Phylogenetic trees

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Phylogenetic trees

• BioE131/231

HIV phylogeny. Simmonds et al (left) Yamamoto et
al (right)
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Star vs hierarchical phylogenies
Hierarchical
Star
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Rooted and unrooted trees
Degree of a node the number of neighbors of
that node. In a binary tree, all nodes have
degree 1 or 3, except the root which has degree
2. (i.e. each node has 0 or 2 children). A binary
tree with N leaf nodes has N-1 internal
nodes(c.f. table-tennis tournament...)
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Rooted directed
Root node
Internal node
Clade, subtree
Leaf node,taxon (pl. taxa)
Synonyms phylogeny, tree, dendrogram, cladogram
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Rooting via outgroups
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Root can be ambiguous
No outgroup... Earliest studies placed LUCA on
eukarya-bacteria branch later studies suggested
bacteria-archaea
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Ultrametric trees molecular clocks
Branch lengths are typically in units of average
number of substitutions per site.Thus, branch
lengths of gt1 have large estimation errors
Ultrametric
Non-ultrametric
(distance)
X
Height of node X
Q. Why are non-ultrametric trees necessary? A.
Mutation rate 1/(generation time) Also
correlated w/other physiological variables (e.g.
metabolic rate) Longitudinal data (e.g. serial
viral sequencing from same host) can also
generate non-ultrametric trees, since leaf nodes
are not contemporaneous
Wen-Hsiung Li, 1985 (2003 Balzan Prize)
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Newick formata.k.a. New Hampshire format

• Rooted tree topologies(A,B,(C,D))
• Branch lengths(A.1,B.2,(C.3,D.4).5)
• Internal node names(A.1,B.2,(C.3,D.4)E.5)F

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Algorithms for phylogenetic reconstruction

• Start with a multiple alignment
• use substitutions to evaluate trees
• indels informative, but harder to model
• Parsimony
• find the tree with the fewest substitutions
• Likelihood
• find the tree with the most likely
  substitutions(transition/transversion bias, long
  branches, ...)
• sum probabilities over unseen ancestral states
• enumerating all possible tree topologies is
  sloooooow
• Distance matrix
• Start by computing all pairwise distances
• Quick approximation to likelihood methods

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UPGMA algorithm

• Creates ultrametric trees
• Basic idea
• Two closest nodes must be siblings
• Parent is equidistant between siblings
• Distance from parent to any other node is average
  of distances of siblings to those nodes

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UPGMA algorithm

• Input a distance matrix, Dij
• Let N be the set of nodes to be joined
• Let the height of node i be Hi
• Initialize Hi0 for all the leaf nodes in N
• While N contains gt1 node
• Find i j, the two closest nodes in N
• (i,j) argmini,j Dij
• Create a new node, k, the parent of (i,j)
• Set Hk .5 (Hi Hj Dij)
• Branch length k?i is (Hk-Hi) and similarly for
  k?j
• For all nodes n in N (excluding i j)
• Set Dkn .5 (Din Dkn)
• Add k to N remove i j

N2 entries
N-1 steps
N2 steps
O(N3) timeIf we maintain argminj Dij for each j,
then it is O(N2) O(N2) memory
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UPGMA in Perl

• Questions
• How to represent a tree?
• For each node, need children/parents/both, name,
  branch length to parent...
• How to print a tree in Newick format?
• Recursive (print a particular node)
• Pre-order traversal (parents before children)
• How to represent a distance matrix?
• Can side-step some of these...

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Identify nodes by name, not by number

• Entry Dij of distance matrix is distanceiname
  -gtjnamewhere iname is the name of node i

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Accessing the distance matrix

• Set of all nodes, N keys (distance)
• Removing a node from the set delete
  distanceiname

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Construct the Newick representation on-the-fly

• Siblings (i, j) (iname , jname)
• Branch lengths Branch k?i has length
  ki Branch k?j has length kj
• Name of new node (k) (inameki,jnamekj)
  
• Then, Newick-format tree is just the name of the
  root node (plus a semicolon)

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Other phylogeny algorithms

• Neighbor-joining (e.g. neighbor program)
• Parents not equidistant from siblings
• Weighted neighbor-joining (e.g. weighbor
  program)
• Corrects for long-branch estimation error
• Quartet-puzzling (e.g. tree-puzzle program)
• Looks at sets of 4 nodes, instead of pairs
• MCMC sampling (e.g. MrBayes program)
• Stochastically explores tree space
• Slow, but provides much more information(confiden
  ce limits, etc.)

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Long branch attraction

• Arises because sequences on long branches share
  chance similarities
• Some methods (esp. parsimony) interpret this
  incorrectly as relatedness
• Solutions
• add more taxa to break up the branches
• use more realistic likelihood models

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Confidence estimates

• Bootstrap
• Sample a random subset of alignment columns (with
  replacement) and build a tree from those
• Repeat a large number of times
• Support for a branch
• defined as of trees that include that branch
• identify a branch by its partitioning of the taxa
• MCMC is a more statistically rigorous way to get
  confidence estimates for trees
• because it samples directly from the posterior
  distribution of trees

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Evolutionary linguistics
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How to estimate distances?

• T. Jukes and C. Cantor
• Berkeley, 1969