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An Introduction to Bioinformatics 2. Comparing biological sequences: sequence alignment (cont’d)

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An Introduction to Bioinformatics Outline Global Alignment Scoring Matrices Local Alignment Alignment with Affine Gap Penalties

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An Introduction to Bioinformatics From LCS to Alignment: Change up the Scoring The Longest Common Subsequence (LCS) problem—the simplest form of sequence alignment – allows only insertions and deletions (no mismatches). In the LCS Problem, we scored 1 for matches and 0 for indels Consider penalizing indels and mismatches with negative scores Simplest scoring schema: +1 : match premium -μ : mismatch penalty -σ : indel penalty

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An Introduction to Bioinformatics Simple Scoring When mismatches are penalized by –μ, indels are penalized by –σ, and matches are rewarded with +1, the resulting score is: #matches – μ(#mismatches) – σ (#indels)

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An Introduction to Bioinformatics The Global Alignment Problem Find the best alignment between two strings under a given scoring schema Input : Strings v and w and a scoring schema Output : Alignment of maximum score ↑→ = -б = 1 if match = -µ if mismatch s i-1,j-1 +1 if v i = w j s i,j = max s i-1,j-1 -µ if v i ≠ w j s i-1,j - σ s i,j-1 - σ : mismatch penalty σ : indel penalty

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An Introduction to Bioinformatics Scoring Matrices To generalize scoring, consider a (4+1) x(4+1) scoring matrix δ. In the case of an amino acid sequence alignment, the scoring matrix would be a (20+1)x(20+1) size. The addition of 1 is to include the score for comparison of a gap character “-”. This will simplify the algorithm as follows: s i-1,j-1 + δ (v i, w j ) s i,j = max s i-1,j + δ (v i, -) s i,j-1 + δ (-, w j )

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An Introduction to Bioinformatics Measuring Similarity Measuring the extent of similarity between two sequences Based on percent sequence identity Based on conservation

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An Introduction to Bioinformatics Percent Sequence Identity The extent to which two nucleotide or amino acid sequences are invariant A C C T G A G – A G A C G T G – G C A G 70% identical mismatch indel

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An Introduction to Bioinformatics Making a Scoring Matrix Scoring matrices are created based on biological evidence. Alignments can be thought of as two sequences that differ due to mutations. Some of these mutations have little effect on the protein’s function, therefore some penalties, δ(v i, w j ), will be less harsh than others.

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An Introduction to Bioinformatics Scoring Matrix: Example ARNK A5-2 R-7 3 N--70 K---6 Notice that although R and K are different amino acids, they have a positive score. Why? They are both positively charged amino acids will not greatly change function of protein.

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An Introduction to Bioinformatics Conservation Amino acid changes that tend to preserve the physico-chemical properties of the original residue Polar to polar aspartate glutamate Nonpolar to nonpolar alanine valine Similarly behaving residues leucine to isoleucine

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An Introduction to Bioinformatics Scoring matrices Amino acid substitution matrices PAM BLOSUM DNA substitution matrices DNA is less conserved than protein sequences Less effective to compare coding regions at nucleotide level

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An Introduction to Bioinformatics PAM Point Accepted Mutation (Dayhoff et al.) 1 PAM = PAM 1 = 1% average change of all amino acid positions After 100 PAMs of evolution, not every residue will have changed some residues may have mutated several times some residues may have returned to their original state some residues may not changed at all

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An Introduction to Bioinformatics PAM X PAM x = PAM 1 x PAM 250 = PAM 1 250 PAM 250 is a widely used scoring matrix: Ala Arg Asn Asp Cys Gln Glu Gly His Ile Leu Lys... A R N D C Q E G H I L K... Ala A 13 6 9 9 5 8 9 12 6 8 6 7... Arg R 3 17 4 3 2 5 3 2 6 3 2 9 Asn N 4 4 6 7 2 5 6 4 6 3 2 5 Asp D 5 4 8 11 1 7 10 5 6 3 2 5 Cys C 2 1 1 1 52 1 1 2 2 2 1 1 Gln Q 3 5 5 6 1 10 7 3 7 2 3 5... Trp W 0 2 0 0 0 0 0 0 1 0 1 0 Tyr Y 1 1 2 1 3 1 1 1 3 2 2 1 Val V 7 4 4 4 4 4 4 4 5 4 15 10

