Pairwise sequence alignment

Transcription

1 Dstace ad smlarty Parwse sequece algmet Lecturer: Mara Alexadersso 29 October 2003 Whe faced wth two related bologcal sequeces, we would lke to estmate ther evolutoary dstace That s, the dstace to ther commo acestor There s o uque ad uversal defto of smlarty Whe comparg regular words, for stace, ther soud, spellg ad meag ca be combed dfferet ways hear ad here: smlar soud ad spellg, but totally dfferet meags hear ad bear: eve more smlar spellg, but dfferet souds ad meags hear ad lste: dfferet soud ad spellg, but very smlar meag I a smlar way, the sequece, structure ad fucto of protes ca be combed dfferet ways Fortuately protes are a lttle more regular tha the stuato descrbed above The geeral rule s that sequece determes structure, ad structure determes fucto So whe studyg sequece smlarty we hope to dscover or valdate smlarty structure ad fucto Ths s ofte successful, but there are couter-examples to each rule Protes wth very dfferet sequece fold smlarly ad perform smlar fucto, protes wth very smlar sequece fold up dfferetly, or protes wth very smlar fuctos but stll very dfferet structure Here we oly study the sequece smlarty, ad a dstace measure dcate the degree of smlarty the sequeces compared Defto Mathematcally, a dstace D s a fucto o a set S, where for obects the followg holds for D : symmetrc: D ( v, w) = D( w, v) u, v, w S o-egatve: D( v, w) 0 wth D( v, w) = 0 oly whe v = w tragle equalty: D ( u, w) D( u, v) + D( v, w) u v w

2 Parwse algmet The game parwse algmet s to place the tow sequeces o top of each other ad sert spaces (gaps) varous umbers ad places both sequeces to obta the hghest umber of colums of detcal resdue pars Algmet algorthms strves to model the mutatoal process gvg rse to the two sequeces, startg from a commo acestor The basc mutatoal processes are substtutos: replace resdues sertos: add resdues deletos: remove resdues Isertos ad deletos are the reverse operatos of oe aother ad are usually called dels for short Dstace ad smlarty A dstace measure assg weghts to each mutato, ad the dstace betwee two sequeces s the mmal sum of weghts for a set of mutatos trasformg oe to aother A smlarty measure assgs weghts accordg to the resemblace of the two sequeces The smlarty betwee two sequeces s the maxmal sum of such weghts I parwse algmet we try to combe the smlarty ad dstace to oe sgle algmet score Smplest model: Edt dstace The edt dstace betwee two sequece s the mmal umber of edt operatos (dels ad substtutos) eeded to trasform oe sequece to aother Example Trasformato of acctga to agcta: accgta agctga agcta Edt dstace = 2 substtuto deleto Algmet scores Whe calculatg the total score of a algmet we assume depedece betwee resdues, such that the probablty of the algmet s x : x x x y : y y y Pr( algmet) = Pr( x, y) Pr( x2, y2 ) Pr( x, y 2 2 )

3 or l(pr( algmet )) = l( y)) + l(pr( x2, y2 )) + + l(pr( x, y )) Substtuto matrces We wat to separate algmets of homologous sequeces from algmets of ohomologous sequeces That s, we wat to kow whether a certa algmet fers homology or has occurred by pure chace (ad thus that the sequeces are depedet) Oe way s by usg the relatve lkelhood y M ) where M s the model for homology (e uder the assumpto that x ad y are related), ad R s the radom algmet model ( x ad y are ot related ad the smlarty occurred by chace) Radom model (R): (o homology) The two sequeces, as well as the postos wth each sequece, are assumed depedet The resdues x x ad y y (DNA bases or amo acds) occur wth probabltes q x ad q y respectvely Match model (M): (homology) The two sequeces are assumed to be depedet (or related), postos wth each sequece are stll assumed depedet The resdues x x ad y y occur together wth probablty p x y Example The algmet ADE CDF has the probabltes Radom model: Match model: I geeral, = q y M ) = A p q AC D p q E DD q p C EF q D q F ad Pr( x, y R ) = = y M ) = = q q x y p x y

