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CIS786, Lecture 3

• Usman Roshan

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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(n3k) time
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Faster Greedy MP3-way labeling

• If we can assign optimal labels to each internal
  node rooted in each possible way, we can speed up
  computation by order of n
• Optimal 3-way labeling
• Sort all 3n subtrees using bucket sort in O(n)
• Starting from small subtrees compute optimal
  labelings
• For each subtree rooted at v, the optimal
  labelings of children nodes is already computed
• Total time O(nk)

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Faster Greedy MP3-way labeling

• If we can assign optimal labels to each internal
  node rooted in each possible way, we can speed up
  computation by order of n
• Optimal 3-way labeling
• Sort all 3n subtrees using bucket sort in O(n)
• Starting from small subtrees compute optimal
  labelings
• For each subtree rooted at v, the optimal
  labelings of children nodes is already computed
• Total time O(nk)

With optimal labeling it takes constant Time to
compute MP score for each Edge and so total
Greedy-MP time Is O(n2k)
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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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• Tree Bisection and Reconnection (TBR)

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• Tree Bisection and Reconnection (TBR)

Delete an edge
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• Tree Bisection and Reconnection (TBR)

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• Tree Bisection and Reconnection (TBR)

Reconnect the trees with a new edge that
bifurcates an edge in each tree
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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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Iterated local search escape local optima by
perturbation
Local optimum
Local search
Perturbation
Local search
Output of perturbation
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Ratchet

• Perturbation input alignment and phylogeny
• Sample with replacement p of sites and reweigh
  them to w
• Perform local search on modified dataset starting
  from the input phylogeny
• Reset the alignment to original after completion
  and output the local minimum

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Ratchet escaping local minimaby data
perturbation
Local optimum
Local search
Ratchet search
Local search
Output of ratchet
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Ratchet escaping local minimaby data
perturbation
Local optimum
Local search
Ratchet search
Local search
Output of ratchet
But how well does this perform? We have to
examine this experimentally on real data
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Experimental methodology for MP on real data

• Collect alignments of real datasets
• Usually constructed using ClustalW
• Followed by manual (eye) adjustments
• Must be reliable to get sensible tree!
• Run methods for a fixed time period
• Compare MP scores as a function of time
• Examine how scores improve over time
• Rate of convergence of different methods (not
  sequence length but as a function of time)

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Experimental methodology for MP on real data

• We use rRNA and DNA alignments
• Obtained from researchers and public databases
• We run iterative improvement and ratchet each for
  24 hours beginning from a randomized greedy MP
  tree
• Each method was run five times and average scores
  were plotted
• We use PAUP---very widely used software package
  for various types of phylogenetic analysis

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500 aligned rbcL sequences (Zilla dataset)
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854 aligned rbcL sequences
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2000 aligned Eukaryotes
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7180 aligned 3domain
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13921 aligned Proteobacteria
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Comparison of MP heuristics

• What about other techniques for escaping local
  minima?
• TNT a combination of divide-and-conquer,
  simulated annealing, and genetic algorithms
• Sectorial search (random) construct ancestral
  sequence states using parsimony randomly select
  a subset of nodes compute iterative-improvement
  trees and if better tree found then replace
• Genetic algorithm (fuse) Exchange subtrees
  between two trees to see if better ones are found
• Default search (1) Do sectorial search starting
  from five randomized greedy MP trees (2) apply
  genetic algorithm to find better ones (3) output
  best tree

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Comparison of MP heuristics

• What about other techniques for escaping local
  minima?
• TNT a combination of divide-and-conquer,
  simulated annealing, and genetic algorithms
• Sectorial search (random) construct ancestral
  sequence states using parsimony randomly select
  a subset of nodes compute iterative-improvement
  trees and if better tree found then replace
• Genetic algorithm (fuse) Exchange subtrees
  between two trees to see if better ones are found
• Default search (1) Do sectorial search starting
  from five randomized greedy MP trees (2) apply
  genetic algorithm to find better ones (3) output
  best tree

How does this compare to PAUP-ratchet?
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Experimental methodology for MP on real data

• We use rRNA and DNA alignments
• Obtained from researchers and public databases
• We run PAUP-ratchet, TNT-default, and
  TNT-ratchet each for 24 hours beginning from
  randomized greedy MP trees
• Each method was run five times on each dataset
  and average scores were plotted

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500 aligned rbcL sequences (Zilla dataset)
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854 aligned rbcL sequences
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2000 aligned Eukaryotes
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7180 aligned 3domain
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13921 aligned Proteobacteria
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Can we do even better?

