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The stoichiometric expansion method allows us to determine the best coefficients
which approximate regulators, as combinations of some specified regulators. An-
other two crucial aspects of LGSS algorithm concern i) the determination of which
regressors have to be considered for each flux regulation function, ii) how to choose
those which provide the best approximations. We address these aspects by integrat-
ing the stoichiometric expansion with two mechanisms: i) a way for scoring the re-
gressors of an initial dictionary of functions, by using the Log-gain principle, and ii)
a way for choosing the best regressors for each flux regulator, by using an extension
of the classical stepwise regression method, based on the least square approxima-
tion and on the
F
statistical test. In the following subsections we develop the details
necessary for a complete description of the LGSS algorithm.
3.4.1
Log-Gain Principle
In the previous section, we have explained how stoichiometric expansion approx-
imates the coefficients of regressors which provide the regulators of a given MP
grammar when regressors are given. In the following subsections we will focus on
the problem of finding which regressors provide the best approximation of regula-
tors, among a set of
d
basic functions
g
1
,
g
d
depending on substance quanti-
ties and parameters. Here we introduce a principle, called
Log-gain principle
,which
is a special case of a general criterion usually satisfied by the variations of quantities
involved in biological phenomena.
Given a time series
v
t
g
2
,...,
=(
v
[
i
]
|
0
≤
i
≤
t
)
,
of non-null real values its discrete
log-gain vector is defined by:
v
t
[
i
+
1
]
−
v
[
i
]
v
t
Lg
(
)=(
Lg
(
v
[
i
])
|
0
≤
i
≤
t
)=
|
0
≤
i
<
.
(3.31)
v
[
i
]
The log-gain principle requires that the flux vector
u
l
[
associated to a
reaction
r
l
has to be a linear combination of log-gains of the time series of tuners of
ϕ
l
. Namely, let
s
t
i
]=
ϕ
l
(
s
[
i
])
=(
s
[
i
]
|
0
≤
i
≤
t
)
be the time series of states along some uniformly
distributed observation points 0
≤
i
≤
t
, with a corresponding time series of fluxes
given by the regulator
ϕ
l
:
t
l
ϕ
=(
ϕ
l
(
s
[
i
])
|
0
≤
i
≤
t
)
.
For any substance or parameter
y
, if the corresponding time series of values along
the state time series
s
t
is:
y
t
=(
y
[
i
]
|
0
≤
i
≤
t
)
,
t
then the log-gain principle is satisfied by
l
when the following equation holds for
suitable real coefficients
q
j
,where
T
l
is the set of tuners of
ϕ
ϕ
l
:
t
l
)=
y
j
∈
T
l
q
j
Lg
(
y
t
j
)
.
Lg
(
ϕ
