Phylogenies via Quartets

Transcription

1 Phylogenies vi Qurtets Dvi Brynt LIRMM, Frne CRM, U. e M. U. Cnterury MGill University

2 Bite-size trees There is only one unroote tree for one, two or three tx... But there re four unroote trees for four tx. Qurtets re the smllest informtive unroote trees.

3 Qurtets n trees Qurtets still ontin enough informtion to reonstrut tree. f e e e f e f ef f ef ef e f ef ef ef

4 Qurtet se tree onstrution pe er ow og emu x x x x x x x x x ACCGCTAAAAACCACC A ACT x ACCCCAAAA CACG ATCAACT x x ACGCGAAGAGA C AA CGCACT x x x AGAGAGCCCAC T AA?ACTTT AGACTACTACTAGGGG TT ACCGCTAAAAACCACC A ACT AGACTACTACTAGGGG TT ACGCGAAGAGA C AA CGCACT AGAGAGCCCAC T AA?ACTTT ACCCCAAAA CACG ATCAACT DATA QUARTETS PHYLOGENY

5 Step 1: inferring qurtets Limiting yourself to four tx hs numer of vntges: Computtionlly esier We n use more hrters (e.g. lign more sites) We n inorporte more omplex moels We n nlyse t without ssuming tree

6 Qurtets from istnes Neighor joining D + D < D + D D + D < D + D Orinry lest squres (D + D D D ) 2 < (D + D D D ) 2 (D + D D D ) 2 < (D + D D D ) 2 Also weighte lest squres,...

7 Qurtets using prsimony Determine whih of,, is the most prsimonious. There re only three informtive hrters: I II III I II III Chooses the qurtet tht is supporte y the lrgest numer of hrters.

8 Qurtets y mximum likelihoo Compute the likelihoo sores for eh qurtet Choose the qurtet with highest likelihoo

9 Performne of qurtet inferene methos Performne epens on rnh lengths. Prsimony NJ Lest-squres y y x x x x y x x y

10 Confiene in qurtets Short internl eges re hr euse we n t tell whih wy they re resolve.

11 Confiene in qurtets Long externl eges re hr euse of lrge numers of hien sustitutions.

12 Giving onfiene to qurtets Bootstrpping Anlytil onfiene mesures Estimtes using Byes theorem: P [I] = L[I] L[I] + L[II] + L[III] Differenes in prsimony lengths Aim: qurtet weights

13 Qurtets s generlise istnes e i g An estimte of this rnh length... j f h i g...is n estimte of this pth length e Weighte qurtets n e use to generlise lef to lef istne estimtes to noe to noe istne estimtes.

14 Comining qurtets There re no prolems if the t is perfet: f e e e f e f ef f ef ef e f ef ef ef Esy when given the omplete set of qurtets of tree.

15 Prtil sets of qurtets Suppose tht we know tht the qurtets n e re true. Strt with n look t the possile ples to insert the finl txon e.

16 Prtil sets of qurtets Suppose tht we know tht the qurtets n e re true. Strt with n look t the possile ples to insert the finl txon e. e e e

17 Prtil sets of qurtets Suppose tht we know tht the qurtets n e re true. Strt with n look t the possile ples to insert the finl txon e. e e e

18 Prtil sets of qurtets Suppose tht we know tht the qurtets n e re true. Strt with n look t the possile ples to insert the finl txon e. e e e

19 Prtil sets of qurtets Suppose tht we know tht the qurtets n e re true. Strt with n look t the possile ples to insert the finl txon e. e e e

20 Prtil sets of qurtets Suppose tht we know tht the qurtets n e re true. Strt with n look t the possile ples to insert the finl txon e. e e

21 Qurtet inferene rules Every tree with: must lso ontin: (1), e e (2), e e (3), ef, ef f Exmple: we re given f, e f, e f. f n e f give e f (y rule 2) e f n e f give f (y rule 1)

22 Qurtet inferene rules Qurtet inferene rules first stuie y Dekker There re infinitely mny qurtet inferene rules However only few re neee for mny pplitions Short qurtet metho Willson s qurtet orretion proeure Pthing up X-trees

23 Comining qurtets is hr Given olletion of true qurtets, is there n unroote tree ontining them? ef ef e f e The generl prolem is NP-hr, whih mens there is proly no fst lgorithm for solving it.

