Advanced Algorithms and Models for Computational Biology -- a machine learning approach. Some important dates in history (billions of years ago)

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1 Advanced Algorithm and Model for Computational Biology -- a machine learning approach Molecular Evolution: nucleotide ubtitution model Eric Xing Lecture 20, April 3, 2006 Reading: DTW book, Chap 12 DEKM book, Chap 8 Some important date in hitory (billion of year ago) Origin of the univere 15 ±4 Formation of the olar ytem 4.6 Firt elf-replicating ytem 3.5 ±0.5 Prokaryotic-eukaryotic divergence 1.8 ±0.3 Plant-animal divergence 1.0 Invertebrate-vertebrate divergence 0.5 Mammalian radiation beginning 0.1 (86 CSH Doolittle et al.) 1

The three kingdom Two important early obervation Different protein evolve at different rate, and thi eem more or le independent of the hot organim, including it generation time.

2 The three kingdom Two important early obervation Different protein evolve at different rate, and thi eem more or le independent of the hot organim, including it generation time. It i neceary to adjut the oberved percent difference between two homologou protein to get a ditance more or le linearly related to the time ince their common ancetor. ( Later we offer a rational bai for doing thi.) A triking early verion of thee obervation i next. 2

3 Rate of macromolecular evolution Corrected amino acid change per 100 reidue 220 Mammal Fibrinopeptide 1.1 MY 4 Bird/Reptile Mammal/ Reptile Reptile/Fih Carp/Lamprey 200 abcde f g h i Pliocene Miocene Oligocene Eocene Cretaceou V ertebrate/ Inect Hemoglobin 5.8 MY Evolution of the globin Cytochrome c 20.0 MY Paleocene Juraic Triaic Permian Carboniferou Devonian Silurian Ordovician Cambrian j Algonkian Huronian Million of year ince divergence 1 Separation of ancetor of plant and animal After Dickeron (1971) How doe equence variation arie? Mutation: (a) Inherent: DNA replication error are not alway corrected. (b) External: expoure to chemical and radiation. Selection: Deleteriou mutation are removed quickly. Neutral and rarely, advantageou mutation, are tolerated and tick around. Fixation: It take time for a new variant to be etablihed (having a table frequency) in a population. 3

4 Modeling DNA bae ubtitution Standard aumption (ometime weakened) Site independence. Site homogeneity. Markovian: given current bae, future ubtitution independent of pat. Temporal homogeneity: tationary Markov chain. Strictly peaking, only applicable to region undergoing little election. Some terminology In evolution, homology (here of protein), mean imilarity due to common ancetry. A common mode of protein evolution i by duplication. Depending on the relation between duplication and peciation date, we have two different type of homologou protein. Looely, Orthologue: the ame gene in different organim; common ancetry goe back to a peciation event. Paralogue: different gene in the ame organim; common ancetry goe back to a gene duplication. Lateral gene tranfer give another form of homology. 4

5 Speciation v. duplication Beta-globin (orthologue) BG-human M V H L T P E E K S A V T A L W G K V N V D E V G G E A L G R L L V V Y P W T Q BG-macaque N... T BG-bovine - - M.. A... A.... F.... K BG-platypu -... S G G N I N. L BG-chicken... W. A... Q L I. G A. C. A... A... I BG-hark -.. W S E V. L H E I. T T. K S I D K H S L. A K.. A. M F I..... T BG-human R F F E S F G D L S T P D A V M G N P K V K A H G K K V L G A F S D G L A H L D BG-macaque S N... BG-bovine A.... N D S.. N. M K... BG-platypu.... A..... S A G A... T S. G. A. K N.. BG-chicken... A... N.. S. T. I L... M. R T S. G. A V K N.. BG-hark. Y. G N L K E F T A C S Y G E. A... T.. L G V A V T.. G BG-human N L K G T F A T L S E L H C D K L H V D P E N F R L L G N V L V C V L A H H F G BG-macaque Q K BG-bovine D A K V... R N.. BG-platypu D K N R..... I V... R.. S BG-chicken. I. N.. S Q D I. I I... A.. S BG-hark D V. S Q. T D.. K K. A E E.... V. S. K.. A K C F. V E. G I L L K BG-human K E F T P P V Q A A Y Q K V V A G V A N A L A H K Y H BG-macaque..... Q BG-bovine..... V L.. D F R.. BG-platypu. D. S. E.... W.. L. S... H.. G.... BG-chicken. D... E C... W.. L. R V.. H... R... BG-hark D K. A. Q T.. I W E. Y F G V. V D. I S K E... mean ame a reference equence - mean deletion 5

