Increasingly complex (accurate) searches. DNA2: diversity & continuity. Alignments & Scores. DNA2: Aligning ancient diversity.

Transcription

1 DNA: (Last week) 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 Multi-sequence alignment pace-time-accuracy tradeoffs Finding genes -- motif profiles 2 DNA2: diversity & continuity DNA: the last 5000 human generations Applications of Dynamic Programming To sequence analysis hotgun sequence assembly Multiple alignments Dispersed & tandem repeats Bird song alignments Gene Expression time-warping 3D-structure alignment Integrate&2: ancient genetic code 3 Through HMMs RNA gene search & structure prediction Distant protein homologies peech recognition Alignments & cores Increasingly complex (accurate) searches Global (e.g. haplotype) ACCACACA ::xx::x: ACACCA core= 5(+) + 3(-) = 2 Local (motif) ACCACACA :::: ACACCA core= (+) = Exact (tringearch) Regular expression (Prositeearch) ubstitution matrix (BlastN) Profile matrix (PI-blast) CGCG CGN{0-9}CG = CGAACG CGCG ~= CACG CGc(g/a) ~ = CACG uffix (shotgun assembly) ACCACACA ::: ACACCA core= 3(+) =3 Gaps (Gap-Blast) Dynamic Programming (NW, M) Hidden Markov Models (HMMER) CGCG ~= CGAACG CGCG ~= CAGACG WU 5 6

2 "Hardness" of (multi-) sequence alignment Align 2 sequences of length N allowing gaps. ACCAC-ACA ACCACACA ::x::x:x: :xxxxxx: AC-ACCA, A-----CACCA, etc. Testing search & classification algorithms eparate Training set and Testing sets Need databases of non-redundant sets. Need evaluation criteria (programs) ensistivity and pecificity (false negatives & positives) 2N gap positions, gap lengths of 0 to N each: A naïve algorithm might scale by O(N 2N ). For N= 3x0 9 this is rather large. Now, what about k>2 sequences? sensitivity (true_predicted/true) specificity (true_predicted/all_predicted) Where do training sets come from? More expensive experiments: crystallography, genetics, biochemistry 7 8 or rearrangements other than gaps? Comparisons of homology scores Pearson WR Protein ci 995 Jun;(6):5-60 Comparison of methods for searching protein sequence databases. Methods Enzymol 996;266: Effective protein sequence comparison. Algorithm: FA, Blastp, Blitz ubstitution matrix:pam20, PAM250, BLOUM50, BLOUM62 Database: PIR, WI-PROT, GenPept witch to protein searches when possible x= u c a g Uxu F Y C uxc uxa - - TER uxg - W Cxu H L cxc P R cxa Q cxg axu N axc I T C- axa K R axg M NH+ gxu D gxc V A G O- gxa E gxg H:D/A 9 A Multiple Alignment of Immunoglobulins VTICTGNIGAG-NHVKWYQQLPG VTICTGTNIG--ITVNWYQQLPG LRLCGFIF--YAMYWVRQAPG LLTCTVGTFDD--YYTWVRQPPG PEVTCVVVDVHEDPQVKFNWYVDG-- LVCLIDFYPGA--VTVAWKAD-- AALGCLVKDYFPEP--VTVWNG--- VLTCLVKGFYPD--IAVEWENG-- 0 coring matrix based on large set of distantly related blocks: Blosum62 coring Functions and Alignments % A C D E F G H I K L M N P Q R T V W Y A C D E F G H I K L M N P Q R T V 22 W Y coring function: ω(match) = +; substitution matrix ω(mismatch) = -; ω(indel) = -2; ω(other) = 0. Alignment score: sum of columns. Optimal alignment: maximum score. } 2

