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In-Depth Information
Ta b l e 3 . 1 7
Types of an MP rule
r
of form α
→
β
monic
|
α
|
=
1
|
S
−
(
r
)
|
=
1
monogenic
|
S
−
(
r
)
|≤
1
non-cooperative
|
S
−
(
r
)
| >
1
cooperative
|
S
−
(
r
)
| > |
S
+
(
r
)
|
synthetic
|
S
−
(
r
)
|
=
|
S
+
(
r
)
|
transformative
|
S
−
(
r
)
| < |
S
+
(
r
)
|
dissociative
assimilative
α
=
0
dispersive
β
=
0
S
−
(
r
)
∩
S
+
(
r
)
=
0
catalytic
An MP system is
positive
if, in any state
s
, reaction fluxes do not consume more
matter than the amount available (for any reaction
r
and any substance
x
):
∑
(
)
≤
ϕ
s
x
r
R
−
(
r
∈
x
)
Theorem 3.4.
For any positive MPF system there exists a dynamically equivalent
MPR system.
Proof.
According to Lemma 3.3, we can start by considering a non-cooperative
MPF system
M
which will result
M
.Nowwetransform
M
into an MPR system
M
is given by the same substances, rules,
dynamically equivalent to
M
. The system
parameters,
ν
,
μ
,
τ
of
M
.Moreover,foranyflux
ϕ
r
of
M
, we define the correspond-
M
as
ing reaction map
f
r
of
r
−
(
)=
|
x
)
|
(
·
ϕ
(
)
f
r
s
s
r
x
r
−
(
gives the multiplicity of the reactant
x
of
r
(when
S
−
(
where
|
x
)
|
r
)=
0, we
consider
|
r
−
(
x
)
|
x
=
1). Finally, for each substance
x
, we define its inertia function
ψ
x
as
∑
ψ
x
(
s
)=
1
−
f
r
(
s
)
.
R
−
(
r
∈
x
)
M
equipped with the reaction maps and with the inertia functions
defined above is dynamically equivalent to the MPF system
The MPR system
M
we started from. In
M
, the reactance
p
r
,
x
(
)
fact, for any rule
r
of
s
is given by:
r
−
(
x
/|
x
)
|
p
r
,
x
(
s
)=
ψ
x
(
s
)+
∑
f
r
(
s
)
r
∈
R
−
(
x
)
