Searching for Similarity in Sequences Gary Benson Departments of Computer Science and Biology Boston University

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Searching for Similarity in SequencesGary
BensonDepartments of Computer Science and
BiologyBoston University
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Topic 1 Outline

• Similarity and Alignment
• Define homology, similarity by descent and
  similarity by convergence
• Common mutations and their mathematical models
• Alignments
• Scoring Alignments
• Gap penalty functions
• Computing the best scoring alignment the
  Longest Common Subsequence (LCS) problem

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Similarity and Biomolecules

• Similarity is expected among biomolecules that
  are descended from a common ancestor. Mutations
  cause differences, but survival of the organism
  requires that mutations occur in regions that are
  less critical to function while important
  catalytic, regulatory or structural regions
  remain similar.

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Similarity and Evolution

• Evolution has duplicated and shuffled bits and
  pieces of molecules to produce new linear
  arrangements that combine function in novel ways.
  Regions of similarity often suggest an
  evolutionary tie and/or common functional
  properties between very different molecules.

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Three common similarity problems

1. Start with a query sequence with unknown
   properties and search within a database of
   millions of sequences to find those which share
   similarity with the query.
2. Start with a small set of sequences and identify
   similarities and differences among them.
3. In many sequences or very long sequences, detect
   commonly occurring patterns.

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What is Similarity?How can we measure it?
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Morphology

• Morphology is the form and structure of an
  organism.
• Should shared morphology mean similarity?

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Hands
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Aquatic Shape
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Shared morphology

• Shared morphology does not necessarily imply
  common ancestry.
• The animals with hands have all evolved from a
  common ancester with a hand.
• The ichthyosaur, shark and porpoise each evolved
  sea life adaptations independently.

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Homology

• When similarity is due to common ancestry, we
  call it homology.

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• Modern molecular biology seeks to understand
  cellular processes through the action of DNA,
  RNA, and protein molecules. This will ultimately
  lead to a biochemical understanding of
• The pathogenesis of infectious diseases like
  AIDS, hepatitus and SARS.
• The mutagenic properties of environmental toxins
  and how they lead to diseases like cancer.
• The etiology of human genetic disease.
• Strategies to prevent and treat diseases through
  drug and vaccine design, gene therapy, risk
  reduction, etc.

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How homology helps

• Given molecular sequences X and Y
• X Y AND INFO(Y) gt INFO(X)
• ( means similar)

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Are the Sequences Similar?
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Are the Sequences Similar

• How similar?
• What parts are the most similar?
• Remember, the common ancestor of the two
  sequences may have existed millions of years ago.

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• How can we tell if the two sequences are similar?
• Similarity judgements should be based on
• The types of changes or mutations that occur
  within sequences.
• Characteristics of those different types of
  mutations.
• The frequency of those mutations.

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Common mutations in DNA

• Substitution
• A C G T T G A C
• A C G A T G A C
• Deletion
• A C G T T G A C
• A C G A C
• Insertion
• A C G T T G A C
• A C G C A A G T T G A C

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Common mutations

• Duplication
• A C G T T G A C
• A C G T T G A T T G A C
• Inversion (double stranded DNA shown)
• A C G T T G A C
• T G C A A C T G
• A C T C A A C C
• A C A G T T G G

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Frequency of mutations

• Substitution gt Insertion, Deletion
• gt gt
• Duplication
• gt
• Inversion

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Evolutionary history of sequences
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Alignments

• There are many ways to align two sequences. We
  just saw one way
• T T A C G T A C A G A T T A
• T - - G G A A C A - - - T A
• Here is another
• T T A C G T A C A G A T T A
• T - - - G G A A C - - A T - A
• Which is better? Remember, we can not choose
  based on the evolutionary history, because that
  is unknown.

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Alignments and Paths through the Alignment Array
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Alignments and Paths through the Alignment Array
t a c g - c a a - - - a c g t g a a t t
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Alignments and PathsAn Alternate Alignment
t - - a c g c a - - a a c g t g - - a a t t
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Finding the Best AlignmentRanking Alignments by
Score

• Score an alignment by
• Partitioning it into columns
• Assign a weight to each column
• Sum the column weights

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Distance Scoring

• Distance scoring
• Alignment gets a non-negative score.
• Alignment of identical sequences scores zero,
• all others gt zero.
• Best alignment has smallest score.
• Typical scoring functions are
• d(a,a) 0 identity
• d(a,b) d(b,a) gt 0 a ? b substitution
• g d(a, ) gt 0 indel (gap)

