BLAST / HIDDEN MARKOV MODELS

Transcription

1 CS262 (Winter 2015) Lecture 5 (January 20) Scribe: Kat Gregry BLAST / HIDDEN MARKOV MODELS BLAST CONTINUED HEURISTIC LOCAL ALIGNMENT Use Cmmnly used t search vast bilgical databases (n the rder f terabases/tetrabases) Quickly find all gd lcal alignments f query t databases! Gd = Abve threshld f length (ex 100+) and sequence similarity (ex 60+%) Heuristic! Relies n assumptin that a gd match is likely t cntain exact matching k-mers K-mer = Wrd f length k Why is this helpful? B/c exact wrds can be easily indexed and srted! Faster than full alignment (Smith-Waterman)! But cannt guarantee ptimal alignment Original BLAST (BLAST = Basic Lcal Alignment Search Tl) Altschul et al. (1990) is prbably mst cited paper in biinfrmatics! One f mst cited papers in science in general Fllwed previus tl FASA (1985) 100s t 1000s papers written n this technique Prcedure 1. Create table f all wrd ccurrences in databases! Table length: 4 k Can be in memry! Each lcatin in table: Dentes a specific wrd Pints t an array f all ccurrences f this specific wrd in database 2. Use sliding windw f length k t cnsider every k-mer in query! Find all matches f each query k-mer t database abve alignment scre threshld 3. Fr every match, extend t left and right t create ptential lcal alignment! Original BLAST: Ungapped extensins until scre belw statistical threshld In DP matrix, equates t matching alng diagnal! Mdificatin allws fr gapped extensins Cmpute a dynamic prgramming matrix where yu cannt crss a cell if it wuld make scre crss threshld a given amunt belw the best scre seen s far Take hme messages: BLAST wrks and is used in practice But we pay a cst in memry and time t find lcal alignments And there is a lt f science abut hw t minimize these csts and make BLAST wrk best SENSITIVITY-SPEED TRADEOFF Quantify: Sensitivity:! Of all existing lcal alignments between query and database, hw many d we find?! Defined fr a given length (say 100) sequence similarity (say 60, 80%) threshld Assume this sequence identity is distributed unifrmly acrss lcal alignment EX) Fr 80% similar match, assume each letter independently has 80% chance f matching

2 Speed:! Hw many randm matches f wrds between query and database are generated?! Every match nt assciated with a lcal alignment is csts cmputatin time because must extend alignment t left and right t check Findings frm Kent (2002):! Aside n Jim Kent: Credited with saving the public human genme prject as PhD student in 2000 Cmplex especially because f hetergeneity f different data sets Cmpeting with Celera Genmics, wh might have made data abut human genme prprietary Called Mst famus graduate student in the wrld Als develped a faster versin f BLAST called BLAT Blast Like Alignment Tl! Simulatin paper quantifies sensitivity and speed fr different ways f indexing Setup: A 100 bp regin f given similarity (81 97%) within a 500 bp query Human genme as database Tested different indexing scheme parameters (% similarity and k) Evaluated: Hw ften is this 100 lng alignment fund with this indexing scheme? (Sensitivity) And hw many randm hits d we have? (Speed) Example Interpretatin frm abve table: With k=8, will find an 81% similar match 91.5% f the time, but will find a 97% similar match 100% f the time. The cst f using k=8 is that a 500bp query will generate 2.9 millin spurius 8-mer matches between query and database. With k=15, will find 81% matches nly 30% f the time but will spend 10,000x less time. Results: Shrt wrds favr sensitivity Lng wrds favr speed METHODS TO IMPROVE SENSITIVITY / SPEED 1. Using pairs f wrds Instead f insisting n an exact wrd match f at least length k, pick a k < k and insist n matching 2 exact wrds f length k spaced within certain distance n same diagnal. Imprvement! Pair f 8-mers separated by at mst 40 nucletides at 81% alignment give 68% sensitivity. Cmpared t an inexact 16-mer, twice as fast and slightly mre sensitive And easier t implement, s used mre frequently than inexact wrds

