Defining what a tree means

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Defining what a tree means
unrooted tree (used when the root isnt known)
rooted tree (all real trees are like this)
ancestral sequence
time vaguely radiates out from somewhere near the
center
divergence time is the sum of (horizontal)
branch lengths
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Toms favorite tree (vertebrate Cyp450)
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A tree has topology and distances
Note - topologically, these are the SAME tree. In
general, two trees are the same if they can be
inter-converted by branch rotations.
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The number of tree topologies grows extremely fast
3 leaves 3 branches 1 internal node 1 topology (3
insertions)
4 leaves 5 branches 2 internal nodes 3 topologies
(x3) (5 insertions)
In general, an unrooted tree with N leaves
has 2N 3 branches N 2 internal nodes O(N!)
topologies
5 leaves 7 branches 3 internal nodes 15
topologies (x5) (7 insertions)
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There are many rooted trees for each unrooted tree
For each unrooted tree, there are 2N - 3 times as
many rooted trees, where N is the number of
leaves ( internal branches 2N 3).
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Trees can be enormously complex
This tree has 203 7-pass chemosensory protein
sequences and could adopt gtgtgt10100 topologies.
Even a tree of 20 sequences could adopt 2 x 1020
topologies.
Note that this does NOT include branch length
differences!
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Duplication and Divergence

• Full gene duplication can occur in two patterns
• 1) speciation (orthology)
• 2) duplication within species (paralogy)
• Gene duplication within a species probably
  occurs only during aberrant DNA rearrangements.
• Immediately after duplication, the two copies of
  a gene are fully redundant and one often mutates
  to non-functionality and fixes in the population
  (pseudogene).
• Some of the time (unclear how often) the two
  copies diverge to become functionally distinct
  enough that each function is under negative
  selection and both are retained.

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Orthology and paralogy trees
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How duplications can arise
2N ? 4N (e.g. failure in mitosis)
1) genome duplication
2) unequal crossing-over
3) replication slippage
4) transposon hops run amock
5) other events involving repair of double strand
breaks
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To reconstruct species phylogeny

• sequence orthologs from each species
• construct gene tree it probably corresponds to
  the species tree (question assuming you
  construct the gene tree correctly what could
  violate this? TWO answers)
• do this for more than one gene to be sure you
  should get nearly the same tree every time.

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Species phylogeny (and a footnote on
humanocentrism)
This tree is amazingly misleading
correct tree
Not to mention 1) the straight line to humans,
2) leaving out gt90 of all species, including
three hominoids bonobo, siamang, gibbon, and 3)
the fact that WE are an African ape.
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How do you make a tree from data?

• data typically takes the form of either a
  multiple sequence alignment or a set of all
  pairwise sequence alignments.
• methods for going from the data to a tree can be
  put into two general categories
• - Distance Matrix methods (e.g. UPGMA,
  Neighbor-Joining)
• - Tree-enumerating methods (e.g. parsimony,
  maximum- likelihood)
• in theory, the tree-enumerating methods are
  superior, but they are not always practical.

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Distance matrix methods

• methods based on a set of pairwise distances (no
  multiple alignment needed, though pairwise
  distances often come from one).
• all methods (in essence) build a tree that tries
  to best match the distances.
• usual standard for best match is the least
  squares of the tree distances compared to the
  real pairwise distances

Let Dm be the matrix distances and Dt be the
tree distances. Find the tree (an internally
consistent set of Dt values) that minimizes
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Example distance matrix (ID) and tree
EGL-2 dEag rEag UNC-103 dErg HERG EGL-2 0.000 0.2
76 0.353 0.542 0.547 0.525 dEag 0.276 0.000 0.305
0.512 0.501 0.508 rEag 0.353 0.305 0.000 0.533 0.5
15 0.510 UNC-103 0.542 0.512 0.533 0.000 0.274 0.2
55 dErg 0.547 0.501 0.515 0.274 0.000 0.263 HERG 0
.525 0.508 0.510 0.255 0.263 0.000
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Least-squares solution

• There are methods that directly solve the
  least-squares problem
• However, they are computationally slow and
  rarely used.
• Fortunately, there are more direct
  approximations that work remarkably well, most
  notably Neighbor-Joining.
• The direct methods use various types of
  sequential clustering.

