1/4/13. Outline. Markov Models. Frequency & profile model. A DNA profile (matrix) Markov chain model. Markov chains

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1 /4/3 I529: Mhne Lernng n onformts (Sprng 23 Mrkov Models Yuzhen Ye Shool of Informts nd omputng Indn Unversty, loomngton Sprng 23 Outlne Smple model (frequeny & profle revew Mrkov hn pg slnd queston Model omprson y log lkelhood rto test Mrkov hn vrnts Kth order Inhomogeneous Mrkov hns Interpolted Mrkov models (IMM ppltons Gene fndng (Genemrk & Glmmer Txonom ssgnment n metgenoms (Phymm N profle (mtrx Frequeny & profle model TT TTT TT TT TT TTT TT TG Sprse dt pseudo-ounts T G T G 2 Frequeny model: the order of nuleotdes n the trnng sequenes s gnored; Profle model: the trnng sequenes re lgned the order of nuleotdes n the trnng sequenes s fully preserved Mrkov hn model: orders re prtlly norported Mrkov hn model Sometmes we need to model dependenes etween djent postons n the sequene There re ertn regons n the genome, lke TT wthn the regultory re, upstrem gene. The pttern G s less ommon thn expeted for rndom smplng. Suh dependenes n e modeled y Mrkov hns. Mrkov hns Mrkov hn s sequene of rndom vrles wth Mrkov property,.e., gven the present stte, the future nd the pst re ndependent. fmous exmple of Mrkov hn s the drunkrd's wlk t eh step, the poston my hnge y + or wth equl prolty. Pr(5->4 = Pr(5->6 =.5, ll other trnston proltes from 5 re. these proltes re ndependent of whether the system ws prevously n step 4 or 6.

2 /4/3 st order Mrkov hn n nteger tme stohst proess, onsstng of set of m> sttes {s,,s m } nd. n m dmensonl ntl dstruton vetor ( p(s,.., p(s m 2. n m m trnston proltes mtrx M= ( s s j For exmple, for N sequene: the sttes re {,, T, G} (m=4 p( the prolty of to e the st letter G the prolty tht G follows n sequene. st order Mrkov hn X X 2 X n- X n For eh nteger n, Mrkov hn ssgns prolty to sequenes (x x n s follows: n = = = = p(( x, x2 xn p( X x p( X x X x,... = 2 n x x = 2 = px ( Mtrx representton The trnston proltes mtrx M =( st M s stohst mtrx: = t st The ntl dstruton vetor (u u m defnes the dstruton of X (p(x =s =u. grph (dreted grph representton Eh dreted edge s ssoted wth the postve trnston prolty from to. lssfton of Mrkov hn sttes Sttes of Mrkov hns re lssfed y the dgrph representton (omttng the tul prolty vlues nother exmple of reurrent nd trnsent sttes, nd re reurrent sttes: they re n strongly onneted omponents whh re snks n the grph. s not reurrent t s trnsent stte lterntve defntons: stte s s reurrent f t n e rehed from ny stte rehle from s; otherwse t s trnsent. nd re trnsent sttes, nd re reurrent sttes. One the proess moves from to, t wll never ome k. 2

3 /4/3 3-stte Mrkov model of the wether ssume the wether n e: rn or snow (stte, loudy (stte 2, or sunny (stte 3 ssume the wether of ny dy t s hrterzed y one of the three sttes = The trnston proltes etween the three sttes = { j} = Questons Gven the frst dy s sunny, wht s the prolty tht the wether for the followng 7 dys wll e sun-sun-rn-rn-sun-loudy-sun? The prolty of the wether styng n stte for d dys? Rner (989 pg slnd modelng In mmmln genomes, the dnuleotde G often trnsforms to (methyl-g whh often susequently muttes to TG. Hene G ppers less thn expeted from wht s expeted from the ndependent frequenes of nd G lone. ue to ologl resons, ths proess s sometmes suppressed n short strethes of genomes suh s n the upstrem regons of mny genes. These res re lled pg slnds. Questons out pg slnds We onsder two questons (nd some vrnts: Queston : Gven short streth of genom dt, does t ome from pg slnd? Queston 2: Gven long pee of genom dt, does t ontn pg slnds n t, where, nd how long? We solve the frst queston y modelng sequenes wth nd wthout pg slnds s Mrkov hns over the sme sttes {,,G,T} ut dfferent trnston proltes. Mrkov models for (non pg slnds The + model: Use trnston mtrx + = ( + st, + st = (the prolty tht t follows s n pg slnd postve smples The - model: Use trnston mtrx - = ( - st, - st = (the prolty tht t follows s n non pg slnd sequene negtve smples Wth these two models, to solve Queston we need to dede whether gven short sequene s more lkely to ome from the + model or from the model. Ths s done y usng the defntons of Mrkov hn, n whh the prmeters re determned y trnng dt. Mtres of the trnston proltes Model omprson + (pg slnds: p + (x x - (rows sum to X - - (non-pg slnds: X - X G T G T X G T G T Gven sequene x=(x.x L, now ompute the lkelhood rto p( x + model RTIO = = p( x model L + = L = p ( x p ( x + If RTIO>, pg slnd s more lkely. tully the log of ths rto s omputed. + x x Note: p + (x x s defned for onvenene s p + (x. p - (x x s defned for onvenene s p - (x. 3

