Refined Buneman Trees

A, C, T
1 2 3
T
A, C, T
T
C, T
A, T
T
T
C T A T T C T A T T
C T A T T
Figure 4.1: An illustration of the parsimony tree method.
4.3 Maximum likelihood methods
Using maximum likelihood methods is quite simple in theory. We start out with
e.g. a set of n nucleotide sequences of length m — aligned such that only substitutions
occur, not insertions or deletions. We then postulate some evolutionary
tree topology over the n sequences, giving us a rooted binary tree with n − 1
inner nodes. For each inner node we assume there is some ancestor sequence for
the sequences in the subtree of that node, but we do not know which nucleotides
are in this ancestor sequence.
We assume some substitution model, and there are many to choose from,
ranging from simple to very complicated. Jukes-Cantor ([JC69]), Kimura ([Kim80])
and Hasegawa-Kishino-Yano ([HKY85]) are names of well known substitution
models, and there is a long hierarchy of increasingly complex models using more
and more biologically founded assumptions. Now, to find the likelihood of a single
nucleotide site in the n leaf sequences we have to multiply probabilities of
substitutions through the tree, for all possible substitutions, i.e. for all possible
assignments of nucleotides to sites in the n − 1 inner nodes in the tree. And
then we would have to sum over all these likelihoods to find the likelihood of
the entire sequences, i.e. a sum over m terms.
Now, this expression would have to be evaluated for all possible tree topologies,
and of course this results in a very time consuming algorithm. But, since
the substitution model is actually based in biology, we have a high confidence
that the resulting tree with maximum likelihood is the real tree.
One way of making this method useful in practise would be to search for
tree topologies in some clever way, so that the search would be able to skip
large parts of the search space, which would of course limit the accuracy of the
method. The ML method is also useful for evaluating trees found by other tree
reconstruction method, since we assume the method captures a lot of biological
meaning, depending on the model.
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