Phylogenetics%20I

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Phylogenetics I
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Evolution

• Evolution of new organisms is driven by
• Mutations
• The DNA sequence can be changed due to single
  base changes, deletion/insertion of DNA segments,
  etc.
• Selection bias

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Theory of Evolution

• Basic idea
• speciation events lead to creation of different
  species.
• Speciation caused by physical separation into
  groups where different genetic variants become
  dominant
• Any two species share a (possibly distant) common
  ancestor

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The Tree of Life
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Primate evolution
A phylogeny is a tree that describes the sequence
of speciation events that lead to the forming of
a set of current day species also called a
phylogenetic tree.
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Morphological vs. Molecular

• Classical phylogenetic analysis morphological
  features number of legs, lengths of legs, etc.
• Modern biological methods allow to use molecular
  features
• Gene sequences
• Protein sequences

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Morphological topology
(Based on Mc Kenna and Bell, 1997)
Archonta
Ungulata
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From sequences to a phylogenetic tree
Rat QEPGGLVVPPTDA Rabbit QEPGGMVVPPTDA Gorilla QE
PGGLVVPPTDA Cat REPGGLVVPPTEG
There are many possible types of sequences to use
(e.g. Mitochondrial vs Nuclear proteins).
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Mitochondrial topology
(Based on Pupko et al.,)
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Nuclear topology
(Based on Pupko et al. slide)
(tree by Madsenl)
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Phylogenenetic trees

• Leaves - current day species (or taxa plural of
  taxon)
• Internal vertices - hypothetical common ancestors
• Edges length - time from one speciation to the
  next

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Twists in molecular phylogenies

• We have to emphasize that gene/protein sequence
  can be homologous for several different reasons
• Orthologs -- sequences diverged after a
  speciation event
• Paralogs -- sequences diverged after a
  duplication event
• Xenologs -- sequences diverged after a horizontal
  transfer (e.g., by virus)

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Paralogs
Consider evolutionary tree of three taxa
Gene Duplication
and assume that at some point in the past a gene
duplication event occurred.
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Paralogs
The gene evolution is described by this tree (A,
B are the copies of the same gene).
Gene Duplication
Speciation events
2B
1B
3A
3B
2A
1A
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Paralogs

• If we happen to consider genes 1A, 2B, and 3A of
  species 1,2,3, we get a wrong tree that does not
  represent the phylogeny of the host species

Gene Duplication
S
S
S
Speciation events
2B
1B
3A
3B
2A
1A
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Types of Trees

• A natural model to consider is that of rooted
  trees

Common Ancestor
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Types of trees

• Unrooted tree represents the same phylogeny
  without the root node

Depending on the model, data from current day
species does not distinguish between different
placements of the root.
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Rooted versus unrooted trees
Tree c
b
a
c
Represents the three rooted trees
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Total numbers of trees

• For N taxa,
• Rooted bifurcating trees
• (2n-3)!! (2n-3)!/2n-2(n-2)!
• Unrooted bifurcating trees
• (2n-5)!!
• Tree shapes

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Positioning Roots in Unrooted Trees

• We can estimate the position of the root by
  introducing an outgroup
• a set of species that are definitely distant from
  all the species of interest

Proposed root
Falcon
Aardvark
Bison
Chimp
Dog
Elephant
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Type of Data

• Distance-based
• Input is a matrix of distances between species
• Can be fraction of residue they disagree on, or
  alignment score between them, or
• Character-based
• Examine each character (e.g., residue) separately

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Two methods of tree Construction

• Distance- A weighted tree that realizes the
  distances between the objects.
• Parsimony A tree with a total minimum number of
  character changes between nodes.

We start with distance based methods, considering
the following question Given a set of species
(leaves in a supposed tree), and distances
between them construct a phylogeny which best
fits the distances.
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Distance Matrix

• Given n species, we can compute the n x n
  distance matrix Dij
• Dij may be defined as the edit distance between a
  gene in species i and species j, where the gene
  of interest is sequenced for all n species.

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The distance between two sequences

• Protein sequences
• PAM
• BLOSUM
• DNA sequences
• Jukes-Cantor
• HGY
• Kimura 2-Parameter

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General Stationary Time-reversible Model
. pCrCA pGrGA pTrTA
pArAC . pGrGC pTrTC
pArAG pCrCG . pTrTG
pArAT pCrCT pGrGT .
R
(Diagonal elements such that rows sum to zero)
Time reversibility pirij pjrji
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General Stationary Time-reversible Model

• P(t) eRt
• Given rates, one can find transition
  probabilities, and vice-versa.

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Jukes-Cantor
. u/3 u/3 u/3
u/3 . u/3 u/3
u/3 u/3 . u/3
u/3 u/3 u/3 .
R
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Jukes-Cantor

• P(no mutation) e-4/3ut
• P(at least one mutation) 1-e-4/3ut
• Ds ¾ (1-e-4/3ut)
• D ? ut -3/4 ln (1-4/3 Ds)

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Kimura 2-Parameter
A C G T
. b a b
b . b a
a b . b
b a b .
R
a/b transition/transversion bias ? R a2b 1
per unit time
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Kimura 2-Parameter

• aR/(R1), b0.5/(R1)

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HKY (Hasegawa, Kishino, Yano)
. mpC mkpG mpT
mpA . mpG mkpT
mkpA mpC . mpT
mpA mkpC mpG .
R
k transversion / transition
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Distances in Trees

• Edges may have weights reflecting
• Number of mutations on evolutionary path from one
  species to another
• Time estimate for evolution of one species into
  another
• In a tree T, we often compute
• dij(T) - the length of a path between leaves i
  and j

