CSCI1950 Z Computa4onal Methods for Biology Lecture 2. Ben Raphael January 26, hhp://cs.brown.edu/courses/csci1950 z/ Outline

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1 CSCI1950 Z Comput4onl Methods for Biology Lecture 2 Ben Rphel Jnury 26, 2009 hhp://cs.brown.edu/courses/csci1950 z/ Outline Review of trees. Coun4ng fetures. Chrcter bsed phylogeny Mximum prsimony Mximum likelihood 1

2 Tree Defini4ons tree: A connected cyclic grph G = (V, E). grph: A set V of vertices (nodes) nd set E of edges, where ech edge (v i, v j ) connects pir of vertices. A pth in G is sequence (v 1, v 2,, v n ) of vertices in V such tht (v i, v i+1 ) re edges in E. A grph is connected provided for every pir v i v j of vertices, there is pth between v i nd v j. A cycle is pth with the sme strting nd ending vertices. A grph is cyclic provided it hs no cycles. Tree Defini4ons degree of vertex v is the number of edges incident to v. A phylogenetic tree is tree with lbel for ech lef (vertex of degree one). A binry phylogenetic tree is phylogenetic tree where every interior (non-lef) vertex hs degree 3; (one prent nd two children). A rooted (*binry) phylogenetic tree is phylogenetic tree with single designted vertex r (* of degree 2). w is prent (ncestor) of v provided (v,w) is on pth to root. In this cse v is child (descendnt) of w. 2

3 Tree Defini4ons tree: A connected cyclic grph G = (V, E). degree of vertex v is the number of edges incident to v. A phylogenetic tree is tree with lbel for ech lef (vertex of degree one). Leves represent existing species Other vertices represent most recent common ncestor. Length of brnches represent evolutionry time. Root (if present) represents the oldest evolutionry ncestor. Coun4ng nd Trees A tree with n ver4ces hs n 1 edges. (Proof?) A rooted binry phylogene4c tree with n leves hs n 1 internl ver4ces; nd thus 2n 1 totl ver4ces. How mny rooted binry phylogene4c trees with n leves? 3

4 Chrcter bsed Phylogene4c Tree Reconstruc4on Input Chrcters Moleculr Morphologicl Algorithm Output 1. Wht is chrcter dt? 2. Wht is the criteri for evlu6ng tree? 3. How do we op6mize this criteri: 1. Over ll possible trees? 2. Over restricted clss of trees? Op6ml phylogene4c tree Chrcter Bsed Tree Reconstruc4on Chrcters my be morphologicl fetures # of eyes or legs or the shpe of bek or fin. Chrcters my be nucleo4des of DNA (A, G, C, T) or mino cids (20 leher lphbet). Vlues re clled sttes of chrcter. Gorill: CCTGTGACGTAACAAACGA Chimpnzee: CCTGTGACGTAGCAAACGA Humn: CCTGTGACGTAGCAAACGA Non inform4ve chrcter 2 stte chrcter 4

5 Chrcter Bsed Tree Reconstruc4on GOAL: determine wht chrcter strings t internl nodes would best explin the chrcter strings for the n observed species An Exmple Vlue1 Vlue2 Mouth Smile Frown Eyebrows Norml Pointed 5

6 Chrcter Bsed Tree Reconstruc4on Which tree is beaer? Chrcter Bsed Tree Reconstruc4on Count chnges on tree 6

7 Chrcter Bsed Tree Reconstruc4on Mximum Prsimony: minimize number of chnges on edges of tree Mximum Prsimony Ockhm s rzor: simplest expln4on for the dt Assumes tht observed chrcter differences resulted from the fewest possible mut4ons Seeks tree with the lowest possible prsimony score, defined sum of cost of ll mut4ons found in the tree 7

8 Chrcter Mtrix Given n species, ech lbeled by m chrcters. Ech chrcter hs k possible sttes. Gorill: CCTGTGACGTAACAAACGA Chimpnzee: CCTGTGACGTAGCAAACGA Humn: CCTGTGACGTAGCAAACGA n x m chrcter mtrix Assume tht chrcters in chrcter string re independent. Prsimony Score Gorill: CCTGTGACGTAACAAACGA Chimpnzee: CCTGTGACGTAGCAAACGA Humn: CCTGTGACGTAGCAAACGA Assume tht chrcters in chrcter string re independent. Given chrcter strings S=s 1 s m nd T=t 1 t m : #chnges (S T) = Σ i d H (s i, t i ) where d H = Hmming distnce d H (v, w) = 0 if v=w d H (v, w) = 1 otherwise prsimony score of the tree s the sum of the lengths (weights) of the edges 8

9 Prsimony nd Tree Reconstruc4on Mximum Prsimony Two comput4onl sub problems: 1. Find the prsimony score for fixed tree. Smll Prsimony Problem (esy) 2. Find the lowest prsimony score over ll trees with n leves. Lrge prsimony problem (hrd) 9

