Phylogeny and Molecular Evolution

Transcription

1 Phylogeny nd Moleculr Evolution Chrcter Bsed Phylogeny 1/50

2 Credit Ron Shmir s lecture notes Notes by Nir Friedmn Dn Geiger, Shlomo Morn, Sgi Snir nd Ron Shmir Durbin et l. Jones nd Pevzner s presenttion Bioinformtics Algorithms book by Phillip Compeu nd Pvel Pvzner ll book photos shown in this lecture re from there. 2/50

3 Type of Tree Reconstruction Chrcter-bsed Input is multiple lignment of sequences (one sequence per species). Distnce-bsed Input is mtrix of distnces between species Distnce cn be the reltive length of the sequence which the two sequences disgree on, or lignment score between them, or (whtever) 3/62

4 Distnce Bsed Tree Reconstruction 4/62

5 Distnce Bsed Tree Reconstruction 5/62

6 UPGMA Algorithm (cont d) /62

7 Type of Tree Reconstruction Chrcter-bsed Input is multiple lignment of sequences (one sequence per species). Distnce-bsed Input is mtrix of distnces between species Distnce cn be the reltive length of the sequence which the two sequences disgree on, or lignment score between them, or (whtever) 8/62

8 Chrcter-bsed tble of winged nd wingless insects 9/62

9 Evolutionry Tree of Winged (red) nd Wingless (blue) Stick Insects bsed on 18S RiboxomlsRNA 10/50

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11 Chrcter-Bsed Tree Reconstruction Chrcters my be nucleotides, where A, G, C, T re sttes of this chrcter. Other chrcters my be the # of eyes or legs or the shpe of bek or mouth By setting the length of n edge in the tree to the Hmming distnce, we my define the score of the tree s the sum of the lengths (weights) of the edges 12/50

12 Exmple With Single Letter Sequences Suppose we hve five species, such tht three hve C nd two T t specified position Miniml tree hs only one evolutionry chnge: C C C C C T T T T C 13/50

13 Prsimony Approch to Evolutionry Tree Reconstruction Applies Occm s rzor principle to identify the simplest explntion for the dt Assumes observed chrcter differences resulted from the fewest possible muttions Seeks the tree tht yields lowest possible prsimony score - sum of cost of ll muttions found in the tree

14 Prsimony nd Tree Reconstruction

15 Prsimony nd Tree Reconstruction

16 Prsimony nd Tree Reconstruction

17 Smll Prsimony Problem Input: Tree T with ech lef lbeled by n m- chrcter string. Output: Lbeling of internl vertices of the tree T minimizing the prsimony score. We cn ssume tht every lef is lbeled by single chrcter, becuse the chrcters in string re independent. 18/50

18 Phylogenetic Trees 19/62

19 Chrcter-Bsed Smll Prsimony 20/50

20 Chrcter-Bsed Smll Prsimony 21/50

21 Chrcter-Bsed Smll Prsimony 22/50

22 Chrcter-Bsed Smll Prsimony 23/50

23 Extension to Mny Letters Wht is the prsimony score of Ardvrk Bison Chimp Dog Elephnt A: CAGGTA B: CAGACA C: CGGGTA D: TGCACT E: TGCGTA When the tree is known, we cn do it chrcter fter chrcter; ech score is computed independently of the others. 24/50

24 Smll Prsimony Minimizing Totl Hmming Distnce A T G C /50

25 Fitch Algorithm (Tree is Given) Work on ech position in string independently. Strt t the leves. If two children hve common chrcter, prent inherits it. Record union nd go up. After reching root, go down to fix sets of size > 1. A A A/C A/T A A C T A 26/50

26 Fitch s Algorithm, More Formlly trverse tree from leves to root determining set of possible sttes (e.g. nucleotides) for ech internl node trverse tree from root to leves picking ncestrl sttes for internl nodes 27/50

27 Fitch s Algorithm Step 1 do post-order (from leves to root) trversl of tree Determine possible sttes R i of internl node i with children j nd k R i R R j j R R k k if R j R otherwise k 28/50

28 C T G T A T 29/50

29 Fitch s Algorithm Step 1 # of chnges = # union opertions C T G T A T 30/50

30 Fitch s Algorithm Step 1 # of chnges = # union opertions T T AGT CT GT C T G T A T 31/50

31 Fitch s Algorithm Step 2 do pre-order (from root to leves) trversl of tree select stte r j of internl node j with prent i s follows: r j ri if ri rbitrry R j stte R j otherwise 32/50

32 Fitch s Algorithm Step 1 # of chnges = # union opertions T T AGT CT GT C T G T A T 33/50

33 Fitch s Algorithm Step 2 T T AGT CT GT C T A G T T 34/50

34 Fitch s Algorithm (cont d) Another exmple: c t 35/50

35 Fitch s Algorithm (cont d) Another exmple: c t {,c} {t,} c t

36 Fitch s Algorithm (cont d) Another exmple: c t {,c} {t,} {,c} c t {t,} c t

37 Fitch s Algorithm (cont d) Another exmple: c t {,c} {t,} {,c} c t {t,} c t c t 38/50

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39 Weighted Prsimony Weighted Prsimony score: Ech chnge is weighted by score c(,b). The weighted prsimony score reduces to the prsimony score when c(,)=0 nd c(,b)=1 for ll b. A T G C /50

40 Weighted Prsimony on Given Tree (Snkoff) Ech position is independent nd computed by itself. Use Dynmic progrmming on given tree. if k is node with children i nd j, then S(k,) = min b (S(i,b)+c(,b)) + min d (S(j,d)+c(,d)) S(i,b) i k S(k,) j S(j,d) S(j,d) the score of subtree rooted t j when j hs the chrcter d. 41/50

41 Evluting Prsimony Scores Dynmic progrmming on given tree Initiliztion: For ech lef i set S(i,) = 0 if i is lbeled by, otherwise S(i,) = Itertion: if k is node with children i nd j, then S(k,) = min x (S(i,x)+c(,x)) + min y (S(j,y)+c(,y)) Termintion: The cost of the tree is min x S(r,x) where r is the root Comment: If we keep in ech node for ech chrcter the two chrcters x, y tht bring bout the minimum, then we cn trce the best ssignment to ll internl nodes. 42/50

42 Snkoff s Algorithm An exmple A T G C A C T G For ech lef i set S(i,) = 0 if i is lbeled by, otherwise S(i,) =

43 Snkoff s Algorithm An exmple 0 A T G C A C T G if k is node with children i nd j, then S(k,) = min x (S(i,x)+c(,x)) + min y (S(j,y)+c(,y)) /50

44 Snkoff s Algorithm An exmple 0 A T G C if k is node with children i nd j, then S(k,) = min x (S(i,x)+c(,x)) + min y (S(j,y)+c(,y)) T T G A C T G /50

45 Complexity of Evluting (Smll) Prsimony If there re n nodes, m chrcters, nd s possible vlues for ech chrcter, then wht is the complexity of Snkoff s lgorithm for Smll Prsimony? 46/50

46 Complexity of Evluting (Smll) Prsimony If there re n nodes, m chrcters, nd s possible vlues for ech chrcter, then complexity is O(nms 2 ). Of course, in Lrge Prsimony we still need to serch over possible trees nd find the best one. One usully resorts to heuristic serch techniques. 47/50

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