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An Introduction to Bioinformatics BLOSUM Blocks Substitution Matrix Scores derived from observations of the frequencies of substitutions in blocks of local alignments in related proteins Matrix name indicates evolutionary distance BLOSUM62 was created using sequences sharing no more than 62% identity

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An Introduction to Bioinformatics The Blosum50 Scoring Matrix

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An Introduction to Bioinformatics Local vs. Global Alignment The Global Alignment Problem tries to find the longest path between vertices (0,0) and (n,m) in the edit graph. The Local Alignment Problem tries to find the longest path among paths between arbitrary vertices (i,j) and (i’, j’) in the edit graph.

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An Introduction to Bioinformatics Local vs. Global Alignment The Global Alignment Problem tries to find the longest path between vertices (0,0) and (n,m) in the edit graph. The Local Alignment Problem tries to find the longest path among paths between arbitrary vertices (i,j) and (i’, j’) in the edit graph. In the edit graph with negatively-scored edges, Local Alignmet may score higher than Global Alignment

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An Introduction to Bioinformatics Local vs. Global Alignment (cont’d) Global Alignment Local Alignment—better alignment to find conserved segment --T—-CC-C-AGT—-TATGT-CAGGGGACACG—A-GCATGCAGA-GAC | || | || | | | ||| || | | | | |||| | AATTGCCGCC-GTCGT-T-TTCAG----CA-GTTATG—T-CAGAT--C tccCAGTTATGTCAGgggacacgagcatgcagagac |||||||||||| aattgccgccgtcgttttcagCAGTTATGTCAGatc

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An Introduction to Bioinformatics Local Alignment: Example Global alignment Local alignment Compute a “mini” Global Alignment to get Local

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An Introduction to Bioinformatics Local Alignments: Why? Two genes in different species may be similar over short conserved regions and dissimilar over remaining regions. Example: Homeobox genes have a short region called the homeodomain that is highly conserved between species. A global alignment would not find the homeodomain because it would try to align the ENTIRE sequence

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An Introduction to Bioinformatics The Local Alignment Problem Goal: Find the best local alignment between two strings Input : Strings v, w and scoring matrix δ Output : Alignment of substrings of v and w whose alignment score is maximum among all possible alignment of all possible substrings

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An Introduction to Bioinformatics The Problem with this Problem Long run time O(n 4 ): - In the grid of size n x n there are ~n 2 vertices (i,j) that may serve as a source. - For each such vertex computing alignments from (i,j) to (i’,j’) takes O(n 2 ) time. This can be remedied by giving free rides

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An Introduction to Bioinformatics Local Alignment: Example Global alignment Local alignment Compute a “mini” Global Alignment to get Local

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An Introduction to Bioinformatics Local Alignment: Example

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An Introduction to Bioinformatics Local Alignment: Example

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An Introduction to Bioinformatics Local Alignment: Example

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An Introduction to Bioinformatics Local Alignment: Example

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An Introduction to Bioinformatics Local Alignment: Example

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An Introduction to Bioinformatics Local Alignment: Running Time Long run time O(n 4 ): - In the grid of size n x n there are ~n 2 vertices (i,j) that may serve as a source. - For each such vertex computing alignments from (i,j) to (i’,j’) takes O(n 2 ) time. This can be remedied by giving free rides

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An Introduction to Bioinformatics Local Alignment: Free Rides Vertex (0,0) The dashed edges represent the free rides from (0,0) to every other node. Yeah, a free ride!

