Biology Reference
In-Depth Information
The amount of information shared between two variables can be quantified by
mutual information:
I
ð
X
;
Y
Þ
H
ð
X
Þ
H
ð
X
j
Y
Þ;
(8.2)
which measures how much the uncertainty on variable
X
is reduced by conditioning
it to another variable
Y
. Notice that because
H
ð
X
j
Y
Þ¼
H
ð
X
;
Y
Þ
H
ð
Y
Þ
, the mutual
information is a symmetrical measure between the two variables
X
and
Y
, i.e.,
I
Þ
In contrast with mutual information, TE is not a symmetrical measure since the
TE from
X
to
Y
is different from the one from
Y
to
X
. This asymmetry comes from the
fact that to compute the TE one has to define the past and the future on one of the
time series
ð
X
;
Y
Þ¼
I
ð
Y
;
X
ð
say
X
Þ
and only the past of the other variable
ð
say
Y
Þ
. The TE is then
defined as
X
future
X
past
X
future
X
past
Y
past
TE
ð
Y
!
X
Þ
H
ð
j
Þ
H
ð
j
;
Þ;
(8.3)
which allows an easy interpretation of TE. The TE from
Y
to
X
measures how much
the uncertainty of the
X
f
uture
is reduced by knowing the
Y
past
and comparing this
number with the situation in which only the
X
past
is reducing the uncertainty of
X
future
. If by adding
Y
past
the uncertainty of
X
future
is reduced more than adding only
X
past
, there is a nonzero TE from
Y
to
X
.
Another interpretation of TE comes from comparing (
8.2
) and (
8.3
); the
TE is the mutual information between
X
future
and
Y
past
conditioning on
Y
past
, i.e.,
TE
X
future
Y
past
X
past
.
A recent application of TE to irreversible enzymatic steps from yeast glycolysis
has shown that functional integration may emerge under certain experimental
conditions (De la Fuente and Cortes
2012
) and in the systemic metabolic structure
(De la Fuente et al.
2011
). Specifically, it was possible to quantify biomolecular
information flows in bits between catalytic elements and determine the emergence
of effective connectivity structures. These effective functional structures are
involved both in the integration of catalytic processes belonging to a single meta-
bolic subsystem and in the functional coordination between different metabolic
subsystems.
The following section addresses these studies related with irreversible enzymes
from glycolysis (Sect.
8.2
) and in dissipative metabolic networks (Sect.
8.3
).
ð
Y
!
X
Þ¼
I
ð
;
j
Þ
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