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3.7
Metabolic Oscillators
Metabolic oscillators are ubiquitous in life phenomena, and life in itself is an oscil-
latory phenomenon reproducing and propagating in space. Many of the metabolic
patterns we analyzed in the previous section provide, in a simple manner, oscilla-
tory behaviors. In this section we present some examples of basic metabolic phe-
nomena. In almost all cases the possibility of dynamical curves exhibiting values
which range in a given interval, without diverging and without terminating, is corre-
lated to the structure of the MP graphs involving cycles of reactions and regulations.
The oscillators which we present are expressed as MPR (systems with reactivity
maps). In this formulation they assume a simpler form and are also easier to be dis-
covered. For any oscillator, its MPR graph (where reaction maps are indicated) is
given together with its dynamics (where inertias and initial values are indicated).
For the sake of simplicity, we maintain the terminology of the previous section
(cis-regulation, trans-regulation, and so on), even if regulators are replaced by re-
action maps, therefore their meanings are somewhat different from the regulation
mechanisms analyzed in the previous section. Dynamics are computed by means of
software simulating MPF systems (Psim, specifically developed in years 2007-2009
[67, 69, 71]). In many cases, oscillations are very robust in dependence of the initial
values and of the algebraic form of regulations, while in other cases systems prove
to be very sensitive and only small changes in the third or fourth decimal digit alters
dramatically the dynamical pattern. Figures 3.19, 3.20, 3.21, and 3.22 provide some
first examples of interesting metabolic oscillators with their dynamics.
The names of many oscillators derive from names of stars. This fact is due to
many reasons. Namely, stars were the first objects of systematical dynamical inves-
tigation (and mathematical formalization). Moreover, the dynamical system Sirius,
the first defined by an MP grammar, exhibits a behavior resembling the double star
Sirius, where the movement of one star is hidden by the other, because in its origi-
nal formulation [92], one substance is transformed in the other two, but the curve of
values (along time) of one substance is quite almost completely hidden by the curve
of the other substance (see Fig. 3.3).
Fig. 3.19
Bicatalyticus
, a very simple MPF oscillator
The metabolic oscillator of Fig. 3.23 is based on a transformation cycle. Its initial
behavior follows a very irregular pattern, however, on a long run, it seems to reach
a regular dynamical pattern. On the contrary, the metabolic oscillator of Fig. 3.24
