Biology Reference
In-Depth Information
Fig. 1. A Petri net representing a chemical reaction of the form
n
A
+
m
B
+
k
C
→ y
D
+
x
E. Circles and squares represent
places and transitions, respectively.
static aspects of the system. We still need a mean to express the dynamical properties of the system. For
that purpose we introduce the notion of
network marking
. A marking is a function that defines the state
of the system, that is the quantity of each of the reactants. The initial marking
M
0
defines the initial
state, and that state undergoes changes as the interactions take place.
Definition 2.
(Network marking):
For a given network
N
, a function
M
:
S
→
N
will be called the
marking of the network.
The definitions that follow, formally describe the mechanics of the network. In a given state, available
transitions fall into two separate categories, active and inactive. Only an active transition can be fired.
Firing of a transition results in change of current state. For ease of expressing that change we introduce
the notion of
transition function
.
Definition 3.
For a given network
N
, and an element
e
∈
T
∪
S
the following two sets are introduced:
1.
•
e
=
{
x
:(
x, e
)
∈
F
}
-
set of predecessors.
2.
e
•
=
{
x
:(
e, x
)
∈
F
}
-
set of successors.
Definition 4.
(Active transition):
For a given network
N
and its marking,
M
a transition
t
∈
T
is
considered
active
if the following condition is met:
∀
s
∈•
tW
(
s, t
)
M
(
s
)
,
denoted by
M
[
t>
.
Definition 5.
(Transition function):
For every transition
t
one can define a corresponding transition
function
t
:
S
→
N
as follows:
⎧
⎨
⎩
−
W
(
s, t
)
s
∈•
t
\
t
•
W
(
t, s
)
s
∈
t
•\•
t
t
(
s
)=
W
(
t, s
)
−
W
(
s, t
)
s
∈•
t
∩
t
•
0
s
∈•
t
∪
t
•
