CS 394C: Computational Biology Algorithms

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CS 394C Computational Biology Algorithms

• Tandy Warnow
• Department of Computer Sciences
• University of Texas at Austin

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DNA Sequence Evolution
3
Molecular Systematics
U
V
W
X
Y
TAGCCCA
TAGACTT
TGCACAA
TGCGCTT
AGGGCAT
X
U
Y
V
W
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Phylogeny estimation methods

• Distance-based (Neighbor joining, NQM, and
  others) mostly statistically consistent and
  polynomial time
• Maximum parsimony and maximum compatibility
  NP-hard and not statistically consistent
• Maximum likelihood NP-hard and usually
  statistically consistent (if solved exactly)
• Bayesian Methods statistically consistent if run
  long enough

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Distance-based methods

• Theorem Let (T,?) be a Cavender-Farris model
  tree, with additive matrix ?(i,j). Let ?gt0 be
  given. The sequence length that suffices for
  accuracy with probability at least 1- ? of NJ
  (neighbor joining) and the Naïve Quartet Method
  is O(log n e(O(max ?(i,j))).

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Neighbor joining (although statistically
consistent) has poor performance on large
diameter trees Nakhleh et al. ISMB 2001

• Simulation study based upon fixed edge lengths,
  K2P model of evolution, sequence lengths fixed to
  1000 nucleotides.
• Error rates reflect proportion of incorrect edges
  in inferred trees.

0.8
NJ
0.6
Error Rate
0.4
0.2
0
0
400
800
1600
1200
No. Taxa
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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
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Solving NP-hard problems exactly is unlikely
leaves trees
4 3
5 15
6 105
7 945
8 10395
9 135135
10 2027025
20 2.2 x 1020
100 4.5 x 10190
1000 2.7 x 102900

• Number of (unrooted) binary trees on n leaves is
  (2n-5)!!
• If each tree on 1000 taxa could be analyzed in
  0.001 seconds, we would find the best tree in
• 2890 millennia

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Approaches for solving MP and ML(and other
NP-hard problems in phylogeny)

1. Hill-climbing heuristics (which can get stuck in
   local optima)
2. Randomized algorithms for getting out of local
   optima
3. Approximation algorithms for MP (based upon
   Steiner Tree approximation algorithms) --
   however, the approx. ratio that is needed is
   probably 1.01 or smaller!

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Problems with techniques for MP and ML
Shown here is the performance of a TNT heuristic
maximum parsimony analysis on a real dataset of
almost 14,000 sequences. (Optimal here means
best score to date, using any method for any
amount of time.) Acceptable error is below 0.01.
Performance of TNT with time
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MP and Cavender-Farris

• Consider a tree (AB,CD) with two very long
  branches leading to A and C, and all other
  branches very short.
• MP will be statistically inconsistent (and
  positively misleading) on this tree.

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Problems with existing phylogeny reconstruction
methods

• Polynomial time methods (generally based upon
  distances) have poor accuracy with large diameter
  datasets.
• Heuristics for NP-hard optimization problems take
  too long (months to reach acceptable local
  optima).

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Warnow et al. Meta-algorithms for phylogenetics

• Basic technique determine the conditions under
  which a phylogeny reconstruction method does well
  (or poorly), and design a divide-and-conquer
  strategy (specific to the method) to improve its
  performance
• Warnow et al. developed a class of
  divide-and-conquer methods, collectively called
  DCMs (Disk-Covering Methods). These are based
  upon chordal graph theory to give fast
  decompositions and provable performance
  guarantees.

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Disk-Covering Method (DCM)
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Improving phylogeny reconstruction methods using
DCMs

• Improving the theoretical convergence rate and
  performance of polynomial time distance-based
  methods using DCM1
• Speeding up heuristics for NP-hard optimization
  problems (Maximum Parsimony and Maximum
  Likelihood) using Rec-I-DCM3

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DCM1 Warnow, St. John, and Moret, SODA 2001
Exponentially converging method
Absolute fast converging method
DCM
SQS

• A two-phase procedure which reduces the sequence
  length requirement of methods. The DCM phase
  produces a collection of trees, and the SQS phase
  picks the best tree.
• The base method is applied to subsets of the
  original dataset. When the base method is NJ,
  you get DCM1-NJ.

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DCM1-boosting distance-based methodsNakhleh et
al. ISMB 2001

• Theorem DCM1-NJ converges to the true tree from
  polynomial length sequences

0.8
NJ
DCM1-NJ
0.6
Error Rate
0.4
0.2
0
0
400
800
1600
1200
No. Taxa
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Rec-I-DCM3 significantly improves performance
(Roshan et al. CSB 2004)
Current best techniques
DCM boosted version of best techniques
Comparison of TNT to Rec-I-DCM3(TNT) on one large
dataset. Similar improvements obtained for RAxML
(maximum likelihood).
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Summary (so far)

• Optimization problems in biology are almost all
  NP-hard, and heuristics may run for months before
  finding local optima.
• The challenge here is to find better heuristics,
  since exact solutions are very unlikely to ever
  be achievable on large datasets.

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Summary

• NP-hard optimization problems abound in phylogeny
  reconstruction, and in computational biology in
  general, and need very accurate solutions
• Many real problems have beautiful and natural
  combinatorial and graph-theoretic formulations