Biology Reference
In-Depth Information
Table 1, continued
T
sync
,
Inequality
Equation (6)
T
sync
,
T
sink
Inequality
Equation (6)
T
sink
(2), (3), (5) (2), (3), (5)
t
54
f
53
f
54
f
110
*2
f
116
t
57
f
55
f
56
f
114
f
116
f
55
f
57
f
115
f
116
t
118
f
114
f
117
f
114
f
118
T
sync
=
{t
3
,
t
5
,
t
8
,
t
9
,
t
11
,
t
13
,
t
17
,
t
24
,
t
25
,
t
30
,
t
50
,
t
72
,
t
75
,
t
82
,
t
86
,
t
116
}
,
T
sink
=
{ t
20
,
t
28
,
t
34
,
t
35
,
t
36
,
t
40
,
t
47
,
t
52
,
t
54
,
t
57
,
t
62
,
t
63
,
t
88
,
t
91
,
t
96
,
t
97
,
t
98
,
t
99
,
t
100
,
t
103
,
t
107
,
t
108
,
t
109
,
t
118
}
,
T
sour
=
{t
1
,
t
2
,
t
4
,
t
6
,
t
7
,
t
10
,
t
12
,
t
16
,
t
21
,
t
22
,
t
23
,
t
29
,
t
37
,
t
41
,
t
49
,
t
53
,
t
55
,
t
58
,
t
64
,
t
69
,
t
74
,
t
77
,
t
85
,
t
89
,
t
92
,
t
101
,
t
104
,
t
110
,
t
114
,
t
115
}
.
stack
. Steps
4
◦
-
7
◦
are the
executed. When
p
is in a conflict state, it will be labeled and pushed to
procedures for the input transitions
t
of
p
. If the condition
t
T
e
holds,
dfs-push
(
t
,
stack
,
PN
) will stop and the other sub-routine
dfs-pop
(
stack
,
PN
) will be invoked; otherwise,
dfs-push
(
t
,
stack
,
PN
) will be invoked recursively.
The sub-routine
dfs-push
(
t
,
stack
,
PN
) is a function used to produce conditional expression with the
use of places saved in
stack
based on expressions (2), (3) and (5). In
3
◦
, if the input transitions of
p
are
marked,
p
will be pulled out from stack. In the case that
p
is a conflict place,
EQ2
(
p
) is used to calculate
conditional expression (in
4
◦
); in the case that
p
is a normal place,
EQ3
(
p
) is used (in
5
◦
).
∈
T
sour
∪
T
sync
∪
Application of proposed algorithm to IL-1 signaling pathway
As a case study, we applied our algorithm to a IL-1 signaling pathway model. Here, we show the
results by applying our algorithm to automatically produce delay times of all the transitions to the IL-1
signaling pathway model shown in Fig. 7. In the IL-1 signaling pathway, it can be found that there
exist several self-loops. We thus break down the IL-1 signaling pathway at the places of self-loops, and
derive all the delay times of the transitions by applying the algorithm
<<
Firing Time Determination
>>
.
Calculated delay times of the transitions are shown in Table 1, which are given in the form of conditional
expression of firing frequency. The simulation can be executed without token-retention by using the
delay time decided obeying the conditional expressions listed in Table 1. If the simulation is conducted
in sufficient time period, it can be observed that firings of the transitions follow the given stochastic
values.
CONCLUSIONS
We have proposed a method of determining the delay time of transitions satisfying derived conditional
expressions for a class of Petri nets, which is acyclic and of no inhibitory arcs. To resolve nondeterministic
firings, we have introduced a stochastic approach to determine firings of transitions in conflict.
In this contribution, we have first presented basic definitions of Petri net and introduced a Petri net
based modeling method for signaling pathways by taking notice on the molecular interactions and
mechanisms. Then we have presented conditional expressions for two cases, conflict-free transitions
and conflict transitions, which express the conditions for smooth token flows in a timed Petri net, i.e.,
smooth signal transductions in a signaling pathway. Based on these expressions, we have proposed an
algorithm to determine the delay times of transitions in a timed Petri net model of a signaling pathway.
