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10. Molecular Phylogeny
most economical to choose an order for the iterated sums consistent with
the graphical structure of the tree. In particular, the tree should be pruned
working upward from the tips. This can be accomplished by a postorder
traversal. Other methods of pruning are possible, but none is as easy to
implement computationally in a recursive computer language.
It is a curious feature of reversible Markov-chain models that the root
can be eliminated and one of the two direct descendants of the root substi-
tuted for it. Suppose that nodes 2 and 3 are the direct descendants of the
,
root. Those arrays of the likelihood (10.16) involving the index b 1 are q b 1
Pr(b 2 | b 1 ), and Pr(b 3 | b 1 ). In the Markov-chain context, it is natural to
as the equilibrium distribution. Furthermore, if t 2 and t 3 are the
take q b 1
times separating the root from nodes 2 and 3, respectively, then
(t 2 )
Pr(b 2 | b 1 )= p b 1 b 2
(t 3 ).
Pr(b 3 | b 1 )= p b 1 b 3
Isolating the sum over b 1 and invoking finite-time detailed balance now give

p (t 3 )= p (t 3 )
q b 1 b 1 b 2 (t 2 )p b 1 b 3 q b 2 b 2 b 1 (t 2 )p b 1 b 3
b 1 b 1

(t 3 )
= q b 2 p b 2 b 1 (t 2 )p b 1 b 3
b 1
p (t 2 + t 3 ).
= q b 2 b 2 b 3
(See Problem 7 for a brief discussion of finite-time detailed balance.) This is
Felsenstein’s pulley principle. The root can be eliminated and moved to
either one of its direct descendants—in this case, node 2. Thus, if only re-
versible chains are considered, then maximum likelihood cannot distinguish
two rooted trees that correspond to the same unrooted tree [6].

10.7 Origin of the Eukaryotes

Eukaryotic organisms differ from prokaryotic organisms in possessing a
nucleus, a cellular organelle housing the chromosomes. The origin of eu-
karyotes from prokaryotic bacteria is one of the most intriguing questions
in evolutionary biology. Bacteria can be subdivided into four broad groups.
The eubacteria are common pathogens of eukaryotes, the halobacte-
ria are found at high salt concentrations, the eocytes metabolize sulfur
and are found at high pressures and temperatures, and the methanogens
metabolize methane. Evolutionary biologists have traditionally classified
the latter three groups in a single phylum, the archebacteria, leaving the
eubacteria as the natural candidates for the ancestors of eukaryotes. In sup-
port of this view is the fact that mitochondria and chloroplasts, important
organelles of eukaryotic cells, derive from eubacteria. Lake [15] upset this