Phylogenetic Tree Reconstruction

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Phylogenetic Tree Reconstruction
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Phylogenetic Tree

• A tree that represents the relationship of a set
  of species or genetic sequences is called a
  phylogenetic tree.

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Phylogenetic tree

• Phylogeny the relationship of species
• Leaves species or sequences (OTUs operational
  taxonomic units)
• Internal nodes ancestors of particular groups of
  the OTUs.
• Branch Length the degree of relatedness between
  the species or sequences corresponding to the
  nodes at the endpoints of the branch.

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Orthologues / Paralogues

• Genes which diverged because of speciation are
  called orthologues. Above a tree of orthologues
  based on a set of alpha haemoglobins.
• Genes which diverged by gene duplication are
  called paralogues. Below a tree of paralogues,
  the alpha, beta, gamma, delta, epsilon, zeta and
  theta chains of human haemoglobins, and
  myoglobin.

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Rooted / Unrooted Tree
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Counting Trees
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Counting Trees
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Rrooting the tree
To root a tree mentally, imagine that the tree is
made of string. Grab the string at the root
and tug on it until the ends of the string (the
taxa) fall opposite the root
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Steps of phylogenetic tree reconstruction

• Choosing a family of homologous sequences
• Aligning the sequences and obtaining a reduced
  multiple alignment by discarding the columns that
  contain gaps.
• Inferring a phylogenetic tree from the reduced
  multiple alignment.

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Methods of Phylogenetic tree reconstruction

• Maximum parsimony methods
• Distance methods
• Probabilistic methods arising from the maximum
  likelihood approach

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Maximum Parsimony Method

• Site
• OTU 1 2 3 4 5 6 7 8 9
• -----------------------
• 1 T C A G A T C A A
• 2 T T A G A A C A A
• 3 T T C G A T C G A
• 4 T T C T A A G G A

• Target find rooted tree topologies, not branch
  lengths.
• Principle search for tree that requires the
  smallest number of character state changes
  between the OTUs (the sequence always evolve in
  the most economic way.)
• Operation at least two different kinds of
  residues at the site, each of which is found in
  at least two of the OUT sequences

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Maximum Parsimony
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Maximum Parsimony
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Traditional Parsimony

• If N is moderate, Fitch's algorithm is realistic.
• If N is large, the branch and bound algorithm
  should be used.

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Distance Methods

• Reconstruct trees (rooted or unrooted, depending
  on the method) from a set of pairwise distances,
  d (d_ij), between the sequences in a fixed
  reduced multiple alignment.
• Given a tree relating the OTUs, obtain a
  tree-generated distance matrix d'.
• Question is dd'? Or, is the distance function
  additive?

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Additivity

• Theorem (Four point condition) Let d be a
  distance function on a M (a set of N OUTs) and
  Ngt4. Then d is additive if and only if the
  following condition holds for every set of four
  distinct numbers 1 lti,j,k,lltN two of the sums
  d_ijd_kl, d_ikdjl, d_ild_jk
  coincide and are greater than or equal to the
  third one. (Saitou and Nei, 1987, Mol. Biol.
  Evol.)

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Additivity
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Distance function

• distance score counted as
• number of mismatched positions in the alignment
• number of sequence positions that must be changed
  to generate the second sequence
• Success depends on degree the distances among a
  set of sequences can be made additive on a
  predicted evolutionary tree

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Example of Distance Analysis

• Distances can be shown as a table
• A ACGCGTTGGGCGATGGCAAC
• B ACGCGTTGGGCGACGGTAAT
• C ACGCATTGAATGATGATAAT
• D ACACATTGAGTGATAATAAT

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Neighbour joining

• Very popular method
• Produces unrooted tree
• Assumes additivity distance between pairs of
  leaves sum of lengths of edges connecting them
  that is, dd'.
• Constructs tree by sequentially joining subtrees

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Neighbor Joining Once we know the correct (i,j)
pair
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Neighbour Joining
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Neighbor joining algorithm
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Neighbour Joining why not pick the smallest
(i,j) pair?
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Example of Distance Analysis

• Using this information, a tree can be drawn
• A ACGCGTTGGGCGATGGCAAC
• B ACGCGTTGGGCGACGGTAAT
• C ACGCATTGAATGATGATAAT
• D ACACATTGAGTGATAATAAT

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Drawbacks of neighbor joining

• In practice, the distance function is often a
  pseudodistance function, which does not satisfy
  the four-point condition. (The triangle
  inequality is hard to satisfy either)
• The algorithm may produce more than one tree,
  these trees may have branches of negative
  lengths, the matrix d' may not coincide with the
  original distance matrix d.

