A space where trees lie?

Transcription

1 A space where trees lie? 1 Susan Holmes Statistics Department, Stanford and INRA- Biométrie, Montpellier,France susan@stat.stanford.edu Funded in part by a grant from NSF-DMS Contains joint work with Karen Vogtmann and Lou Billera

2 Do we care about inferences for phylogenetic trees? 2 Cetacees: recognising what is being sold as Whale meat in Japan? Steve Palumbi, Harvard. Scott Baker, Auckland. Whale Earth Trust Press Release

3 The River without a Paddle? 3 Human immunodeficiency virus: Phylogeny and the origin of HIV-1

4 4 The origin of human immunodeficiency virus type 1 (HIV-1) is controversial. Phylogeny has showed that viruses obtained from the Democratic Republic of Congo in Africa have a

5 quantitatively different phylogenetic tree structure from those sampled in other parts of the world. 5 Quest for the origin of AIDS This indicates that the structure of HIV-1 phylogenies is the result of epidemiological processes acting within human populations alone, and is not due to multiple cross-species transmission initiated by oral polio vaccination. Serial Passage Conversely, phylogenetic analysis of HIV-1 sequences indicates that group M originated before the vaccination campaign, supporting a model of natural transfer from chimpanzees to humans. If this timescale is correct, then the OPV theory remains a viable

6 hypothesis of HIV-1 origins only if the subtypes of group M differentiated in chimpanzees before their transmission to humans. 6

7 Confidence Intervals? 7 Korber and colleagues extrapolated the timing of the origin of HIV-1 group M back to a single viral ancestor in 1931, give or take about 12 years for 95% confidence limits. Because this calendar of events obviously pre-dated the OPV trials, in the revised version of his book, Hooper suggested that group M first began to diverge in chimpanzees, and that there were then several independent transfers of virus to humans via OPV. In that case, several OPV batches should bear evidence

8 of their production in chimpanzee tissue, yet no such evidence has been found. 8

9 Closure: Polio vaccines exonerated Nature 410, (2001) 9 The OPV batch that Hooper considered to be under most suspicion, however, was CHAT 10A-11. An original vial of the batch was found at Britain s National Institute for Biological Standards and Control, and the new tests show that it was prepared from rhesus-macaque cells.

10 How sure are we of the answers? 10 Phylogenetic Trees and Variability. Aggregating/Combining trees, Stability of sets of trees, Comparisons of sets of trees of several kinds. Explanation of one set of trees by another. Combining trees with other data. Confidence Statements for trees.

11 11 A Parameter T :a semi-labeled binary Tree 0 root inner edges inner edges Inner Node leaves

12 Data T 1, T 2,..., T n : many semi-labeled binary Trees 12 0 root 0 root 0 root inner edges inner edges inner edges inner edges inner edges inner edges Inner Node Inner Node Inner Node leaves 0 root leaves 0 root leaves 0 root inner edges inner edges inner edges inner edges inner edges inner edges Inner Node Inner Node Inner Node leaves leaves leaves

13 What to do when data are trees? 13 The data could be considered equivalent to a set of trees. Building a probability distribution on trees is a complex procedure. Most biologists agree that the simplest possible probability distribution on tree space, the uniform distribution is not relevant. Others: Yule process, for coalescent trees: Kingman s model.

14 Another non standard type of data 14 Here we are going to give an analog of the nonstandard splits-and-trees data. The following example involves data that do not belong to R. Commonly called rank data. (1, 4, 3, 2)(3, 4, 1, 2)(3, 2, 1, 4) (4, 2, 1, 3) that we would also like to summarize. This data occurs in genetics: gene order data are available (for a review see chapter 10 of Pezner).