10 Smll Prsimony Problem Input: Tree T with ech lef lbeled by n m chrcter string. Output: Lbeling of internl ver4ces of the tree T minimizing the prsimony score. Since chrcters re independent, every lef is lbeled by single chrcter. Smll Prsimony Problem Input: T: tree with ech lef lbeled by n m chrcter string. Output: Lbeling of internl ver4ces of the tree T minimizing the prsimony score. Lrge Prsimony Problem Input: M: n n x m chrcter mtrix. Output: A tree T with: n leves lbeled by the n rows of mtrix M lbeling of the internl ver4ces of T minimizing the prsimony score over ll possible trees nd ll possible lbelings of internl ver4ces 10

11 Smll Prsimony Problem Input: Binry tree T with ech lef lbeled by n m chrcter string. Output: Lbeling of internl ver4ces of the tree T minimizing the prsimony score. Since chrcters re independent, every lef is lbeled by single chrcter. Weighted Smll Prsimony Problem More generl version of Smll Prsimony Problem Input includes k x k scoring mtrix δ describing the cost of trnsforming ech of k sttes into nother stte. Smll Prsimony Problem is specil cse: δ ij = 0, if i = j, 1, otherwise. 11

12 Scoring Mtrices Smll Prsimony Problem A T G C A T G C Weighted Smll Prsimony Problem A T G C A T G C Unweighted vs. Weighted Smll Prsimony Scoring Mtrix: A T G C A T G C Smll Prsimony Score: 5 12

13 Unweighted vs. Weighted Weighted Prsimony Scoring Mtrix: A T G C A T G C Weighted Prsimony Score: 22 Weighted Smll Prsimony Problem Input: T: tree with ech lef lbeled by n m chrcter string from k leher lphbet. δ: k x k scoring mtrix Output: Lbeling of internl ver4ces of the tree T minimizing the weighted prsimony score. 13

14 Snkoff Algorithm Clculte nd keep trck of score for every possible lbel t ech vertex: s t (v) = minimum prsimony score of the subtree rooted t vertex v if v hs chrcter t s t (v) t.. Snkoff Algorithm s t (v) = minimum prsimony score of the subtree rooted t vertex v if v hs chrcter t The score s t (v) is bsed only on scores of its children: s t (prent) = min i {s i ( leo child ) + δ i, t } + min j {s j ( right child ) + δ j, t } t δ i, t δ j, t s i (leo child) s j (right child) 14

15 Snkoff Algorithm (cont.) Begin t leves: If lef hs the chrcter in ques4on, score is 0 Else, score is Snkoff Algorithm (cont.) s t (v) = min i {s i (u) + δ i, t } + min j {s j (w) + δ j, t } s i (u) δ i, A sum s A (v) = 0 min i {s i (u) + δ i, A } + min j {s j (w) + δ j, A } A T 3 G 4 C 9 15

16 Snkoff Algorithm (cont.) s t (v) = min i {s i (u) + δ i, t } + min j {s j (w) + δ j, t } s j (u) δ j, A sum s A (v) = min 0 i {s i (u) + δ i, A } + + min 9 = j {s 9 j (w) + δ j, A } A 0 T 3 G 4 C Snkoff Algorithm (cont.) s t (v) = min i {s i (u) + δ i, t } + min j {s j (w) + δ j, t } Repet for T, G, nd C 16

17 Snkoff Algorithm (cont.) Repet for right subtree Snkoff Algorithm (cont.) Repet for root 17

18 Snkoff Algorithm (cont.) Smllest score t root is minimum weighted prsimony score In this cse, 9 so lbel with T Snkoff Algorithm: Trveling down the Tree The scores t the root vertex hve been computed by going up the tree Aoer the scores t root vertex re computed the Snkoff lgorithm moves down the tree nd ssign ech vertex with op4ml chrcter. 18

19 Snkoff Algorithm (cont.) 9 is derived from So left child is T, And right child is T Snkoff Algorithm (cont.) And the tree is thus lbeled 19

20 Anlysis of Snkoff s Algorithm A dynmic progrmming problem lgorithm: Op>ml substructure: solu4on obtined by solving smller problem of sme type. s t (prent) = min i {s i ( leo child ) + δ i, t } + min j {s j ( right child ) + δ j, t } t Recurrence termintes t leves, where solu4on is known. s i (leo child) δ i, t δ j, t s j (right child) Anlysis of Snkoff s Algorithm How mny comput6ons do we perform for n species, m chrcters, nd k sttes per chrcter? Forwrd step: At ech internl node of tree: s t (prent) = min i {s i ( leo child ) + δ i, t } + min j {s j ( right child ) + δ j, t } 2k sums nd 2 (k 1) comprisons = 4k 2 n 1 internl nodes. (4k 2)(n 1) sums. Trcebck: one lookup per internl node. (n 1) oper4ons For ech chrcter (4k 2)(n 1) + (n 1) oper4ons C n k Above clcul4on performed once for ech chrcter: C m n k oper4ons O( m n k) 4me. [ big O ] Increses linerly w/ # of species or # of chrcters. 20