• Yes! But first lets look at
• Disk-Covering Methods

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Disk Covering Methods (DCMs)

• DCMs are divide-and-conquer booster methods. They
  divide the dataset into small subproblems,
  compute subtrees using a given base method, merge
  the subtrees, and refine the supertree.
• DCMs to date
• DCM1 for improving statistical performance of
  distance-based methods.
• DCM2 for improving heuristic search for MP and
  ML
• DCM3 latest, fastest, and best (in accuracy and
  optimality) DCM

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DCM2 technique for speeding up MP searches
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DCM2 decomposition

• DCM2
• Input distance matrix d, threshold
  , sequences S
• Algorithm
• 1a. Compute a threshold graph G using q and d
• 1b. Perform a minimum weight triangulation of G

• Find separator X in G which minimizes max
  where are the connected components of G
  X
• Output subproblems as .

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Threshold graph

• Add edges until graph is connected
• Perform minimum weight triangulation
• NP-hard
• Triangulated graphperfect elimination ordering
  (PEO)
• Max cliques can be determined in linear time
• Use greedy triangulation heuristic compute PEO
  by adding vertices which minimize largest edge
  added
• Worst case is O(n3) but fast in practice

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Finding DCM2 separator

1. Find separator X in G which minimizes max
   where are the connected components of G
   X
2. Output subproblems as
3. This takes O(n3) worst case time perform depth
   first search on each component (O(n2)) for each
   of O(n) separators

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DCM2 subsets
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DCM3 decomposition - example
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DCM1 vs DCM2
DCM1 decomposition NJ gets better accuracy on
small diameter subproblems (which we shall return
to later)
DCM2 decomposition Getting a smaller number of
smaller subproblems speeds up solution
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We saw how decomposition takes place, now on to
supertree methods
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Supertree Methods
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Optimization problems

• Subtree Compatibility Given set of trees
• ,does there exist tree
  ,such that, (we
  say contains ).
• NP-hard (Steel 1992)
• Special cases are poly-time (rooted trees, DCM)
• MRP also NP-hard

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Direct supertree methods

• Strict consensus supertrees, MinCutSupertrees

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Indirect supertree methods

• MRP, Average consensus

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MRP---Matrix Representation using Parsimony (very
popular)
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Strict Consensus Merger---faster and used in DCMs
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Strict Consensus Merger compatible subtrees
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Strict Consensus Merger compatible but collision
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Strict Consensus Merger incompatible subtrees
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Strict Consensus Merger incompatible and
collision
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Strict Consensus Merger difference from Gordons
SC method
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We saw how decomposition takes place, now on to
supertree methods
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Tree Refinement

• Challenge given unresolved tree, find optimal
  refinement that has an optimal parsimony score
• NP-hard

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Tree Refinement
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We saw how decomposition takes place, now on to
supertree methods
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Comparing DCM decompositions
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Study of DCM decompositions
Comparison of MP scores
Comparison of running times
DCM2 is faster and better than DCM1
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Best DCM (DCM2) vs Random
Comparison of MP scores
Comparison of running times
DCM2 is better than RANDOM w.r.t MP scores and
running times
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DCM2 (comparing two different thresholds)
Comparison of MP scores
Comparison of running times
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Threshold selection techniques
Biological dataset of 503 rRNA sequences.
Threshold value at which we get two subproblems
has best MP score.
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Comparing supertree methods
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MRP vs. SCM
Comparison of MP scores
Comparison of running times

1. SCM is better than MRP

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Comparing tree refinement techniques
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Study of tree refinement techniques
Comparison of MP scores
Comparison of running times
Constrained tree search had best MP scores but is
slower than other methods
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Next time

• DCM1 for improving NJ
• Recursive-Iterative-DCM3 state of the art in
  solving MP and ML