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

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

Molecular Systematics V W X Y AGGGCAT TAGCCCA TAGACTT TGCACAA TGCGCTT X U Y V W

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

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))).

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

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.

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

Maximum Parsimony ACT GTA ACA ACT GTT ACA GTT GTA GTA ACA ACT GTT

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

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

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

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

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

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.

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).

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.

Disk-Covering Method (DCM)

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

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.

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

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).

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.

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