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BNFO 602 Lecture 1

• Usman Roshan

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Phylogenetics

• Study of how species relate to each other
• Nothing in biology makes sense, except in the
  light of evolution, Theodosius Dobzhansky, Am.
  Biol. Teacher (1973)
• Rich in computational problems
• Fundamental tool in comparative bioinformatics

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Why phylogenetics?

• Study of evolution
• Origin and migration of humans
• Origin and spead of disease
• Many applications in comparative bioinformatics
• Sequence alignment
• Motif detection (phylogenetic motifs,
  evolutionary trace, phylogenetic footprinting)
• Correlated mutation (useful for structural
  contact prediction)
• Protein interaction
• Gene networks
• Vaccine devlopment
• And many more

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Phylogeny Problem
U
V
W
X
Y
TAGCCCA
TAGACTT
TGCACAA
TGCGCTT
AGGGCAT
X
U
Y
V
W
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Bipartitions

• Phylogenies are equivalent to bipartitions

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Topological differences
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Phylogeny Problem

• Two main methodologies
• Alignment first and phylogeny second
• Construct alignment using one of the MANY
  alignment programs in the literature
• Do manual (eye) adjustments if necessary
• Apply a phylogeny reconstruction method
• Fast but biologically not realistic
• Phylogeny is highly dependent on accuracy of
  alignment (but so is the alignment on the
  phylogeny!)
• Simultaneously alignment and phylogeny
  reconstruction
• Output both an alignment and phylogeny
• Computationally much harder
• Biologically more realistic as insertions,
  deletions, and mutations occur during the
  evolutionary process

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First methodology

• Compute alignment (for now we assume we are given
  an alignment)
• Construct a phylogeny (two approaches)
• Distance-based methods
• Input Distance matrix containing pairwise
  statistical estimation of aligned sequences
• Output Phylogenetic tree
• Fast but less accurate
• Character-based methods
• Input Sequence alignment
• Output Phylogenetic tree
• Accurate but computationally very hard

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Distance-based methods
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Evolution on a single edge

• Poisson process
• Number of changes in a fixed time interval t is
  independent of changes in any other
  non-overlapping time interval u
• Number of changes in time interval t is
  proportional to the length of the interval
• No changes in time interval of length 0
• Let X be the number of nucleotide changes on a
  single edge. We assume X is a Poisson process
• Probability dictates that

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Evolution on a single edge

• We want to compute (the probability of a
  nucleotide change on edge e)
• The probability of observing a change is just the
  sum of probabilities of observing k changes over
  all possible values of k (excluding even ones
  because those changes cannot be seen)

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Evolution on a single edge

• Expected number of nucleotide changes on a given
  edge is given by
• Key is additive

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Additivity

• Assume we have a path of k edges and that p1,
  p2,, pk are the probabilities of change on each
  edge of the path
• Using induction we can show that
• Multiplicative term is hard to deal with and does
  not easily decompose into a product or sum of
  pis

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Additivity

• But the expected number of nucleotide changes on
  the path p is elegant

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Evolutionary models

• Simple 0,1 alphabet evolutionary model
• i.i.d. model
• uniformly random root sequence
• Jukes-Cantor
• Uniformly random root sequence
• i.i.d. model

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Evolutionary models

• General Markov Model
• Uniformly random root sequence
• i.i.d. model
• For time reversible models

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Variation across sites

• Standard assumption of how sites can vary is that
  each site has a multiplicative scaling factor
• Typically these scaling factors are drawn from a
  Gamma distribution (or Gamma plus invariant)

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Special issues

• Molecular clock the expected number of changes
  for a site is proportional to time
• No-common-mechanism model there is a random
  variable for every combination of edge and site

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Evolutionary distance estimation
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Estimating evolutionary distances

• For sequences A and B what is the evolutionary
  distance under the Jukes-Cantor model?
• ACCTGTGGGTAACCACCC
• ACCTGAGGGATAGGTCCG
• But we dont know what is

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Estimating evolutionary distances

• Assume nucleotide changes are Bernoulli trials
  (i.i.d. trials of success or failure)
• is probability of head in n Bernoulli trials
  (n is sequence length)
• Compute a maximum likelihood estimate for

ACCTGTGGGTAACCACCC ACCTGAGGGATAGGTCCG
0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 1
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Estimating evolutionary distance

• We want to find the value of p that maximizes the
  probability
• Set dP/dp to 0 and solve for p to get

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Estimating evolutionary distances

• 5/18
• Continuing in this manner we estimate for
  all pairs of sequences in the alignment
• We now have a distance matrix under a
  biologically sound evolutionary model

ACCTGTGGGTAACCACCC ACCTGAGGGATAGGTCCG
0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 1
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Distance methods
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Distance methods

• UPGMA similar to hierarchical clustering but not
  additive
• Neighbor-joining more sophisticated and additive
• What is additivity?

