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					U NIVERSITY

OF THE

W ITWATERSRAND , J OHANNESBURG

School of Computer Science

Searching
Scott Hazelhurst

February/March 2005

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1 Introduction
Searching and matching is fundamental to much of bioinformatics: • function follows shape; • shape follows sequence; • similar sequence means similar form Understanding similarities and differences gives insight • can be visual or numeric • can be global or local

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Matching can be exact: • are these sequences the same? • does this (short) sequence occur in this (long) sequence? Matching can be approximate: • are these sequences similar? (what does it mean to be similar?) • how similar (or different) are these sequences? (how do we measure) And what about 3D matching?

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Exact matching: • can be performed very efficiently (linear or even better) e.g consider the performance of Google. Inexact matching: • different models; • need to take into account the biology • computationally more challenging

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Will look at • Dot Plots Good tool for visualisation • Edit distance, alignment and similarity Smith-Waterman, Needleman-Wunsch • Heuristic tools Blast and others

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Why do we do approximate matching? • evolution The DNA of organisms change Individual bases change, are deleted, inserted. Can be gross changes: gene re-ordering, chromosomal splitting joining (much rarer, but very important) • lab errors, contamination, sequencing errors

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2 Dot Plots
• Find regions of similarity between two sequences • Find repeated regions within a sequence Two dimensional picture: graph sequence r1 against sequence r2 as follows: • r1 is represented on the x-axis; • r2 is represented on the y-axis; • put a dot at position i, j iff i-th nucleotide/amino-acid are the same. r1 [i] == r2 [j]

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Plotting at the nucleotide/amino-acid level is too fine! Probability high that two positions match by chance: • choose a sliding window k, threshold θ • put a dot at position (i, j) if a word of length k starting at position i matches a word of length k at position j with at least a score θ. Problem of selectivity versus sensitivity: • make k too small, θ too small matches happen by chance; • make k too big, θ too big, miss real matches

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Can also do an n versus n comparison

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3 Alignments
Want to compare two sequences: We hypothesise that they came from a common ancestor through: • mutations • insertions, deletions How similar are they? Try to align them • ACGACTTTACTTAACCGAGGGTAGTC • AGACTTTACTATGAAACGAGTGGCAGTC ACGACTTTACT-T-AACCGA-GGGTAGTC A-GACTTTACTATGAAACGAGTGGCAGTC

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Once you have an alignment you can score it: • reward matches • penalise mismatches • have costs for indels • usually have some cost model – transitions, transversions – gap creation, extension penalties • give a scoring matrix – more later Goal: find the optimal alignment – find fewest number of edit operations required to transform one into the other.

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3.1

Global alignment

Find the best overall alignment between two sequences overall • known by dynamic programming algorithm used to solve it: Needleman-Wunsch • basic idea: – trivial to compute optimal alignment of simple sequences – build optimal alignment of bigger sequences from smaller sequences. Illustrate using unit cost model: reward of 1 for a match, −1 for a mismatch or indel.

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Suppose you have two empty sequences: • optimal alignment cost is 0 Suppose you have two sequences each with exactly one character • If the two characters are the same, get the reward for the match • If the two charactes are different, pay the penalty for a mismatch. Look at the general case: • sequence x = c0 c1 . . . cn−1 • sequence y = d0 d1 . . . dm−1

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To find the score of the optimal alignment for x and y consider each of the following three cases: • Reward/Penalty for matching/mismatching x0 with y0 plus Score of alignment of x1 x2 . . . xn−1 with y1 y2 . . . ym−1 • Penalty of a deletion of x0 from x plus Score of alignment of x1 x2 . . . xn−1 with y0 y1 . . . ym−1 • Penalty of deletion of y0 from y plus Score of alignment of x0 x1 x2 . . . xn−1 with y1 . . . ym−1 Choose the best.
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ed (ATAT, CAT) = best of A −1 + ed (TAT, AT) (mismatch) B −1 + ed (TAT, CAT) (insert in ATAT, gap in CAT) C −1 + ed (ATAT, AT) (gap in ATAT, insert in CAT)

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(A) ed (TAT, AT) = best of A1 −1 + ed (AT, T) A2 −1 + AT , AT ed (AT, AT) = best of – 1 + ed (T, T) – −1 + ed (T, AT) – −1 + ed (AT, T)

A3 −1 + ed (T, AT) And so and so forth: caclulate all, subsitute back.

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Dynamic programming approach: • e[i, j] is the score of aligning xi xi+1 . . . xn−1 and yj yj+1 . . . ym−1 . • c(i, j) is reward/penalty for matching xi with yj • δ cost for a deletion • Represent e as an n + 1 × m + 1 matrix. • Fill in the matrix in stripes • e[n, m] is easy to compute: 0 • For each other element in row n or column m e[n, j] = δ + e[n, j + 1], e[j, m] = δ + e[j + 1, m]

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• For each other element:   c(i, j) + e[i + 1, j + 1]   e[i, j] = max δ + e[i + 1, j]    δ + e[i, j + 1]

      

Fill in the matrix from bottom-right to top-left • can compute the overall score; • can reconstruct what the alignment is depending on choice at each stage;

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Work out best optimal alignment of ATTT and ACTAT. • +1 for a match, 0 for mismatch or gap A A T T T C T A T

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3.2

Semi-global alignments

NW algorithm above does global alignment: compare sequence x to sequence y. Sometimes have a short sequence x and a long sequence y and want to find the best alignment of x to some substring of y. • semi-global alignment • allow free deletions at either end of the long sequence

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3.3

Local alignment

Known as Smith-Waterman algorithm. Given two sequences x and y find the optimal alignment of some sub-string of x with some substring of y. Need to balance: • Want as few errors, deletions as possible; • Want the match to be as long as possible. Approach: amend our DP algorithm above so that at each stage there is a new choice • can throw away the match found so far, and can start again

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 c(i, j) + e[i + 1, j + 1]     δ + e[i + 1, j] e[i, j] = δ + e[i, j + 1]     0

Match/mismatch Gap in y Gap in x Start again

Can always avoid errors, but then may only have short local alignments.

