Phylogeny II : Parsimony, ML, SEMPHY

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Phylogeny II Parsimony, ML, SEMPHY
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Phylogenetic Tree

• Topology bifurcating
• Leaves - 1N
• Internal nodes N12N-2

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Character Based Methods

• We start with a multiple alignment
• Assumptions
• All sequences are homologous
• Each position in alignment is homologous
• Positions evolve independently
• No gaps
• We seek to explain the evolution of each position
  in the alignment

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Parsimony

• Character-based method
• A way to score trees (but not to build trees!)
• Assumptions
• Independence of characters (no interactions)
• Best tree is one where minimal changes take place

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A Simple Example

• What is the parsimony score of

A CAGGTA B CAGACA C CGGGTA D TGCACT E TGCGTA
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A Simple Example
A CAGGTA B CAGACA C CGGGTA D TGCACT E TGCGTA

• Each column is scored separately.
• Lets look at the first column
• Minimal tree has one evolutionary change

C
T
C
T
C
C
C
T
T ? C
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Evaluating Parsimony Scores

• How do we compute the Parsimony score for a given
  tree?
• Traditional Parsimony
• Each base change has a cost of 1
• Weighted Parsimony
• Each change is weighted by the score c(a,b)

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Traditional Parsimony
a
a

• Solved independently for each position
• Linear time solution

a,g
a
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Evaluating Weighted Parsimony

• Dynamic programming on the tree
• S(i,a) cost of tree rooted at i if i is labeled
  by a
• Initialization
• For each leaf i set S(i,a) 0 if i is labeled by
  a, otherwise S(i,a) ?
• Iteration
• if k is a node with children i and j, then
  S(k,a) minb(S(i,b)c(a,b))
  minb(S(j,b)c(a,b))
• Termination
• cost of tree is minaS(r,a) where r is the root

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Cost of Evaluating Parsimony

• Score is evaluated on each position independetly.
  Scores are then summed over all positions.
• If there are n nodes, m characters, and k
  possible values for each character, then
  complexity is O(nmk)
• By keeping traceback information, we can
  reconstruct most parsimonious values at each
  ancestor node

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Maximum Parsimony
1 2 3 4 5 6 7 8 9 10 Species 1
- A G G G T A A C T G Species 2 - A C G A T T A
T T A Species 3 - A T A A T T G T C T Species 4
- A A T G T T G T C G
How many possible unrooted trees?
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Maximum Parsimony
How many possible unrooted trees?
1 2 3 4 5 6 7 8 9
10 Species 1 - A G G G T A A C T G Species 2 - A
C G A T T A T T A Species 3 - A T A A T T G T C
T Species 4 - A A T G T T G T C G
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Maximum Parsimony
How many substitutions?
MP
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Maximum Parsimony
1 2 3 4 5 6 7 8 9 10 1 - A G G G T A A
C T G 2 - A C G A T T A T T A 3 - A T A A T T G
T C T 4 - A A T G T T G T C G
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Maximum Parsimony
1 2 3 4 5 6 7 8 9 10 1 - A G G G T A A C
T G 2 - A C G A T T A T T A 3 - A T A A T T G T
C T 4 - A A T G T T G T C G
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Maximum Parsimony
4 1 - G 2 - C 3 - T 4 - A
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Maximum Parsimony
1 2 3 4 5 6 7 8 9 10 1 - A G G G T A A C
T G 2 - A C G A T T A T T A 3 - A T A A T T G T
C T 4 - A A T G T T G T C G
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Maximum Parsimony
1 2 3 4 5 6 7 8 9 10 1 - A G G G T A A C
T G 2 - A C G A T T A T T A 3 - A T A A T T G T
C T 4 - A A T G T T G T C G
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Maximum Parsimony
4 1 - G 2 - A 3 - A 4 - G
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Maximum Parsimony
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Maximum Parsimony
1 2 3 4 5 6 7 8 9 10 1 - A G G G T A A C
T G 2 - A C G A T T A T T A 3 - A T A A T T G T
C T 4 - A A T G T T G T C G
0 3 2 2 0 1 1 1 1 3 14
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Searching for Trees
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Searching for the Optimal Tree

• Exhaustive Search
• Very intensive
• Branch and Bound
• A compromise
• Heuristic
• Fast
• Usually starts with NJ

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Phylogenetic Tree Assumptions

• Topology bifurcating
• Leaves - 1N
• Internal nodes N12N-2
• Lengths t ti for each branch
• Phylogenetic tree (Topology, Lengths) (T,t)

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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 2-parameter model
• Such models are used to derive the probabilities

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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 a.

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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

• Both models satisfy the following properties
• Lack of memory
• 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

x5
t4
x4
t2
t3
t1
x1
x2
x3
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Computing the Tree Likelihood

• We are interested in the probability of observed
  data given tree and branch lengths
• Computed by summing over internal nodes
• This can be done efficiently using a tree upward
  traversal pass.

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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 search
• Sampling methods (Metropolis)

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Optimal Tree Search

• Perform search over possible topologies

Parameter space
Parametric optimization (EM)
Local Maxima
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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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Structural EM

• Idea Use parameters found for current topology
  to help evaluate new topologies.
• Outline
• Perform search in (T, t) space.
• Use EM-like iterations
• E-step use current solution to compute expected
  sufficient statistics for all topologies
• M-step select new topology based on these
  expected sufficient statistics

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The Complete-Data Scenario

• Suppose we observe H, the ancestral sequences.

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Expected Likelihood

• Start with a tree (T0,t0)
• Compute
• Formal justification
• Define
• Theorem
• Consequence improvement in expected score ?
  improvement in likelihood

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Proof

• Theorem
• Simple application of Jensens inequality

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Algorithm Outline
Unlike standard EM for trees, we compute all
possible pairwise statistics Time O(N2M)
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Algorithm Outline
Pairwise weights
This stage also computes the branch length for
each pair (i,j)
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Algorithm Outline
Max. Spanning Tree
Fast greedy procedure to find tree By
construction Q(T,t) ? Q(T0,t0) Thus,
l(T,t) ? l(T0,t0)
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Algorithm Outline
Fix Tree
Remove redundant nodes Add nodes to break large
degree This operation preserves likelihood
l(T1,t) l(T,t) ? l(T0,t0)
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Assessing trees the Bootstrap

• Often we dont trust the tree found as the
  correct one.
• Bootstrapping
• Sample (with replacement) n positions from the
  alignment
• Learn the best tree for each sample
• Look for tree features which are frequent in all
  trees.
• For some models this procedure approximates the
  tree posterior P(T X1,,Xn)

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Algorithm Outline
Construct bifurcation T1
New Tree
Thm l(T1,t1) ? l(T0,t0)
These steps are then repeated until convergence