Biomedical Engineering Reference
In-Depth Information
.˘
1
O
and the corresponding logarithm matrix function log
.t //
is:
A
C
GT
0
@
1
A
0:0524 0:0042
0:0466
0:0016
A
C
G
T:
0:0042 0:1008 0:0002
0:0963
˘
1
O
.t / D
0:1078
0:0006 0:1091 0:0007
0:00019
0:1154
0:0004 0:1176
˘
.˘
1
O
.t //
The product
log
is:
A
C
GT
0
1
0:0159 0:0013
0:0142
0:0005
A
C
G
T
@
A
0:0013 0:0310 0:0001
0:0297
.˘
1
O
˘
log
.t // D
0:0142
0:0001 0:0143 0:0001
0:0005
0:0293
0:0001 0:0302
.˘
1
whose negative trace provides the evolutionary distance
d DtrŒ˘
log
O
.
The reader interested in more sophisticated applications of the GTR distance will
findusefulexamplesinLanaveetal.[
48
], Rodriguez et al. [
63
], and Cantanzaro
et al. [
10
,
11
].
.t // D 0:09152
8.3
The Least-Squares Paradigm of Phylogenetic Estimation
A paradigm of phylogenetic estimation is a quantitative criterion used to discern
a phylogeny from among plausible alternatives. One of the earliest paradigms was
introduced by Cavalli-Sforza and Edwards [
15
] and is known as
the additive model
or the
the least-squares model
of phylogenetic estimation [
9
].
Cavalli-Sforza and Edwards observed that as molecular data provide the most
detailed anatomy possible for any organism, the diversity of life on Earth must be
reflected in them. Hence, if evolution of a set of molecular data from taxa could be
seen as a tree, then it could be described through a process that changes nucleotides
over time. The trajectories described by such a process would split as taxa diverges,
unite as taxa hybridize, end as taxa become extinct, and living taxa would be repre-
sented by the intercept of the process and the “now” plane (Fig.
8.2
).
In general, we do not have a sampling of such a process over time but only
the knowledge of the living taxa. Hence, in absence of further information, one
may be able only to reconstruct the projection of the process onto the “now” plane
rather than the process itself. Note that altough the evolutionary process over time is
“directed,” its projection is not (Fig.
8.2
). Thus, when the projection is consid-
ered, the direction of evolution is definitely missed. Nevertheless, the projection
of the evolutionary process constitutes still an important piece of information for
the analyzed taxa; for this reason, Cavalli-Sforza and Edwards proposed a possible
paradigm to recover it.
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