Biology Reference
In-Depth Information
evolution is efficient. Since branch lengths represent amount of evolutionary change
leading from one, say, species to the next, the total tree length
is a measure of
that efficiency across the whole tree, and hence should be minimal under a minimum
evolution perspective.
There are many different minimum evolution methods; for all, it is necessary to
start from
D
and come up with not only with the phylogenetic tree
T
, but also the edge
weighting (branch lengths)
ω(
T
)
ω
. Since there are finitely many possible phylogenetic
n
2
∈
R
X
-trees, one approach is, starting with any dissimilarity map
D
and starting
∈
T
n
, to come up with a branch length estimation scheme
ω
with any tree
T
for
T
based on the data
D
. One then computes the resulting tree length
ω(
T
)
and, looking
over all possible
T
, seeks
T
so that
ω(
T
)
in minimized.
from
D
, the Balanced Minimum Evolution (BME)
method, uses an approach of Pauplin's [
14
]. Pauplin's scheme for branch length
estimation from
D
, applied to the case of just a quartet, should now be very familiar
to us from our previous exercises on the four-point condition and the NJ Algorithm.
Specifically, if
e
One approach for producing
ω
∈
E
is the interior edge joining
u
and
v
in the (unrooted) quartet
tree
((A,B), (C,D));
as in Example
10.3
, then
ω(
e
)
is given by setting
ω(
e
)
=
1
1
4
(
d
(
A
,
C
)
+
d
(
A
,
D
)
+
d
(
B
,
C
)
+
d
(
B
,
D
))
−
2
(
d
(
A
,
B
)
+
d
(
C
,
D
))
. This idea
is extended in [
14
] to the case when
A
,
B
,
C
,
D
are replaced by entire subtrees in an
arbitrary tree
T
for any interior
edge in
T
that accounts for the relative size (say, number of leaves) in each subtree.
(This is the “balance” of the BMEmethod.) The (easier) case of edges that are external
is also handled there. Rather than say more about these formulas, however, we instead
make use of a very nice result by which the total tree length
∈
T
n
. The result gives an iterative formula for
ω(
e
)
ω(
T
)
can be calculated
more easily, without first finding all the branch lengths
ω(
e
)
.
T
Starting from
T
∈
T
n
, define
w
(
i
,
j
)
to be one divided by the product of deg
(
x
)
−
1 for every interior node
x
encountered on the path
P
i
,
j
in
T
from
i
to
j
. Equivalently,
set
y
T
(
i
,
j
)
to be the number of edges between leaves
i
and
j
on the path
P
i
,
j
.Inthis
2
1
−
y
T
T
(
i
,
j
)
.Nowset
case, we can see that
w
(
i
,
j
)
:=
T
T
T
T
T
T
w
:=
(w
(
1
,
2
), w
(
1
,
3
),...,w
(
1
,
n
), w
(
2
,
3
), w
(
2
,
4
),...,
n
2
T
T
w
(
2
,
n
),...,w
(
n
−
1
,
n
))
∈
R
.
T
only depends on the topology of
T
, so we may call it the
BME vector
Notice that
w
of
T
.
Example 10.8.
For
T
∈
T
5
showninFigure
10.8
, below, one obtains the BME vector
1
4
,
1
8
,
1
8
,
1
2
,
1
4
,
1
4
,
1
4
,
1
8
,
1
8
T
w
=
.
T
Exercise 10.20.
Find the BME vectors
w
for each
T
∈
T
4
.
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