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Special distance function Ultrameric distance

• Definition A distance function d on a set M of
  OTUs is called ultrameric, if for any three
  distinct elements x_i, x_j, x_k, two of the
  distances d_ij, d_ik, d_jk concide and are
  greater than or equal to the third.
• It satisfies the four-point condition.
• It is additive, and can be recovered by the
  generated phylogenetic tree.

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UPUPGMA -- Unweighted Pair Group Method with
Arithmetic meanGMA(sequential clustering method)
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UPGMA distance function between two clusters
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UPGMA
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UPGMA Step 1combine B and C
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UPGMA step 2combine BC and D
(1012)/2
(46)/2
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UPGMA step 3combine A and E
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UPGMA step 4combine AE and BCD
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UPGMA Result
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(No Transcript)
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When UPGMA fails
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Maximum Likelihood method

• Assumption Maximum likelihood supposes a model
  of evolution along tree branches.
• Strategy Find parameters (tree, branch lengths,
  substitution rate) that maximizes the likelihood
  assigned to the data.
• Note Model of evolution does not include
  insertion and deletion of the nucleotides.
• In Phylip package program PROTML

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Probabilistic Methods

• The phylogenetic tree represents a generative
  probabilistic model (like HMMs) for the observed
  sequences.
• Background probabilities q(a)
• Mutation probabilities P(ab, t)
• Models for evolutionary mutations
• Jukes Cantor
• Kimura model
• Felsenstein model
• Hasegawa-Kishino-Yano model

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Jukes Cantor model

• A model for mutation rates

• Mutation occurs at a constant rate
• Each nucleotide is equally likely to mutate into
  any other nucleotide with rate alpha.

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Kimura 2-parameter model

• Allows a different rate for transitions and
  transversions.

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Mutation Probabilities

• The rate matrix R is used to derive the mutation
  probability matrix S
• S is obtained by integration. For Jukes Cantor
• q can be obtained by setting t to infinity

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Mutation Probabilities

• All models satisfy the following properties
• Markovian property
• Reversibility
• Exist stationary probabilities Pa s.t.

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Probabilistic Approach

• Given P,q, the tree topology and branch lengths,
  we can compute

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Computing the Tree Likelihood
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Tree Likelihood Computation

• Define P(Lka) prob. of leaves below node k
  given that xka
• Init for leaves P(Lka)1 if xka 0 otherwise
• Iteration if k is node with children i and j,
  then
• TerminationLikelihood is

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Maximum Likelihood (ML)

• Score each tree by
• Assumption of independent positions
• Branch lengths t can be optimized
• Gradient ascent
• EM
• We look for the highest scoring tree
• Exhaustive
• Sampling methods (Metropolis)

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Optimal Tree Search

• Perform search over possible topologies

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Computational Problem

• Such procedures are computationally expensive!
• Computation of optimal parameters, per candidate,
  requires non-trivial optimization step.
• Spend non-negligible computation on a candidate,
  even if it is a low scoring one.
• In practice, such learning procedures can only
  consider small sets of candidate structures

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Max Likelihood versus Parsimony

• (Example from BSA p. 225)
• Choose tree T, with unequal branch lengths.
• Generate 1000 sequences of length N according to
  probabilistic model
• (A) Reconstruction by ML (B)
  Reconstruction by Parsimony

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Max Likelihood versus NJ

• (Example from BSA p. 225)
• Choose tree T, with unequal branch lengths.
• Generate 1000 sequences of length N according to
  probabilistic model
• (A) Reconstruction by ML (B)
  Reconstruction by NJ

Conclusion ML infers right tree as N gets
largerl. If the probabilistic model is correct,
the ML distances shall be very close to additive,
therefore the NJ method predicts the correct
tree.
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Phylip - practicalities

• Menu-driven, no command line
• Input file format
• First line ltnumber of sequencesgt ltnumber of
  letters per sequencegt
• Next lines Sequences
• First ten characters is the sequence name
• Then sequence follows. Spaces and newlines are
  allowed.
• Dashes (-) signify gaps
• Example

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The End