1. BLAST (Karlin Altschul) Statistics

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1 Parwse seuece algmet global ad local Multple seuece algmet Substtuto matrces Database searchg global local BLAST Seuece statstcs Evolutoary tree recostructo Gee Fdg Prote structure predcto RNA structure predcto Computatoal geomcs 1. BLAST (Karl Altschul) Statstcs Expected umber of ugapped algmets wth score S foud wth radom seueces s: E = Km e λs where K s a costat that depeds o S[, ad ca be computed from the theory for ay scorg fucto. The parameter λ s specfed by the euato 1=Σp p e λs[ Note that E s proportoal to the sze of the search space, m, ad decreases expoetally wth the score, S 1

2 2. Normalzed bt scores E values deped o K ad λ, E Kme S whch tur deped o the scorg matrx, S[,. By ormalzg the algmet scores S ' S l K l 2 we obta E values that are depedet of K, λ ad S[,. E m2 S ' Wth ormalzed bt scores, E values from database searches wth dfferet scorg matrces ca be compared. 3. Target freueces Gve seueces a ad b: Alterate hypothess (Ĥ a ): a ad b are related at PAMs dvergece. Resdues ad are alged wth target freueces, Null hypothess (Ĥ 0 ): a ad b are urelated. Resdues ad are alged wth backgroud freueces, p p Note that the PAM ad BLOSUM matrces were costructed by estmatg from data. However, ay scorg matrx (that satsfes the approprate assumptos for Karl Altschul statstcs) ca be expressed as a log odds matrx of the form S [, log 2 p p 2

3 The freueces the above euato are the characterstc target freueces of the matrx S[]. I other words, s the freuecy wth whch s alged wth Maxmal Segmet Pars (MSPs) obtaed wth S[]. Recall that a MSP s the hghest scorg par of detcal legth segmets chose from two seueces. The boudares of a MSP are chose to maxmze ts score, so a MSP may be of ay legth. 1 1 Altschul et al. J Mol Bol 215: (1990) Target freueces for substtuto matrx S[]ca be estmated emprcally as follows: Geerate radom seueces from backgroud probabltes Fd MSPs pars of radom seueces usg S[] to score algmets Cout target freueces those MSPs Target freueces ca also be estmated theoretcally usg the euato: S [, p p e Theorem (Karl ad Altschul, 1990) The best scorg matrx for dstgushg sgfcat algmets from chace algmets s the scorg matrx that gves the greatest dfferece scores betwee related algmets ad chace algmets. For seueces dverged by PAMs, the best dscrmato s obtaed by S [, log 2 p p the matrx correspodg to the from related seueces at the evolutoary dstace of terest. 3

4 Proof by cotradcto: Suppose 1. S*[ ] s the matrx that best dstgushes chace algmets from * S[, related algmets at a gve evolutoary dstace ad let p p e 2. the freueces of observg pared wth MSPs (locally maxmal ugapped algmets) obtaed wth S*[ ] are ot *. The there exsts some x ad y Σ that are alged MSPs wth a freuecy greater tha *. We ca crease the score of the MSPs by creasg the score for algg x wth y, dcatg that S * [x,y] does ot have the best dscrmatory power, leadg to a cotradcto. Implcatos BLAST wll gve reasoable accuracy as log as the emprcal target freueces,,, the algmets of terest do ot devate too far from the theoretcal target freueces: S[, p p e Reasoable accuracy ca be acheved wth two or three matrces. 4

5 The average score ( bts) per algmet posto whe usg a PAM Mmatrx to compare seueces fact separated by D PAMs (Calculated by smulato) Effcecy = Score wth PAM M Score wth PAM D = 94% effcecy Choosg your scorg matrx 1. BLAST wll gve reasoable accuracy as log as the emprcal target freueces do ot devate too far from the theoretcal target freueces Use PAM40, PAM120 & PAM240, or PAM120 & PAM The lower the relatve etropy, H, the loger the mmum algmet that s dstgushable from chace. 5