24 Comining qurtets is hr There re esy speil ses of qurtet omptiility: When ll qurtets ontins prtiulr lef; When we hve one qurtet for eh set of four leves; When the set of qurterts ontins minimum efining suset

25 Experiments moel An orle will tell you how ny four tx re relte. How mny queries o you nee to mke to etermine the evolutionry tree for N tx? When you hve N tx, O(N log N) queries re neessry n suffiient (Knnn n Wrnow)

26 From qurtets to trees REALITY CHECK: We o not hve qurtet orle We o not hve olletion of gurntee qurtets Alterntive: fin tree tht grees with s mny qurtets s possile.

27 Qurtets tht lmost fit e e e e 80% (not too ) e

28 Heuristis - ATree ATree: First onvert the qurtets into istnes, then pply lustering (Sttth n Tversky 1977). D[, ] = numer of qurtets of the form x y Version of Ben-Dor et l.: first eme points in R n using semi-efinte progrmming, then ompute istnes n use lustering.

29 Heuristis - Qurtet Puzzling Strimmer n von Heseler (1996) h g f e x

30 Heuristis - Qurtet Puzzling Strimmer n von Heseler (1996) h g f e x

31 Heuristis - Qurtet Puzzling Strimmer n von Heseler (1996) 1 h x f e fg x g

32 Heuristis - Qurtet Puzzling Strimmer n von Heseler (1996) 1 h x f e fg x g

33 Heuristis - Qurtet Puzzling Strimmer n von Heseler (1996) 1 h g f x 1 e fg x

34 Heuristis - Qurtet Puzzling Strimmer n von Heseler (1996) 1 h g f 1 1 x 2 2 x 1 e fg x

35 Heuristis - Qurtet Puzzling Insertion orer is rnom Ties re roken rnomly The onstrution is repete mny times, n mjority rule onsensus tree returne.

36 Moifitions of Qurtet Puzzling Vinent Rnwez (n others) hve notie glith in qurtet puzzling: h g f e x

37 Moifitions of Qurtet Puzzling Vinent Rnwez (n others) hve notie glith in qurtet puzzling: h g f e x Also moifie insertion sheme (W O metho).

38 Heuristis - AQu An gglomertive lgorithm for qurtet trees. An vntge of this pproh is tht estimtes of qurtet support n e improve uring gglomertion (nlogous to NJ n Bio-NJ).

39 Ext lgorithms Dynmil progrmming ext lgorithm (Ben-Dor et l. 1998)... goo for up to 20 tx Restrite version: restrit to those trees with eges (splits) in given set. (Brynt n Steel 1999) e PTAS (Jing et l. 1998). e

40 Qurtets ross split e e f f g g f

41 Q tree Fin splits A B suh tht ll the qurtets ross A B re in Q. These splits form tree, lle the Q tree. The Q tree is the mximum resolve tree with ll its qurtets ontine in Q.

42 Extensions to Q tree Prolem: The Q tree is often str. Qurtet lening tree: inlue ll splits A B with less thn 1 2 ( A 1)( B 1) qurtets not in Q. With 100 tx, up to 1200 qurtets my not e in Q......ut this is out of 1, 500, 625 qurtets ross the split!

43 Hyperlening Hyperlening tree: inlue ll splits A B with less thn m 2 ( A 1)( B 1) qurtets not in Q (for some m 1). It tkes O(n 5 f(m)) time (for some ugly funtion f) ut oesn t lwys return tree.

44 The prolem with eing isrete Choosing one qurtet only ignores ineision. OR? Ignore unrelile qurtets? Keep ll, ut own-weight unrelile qurtets? A mixture?

45 Weighte qurtet reonstrution Qurtet Puzzling: L L 1 L 2 L 3 Sore L 1 L 2 L 3 L 1 +L 2 +L 3 L 1 +L 2 +L 3 L 1 +L 2 +L 3 Dynmil progrmming lgorithms, AQu, ll llow weighte qurtets. Sore of tree equls the sum of the sores of its qurtets.

