Understanding sets of trees

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Understanding sets of trees

• CS 394C
• September 10, 2009

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Basic challenge

• Phylogenetic analyses are sometimes based upon a
  single marker, but often based upon many markers
• Each marker can be analyzed separately, or the
  entire set can be combined into one
  super-matrix
• Each matrix (each dataset) can result in many
  trees (almost no matter how you analyze the
  matrix)
• What to do with huge numbers of trees?

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What to do?

• How to estimate evolutionary history from many
  trees
• How to efficiently store large sets of trees
• How to enable efficient queries of the set of
  trees

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What to do?

• How to estimate evolutionary history from many
  trees
• How to efficiently store large sets of trees
• How to enable efficient queries of the set of
  trees

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First, a few questions

• Why are gene trees different from the species
  tree?
• Why are estimated gene trees different from the
  true gene tree?
• Under what conditions is the true evolutionary
  history not a tree? (i.e., what is
  reticulation?)

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Reticulation

• Evolutionary histories can be reticulate (meaning
  non-treelike)
• Horizontal Gene Transfer (HGT)
• Hybrid speciation
• Recombination
• Most phylogeny estimation methods produce trees.
• Good resource about reticulate phylogenies book
  chapter by Luay Nakhleh (see 394C webpage for the
  link)

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• We will assume that all evolutionary histories
  are treelike for the remainder of todays
  presentation.
• Later in the course well discuss reticulate
  evolution

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Estimated Gene Trees can differ from Species Trees

• Biological reasons
• Deep coalescent events (alleles)
• Gene duplication and loss (gene families)
• Computational reasons
• Insufficient time
• Poor methods (e.g., UPGMA)
• Poor models (e.g., ML using Jukes-Cantor)
• Data issues
• Insufficient data (meaning not enough sites)
• Poor alignments

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Examples of problems

• When true gene trees can differ from species
  tree
• Given a collection of gene trees, find a species
  tree that minimizes the number of deep
  coalescent events
• When true gene trees should equal the species
  tree
• Given a collection of gene trees, find a species
  tree that minimizes the total distance to the
  gene trees

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When gene trees can differ from species tree

• Software/Algorithms for deep-coalescent (see
  PhyloNet from Nakhlehs webpage at Rice)
• GLASS (Roch and Mossel) - distance-based
• MDC (Than and Nakhleh) - parsimony
• STEM (Kubatko) - ML
• BEST (Liu et al.) - Bayesian
• BUCKy (Ané et al.) - Bayesian
• Software/Algorithms for duplication-loss
• NOTUNG (Durand)
• Duptree (Bansal et al.)
• Hallet and Lagergren - algorithms/complexity

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When gene trees should equal the species tree

• The problem here is that estimated gene trees can
  differ from the true gene trees.
• Although the problem is simple, it is still
  interesting -- computationally and
  mathematically.
• Plus, we can still make novel contributions.

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The very simplest problem

• Easiest case
• One species tree, true gene trees will agree with
  the species tree,
• Estimated trees are on the full set of taxa
• Approaches
• Consensus methods return a tree on the entire
  set S of taxa summarizing the input trees
• Agreement methods return a tree on a subset of
  the taxa on which the trees agree
• Clustering, then consensus/agreement

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Consensus methods

• These are the most usual ways of analyzing
  datasets of trees
• Examples
• Strict consensus
• Majority consensus
• Greedy consensus (aka extended majority)
• Others less frequently used include Gordons,
  Adams, the Strict Consensus Supertree, Local
  Consensus methods, and more.
• Survey paper by David Bryant for some of these

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Simplest problems, cont.

