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Evolution informs about everything in biology

- Big genome sequencing projects just produce data – so what?
- Evolutionary history relates all organisms and genes, and helps us understand and predict
- interactions between genes (genetic networks)
- drug design
- predicting functions of genes
- influenza vaccine development
- origins and spread of disease
- origins and migrations of humans

Reconstructing the “Tree” of Life

Handling large datasets: millions of species and

NP-hard optimization problems

NSF funds many projects

towards this goal, under

the Assembling the Tree of Life program

Cyber Infrastructure for Phylogenetic Research

Purpose: to create a national infrastructure of hardware,

algorithms, database technology, etc., necessary to infer

the Tree of Life.

Group: approx. 36 biologists, computer scientists, and

mathematicians from 18 institutions.

Funding: $11.6 M (large ITR grant from NSF).

Bernard Moret

Georgia Tech

David Bader

UCSD/SDSC

Fran Berman

Alex Borchers

John Huelsenbeck

Terri Liebowitz

Mark Miller

University of Connecticut

Paul O Lewis

University of Pennsylvania

Junhyong Kim

Susan Davidson

Sampath Kannan

Val Tannen

Texas A&M

Tiffani Williams

UT Austin

Tandy Warnow

David M. Hillis

Warren Hunt

Robert Jansen

Randy Linder

Lauren Meyers

Daniel Miranker

University of Arizona

David R. Maddison

University of British Columbia

Wayne Maddison

North Carolina State University

Spencer Muse

American Museum of Natural History

Ward C. Wheeler

NJIT

Usman Roshan

UC Berkeley

Satish Rao

Steve Evans

Richard M Karp

Brent Mishler

Elchanan Mossel

Eugene W. Myers

Christos M. Papadimitriou

Stuart J. Russell

Rice

Luay Nakhleh

SUNY Buffalo

William Piel

Florida State University

David L. Swofford

Mark Holder

Yale

Michael Donoghue

Paul Turner

CIPRES Members

CIPRES algorithms research (sample)

- Improved heuristics for NP-hard optimization problems (MP, ML, tree alignment)
- Obtaining better mathematical theory for phylogeny reconstruction methods under Markov models of evolution
- Supertree methods
- Constructing networks rather than trees (detecting and reconstructing reticulate evolution)
- Whole genome phylogeny

This talk

- Phylogeny reconstruction through a divide-and-conquer strategy using chordal graph theory
- “Absolute fast converging” methods
- Improved heuristics for NP-hard optimization problems

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DNA Sequence EvolutionCost

Global optimum

Phylogenetic trees

Phylogenetic reconstruction methods- Heuristics for NP-hard optimization criteria (Maximum Parsimony and Maximum Likelihood)

- Polynomial time distance-based methods: Neighbor Joining, FastME, etc.

3. Bayesian MCMC methods.

Evaluating phylogeny reconstruction methods

- In simulation: how “topologically” accurate are trees reconstructed by the method?
- On real data: how good are the “scores” (typically either MP or ML scores) obtained by the method, as a function of time?

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DNA Sequence EvolutionMarkov models of DNA sequence evolution

General Time Reversible (GTR) Markov Model:

- The state at the root is random
- The model tree is a pair consisting of a rooted binary tree T with edge lengths, where w(e) indicates the number of times a site changes on edge e.
- There is a 4x4 symmetric substitution matrix for the sites, so that if a site changes on an edge, it uses the matrix to determine the probability of each change.
- The evolutionary process is Markovian
- All sites evolve identically and independently

Quantifying Error

FN

FN: false negative

(missing edge)

FP: false positive

(incorrect edge)

50% error rate

FP

Neighbor joining has poor accuracy on large diameter model 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

1200

1600

No. Taxa

Problems with current techniques for MP

Shown here is the performance of the TNT software for maximum parsimony on a real dataset of almost 14,000 sequences. The required level of accuracy with respect to MP score is no more than 0.01% error (otherwise high topological error results).

(“Optimal” here means best score to date, using any method for any amount of time.)

Performance of TNT with time

Empirical problems with existing methods

- Polynomial time methods have poor topological accuracy on large datasets – we need better polynomial time methods.
- Heuristics for Maximum Parsimony (MP) and Maximum Likelihood (ML) and Bayesian MCMC methods cannot handle large datasets (take too long!) – we need new heuristics that can analyze large datasets.

“Boosting” phylogeny reconstruction methods

- DCMs “boost” the performance of phylogeny reconstruction methods.

DCM

Base method M

DCM-M

Graph-theoretic divide-and-conquer (DCM’s)

- Define a triangulated (i.e. chordal) graph so that its vertices correspond to the input taxa
- Compute a decomposition of the graph into overlapping subgraphs, thus defining a decomposition of the taxa into overlapping subsets.
- Apply the “base method” to each subset of taxa, to construct a subtree
- Merge the subtrees into a single tree on the full set of taxa.

Some properties of chordal graphs

- Every chordal graph has at most n maximal cliques, and these can be found in polynomial time: Maxclique decomposition.
- Every chordal graph has a vertex separator which is a maximal clique: Separator-component decomposition.
- Every chordal graph has a perfect elimination scheme: enables us to merge correct subtrees and get a correct supertree back, if subtrees are big enough.

