DNA1: Last week's take-home lessons

Transcription

1 Harvard-MIT Division of Health Sciences and Technology HST.508: Genomics and Computational Biology DNA1: Last week's take-home lessons Types of mutants Mutation, drift, selection Binomial for each Association studies χ 2 statistic Linked & causative alleles Alleles, Haplotypes, genotypes Computing the first genome, the second... New technologies Random and systematic errors 1

2 DNA2: Today's story and goals Motivation and connection to DNA1 Comparing types of alignments & algorithms Dynamic programming Multi-sequence alignment Space-time-accuracy tradeoffs Finding genes -- motif profiles Hidden Markov Model for CpG Islands 2

3 DNA 2 DNA1: the last 5000 generations Intro2: Common & simple Figure ( 3

4 Applications of Dynamic Programming ato sequence analysis Shotgun sequence assembly Multiple alignments Dispersed & tandem repeats Bird song alignments Gene Expression time-warping athrough HMMs RNA gene search & structure prediction Distant protein homologies Speech recognition 4

5 Alignments & Scores Global (e.g. haplotype) Local (motif) ACCACACA ACCACACA ::xx::x: :::: ACACCATA ACACCATA Score= 5(+1) + 3(-1) = 2 Score= 4(+1) = 4 Suffix (shotgun assembly) ACCACACA ::: ACACCATA Score= 3(+1) =3 5

6 Increasingly complex (accurate) searches Exact (StringSearch) Regular expression (PrositeSearch) Substitution matrix (BlastN) Profile matrix (PSI-blast) Gaps (Gap-Blast) Dynamic Programming (NW, SM) CGCG CGN{0-9}CG = CGAACG CGCG ~= CACG CGc(g/a) ~ = CACG CGCG ~= CGAACG CGCG ~= CAGACG Hidden Markov Models (HMMER) WU ( 6

7 "Hardness" of (multi-) sequence alignment Align 2 sequences of length N allowing gaps. ACCAC-ACA ACCACACA ::x::x:x: :xxxxxx: AC-ACCATA, A-----CACCATA, etc. 2N gap positions, gap lengths of 0 to N each: A naïve algorithm might scale by O(N 2N ). For N= 3x10 9 this is rather large. Now, what about k>2 sequences? or rearrangements other than gaps? 7

8 Testing search & classification algorithms Separate Training set and Testing sets Need databases of non-redundant sets. Need evaluation criteria (programs) Sensitivity and Specificity (false negatives & positives) sensitivity (true_predicted/true) specificity (true_predicted/all_predicted) Where do training sets come from? More expensive experiments: crystallography, genetics, biochemistry 8

9 Comparisons of homology scores Pearson WR Protein Sci 1995 Jun;4(6): Comparison of methods for searching protein sequence databases. Methods Enzymol 1996;266: Effective protein sequence comparison. Algorithm: FASTA, Blastp, Blitz Substitution matrix:pam120, PAM250, BLOSUM50, BLOSUM62 Database: PIR, SWISS-PROT, GenPept 9

10 Switch to protein searches when possible M F x= u c a g 3 uac 3 aag 5'... aug uuu... Adjacent mrna codons Uxu F Y C uxc S uxa - - uxg - W Cxu L H cxc P R cxa Q cxg axu N S axc I T axa K R axg M TER C-S NH+ gxu D gxc V A G O gxa E gxg H:D/A 10

11 A Multiple Alignment of Immunoglobulins VTISCTGSSSNIGAG-NHVKWYQQLPG VTISCTGTSSNIGS--ITVNWYQQLPG LRLSCSSSGFIFSS--YAMYWVRQAPG LSLTCTVSGTSFDD--YYSTWVRQPPG PEVTCVVVDVSHEDPQVKFNWYVDG-- ATLVCLISDFYPGA--VTVAWKADS-- AALGCLVKDYFPEP--VTVSWNSG--- VSLTCLVKGFYPSD--IAVEWESNG-- 11