• Agreement methods return trees on subsets of S,
  on which the trees are the same (or compatible)
• MAST maximum agreement subtree (used in
  practice, sometimes)
• MCST maximum compatible subtree (Ganapathy et
  al., not used in practice)
• The difference between these is how polytomies
  are handled

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Soft vs. hard polytomies

• Polytomy node of high degree (greater than three
  for an unrooted tree)
• Polytomies arise in estimations when consensus
  methods are used
• Polytomies also arise when contracting short
  branches in estimated trees
• Polytomies can be hard (representing true
  radiations) or soft (representing lack of
  information)

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Compatible source trees

• Estimated trees can be compatible when we
  interpret polytomies as soft
• Compatible means that there is a tree which is
  a common refinement.
• Example 123456, 123456, 123546.
• We can compute the compatibility tree (when it
  exists) in O(nk) time, where nS and there are
  k source trees

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Computational complexity

• Most consensus methods (which return a tree on
  the entire set S of taxa) are polynomial time.
• Most agreement methods (which return a tree on
  the largest subset of the taxa on which the
  source trees agree) are based upon NP-hard
  problems. Some (e.g., MAST) have fixed-parameter
  polynomial time solutions.

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Supertree problems

• Realistic complexity not all the source trees
  are on the same set of taxa.
• Obvious problems
• Find the tree on which all the source trees agree
  (if it exists).
• Find the tree on which a maximum number of the
  source trees agree.
• Both are NP-hard.

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Quartet compatibility

• Simple case all the source trees are on four
  taxa.
• We ask does there exist a tree which agrees with
  all the source trees?
• NP-hard!

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Quartet tree amalgamation

• Given collection of quartet trees, find a tree
  which agrees with a maximum number of these
  quartet trees
• NP-hard, since compatibility is NP-hard
• Hard to approximate, but PTAS if you have a tree
  on every quartet of taxa (Jiang et al.)

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Quartet amalgamation algorithms

• Quartet Puzzling (Strimmer and von Haeseler)
• Q (Berry et al.)
• Quartet Cleaning (Berry et al.)
• Weight Optimization (Ranwez and Gascuel)
• Quartets MaxCut (Snir and Rao)
• But see also the paper (St. John et al.)
  evaluating early quartet methods on the CS 394C
  webpage

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What about rooted trees?

• Given set of rooted source trees, we ask
• Is there a tree on which all the rooted source
  trees are correct?

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Rooted tree compatibility

• Aho, Sagiv, Szymanski, and Ullman polynomial
  time, recursive algorithm
• If n1, return the singleton tree.
• If ngt1, then compute an equivalence relation on
  the set of taxa as follows.
• For each rooted triple ((a,b),c) in the set, put
  a and b in the same equivalence class.
• Compute transitive closure.
• If only one equivalence class, reject (set is
  incompatible). Otherwise, recurse on each subset,
  and return tree obtained by making all
  recursively computed trees sibling subtrees.

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Subtree compatibility

• If source trees are rooted, then compatibility
  can be tested in polynomial time. Optimization
  problems are NP-hard, however.
• If source trees are unrooted, then compatibility
  is NP-hard. And so optimization problems are
  also NP-hard.

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Supertree problems, in practice

• In practice, the most frequently used supertree
  method is MRP, for Matrix Representation with
  Parsimony.
• There are, however, many other supertree methods!

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Many Supertree Methods

• MRP
• weighted MRP
• Min-Cut
• Modified Min-Cut
• Semi-strict Supertree
• MRF
• MRD
• QILI

• SDM
• Q-imputation
• PhySIC
• Majority-Rule Supertrees
• Maximum Likelihood Supertrees
• and many more ...

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MRP

• Idea take every sourcetree, and replace it with
  a matrix of 0,1,?.
• Concatenate the matrices.
• Apply Maximum Parsimony.
• If all the source trees are compatible, then an
  exact solution to MRP will return the
  compatibility trees.

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Homework, due 9/15

• Read two papers (linked on the webpage)
• St. John et al., about quartet-based methods
• Moret et al., about sequence-length requirements
• Pick one, write summary, and include questions

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Question!

• How do you feel about occasionally having class
  on some Monday or Friday, so we can have guest
  lectures?