6 Beta-globin: uncorrected pairwie ditance DISTANCES between protein equence (calculated over: 1 to 147) Below diagonal: oberved number of difference Above diagonal: number of difference per 100 amino acid hum mac bov pla chi ha hum mac bov pla chi ha Beta-globin: corrected pairwie ditance DISTANCES between protein equence (calculated over: 1 to 147) Below diagonal: oberved number of difference Above diagonal: number of difference per 100 amino acid Correction method: Juke-Cantor hum mac bov pla chi ha hum mac bov pla chi ha

7 Human globin (paralogue) alpha-human beta-human delta-human epilon-human gamma-human myo-human - V L S P A D K T N V K A A W G K V G A H A G E Y G A E A L E R M F L S F P T T V H. T. E E. S A. T. L N V D. V. G... G. L L V V Y. W. V H. T. E E.. A. N. L N V D A V. G... G. L L V V Y. W. V H F T A E E. A A. T S L. S. M - - N V E. A. G... G. L L V V Y. W. G H F T E E.. A T I T S L N V E D A. G. T. G. L L V V Y. W. - G.. D G E W Q L. L N V.... E. D I P G H. Q. V. I. L. K G H. E alpha-human beta-human delta-human epilon-human gamma-human myo-human K T Y F P H F - D L S H G S A Q V K G H G K K V A D A L T N A V A H V Q R F. E S. G... T P D. V M G N P K.. A..... L G. F S D G L.. L Q R F. E S. G... S P D. V M G N P K.. A..... L G. F S D G L.. L Q R F. D S. G N.. S P.. I L G N P K.. A..... L T S F G D. I K N M Q R F. D S. G N.. S A.. I M G N P K.. A..... L T S. G D. I K. L L E K. D K. K H. K S E D E M K A S E D L. K.. A T. L T.. G G I L K K K alpha-human beta-human delta-human epilon-human gamma-human myo-human D D M P N A L S A L S D L H A H K L R V D P V N F K L L S H C L L V T L A A H L. N L K G T F A T.. E.. C D.. H... E.. R.. G N V. V C V.. H. F. N L K G T F. Q.. E.. C D.. H... E.. R.. G N V. V C V.. R N F. N L K P. F A K.. E.. C D.. H... E..... G N V M V I I.. T. F.. L K G T F A Q.. E.. C D.. H... E..... G N V. V T V.. I. F G H H E A E I K P. A Q S.. T. H K I P V K Y L E F I. E. I I Q V. Q S K H alpha-human beta-human delta-human epilon-human gamma-human myo-human P A E F T P A V H A S L D K F L A S V S T V L T S K Y R G K.... P. Q. A Y Q. V V. G. A N A. A H.. H G K.... Q M Q. A Y Q. V V. G. A N A. A H.. H G K.... E. Q. A W Q. L V S A. A I A. A H.. H G K.... E. Q.. W Q. M V T A. A S A. S. R. H G D. G A D A Q G A M N. A. E L F R K D M A. N. K E L G F Q G Human globin: corrected pairwie ditance DISTANCES between protein equence (calculated over 1 to 141) Below diagonal: oberved number of difference Above diagonal: etimated number of ubtitution per 100 amino acid Correction method: Juke-Cantor alpha beta delta epil gamma myo alpha beta delta epil gamma myo

