Page 237 - Applied Probability

10. Molecular Phylogeny
223
TABLE 10.4. Maximum Loglikelihoods of Various Hemoglobin Models
Model
Rate
Penalty
Maximum
Spatial
Type
-1918.6
1
0
0
Nucleotide Classes Parameters Parameters Loglikelihood
Codon 1 0 0 -1889.8
Codon 1 1 0 -1853.2
Codon 1 4 0 -1847.4
Codon 2 4 2 -1825.4
10.11 Problems
1. Compute the number of unrooted evolutionary trees possible for n
contemporary taxa. (Hint: How does this relate to the number of
rooted trees?)
2. In the notation of Section 10.2, let S n = T 2 + ··· + T n . Prove the
inequalities

n − 2 1
T n 1+ ≤ S n ≤ T n 1+
2n 2 n − 1
for all n ≥ 2.
3. Consider four contemporary taxa numbered 1, 2, 3, and 4. A total of
n shared DNA sites are sequenced for each taxon. Let N wxyz be the
number of sites at which taxon 1 has base w, taxon 2 base x, and so
forth. If we denote the three possible unrooted trees by E, F, and G,
then we can define three statistics

=
N E N rrss
r∈{A,G,C,T } s=r

N F = N rsrs
r∈{A,G,C,T } s=r

=
N G N rssr
r∈{A,G,C,T } s=r

for discriminating among the unrooted trees. Show that maximum
parsimony selects the unrooted tree E, F, or G with largest statis-
tic N E , N F,or N G . Draw the unrooted tree corresponding to each
statistic.
4. Let u and v be column vectors with the same number of components.
Applying the definition of the matrix exponential (10.7), show that
t
, I + suv t if v u =0
e suv t = e sv u −1 t
t
I + uv otherwise.
v t u