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Fig. 3.2
The MP graph of MP grammar of Table 3.5
3. by the set of Eqs. (3.5) of Definition 3.1;
4. by the vector equation
EMA
(see 3.10).
The MP grammar given in Table 3.5 has the dynamics given in Fig. 3.3. It defines a
synthetic oscillator, very often considered in the MP theory, called Sirius (see [92]).
The oscillator is made of three substances
A
,
B
,and
C
and five reactions.
Ta b l e 3 . 5
The MP grammar of Sirius oscillator, corresponding to the MP graph of Fig. 3.2.
Its dynamics with initial values
A
[
0
]=
100,
B
[
0
]=
100,
C
[
0
]=
0
.
02, is given in Fig. 3.3.
→
A
ϕ
1
=
.
+
.
r
1
:0
0
047
0
087
A
r
2
:
A
→
B
ϕ
2
=
0
.
002
A
+
0
.
0002
AC
r
3
:
A
→
C
ϕ
3
=
0
.
002
A
+
0
.
0002
AB
r
4
:
B
→
0
ϕ
4
=
0
.
4
B
r
5
:
C
→
0
ϕ
5
=
0
.
4
C
Ta b l e 3 . 6
The stoichiometric matrix of MP grammar of Table 3.5
⎛
⎝
⎞
⎠
1
−
1
−
100
010
−
10
0010
−
1
A =
Table 3.7 reports a useful notation concerning MP rules which reveals an intrin-
sic duality between substances and reactions of an MP grammar. This notation can
be easily extended to a set of substances and to a set of reactions. The
closure prop-
erty
of a reaction graph of substances
X
and reactions
Y
can be expressed by the
condition
R
o
S
o
Y
and
S
o
R
o
(
(
Y
)) =
(
(
X
)) =
X
.