46 Willson Prsimony Compute prsimony sores for,,. The inonsisteny of qurtet is the length of the qurtet minus the shortest length of the other two qurtets. Exmple: if hs length 20, hs length 14 n hs length 25, the inonsisteny of is = 6.

47 Willson Prsimony. The inonsisteny of tree is equl to the inonsisteny of its most inonsistent qurtet. We wnt to fin the lest inonsistent tree. Willson suggests tx insertion heuristi similr to qurtet puzzling.

48 Bunemn Tree w( ) = 1 2 (min(d + D, D + D ) D D ) w( ) is n estimte of the internl rnh length.

49 Bunemn tree Minimum of w( ) over ll qurtets ross rnh is n estimte of the rnh length. Bunemn tree: tke splits A B whih hve ll positive qurtets.

50 Clen tree To get the most urte estimte of n rnh length, we wnt to use s mny qurtets s possile. Clen tree: tke splits A B suh tht the verge of the minimum ( A 1)( B 1) weight qurtets ross the split is positive. The Clen tree is the weighte equivlent of the Qurtet lening tree.

51 WLS with qurtets k m n e f g Length of pth from k to q estimte from gi, gj, hi, hj. WLS fit of tree is v xy (length pth xy in tree length estimte) 2 xy o p q r s j i h

52 How well o qurtet methos work? Philippe n Douzery (1994) The Pitflls of Moleulr Phylogeny Bse on Four Speies, s Illustrte y the Cete/Artiotyl Reltionships. Use NJ n Prsimony with ootstrpping One tx from eh of four groups Different smples le to ifferent resolutions

53 How well o qurtet methos work? Strimmer n von Hessler (1996) Qurtet puzzling. Two 8-tx trees, one with moleulr lok, one without. Dimeter rnge from 0.17 to 1.4. Jukes-Cntor n K2P moels. Performne of QP inferior to Mx Likelihoo. Generlly etter thn NJ, espeilly when t non-loklike.

54 How well o qurtet methos work? Berry n Gsuel (1997) Reonstruting phylogenies from resolve 4-trees. 10-tx trees generte using Poissonin proess Dimeter rnge from 0.04 to 0.4. K2P moel, with mixe sustitution rtes over sites/eges Q metho mkes few errors Q + AQu generlly outperforme NJ (w.r.t. num flse s n flse + s

55 How well o qurtet methos work? Berry et l. Hyperlening pper. Use rel 10-tx tree (with ege sling). Hyperlening reovering most rnhes in tree, even with quite short (100 site) sequenes. Resonly urte when returning unresolve tree.

56 How well o qurtet methos work? Rnwez n Gsuel. Qurtet se phylogeneti inferene. Six ifferent 12-tx trees, with vrying rtes ross sites n lineges. Dimeter vrie from 0.1 to 2.0. FstDNAML n BioNJ onsistently outperform qurtet methos, inluing weighte vritions of QP n WO Qurtet puzzling outperforme y prsimony (exept when sequenes iverge lot

57 Do qurtet methos work? St. John et l. Performne stuy of Phylogeneti Methos (Qurtets n NJ). 5,10,20,40 tx trees, hoosen uniformly. Brnh lengths hosen uniformly. Jukes-Cntor sequenes. NJ generlly infers qurtets s well s ML Most qurtet methos re lousy t reonstruting trees n re highly sensitive to errors in qurtet t The more the tree infltes, the worse Q-methos perform

58 Wht then for qurtet methos? Theoretilly, Qurtet methos shoul e le to out-perform istne methos. Convergene results shoul e t lest s goo s existing methos. Currently we ignore omplex reltionships etween qurtets. Simultneous lignment n qurtet onstrution?

59 Questions... to o... isussion... Experimentl stuies to explin ifferenes etween experimentl stuies. How o we est infer qurtets n onfiene mesures for qurtets? Whih type of qurtets re using the prolems? How to est omine gglomertive pproh with qurtets? Is it neessry to use ll ( n 4) qurtets? Moel of epenenies etween qurtets Extension of qurtet methos to more generl supertree methos.