1. The Disk-Covering Method for Phylogenetic Tree Reconstruction
Tandy Warnow
The Program in Evolutionary Dynamics at Harvard University
The University of Texas at Austin

2. Phylogeny Orangutan Gorilla Chimpanzee Human
From the Tree of the Life Website, University of Arizona
Orangutan
Gorilla
Chimpanzee
Human
A phylogeny is a tree representation for the evolutionary history relating the species we are interested in.
This is an example of a 13-species phylogeny.
At each leaf of the tree is a species – we also call it a taxon in phylogenetics (plural form is taxa). They are all distinct.
Each internal node corresponds to a speciation event in the past.
When reconstructing the phylogeny we compare the characteristics of the taxa, such as their appearance, physiological features, or the composition of the genetic material.

3. 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

4. Reconstructing the “Tree” of Life
Handling large datasets: millions of species

5. Purpose: to create a national infrastructure of hardware,
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: 40 biologists, computer scientists, and
mathematicians from 13 institutions.
Funding: $11.6 M (large ITR grant from NSF).

6. CIPRes Members University of New Mexico Bernard Moret David Bader
Tiffani Williams
UCSD/SDSC
Fran Berman
Alex Borchers
Phil Bourne
John Huelsenbeck
Terri Liebowitz
Mark Miller
University of Connecticut
Paul O Lewis
University of Pennsylvania
Susan Davidson
Junhyong Kim
Sampath Kannan
UT Austin
Tandy Warnow
David M. Hillis
Warren Hunt
Robert Jansen
Randy Linder
Lauren Meyers
Daniel Miranker
Usman Roshan
Luay Nakhleh
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
UC Berkeley
Satish Rao
Steve Evans
Richard M Karp
Brent Mishler
Elchanan Mossel
Eugene W. Myers
Christos M. Papadimitriou
Stuart J. Russell
SUNY Buffalo
William Piel
Florida State University
David L. Swofford
Mark Holder
Yale
Michael Donoghue
Paul Turner

7. 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

8. Steps in a phylogenetic analysis
Gather data
Align sequences
Reconstruct phylogeny on the multiple alignment - often obtaining a large number of trees
Compute consensus (or otherwise estimate the reliable components of the evolutionary history)
Perform post-tree analyses.

9. Phylogeny Problem U V W X Y X U Y V W AGGGCAT TAGCCCA TAGACTT TGCACAA
TGCGCTT X U Y V W

10. CIPRES research in algorithms
Heuristics for NP-hard problems in phylogeny reconstruction
Compact representation of sets of trees
Reticulate evolution reconstruction
Performance of phylogeny reconstruction methods under stochastic models of evolution
Gene order phylogeny
Genomic alignment
Lower bounds for MP
Distance-based reconstruction
Gene family evolution
High-throughput phylogenetic placement
Multiple sequence alignment

11. CIPRES Algorithms Group
Tandy Warnow (focus leader)
David Bader, Steve Evans, John Huelsenbeck, Warren Hunt, Sampath Kannan, Dick Karp, Paul Lewis, Bernard Moret, Elchanan Mossel, Luay Nakhleh, Christos Papadimitriou, Satish Rao, Usman Roshan, Jijun Tang, Li-San Wang, Tiffani Williams

12. Phylogeny ReconstructionV W X Y
AGGGCAT
TAGCCCA
TAGACTT
TGCACAA
TGCGCTT X U Y V W

13. Phylogenetic reconstruction methods
Hill-climbing heuristics for hard optimization criteria (Maximum Parsimony and Maximum Likelihood)
Phylogenetic trees
Cost
Global optimum
Local optimum
Polynomial time distance-based methods: Neighbor Joining, FastME, Weighbor, etc.

14. Performance criteria Running time. Space.
Statistical performance issues (e.g., statistical consistency) with respect to a Markov model of evolution.
“Topological accuracy” with respect to the underlying true tree. Typically studied in simulation.
Accuracy with respect to a particular criterion (e.g. tree length or likelihood score), on real data.

15. Markov models of site evolution
Simplest (Jukes-Cantor):
The model tree is a pair (T,{e,p(e)}), where T is a rooted binary tree, and p(e) is the probability of a substitution on the edge e
The state at the root is random
If a site changes on an edge, it changes with equal probability to each of the remaining states
The evolutionary process is Markovian
More complex models (such as the General Markov model) are also considered, with little change to the theory.
Variation between different sites is either prohibited or minimized, in order to ensure identifiability of the model.

16. Distance-based Phylogenetic Methods

17. 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.

18. Maximum Likelihood
Input: Set S of n aligned sequences of length k, and a specified parametric model
Output:
A phylogenetic tree T leaf-labeled by sequences in S
With additional model parameters (e.g. edge “lengths”)
such that Pr[S|(T, params)] is maximized.