4 x ad y really homologous y M ) y M ) y M ) S = l l() = 0 If S s large eough we reect the radom model ad assume x ad y to be homologous y M ) p S = l = l q x y x p q x2 y2 where s x, y ) s the score for par x, y ) ( The scores should be such that ( x q p y x y q y = = p l q q dettes ad coservatve substtutos get a postve score, eutral substtutos get score 0, p = q q x y o-coservatve substtutos get egatve scores, x y x y x x y y x = = x y p < q q s( x, y ) p > q q A substtuto matrx for amo acds s a matrx of the scores for each resdue combato If for all pars x, y ) we kew the probabltes p, q ad q we could ( calculate all 20 dstctve etres the matrx (20 sce s x, y ) = s( y, x ) ), ad the use t to check f S s large eough x y y x x y ( But we do t! Aother way s to estmate them from cofrmed algmets Problems wth ths: Hard to fd a radom sample Protes come famles, so the kow protes are hghly based Dfferet pars or protes have dfferet dstaces to ther commo acestor, ad thus ther scorg matrces would be dfferet We do t wat to estmate a scorg matrx for each possble evolutoary dstace, but rather ust oe matrx altogether If the commo acestor s very recet we would expect p ab to be small for a b, ad s ( a, b) should be strogly egatve If log tme has passed sce the sequeces dverged we expect p ab to ted to q q q b, ad so s ( a, b ) should be close to zero for all a, b PAM matrces (Dayhoff) PAM = percet accepted pot mutato The PAM matrx was costructed from 7 prote famles Sequeces of at least 85% detty were alged Ths requremet was because resulted uambguous algmet chaces of two substtutos the same posto s small y

5 The PAM matrx cossts of trasto probabltes Pr( b a) of a substtuto of amo acd a to amo acd b prote sequeces of such a evolutoary dstace that the average umber of substtutos s % of the umber of postos That s, exposg a prote to the evolutoary chage such a tme terval results o average substtutos 00 amo acds To extrapolate to loger evolutoary tmes we assume that the substtutos occur as a tme-reversble Markov process, ad wth PAM cosstg of etres Pr( b a, t = ) the a PAM2 s costructed by PR ( b a, t = 2) = Pr( c a, t = ) Pr( b c, t = ) c that s, substtutos from a to b va some arbtrary amo acd c (summg over all possble amo acds) A PAMx s costructed by terato Pr( b a, t = x) = Pr( c a, t = x ) Pr( b c, t = ) The score matrx s the obtaed by c Pr( b a, t) s( a, b t) = l Oe dsadvatage wth the PAM matrces s that whle there s oly a small estmato error PAM t grows wth dstace ad s rather large PAM250 To get aroud ths problem the BLOSUM (BLOcks SUbsttuto Matrx) matrx famly was costructed Gap pealtes Whe algg two sequeces we place them o top of each other ad sert spaces varous umbers ad places to obta the maxmal umber of dettes alged These spaces fer gaps, ad the legth of a gap s smply the umber of spaces, or del operatos, t Stadard gap pealtes for a gap of legth g are Lear pealtes: γ ( g) = gd e the gap resdues are depedet wth score q b (or pealty d ) Affe pealtes: γ ( g) = d e( g ) e opeg a gap gets score ad the, whe extedg t, every succeedg gap resdue gets score e Usually e < d Usg lear gap pealtes results a smpler model, but t s less relevat bologcally tha the affe gap Usg lear gaps s to say that havg a serto of say 0 resdues s as dffcult as havg 0 sertos of resdue each It makes more sese that 0 cosecutve gap resdues should be pealzed less tha 0 separate oes

6 Algmet algorthms Global algmet: Needlema-Wusch We wat to alg two etre sequeces, ths tme ot ecessarly of the same legth sce we re allowed to sert gaps We let x : x x y : y 2 x 2 y y, ) = score of the best algmet of subsequeces x x2 x ad y y2 y m We arrage these scores a m matrx F F s flled by usg the dyamc programmg method If we use a lear gap pealty, we start wth F ( 0,0) = 0 ad the fll the frst row wth,0) = d, ad the the frst colum wth 0, ) = The rest of the matrx s flled from top-left to bottom-rght usg, ) + s( x, y ), ) = max, ), ) (match/msmatch) (gap y) (gap x) That s, each posto (, ) the score s bult upo oe of three possble prevous postos, ) + s( x, y ), ) F (, ), ) Ths way the last cell the matrx, F (, m), cotas the score of the optmal algmet To ot ust get the score, but the actual algmet tself, we keep poters each posto the matrx to the best prevous posto That s, for posto (, ) we remember whch of the postos (, ), (, ) or (, ) that gave rse to the score F (, ) The we backtrack through F, startg (, m), to obta the best global algmet

7 Example The dyamc programmg matrx for the two sequeces ANVDR VCNDR usg lear gap pealtes d = 8 ad the BLOSUM80 scorg matrx (see below), becomes A N V D R V C N D R Thus, the score of the optmal algmet s, ad the optmal algmet tself s A-NVDR VCN-DR or -ANVDR VCN-DR e there are two algmets wth the exact same score Local algmet: Smth-Waterma Sometmes we wat to fd a coserved rego, eg a coserved prote doma, a ot alg the etre sequeces CTCCCCCCTTCAGGCTCGCCAC -T-----CTTCAGGC-----A- The local algmet algorthm s very smlar to the global, t oly eeds a slght modfcato We talze F ( 0,0) = 0 as before, but ow the frst row ad the frst colum also become 0, that s F (,0) = 0, ) = 0

