Biology Reference
In-Depth Information
Definition 3
: Let
PN
be a Petri net. A timed Petri net
TPN
is defined by
TPN
=
(
PN
,
D
), where
D
is a
set of firing delay times of each transition in
T
.
The firing rule of a timed Petri net
TPN
is defined as follows: (i) If the firing of a transition
t
i
is
decided, tokens required for the firing are reserved. We call these tokens
reserved tokens
. (ii) When
the delay time
d
i
of
t
i
passed,
t
i
fires to remove the reserved tokens from the input places of
t
i
and put
non-reserved tokens into the output places of
t
i
. In a timed Petri net, firing times of a transition
t
i
per
time unit is called
firing frequency
f
i
.
f
i
represents the maximum firing frequency of
t
i
. The delay time
d
i
of
t
i
is given as the reciprocal of
f
i
.
Retention-free Petri net
Signaling pathways are composed of consecutive signaling events that are mediated by intracellular
signaling proteins (usually enzymes) that relay the signal into the cell by activating the next enzyme from
inactive state to active state on receipt of up-stream signal. With this feature, it can be considered that the
signaling pathway will be in an abnormal state if the accumulation of the substances occurs per time unit.
In Petri nets, such accumulation of a substance is represented by the token retention of the corresponding
place. The token number of the place grows infinitely with the firing of its input transitions, and the lack
of firing by the output transition. In this paper, such a Petri net in which signal flows can be steadily and
smoothly propagated without the token retention at any place is called a
retention-free
Petri net.
Retention-free
Petri nets are a subclass of timed Petri nets, where the total amount of in-flowing tokens
(
i
=1
K
I
i
) is not larger than the possible maximum number of out-flowing tokens (
j
=1
K
O
j
) for each
place
p
per time unit, which is represented by following inequality:
m
n
K
I
i
K
O
j
(1)
i
=1
j
=1
MODELING OF SIGNALING PATHWAYS
We give here a modeling method with a series of modeling rules for signaling pathways based
on Petri net representation. As an example, the interleukin-1 (IL-1) signaling pathway is used to
demonstrate our modeling method. Petri net based modeling of signaling pathways provides us with an
intuitive understanding of the intrinsic structure and features of signaling pathways, and further enables
computational experiments on the constructed Petri net model as being demonstrated in the following
sections.
Modeling rules
The structural characteristics of signaling pathways can be naturally and explicitly expressed by Petri
nets according to following rules.
1. Places denote static elements including chemical compounds, conditions, states, substances and
cellular organelles participating in the biological pathways. Tokens indicate the presence of these
elements. The number of tokens is given to represent the amount of chemical substances.
2. Transitions denote active elements including chemical reactions, events, actions, conversions and
catalyzed reactions.