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16 Sufficiency 16

17 Confidence Statements 17 R 5 S J R boundary R 2 ^ π. ^ π π. R 1 R 4 R 6

18 Confidence Statements 17 R 5 S J R boundary R 2 ^ π. ^ π π. R 1 R 4 R 6 From Efron, Halloran, Holmes, (1996).

19 Confidence Statements 17 R 5 S J 9-10 boundary R 2 ^ π. ^ π R 3 π. R 1 R 4 R 6 What is the curvature of the boundary? From Efron, Halloran, Holmes, (1996).

20 Confidence Statements 17 R 5 S J 9-10 boundary R 2 ^ π. ^ π R 3 π. R 1 R 4 R 6 What is the curvature of the boundary? How many neighbors does a region have? From Efron, Halloran, Holmes, (1996).

21 Confidence Statements 17 R 5 S J 9-10 boundary R 2 ^ π. ^ π R 3 π. R 1 R 4 R 6 What is the curvature of the boundary? How many neighbors does a region have? From Efron, Halloran, Holmes, (1996).

22 Bootstrap Confidence Values Macaca mul Macaca fus!! Macaca fas!!! Macaca syl !! Hylobates!!!!!! +----Homosapien! ! Pan!!!!!!! Gorilla!!! ! Pongo!!!!! Saimiri sc! Tarsius sy Lemur catt Do these values mean anything?

23 Simple confidence values 19 Univariate. Multiple Testing. Composite Statements.

24 Simple confidence values 19 Univariate. Multiple Testing. Composite Statements. In general, the clade frequencies are not sufficient statistics for the data on trees.

25 Confidence Statements for trees 20

26 Aims 21 Fill Tree Space and make meaningful boundaries. Define distances between trees. Define neighborhoods, meaningful measures. Principal directions of variations in tree space, summarizing : structure + noise. Confidence statements, convex hulls.

27 Rotation Moves

28 Rotation Moves

29 Rotation Moves These are biologists NNI moves.

30 Boundary for trees with 3 leaves

31 Extension to 4 24

32 The quadrant for one tree 25 (0,1) (1,1) (0,0) (1,0)

33 The quadrant for one tree 25 (0,1) (1,1) (0,0) (1,0)

34 The quadrant for one tree (0,1) (1,1) 1 { { (0,0) (1,0)

35 The quadrant for one tree (0,1) (1,1) 1 { { (0,0) (1,0)

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38 The cube complex 27 A binary n-tree has the maximal possible number of interior edges (n 2). It determines the largest possible dimensional quadrant which is n 2-dimensional. The quadrant corresponding to each tree which is not binary appears as a boundary face of at least three binary trees; in particular the origin of each quadrant corresponds to the (unique) tree with no interior edges.

39 The cube complex 27 A binary n-tree has the maximal possible number of interior edges (n 2). It determines the largest possible dimensional quadrant which is n 2-dimensional. The quadrant corresponding to each tree which is not binary appears as a boundary face of at least three binary trees; in particular the origin of each quadrant corresponds to the (unique) tree with no interior edges. T n is built by taking one n 2-dimensional quadrant for each of the (2n 3)!! = (2n 3) (2n 5) possible binary trees, and gluing them together along their common faces.

40 For n = 3 there are three binary trees, each with 1 interior edge. Each tree thus determines a 1-dimensional quadrant, i.e. a ray from the origin. The three rays are identified at their origins. Figure for n=3. 28

41 Three quadrants sharing a ray for n=4 Boundary

42 Three quadrants sharing a ray for n=4 Boundary Note that the bottom boundary rays form a copy of T 3 embedded in T 4.

43 Three quadrants sharing a ray for n=4 Boundary Note that the bottom boundary rays form a copy of T 3 embedded in T 4. In general, T n contains many embedded copies of T k for k < n.

44 CAT(0) space, (Gromov) c c 30 a b a b

45 CAT(0) space, (Gromov) c c 30 a Triangles are thin. b a b

46 Consequences 31 Averaging works better than it should, (an argument against total evidence computation without decomposing??). We can build Bayesian priors based on distances. We can make a useful bootstrap statement. We can make convex hulls. Confidence regions. We can use Mallow s model. We know how many neighbors any tree has.

