Tree Space: Algorithms & Applications Part I. Megan Owen University of Waterloo

Transcription

1 Tree Space: lgorithms & pplications Part I Megan Owen University of Waterloo

2 Phylogenetic Trees a phylogenetic tree: GTTTGT GTTGTT GTGT GGTTGTT? questions: how do we infer a tree from data? how do we compare two trees? how do we compute meaningful statistics?

3 Statistics on Trees to estimate a phylogenetic tree, the parameters are: tree topology edge lengths this is not a standard statistical problem! R d (usual parameter space is ) need to develop new statistical techniques

4 (My) Ultimate Goal develop statistical theory for a space of phylogenetic trees analogous to statistical theory for uclidean space challenges: which space should we use? how to verify theories and algorithms? non-uclidean behaviour (i.e. sticky means) algorithms need to be practical or biologists won t use them

5 Outline 1. Tree spaces, including description of HV tree space. Polynomial time algorithm for computing distance in HV tree space 3. Open problems. Mean and variance in HV tree space 5. pplications of HV tree space 6. More open problems

6 Tree Space a tree space is a metric space such that the points of the space are in bijection with some well-defined set of trees the metric of the tree space induces a distance between trees i.e. HV tree space, tropical tree space OR given a distance measure between trees that is a metric, it induces a tree space i.e. NNI distance, Robinson-Foulds distance

7 xamples of Tree Spaces xample 1: a discrete tree space vertices = tree topologies edge between vertices iff topologies differ by NNI move shortest paths are not unique NP-hard to compute shortest path

8 xample dissimilarity map tropical tree space = set of dissimilarity maps in R (n ) that are realizable as trees geodesics are not unique no algorithm for computing geodesics

9 HV Tree Space constructed by illera, Holmes, Vogtmann, 1 T n parametrizes all trees with n leaves and edge lengths includes degenerate trees all interior edges must have length/weight F

10 Splits each tree edge induces a split e 3 e e 1 a split is a partition of the set of leaves: e 3 = { {,}, {,,,} } or e 3 =

11 Tree Space represent each tree as a vector coordinates = splits (,,,, 3,,,,,,,...)... (,,,, 3,,,,,,,...)......

12 Split ompatibility e x = X X is compatible with e y = Y Y if there exists a tree containing both splits e 1 e 3 e ex. e 3 = is compatible with e = but not with f =

13 Tree Space not all sets of splits form a tree not all vectors are possible not a uclidean space... (,,,, 3,, 3,,,,,...)...

14 Tree Space 5 1

15 Tree Space = geodesic = cone path

16 Tree Space T n T T 1 T T 1 = shortest path (geodesic)

17 T n Structure of x

18 Geodesics

19 Tree Space Properties Theorem (illera, Holmes,Vogtmann, 1): Tree space has global non-positive curvature (T()). unique geodesics (shortest paths) well-defined mid-point tree geodesic distance = length of shortest path (geodesic) between two trees T 1 and T computable in polynomial time via GTP algorithm (O., Provan, 11)

20 Non-positive urvature non-positive curvature (NP) = triangles are at least as thin as in uclidean space global non-positive curvature = all triangles are at least as thin as in uclidean space = T() triangle in a NP space: uclidean comparison triangle: X d(,x) d (,X ) X T() unique shortest paths (geodesics)

21 T() ubical omplexes Theorem (Gromov, 1987): cubical complex is T() xxit is simply connected and the link of any vertex is a flag simplicial complex it is simply connected and if a vertex is incident to K edges, any pair of which specify a square, then these K edges also specify a K-dimensional cube. not T(): T():

22 Thin Triangle = geodesic

23 Outline 1. Tree spaces, including description of HV tree space. Polynomial time algorithm for computing distance in HV tree space 3. Open problems. Mean and variance in HV tree space 5. pplications of HV tree space 6. More open problems

24 Size of T n if trees have n leaves, then: orthants have (n-)-dimensions (n - 3)!! tree topologies = (n - 3)!! orthants orthant = non-negative part of R n

25 ommon dges Lemma (illera, Holmes, Vogtmann, 1): If e is a common edge, then every tree on the geodesic also contains e = =

26 Geodesic istance so can restrict problem to computing geodesic distance between two trees T 1 and T with no common edges two previous exponential algorithms: GeoMeTree (Kupczok et al., 8) GeodeMaps (O., 11) - approx. algorithm (menta et al, 7)

27 Geodesic ombinatorics e e 3 3 e 1 drop e 1 add f 1 1 e 3 e f 1 5 drop e 3 f 1 add f 3 e 5 drop e f 3 1 add f f 1 3 f f drop T T 1 e 1 e 3 e 3 5 add f 1 1

28 Geodesic ombinatorics e 1 e 1 e drop e 1 add f 1 f 1 1 e 3 e 5 3 drop e 3 add f 3 e f 3 drop e 3 f 1 add f 3 e 5 drop e f 3 1 add f f 1 3 f T T drop e 1 and e 1 3 e 1 5 add f 1 and f f 3 5

29 Path Spaces e 1 f 1 e 1 e e T 1 1 e 3 e f drop e and e 3 e 1 add f 3 f drop e 1 3 f 3 add f 1 and f f f 1 f T

30 Geodesic ombinatorics f 1 e e f 1 f f 3 f 1 e f 3 f 1 f f T e f 1 5 f 3 1 f 1 e e 3 e 1 e f 3 3 e 1 e 1 T 1 e 3 e 5 1 e 1 e e 3 f 3 e