3 Calculating Alignment cores GA A-TGA () :xx: (2) : : : ACTA ACT-A A T G A () core = ω = + = 0. A C T A A T G A (2) core = ω = =. A C T A if ω ( indel ) =, core = + + = +. Multi-sequence alignment pace-time-accuracy tradeoffs Finding genes -- motif profiles 3 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. Recursion of Optimal Global Alignments u : optimal global alignment score of u and v. v GA ACT A GA G A = max ACTA ACT A G A. ACTA 5 6 Recursion of Optimal Local Alignments Computing Row-by-Row u : optimal local alignment score of u and v. v uu2... ui vv 2... vj vj uu2... ui ui uu2... ui = max vv 2... vj vj vv 2... vj uu2... ui ui vv 2... vj 0. 7 T min G min A min T min G min A min min = T min G min A min T min G min A min

4 Traceback Optimal Global Alignment Local and Global Alignments T min G min A min A G : A T T A : C A 9 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 Time and pace Complexity of Computing Alignments Time and pace Problems For two sequences u=u u 2 u n and v=v v 2 v m, finding the optimal alignment takes O(mn) time and O(mn) space. An O()-time operation: one comparison, three multiplication steps, computing an entry in the alignment table An O()-space memory: one byte, a data structure of two floating points, an entry in the alignment table Comparing two one-megabase genomes. pace: An entry: bytes; Table: * 0^6 * 0^6 = G bytes memory. Time: 000 MHz CPU: M entries/second; 0^2 entries: M seconds = 0 days Time & pace Improvement for w-band Global Alignments ummary Two sequences differ by at most w bps (w<<n). w-band algorithm: O(wn) time and space. Example: 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 tatistical interpretation of alignments Computing optimal global alignment Computing optimal local alignment Time and space complexity Improvement of time and space coring functions 23 2

5 A Multiple Alignment of Immunoglobulins Multi-sequence alignment pace-time-accuracy tradeoffs Finding genes -- motif profiles VTICTGNIGAG-NHVKWYQQLPG VTICTGTNIG--ITVNWYQQLPG LRLCGFIF--YAMYWVRQAPG LLTCTVGTFDD--YYTWVRQPPG PEVTCVVVDVHEDPQVKFNWYVDG-- LVCLIDFYPGA--VTVAWKAD-- AALGCLVKDYFPEP--VTVWNG--- VLTCLVKGFYPD--IAVEWENG A multiple alignment <=> Dynamic programming on a hyperlattice Multiple Alignment vs Pairwise Alignment From G. Fullen, 996. A- -T Optimal Multiple Alignment A- -T Non-Optimal Pairwise Alignment Computing a Node on Hyperlattice N N - N- N N N N N - NA A V NV N- N- - + δ A NV NA N NV N- NA N V N + δ N A NV - + N N - - δ A V N N - + δ - N - A NV N - V = N + NA N max s δ NĀ - N - V N + δ - NA - NV - s N + δ NĀ - NV - N + δ - N - A 29 k=3 2 k =7 Challenges of Optimal Multiple Alignments pace complexity (hyperlattice size): O(n k ) for k sequences each n long. Computing a hyperlattice node: O(2 k ). Time complexity: O(2 k n k ). Find the optimal solution is exponential in k (non-polynomial, NP-hard). 30

6 Methods and Heuristics for Optimal Multiple Alignments Optimal: dynamic programming Pruning the hyperlattice (MA) Heuristics: tree alignments(clustalw) star alignments sampling (Gibbs) (discussed in RNA2) local profiling with iteration (PI-Blast,...) ClustalW: Progressive Multiple Alignment 2 All Pairwise Alignments imilarity Matrix Cluster Analysis From Higgins(99) and Thompson(99). Multiple Alignment tep:. Aligning and 3 2. Aligning 2 and 3. Aligning (, 3 ) with ( 2, ). Dendrogram 3 Distance 2 32 tar Alignments s = TGCCT s = GGCCT 2 s = CCATTT 3 s = CTTCTT s5 = ACTGACC Pairwise Alignment imilarity Matrix s s2 s3 s s2 7 s3 2 2 s s Find the Central equence s Multiple Alignment TGCCT-- GGCCT-- C-CATTT CTTC-TT-- ACTGACC---- *GCCTTT Combine into Multiple Alignment Pairwise Alignment TGCCT GGCCT TGCCT-- C-CATTT TGCCT CTTC-TT TGCCT ACTGACC 33 Multi-sequence alignment pace-time-accuracy tradeoffs Finding genes -- motif profiles 3 Accurately finding genes & their edges % Annotated "Protein" izes Yeast & Mycoplasma What is distinctive? Failure to find edges? 0. Promoters & CGs islands Variety & combinations. Preferred codons Tiny proteins (& RNAs) 2. RNA splice signals Alternatives & weak motifs 3. Frame across splices Alternatives. Inter-species conservation Gene too close or distant 5. cdna for splice edges Rare transcript 2% 0% Largest proteins: C 8% Mycoplasma Gli Yeast Midasin MG 90 6% Human Titin 3,350 % Yeast 2% 0% % of proteins at length x 35 smallest proteins: Mycoplasma L36 37 x = "Protein" size in #aa Yeast yjr5wa 6 2:20[equence Length] AND cerevisiae 36

