Supertrees: Algorithms and Databases Roderic Page University of Glasgow [email protected] DIMACS Working Group Meeting on Mathematical and Computational.

Supertrees: Algorithms and Databases

Roderic PageUniversity of Glasgow

[email protected]

DIMACS Working Group Meeting on Mathematical

and Computational Aspects Related to the Study of

The Tree of Life

What do we mean by the “Tree of Life”

or

Supertrees, datatypes, databases, taxonomy

Tree algorithms, models, genomics,lateral gene transfer

Our perception of what the tree is may affect what we view as being the “interesting” problems

Topics

• Supertrees (MinCut)

• Phylogenetic databases

Tree terminology

a b c d

{a,b}

{a,b,c}

{a,b,c,d} root

leaf

internal nodecluster

edge

Nestings and triplets

a b c d

{a,b} <T {a,b,c,d}

{b,c} <T {a,b,c,d}

(bc)d

bc|d

Nestings

Triplets

Supertree

a b c b c da b c d

supertree

T1 T2

+ =

Some desirable properties of a supertree method

(Steel et al., 2000)

• The supertree can be computed in polynomial time

• A grouping in one or more trees that is not contradicted by any other tree occurs in the supertree

Aho et al.’s algorithm (OneTree)Aho, A. V., Sagiv, Y., Syzmanski, T. G., and Ullman, J. D. 1981. Inferring a

tree from lowest common ancestors with an application to the optimization of relational expressions. SIAM J. Comput. 10: 405-421.

Input: set of rooted trees

1. If set is compatible (i.e., will agree on a tree), output that tree.

2. If set is not compatible, stop!

a b c b c d

T1 T2

a b

cd

a, b

d

a, b, c, d

a b

ca, b, c

a b

c

Aho et al.’sOneTree algorithm

supertree

Mincut supertreesSemple, C., and Steel, M. 2000. A supertree method for

rooted trees. Discrete Appl. Math. 105: 147-158.

• Modifies OneTree by cutting graph

• Requires rooted trees (no analogue of OneTree for unrooted trees)

• Recursive

• Polynomial time

a b c d e a b c d

T1 T2

a

b

c

de

{T1,T2}S

Semple and Steel (2000)

a

b

c

de

a,b

c

de

1

1 1

1

11

1

2

{T1,T2}Smax

S /E{T1,T2} {T1,T2}

Collapsing the graph(Semple and Steel mincut algorithm)

This edge has

maximum weight

Cut the graph to get supertree

a b c d e

supertree

a,b

c

de

1

1

1max

S /E{T1,T2} {T1,T2}

My mincut supertree implementationdarwin.zoology.gla.ac.uk/~rpage/supertree

• Written in C++

• Uses GTL (Graph Template Library) to handle graphs (formerly a free alternative to LEDA)

• Finds all mincuts of a graph faster than Semple and Steel’s algorithm

A counter example: two input trees...

a

b

c

x1

x2

x3

c

b

a

y1

y2

y3

y4

Mincut gives this (strange) result

cx1x2x3bay1y2y3y4

• Disputed relationships among a, b, and c are resolved

• x1, x2, and x3 collapsed into polytomy

Problem:Cuts depend on connectivity(in this example it is a function of tree size)

a

x1

x2 y1

y3

y4x3

y2

c

b

{T1,T2}S

So, mincut doesn’t work

• But, Semple and Steel said it did

• My program seems to work

• Argh!!! What is happening….?

What mincut does… …and does not do

• Mincut supertree is guaranteed to include any nesting which occurs in all input trees

• Makes no claims about nestings which occur in only some of the trees

• “Does exactly what it says on the tin™”

Modifying mincut supertree

• Can we incorporate more of the information in the input trees?

• Three categories of information• Unanimous (all trees have that grouping)• Contradicted (trees explicitly disagree)• Uncontradicted (some trees have information

that no other tree disagrees with)

Uncontradicted informationassume we have k input trees

a b

a and b co-occurin a tree

a and b nestedin a tree

a b

c n

c - n = 0 uncontradicted (if c = k then unanimous)

c - n > 0 contradicted

Uncontradicted informationassume we have k input trees

a b

a and b co-occurin a tree

a and b nestedin a tree

a b

c n

c - n -f = 0 uncontradicted (if c = k then unanimous)

c - n - f > 0 contradicted

a b

a and b in a fan

f

a

b

c

x1 x

x3

y1 y2y3 y4

2

a

b

c

y1

y3

y4

x1

x2

x3

y2

Uncontradicted

Uncontradicted but adjacent to contradictedContradicted

Classifying edges

{T1,T2}S

Modified mincut

• Species a, b, and c form a polytomy

• x1, x2, and x3 resolved as per the input tree

modified mincut

abcx1x2x3y1y2y3y4

1 2 3 4 5

1 2 3 4 5 1 2 3 4 5

1 2 3 4 5

(12)5

(45)1

(23)5

(34)1

If no tree contradicts an item of information, is that information always in the supertree?