12 Scoring matrix based on large set of distantly related blocks: Blosum % A C D E F G H I K L M N P Q R S T V W Y A C D E F G H I K L M N P Q R S T V 22 4 W 14 Y 12

13 Scoring Functions and Alignments ascoring function: ω(match) = +1; ω(mismatch) = -1; ω(indel) = -2; } ω(other) = 0. aalignment score: sum of columns. aoptimal alignment: maximum score. substitution matrix 13

14 Calculating Alignment Scores ATGA A-TGA (1) :xx: (2) : : : ACTA ACT-A A T G A (1)Score = ω + ω + ω + ω = = 0. A C T A A T G A (2) Score =ω +ω + ω + ω + ω = = 1. A C T A if ω ( indel ) = 1, Score = =

15 DNA2: Today's story and goals Motivation and connection to DNA1 Comparing types of alignments & algorithms Dynamic programming Multi-sequence alignment Space-time-accuracy tradeoffs Finding genes -- motif profiles Hidden Markov Model for CpG Islands 15

16 What is dynamic programming? A dynamic programming algorithm solves every subsubproblem just once and then saves its answer in a table, avoiding the work of recomputing the answer every time the subsubproblem is encountered. -- Cormen et al. "Introduction to Algorithms", The MIT Press. 16

17 Recursion of Optimal Global Alignments u s : optimal global alignment score of u and v. v ATGA s + ω ; ACT A ATGA ATG A s = max s + ω ; ACTA ACT A ATG A s + ω. ACTA 17

18 Recursion of Optimal Local Alignments u s : optimal local alignment score of u and v. v u 1 u 2...ui s +ω ; 1 v v 2...vj 1 vj u 1u 2...ui 1 ui u 1 u 2...ui s +ω ; s = max v 1 v 2...vj 1 vj v 1 v 2...vj 1 s u u 2...ui 1 ui +ω ; v 1 v 2...vj 0. 18

19 Computing Row-by-Row A C T A 0 min min min min A min T min G min A min A C T A 0 min min min min A min T min G A min min A C T A 0 min min min min A min T min G min A min A C T A 0 min min min min A min T min G min A min min =

20 Traceback Optimal Global Alignment A C T A 0 min min min min A min T min G min A min A G T A : : A T C A 20

21 Local and Global Alignments A C C A C A C A A C A C C A T A A C C A C A C A 0 m m m m m m m m A m C m A m C m C m A m T m A m

22 Time and Space Complexity of Computing Alignments For two sequences u=u 1 u 2 u n and v=v 1 v 2 v m, finding the optimal alignment takes O(mn) time and O(mn) space. An O(1)-time operation: one comparison, three multiplication steps, computing an entry in the alignment table An O(1)-space memory: one byte, a data structure of two floating points, an entry in the alignment table 22

23 Time and Space Problems acomparing two one-megabase genomes. aspace: An entry: 4 bytes; Table: 4 * 10^6 * 10^6 = 4 G bytes memory. atime: 1000 MHz CPU: 1M entries/second; 10^12 entries: 1M seconds = 10 days. 23

24 Time & Space Improvement for w-band Global Alignments atwo sequences differ by at most w bps (w<<n). aw-band algorithm: O(wn) time and space. aexample: w=3. A C C A C A C A 0 m m m A m C m A m C C A T A

25 Summary Dynamic programming Statistical interpretation of alignments Computing optimal global alignment Computing optimal local alignment Time and space complexity Improvement of time and space Scoring functions 25

26 DNA2: Today's story and goals Motivation and connection to DNA1 Comparing types of alignments & algorithms Dynamic programming Multi-sequence alignment Space-time-accuracy tradeoffs Finding genes -- motif profiles Hidden Markov Model for CpG Islands 26