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An Introduction to Bioinformatics The Local Alignment Recurrence The largest value of s i,j over the whole edit graph is the score of the best local alignment. The recurrence: 0 s i,j = max s i-1,j-1 + δ (v i, w j ) s i-1,j + δ (v i, -) s i,j-1 + δ (-, w j ) Notice there is only this change from the original recurrence of a Global Alignment

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An Introduction to Bioinformatics The Local Alignment Recurrence The largest value of s i,j over the whole edit graph is the score of the best local alignment. The recurrence : 0 s i,j = max s i-1,j-1 + δ (v i, w j ) s i-1,j + δ (v i, -) s i,j-1 + δ (-, w j ) Power of ZERO: there is only this change from the original recurrence of a Global Alignment - since there is only one “free ride” edge entering into every vertex

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An Introduction to Bioinformatics The Smith-Waterman algorithm Idea: Ignore badly aligning regions Modifications to Needleman-Wunsch: Initialization:F(0, j) = F(i, 0) = 0 0 Iteration:F(i, j) = max F(i – 1, j) – d F(i, j – 1) – d F(i – 1, j – 1) + s(x i, y j )

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An Introduction to Bioinformatics Scoring Indels: Naive Approach A fixed penalty σ is given to every indel: -σ for 1 indel, -2σ for 2 consecutive indels -3σ for 3 consecutive indels, etc. Can be too severe penalty for a series of 100 consecutive indels

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An Introduction to Bioinformatics Affine Gap Penalties In nature, a series of k indels often come as a single event rather than a series of k single nucleotide events: Normal scoring would give the same score for both alignments This is more likely. This is less likely.

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An Introduction to Bioinformatics Accounting for Gaps Gaps- contiguous sequence of spaces in one of the rows Score for a gap of length x is: -(ρ + σx) where ρ >0 is the penalty for introducing a gap: gap opening penalty ρ will be large relative to σ: gap extension penalty because you do not want to add too much of a penalty for extending the gap.

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An Introduction to Bioinformatics Affine Gap Penalties Gap penalties: -ρ-σ when there is 1 indel -ρ-2σ when there are 2 indels -ρ-3σ when there are 3 indels, etc. -ρ- x · σ (-gap opening - x gap extensions) Somehow reduced penalties (as compared to naïve scoring) are given to runs of horizontal and vertical edges

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An Introduction to Bioinformatics Compromise: affine gaps ρ σ Gap cost

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An Introduction to Bioinformatics Affine Gap Penalties and Edit Graph To reflect affine gap penalties we have to add “long” horizontal and vertical edges to the edit graph. Each such edge of length x should have weight - - x *

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An Introduction to Bioinformatics Adding “Affine Penalty” Edges to the Edit Graph There are many such edges! Adding them to the graph increases the running time of the alignment algorithm by a factor of n (where n is the number of vertices) So the complexity increases from O(n 2 ) to O(n 3 )

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An Introduction to Bioinformatics Manhattan in 3 Layers ρ ρ σ σ δ δ δ δ δ

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An Introduction to Bioinformatics Affine Gap Penalties and 3 Layer Manhattan Grid The three recurrences for the scoring algorithm creates a 3-layered graph. The top level creates/extends gaps in the sequence w. The bottom level creates/extends gaps in sequence v. The middle level extends matches and mismatches.

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An Introduction to Bioinformatics Switching between 3 Layers Levels: The main level is for diagonal edges The lower level is for horizontal edges The upper level is for vertical edges A jumping penalty is assigned to moving from the main level to either the upper level or the lower level (- ) There is a gap extension penalty for each continuation on a level other than the main level (- )

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An Introduction to Bioinformatics The 3-leveled Manhattan Grid

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An Introduction to Bioinformatics Affine Gap Penalty Recurrences s i,j = s i-1,j - σ max s i-1,j –(ρ+σ) s i,j = s i,j-1 - σ max s i,j-1 –(ρ+σ) s i,j = s i-1,j-1 + δ (v i, w j ) max s i,j s i,j Continue Gap in w (deletion) Start Gap in w (deletion): from middle Continue Gap in v (insertion) Start Gap in v (insertion):from middle Match or Mismatch End deletion: from top End insertion: from bottom

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An Introduction to Bioinformatics Bounded Dynamic Programming Initialization: F(i,0), F(0,j) undefined for i, j > k Iteration: For i = 1…M For j = max(1, i – k)…min(N, i+k) F(i – 1, j – 1)+ s(x i, y j ) F(i, j) = maxF(i, j – 1) – d, if j > i – k(N) F(i – 1, j) – d, if j < i + k(N) Termination:same Easy to extend to the affine gap case x 1 ………………………… x M y 1 ………………………… y N k(N)

