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Proof.
If the system is closed, then it is matter-bounded. From the cycle dissipation
principle no reaction cycles are present in the system. Therefore, after a number of
steps reactants disappear. Therefore, reactions in the system cannot continue to be
applied.
Lemma 3.6.
Any open but non-dispersive MP system which is matter-bounded can-
not be time-unbounded.
Lemma 3.7.
Any open, but non-assimilative MP system which is matter-bounded
cannot be time-unbounded.
In the context of MP systems we say that a system oscillates if its substance quanti-
ties vary in time, but their values are always within some fixed ranges.
Proposition 3.8.
An MP system having a reaction cycle and which is matter-
bounded but time-unbounded, has to be assimilative, dissipative, and oscillating.
Proof.
(
outline
) An MP system with a reaction cycle needs to feed some substance
from outside, therefore it is assimilative. But if it is also matter-bounded and time-
unbounded, then it has to be dissipative. These requirements imply that surely its
substance quantities have to vary within some intervals, therefore the system is
oscillating.
Cells host complex metabolic processes and are bounded in matter and unbounded
in time (when considered with their genealogical lineages). In fact, their sizes can-
not be greater than some values and their reactions need to last as long as possible,
because reproduction provides the reiteration of their internal metabolic processes
in time. Of course, they are special conservative metabolic systems. This means
that what was abstractly deduced for conservative MP systems definitely applies to
cells, which need to realize oscillatory phenomena. In conclusion,
life is oscillation
and the analysis of metabolic oscillations is crucial to understanding life. In this
regard, it is interesting to quote the words of the Nobel laureate Ilia Prigogine in
his foreword to Goldbeter's topic [82] (the oscillatory phenomenon to which Pri-
gogine refers is the Belousov-Zhabotinsky phenomenon, which we considered in
Sect. 3.1.2, about time asymmetry see also [147], Sect. 7 of Chap. 4, and Sect. 8 of
Chap. 5):
I remember my astonishment when I was shown for the first time a chemical oscillatory
reaction, more than 20 years ago. I still think that this astonishment was justified, as the
existence of chemical oscillations illustrates a quite unexpected behavior. We are used
to thinking of molecules as traveling in a disordered way through space and colliding
with each other according to the laws of chance. It is clear, however, that this molecular
disorder may not give rise by itself to supramolecular coherent phenomena in which
millions and millions of molecules are correlated over macroscopic dimensions. ...It
is this synchronization that breaks temporal symmetry.