8 Correcting ditance between DNA and protein equence Why it i neceary to adjut oberved percent difference to get a ditance meaure which cale linearly with time? Thi i becaue we can have multiple and back ubtitution at a given poition along a lineage. All of the correction method (with name like Juke-Cantor, 2- parameter Kimura, etc) are jutified by imple probabilitic argument involving Markov chain whoe bai i worth matering. The ame molecular evolutionary model can be ued in coring equence alignment. Markov chain State pace = {A,C,G,T}. p(i,j) = pr(next tate S j current tate S i ) Markov aumption: p(next tate S j current tate S i & any configuration of tate before thi) = p(i,j) Only the preent tate, not previou tate, affect the prob of moving to next tate. 8

9 The multiplication rule pr(tate after next i S k current tate i S i ) = j pr(tate after next i S k, next tate i S j current tate i S i ) [addition rule] = j pr(next tate i S j current tate i S i ) x pr(tate after next i S k current tate i S i, next tate i S j ) = j p i,j x p j,k [multiplication rule] [Markov aumption] = (i,k)-element of P 2, where P=(p i,j ). More generally, pr(tate t tep from now i S k current tate i S i ) = i,k element of P t Continuou-time verion For any (, t): Let p ij (t) = pr(s j at time t+ S i at time ) denote the tationary (time-homogeneou) tranition probabilitie. Let P(t) = (p ij (t)) denote the matrix of p ij (t). Then for any (t, u): P(t+u) = P(t) P(u). It follow that P(t) = exp(qt), where Q = P (0) (the derivative of P(t) at t = 0 ). Q i called the infiniteimal matrix (tranition rate matrix) of P(t), and atifie P (t) = QP(t) = P(t)Q. Important approximation: when t i mall, P(t) I + Qt. 9

10 Interpretation of Q Roughly, q ij i the rate of tranition of i to j, while q ii = Σ j i q ij, o each row um i 0 (Why?). Now we have the hort-time approximation: p ( t + h) = q h o( h) i j ij + pi = j ( t + h) = 1 + qiih + o( h) where p ij (t+h) i the probability of tranitioning from i at time t to j at time t+h Now conider the Chapman-Kolmogorov relation: (auming we have a continuou-time Markov chain, and let p j (t) = pr(s j at time t)) i.e., h 1 p ( t + h) = = j i pr ( S at t ) pr ( S at t + h S at t ) = p ( t ) ( 1 + q h) + j i i pr ( S at t, S at t + h) jj i i p ( t ) hq i i j ( pj ( t + h) pj ( t )) = pj ( t ) qjj + pi ( t ) qij, which become: i j j j ij P = QP a h 0. Probabilitic model for DNA change Orc: Elf: Dwarf: Hobbit: Human: ACAGTGACGCCCCAAACGT ACAGTGACGCTACAAACGT CCTGTGACGTAACAAACGA CCTGTGACGTAGCAAACGA CCTGTGACGTAGCAAACGA 10

11 The Juke-Cantor model (1969) Subtitution rate: -µ µ/3 -µ A G µ/3 µ/3 µ/3 µ/3 -µ C µ/3 T -µ the implet ymmetrical model for DNA evolution Tranition probabilitie under the Juke-Cantor model IID aumption: All ite change independently All ite have the ame tochatic proce working at them Equiprobablity aumption: Make up a fictional kind of event, uch that when it happen the ite change to one of the 4 bae choen at random equiprobably Equilibrium condition: No matter how many of thee fictional event occur, provided it i not zero, the chance of ending up at a particular bae i 1/4. Solving differentially equation ytem P = QP 11

12 Tranition probabilitie under the Juke-Cantor model (cont.) Prob tranition matrix: A C G T A r(t) (t) (t) (t) P(t) = C (t) r(t) (t) (t) G (t) (t) r(t) (t) T (t) (t) (t) r(t) Where we can derive: r ( t ) = 4 t ( 1 + 3e ) µ t ( 1 e ) 1 3 µ 4 ( t ) = Homework! 4 Juke-Cantor (cont.) Fraction of ite difference difference per ite time 12

