Methods for Determining Trees

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Methods for Determining Trees

• Sequence Based methods
• Maximum Parsimony
• Maximum Likelihood
• Distance Based methods
• UPGMA
• Neighbor Joining

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Maximum Parsimony

• A phylogeny constructed with the method of
  maximum parsimony explains evolution with the
  fewest evolutionary changes.
• Multiple sequence alignment must first be
  obtained. Each aligned column is a site.

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Maximum Parsimony Example

• 1 A A G A G T G C A
• 2 A G C C G T G C G
• 3 A G A T A T C C A
• 4 A G A G A T C C G
• four sequences, nine sites, three possible
  unrooted trees

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Maximum Parsimony Example

• Possible Trees I

Number of Mutations 10
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Maximum Parsimony Example

• Possible Trees II

(1)AAGAGTGCA
(2)AGCCGTGCG
1
3
5
AGGAGTGCA
AGAGGTCCG
4
1
(3)AGATATCCA
(4)AGAGATCCG
Number of Mutations 14
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Maximum Parsimony Example

• Possible Trees III

(1)AAGAGTGCA
(2)AGCCGTGCG
1
3
5
AGGAGTGCA
AGATGTCCG
5
2
(3)AGATATCCA
(4)AGAGATCCG
Number of Mutations 16
Tree I has the topology with the least number of
mutations and thus is the most parsimonious tree.
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Maximum Parsimony Example

• Some sites are informative, others are not
• Informative site there are at least two
  different kinds of nucleotides at the site, each
  of which is represented in at least two of the
  sequences under study.
• Only informative sites are considered

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Maximum Parsimony Example

• 1 A A G A G T G C A
• 2 A G C C G T G C G
• 3 A G A T A T C C A
• 4 A G A G A T C C G
• Three informative columns

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Maximum Parsimony Example

• 1 G G A
• 2 G G G
• 3 A C A
• 4 A C G
• Tree 1 4
• Tree 2 5
• Tree 3 6

Column 1
Column 2
Column 3
Is a substitution
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Maximum Parsimony Problems

• Small Parsimony Problem
• Given the phylogeny topology, compute the
  internal nodes to minimize the total number of
  mutations
• Used to evaluate the phylogeny
• Polynomial time solvable.
• Large Parsimony Problem
• Given that we have a way of determining the score
  of a given phylogeny, search through all possible
  phylogenies to find the best one
• Proved to be NP-complete.

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Fitchs Algorithm for Small Parsimony Problem

• Consider each site separately
• Dynamic programming style
• Constructs a set of possible states (possible
  nucleotides) for each internal node
• Start at the leaves of the phylogeny. Each leaf
  is labeled with the singleton set containing the
  nucleotide at that particular site.
• Traverse in a postorder manner (all of the
  children of the current node have been visited
  before the current node).

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Fitchs Algorithm for Small Parsimony Problem

• If m is an internal node with children l and r
  having states Sl and Sr respectively. The state
  of m, Sm , is computed as follows
• if
  is empty
• otherwise
• Each application of the first rule
  contributes one count to the number of changes.

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Fitchs Algorithm for Small Parsimony Problem
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Exhaustive Search Number of Trees
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Branch and Bound for Large Parsimony Problem

• Consider trees of increasing size (starting from
  3 species)
• Branch add one species, check all possible
  phylogeny topologies
• Bound one solution as the first bound, update
  the bound while finding a better one
• Abort an extension if score already exceeds
  current best.

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Branch and Bound for Large Parsimony Problem

• The worst case time complexity is the same as the
  complexity of exhaustive search
• With a wisely chosen bound, many subtrees will be
  cut and therefore the running time will decrease
• Sometimes a special traversal order finds better
  solutions faster
• A algorithm ?

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Maximum Parsimony

• Time consuming algorithm
• Only works well if the sequences have a strong
  sequence similarity

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Maximum Likelihood

• Evaluate the topologies of different trees and
  picks the best one according to an optimality
  criterion, the likelihood score
• Require a specified model of the evolutionary
  process that can account for the conversion of
  one sequence into another.