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Similarity Scoring

• Similarity scoring
• Alignment scores may be positive, zero, or
  negative.
• More similar means larger positive score.
• The best alignment has largest score.
• Typical scoring functions are
• s(a,b) is gt 0 if a and b are similar in one or
  more characteristics or are observed to
  substitute frequently for each other
• 0 otherwise substitution
• g s(a, ) lt 0 indel (gap)

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Gap penalty functions

• Single character gap penalty
• g(a, ) c
• (c a constant or a value dependent on a)
• Affine (linear) gap penalty
• g(k) a ßk
• (a is a gap opening penalty, ß is a gap
  extension penalty)
• Concave gap penalty
• g(k) a ß(m(k))
• m(k) is a function like log(k) which grows more
  slowly as k increases.

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Distance Scoring

• Alignment parameters
• d(a, a) 0 d(a, b) 2, g 4
• A G C C G T A T
• A C G A - - T - T
• 0 4 0 2 4 4 0 4 0

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Similarity Scoring

• Scoring parameters
• s(a, a) 5, s(a, b) - 3, g - 8
• A G C C G T A T
• A C G A - - T - T
• 5 5 5 5
• 8 3 8 8 8

- 15
-
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Similarity scoring with affine gap

• Alignment parameters
• s(a, a) 5, s(a, b) - 3,
• g(k) a ßk, a - 5, ß - 4
• A G C C G T A T
• A C G A - - T - T
• 5 5 5 5
• 8 3 8 4 8

- 11
-
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Computing the Optimal AlignmentThe LCS Problem
as Prototype

• The Longest Common Subsequence (LCS) problem is a
  method for comparing sequences. Although the
  solution does not produce an alignment, it
  illustrates a method of dynamic programming that
  is very similar to that used by alignment
  algorithms.

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Longest Common Subsequence Problem

• Let X be a string of characters. A subsequence
  X of X is formed by discarding zero or more
  letters of X. Note that the letters in X
  maintain their same order as in X.
• Let X and Y be two strings. A common subsequence
  Z is a subsequence of both. A longest common
  subsequence (LCS) is the longest such Z.
• Examples

X a b c d e b a Y b e b d c e a c d Z b d
e a
X a b c d e b a X a b d b
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LCS Problem

• Given Two sequences X and Y.
• Find An LCS for X and Y.
• A divide and conquer solution can be developed by
  looking at what happens to the last letters in
  each sequence. That is, are they part of the LCS
  solution or not?

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Possible ways to split the problem
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LCS recursion
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Filling the dynamic programming array
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Filling the dynamic programming array
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Necessary values in adjacent cells
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Completed LCS array
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Tracing back for a solution
LCS bdea
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LCS time complexity

• There are (n 1)(m 1) cells in the LCS score
  array. Each cell is filled by examining 3 other
  cells in constant time. The time complexity to
  fill the array is O(nm).
• Tracing back for an LCS solution takes at most n
  m steps.
• The total time complexity is therefore O(nm).

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Topic 2 Outline

• Types of Alignment
• Substitution Matrices
• Global vs Local Alignment
• Recursions for Global, time complexity
• Global alignment with affine gap penalty, time
  complexity
• Similarity scoring and local alignment
• Recursion for local, time complexity
• Finding suboptimal local alignments declumping
• Substitution Matrices

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Global vs Local Alignment

• Given two strings, X and Y
• global alignment produces an alignment that
  contains all of X and all of Y.
• local alignment produces an alignment that
  contains only the best matching substrings, one
  from X and one from Y.

X
Y
X
Y
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Global vs Local Alignment

• Global alignment is useful when
• The sequences are known to be related throughout
  their length, for example, similar protein
  sequences from close species.
• Local alignment is useful when
• The sequences are believed to contain parts that
  are closely related.

