Maximum Likelihood Estimation

Transcription

1 Multple sequence algnment Parwse sequence algnment ( and ) Substtuton matrces Database searchng Maxmum Lelhood Estmaton Observaton: Data, D (HHHTHHTH) What process generated ths data? Alternatve hypothess: Ha (p 0.5) Null hypothess: Ho (p=0.5) Evolutonary tree reconstructon Gene Fndng Proten structure predcton RNA structure predcton BLAST Sequence statstcs Computatonal genomcs Ha D): posteror probablty Ha): pror probablty D Ha): lelhood of the data gven the hypothess Maxmum Lelhood Estmaton What process generated ths data? Model wth parameters: e.g., bnomal wth parameter p n n ( ) p (1 p n,, p) = ) te that the lelhood s not the probablty of the hypothess Probablty of observng HHHHTHHTH as a functon of p Probablty of observng heads n 8 flps when p=0.75 The best estmate of p s the value that maxmzes the lelhood of the data. To obtan p, solve: d( D H ) = 0 dp d( 6 ( ) p (1 p) ) = 0 dp p = 0.75 Lelhood: D H(p)) Probablty: Ha D()) te that the lelhood s not the probablty of the hypothess Probablty H(0.5) D(/8)) Probablty H(0.75) D(/8)) Lelhood D H(0.75)) Hypothess testng What s the probablty that HHHTHHTH was generated by a far con? Null hypothess: H 0 (p=0.5) Test statstc: a quantty calculated from the data that allows us to test H a aganst H 0 Y = Number of heads,, n eght con flps Y 8 n ( ) 0.5 (1 0.5 = ) 1

2 Hypothess testng Hypothess testng p-value: Probablty of observng a test statstc at least as extreme as Y under H 0. Y 6 p = 0.5) 8 8 n ( ) 0.5 (1 0.5) Y 6 H 0 ) = = Reect H 0 at the sgnfcance level. p-value: Probablty of observng a test statstc at least as extreme as Y under H 0. Hypothess testng Lelhood rato: 6 heads n 8 flps p=0.75) 6 heads n 8 flps p=0.5) D H a ) D ) Observng 6 heads n 8 con flps s 2.85 tmes as lely f p = 0.75 than f p= 0.5. H 0 H a (p=0.75), H0 (p=0.5) (0.75) 6 (0.25) 2 (0.5) 6 (0.5) 2 = 2.85 Maxmum Lelhood Estmaton for Phylogeny Reconstructon Data: Multple sequence algnment, n stes, taxa Model: sequence evoluton, e.g. Jues Cantor Parameters:α Internal labels, ľ = (l 1, l 2 l m ) Branch lengths, d = (l 1, l 2 l ) Gven a topology, T Select l,d such that MSA T, l,d ) s maxmum Maxmum Lelhood Estmaton for Phylogeny Reconstructon Assumptons: stes are ndependent: score each ste separately MSA H ) = ste H ) lneages are ndependent: compute each branch separately P ste H ) = ste d ) ( Maxmum Lelhood Estmaton for Phylogeny Reconstructon Gven a topology, T, Select l,d such that P ( MSA H ) ste T, l, d ) = s maxmum 2

3 r= {C, G, A, T} Maxmum Lelhood Estmaton for Phylogeny Reconstructon C d 1 d 2 ste T) = r=c)d 1 ) CC d 1 ) CG + r=g)d 1 ) GC d 1 ) GG + 2r=)d 1 ) C d 1 ) G Probabltes gven by Jues Cantor model G Consstent (more data, better estmaton) (2 5)! Computatonally ntensve T ( ) = 3 2 ( 3)! Consder T() trees For each nternal node, Σ labels. MLE used more often for DNA than for proten sequences Branch lengths are typcally determned numercally. If evolutonary model s a reversble Marov chan then the MLE dstance matrx converges to addtve. Neghbor Jonng s a consstent method te that parsmony s not consstent. Selectng data for tree reconstructon For reconstructng recent events, use DNA sequences For reconstructng dstant events, use amno acd sequences Select sequences that Are present n all taxa Contan a conserved regon Exhbt varaton wthn that regon e.g., Rbosomal (16xRNA) genes were used to reconstruct the tree of lfe. These genes encode products use n all organsms from bactera to mammals. Ptfalls: duplcated genes, horzontal gene transfer, mosac genes. Comparson of Phylogeny Reconstructon Methods Parsmony Selecton domnates, e.g., rbosomal genes Exhaustve search Dstance Neutral mutaton domnates, e.g., mmunoglobuln sequences Exhaustve search, greedy heurstcs. Neghbor Jonng fnds correct tree n quadratc tme f data s addtve. UPGMA fnds correct tree n quadratc tme f data s ultrametrc. Maxmum Lelhood Neutral mutaton domnates, e.g., mmunoglobuln sequences Exhaustve search Data Parsmony Character Dstance Dstance Max Lelhood Character Parwse sequence algnment ( and ) NP-complete Topology Multple sequence algnment Branch lengths Ancestral states DNA Amno acds Consstent Selectve pressure Model of mutatonal change Prob Prob Very slow Evolutonary tree reconstructon Substtuton matrces Gene Fndng Proten structure predcton RNA structure predcton Database searchng BLAST Sequence statstcs Computatonal genomcs 3

