1. CS 394C: Computational Biology Algorithms
Tandy Warnow
Department of Computer Sciences
University of Texas at Austin

2. DNA Sequence Evolution
-3 mil yrs
-2 mil yrs
-1 mil yrs
today
AAGACTT
TGGACTT
AAGGCCT
AGGGCAT
TAGCCCT
AGCACTT
AGCGCTT
AGCACAA
TAGACTT
TAGCCCA
AAGACTT
TGGACTT
AAGGCCT
AGGGCAT
TAGCCCT
AGCACTT
AAGGCCT
TGGACTT
TAGCCCA
TAGACTT
AGCGCTT
AGCACAA
AGGGCAT
TAGCCCT
AGCACTT

3. Molecular SystematicsV W X Y
AGGGCAT
TAGCCCA
TAGACTT
TGCACAA
TGCGCTT X U Y V W

4. 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

5. Distance-based methods
Theorem: Let (T,) be a Cavender-Farris model tree, with additive matrix [(i,j)]. Let >0 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))).

6. 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
400
800
1200
1600
No. Taxa

7. 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.

8. Maximum parsimony (example)
Input: Four sequences
ACT
ACA
GTT
GTA
Question: which of the three trees has the best MP scores?

9. Maximum Parsimony
ACT
GTA
ACA
ACT
GTT
ACA
GTT
GTA
GTA
ACA
ACT
GTT

10. Maximum Parsimony ACT GTA ACA ACT GTT GTA ACA ACT 2 1 1 2 GTT 3 3 GTT
MP score = 7
MP score = 5
GTA
ACA
ACA
GTA 2 1 1
ACT
GTT
MP score = 4
Optimal MP tree

11. Maximum Parsimony Optimal labeling can be computed in polynomial
ACT
ACA
GTT
GTA 1 2
MP score = 4
Finding the optimal MP tree is NP-hard
Optimal labeling can be computed in polynomial
time using Dynamic Programming

12. Solving NP-hard problems exactly is … unlikely
#leaves
#trees 4 3 5 15 6
105 7
945 8
10395 9
135135 10 20
2.2 x 1020
100
4.5 x 10190
1000
2.7 x
Number of (unrooted) binary trees on n leaves is (2n-5)!!
If each tree on 1000 taxa could be analyzed in seconds, we would find the best tree in
2890 millennia

13. Approaches for “solving” MP and ML (and other NP-hard problems in phylogeny)
Hill-climbing heuristics (which can get stuck in local optima)
Randomized algorithms for getting out of local optima
Approximation algorithms for MP (based upon Steiner Tree approximation algorithms) -- however, the approx. ratio that is needed is probably 1.01 or smaller!
Phylogenetic trees
Cost
Global optimum
Local optimum

14. 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

15. 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.

16. 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).

17. 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.

18. Disk-Covering Method (DCM)

19. 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

20. 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.

21. DCM1-boosting distance-based methods [Nakhleh 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
400
800
1200
1600
No. Taxa

22. 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).

23. 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.

24. 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