4 /4/3 Log lkelhood rto test Tkng logrthm yelds p(x...x L + log Q = log = p(x...x L If logq >, then + s more lkely (pg slnd. If logq <, then - s more lkely (non-pg slnd. p+ (x x log p (x x toy exmple Sequene: GTGG P(GTGG + =? P(GTGG - =? Log lkelhood rto? Where do the prmeters (trnston proltes ome from? Lernng from trnng dt. Soure: olleton of sequenes from pg slnds, nd olleton of sequenes from non-pg slnds. Input: Tuples of the form (x,, x L, h, where h s + or - pg slnd: queston 2 Queston 2: Gven long pee of genom dt, does t ontn pg slnds n t, nd where? For ths, we need to dede whh prts of gven long sequene of letters s more lkely to ome from the + model, nd whh prts re more lkely to ome from the model. We wll defne Mrkov hn over 8 sttes. Output: Mxmum Lkelhood prmeters (MLE ount ll prs (X =, X - = wth lel +, nd wth lel -, sy the numers re N,+ nd N, G + T + - G - T - The prolem s tht we don t know the sequene of sttes (hdden whh re trversed, ut just the sequene of letters (oservton. Hdden Mrkov Model! Mrkov model vrtons kth order Mrkov hns (Mrkov hns wth memory Inhomogeneous Mrkov hns (vs homogeneous Mrkov hns Interpolted Mrkov hns kth order Mrkov hn ( Mrkov hn wth memory k kth Mrkov hn ssgns prolty to sequenes (x x n s follows: n p ( x... xn = p( X = x,..., X k = xk p( X = x X = x, X 2 = x 2,..., X k = x k Intl dstruton = k Trnston proltes 4

5 /4/3 Inhomogeneous Mrkov hn for gene fndng Inhomogeneous Mrkov hn: predton X X 2 X 3 X 4 X 5 X 6 X 7 X X 2 X 3 X 4 X 5 X 6 X 7 Redng frme Redng frme 2 gn, the prmeters (the trnston proltes,,, nd need to e lerned from trnng smples Redng frme 3 Gene fndng usng nhomogeneous Mrkov hn onsder sequene x x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9. where x s nuleotde let p = xx2 x2x3 x3x4 x4x5 x5x6 x6x7. p 2 = xx2 x2x3 x3x4 x4x5 x5x6 x6x7. p 3 = xx2 x2x3 x3x4 x4x5 x5x6 x6x7. then prolty tht th redng frme s the odng frme s: p P = p + p 2 + p 3 Genemrk (gene fnder for terl genomes Seletng the order of Mrkov hn For Mrkov models, wht order to hoose? Hgher order, more memory (hgher predtve vlue, ut mens more prmeters to lern The hgher the order, the less relle the prmeter estmtes. E.g., we hve N sequene of kp 2 nd order Mrkov hn, 4 3 =64 prmeters, 562 tmes on verge for eh hstory 5 th order, 4 6 =496 prmeters, 24 tmes on verge 8 th order, 4 9 =65536 prmeters,.5 tmes on verge Interpolted Mrkov models (IMMs IMMs re lled vrle-order Mrkov models IMM uses vrle numer of sttes to ompute the prolty of the next stte smple lner nterpolton P (x x n,, x = P (x+ P (x x + + np (x x n,, x generl lner nterpolton P (x x n,,x = P (x+ (xp (x x + + n(x n,,x P (x x n,,x GLIMMER Glmmer s system for fndng genes n mrol N, espelly the genomes of ter, rhe, nd vruses eukryot verson of Glmmer: GlmmerHMM Glmmer (Gene Lotor nd Interpolted Mrkov ModelER uses IMMs to dentfy the odng. Glmmer verson 3.2 s the urrent verson of the system ( glmmer/ Glmmer3 mkes severl lgorthm hnges to redue the numer of flse postve predtons nd to mprove the ury of strt-ste predtons 5

6 /4/3 IMM n GLIMMER lner omnton of 8 dfferent Mrkov hns, from st through 8th-order, weghtng eh model ordng to ts predtve power. Glmmer uses 3-perod nonhomogenous Mrkov models n ts IMMs. Sore of sequene s the produt of nterpolted proltes of ses n the sequene IMM trnng Longer ontext s lwys etter; only reson not to use t s undersmplng n trnng dt. If sequene ours frequently enough n trnng dt, use t,.e., λ = Otherwse, use frequeny nd χ 2 sgnfne to set λ. lusterng metgenom sequenes wth IMMs IMMs re used to lssfy metgenom sequenes sed on ptterns of N dstnt to lde ( spees, genus, or hgher-level phylogenet group. urng trnng, the IMM lgorthm onstruts prolty dstrutons representng oserved ptterns of nuleotdes tht hrterze eh spees. Nt Methods 29, 6(9: