CS 394C: Computational Biology Algorithms

Tandy WarnowDepartment of Computer Sciences

University of Texas at Austin

DNA Sequence Evolution

AAGACTT

TGGACTTAAGGCCT

-3 mil yrs

-2 mil yrs

-1 mil yrs

today

AGGGCAT TAGCCCT AGCACTT

AAGGCCT TGGACTT

TAGCCCA TAGACTT AGCGCTTAGCACAAAGGGCAT

AGGGCAT TAGCCCT AGCACTT

AAGACTT

TGGACTTAAGGCCT

AGGGCAT TAGCCCT AGCACTT

AAGGCCT TGGACTT

AGCGCTTAGCACAATAGACTTTAGCCCAAGGGCAT

Molecular Systematics

TAGCCCA TAGACTT TGCACAA TGCGCTTAGGGCAT

U V W X Y

U

V W

X

Y

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.

NJ

0 400 800 16001200No. Taxa

0

0.2

0.4

0.6

0.8

Err

or R

ate

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. ∑∈ )(),(

),(TEji

jiH

Maximum parsimony (example)

• Input: Four sequences– ACT– ACA– GTT– GTA

• Question: which of the three trees has the best MP scores?

Maximum Parsimony

ACT

GTT ACA

GTA ACA ACT

GTAGTT

ACT

ACA

GTT

GTA

Maximum Parsimony

ACT

GTT

GTT GTA

ACA

GTA

12

2

MP score = 5

ACA ACT

GTAGTT

ACA ACT

3 1 3

MP score = 7

ACT

ACA

GTT

GTAACA GTA

1 2 1

MP score = 4

Optimal MP tree

Maximum Parsimony

ACT

ACA

GTT

GTAACA GTA

1 2 1

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

• 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

#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

1. Hill-climbing heuristics (which can get stuck in local optima)2. Randomized algorithms for getting out of local optima3. Approximation algorithms for MP (based upon Steiner Tree approximation

algorithms) -- however, the approx. ratio that is needed is probably 1.01 or smaller!

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

Phylogenetic trees

Cost

Global optimum

Local optimum

Problems with techniques for MP and ML

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 4 8 12 16 20 24

Hours

Average MP score above

optimal, shown as a percentage of

the optimal

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

• 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.

DCM SQSExponentiallyconvergingmethod

Absolute fast convergingmethod

DCM1-boosting distance-based methods[Nakhleh et al. ISMB 2001]

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

NJ

DCM1-NJ

0 400 800 16001200No. Taxa

0

0.2

0.4

0.6

0.8

Err

or R

ate

Rec-I-DCM3 significantly improves performance (Roshan et al. CSB 2004)

Comparison of TNT to Rec-I-DCM3(TNT) on one large dataset.Similar improvements obtained for RAxML (maximum likelihood).

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 4 8 12 16 20 24

Hours

Average MP score above

optimal, shown as a percentage of

the optimal

Current best techniques

DCM boosted version of best techniques

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