13 Kimura' K2P model (1980) Subtitution rate: --2β A G --2β β β β β --2β C T --2β which allow for different rate of tranition and tranverion. Tranition (rate ) are much more likely than tranverion (rate β). Kimura (cont.) Prob tranition matrix: r(t) (t) u(t) (t) P(t) = (t) r(t) (t) u(t) u(t) (t) r(t) (t) (t) u(t) (t) r(t) Where (t) = ¼ (1 e -4βt ) u(t) = ¼ (1 + e -4βt e -2(+β)t ) r(t) = 1 2(t) u(t) By proper choice of and one can achieve the overall rate of change and T=Tn ratio R you want (warning: terminological tangle). 13

A; C; G; T and they are et up to have an equilibrium

14 Kimura (cont.) Tranition, tranverion expected under different R: Other commonly ued model Two model that pecify the equilibrium bae frequencie (you provide the frequencie A; C; G; T and they are et up to have an equilibrium which achieve them), and alo let you control the tranition/tranverion ratio: The Haegawa-Kihino-Yano (1985) model: 14

Other commonly ued model The F84 model (Felentein) where π R = π A + π G and π Y = π C + π T (The equilibrium frequencie of purine and pyrimidine) The general time-reverible model It maintain

15 Other commonly ued model The F84 model (Felentein) where π R = π A + π G and π Y = π C + π T (The equilibrium frequencie of purine and pyrimidine) The general time-reverible model It maintain "detailed balance" o that the probability of tarting at (ay) A and ending at (ay) T in evolution i the ame a the probability of tarting at T and ending at A: A C G T A C G T (t) πc (t) βπg (t) γπ π (t) (t) δπ A G (t) επ βπ (t) (t) δπ A C (t) νπ γπ (t) (t) επ (t) νπ A C G T T T And there i of coure the general 12-parameter model which ha arbitrary rate for each of the 12 poible change (from each of the 4 nucleotide to each of the 3 other). (Neither of thee ha formula for the tranition probabilitie, but thoe can be done numerically.) 15

16 Relation between model Adjuting evolutionary ditance uing bae-ubtitution model 16

17 The Juke-Cantor model -3 Common ancetor of human and orang Q = t time unit r Human (now) P = r r r Conider e.g. the 2nd poition in a-globin2 Alu1. r = (1+3e 4t )/4, = (1 e 4t )/4. Definition of PAM Let P(t) = exp(qt). Then the A,G element of P(t) i pr(g now A then) = (1 e 4t)/4. Same for all pair of different nucleotide. Overall rate of change k = 3t. PAM = accepted point mutation When k =.01, decribed a 1 PAM Put t =.01/3 = 1/300. Then the reulting P = P(1/300) i called the PAM(1) matrix. Why ue PAM? 17

18 Evolutionary time, PAM Since equence evolve at different rate, it i convenient to recale time o that 1 PAM of evolutionary time correpond to 1% expected ubtitution. For Juke-Cantor, k = 3t i the expected number of ubtitution in [0,t], o i a ditance. (Show thi.) Set 3t = 1/100, or t = 1/300, o 1 PAM = 1/300 year. Ditance adjutment For a pair of equence, k = 3t i the deired metric, but not obervable. Intead, pr(different) i oberved. So we ue a model to convert pr(different) to k. Thi i completely analogou to the converion of θ = pr(recombination) to genetic (map) ditance (= expected number of croover) uing the Haldane map function θ = 1/2 (1 e -2d ), auming the no-interference (Poion) model. 18

19 Toward Juke-Cantor adjutment E.g., 2nd poition in a-globin Alu 1 common ancetor Aume that the common ancetor ha A, G, C or T with probability 1/4. G orang C human Then the chance of the nt differing p = 3/4 (1 e 8t ) = 3/4 (1 e 4k/3 ), ince k =2 3t 3/4 t Juke-Cantor adjutment If we uppoe all nucleotide poition behave identically and independently, and n differ out of n, we can invert thi, obtaining ) 3 4 k = log 1 n / n 4 3 Thi i the corrected or adjuted fraction of difference (under thi imple model). 100 to get PAM The analogou imple model for amino acid equence ha 100 for PAM. ) k = log 1 n / n