47 How many neighbors for a given tree?(w.h.li,1993) 32 We know the number of neighbors of each tree.

48 How many neighbors for a given tree?(w.h.li,1993) 32 We know the number of neighbors of each tree.

49 How many neighbors for a given tree?(w.h.li,1993) 32 We know the number of neighbors of each tree. For a tree with only two inner edges, there is the only one way of having two edges small: to be close to the origin-star tree: 15 neighbors. This same notion of neighborhood

50 containing 15 different branching orders applies to all trees on as many leaves as necessary but who have two contiguous small edges and all the other inner edges significantly bigger than This picture of treespace frees us from having to use simulations to find out how many different trees are in a neighborhood of a given radius r around a given tree. All we have to do is check the sets of continguous edges in the tree smaller than r, say there is only one set of size k, then the neighborhood will contain (2k 3)!! = (2k 3) (2k 5) 3 different trees. If there are m sets of sizes (n 1, n 2,..., n m )

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55 In this case the number of trees within r will be = 4725, in general: (2n 1 3)!! (2n 2 3)!! (2n 3 3)!! (2n m 3)!!

56 A tree near the star tree at the origin will have an exponential number of neighbors. This explosion of the volume of a neighborhood at the origin provides for interesting math problems. 36

57 Importance of distance for the bootstrap: 37 X original data ˆT estimate. Call X bootstrap samples consistent with the model used for estimating the tree: Non parametric multinomial resampling for a parsimony tree. Seqgen parametric type resampling with the same parameters for a ML. Bayesian GAMMA prior on rates and generation (Yang 2000) for random sequences according to ˆT

58 Bootstrap Theory 38 Distribution(d( ˆT, T ) = Distribution(d( ˆT, ˆT ))

59 Convex Hulls and confidence regions 39

60 39 Convex Hulls and confidence regions

61 39 Convex Hulls and confidence regions

62 Perspectives, problems 40 How tree-like are the data? Translates into how well are the data projected onto tree space. Can be solved empirically by making a graph and asking how treelike it is. Mathematical analysis: hyperbolicity. al. 2001). (Moulton et

63 Tree trajectories 41...

64 Tree trajectories

65 Tree trajectories

66 Tree trajectories Regression in treespace. Extension from trees to general graphs (median networks,...)

67 How can mathematical statistics help? 42 Decompositions that can be generalisable. Geometric Picture of Tree Space A space for comparisons. Ways of projecting. Follow trees as they change, (paths of trees) Aggregating trees, expectations for various measures. Neighborhoods (convex hulls of trees)... Justification of commonsense, ground for generalizations.

68 Proof by direct decomposition 43 Call B n 1 the subgroup of S 2n 2 that fixes the pairs then {1, 2}{3, 4}... {2n 3, 2n 2} and M n 1 = (2n 2)! 2 n 1 (n 1)! M n 1 = S 2n /B n 1 = (2n 3)!! = (2n 3) (2n 5) This formula for the number of trees was first proved using generating functions by Schroder (1873)[?].

69 (S 2n 2, B n 1 ) form a Gelfand pair Diaconis and Shahshahani (1987). 44 L(M n 1 ) = V 1 V 2... V λ A multiplicity free representation. L(M n 1 ) = S 2λ λ n where the direct sum is over all partitions λ of m, 2λ = (2λ 1, 2λ 2,..., 2λ k ) and S 2λ is associated irreducible representation of the symmetric group S 2m. Just to take the first few: for λ = n 1 S λ are the constants, and this gives the sample size. for λ = (n 2, 1), S λ are the number of times each pair

70 appears. for λ = (n 3, 2), S λ are the number of times partition of 4 appears in the tree. for λ = (n 3, 1, 1), S λ are the number of times 2 pairs appear simultaneously. This decomposition is similar to what was done by Diaconis for permutation data.[?] 45