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Ta b l e 3 . 1 6
The MP mitotic oscillator with the minimum total number of monomials (
τ
=
173
·
10
−
3
min,
RMSE
≈
2
.
67
·
10
−
2
). Constants and initial values:
v
i
=
0
.
025,
k
1
=
0
.
0209,
k
2
=
0
.
0149329,
k
3
=
0
.
0351323,
k
4
=
0
.
0200062,
k
5
=
0
.
000662743,
k
6
=
0
.
215816,
k
7
=
0
.
0696881,
k
8
=
0
.
0911799,
k
9
=
0
.
166106,
k
10
=
0
.
569463,
k
11
=
0
.
00823672,
k
12
=
0
.
252676,
k
13
=
0
.
404647,
k
14
=
0
.
668527,
C
[
0
]=
M
[
0
]=
X
[
0
]=
0
.
01,
M
+
[
0
]=
X
+
[
0
]=
0
.
99.
r
1
:0
→
C
ϕ
1
=
v
i
r
2
:
C
→
0
ϕ
2
=
k
1
+
k
2
M
+
k
3
X
−
k
4
CM
r
3
:
M
+
→
M
ϕ
3
=
k
5
+
k
6
CM
r
4
:
M
→
M
+
ϕ
4
=
k
7
M
+
k
8
X
r
5
:
X
+
→
X
ϕ
5
=
k
9
C
+
k
10
M
r
6
:
X
→
X
+
ϕ
6
=
k
11
+
k
12
X
+
k
13
C
2
+
k
14
CM
Two MP sy stem s ar e
dynamically equivalent
when they have the same substances
and parameters, the same parameter evolution functions, the same scale factors
ν
,
μ
,
τ
, and, starting from the same initial state, provide the same dynamics.
Any MPR system is a special case of an MPF system, but we will show that, for
some MPF systems, equivalent MPR systems exist. Lemma 3.3 and Theorem 3.4
explain the equivalence between MP systems based on flux functions and MP sys-
tem based on reactivity and inertia parameters. In this lemma, and in the related
theorem, we consider the notion of
non-cooperative
MP rule. Table 3.17 considers
this and other properties of general interest for an MP rule
r
given by the multiset
rewriting
.
An MP system is
non-cooperative
if each rule
r
defined in the system is non-
cooperative:
α
→
β
S
−
(
|
r
)
|≤
1
.
Lemma 3.3.
For any MPF system there exists a non-cooperative MPF system which
is dynamically equivalent to it.
Proof.
If a rule, for example
r
:
a
+
c
→
b
, has more than one reactant, then we can
split it into two rules:
r
1
:
a
→
b
,
r
2
:
c
→
0 by requiring that the fluxes
ϕ
1
,
ϕ
2
of the
two rules are equal to the flux
ϕ
r
of
r
. We can proceed in this way for all the rules
which are cooperative. The non-cooperative MP system which we get in this manner
is dynamically equivalent to the original one.