8 The rest of the matrx s flled usg, ) + s( x, y ), ), ) = max, ) 0 The traceback starts the (, ) cell wth the hghest score whch s ot ecessarly (but of course stll could be) cell (, m) The we follow the poters back utl a cell wth score 0 s reached or utl we reach cell ( 0,0) For ths to work the expected score of a radom match must be egatve Otherwse log radom matches wll get a hgh score ust based o ther legth Also we have to have s ( a, b) > 0 for at least some pars ( a, b), otherwse the algorthm wll ot fd ay algmets at all Heurstc algmet algorthms Usg dyamcal programmg algorthms as Needlema-Wusch ad Smth- Waterma we are guarateed to fd the optmal algmet All possble algmets are searched through ad the oe wth the hghest score s selected Ufortuately though, these algorthms are usually computatoally complex ( speed ad memory usage) Heurstc searches are less sestve, but pretty good stll ad much faster The dea s that most good algmets have short stretches of ugapped very hgh scorg matches BLAST (Basc Local Algmet Search Tool) BLAST s oe of the most wdely used tool boformatcs, ad perhaps bology all together BLAST takes a query sequece (DNA or prote) ad searches for local algmet matches a database The procedure s as follows Fd all substrgs of legth w ( w -legth words) the database that algs wth words the query sequece, where the (ugapped) algmet has score hgher tha some threshold t These words are called hts Exted each ht to see f t s cotaed a (ugapped) algmet segmet of score hgher tha some other threshold S Usually w s about 3-5 for amo acds ad ~2 for DNA Example Let w = 2, t = 8 ad use a PAM20 scorg matrx (see below) Assume that we wat to compare the followg query sequece to a database of kow protes Query: QLNFSAGW The possble w -legth words the query are the QL, LN, NF, FS, SA, AG, GW Frst we wat to extract all protes the database that cota at least oe w -legth word that algs to oe of the w -legth words the query such that the score (usg

9 PAM20) s at least t = 8 These w -legth words are the called hts For stace, for QL, the possble hts are QL, QM ad HL: QL: QL =, QL = 9, QL = 8 QL,QM,HL QL QM HL I the same maer hts for all the other w -legth words the query are determed, ad protes cotag these extracted from the database LN: LN = 9 LN LN NF: NF = 2, NF = 8, NF = 8 NF,AF,NY NF AF NY FS: FS = FS = FS = FS = 9, FS = 8 FS,FA,FN,FG,FQ FS FA FN FG FQ SA: - AG: AG GW: GW,AW,SW,NW,DW,EW,ET,RW,VW,QW,KW,RW,CW,HW,IW,MW For the protes extracted from the database, try to exted aroud each ht to see f t s part of a larger algmet of score S 20 For stace, the database etry NLNYTRW cotas hts NL, LN ad RW Q LN FSAGW N LN YTPW BLAST reports the score ad the E-value (see below) FASTA For ucleotde searches, FASTA may be more sestve tha BLAST The procedure s as follows: A lookup table s created for all detcal matchg words of legth ktup (-2 for protes, 4-6 for DNA) betwee the query sequece ad the database The comparso of the query sequece ad the database ca be vewed as a set of dotplots, wth the query as the vertcal sequece ad the database sequeces as the horzotal Dagoals wth the largest umber of words are regstered The best regos are rescored, usg a scorg matrx, tryg to exted the match for as log as possble ad stll have a score above a gve threshold Ugapped regos are oed f the total score s S The hghest scorg caddates are realged usg a dyamcal programmg algorthm, but restrctg t to a bad aroud the caddate match

10 Sgfcace of scores Whe searchg prote databases usg search tools such as BLAST or FASTA the best ht s reported alog wth a E-value ad a P-value These are used to somehow dcate how covcg the smlarty betwee your query sequece ad the top ht the database s The statstcal theory for optmal algmet scores s very complex, ad there s a rgorous theory developed oly for local algmets wthout gaps It s ths theory that s used whe reportg the E- ad P-values database searches The classcal approach uses the extreme value dstrbuto to calculate the probablty that the best match from a search of N urelated sequeces has score S If ths probablty s very small we do t beleve that the sequeces are urelated, ad so they are lkely to be homologous HSP = hgh scorg par If two radom sequeces of legths ad m are alged, the probablty of fdg at least oe segmet par wth score S s where Pr( at least oe HSP wth score S ) exp{ Kme λs } K, λ depeds o the scorg scheme used We call ths the P-value The expected umber of segmet pars havg score We call ths the E-value E[# HSPs wth score S S the radom model s ] = Kme λs Scores are ofte ormalzed to get rd of the depedece o the scorg system ad the E-value / 2 S ' m S l K S' = λ l 2 BLAST reports the ormalzed score ad the E-value where m s the legth of the query sequece ad the legth of the etre database (the sum of all sequece legths the database)

11 BLOSUM80 PAM20 A R N D C Q E G H I L K M F P S T W Y V A R N D C Q E G H I L K M F P S T W Y V