Constructing Phylogenies from Quartets: Elucidation of Eutherian Superordinal Relationships

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Constructing Phylogenies from Quartets
Elucidation of Eutherian Superordinal
Relationships
A. Ben-Dor, B. Chor, D. Graur, R. Ophir, D.
Pelleg Journal of Computational Biology, Vol. 5,
1998, pp. 377?390.

• Speaker Chuang-Chieh Lin
• National Chung Cheng University

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Outline

• Introduction and preliminaries
• Problem description
• The dynamic programming algorithm
• The space complexity and the time complexity

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Evolutionary trees

• Let S be a set of taxa and S n.
• An evolutionary tree T on S is an unrooted,
  leaf-labeled tree such that the leaves of T are
  bijectively labeled by the taxa in S, and each
  internal node of T has degree 3.

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Evolutionary trees

• For 4 taxa a, b, c, d, we have 3 possible
  topologies

a
c
a
b
a
c
b
d
d
c
d
b
adbc
abcd
acbd
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Evolutionary trees (contd.)

• For 5 taxa a, b, c, d, e, how many possible
  evolutionary trees can we derive?
• The answer is 5 ? 3 15.

a
c
There are 5 possible positions for e to be
inserted.
b
d
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Evolutionary trees (contd.)

• For n taxa, how many possible evolutionary trees
  can we derive?
• The answer is (2n ? 5)!!
• This observation can be verified by induction on
  n.
• For an odd positive integer n, it is defined that
  n!! n?? (n ? 2) ? (n ? 4) ? ? 3 ? 1.
• If n 15, (2n ? 5)!! is approximately 8 ? 1012.

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n!!

• Let us analyze n!! in another way.
• For a nonnegative integer m ? 0, let n 2m 1.
• Then we have

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(2n-5)!! O(nn-2)

• For n taxa, we have (2n ? 5)!! O((n ? 3)n?2)
• O(nn-2) possible evolutionary trees.

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Quartet topologies

• A set of four taxa is called a quartet.
• Given an evolutionary tree T and a quartet a, b,
  c, d, the quartet topology of a, b, c, d
  induced by T is obtained by the following
  procedure.

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Step 1 All leaves but a, b, c and d are deleted
from the tree. Edges adjacent to these leaves are
also deleted.
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Step 2 Internal nodes with degree two are
contracted and deleted, so their two adjacent
nodes become connected. This process is repeated
until no internal nodes of degree two are left.
For simplicity, we denote the quartet topology
above by bcad, which is a kind of bipartition
of a, b, c, d.
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• For simplicity, we denote the quartet topology
  above by bcad, which is a kind of bipartition
  of a, b, c, d.
• Note that each input quartet topology t is
  accom-panied by a positive weight Ct .

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Problem description

• Input
• A list of weighted quartet topologies over n
  taxa.
• Output
• A binary tree with n leaves such that the total
  weight of the satisfied quartet topologies is
  maximized.
• This problem was shown to be NP-hard.

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Quartet method

• The fact that small phylogenies are easier to
  infer than large ones leads to another approach
  the quartet method.
• First, consider subsets of 4 taxa, one at a time,
  and infer the phylogenies (i.e., quartet
  topologies) for these subsets.
• The next stage combines the multiple quartet
  topologies into a single phylogeny.

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• Given a set of quartet topologies Q, how to
  determine whether an evolutionary tree T is
  good or bad?

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• Given an evolutionary tree T and a set of quartet
  topologies Q.
• We say that T satisfies a quartet topology tq of
  a quartet q if the induced quartet topology of q
  by T is exactly tq.

For example, T satisfies abdg, cefg,
adbc, etc.
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Score

• We denote by S, where S ? Q, the set of quartet
  topologies that are satisfied by T, and let U Q
  ? S.
• We define the score of the evolutionary tree T as
  follows.

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Score (contd.)

• The latter term was chosen because there are
  three possible topologies for every quartet.
• Therefore this term equals the expected increase.
• In a variant of the same method, the latter term
  is zeroed, so the quartet topologies which are
  not satisfied by T do not contribute to the score.

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Score (contd.)

• It can be easily derived that is
  an upper bound on the score of any evolutionary
  tree T.