4 Multple sequence algnment Parwse sequence algnment ( and ) Local MSA Methods Dscovery: HMM s Gbb s sampler PSI BLAST Dscovery: dentfyng conserved patterns n multple sequences Representaton: Constructng probablstc models of MSA s Recognton: fndng new nstances of nown patterns (usng those models) Modelng Poston Specfc Scorng Matrces (PSSMs) HMM s But frst, some motvatng examples Applcatons of Local MSA tf polymerase Conserved patterns n bologcal sequences Example: Transcrpton factor bndng stes SP gcttt AATTTTCACTATATACTATAA cgatt ST cagat ATAAATGATATAGTGGTTATA gttaa ST atctt TTTTATTATTAAATCGTATTA gcagc EC aggct ATAAATGATATAGTGGTTATA gttag EC acctt TTTTATTATTAAATCGTATTA gtcac VC ttata ACTAATAATTATAAAATATGT gtgtc YP gctga TGAAATGATATAATCGTTATA taaga transcrpton factor bndng stes promotor ntrons Some nown bndng ste motfs agcgagcctgagcactcgaggcatctctgcacattcagcatgggatgggcctcctgtccctgtatgcgcctgatga Applcatons of Local MSA Conserved patterns n bologcal sequences Example: Proten domans recurrng sequence or structure sub-unts fold ndependent of context found n varous contexts (.e., protens that do dfferent thngs) Multdoman protens: PAX gene famly developmental regulatory genes that encode transcrpton factors early expressed durng embryogeness role n morphologcal boundares and early regonalsaton nase nase Insuln receptor-related receptor Neurotrophn receptor 4

5 Pax structure homeodoman 1 mphnsrsgh gglnqlggaf vngrplpevv rqrvdlahq gvrpcdsrq lrvshgcvs 61 lgryyetgs rpgvggs pvatpvve gdyrqnp tmfawerdr llaegvcdnd 121 tvpsvssnr rtvqqpf nlpmdscvat slspghtl pssavtppes pqsdslgsty 181 sngllgaq pgndnrmd dsdqdscrls dsqssssgp rhlrtdtfs qhhlealecp 241 ferqhypeay aspshtgeq glyplpllns alddgatlt ssntplgrnl sthqtypvva 301 dphspfaq etpelsssss tpsslsssaf ldlqqvgsgg pagasvppfn afphaasvyg 361 qftgqallsg remvgptlpg ypphptsgq gsyassaag mvagseysgn ayshtpyssy 421 seawrfpnss llsspyyyss tsrpsappts atafdhl pared box gene 8 [Mus musculus] g ref NP_ [ ] pared doman Pax doman archtecture CDART: Conserved Doman Archtecture Retreval Tool Local Multple Sequence Algnment Probablstc Framewor Example: PAX gene famly Representaton Gven a MSA for the pax doman, construct probablstc model Recognton (usng model) Gven a new sequence, does t contan the Pax doman? Fnd all sequences wth Pax domans n the data base. Proten doman databases Conserved Doman Database (CDD) Representaton: Poston specfc scorng matrces (PSSMs) Structurally corrected MSAs CDART: Conserved Doman Archtecture Retreval Tool PFAM, SMART Representaton: Hdden Marov Models (HMM s) Curated MSA s More: see Mount, Table 9.5 5