21 Anlysis of Snkoff s Algorithm How mny comput6ons do we perform for n species, m chrcters, nd k sttes per chrcter? Trcebck: 2k sums Above clcul4on performed once for ech chrcter O( m n k) 4me. [ big O ] Increses linerly w/ # of species or # of chrcters. Fitch s Algorithm Solves Smll Prsimony problem Published 4 yers before Snkoff (1971) Mkes two psses through tree: Leves root. Root leves. 21

22 Fitch Algorithm: Step 1 Assign set S(v) of lehers to every vertex v in the tree, trversing the tree from leves to root S(l) = observed chrcter for ech lef l For vertex v with children u nd w: S(v) = S(u) S(w) if non empty intersec4on S(u) S(w), otherwise E.g. if the node we re looking t hs leo child lbeled {A, C} nd right child lbeled {A, T}, the node will be given the set {A, C, T} Fitch s Algorithm: Exmple c t {,c} {t,} {,c} c t {t,} c t 22

23 Fitch Algorithm: Step 2 Assign lbels to ech vertex, trversing the tree from root to leves. Assign root r rbitrrily from its set S(r) For ll other ver4ces v: If its prent s lbel is in its set S(v), ssign it its prent s lbel Else, choose n rbitrry leher from its set S(v) s its lbel Fitch s Algorithm: Exmple c t {,c} {t,} c t {,c} {t,} c t c t 23

24 Fitch Algorithm (cont.) Fitch vs. Snkoff Both hve n O(nk) run4me Are they ctully different? Let s compre 24

25 Fitch As seen previously: Comprison of Fitch nd Snkoff As seen erlier, the scoring mtrix for the Fitch lgorithm is merely: A T G C A T G C So let s do the sme problem using Snkoff lgorithm nd this scoring mtrix 25

26 Snkoff Snkoff vs. Fitch The Snkoff lgorithm gives the sme set of op4ml lbels s the Fitch lgorithm For Snkoff lgorithm, chrcter t is op4ml for vertex v if s t (v) = min 1<i<k s i (v) Let S v = set of op4ml lehers for v. Then S v = S u S w if S u S v, S u S w, This is lso the Fitch recurrence The two lgorithms re iden4cl otherwise. 26

27 A Problem with Prsimony Ignores brnch lengths on trees A A A A A A A C A A A A Sme prsimony score. Mut4on more likely on longer brnch. A C x Probbilis4c Model y t Pr[ x y, t] = probbility tht y muttes to x in 4me t Given tree T with leves lbeled by present chrcters, wht is the probbility of lbeling of ncestrl nodes? Assume: 1. Chrcters evolve independently. 2. Constnt rte of mut4on on ech brnch. 3. Stte of vertex depends only on prent nd brnch length: i.e. Pr[ x y, t] depends only on y nd t. (Mrkov process) 27

28 Probbilis4c Model y x t Pr[ x y, t] = probbility tht y muttes to x in 4me t Two species t 1 t 2 x 1 x 2 T = tree topology x 1, x 2 : chrcters for ech species : chrcter for ncestor Pr[ x 1, x 2, T, t 1, t 2 ] = q Pr[x 1, t 1 ] Pr[x 2, t 2 ] q = Pr[ ncestor hs chrcter ] Probbilis4c Model n species: x 1, x 2,, x n Let α(i) = ncestor of node i. Let n+1, n+2,, 2n 1 = chrcters on internl nodes, where nodes re number from internl ver4ces up to root. Pr[x 1,..., x n T, t 1,..., t 2n 2 ]= n+1, n+2,.., 2n 1 q 2n 1 2n 2 i=n+1 Pr[ i α(i),t i ] n Pr[x i α(i),t i ] i=1 Follows from Lw of Totl Probbility: P(X) = Σ P(X Y i ) P(Y i ). 28

29 Felsenstein s Algorithm Let Pr[T k ] = probbility of lef nodes below node k, given k =. Compute vi dynmic progrmming b c Pr[T k ] = b Pr[b, t i ]Pr[T i b] c Pr[c, t j ]Pr[T j c] Ini4l condi4ons. For k = 1,, n (lef nodes) Pr[T k ] = 1, if = x k 0, otherwise. Compu4ng the Likelihood Let Pr[T k ] = probbility of lef nodes below node k, given k =. Pr[x 1,..., x n T, t ]= Pr[T 2n 1 ]q Note: Root is node 2n 1 29

30 Mximum Likelihood Let Pr[T k ] = probbility of lef nodes below node k, given k =. ( Pr[T k ] = mx b ( ) Pr[b, t i ]Pr[T i b]) mx Pr[c, t j ]Pr[T j c] c Trcebck s before with Snkoff s lgorithm. Mx. Prsimony vs. Mx. Likelihood Set δ ij = log P(j i) in weighted prsimony (Snkoff lgorithm) Weighted prsimony produces mximum probbility ssignments, ignoring brnch lengths 30