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Sequential clustering approach
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Neighbor-Joining Algorithm (NJ)
Essentially as on previous slide, but correction
for distance to other leaves is made.
Specifically, for two leaves i and j, we denote
the set of all other leaves as L, and the size of
that set as , and we compute the corrected
distance Dij as
heres an intuitive rationale (consider
clustering the first two leaves)
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Neighbor-Joining corrects for different rates of
evolution on branches
these two branches have changed faster
The corrective terms change the distance from
leaf 1 and 3 so that they are clustered before
leaf 1 and 2 (and similarly for leaves 2 and 4).
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Parsimony method

• intuitively appealing find the tree that can
  explain the observed sequences with the smallest
  number of changes.
• BUT requires enumeration of tree topologies, so
  not feasible except for large data sets. (also
  inferior to ML methods, which are not much more
  computationally intensive).

1 AAG 2 AAA 3 GGA 4 AGA
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Maximum Likelihood Method

• First developed by Joe Felsenstein here at UW.
• Formulate a specific model of evolution, with
  rates and probabilities of various changes.
• Assess the likelihood of the data set given a
  particular tree topology and branch length set
  (and your model).
• Use a tree-rearrangement/hill-climbing method to
  move from reasonably good trees toward the best
  tree.
• Even more computationally demanding than
  parsimony, but probably the best method
  especially when statistical analysis is critical.

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Tree topology enumeration is very problematic
3 leaves 3 branches 1 internal node 1 topology (3
insertions)
4 leaves 5 branches 2 internal nodes 3 topologies
(x3) (5 insertions)
In general, a tree with N leaves has 2N 3
branches N 2 internal nodes O(N!) topologies
5 leaves 7 branches 3 internal nodes 15
topologies (x5) (7 insertions)
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Speed up in topology finding

• Used in both parsimony and maximum-likelihood
  methods.
• Start with an NJ distance tree, assuming that it
  approximates the correct tree.
• Assess the likelihood of the tree, then test
  multiple simple rearrangements of the tree
  looking for better likelihoods.
• Repeat until nothing gets better.

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Topology statistics the bootstrap

• we can say that the data are most consistent
  with a particular tree (with one or another
  method) but how confident are branch points?
• for example

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Bootstrap and jack-knife calculation (resampling)

• General principal is to stress the data
  repeatedly and recompute the tree each time,
  looking for robust features.
• Jack-knife drop each of N sequences from the
  alignment and recompute the resulting N trees,
  testing whether they are compatible with
  original.
• Bootstrap recompute pairwise distances from a
  random sample of alignment columns (with
  replacement). Recompute the tree for each new
  distance set and see how often a particular tree
  branch is positioned the same.

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Bootstrap (cont.)
real alignment
Compute relative distances among pairs by
comparing scores (not discussed in class).
Similar resampling and distance derivation done
for each sequence pair, then a new tree
calculated. Repeat ad nauseum.
EGL-2 dEag rEag UNC-103 dErg HERG EGL-2 0.000 0.2
76 0.353 0.542 0.547 0.525 dEag 0.276 0.000 0.305
0.512 0.501 0.508 rEag 0.353 0.305 0.000 0.533 0.5
15 0.510 UNC-103 0.542 0.512 0.533 0.000 0.274 0.2
55 dErg 0.547 0.501 0.515 0.274 0.000 0.263 HERG 0
.525 0.508 0.510 0.255 0.263 0.000
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The molecular clock concept

• divergence distance vs. time NOT the same!
• molecular clock is common default hypothesis
  for translating distance into time (assume that
  divergence time).
• known to be violated whenever sufficiently broad
  groups are considered.
• however, frequently approximately valid for a
  particular sequence family over relatively short
  times (e.g. most proteins in primates).

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Things that tend to invalidate the molecular clock

• differential changes in generation time on some
  branches.
• differential changes in selective constraints on
  some branches (extreme example would be positive
  selection).
• depth of divergence - though correctable unless
  distances are too long.

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CAUTIONARY NOTES ON VARIOUS METHODS

• All tree building methods depend strongly on
  high quality alignments. To the degree that
  alignments are problematic (deep divergence) the
  trees are problematic. Bootstrap and other
  methods do NOT fully account for this.
• For good alignments of sequences that are not
  excessively divergent, all tree-building methods
  should give fairly similar trees. (If they dont,
  be skeptical)
• Trees that LOOK very different often are not
  you may have to reroot the tree and rotate tree
  branches to get two trees to look similar to the
  human eye.