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Global Alignment Problem

• Given two sequences X and Y and alignment
  scoring functions,
• Find the best scoring alignment that includes
  all of X and all of Y.
• Solution Dynamic Programming

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Global Alignment

• Analysis of global alignment is similar to the
  LCS. Alignments can end in one of three ways.
  In terms of the prefix strings x1xi and y1yj,
  we have
• 1. xi and yj are aligned with each other. (Here
  it makes no difference whether xi and yj are the
  same.)
• Gi,j Gi 1, j 1 s(xi, yj)
• X C G T
• Y C G C

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Global Alignment

• xi is deleted (aligned against a dash).
• Gi, j Gi 1, j g
• X C A T
• Y C A -
• yj is deleted (aligned against a dash).
• Gi, j Gi, j 1 g
• X C A
• Y C A A

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Global alignment recursion(similarity scoring)
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Global alignment example
match 2, mismatch - 3, gap - 4
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Global Alignment Example
match 2, mismatch - 3, gap - 4
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Global Alignment Example
Tracing back for an alignment
C G T A G C C T A G A
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Global alignment time complexity

• As with the LCS problem, there are (n 1) (m
  1) cells in the dynamic programming array. Each
  is filled by examining 3 other cells in constant
  time. The time complexity to fill the array is
  O(nm).
• Tracing back for a global alignment takes at most
  n m steps.
• The total time complexity is therefore O(nm).

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Global alignment and affine gap penalty

• Recall the affine gap penalty function
• g(k) a ßk
• When xi or yj is deleted, we have to consider
  that it could be the last of a string of
  characters that is deleted as one unit. And the
  size of that unit will affect the deletion cost.

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Time Complexity (naïve)with Affine Gap Cost

• For each (i, j) in the alignment matrix, there
  are O(n m) posible deletion costs that must be
  considered in order to choose the optimal cost.
  Without any improvements, the time complexity
  grows to O(nm(n m)) or cubic O(n3) time.

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Refining the Affine Gap Computation

• The regularity of deletion costs helps reduce the
  time complexity. Observe the two tables.

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Auxillary Functions for Affine Gap
Ei,j is max of all possibilities
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Auxillary Functions for Affine Gap
Fi,j is the maximum of all possibilities
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Global Alignment with Affine Gap
recursion(similarity scoring)
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Time Complexity with affine gap cost

• A total of (n 1)(m 1) cells must be computed.
  For each cell, E, F, and G values must be
  computed. E and F both require looking up 2
  values. G requires looking up 3 values. Time to
  compute scores is O(nm).
• Tracing back can be done in O(n m) if the E and
  F values are retained. This triples the memory
  space required for scoring arrays (E, F, and G).
• Total time O(nm).

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Local Alignment Problem

• Given two sequences X and Y and alignment
  scoring functions,
• Find the best scoring alignment over all
  substring pairs, one from X and one from Y.
• Solution Dynamic Programming

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Local Alignment Looks Harder than Global
Alignment

• Where global alignment asks for the solution to
  one problem, the best alignment of X1m versus
  Y1n, local alignment asks for the best
  alignment out of O(n4) subproblems, any substring
  in X versus any substring in Y
• Xhi versus Ykj
• for 1 h i m, 1 k j n
• Instead, we solve O(n2) subproblems, the best
  alignment of any substring ending at xi versus
  any substring ending at yj.

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Local Alignment and Similarity Scoring

• Local alignment uses similarity scoring for the
  following reason. When an alignment score is
  negative, the alignment is worse than no
  alignment at all. For an (i, j) pair, it often
  happens that the best alignment of every
  substring ending at xi with every substring
  ending at yj has a negative score.
• Similarity scoring detects these bad alignments
  and local alignment discards them. If every
  alignment score for an (i, j) cell is negative,
  then the score is reset to zero.

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Local Alignment Recursion
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Local Alignment Time Complexity

• Proportional to the product of the sequence
  lengths O(nm).

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Finding Suboptimal Alignments

• When computing local alignment, we may want to
  know optimal and suboptimal alignments. This
  can be important in the case where the sequences
  contain several parts that are similar.

X
Y
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High Alignment Scores may not be Independent
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Declump scores by prohibiting match and
substitution pairings from realignment
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Source Michael S. Waterman, 1994
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Source Michael S. Waterman, 1994
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Substitution Matrices

• Used for protein alignments
• Substitution rates for amino acid pairs are
  determined from known similar sequences
• Matrices contain log-odds scores
• First matrices designed by Margaret Dayhof are
  called PAM matrices (Point Accepted Mutation)
• Current matrices designed by Henikoff and
  Henikoff are called BLOSUM matrices (Blocks
  Substitution Matrices)

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Odds Ratios

• Odds is a ratio of probabilities for two events
  which are mutually exclusive.
• In horse racing, for example, odds is the ratio
  of the betting that a horse will lose to the
  betting that the horse will win. So, for a horse
  with odds of 20 to 1, the betting is 20 times
  higher that the horse will lose than it will win,
  while for a horse with odds of 3 to 2, the
  betting is only 1.5 times higher that the horse
  will lose than win.