27 A Multiple Alignment of Immunoglobulins VTISCTGSSSNIGAG-NHVKWYQQLPG VTISCTGTSSNIGS--ITVNWYQQLPG LRLSCSSSGFIFSS--YAMYWVRQAPG LSLTCTVSGTSFDD--YYSTWVRQPPG PEVTCVVVDVSHEDPQVKFNWYVDG-- ATLVCLISDFYPGA--VTVAWKADS-- AALGCLVKDYFPEP--VTVSWNSG--- VSLTCLVKGFYPSD--IAVEWESNG-- 27

28 A multiple alignment <=> Dynamic programming on a hyperlattice See G. Fullen,

29 Multiple Alignment vs Pairwise Alignment AT AT AT A- -T AT AT A- -T Optimal Multiple Alignment Non-Optimal Pairwise Alignment 29

30 - - + δ s Computing a Node on Hyperlattice k=3 2 k 1=7 - S A +δ NV N N V - A +δ - - NS N N - S - +δ - NA NV N V S NA N N V - - V S A N N N + δ + δ - NS N NA s s s s s = max NA NS s NA - N NV s V S A NA NS - N s - N NS - N s - N NV N N N s A S - +δ NV NS NA s NV - N s - - A + δ - NS N NV s 30

31 Challenges of Optimal Multiple Alignments aspace complexity (hyperlattice size): O(n k ) for k sequences each n long. acomputing a hyperlattice node: O(2 k ). atime complexity: O(2 k n k ). afind the optimal solution is exponential in k (non-polynomial, NP-hard). 31

32 Methods and Heuristics for Optimal Multiple Alignments aoptimal: dynamic programming Pruning the hyperlattice (MSA) aheuristics: tree alignments(clustalw) star alignments sampling (Gibbs) (discussed in RNA2) local profiling with iteration (PSI-Blast,...) 32

33 ClustalW: Progressive Multiple Alignment S 1 S 2 S 3 S 4 S S S S 1 S 2 4 All Pairwise Alignments Similarity Matrix S S S4 See Higgins(1991) and Thompson(1994) Cluster Analysis Multiple Alignment Step: 1. Aligning S 1 and S 3 2. Aligning S 2 and S 4 3. Aligning (S 1,S 3 ) with (S 2,S 4 ). Dendrogram S 1 S 3 Distance S 2 S 4 33

34 Star Alignments s s s s s = = = = = s s s s ATTGCCATT ATGGCCATT ATCCAATTTT ATCTTCTT ACTGACC Similarity s2 7 s3 2 2 Pairwise Alignment Matrix s s Find the Central Sequence s 1 Multiple Alignment ATTGCCATT-- ATGGCCATT-- ATC-CAATTTT ATCTTC-TT-- ACTGACC---- AT*GCCATTTT Combine into Multiple Alignment Pairwise Alignment ATTGCCATT ATGGCCATT ATTGCCATT-- ATC-CAATTTT ATTGCCATT ATCTTC-TT ATTGCCATT ACTGACC 34

35 DNA2: Today's story and goals Motivation and connection to DNA1 Comparing types of alignments & algorithms Dynamic programming Multi-sequence alignment Space-time-accuracy tradeoffs Finding genes -- motif profiles Hidden Markov Model for CpG Islands 35

36 Accurately finding genes & their edges What is distinctive? 0. Promoters & CGs islands 1. Preferred codons 2. RNA splice signals 3. Frame across splices 4. Inter-species conservation 5. cdna for splice edges Failure to find edges? Variety & combinations Tiny proteins (& RNAs) Alternatives & weak motifs Alternatives Gene too close or distant Rare transcript 36

37 Annotated "Protein" Sizes in Yeast & Mycoplasma 14% 12% 10% 8% 6% 4% 2% Yeast SC MG % of proteins at length x 0% x = "Protein" size in #aa 37

38 Predicting small proteins (ORFs) #codons giving random ORF expectation=1 per kb (vs %GC) min max Yeast ORF length giving random expectation=1 per kb (vs %GC) 1 0% 50% 100% 38