3 2. Using inexact wrds Increase k but allw fr 1 r 2 mismatches in the k- mers Index scheme mdificatin fr building dictinary f database k-mers:! Fr every k-mer in database, create all k-mers with at least ne mismatch and recrd each ccurrence in the table f all wrd ccurrences Imprvement:! Sensitivity f 13-mer with ne mismatch is better than that f 9-mer with n mismatch. Generates nly 68k spurius alignments (vs 635k) Mre sensitive and ~10x faster! In all cases, can d much better mving t inexact k-mers Drawback: Harder t implement 3. Using wrd patterns Elegant cntributin frm ~2000! Nthing necessitates that in indexing we have all psitins be cnsecutive! Lks fr same pattern with same shift between letters thrughut Imprvement: (Example: 100-lng, 70% cnserved regin)! Decreases expected number f hits (1.07 t 0.97)! Increases prbability f 1 match within a lng cnserved regin (0.3 t 0.47)! When plt sensitivity as a functin f sequence similarity n 100 psitins (graph t right): Nncnsecutive 11-mers (red) mre sensitive than cnsecutive 10-mers while being 4x faster because generates 4x fewer randm hits Multiple Patterns! Multiple patterns can imprve sensitivity even further.! But hw many patterns is best? Becmes NP hard prblem given mdel (prbability distributins) fr hmlgy between 2 psitins t find the best K t ptimize bth speed and sensitivity K patterns takes k times lnger t scan But patterns cmpliment each ther HIDDEN MARKOV MODELS DISHONEST CASINO MODEL Dishnest Casin Mdel: a. Yu bet $1 b. Yu rll a fair die c. Dishnest casin player rlls his die. It can be either fair r laded, and yu can t tell which he is using r when he switches d. Highest number wins 2$ Analysis: Fair/Laded are hidden states f system Sequence f rlls is bservable variables Memry-less because nt necessary t cnsider previus runds

4 3 MAIN QUESTIONS WE CAN ANSWER Questin 1: EVALUATION Given a sequence f rlls by the casin player, ask Hw likely is this sequence given hw ur mdel f the casin wrks? Fr example, if we suspect that the dishnest casin player chse Fair Laded Fair and then bserve the sequence we might suspect it fits ut mdel but wuld need a way t frmulate this precisely. Questin 2: DECODING Given a sequence f rlls by the casin player, ask What prtin f the sequence was generated with fair die and what with laded die?! Nt a priri an easy questin Must knw hw fair laded die is and hw ften the player can switch between dice! If he can change die each rll, wuld be mst likely t cnsider each 6 in the sequence t be frm the laded die! But if he cannt change die at all, we shuld cnsider either the whle sequence fair r laded Questin 3: LEARNING:! Given a sequence f rlls by the casin player, ask Hw laded is the laded die? Hw fair is the fair die? Hw ften des the casin player change frm fair t laded and back?! Lwer threshhl = cunt # sizes acrss sequence! Maybe d smething fancy and lk arund middle! Precisely mathematically is nt s blear! Will find ptimal parameters (r try NP hard prblem) HIDDEN MARKOV MODELS (HMMS) Hidden Markv Mdel: Statistical Markv mdel where the state is nt directly visible but the utput is visible Simple t implement and wrks like magic Cnsist f: Alphabet: Σ={b 1,b 2,,b M} Set f States: Q = {1,, K} Transitin Prbabilities between any 2 states! a ij = Transitin prbability frm state I t state j! a i1 + + a ik = 1 fr all states i = 1 K Start prbabilities a i! a a 0K = 1 Emissin prbabilities fr each state! e i(b) = P(x i π i = k)! e i(b 1) + e i(b M) = 1 fr all states i = 1 [We will nt use End Prbabilities a i0 as defined in Durbin]

