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In-Depth Information
An MP model assumes that some species are related by means of some trans-
formations, and that the variations are due to the global action of these transfor-
mations. Of course, their execution could involve very complex underlying sub-
transformations, but this is outside the objective of the model. It only tries to ex-
plain what is observed in terms of the chosen species and transformations. If the
choices are not the right one, this means that the model was not adequately defined,
but this is independent from the methodology, it is only a matter of modeling de-
sign. Therefore, MP modeling is deliberately at a different, more abstract, level with
respect to ODE models. This does not means that it is less adherent to the reality,
but simply that it is focused on a different level of reality. Of course, in this way
many important dynamical details can be lost, but in many cases what is relevant
is the global pattern of a behavior and the main correlations among the involved
quantities. Moreover, in many cases, a higher level of analysis is the only possible
approach, because we know only some time series related to a given phenomenon,
and no idea is available about the internal laws ruling the observed phenomenon.
This is the case that we consider above, concerning a situation of gene expressions
network, where thousands of variables are involved.
3.8.1
Metabolic Computing
We have already shown (see Sect. 3.3) that MP systems can be used for approxi-
mating real functions. But they can also generate time series which correspond to
exact values of real functions along sequences of values. In the following, we give
some examples of MP grammars computing (or generating) the values of linear (Ta-
ble 3.22), square (Table 3.23), root square (Table 3.25), and exponential functions
(Fig. 5.13). It is remarkable that in all these cases fluxes are computed by linear
regulators.
In these computations, we assume a criterion of
lack of reactant block
,thatis,
the flux of a rule becomes null when the regulator assumes a value greater than the
current quantity of one of its reactants (there is an insufficient amount of matter for
applying the rule). In this case, also all the rules which share that reactant are forced
to have a null flux.
Elaborating on the idea of fluxes that become null when they exceed the availabil-
ity of reactants, a computation model on natural numbers can be developed which
is computationally universal (equivalent to register machines [99]).
The square root of positive integers is easily computed by the algorithm given in
Table 3.24, where the input value
n
>
0 is decreased at any step
i
by odd number
2
i
1, thus by reducing the initial value of increasing squares. This reduction is
continued while the reduced value remains positive (or null) and, at same time, a
counter (
B
) memorizes the number of steps. When the reduction process stops, the
counter provides the approximate value of the square root of
n
. This algorithm is
expressed by the MP grammar of Table 3.25. In fact, the number
n
is put as value
of
A
, and at any step
A
is decreased by increasing odd numbers, according to rules
+