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An Introduction to Bioinformatics Computing Alignment Path Requires Quadratic Memory Alignment Path Space complexity for computing alignment path for sequences of length n and m is O(nm) We need to keep all backtracking references in memory to reconstruct the path (backtracking) n m

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An Introduction to Bioinformatics Computing Alignment Score with Linear Memory Alignment Score Space complexity of computing just the score itself is O(n) We only need the previous column to calculate the current column, and we can then throw away that previous column once we’re done using it 2 n n

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An Introduction to Bioinformatics Computing Alignment Score: Recycling Columns memory for column 1 is used to calculate column 3 memory for column 2 is used to calculate column 4 Only two columns of scores are saved at any given time

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An Introduction to Bioinformatics Crossing the Middle Line m/2 m n Prefix(i) Suffix(i) We want to calculate the longest path from (0,0) to (n,m) that passes through (i,m/2) where i ranges from 0 to n and represents the i-th row Define length(i) as the length of the longest path from (0,0) to (n,m) that passes through vertex (i, m/2)

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An Introduction to Bioinformatics m/2 m n Prefix(i) Suffix(i) Define (mid,m/2) as the vertex where the longest path crosses the middle column. length(mid) = optimal length = max 0 i n length(i) Crossing the Middle Line

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An Introduction to Bioinformatics Computing Prefix(i) prefix(i) is the length of the longest path from (0,0) to (i,m/2) Compute prefix(i) by dynamic programming in the left half of the matrix 0 m/2 m store prefix(i) column

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An Introduction to Bioinformatics Computing Suffix(i) suffix(i) is the length of the longest path from (i,m/2) to (n,m) suffix(i) is the length of the longest path from (n,m) to (i,m/2) with all edges reversed Compute suffix(i) by dynamic programming in the right half of the “reversed” matrix 0 m/2 m store suffix(i) column

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An Introduction to Bioinformatics Length(i) = Prefix(i) + Suffix(i) Add prefix(i) and suffix(i) to compute length(i): length(i)=prefix(i) + suffix(i) You now have a middle vertex of the maximum path (i,m/2) as maximum of length(i) middle point found 0 m/2 m 0i0i

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An Introduction to Bioinformatics Finding the Middle Point 0 m/4 m/2 3m/4 m

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An Introduction to Bioinformatics Finding the Middle Point again 0 m/4 m/2 3m/4 m

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An Introduction to Bioinformatics And Again 0 m/8 m/4 3m/8 m/2 5m/8 3m/4 7m/8 m

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An Introduction to Bioinformatics Time = Area: First Pass On first pass, the algorithm covers the entire area Area = n m

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An Introduction to Bioinformatics Time = Area: First Pass On first pass, the algorithm covers the entire area Area = n m Computing prefix(i) Computing suffix(i)

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An Introduction to Bioinformatics Time = Area: Second Pass On second pass, the algorithm covers only 1/2 of the area Area/2

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An Introduction to Bioinformatics Time = Area: Third Pass On third pass, only 1/4th is covered. Area/4

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An Introduction to Bioinformatics Geometric Reduction At Each Iteration 1 + ½ + ¼ +... + (½) k ≤ 2 Runtime: O(Area) = O(nm) first pass: 1 2 nd pass: 1/2 3 rd pass: 1/4 5 th pass: 1/16 4 th pass: 1/8

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An Introduction to Bioinformatics Indexing-based Local Aligners BLAST, WU-BLAST, BlastZ, MegaBLAST, BLAT, PatternHunter, ……

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An Introduction to Bioinformatics Some useful applications of alignments Given a newly discovered gene, Does it occur in other species? How fast does it evolve? Assume we try Smith-Waterman: The entire genomic database Our new gene 10 4 10 10 - 10 11

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An Introduction to Bioinformatics Some useful applications of alignments Given a newly sequenced organism, Which subregions align with other organisms? Potential genes Other biological characteristics Assume we try Smith-Waterman: The entire genomic database Our newly sequenced mammal 3 10 9 10 10 - 10 11

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An Introduction to Bioinformatics Indexing-based local alignment (BLAST- Basic Local Alignment Search Tool) Main idea: 1.Construct a dictionary of all the words in the query 2.Initiate a local alignment for each word match between query and DB Running Time: O(MN) However, orders of magnitude faster than Smith-Waterman query DB