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4 Heuristic searches
DP approaches are seen as the most accurate. • Expensive: quadratic in the size of the
10000 x x*log(x) x*x

8000

6000

4000

2000

0 0 20 40 60 80 100

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Balance performance against accuracy. Basic idea: • inexact matching is slow, exact matching is fast • for one sequence to match another approximately: – parts of the two sequences must match exactly • Find possible places of matching, refine further • Different approaches with different trade-offs

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4.1

FASTA

User chooses a paramter ktup – substring length (commonly 2 or 1 for amino acids, 6 for nucleotides0. Build an approximate DP table as follows: • Find all hotspots (i, j) in the table where a ktup length substring of x starting at i matches a substring of y starting at j. • Find and evaluate runs of hotspots in the diagonals — Weight for length, penalise for gaps. • Return the best 10. • Score using biological information (e.g. PAM, BLOSUM): get sub-alignments • Try to combine sub-alignments to allow gaps. • Apply Smith-Waterman on the sub-matrices of interest.
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4.2

BLAST

Basic Local Alignment Search Tool (Altschul, Gish, Miller, Myers, Lipman, 1990) Generic algorithm: • Fix cut-off score C, length w (about 12) threshold θ. • To compare strings S1 and S2 , • Find all sub-strings of length w that match in both and have a score above θ This can be done very efficiently. • Extend these matches as much as possible: find locally maximal matching pairs. • Return matches above the cut-off score. Original BLAST did not deal with gaps. Latest versions of BLAST do
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Only generic versions of algorithm described – have been many extensions and improvements. BLAST is seen as faster & better than FASTA but mileage differs. NB: Both are heuristics — can miss interesting matches

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5 Substitution matrices
Important to incorporate biological information. • Some differences are more biologically important than others. Must apply mind to the problem: • What are you comparing? nucleotides or amino acids? • Do you know how similar the sequences are?

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5.1

Basic nucleotide substitution models

Will explore evolutionary models in more detail later. Here are three basic approaches.
Identity matrix A A T C G 1 0 0 0 T 0 1 0 0 C 0 0 1 0 G 0 0 0 1 A T C G BLAST matrix A 5 −4 −4 −4 T −4 5 −4 −4 C −4 −4 5 −4 G −4 −4 −4 5 A T C G Transition/transversion matrix A 1 −5 −5 −1 T −5 1 −1 −5 C −5 −1 1 −5 G −1 −5 −5 1

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5.2

Amino-acid models

PAM – point/percent accepted mutation • obtained by studies of similar sequences • different variants depending on how closely related the sequences are PAM-x: expect there to be x mutations per 100 characters. Use larger x values for more distantly related matrices. Empirical studies done to produce. Many tools allow you to select.

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Problems: • determining how far this is • have extant sequences only (particularly for amino-acids) • how to handle gaps do we allow? how do we weight opening a gap? how do we weight exending? PAM-250 often used for evolutionary studies, but you must apply your mind to the data.

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BLOSUM (blocks substiution matrix) Obtained from studies of conserved regions in distantly related matrix. Also parameterised: BLOSUM-x • Roughly: BLOSUM-x is useful for comparing sequences with x% similarity. Lower numbered matrices are used for more distantly related sequences. Good when there are blocks of conserved amino acids. BLOSUM-62 is a commonly used sequence Again: you must apply your mind to the problem, and understand the biology (e.g. even understand the 3D structure).
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What are you comparing? Nucleotide vs amino-acids Good rule of thumb: using standard tools it is better to compare amino-acids than nucleotides • but must understand the biology Why? • amino acids scoring matrices allow more meaninful alignments e.g. a change of A −→ C may have very different effects depending on where it is • do need to know ORF or try all 6 reading frames. • sometimes nucleotide comparison is the best

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Example tools: • BLASTP: source protein in a protein database; • BLASTN: source nucleotide sequence in a nucleotide sequence DB best for high-scoring matches • BLASTX: source nucleotide sequence in a protein database • TBLASTN: source protein sequence in a nucleotide database • TBLASTX: source nucleotide sequence in a nucleotide database first translating to amino acids.

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5.3

Interpretation of results

A match can happen because: • biologically relevant and interesting; • chance: e.g. small query sequences will occur by chance in any reasonable size database with absolutely no biological evidence Should decide whether statistically interesting. Topic of another lecture.

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6 Distance and similarity
Duals of each other • large distance implies low similarity and vice-versa • scores in matrices change their roles Have already seen Edit or Levenshtein distance • score of the optimal alignment NB: typically when seen as a distance, penalties are positive numbers, rewards are 0. Another simple one is Hamming distance: • number of places two sequences match after an alignment is done.
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There are many other distance functions, based upon both empirical and statistical studies. Many are word statistic based. • we can either fix a word size or consider all words. Suppose we fix a word length k • Let w be be all the possible words of length k • Let cw (y) be the number of times that the string w appears in the string y.

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Given two sequences x and y Manhattan distance: d(x, y) = abs( d2 distance: d2 (x, y) =
w∈W (cw (x) w∈W

cw (x) − cw (y))

− cw (y))2 xs

Often used in clustering applications: • sensitive to repeats

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