19. 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).
Phylogenetic trees
Cost
Global optimum
Local optimum

20. Theoretical results
Neighbor Joining is polynomial time, and statistically consistent under typical models of evolution.
Maximum Parsimony is NP-hard, and even exact solutions are not statistically consistent under typical models.
Maximum Likelihood is of unknown computational complexity, but statistically consistent under typical models.

21. Quantifying Error FN: false negative (missing edge) FP: false positive
(incorrect edge)
50% error rate FP

22. Neighbor joining 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

23. Theoretical convergence rates
Atteson: Let T be a 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).
Proof: Show NJ accurate on input matrix d such that max{|Dij-dij|}<f/2, for f the minimum edge length.

24. Absolute fast convergence vs. exponential convergence

25. Other standard polynomial time methods don’t improve substantially on NJ (and have the same problem with large diameter datasets).
What about trying to “solve” maximum parsimony or maximum likelihood?

26. Problems with NJ
Theory: The convergence rate is exponential: the number of sites needed to obtain an accurate reconstruction of the tree with high probability grows exponentially in the evolutionary diameter.
Empirical: NJ has poor performance on datasets with some large leaf-to-leaf distances.

27. 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

28. How good an MP analysis do we need?
Our research shows that we need to get within 0.01% of optimal (or better even, on large datasets) to return reasonable estimates of the true tree’s “topology”

29. Problems with current techniques for MP
Average MP scores above optimal of best methods at 24 hours across 10 datasets
Best current techniques fail to reach 0.01% of optimal at the end of 24 hours, on large datasets

30. Problems with current techniques for MP
Shown here is the performance of a 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

31. Empirical problems with existing methods
Heuristics for Maximum Parsimony (MP) and Maximum Likelihood (ML) cannot handle large datasets (take too long!) – we need new heuristics for MP/ML that can analyze large datasets
Polynomial time methods have poor topological accuracy on large diameter datasets – we need better polynomial time methods

32. Using divide-and-conquer
Conjecture: better (more accurate) solutions will be found if we analyze a small number of smaller subsets and then combine solutions
Note: different “base” methods will need potentially different decompositions.
Alert: the subtree compatibility problem is NP-complete!

33. Using divide-and-conquer
Conjecture: better (more accurate) solutions will be found if we analyze a small number of smaller subsets and then combine solutions
Note: different “base” methods will need potentially different decompositions.
Alert: the subtree compatibility problem is NP-complete!

34. Using divide-and-conquer
Conjecture: better (more accurate) solutions will be found if we analyze a small number of smaller subsets and then combine solutions
Note: different “base” methods will need potentially different decompositions.
Alert: the subtree compatibility problem is NP-complete!

35. Strict Consensus Merger (SCM)

36. DCMs: Divide-and-conquer for improving phylogeny reconstruction

37. “Boosting” phylogeny reconstruction methods
DCMs “boost” the performance of phylogeny reconstruction methods.
DCM
Base method M
DCM-M

38. 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)

39. DCM1-boosting distance-based methods [Nakhleh et al. ISMB 2001]
DCM1-boosting makes distance-based methods more accurate
Theoretical guarantees that 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

40. DCM-Boosting [Warnow et al. 2001]
DCM+SQS is a two-phase procedure which reduces the sequence length requirement of methods.
Exponentially
converging
method
Absolute fast converging
method
DCM
SQS
(Will present this on Monday in Applied Math colloquium.)

41. DCM1 Decompositions
Input: Set S of sequences, distance matrix d, threshold value
1. Compute threshold graph
2. Perform minimum weight triangulation
DCM1 decomposition : compute the maximal cliques

42. Major challenge: MP and ML
Maximum Parsimony (MP) and Maximum Likelihood (ML) remain the methods of choice for most systematists
The main challenge here is to make it possible to obtain good solutions to MP or ML in reasonable time periods on large datasets

43. 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.

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

45. Maximum Parsimony
ACT
GTA
ACA
ACT
GTT
ACA
GTT
GTA
GTA
ACA
ACT
GTT

46. 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

47. Maximum Parsimony: computational complexity
ACT
ACA
GTT
GTA 1 2
MP score = 4
Finding the optimal MP tree is NP-hard
Optimal labeling can be
computed in linear time O(nk)

48. 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 datasets in 24 hours.
Performance of TNT with time

49. 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.

50. How can we improve upon existing techniques?

51. Our objective: speed up the best MP heuristics
Fake study
Performance of hill-climbing heuristic
MP score
of best trees
Desired Performance
Time

52. Divide-and-conquer technique for speeding up MP/ML searches

53. DCM Decompositions
Input: Set S of sequences, distance matrix d, threshold value
1. Compute threshold graph
2. Perform minimum weight triangulation
DCM2 decomposition:
Clique-separator plus component
DCM1 decomposition :

54. But: it didn’t work!
A simple divide-and-conquer was insufficient for the best performing MP heuristics -- TNT by itself was as good as DCM(TNT).