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An Introduction to Bioinformatics Indexing-based local alignment Dictionary: All words of length k (~10) Alignment initiated between words of alignment score T (typically T = k) Alignment: Ungapped extensions until score below statistical threshold Output: All local alignments with score > statistical threshold …… query DB query scan

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An Introduction to Bioinformatics Indexing-based local alignment— Extensions A C G A A G T A A G G T C C A G T C C C T T C C T G G A T T G C G A Example: k = 4 The matching word GGTC initiates an alignment Extension to the left and right with no gaps until alignment falls < T below best so far Output: GTAAGGTCC GTTAGGTCC

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An Introduction to Bioinformatics Indexing-based local alignment— Extensions A C G A A G T A A G G T C C A G T C T G A T C C T G G A T T G C G A Gapped extensions Extensions with gaps in a band around anchor Output: GTAAGGTCCAGT GTTAGGTC-AGT

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An Introduction to Bioinformatics Indexing-based local alignment— Extensions A C G A A G T A A G G T C C A G T C T G A T C C T G G A T T G C G A Gapped extensions until threshold Extensions with gaps until score < T below best score so far Output: GTAAGGTCCAGT GTTAGGTC-AGT

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An Introduction to Bioinformatics Indexing-based local alignment—The index Sensitivity/speed tradeoff long words (k = 15) short words (k = 7) Sensitivity Speed Kent WJ, Genome Research 2002 Sens. Speed

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An Introduction to Bioinformatics Indexing-based local alignment—The index Methods to improve sensitivity/speed 1.Using pairs of words 2.Using inexact words 3.Patterns—non consecutive positions ……ATAACGGACGACTGATTACACTGATTCTTAC…… ……GGCACGGACCAGTGACTACTCTGATTCCCAG…… ……ATAACGGACGACTGATTACACTGATTCTTAC…… ……GGCGCCGACGAGTGATTACACAGATTGCCAG…… TTTGATTACACAGAT T G TT CAC G

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An Introduction to Bioinformatics Measured improvement Kent WJ, Genome Research 2002

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An Introduction to Bioinformatics Non-consecutive words—Patterns Patterns increase the likelihood of at least one match within a long conserved region 3 common 5 common 7 common Consecutive PositionsNon-Consecutive Positions 6 common On a 100-long 70% conserved region: Consecutive Non-consecutive Expected # hits: 1.070.97 Prob[at least one hit]:0.300.47

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An Introduction to Bioinformatics Advantage of Patterns 11 positions 10 positions

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An Introduction to Bioinformatics Multiple patterns K patterns Takes K times longer to scan Patterns can complement one another Computational problem: Given: a model (prob distribution) for homology between two regions Find: best set of K patterns that maximizes Prob(at least one match) TTTGATTACACAGAT T G TT CAC G T G T C CAG TTGATT A G Buhler et al. RECOMB 2003 Sun & Buhler RECOMB 2004 How long does it take to search the query?

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An Introduction to Bioinformatics Variants of BLAST NCBI BLAST: search the universe http://www.ncbi.nlm.nih.gov/BLAST/ http://www.ncbi.nlm.nih.gov/BLAST/ MEGABLAST: Optimized to align very similar sequences Works best when k = 4i 16 Linear gap penalty WU-BLAST: (Wash U BLAST) Very good optimizations Good set of features & command line arguments BLAT Faster, less sensitive than BLAST Good for aligning huge numbers of queries CHAOS Uses inexact k-mers, sensitive PatternHunter Uses patterns instead of k-mers BlastZ Uses patterns, good for finding genes