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Distance in Trees an Exampe
d1,4 12 13 14 17 12 68
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Fitting Distance Matrix

• Given n species, we can compute the n x n
  distance matrix Dij
• Evolution of these genes is described by a tree
  that we dont know.
• We need an algorithm to construct a tree that
  best fits the distance matrix Dij

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Reconstructing a 3 Leaved Tree

• Tree reconstruction for any 3x3 matrix is
  straightforward
• We have 3 leaves i, j, k and a center vertex c

Observe dic djc Dij dic dkc Dik djc
dkc Djk
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Reconstructing a 3 Leaved Tree
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Trees with gt 3 Leaves

• An tree with n leaves has 2n-3 edges
• This means fitting a given tree to a distance
  matrix D requires solving a system of n choose
  2 equations with 2n-3 variables
• This is not always possible to solve for n gt 3

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Additive Distance Matrices
Matrix D is ADDITIVE if there exists a tree T
with dij(T) Dij
NON-ADDITIVE otherwise
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Distance Based Phylogeny Problem

• Goal Reconstruct an evolutionary tree from a
  distance matrix
• Input n x n distance matrix Dij
• Output weighted tree T with n leaves fitting D
• If D is additive, this problem has a solution and
  there is a simple algorithm to solve it

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Using Neighboring Leaves to Construct the Tree

• Find neighboring leaves i and j with parent k
• Remove the rows and columns of i and j
• Add a new row and column corresponding to k,
  where the distance from k to any other leaf m can
  be computed as

Dkm (Dim Djm Dij)/2
Compress i and j into k, iterate algorithm for
rest of tree
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Finding Neighboring Leaves

• To find neighboring leaves we simply select a
  pair of closest leaves.

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Finding Neighboring Leaves

• To find neighboring leaves we simply select a
  pair of closest leaves.
• WRONG

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Finding Neighboring Leaves

• Closest leaves arent necessarily neighbors
• i and j are neighbors, but (dij 13) gt (djk 12)

• Finding a pair of neighboring leaves is
• a nontrivial problem!

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Neighbor Joining Algorithm

• In 1987 Naruya Saitou and Masatoshi Nei developed
  a neighbor joining algorithm for phylogenetic
  tree reconstruction
• Finds a pair of leaves that are close to each
  other but far from other leaves implicitly finds
  a pair of neighboring leaves
• Advantages works well for additive and other
  non-additive matrices, it does not have the
  flawed molecular clock assumption

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Constructing additive treesThe neighbor joining
algorithm

• Let i, j be neighboring leaves in a tree, let k
  be their parent, and let m be any other vertex.
• The formula
• shows that we can compute the distances of k to
  all other leaves. This suggest the following
  method to construct tree from a distance matrix
• Find neighboring leaves i,j in the tree,
• Replace i,j by their parent k and recursively
  construct a tree T for the smaller set.
• Add i,j as children of k in T.

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Neighbor Finding

• How can we find from distances alone a pair of
  nodes which are neighboring leaves?
• Closest nodes arent necessarily neighboring
  leaves.

Next we show one way to find neighbors from
distances.
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Neighbor Finding Seitou Nei algorithm
Definitions
Theorem (Saitou Nei) Assume all edge weights
are positive. If D(i,j) is minimal (among all
pairs of leaves), then i and j are neighboring
leaves in the tree.
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Complexity of Neighbor Joining Algorithm

• Naive Implementation
• Initialization ?(L2) to compute d(r,i) and
  C(i,j) for all i,j?L.
• Each Iteration
• O(L2) to find the maximal C(i,j).
• O(L) to compute C(m,k)m? L for the new node k.
• Total of O(L3).

r
C(m,k)
m
k
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Complexity of Neighbor Joining Algorithm

• Using Heap to store the C(i,j)s
• Input Distance matrix D d(i,j), and an
  arbitrary object r.
• Initialization ?(L2) to compute and heapify the
  C(i,j)s in a heap H.
• Each Iteration
• O(log L) to find and delete the maximal C(i,j)
  from H.
• O(L) to add the values d(k,m) to D, for all
  objects m.
• O(L) to delete d(m,i), d(m,j) from D (for all
  m).
• O(L log L) to delete C(i,m), C(j,m) and add
  C(k,m) from H, for all objects m.
• Total of O(L2 log L).
• (implementation details are omitted)

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Neighbor Joining Algorithm

• Applicable to matrices which are not additive
• Known to work good in practice
• The algorithm and its variants are the most
  widely used distance-based algorithms today.

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The Four Point Condition
Compute 1. Dij Dkl, 2. Dik Djl, 3. Dil Djk
2
3
1
2 and 3 represent the same number the length of
all edges the middle edge (it is counted twice)
1 represents a smaller number the length of all
edges the middle edge
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The Four Point Condition Theorem

• The four point condition for the quartet i,j,k,l
  is satisfied if two of these sums are the same,
  with the third sum smaller than these first two
• Theorem An n x n matrix D is additive if and
  only if the four point condition holds for every
  quartet 1 i,j,k,l n

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Least Squares Distance Phylogeny Problem

• If the distance matrix D is NOT additive, then we
  look for a tree T that approximates D the best
• Squared Error ?i,j (dij(T)
  Dij)2
• Squared Error is a measure of the quality of the
  fit between distance matrix and the tree we want
  to minimize it.
• Least Squares Distance Phylogeny Problem finding
  the best approximation tree T for a non-additive
  matrix D (NP-hard).