Talk Outline. Taxon coverage pattern. Part I: Partial taxon coverage. Lassoing a tree: Phylogenetic theory for sparse patterns of taxon coverage

Transcription

1 Lssoing tr: Phylognti thory for sprs pttrns of ton ovrg Joint work with Tlk Outlin Prt 1: Disivnss Prt 2: Lssoing tr Mik Stl Anrs Drss Kthrin Hur Prt 3: Quntifying LGT [if tim?] Phylomni Novmr 10, 2011 Mihll MMhon Mihl Snrson 2 Prt I: Prtil ton ovrg Ton ovrg pttrn Group T Loi % Cittion Missing Mtzo Dunn t l Ppilionoi MMhon n lgums Snrson 2006 Gn1 Gn2 Two ton sts: {,,,} n {,,,} = {{,,,},{,,,}} Astrls Smith t l Eukryots Golooff t l Not: this os not ommit to ny pr4ulr gn tr topologis, just th ton sts

2 Disivnss Dfinition: Lt S olltion of ton sts. Thn: S is isiv for phylognti tr T provi T is uniquly trmin y how it rsolvs h of th ton sts in S. i.. if S= {Y 1,,Y k } thn T is th only tr tht isplys T Y 1,.,T Y k S is phylogntilly isiv for ll trs if this hols for ll hois of T. T 5 must inry Not phylogntilly isiv (for ll trs) Of th 15 possil inry unroot trs for this t st... Thr r 6, lik tht low, whr th ton ovrg is isiv...n 9, lik tht low, whr it is not isiv Phylogntilly isiv (for ll trs) Tsting whthr ton ovrg is phylogntilly isiv (for ll trs) Thorm [S+Snrson, 2010] S is phylogntilly isiv for ll trs if n only if: For h 4-wy prtition of th full ton st, thr r four t in on of th ton sts in S tht intrst h lok of th prtition. Complity? Mnul Biirsky s `No 8 rinow olouring prolm 8 2

3 A lowr oun on numr of loi for isivnss for ll trs Molling rnom ton ovrg Thorm: If olltion S of ton sts of siz n 1, n 2,, n k is phylogntilly isiv for ll trs with n lvs thn: k $ j=1 n j (n j "1)(n j " 2) # n(n "1)(n " 2) So, if h ton st in S hs siz t most m thn: n(n #1)(n # 2) k " " (n /m)3 m(m #1)(m # 2) Empls: m ol nsity (m/n) mi num loi (k) 33% 27 80% 2 p(,j) = proility ton is prsnt t loi j. Chois m inpnntly for k loi to gt pttrn S of ton ovrg (n, p j, k). Contt: whol gnom squning t low ovrg tht gnrts prtil ssmlis. Or omplt gnom squns t p phylognti pths whr loss of homology is n issu in ssmling t sts 9 Simplst mol: Uniform ovrg: p j =p Thorm For ny root inry tr T with n lvs, with ovrg proility p, th proility tht st S of k (rnom) ton sts is phylogntilly isiv for T is t lst 1 - if ( ) #log(1# p 3 ) log (n # 2) /$ k " Morovr, if k is muh lss thn this, thn S is not (w.p. 1- ) phylogntilly isiv ~ log( n /) p 3 # n / 3 & log% ( $ "log(1") ' k "log(1" p 3 ) Thr tnsions Allow h g in h T Y i to prsnt only with proility q (ls th g is ollps) p 3 p 3 q Vril ovrg (VC): p ( ) p for ll t in l C, n p ( ) p' for ny ton not in C (whr p ' <p ). Unroot trs p 3 p 4 p 3 q p 2 p'q 3

