
functional Petri net
model for the reaction. Note that, at the transition T, the reaction speed is assigned.
P
d
and
P
m
are the
same as ones in (b). Different weights “1” and “2” are assigned to two arcs.
From the definition of HPN/HDN, the firing speed of a continuous place must be the same as the
consuming speed through each arc from its source place and the contents of all source places are
consumed with the same speed. This speed is also the same as the production speed through each
arc from the transition. This is the unfavorable feature of HPN/HDN for biopathway simulation. For
example, consider a reaction in which a dimmer is cleaved to two monomers (Fig. 2(a)). This reaction
in the HDN model could be represented as shown in Fig. 2 (b) by using a test arc and a transition for
amplification (note that the amounts consumed and produced in places by continuous transition firing
are the same by definition while the amount of monomers is twice as large as that of dimers). But it is
neither intuitive nor natural at all. It may be obvious that this feature of HPN/HDN is a severe drawback
in modeling biopathways.
On the other hand, some favorable features have been also introduced in Petri net theory. In addition
to normal arc explained so far,
inhibitory arc
and
test arc
have been defined for convenience (Fig. 1). An
inhibitory arc with weight
r
enables the transition to fire only if the content of the place at the source of
the arc is less than or equal to
r
. For example, an inhibitory arc can be used to represent the function of
“repress” in gene regulation. A test arc does not consume any content of the place at the source of the
arc by firing. For example, test arcs can be used to represent the transcription process since nothing is
consumed by this process except for degradation.
Definition 1
A
hybrid functional Petri net
(HFPN) is defined by extending the notion of transition of
HPN/HDN [5,7,8] in the following way: HFPN has five kinds of arcs;
discrete input arc
,
continuous
input arc
,
test input arc
,
discrete output arc
, and
continuous output arc
. A discrete input arc (continuous
input arc) is directed to a discrete transition (continuous transition) from a discrete/continuous place
(continuous place) from which it consumes the content of the source place by firing. A test input arc
is directed from a place of any kind to a transition of any kind. It does not consume the content of the
source place. These three arcs are called
input arcs
. A discrete output arc is directed from a discrete
transition to a place of any kind. A continuous output arc is directed from a continuous transition to a
continuous place. These two arcs are called
output arcs
.
1.
Continuous transition
:A
continuous transition
T
of HFPN consists of continuous/test input arcs
a
1
,
...
,
a
p
from places
P
1
,
...
,
P
p
to
T
and continuous output arcs
b
1
,
...
,
b
q
from
T
to continuous
places
Q
1
,
...
,
Q
q
. Let
m
1
(
t
)
,
...
,
m
p
(
t
)
and
n
1
(
t
)
,
...
,
n
q
(
t
)
be the contents of
P
1
,
...
,
P
p
and
Q
1
,
...
,
Q
q
at time
t
, respectively. The continuous transition
T
specifies the following:
a. The
firing condition
is given by a predicate
c
(
m
1
(
t
)
,
...
,
m
p
(
t
))
. As long as this condition is
true,
T
fires continuously.