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Alignment and Odds

• A substitution score can be interpreted as an
  odds ratio. For an individual pair of aligned
  amino acids, the events are
• The pair are aligned because they are
  evolutionarily related
• The pair are aligned merely by chance.
• For each pair, the relevant question is
• What are the odds that amino acid i would be
  substituted for amino acid j if they were
  evolutionarily related?
• If the odds are good, then the pair supports the
  alignment.
• If the odds are bad, then the pair reduces
  confidence in the alignment.

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Log Odds

• The odds for the entire alignment
• Aligned because the sequences are
  evolutionarily related or aligned by chance
  alone
• can be obtained by multiplying the odds for each
  aligned pair of amino acids.
• Multiplication is expensive computationally, so
  logarithms are used because they can be added.

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Log-Odds Substitution Matrices

• The BLOSUM and PAM substitution matrices contain
  log-odds values. The ratios have the basic form
• Observed probability of pairing amino acids i
  and j in related sequences
• Expected probability of pairing at random

Oij Eij

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BLOSUM Data

• DATA
• Ungapped multiple alignments (blocks) taken from
  504 families of known related protein sequences
  in the PROSITE database. In the original paper
  (1992), this produced 2100 blocks.

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BLOSUM Observed Frequencies

• A typical block
• R T S C L H
• K T S A N P
• K W D C L P
• K
• R
• E

First column pairs RK 6 RR 1 RE 2 KK
3 KE 3 Repeat and accumulate for every column.
Fij Pair (i, j) counts Oi,j Fij / total
number of pairs observed pair frequencies
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BLOSUM Background Frequencies

• Pi probability of amino acid i occurring in an
  amino acid pair
• Pi Oii Sj ? i Oij / 2
• Eij expected probability of random pairs
• Eij

Pi Pj if i j Pi Pj Pj Pi
if i ? j
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BLOSUM Log-Odds Values

• Sij Log-odds ratio for aligning amino acids i
  and j.
• Sij 2 log 2 (Oij / Eij)
• If observed frequencies are
• as expected, Sij 0
• greater than expected, Sij gt 0 (positive)
• less than expected, Sij lt 0 (negative)

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Related Sequences Must Be Clustered

• Overrepresentation of closely related sequences
  can bias the matrices.
• K
• K 4 sequences more than
• K 80 identical
• K
• R
• E
• The overrepresentation of the closely related
  sequences increases the observed K to K pairing,
  falsely increasing the log-odds score for that
  pair.

Other related sequences
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Clustering Sequences

• Closely related sequences are clustered and the
  letters in any cluster are given fractional
  values. In the cluster below, amino acids K in
  the first column counts as ¼ each rather than 1.
  R and T in the fourth column count as ½ each.
• K R
• K R 4 sequences more than
• K T 80 identical clustered
• K T
• R L
• E K

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The BLOSUM Matrix Family

• Blosum 80 clusters sequences if they are 80
  identical (clustering is not transitive).
• Blosum 62 clusters sequences if they are 62
  identical.
• Lower numbers yield matrices that give higher
  scores to alignments of distantly related
  sequences.

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Topic 3 Outline

• Database Searching Algorithms
• Sensitivity vs Selectivity vs Running Time
• Dot Plots
• Blast
• High Scoring Segment Pairs (HSP)
• Target Words
• Extending a Hit
• Statistics of HSP scores
• Gapped Blast modifications

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Alternatives to Alignment

• Alignment is fine when the sequences are
  relatively short, but is unusable for longer
  sequences such as are encountered in
• database searches
• comparison of genomes
• large repetition identification
• because it takes too long. For these, we need
  alternate methods.

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Dot Plots

• Two dimensional array like an alignment array.
  Put a dot in each cell where the sequence
  characters match. Long diagonal runs of dots
  indicate sequence similarity.
• Match rule for proteins might be positive
  substitution value in BLOSUM matrix.

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Extended Dot Plots (Nature, 19 June 2003, Vol
423, p 831)
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Tandem repeat
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Database Search AlgorithmsSensitivity,
Selectivity, Running Time

• Sensitivity the ability to detect weak
  similarities between sequences (often due to long
  evolutionary separation). Increasing sensitivity
  reduces false negatives, those database sequences
  similar to the query, but rejected.
• Selectivity the ability to screen out
  similarities due to chance. Increasing
  selectivity reduces false positives, those
  sequences recognized as similar when they are not.