31 haracterizing Geodesics at i th transition between orthants: edges i are dropped edges i are added ( 1,..., k ) partitions (T 1 ) and ( 1,..., k ) partitions (T ) geodesic characterized by 3 properties Property 1: xx i and j compatible for all i > j

32 Property 1 Property 1: xx i and j compatible for all i > j e 1 e T 1 e 1 drop 1 3 e 1 f 1 f add 1 f drop f add T

33 Property (e 1,e,e 3 ) x 3 1 = { e 1, e } 1 = { f 1, f } x 1 = { e 3 } = { f 3 } x (-f 1,-f,-f 3 )

34 R k Isometric to part of R k (e 1,e,e 3 ) x 3 (-f 1,-f,e 3 ) ( 1, ) x 3 ( e 1 + e, e 3) ( f1 + f, e 3) = x 1 (,,e 3 ) v 1 v (-f 1,-f,) v 1 v x (-f 1,-f,-f 3 ) = geodesic ( f1 + f, f 3) = ( 1, )

35 Property line from ( 1,..., k ) to (- 1,..., - k ) is the geodesic in our region of k k geodesic distance = uclidean distance xx = iff k i + i i=1 Property : k k R k

36 Property 3 1 = {e, e 5 } 1 = {e 1, e 3 } e 5 e 1 e 1 e 3 1 e e 5 3 T 1 T e e 3 e e 3

37 Property 3 1 = {e, e 5 } e 5 drop e, e 5 add e 1, e 3 e 1 1 = {e 1, e 3 } e 1 e 3 1 e e 5 3 T 1 drop e 5 add e 3 drop e add e 1 T e e For ( i, i ), partition 1 of i, and partition 1 xxxxxxxof i, such that is compatible with 1.

38 Property 3 1 = {e, e 5 } 1 = {e 1, e 3 } 1 Want e e 5 3 e < T 1 T drop 1 add 1 drop add T e 1 e e 3 e 3 e 1 3 1

39 Property 3 Property 3: xx For each pair ( i, i ), partition 1 of i, and partition 1 1 of i, such that is compatible with 1 and. 1 < Theorem: Partitions ( 1,..., k ) and ( 1,..., k ) represent the geodesic iff Properties 1,, and 3 hold.

40 Geodesic lgorithm Initialize: 1 = (T 1 ), 1 = (T ) (cone path) P1 and P hold. Iterative Step: P1 and P hold for ( 1,..., r ) xxand ( 1,..., r ). oes ( i, i ) satisfies Property 3 for every i? No: split blocks i and i, and re-index the new partition to get ( 1,..., r+1 ) and ( 1,..., r+1 ). Yes: we are done.

41 hecking Property 3 Property 3: For each pair ( i, i ), partition 1 of i, and partition 1 of i, such that is 1 1 < compatible with 1 and. G( i, i ) { i 1 { { } 1 } { i dges between incompatible vertices

42 hecking Property 3 For each pair ( i, i ), partition 1 of i, and partition 1 with 1 and. of i, such that is compatible 1 1 < i = i = 1 an assume. Then 1 1 < 1 1 or + 1 = e e + f 1 f > 1

43 { Property 3 G( i, i ) i { i Weight each vertex: edge i Weight each vertex: edge i dges between incompatible vertices We can add an orthant if there is a min. weight vertex cover of G( i, i ) with weight < 1. omplexity: O(n 3 ) (Solve as max. flow problem.)

44 Geodesic lgorithm Initialize: 1 = all splits of T 1, 1 = all splits of T xxxxxxxxxxx(cone path) Iterative Step: urrent orthant sequence given xxby ( 1,..., r ) and ( 1,..., r ). oes ( i, i ) satisfy the Shortcut Property for any i? Yes: split blocks i and i, and re-index the new partition to get ( 1,..., r+1 ) and ( 1,..., r+1 ). No: we are done. Total time: O(n )

45 Open Problems HV space: put L 1 metric on orthants, instead of L metric geodesic distance is the weighted Robinson- Foulds distance geodesics are not unique so not T() what if put L metric on orthants?

46 how often are geodesics not unique? OP: Tropical Tree Space dissimilarity map tropical tree space = set of dissimilarity maps in that are realizable as trees R (n ) open problems: algorithm for computing geodesic

47 OP: Tropical Tree Space tropical and HV versions of tree space: same combinatorics different geodesics e 1 HV tropical e 1 e e e 3 e 3

48 OP: Tropical Tree Space tropical and HV versions of tree space: same combinatorics different geodesics e 1 HV tropical e 1 e e = HV geodesic e 3 = tropical geodesic e 3

49 OP: Phylo Orange Space phylogenetic orange/edge-product space: compatification of HV tree space at all trees with all edge lengths are identified space where trees are identified iff they induce the same Markov process on their leaves what is a natural metric for this space? properties of the space under this metric? how to compute distances?

50 OP: Other Tree Spaces space of unlabelled trees with n-leaves? (Feragen et al. 1, 11; Hultman 7) space of trees with different, but overlapping taxa sets (i.e. for supertrees) what about just one potentially missing taxon? (i.e. rogue taxon) space of phylogenetic networks?

51 OP: Visualization what is the best way to visualize a set of points in some tree space? Multi-imensional Scaling (Hillis, Heath, St. John, 5) tree of trees (Nye, 8; hakerian and Holmes, 1)

52 nd of Part I L. illera, S. Holmes, and K. Vogtmann. Geometry of the space of phylogenetic trees. dvances in pplied Mathematics, 7: , 1. M. Owen and S. Provan. fast algorithm for computing geodesic distances in tree space. I/M Trans. omputational iology and ioinformatics, 8:-13, 11. GTP code: miscellaneous/provan/treespace