20 Illutration 1. Human and bovine beta-globin are aligned with no deletion at 145 out of 147 ite. They differ at 23 of thee ite. Thu n /n = 23/145, and the corrected ditance uing the Juke- Cantor formula i (natural log) 19/20 log(1 20/19 23/145) = The human and gorilla equence are aligned without gap acro all 300 bp, and differ at 14 ite. Thu n /n = 14/300, and the corrected ditance uing the Juke-Cantor formula i 3/4 log(1 4/3 14/300) = Correpondence between oberved a.a. difference and the evolutionary ditance (Dayhoff et al., 1978) Oberved Percent Difference Evolutionary Ditance in PAM

Scoring matrice for alignment How coring matrice work 134 LQQGELDLVMTSDILPRSELHYSPMFDFEVRLVLAPDHPLASKTQITPEDLASETLLI 137 LDSNSVDLVLMGVPPRNVEVEAEAFMDNPLVVIAPPDHPLAGERAISLARLAEETFVM C 9 D:D = +6 S -1 4

21 Scoring matrice for alignment How coring matrice work 134 LQQGELDLVMTSDILPRSELHYSPMFDFEVRLVLAPDHPLASKTQITPEDLASETLLI 137 LDSNSVDLVLMGVPPRNVEVEAEAFMDNPLVVIAPPDHPLAGERAISLARLAEETFVM C 9 D:D = +6 S -1 4 T P A G D:R = -2 N D E Q H R K M BLOSUM62 I L V F Y W C S T P A G N D E Q H R K M I L V F Y W From Henikoff

22 Statitical motivation for alignment core Alignment: AGCTGATCA... AACCGGTTA... Hypothee: H = homologou (indep. ite, Juke-Cantor) R = random (indep. ite, equal freq.) pr( data H ) = pr(aa H )pr(ga H )pr(cc H )... a d 3 = ( 1 p) p,where a = # agreement, d = #diagreement, p = ( 1 e 4 pr( data R ) = pr(aa R )pr(ga R )pr(cc R )... 1 a 3 = ( ) ( ) 4 4 d pr( data H ) 1 p p log{ } = a log + d log = a σ + d ( µ ). pr( data R ) 1 / 4 3/ 4 Since p<3/4, σ = log((1-p)/(1/4))>0, while -µ= log(p/(3/4))<0. Thu the alignment core = a σ + d (-µ), where the match core σ > 0, and the mimatch penalty i -µ < 0. 8t ). Large and mall evolutionary ditance Recall that p = (3/4)(1-e -8t ), σ = log((1-p)/(1/4)), -µ = log(p/(3/4)). Now note that if t 0, then p 6t, and 1-p 1, and o σ log4, while -µ log8t i large and negative. That i, we ee a big difference in the two value of σ and µ for mall ditance. Converely, if t i large, p = (3/4)(1-ε), hence p/(3/4) = 1- ε, giving µ = -log(1- ε) ε, while 1-p = (1+3ε)/4, (1-p)/(1/4) = 1+3ε, and o σ = log(1+3ε) 3ε. Thu the core are about 3 (for a match) to 1 (for a mimatch) for large ditance. Thi make ene, a mimatche will on average be about 3 time more frequent than matche. the matrix which perform bet will be the matrix that reflect the evolutionary eparation of the equence being aligned. 22

23 What about multiple alignment Phylogenetic method: a tree, with branch length, and the data at a ingle ite. t 7 t 5 t 8 t 6 t 2 t 1 t 3 t 4 ACAGTGACGCCCCAAACGT ACAGTGACGCTACAAACGT CCTGTGACGTAACAAACGA CCTGTGACGTAGCAAACGA CCTGTGACGTAGCAAACGA See next lecture for how to compute likelihood under thi hypothei Acknowledgment Terry Speed: for ome of the lide modified from hi lecture at UC Berkeley Phil Green and Joe Felentein: for ome of the lide modified from hi lecture at Univ. of Wahington 23

Molecular Evolution and Comparative Genomics

Molecular Evolution and Comparative Genomics --- the phylogenetic HMM model 10-810, CMB lecture 5---Eric Xing Some important dates in history (billions of years ago) Origin of the universe 15 ±4 Formation