Running time
1 Running time
Sensitivity
Selectivity
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BLAST

• The BLAST program is designed for fast sequence
  to database search. The basic idea is that a
  high scoring alignment between the query and a
  database sequence will almost always contain a
  core part that is well conserved. This core is
  called a High-scoring Segment Pair (HSP). The
  statistical theory behind BLAST deals with the
  probability of finding HSPs.

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High-scoring Segment Pair (HSP)

• An HSP consists of two equal length substrings,
  one from the query and one from the database
  sequence. When aligned without gaps, the score,
  S, is above a specified threshold and is locally
  maximal, meaning that extending the substrings on
  either end to lengthen the alignment produces a
  smaller score.
• Suppose we have the following two sequences

X LKFSFALCCTIG Y ADQHLSRPTWAFYC S F A L
C C T W A F Y C 1 1 4 0 -2 9 13
One of many segment pairs is (BLOSUM scoring)
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High Scoring Segment Pairs
X LKFSFALCCTIG Y ADQHLSRPTWAFYC S F
A L C C T W A F Y C 1 1 4
0 -2 9 13 This pair is locally maximal
because shortening or lengthening the alignment
reduces the score. L K F S F A L C
C S R P T W A F Y C -2 2 -4 1
1 4 0 -2 9
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Assumptions for the Statistical Theory

• We assume
• a simple protein model the probabilities for
  the amino acids appearing at each position in a
  protein are independent and identically
  distributed (iid) and amino acid i occurs
  randomly with probability Pi.
• the substitution matrix (such as BLOSUM 62) has
  at least one positive score.
• the expected score for amino acid pairs
• S pipj sij lt 0
• is negative.

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Normalized Scores of HSPs

• Given Pi and Sij, the statistical theory yields
  two parameters, ? and K which can be used to
  normalize an HSP score S with the formula
• S'
• The units of S' are bits. When two random
  proteins sequences are compared, the expected
  number of HSPs with a normalized score of S' is
• E N / 2S'
• where N is the product of the sequence lengths.

?S ln K ln 2
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Score for a Given Statistical Significance

• Solving for S' yields
• S' log2 (N / E)
• For example, if
• the query protein has length 250.
• the database has length 50 000 000
• the E-value is .05
• then the score required to reach that level of
  statistical signficance is 38 bits.

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Deficiencies in the BLAST theory

• The result is asymptotic, meaning there is some
  error for finite sequences.
• Local variations in residue composition in real
  sequences often do not match the model.
• The model assumes that all mutations at all sites
  in a protein sequence can be described by a
  single substitution matrix.
• It does not apply to gapped alignments.

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Finding HSPs

• Blast finds HSPs by first looking for well
  conserved core alignments, i.e., ungapped
  alignments between short query words of length
  3 and database words. Each core alignment must
  score greater than a threshold score, T, which is
  typically 13 using the BLOSUM matrix.
• Core alignments or hits are typically found
  using pattern matching finite automata, i.e.
  linear search through the database.

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Query and Target Words

• Suppose the query sequence is LVNRKPVVP.
• Chop the query into all possible words of length
  3
• LVN VNR NRK RKP KPV PVV VVP
• Collect target words associated with each query
  word. For example, the word RKP has six target
  words that score at least 13 (using BLOSUM 62)
  when aligned with RKP
• QKP KKP RQP REP RRP RKP
• In general, there will be a large set of target
  words derived from the query sequence. Each
  target word and its associated query word could
  form the core of an HSP.

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Extending a Hit

• A hit is extended in either direction to find its
  locally maximal segment pair. Extension
  terminates when the score drops too far below the
  highest score found.
• For example, Suppose the target RRP is found and
  it occurs in the sequence
• EPGVCRRPLKCTAS
• When trying to extend the core against LVNRKPVVP
  the HSP has 6 letters and a score of 16
• L V N R K P V V P
• G V C R R P L K C
• -3 4 -3 5 2 7 1 -2 -3

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Gapped Blast

• Gapped BLAST (1997) has two improvements over the
  original BLAST (1990).
• Two hits Only extends core alignments when two
  occur nearby on the same diagonal. Involves
  lowering the threshold T to retain sensitivity,
  but reduces extension which is the most costly.
• Gapped alignments are computed. Allows raising
  the threshold T while retaining selectivity.
  Speeds initial database scan.