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An Introduction to Bioinformatics Example Query: gattacaccccgattacaccccgattaca (29 letters) [2 mins] Database: All GenBank+EMBL+DDBJ+PDB sequences (but no EST, STS, GSS, or phase 0, 1 or 2 HTGS sequences) 1,726,556 sequences; 8,074,398,388 total letters >gi|28570323|gb|AC108906.9| Oryza sativa chromosome 3 BAC OSJNBa0087C10 genomic sequence, complete sequence Length = 144487 Score = 34.2 bits (17), Expect = 4.5 Identities = 20/21 (95%) Strand = Plus / Plusgi|28570323|gb|AC108906.9| Query: 4 tacaccccgattacaccccga 24 ||||||| ||||||||||||| Sbjct: 125138 tacacccagattacaccccga 125158 Score = 34.2 bits (17), Expect = 4.5 Identities = 20/21 (95%) Strand = Plus / Plus Query: 4 tacaccccgattacaccccga 24 ||||||| ||||||||||||| Sbjct: 125104 tacacccagattacaccccga 125124 >gi|28173089|gb|AC104321.7| Oryza sativa chromosome 3 BAC OSJNBa0052F07 genomic sequence, complete sequence Length = 139823 Score = 34.2 bits (17), Expect = 4.5 Identities = 20/21 (95%) Strand = Plus / Plusgi|28173089|gb|AC104321.7| Query: 4 tacaccccgattacaccccga 24 ||||||| ||||||||||||| Sbjct: 3891 tacacccagattacaccccga 3911

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An Introduction to Bioinformatics Example Query: Human atoh enhancer, 179 letters[1.5 min] Result: 57 blast hits 1. gi|7677270|gb|AF218259.1|AF218259 Homo sapiens ATOH1 enhanc... 355 1e-95 gi|7677270|gb|AF218259.1|AF218259355 2.gi|22779500|gb|AC091158.11| Mus musculus Strain C57BL6/J ch... 264 4e-68gi|22779500|gb|AC091158.11|264 3.gi|7677269|gb|AF218258.1|AF218258 Mus musculus Atoh1 enhanc... 256 9e-66gi|7677269|gb|AF218258.1|AF218258256 4.gi|28875397|gb|AF467292.1| Gallus gallus CATH1 (CATH1) gene... 78 5e-12gi|28875397|gb|AF467292.1|78 5.gi|27550980|emb|AL807792.6| Zebrafish DNA sequence from clo... 54 7e-05gi|27550980|emb|AL807792.6|54 6.gi|22002129|gb|AC092389.4| Oryza sativa chromosome 10 BAC O... 44 0.068gi|22002129|gb|AC092389.4|44 7.gi|22094122|ref|NM_013676.1| Mus musculus suppressor of Ty... 42 0.27gi|22094122|ref|NM_013676.1|42 8.gi|13938031|gb|BC007132.1| Mus musculus, Similar to suppres... 42 0.27gi|13938031|gb|BC007132.1|42 gi|7677269|gb|AF218258.1|AF218258gi|7677269|gb|AF218258.1|AF218258 Mus musculus Atoh1 enhancer sequence Length = 1517 Score = 256 bits (129), Expect = 9e-66 Identities = 167/177 (94%), Gaps = 2/177 (1%) Strand = Plus / Plus Query: 3 tgacaatagagggtctggcagaggctcctggccgcggtgcggagcgtctggagcggagca 62 ||||||||||||| ||||||||||||||||||| |||||||||||||||||||||||||| Sbjct: 1144 tgacaatagaggggctggcagaggctcctggccccggtgcggagcgtctggagcggagca 1203 Query: 63 cgcgctgtcagctggtgagcgcactctcctttcaggcagctccccggggagctgtgcggc 122 |||||||||||||||||||||||||| ||||||||| |||||||||||||||| ||||| Sbjct: 1204 cgcgctgtcagctggtgagcgcactc-gctttcaggccgctccccggggagctgagcggc 1262 Query: 123 cacatttaacaccatcatcacccctccccggcctcctcaacctcggcctcctcctcg 179 ||||||||||||| || ||| |||||||||||||||||||| ||||||||||||||| Sbjct: 1263 cacatttaacaccgtcgtca-ccctccccggcctcctcaacatcggcctcctcctcg 1318

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Sequence Alignment Cont’d. Needleman-Wunsch with affine gaps Initialization:V(i, 0) = d + (i – 1) e V(0, j) = d + (j – 1) e Iteration: V(i, j) = max{

Sequence Alignment Cont’d. Needleman-Wunsch with affine gaps Initialization:V(i, 0) = d + (i – 1) e V(0, j) = d + (j – 1) e Iteration: V(i, j) = max{

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