4 Etnsion: Trrs" Disivnss in rl t (rnom smpling) (Sin, 2011)# # n # Construt optiml tr for ll th loi (MP, or ML/Bysin with prtition nlysis) Lmm: If T is n optiml tr, n Y1,,Yk r th t sts thn ANY othr tr tht isplys T Y1,.,T Yk is optiml. Ls to lrg trrs of qully optiml trs. Eg. Bouhnk- Khlli t l [Grsss] 298 t, 3 loi. 61 million trs By rmoving 12 of 298 originl t, trr siz ru from 61 million trs to 1 tr. Thorm [MB]: All trs on trr long to th sm tr isln p 13 Tlk Outlin NJ n ll tht Prt 1: Disivnss Prt 2: Lssoing tr ( T,l ) Prt 3: Quntifying LGT (T, l ) f f NJ Clssi rsults (1960s) (T,l ) = (T ʹ,l ʹ ) T = T ʹ & l = l ʹ Unr pthy ton smpling, rlil (,y) vlus my not vill for ll pirs. Empl: Thr my no lous tht ontins oth n y

5 Do w n ll th -vlus? " Givn L X % $ ' n = # 2 & Dos trmin T? l? Empl: ( T, l) L trmins T ut not l. L L = L "{} trmins T n l L = {,,, } L = {,,,, } trmins nithr Dfinition (Lssos) " L X % $ ' # 2 & L lssos th rnh lngths of T iff L = ʹ L l = lʹ ( T, l) ( T, l ) L lssos th shp of T iff L Lʹ T = Tʹ ( T, l) = ( T ʹ, lʹ ) L is strong lsso for T if oth hol. W ll th lmnt of L th hors of th lsso Th grph (X, L) L = {,,, } Qustion 1: Is vry rnh-lngth lsso strong lsso? Lmm: v If L lssos th shp of T thn (X,L) is onnt. ' v If L lssos th rnh lngths of T thn vry onnt omponnt of (X, L) hs n o yl. ' '

6 Is n rnh-lngth lsso strong lsso? Qustion 2: How smll n lssos? ' Osrvtion: Any rnh-lngth lsso of T hs siz t lst E(T). 2 2 ' ' ' No Thorm: If L lssos th shp of T thn L > E -1. Why? Str tr T* If L lssos th shp of str tr iff L= X$ # & " 2 % 2 2 ' ' Binry tr T Thr ist topologil lssos of siz 4(n-3). Why? Cn w o ttr? Qustion 2: How smll n lssos? A ky i: Lssos s Covrs Thorm Any inry tr T with n lvs hs strong lsso of siz 2n-3 [Yushmnov, 1984] Morovr, T hs topologil lsso of siz 2n-4. Lmm: If L lssos th shp or th rnh lngths of (inry) T thn L is ovr for T

7 Point ovrs Thorm: If L is point ovr for T thn L is strong lsso for T. Th Triplt Covr onjtur Lmm: If L ontins triplt ovr for T thn L lssos T s rnh lngths Thorm: If L ontins triplt ovr for T n on for T thn T=T Corollry: Thr r strong lssos for T of siz 2n-3 {,, } L Conjtur: If L ontins triplt ovr for T thn L lssos T s shp Finlly miniml rnh lngth lssos M(T) := st of rnh lngth lssos of T of miniml siz. Furthr tils Snrson, M.J., MMhon, M.M. n Stl, M. (2010). Phylognomis with inomplt ton ovrg: th limits to infrn. BMC Evolutionry Biology 10: 155 Simpl mpl: Str tr Empl of lmnt of M(T) for. n = 23 hr is on: Lmm: M(T) is mtroi tht trmins th shp of T (i.. M(T ) = M(T ') T = T ') Stl, M. n Snrson, M.J. (2010). Chrtrizing phylogntilly isiv ton ovrg. Appli Mthmtis Lttrs 23, Snrson, M.J., MMhon, M.M. n Stl, M. Trrs in phylognti tr sp. Sin 333: Drss, A.W.M., Hur, K.T. n Stl, M. 'Lssoing' phylognti tr (I): Bsi proprtis, shllings, n ovrs. Journl of Mthmtil Biology (in prss). Thorm: M(T ) is inry mtroi if n only if T is trpillr tr Lsso (II), (III) in prp. Th Annul Nw Zln Phylogntis Mting Sun. 29th Jn. - Fri. 3r F ABCASS F 4-12 th 1 st NZ phylogntis workshop F