Distance-Based Approaches to Inferring Phylogenetic Trees

Transcription

1 Dstance-Base Approaches to Inferrng Phylogenetc Trees BMI/CS Mark Craven Fall 0 Representng stances n roote an unroote trees st(a,c) = 8 st(a,d) = 5 B C A E D B C D.5.5 A E stances represente by summe heght of eges to reach common ancestor stances represente by summe length of eges to reach common ancestor

2 Dstance-base approaches n! n M M! gven: an matrx where s the stance between taxa an!! o: bul an ege-weghte tree such that the stances between leaves an correspon to M A B C D E A B C D 0 5 E 0 A E D B C Where o we get stances?! commonly obtane from sequence algnments f = #msmatches #matches + #msmatches n algnment of sequence wth sequence! st(, ) = f! to correct for multple substtutons at a sngle poston: st Jukes-Cantor (, ) = " ln # " f & % ( $ '

3 Dstance metrcs! propertes of a stance metrc st( x, x )! 0 st( x, x ) = 0 st( x, x ) = st( x, x ) st( x, x )! st( x, x ) + st( x, x ) k k The molecular clock hypothess! In the 960s, sequence ata were accumulate for small, abunant protens such as globns, cytochromes c, an fbrnopeptes. Some protens appeare to evolve slowly, whle others evolve raply.! Lnus Paulng, Emanuel Margolash an others propose the hypothess of a molecular clock: For every gven proten, the rate of molecular evoluton s approxmately constant n all evolutonary lneages

4 Fgure from Bonformatcs an Functonal Genomcs by J Pevsner. Copyrght 009 by Wley. correcte amno ac changes per 00 resues (m) Mllons of years snce vergence The molecular clock assumpton & ultrametrc ata! the molecular clock assumpton s not generally true: selecton pressures vary across tme peros, organsms, genes wthn an organsm, regons wthn a gene! f t oes hol, then the ata s sa to be ultrametrc

5 The molecular clock assumpton & ultrametrc ata! ultrametrc ata: for any trplet of sequences,,, k, the stances are ether all equal, or two are equal an the remanng one s smaller A B C D E A B C D 0 5 E 0 A E D B C The UPGMA metho (Unweghte Par Group Metho usng Arthmetc Averages)! gven ultrametrc ata, UPGMA wll reconstruct the tree T that s consstent wth the ata! basc ea:! teratvely pck two taxa/clusters an merge them! create new noe n tree for merge cluster! stance between clusters an of taxa s efne as = C C # pq p "C, q "C! (avg. stance between pars of taxa from each cluster) C C

6 UPGMA algorthm assgn each taxon to ts own cluster efne one leaf for each taxon; place t at heght 0 whle more than two clusters etermne two clusters, wth smallest efne a new cluster C = C! C efne a noe k wth chlren an ; place t at heght replace clusters an wth k! compute stance between k an other clusters on last two clusters, an, by root at heght k / / UPGMA! gven a new cluster Ck forme by mergng C an! we can calculate the stance between C k an any other cluster as follows C l C kl = l C C + l C + C

7 UPGMA example ntal state A B C D E A B C D 0 5 E 0 A E D B C after one merge AE B C D AE B 0 8 C 0 8 D 0 A E D B C after two merges after three merges fnal state UPGMA example (cont.) AE BC D AE BC 0 8 A E D B C D 0 AED BC AED 0 8 BC 0 A E D B C A E D B C

8 Neghbor onng! unlke UPGMA! oesn t make molecular clock assumpton! prouces unroote trees! oes assume atvty: stance between par of leaves s sum of lengths of eges connectng them! lke UPGMA, constructs a tree by teratvely onng subtrees! two key fferences! how par of subtrees to be merge s selecte on each teraton! how stances are upate after each merge Pckng pars of noes to on n NJ! at each step, we pck a par of noes to on; shoul we pck a par wth mnmal?! suppose the real tree looks lke ths an we re pckng the frst par of noes to on? A B AB = AC = 0.5 C! wrong ecson to on A an B: nee to conser stance of par to other leaves D

9 Pckng pars of noes to on n NJ! to avo ths, pck par to on base on [Satou & Ne 87; Stuer & Keppler 88] D =! + ( r r ) D r = L " $ k#l k where L s the set of leaves Upatng stances n neghbor onng! gven a new nternal noe k, the stance to another noe m s gven by:!! km = ( m + m! k! ) m! m m

10 Upatng stances n neghbor onng! can calculate the stance from a leaf to ts parent noe n the same way k = ( + m!!! k! m ) m! =! k k Upatng stances n neghbor onng! we can generalze ths so that we take nto account the stance to all other leaves where k = ( + r! r r = L " an L s the set of leaves $ m#l m! ths s more robust f ata aren t strctly atve )

11 Neghbor onng algorthm efne the tree T = set of leaf noes L = T! whle more than two subtrees n T! pck the par, n L wth mnmal D a to T a new noe k onng an! etermne new stances k = ( + r " r ) k = " k km = ( m + m " ) for all other m n L remove an from L an nsert k (treat t lke a leaf) on two remanng subtrees, an wth ege of length Testng for atvty! for every set of four leaves,,, k, an l, two of the stances + kl, k + l an l + k must be equal an not less than the thr

12 Rootng trees! fnng a root n an unroote tree s sometmes accomplshe by usng an outgroup! outgroup: a speces known to be more stantly relate to remanng speces than they are to each other! ege onng the outgroup to the rest of the tree s best canate for root poston outgroup canate root Rootng trees chmpanzee lce use as outgroup n human lce stuy

13 Comments on stance-base methos! f the gven stance ata s ultrametrc (an these stances represent real stances), then UPGMA wll entfy the correct tree! f the ata s atve (an these stances represent real stances), then neghbor onng wll entfy the correct tree! otherwse, the methos may not recover the correct tree, but they may stll be reasonable heurstcs! neghbor onng s commonly use