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Additivity
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UPGMA

• UPGMA is not additive but works for
• ultrametric trees. Takes O(n2) time

B
A
C
D
A
6
26
26
10
10
26
26
B
6
C
3
3
3
3
D
A
D
C
B
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UPGMA

• Initialize n clusters where each cluster i
  contains the sequence i
• Find closest pair of clusters i, j, using
  distances in matrix D
• Make them neighbors in the tree by adding new
  node (ij), and set distance from (ij) to i and j
  as Dij/2
• Update distance matrix D for all clusters k do
  the following (ni and nj are size of clusters i
  and j respectively)
• Delete columns and rows for i and j in D and add
  new ones corresponding to cluster (ij) with
  distances as computed above
• Goto step 2 until only one cluster is left

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UPGMA
B
A
C
D
13
13
A
6
26
26
26
26
B
3
6
3
C
3
3
D
A
D
C
B
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UPGMA

• Doesnt work (in general) for non-ultrametric
• trees

B
A
C
D
3
3
A
13
16
26
3
3
12
19
B
10
10
C
B
13
C
D
D
A
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UPGMA

• UPGMA constructs incorrect tree here

7.25
B
A
C
D
7.25
A
13
16
26
7.25
7.25
12
19
B
6
6
13
C
B
D
A
C
D
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UPGMA

• Bipartition (BC,AD) is not in true tree

7.25
3
3
3
3
7.25
7.25
7.25
10
10
C
B
6
6
D
A
B
D
A
C
True tree
UPGMA tree
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Neighbor joining

• Additive and O(n2) time
• Initialization same as UPGMA
• For each species compute
• Select i and j for which
  is minimum
• Make them neighbors in the tree by adding new
  node (ij), and set distance from (ij) to i and j
  as

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Neighbor joining

1. Update distance matrix D for all clusters k do
   the following
2. Delete columns and rows for i and j in D and add
   new ones corresponding to cluster (ij) with
   distances as computed above
3. Go to 3 until two nodes/clusters are left

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NJ

• NJ constructs the correct tree for additive
• matrices

B
A
C
D
3
3
A
13
16
26
3
3
12
19
B
10
10
C
B
13
C
D
D
A
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Simulation studies
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Simulation studies

• The true evolutionary tree is never known in
  practice. Simulation allows us to study accuracy
  of methods under biologically realistic scenarios
• Mathematics behind the phylogenetics is often
  complex and challenging. Simulation allows us to
  study algorithms when not possible theoretically
  and also examine algorithm performance under
  various conditions such as different evolutionary
  rates, sequence lengths, or numbers of taxa

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Statistical consistency

• As sequence lengths tend to infinity the distance
  estimation improves and eventually leads to the
  true additive matrix
• If a method like NJ is then applied we get the
  true tree.
• In practice, however, we have limited sequence
  length. Therefore we want to know how much
  sequence length a method requires to achieve low
  error

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Convergence rates

• Can be studied experimentally or theoretically
• Theoretical results offer loose bounds
• Experiments (under simulation) provide more
  realistic bounds on sequence lengths

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Sequence length requirements
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Sequence length requirements
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Typical performance study
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Sequence lengths for NJ
Sequence lengths required to obtain 90 accuracy
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Error rate of NJ
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Improving sequence length requirements

• Later we will look at Disk-Covering Methods and
  study sequence length requirements of other
  methods (in addition to NJ)

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Maximum Parsimony

• Character based method
• NP-hard (reduction to the Steiner tree problem)
• Widely-used in phylogenetics
• Slower than NJ but more accurate
• Faster than ML
• Assumes i.i.d.

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Maximum Parsimony

• Input Set S of n aligned sequences of length k
• Output A phylogenetic tree T
• leaf-labeled by sequences in S
• additional sequences of length k labeling the
  internal nodes of T
• such that is minimized.

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Maximum parsimony (example)

• Input Four sequences
• ACT
• ACA
• GTT
• GTA
• Question which of the three trees has the best
  MP scores?

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Maximum Parsimony
ACT
ACT
ACA
GTA
GTT
GTT
ACA
GTA
GTA
ACA
ACT
GTT
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Maximum Parsimony
ACT
ACT
ACA
GTA
GTT
GTA
ACA
ACT
2
1
1
3
3
2
GTT
GTT
ACA
GTA
MP score 7
MP score 5
GTA
ACA
ACA
GTA
2
1
1
ACT
GTT
MP score 4
Optimal MP tree
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Maximum Parsimony computational complexity
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Local search strategies
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Local search for MP

• Determine a candidate solution s
• While s is not a local minimum
• Find a neighbor s of s such that MP(s)ltMP(s)
• If found set ss
• Else return s and exit
• Time complexity unknown---could take forever or
  end quickly depending on starting tree and local
  move
• Need to specify how to construct starting tree
  and local move

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Starting tree for MP

• Random phylogeny---O(n) time
• Greedy-MP

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Greedy-MP
Greedy-MP takes O(n2k2) time
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Local moves for MP NNI

• For each edge we get two different topologies
• Neighborhood size is 2n-6

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Local moves for MP SPR

• Neighborhood size is quadratic in number of taxa
• Computing the minimum number of SPR moves between
  two rooted phylogenies is NP-hard

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Local moves for MP TBR

• Neighborhood size is cubic in number of taxa
• Computing the minimum number of TBR moves between
  two rooted phylogenies is NP-hard

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Local optima is a problem
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Iterated local search escape local optima by
perturbation
Local optimum
Local search
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Iterated local search escape local optima by
perturbation
Local optimum
Local search
Perturbation
Output of perturbation
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Iterated local search escape local optima by
perturbation
Local optimum
Local search
Perturbation
Local search
Output of perturbation
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ILS for MP

• Ratchet
• Iterative-DCM3
• TNT

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Next time

• Performance studies on local search for MP
• Maximum likelihood
• Alignment