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Let us consider
conservative
MP systems, which are close to the real metabolic
system, because they satisfy two essential requirements of metabolic phenomena:
matter conservation
and
energy orientation
of metabolic transformations. Let us
give some notation and definitions for expressing these requirements.
Given a reaction
α
→
β
α
,
β
are not empty, the matter conser-
vation principle requires that the whole matter of reactants has to equate the whole
matter of products, that is, neither loss nor gain of matter can occur in transforma-
tion reactions. This implies that in such a system a gain of matter can be realized
only with input rules of kind 0
where multisets
→
x
, and a loss of matter can occur only with out-
put rules of kind
x
→
0. In symbols, according to notation of MP systems, we have
(
μ
(
x
)
as the mass of a mole of
x
):
x
∈
α
μ
(
x
)=
y
∈
β
μ
(
y
)
.
A
reaction chain
of substances
x
1
,
x
2
,...,
x
n
is given by reactions
r
1
,
r
2
,...,
r
n
−
1
S
−
(
S
+
(
such that
x
1
∈
r
1
)
(
x
1
is a reactant of
r
1
)
x
n
∈
r
n
−
1
)
(
x
n
is a product of
r
n
−
1
)
S
+
(
S
−
(
and, for 1
<
i
<
n
,
x
i
∈
r
i
−
1
)
and
x
i
∈
r
i
)
.A
reaction cycle
is a reaction chain
where
x
n
=
x
1
.
An MP system is
time-bounded
if there is a bound to the maximum number
of times its reactions can be applied. This means that, after a number of steps, its
reactions do not have the amounts of substances necessary to them. A system is
time-unbounded
if it is not time-bounded. The system is
matter-bounded
if all
the matter available in the system is not greater than a given quantity. A system
is
matter-unbounded
if it is not matter-bounded. Of course, no physical system
can last an infinite time, or can have an infinite quantity of matter, therefore, when
speaking of time or matter unboundedness, we have to think of the term
unbounded
as equivalent to having
extreme
values (which we avoid to fix now) of duration or
size (in relation to a certain time and matter availability, respectively).
An MP system is
assimilative
when there is at least one input rule. It is
dis-
persive
if there is at least one output rule. It is
dissipative
if it is simultaneously
assimilative
and
dispersive
, and it is
open
if it is either
assimilative
or
dispersive
,
while it is
closed
if it is not open.
A simple way for providing oscillations in a closed MP system is given by only
two substances
A
A
with flux maps
A
and
B
respectively. However, no real chemical or biochemical system of this kind
can exist. This impossibility is connected to a crucial aspect of chemical transfor-
mations which have to obey a principle based on Gibbs' variation of free energy.
In simpler terms, the natural orientation of a chemical transformation is from reac-
tants that are globally
energetically richer
to correspondent products
energetically
poorer
. This verse can be inverted only by providing an additional energy, for ex-
ample, by coupling the reaction with another one releasing the necessary energy for
this inversion.
,
B
and two reactions:
r
1
:
A
→
B
and
r
2
:
B
→