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Preliminaries for the dynamic programming
algorithm

• For technical reasons, the following discussion
  deals with rooted evolutionary trees.
• For a node v, its left and right children are
  denoted by vl and vr respectively.

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Preliminaries for the dynamic programming
algorithm (contd.)

• Given a rooted evolutionary tree T and a node v
  in it we denote by T(v) the subtree of T rooted
  at v.

u
w
v
T(v)
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Preliminaries for the dynamic programming
algorithm (contd.)

• We denote by L(T) the set of leaves (i.e., taxa)
  of the tree T.

u
w
v
L(Tv)

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Preliminaries for the dynamic programming
algorithm (contd.)

• For a pair of nodes u, v, the least common
  ancestor of u and v, lca(u, v), is defined as an
  ancestor p of both u and v such that no node in
  T(p) other than p is an ancestor of both u and v.

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Preliminaries for the dynamic programming
algorithm (contd.)
The lca of a and c.
a
c
a
c
b
d
b
d
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Preliminaries for the dynamic programming
algorithm (contd.)

• Definition Given a quartet topology t abcd
  and an evolutionary tree T, the quartet least
  common ancestor of t, qlca(t) is defined as a
  node p that is the lca of two or more pairs of
  elements from a, b, c, d, and no node in T(p)
  except p is the lca of two or more pairs of
  elements from a, b, c, d.

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Preliminaries for the dynamic programming
algorithm (contd.)
The qlca for abcd.
a
a
b
c
b
c
d
d
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Another equivalent definition for the quartet
least common ancestor

• Definition Given a quartet topology t abcd
  and an evolutionary tree T, the qlca of t is a
  node p such that
• L(T(p))??a, b, c, d ? 3.
• For any child s of p, L(T(s))??a, b, c, d ? 2.

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Some observations

• Every quartet topology t has a unique qlca(t).
• Given a tree T and a quartet topology t, the
  subtree rooted at qlca(t) determines whether t is
  satisfied in the evolutionary tree T.
• Let t abcd and v qlca(t). We look at vl ,
  vr , T(vl) and T(vr).
• At least one of these subtrees contains exactly
  two taxa e, f from a, b, c, d.
• Then t is satisfied iff the pair e, f is either
  a, b or c, d.

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Some observations (contd.)

• Given a quartet topology t abcd and an
  evolutionary tree T, let v qlca(t). Then T
  satisfies t if and only if at least one of the
  following holds
• a, b ? L(T(s)).
• c, d ? L(T(s)).
• where s vl or s vr.

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The algorithm

• We denote by SATQ(T(v)) the set of quartet
  topologies t ? Q such that t is satisfied by T,
  and qlca(t) is a node in T(v).
• Let TOPQ(T(v))?? SATQ(T(v)) be the set of quartet
  topologies in Q that have v as their qlca and are
  satisfied by T.

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The algorithm (contd.)

• For a set A ? Q of quartet topologies, let
  denote the sum of their weights.
  
• The score of the subtree T(v) (with respect to Q)
  is defined as

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The algorithm (contd.)

• By the above equation, we have

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The algorithm (contd.)

• Let S be a set of three or more taxa.
• Denote by opt_scoreQ(S) the maximum score with
  respect to Q among all trees that have S as their
  set of leaves.
• We denote by opt_treeQ(S) a tree which attains
  the maximum score.

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The algorithm (contd.)

• For every proper partition of S into two subsets
  S1 and S2, let T(S1, S2) denote a tree whose left
  subtree equals opt_treeQ(S1) and its right
  subtree equals opt_treeQ(S2).
• We then have

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The algorithm (contd.)

• This implies that
• By employing the dynamic programming paradigm, we
  can avoid wasteful repetitions.
• To do this, we scan the subsets S?? 1 ,2 , n
  by increasing size of S.

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The algorithm (contd.)

• For simplicity, the details of implementing the
  dynamic programming algorithm are omitted.

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The space complexity and the time complexity

• The time complexity
• The space complexity ?
• ?(2n).

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Thank you.
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References

• S92 M. Steel The complexity of reconstructing
  trees from qualitative characters and subtrees.
  Journal of Classification, 9 (1992), pp. 91-116.