7 Predicting small proteins (ORFs) #codons giving random ORF expectation= per kb (vs %GC) min max mall coding regions Mutations in domain II of 23 rrna facilitate translation of a 23 rrna-encoded pentapeptide conferring erythromycin resistance. Dam et al. 996 J Mol Biol 259:-6 Trp (W) leader peptide, codons: MKAIFVLKGWWRT TOP Yeast ORF length giving random expectation= per kb (vs %GC) Phe (F) leader peptide, 5 codons: MKHIPFFFAFFFTFP TOP His (H) leader peptide, 6 codons: TOP MTRVQFKHHHHHHHPD 0% 50% 00% 37 Other examples in proteomics lectures 38 Motif Matrices Protein starts GeneMark a a t g c a t g g a t g t g t g a c g 0 t 0 0 Align and calculate frequencies. Note: Higher order correlations lost Motif Matrices a a t g +3++ = 2 c a t g +3++ = 2 g a t g +3++ = 2 t g t g +++ = 0 a c g 0 t 0 0 Align and calculate frequencies. Note: Higher order correlations lost. core test sets: Multi- sequence alignment pace- time- accuracy tradeoffs Finding genes - - motif profiles a c c c = 2

8 Why probabilistic models in sequence analysis? A Basic idea Recognition - Is this sequence a protein start? Discrimination - Is this protein more like a hemoglobin or a myoglobin? Database search - What are all of sequences in wissprot that look like a serine protease? Assign a number to every possible sequence such that s P(s M) = P(s M) is a probability of sequence s given a model M. 3 equence recognition Database search 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. N = null model (random bases or AAs) Report all sequences with logp(s M) - logp(s N) > logp(n) - logp(m) Example, say α/β hydrolase fold is rare in the database, about 0 in 0,000,000. The threshold is 20 bits. If considering 0.05 as a significant level, then the threshold is 20+. = 2. bits. 5 6 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. CpG Island + in a ocean of - First order Hidden Markov Model MM=6, HMM= 6 transition probabilities (adjacent bp) P(A+ A+) A+ T+ A- T- AmAcid Codon Number /000 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 7 C+ P(G+ C+) > G+ P(C- A+)> C- G- 8

9 Estimate transistion probabilities -- an example Estimated transistion probabilities from 8 "known" islands Training set P(G C) = 3/7 = #(CG) / N #(CN) = aaacagcctgacatggttccgaacagcctcgacggcgtt Isle A+ C+ G+ T+ A- C- G- T- A C G T ums Ocean A- C- G- T- A+ C+ G+ T+ A C G T ums Training set P(G C) = #(CG) / N #(CN) (+) A C G T A C G T (-) A C G T A C G T Laplace pseudocount: Add + count to each observed. (p.9,08,32 Dirichlet) ums Viterbi: dynamic programming for HMM Most probable path l,k=2 states Recursion: v l (i+) = e l (x i +) max(v k (i)a kl ) /8*.27 v s i = C G C G begin A C G T A C G T a= table in slide 50 e= emit s i in state l (Durbin p.56) 5 Multi- sequence alignment pace- time- accuracy tradeoffs Finding genes - - motif profiles 52