39 Small coding regions Mutations in domain II of 23 S rrna facilitate translation of a 23 S rrna-encoded pentapeptide conferring erythromycin resistance. Dam et al J Mol Biol 259:1-6 Trp (W) leader peptide, 14 codons: MKAIFVLKGWWRTS STOP Phe (F) leader peptide, 15 codons: MKHIPFFFAFFFTFP STOP His (H) leader peptide, 16 codons: STOP MTRVQFKHHHHHHHPD Other examples in proteomics lectures 39

40 Motif Matrices a a t g c a t g g a t g t g t g a c g t Align and calculate frequencies. Note: Higher order correlations lost. 40

41 Protein starts See GeneMark 41

42 Motif Matrices a a t g c a t g g a t g t g t g = = = = 10 a c g t Align and calculate frequencies. Note: Higher order correlations lost. Score test sets: a c c c = 1 42

43 DNA2: Today's story and goals Motivation and connection to DNA1 Comparing types of alignments & algorithms Dynamic programming Multi-sequence alignment Space-time-accuracy tradeoffs Finding genes -- motif profiles Hidden Markov Model for CpG Islands 43

44 Why probabilistic models in sequence analysis? a Recognition - Is this sequence a protein start? a Discrimination - Is this protein more like a hemoglobin or a myoglobin? a Database search - What are all of sequences in SwissProt that look like a serine protease? 44

45 A Basic idea Assign a number to every possible sequence such that s P(s M) = 1 P(s M) is a probability of sequence s given a model M. 45

46 Sequence recognition Recognition question - What is the probability that the sequence s is from the start site model M? P(M s) = P(M)* P(s M) / P(s) (Bayes' theorem) P(M) and P(s) are prior probabilities and P(M s) is posterior probability. 46

47 Database search a N = null model (random bases or AAs) a Report all sequences with logp(s M) - logp(s N) > logp(n) - logp(m) a Example, say α/β hydrolase fold is rare in the database, about 10 in 10,000,000. The threshold is 20 bits. If considering 0.05 as a significant level, then the threshold is = 24.4 bits. 47

48 Plausible sources of mono, di, tri, & tetra- nucleotide biases C rare due to lack of uracil glycosylase (cytidine deamination) TT rare due to lack of UV repair enzymes. CG rare due to 5methylCG to TG transitions (cytidine deamination) AGG rare due to low abundance of the corresponding Arg-tRNA. CTAG rare in bacteria due to error-prone "repair" of CTAGG to C*CAGG. AAAA excess due to polya pseudogenes and/or polymerase slippage. AmAcid Codon Number /1000 Fraction Arg AGG Arg AGA Arg CGG Arg CGA Arg CGT Arg CGC ftp://sanger.otago.ac.nz/pub/transterm/data/codons/bct/esccol.cod 48

49 CpG Island + in a ocean of - First order Hidden Markov Model P(A+ A+) A+ MM=16, HMM= 64 transition probabilities (adjacent bp) T+ A- T- C+ P(G+ C+) > G+ P(C- A+)> C- G- 49

50 Estimate transistion probabilities -- an example Training set S= aaacagcctgacatggttccgaacagcctcgacatggcgtt Isle A+ C+ G+ T+ A C G T- A C G T Sums Ocean A C G T A+ C+ G+ T+ A C G T Sums Laplace pseudocount: Add +1 count to each observed. (p.9,108,321 P(G C) = #(CG) / N #(CN) 50 Dirichlet) (

51 Estimated transistion probabilities from 48 "known" islands Training set (+) A C G T A C G T (-) A C G T P(G C) = A #(CG) / N #(CN) C G T Sums

52 Viterbi: dynamic programming Most probable path l,k=2 states Recursion: v l (i+1) = e l (x i +1) max(v k (i)a kl ) for HMM 1/8*.27 v s i = C G C G begin A C G T A C G T a= table in slide 51 e= emit s i in state l (Durbin p.56) 52

53 DNA2: Today's story and goals Motivation and connection to DNA1 Comparing types of alignments & algorithms Dynamic programming Multi-sequence alignment Space-time-accuracy tradeoffs Finding genes -- motif profiles Hidden Markov Model for CpG Islands 53