DCMs (Disk-Covering Methods)

- DCMs for polynomial time methods improve topological accuracy (empirical observation), and have provable theoretical guarantees under Markov models of evolution
- DCMs for hard optimization problems reduce running time needed to achieve good levels of accuracy (empirically observation)

Statistical consistency, convergence rates, and absolute fast convergence

Neighbor Joining’s sequence length requirement is exponential!

- Atteson: Let T be a General Markov model tree defining additive matrix D. Then Neighbor Joining will reconstruct the true tree with high probability from sequences that are of length at least O(lg n emax Dij).

DCM1-Boosting [Warnow et al. SODA 2001]

- DCM1+SQS is a two-phase procedure which reduces the sequence length requirement of methods.

Exponentially

converging

method

Absolute fast converging

method

DCM1

SQS

Improving upon NJ

- Construct trees on a number of smaller diameter subproblems, and merge the subtrees into a tree on the full dataset.
- Our approach:
- Phase I: produce O(n2) trees (one for each diameter)
- Phase II: pick the “best” tree from the set.

DCM1 Decompositions

Input: Set S of sequences, distance matrix d, threshold value

1. Compute threshold graph

2. Perform minimum weight triangulation (note: if d is an additive matrix, then

the threshold graph is provably chordal).

DCM1 decomposition :

Compute maximal cliques

DCM1-boosting distance-based methods[Nakhleh et al. ISMB 2001 and Warnow et al. SODA 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

1200

1600

No. Taxa

What about solving MP and ML?

- Maximum Parsimony (MP) and maximum likelihood (ML) are the major phylogeny estimation methods used by systematists.

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.

computed in linear time O(nk)

GTA

ACA

ACA

GTA

2

1

1

ACT

GTT

MP score = 4

Finding the optimal MP tree is NP-hard

Maximum Parsimony: computational complexityCost

Global optimum

Phylogenetic trees

Approaches for “solving” MP/ML- 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).

Problems with current techniques for MP

Best methods are a combination of simulated annealing, divide-and-conquer and genetic algorithms, as implemented in the software package TNT. However, they

do not reach 0.01% of optimal on large datasetsin 24 hours.

Performance of TNT with time

Observations

- The best MP heuristics cannot get acceptably good solutions within 24 hours on most of these large datasets.
- Datasets of these sizes may need months (or years) of further analysis to reach reasonable solutions.
- Apparent convergence can be misleading.

Our objective: speed up the best MP heuristics

Fake study

Performance of hill-climbing heuristic

MP score

of best trees

Desired Performance

Time

DCM Decompositions

Input: Set S of sequences, distance matrix d, threshold value

1. Compute threshold graph

2. Perform minimum weight triangulation

DCM2 decomposition:

Separator plus components

DCM1 decomposition :

Max cliques

Empirical observation

- No DCM based upon the threshold graphs gave us an improvement over the best heuristics for MP!

A conjecture as to why current techniques are poor:

- Our studies suggest that trees with near optimal scores tend to be topologically close (RF distance less than 15%) from the other near optimal trees.
- The best heuristics for MP are based upon the TBR move to explore tree space: there are O(n3) neighbors of every tree, most of which have large RF distances.
- So TBR may be useful initially (to reach near optimality) but then more “localized” searches are more productive.

Using DCMs differently

- Observation: DCMs make small local changes to the tree
- New algorithmic strategy: use DCMs iteratively and/or recursively to improve heuristics on large datasets
- We needed a decomposition strategy that produces small subproblems quickly.

Input: Set S of sequences, and guide-tree T

We use a new graph (“short subtree graph”) G(S,T))

Note: G(S,T) is chordal!

2. Find clique separator in G(S,T) and form subproblems

New DCM3 decomposition- DCM3 decompositions
- can be obtained in O(n) time
- (2) yield small subproblems
- (3) can be used iteratively

Comparison of DCMs (13,921 sequences)

Base method is the TNT-ratchet. “Optimal” refers to the best score found by any method using any amount of time, to date.

Rec-I-DCM3 significantly improves performance

Current best techniques

DCM boosted version of best techniques

Comparison of TNT to Rec-I-DCM3(TNT) on one large dataset

Conclusions (and comments)

- Rec-I-DCM3 improves upon the best performing heuristics for MP.
- The improvement increases with the difficulty of the dataset.
- DCMs also boost the performance of ML heuristics (not shown).
- Rec-I-DCM3 will be in the first software release from the CIPRES project

Other research projects

- Simultaneous estimation of tree and multiple sequence alignment
- Supertree methods
- Constructing networks rather than trees (detecting and reconstructing reticulate evolution)
- Obtaining better bounds on sequence length requirements of phylogeny reconstruction methods
- Whole genome phylogeny
- Constructing forests rather than trees

Acknowledgments

- The CIPRES project www.phylo.org
- The National Science Foundation
- The David and Lucile Packard Foundation
- The Program for Evolutionary Dynamics at Harvard, and the Radcliffe Institute for Advanced Research
- The Institute for Cellular and Molecular Biology at UT-Austin
- Collaborators: Bernard Moret, Usman Roshan, Tiffani Williams, and Daniel Huson.

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