55. Empirical observation
DCM1 not as good as DCM2 for MP
DCM2 decompositions too large, too slow to compute.
Neither improved the best MP heuristics.

56. How can we improve upon existing techniques?

57. Tree Bisection and Reconnection (TBR)

58. Tree Bisection and Reconnection (TBR)
Delete an edge

59. Tree Bisection and Reconnection (TBR)

60. Tree Bisection and Reconnection (TBR)
Reconnect the trees with a new edge
that bifurcates an edge in each tree

61. 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 standard technique (TBR) for moving around tree space explores O(n3) trees, which are mostly topologically distant.
So TBR may be useful initially (to reach near optimality) but then more “localized” searches are more productive.

62. 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
However, the initial DCMs for MP
produced large subproblems and
took too long to compute
We needed a decomposition strategy that produces small subproblems quickly.

63. 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
However, the initial DCMs for MP
produced large subproblems and
took too long to compute
We needed a decomposition strategy that produces small subproblems quickly.

64. 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
However, the initial DCMs for MP
produced large subproblems and
took too long to compute
We needed a decomposition strategy that produces small subproblems quickly.

65. New DCM3 decomposition Input: Set S of sequences, and guide-tree T
1. Compute short subtree graph G(S,T), based upon T
2. Find clique separator in the graph G(S,T) and form subproblems
DCM3 decompositions
can be obtained in O(n) time
(2) yield small subproblems
(3) can be used iteratively

66. Iterative-DCM3 T
DCM3
Base method T’

67. New DCMs DCM3 Recursive-DCM3 Iterative DCM3 Recursive-Iterative DCM3
Compute subproblems using DCM3 decomposition
Apply base method to each subproblem to yield subtrees
Merge subtrees using the Strict Consensus Merger technique
Randomly refine to make it binary
Recursive-DCM3
Iterative DCM3
Compute a DCM3 tree
Perform local search and go to step 1
Recursive-Iterative DCM3

68. Datasets Obtained from various researchers and online databases
1322 lsu rRNA of all organisms
2000 Eukaryotic rRNA
2594 rbcL DNA
4583 Actinobacteria 16s rRNA
6590 ssu rRNA of all Eukaryotes
7180 three-domain rRNA
7322 Firmicutes bacteria 16s rRNA
8506 three-domain+2org rRNA
11361 ssu rRNA of all Bacteria
13921 Proteobacteria 16s rRNA

69. Comparison of DCM decompositions (Maximum subset size)
DCM2 subproblems are almost as large as the full dataset size on datasets 1 through 4.
On datasets 5-10 DCM2 was too slow to compute a decomposition within 24 hours.

70. Comparison of DCMs (4583 sequences)
Base method is the TNT-ratchet. DCM2 tree takes almost 10 hours to produce a tree and is
too slow to run on larger datasets. Rec-I-DCM3 is the best method at all times.

71. Comparison of DCMs (13,921 sequences)
Base method is the TNT-ratchet. Note the improvement in DCMs as we move from the default
to recursion to iteration to recursion+iteration. On very large datasets Rec-I-DCM3 gives
significant improvements over unboosted TNT.

72. 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

73. Rec-I-DCM3(TNT) vs. TNT (Comparison of scores at 24 hours)
BASE METHOD????
Base method is the default TNT technique, the current best method for MP. Rec-I-DCM3
significantly improves upon the unboosted TNT by returning trees which are at most 0.01%
above optimal on most datasets.

74. Observations
Rec-I-DCM3 improves upon the best performing heuristics for MP.
The improvement increases with the difficulty of the dataset.

75. Other DCMs
DCM for NJ and other distance methods produces absolute fast converging (afc) methods, which improve upon NJ in simulation studies
DCMs have been used to scale GRAPPA (software for whole genome phylogenetic analysis) from its maximum of about genomes to 1000 genomes.
Current projects: DCM development for maximum likelihood and multiple sequence alignment.

76. Questions
Tree shape (including branch lengths) has an impact on phylogeny reconstruction - but what model of tree shape to use?
What is the sequence length requirement for Maximum Likelihood? (Result by Szekely and Steel is worse than that for Neighbor Joining.)
Why is MP not so bad?

77. General comments
There is interesting computer science research to be done in computational phylogenetics, with a tremendous potential for impact.
Algorithm development must be tested on both real and simulated data.
The interplay between data, stochastic models of evolution, optimization problems, and algorithms, is important and instructive.

78. Acknowledgements NSF The David and Lucile Packard Foundation
The Program in Evolutionary Dynamics at Harvard
The Institute for Cellular and Molecular Biology at UT-Austin
Collaborators: Usman Roshan, Bernard Moret, and Tiffani Williams

79. Reconstructing the “Tree” of Life
Handling large datasets:
millions of species
The “Tree of Life” is not
really a tree:
reticulate evolution