5 Memry-Less: At each time step t, the nly thing that affects future is current state π t Fr states (π t)! P(π t+1 = k whatever happened s far ) =! P(π t+1 = k π 1, π 2,, π t, x 1, x 2,, x t) =! P(π t+1 = k π t) Fr emissin (x t)! P(π t+1 = k whatever happened s far ) =! P(π t+1 = k π 1, π 2,, π t, x 1, x 2,, x t) =! P(π t+1 = k π t) On making the memry-less assumptin:! T determine if this memry-less assumptin hlds, we culd bserve and prve rules t shw that assumptin fails Ex) Shw that every time the casin player rlls 6 times, he definitely switches dice! The assumptin is a pwerful tl because great in mdeling! Cnvenient t train, dcument, and run but in general, it is true that assumptins d fail! In a way, HMMs are very simplistic ways t mdel sequences (especially prteins) We will cver pwerful, mre recent technique called Cnditinal Randm Fields PARSES Parse f a sequence: Given a sequence x = x 1 t x n, a parse f sequence x is the underlying sequence f states π = π 1,, π n Calculate precise likelihd f parse as the prduct f transmissin and emissin prbabilities P(x=x 1,, x n, π = π 1,, π n) = P(x N π n) P(π N π N-1) P(x 2 π 2) P(π 2 π 1) * P(x 1 π 1) P(π 1) = a 0π1 a π1π2 a πn-1πn * e π1 (x 1) e πn (x N) Useful alternative apprach:! Observe that in ur casin example HMM, we have a finite, ften small number f parameters. In this mdel, we have nly 4 transitin parameters (2 independent). We als have 12 emissin prbabilities (1-6 frm each f 2 states) and 2 start prbabilities = 18 parameters called θ 1 θ 18! Imagine that yu play 1 millin rlls. That underlying sequence has 2 millin arrws. P i (az a1000) * P i (e1 e )! Anther way t apprach is having these expnentiated by the number times they ccur. In the parse, create an indicatr (called the feature cunts): F(j, x, π) = # parameter θ j ccurs in (x, π)! Then: P(x, π) = Π j=1 n θ F(j, x, π ) j = exp[σ j=1 n lg(θj) * F(j, x, π)]! Culd think f this as a scre We will cme back t this and talk abut ptimizing with respect t parameters If casin player culdn t switch dice, is it mre likely all the rlls were dne with the fair die r the laded ne? Given: Sequence f rlls is x = {1, 2, 1, 5, 6, 2, 1, 5, 2, 4}! P(Fair) = ½ * P(1 Fair) P (Fair Fair) * P(2 Fair)P(Fair Fair) * * P(4 Fair) = ½ * (1/6) 10 * (0.95) 9 = * 10-9! P(Laded) = ½ * P(1 Laded) P(Laded Laded) P(4 Laded) = ½ * (1/10) 9 * (1/2) 1 * (0.95) 9 = 0.16 * 10-9! P(Fair) > P(Laded) Given: Sequence f rlls is x = {1, 6, 6, 5, 6, 2, 6, 6, 3, 6}! P(Fair) = ½ * (1/6) 10 * (0.95) 9 = * 10-9! P(Laded) = ½ * (1/10) 4 * (1/2) 6 * (0.95) 9 = 0.5 * 10-7! P(Fair) < P(Laded)

6 DECODING APPLICATION IN BIOLOGY Given a sequence f utputs, find the mst likely sequence f states that prduced it Bilgical applicatin: If we have a mdel f gene structure, may ask what is the mst likely ptential gene structure in a new gene sequence. Wrk n gene structure first fcused n finding genes cmputatinally Hwever, at cnference f gene finding, Chris Burge presented GENSCAN! Used a HMM! Wrked 10% better than ther methds! Thugh later imprved upn, this marks the intrductin f HMMs int cmputatinal bilgy. They have since becme very ppular, trending alng with ther graphical/prbabilistic mdels f mdeling data