1 2

3

4

5

No!Steel, Dress, & Böcker 2000

• The four trees display (12)5, (23)5, (34)1, and (45)1

• No tree displays (IK)J or (JK)I for any (IJ)K above

• Triplets are uncontradicted, but cannot form a tree

Future directions for supertrees

• Improve handling of uncontradicted information

• Add support for constraints

• Visualising very big trees

• Better integration into phylogeny

databases (www.treebase.org)

darwin.zoology.gla.ac.uk/~rpage/supertree

Supertree Challenge (proposed by Mike Sanderson [email protected])

The TreeBASE database currently contains over 1000 phylogenies with over 11,000 taxa among them. Many of these trees share taxa with each other and are therefore candidates for the construction of composite phylogenies, or "supertrees", by various algorithms. A challenging problem is the construction of the largest and "best" supertree possible from this database. "Largest" and "best" may represent conflicting goals, however, because resolution of a supertree can be easily diminished by addition of "inappropriate" trees or taxa.

It’s a scandal

• We cannot answer even the most basic question: “what is the phylogeny for group x?”

• GenBank is currently the best phylogenetic database (!)

• Can't even say how many species are in a given group

• Little idea of who is doing what

Tree of Lifetolweb.org

• Provides text and images

• Relies on extensive manual effort (e.g., writing text)

• Can’t do any computations with it

• Limited research value

TreeBASEwww.treebase.org

• Relational database

• Query by author, taxon, study number

• Compute supertrees

• Submit NEXUS data files

TreeBASE

TreeBASE and mincut supertrees

• User selects two or more trees

• Clicks on button

and script on darwin.zoology.gla.ac.uk is run to create supertree

• Can view as PS, PDF, treefile, or in Java applet (ATV)

What’s wrong with TreeBASE?

• No consistency of taxon names

• (e.g., Human, Homo sapiens,

Homo sapiens X54666-1)

• No consistency of data names (e.g., gene names, morphological characters, etc.)

The same organism may have multiple names

Starting December 1, the ALL Species Foundation will close its San Francisco office because of a lack of funding for the Foundation.

www.all-species.org

Press Release: November 13, 2002

“The ALL Species Foundation is a non-profit organization dedicated to the complete inventory of all species of life on Earth within the next 25 years - a human generation.”

The first challenge

• We need a taxonomic name server that can resolve the name of any organism

• This server needs to reconcile multiple classifications (e.g., GenBank, ITIS, etc.)

• Must handle at least 1 million names, perhaps 100 million

• How do we query trees?

• Trees can be classifications or phylogenies

Second Challenge

SQL Queries on Trees

• Oracle SQL Transitive Closure Query (recursion)

• Nested queries

• Node path queries

1. All ancestors of node A

A

2. Least Common Ancestor (LCA) of A and B

A B

3. Spanning Clade of A and B

A B

4. Path Length from A and B

A B

5

Node paths

/1

/1/1

/1/1/2/1/1/1/2 /1/2/2

/1/2

/1/2/1/2

/1/1/1

/1/1/1/1

Node paths - selecting subtree

/1

/1/1

/1/1/2/1/1/1/2 /1/2/2

/1/2

/1/2/1/2

/1/1/1

/1/1/1/1

SELECT node WHERE (path LIKE “/1/1/%”) AND (path < “/1/10/%”);

Node paths - selecting subtree

/1

/1/1

/1/1/2/1/1/1/2 /1/2/2

/1/2

/1/2/1/2

/1/1/1

/1/1/1/1

SELECT node WHERE (path LIKE “/1/1/%”) AND (path < “/1/10/%”)AND (num_children IS 0);

Node paths - LCA

/1

/1/1

/1/1/2/1/1/1/2 /1/2/2

/1/2

/1/2/1/2

/1/1/1

/1/1/1/1

Common substring starting from left

What do we do now…?

• Setup a taxonomic name server (TNS)• Develop a phylogenetic genetic database

linked to TNS, PubMed, GenBank, etc.• Develop easy ways to populate database (e.g.,

from TreeBASE, GenBank, journal databases)

• Develop standard set of tree queries• Deploy