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Maximum Likelihood Model

• The model is composed of the composition and the
  substitution process
• Composition Frequencies of the character states
• Substitution Process Rate of change from one
  character state to another character state

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Maximum Likelihood Model

• For DNA sequences, A simple model is that the
  rate of change from a to c or vice versa is 0.4,
  the composition of a is 0.25 and the composition
  of c is 0.25

P
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Maximum Likelihood Model

• For nucleotide sequences, there are 16 possible
  ways to describe substitutions - a 4x4 matrix.
  Each entry in the matrix represents the
  substitution rate from nucleotide i to nucleotide
  j (rows, and columns, follow the order A, C, G,
  T).

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Maximum Likelihood Model

• In this matrix, the probability of an a changing
  to a c is 0.01 and the probability of a c
  remaining the same is 0.979, etc.
• The rows of this matrix sum to 1 - meaning that
  for every nucleotide, we have covered all the
  possibilities of what might happen to it. The
  columns do not sum to anything in particular

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Maximum Likelihood Model

• This matrix corresponds to one Certain
  Evolutionary Distance
• In the computation of likelihood, we need a
  matrix that can describe the branch lengths.
• Normally, a model gives another rate matrix, Q,
  which gives branch lengths in substitutions per
  site. For a branch length of v

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Maximum Likelihood Example

• A ccat
• B ccgt
• C gcat

What is the likelihood of alignment
given a tree topology with branch lengths,
a rate matrix,
and a composition,
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Maximum Likelihood Example
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Maximum Likelihood Example

• If we calculate the other three sites in a
  similar way, we get site likelihoods 0.245,
  0.00368, and 0.166. If we multiply them together,
  we get a likelihood for the tree of 3.0410-6.

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Maximum Likelihood Example
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Maximum Likelihood Search

• Input
• Alignment of sequences
• Model rate matrix, base frequencies
• Search
• Go through all possible trees, For each tree
  calculate branch lengths, and then likelihood
  value
• Output the tree with maximum likelihood value

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Heuristic Search in PAUP

• Stepwise addition
• As is
• Random
• Closest
• Simple
• Branch swapping
• Nearest Neighbor Interchange(NNI)
• Subtree Pruning Regrafting (SPR)
  
• Tree Bisection Reconnection(TBR)

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Heuristic Search in PAUP

• Stepwise addition

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Addition Order

• As is Input sequences order
• Random In each step, select a random one
• Closest In each step, all remaining taxa are
  considered
• Simple Assume a reference taxon, calculate the
  distance between this reference taxon and all the
  other taxa, add the taxon in the increase order
  of the distances

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Nearest Neighbor Interchange(NNI)
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Subtree Pruning Regrafting (SPR)
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Tree Bisection Reconnection(TBR)
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Heuristic Search in PHYLIP

• Stepwise addition in As Is or Random order
• In each step, do Local Rearrangement by using
  Nearest Neighbor Interchange (NNI)
• After finish adding, do Global Rearrangement by
  using Subtree Pruning Regrafting (SPR)

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Distance Method
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Distance Method

• Distance table used, a symmetric matrix M that
  gives the pairwise distances
• Goal Build an edge-weighted tree where each
  leaf (external node) corresponds to one object of
  M and so that distances measured on the tree
  between leaves i and j correspond to Mij

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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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Example of Distance Analysis

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

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UPGMA

• Unweighted Pair Group Method Using Arithmetic
  Averages
• Works by clustering sequences

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UPGMA

• distance dij between clusters Ci and Cj is
  average distance between pairs of sequences from
  each cluster
•  
• Ci and Cj are the number of sequences in
  clusters i and j

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UPGMA

• Initialization Steps
• Assign each sequence i to its own cluster Ci
• Define one leaf of the tree T for each sequence

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UPGMA

• Iteration steps
• 3. Determine the two clusters, i and j for which
  dij is minimal
• 4. Define a new cluster k by Ck Ci ? Cj, and
  define dkl for all l
• 5. Add k to the current clusters and remove i and
  j.
• 6. Continue steps 3-6 until only two clusters i
  and j remain.

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UPGMA

• Example using 5 Sequences

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Neighbor Joining

• Begin with star topology no neighbors have been
  joined

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Neighbor Joining

• Tree modified by joining pairs of sequences
• Pair is chosen by calculating sum of branch
  lengths for the corresponding tree

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Neighbor Joining

• If A and B are joined

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Neighbor Joining

• Pair with smallest branch length chosen to be
  joined
• Calculate new branch lengths

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Neighbor Joining

• A new distance table is created with joined
  sequences entered as a composite
• Repeat process to select next pair to join
• Process continues until correctly branched tree
  and distances identified

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Comparison of Methods