6 The Twlght Zoe % Idetty The scale dcates % detty betwee alged seueces 100 The Twlght Zoe Aroud 20%-35% detty Radom algmets beg to appear Twlght zoe Parwse searches PSSM/HMM searches 0 Structure predcto 4. Iformato cotet of substtuto matrces ad algmets Recall that the expected umber of ugapped algmets wth score S foud wth radom seueces s: E = Km e λs. Itutvely, (for a fxed database) the probablty of fdg a sgfcat match depeds o the legth of the uery, m, ad the score of the algmet, S. S, tur, depeds o the rato betwee ad p p. Aother way to thk about ths: the probablty of fdg a sgfcat match depeds the amout of formato that ca be derved from a algmet order to dscrmate betwee related seueces (Ĥ a ) ad chace algmets (Ĥ 0 ). To ask ths uesto, we ca take advatage of a formalsm from formato theory: Relatve Etropy. 6

7 Relatve Etropy also called the Kullback Lebler Dvergece Suppose you have a geeral framework wth a alterate hypothess Ĥ a ad a ull hypothess Ĥ 0. Evet occurs wth probablty p uder the ull hypothess ad probablty uder the alterate hypothess The relatve etropy, H log p s the expected dscrmato formato: the formato avalable to dscrmate favor of hypothess Ĥ a agast hypothess Ĥ 0, gve Ĥ a s true. I the BLAST cotext, the relatve etropy gves formato avalable to dstgush related algmets at PAMs (Ĥ a ) from chace algmets (Ĥ 0 ) gve a partcular substtuto matrx S [] H log, p p, S [, The relatve etropy of a substtuto matrx s gve bts per posto ad ca be calculated from S[] usg the euatos H S [, S [, ad p p e, BLOSUM PAM Seuece bts/ste bts/ste detty % % % % % % % 7

8 How log does a seuece have to be order to fd a statstcally sgfcat match at a gve evolutoary dvergece? Oe way to ask ths uesto s to ask what value of m wll gve us Ê, the largest umber of false postves that we are wllg to tolerate. We start wth the expected umber of false postves: E = Km e λs. If we choose λ = l 2 ad assume that Ês roughly eual to K, the wth some algebrac mapulato, we ca show that S log 2 m whe E= Ê. We ca terpret ths result as follows: the mmum score eed to dstgush MSPs from chace s euvalet to the umber of bts reured to specfy the startg posto of the algmet. To see ths, ote that to specfy the startg posto of ay algmet a asearch space of sze m, the worst case we could eed a umber as bt as m.. To express that umber, bary, could reure as may as log 2 m 0 s ad 1 s, or log 2 m bts. How may bts are reured to fd meagful algmets a databases of 1 bllo resdues? For a typcal amo acd seuece of legth m = 250 ad a database of sze = 10 9, log 2 m = 38 bts are reured to dstgush sgfcat MSP s from chace. Implcatos The lower the relatve etropy, H, the loger the mmum algmet that s dstgushable from chace. mmum umber of bts bts per posto mmum uery seuece = legth I a data base of legth 1 bllo, 38 bts are reured. A uery seuece must be at least 38/2.57 = 15 resdues log at 30 PAMs 38/0.70 = 54 resdues log at 160 PAMs 38/0.36 = 105 resdues log at 250 PAMs to dstgush sgfcat HSP s from chace. PAM Se Id % % % % % 8

9 Implcatos, cotued Suppose you wsh to fd seueces related to a 40 resdue log uery seuece. To obta the 38 bts you eed order to fd sgfcat matches, you wll eed to search wth the PAM 30 matrx, whch has hgher relatve etropy. If you have reaso to beleve that your uery seuece s a member of a hghly dverged gee famly, the you have a problem. We kow from the theorem that we wll obta the best sestvty f we use a matrx that correspods to the evolutoary dvergece of the matches we seek. If the famly s hghly dverged, the best sestvty wll be obtaed wth the PAM250 matrx, but PAM250 wll oly gve you 40*.36=14 bts. Bummer! I ths case, you could try to fd a loger uery seuece. Or, you could cosder searchg a smaller database, whch reures fewer bts. Perhaps fdg related seueces, say, mammals oly would be suffcet for your study. If you have reaso to beleve that your uery seuece s a member of a hghly coserved gee famly, the you re luck! The PAM 30 matrx s a sutable matrx for ths famly ad t wll provde eough formato to fd matches a database of 1bllo resdues.. 9

Lecture 9: Tolerant Testing

Lecture 9: Tolerant Testing Lecture 9: Tolerat Testg Dael Kae Scrbe: Sakeerth Rao Aprl 4, 07 Abstract I ths lecture we prove a quas lear lower boud o the umber of samples eeded to do tolerat testg for L dstace. Tolerat Testg We have