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Optimal Substitution Matricesfor Distant
Homologies

• Among HSPs (ungapped) from the comparison of
  random sequences, amino acids ai and aj are
  aligned with frequency approaching
• qij pipje?sij
• The qij are called target frequencies for the
  given substitution matrix sij. Among alignments
  of distantly related proteins, amino acids tend
  to be paired with certain characteristic
  frequencies. Only if these correspond to a
  matrixs target frequencies ... can the matrix be
  optimal for distinguishing the distant local
  homologies. (Altschul 1991)

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Log-Odds Matrices Again

• Rearranging qij pipje?sij yields
• sij ln(qij /pipj)/?
• where ? acts as a scaling factor for the
  logarithm. This shows
• that the substitution scores are inherently
  log-odds scores for the target and background
  frequencies, and
• scores may be chosen that correspond to any
  desired set of target frequencies.

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Topic 4 Outline

• Specialized Sequence Alignment Algorithms
• Sim4
• Blat

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sim4

• Sim4 is a program for aligning a cDNA sequence to
  a genomic sequence.
• cDNA is complementary to a messenger RNA which is
  the RNA molecule after introns have been cut out
  and the exons spliced together. The difference
  between cDNA and genomic DNA is the absence of
  the intron sequences in cDNA.
• Sim4 assumes that the differences between the two
  sequences are limited to
• introns in the genomic sequence
• sequencing errors in either sequence

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sim4 Find HSPs

• In the first step, sim4 finds HSPs which must
  have an exact matching core of 12 nucleotides.
  The core is extended on both ends with a score of
  1 for match and -5 for mismatch.

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sim4 Select chains of HSPs

• HSps are chained with the constraints that
• starting positions in the cDNA are increasing
• HSPs are in nearby diagonals or are in diagonals
  separated by plausible intron distances

Genomic
cDNA
starting positions increase
gap typical for intron
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sim4 Trim overlaps in cDNA

• Exon cores that overlap in the cDNA are trimmed
  to find
• GT ... AG, a common intron signal, in the
  genomic DNA.

GT
AG
overlap
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sim4 Trim overlaps in cDNA

• Exon cores that overlap in the cDNA are trimmed
  to find
• GT ... AG, a common intron signal, in the
  genomic DNA.

GT
AG
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sim4 Filling Gaps in cDNA

• Gaps are filled by recursively finding HSPs with
  smaller cores (starting with exact matches of
  length 8)

smaller cores are chained as before
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BLAT The BLAST-Like Alignment Tool

• BLAT is a specialized program for mRNA to DNA
  alignments and cross-species protein alignments.
  It is faster than BLAST and sim4 for these tasks.
  It is not appropriate for distant homology
  searching.
• BLAT is available at the UC Santa Cruz human
  genome website for database searches against the
  human genome.

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BLAT vs BLAST

• Both BLAT and BLAST first look for high-scoring
  pairs
• BLAST collects short words in the query and does
  a linear scan through the database for those
  words.
• BLAT builds an index of the database (a data
  structure suited for rapid detection of word
  matches) and then scans linearly through the
  query. Since the query is much smaller than the
  database, the scan phase is very rapid. The data
  structure is persistent, meaning that it is built
  once and then reused for every query search.

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BLAT Hits

• To form the database index, BLAT cuts the
  database sequences into non-overlapping words of
  length k.

Database
Query
A hit is an exact match or a near perfect match
(single character mismatch) with any query
subword.
Query k-words
116
BLAT Criteria for Extending Hits

• BLAT allows different criteria for extending
  hits
• Single exact match
• Single near-exact match (one difference)
• C A G T G C G A T G A
• C A G T A C G A T G A

• Two exact matches on the same diagonal
  separated by a small distance

117
BLAT Statistics

• The size, k, of matching words is selected to
  balance sensitivity and selectivity with running
  time. BLAT makes flexible assumptions about the
  query and database sequences to select k, the
  word match size
• M the percent matching between homologous
  regions (98 for cDNA/genomic alignments, 89 for
  cross-species protein alignments)
• H the size of the homologous regions
• G the size of the database (3 000 000 000
  bases)
• Q the size of a query
• A the alphabet size (4 for DNA, 20 for protein)

118
BLAT Statistics for k Selection (Exact Match)

• The number of non-overlapping k-words in a
  homologous region is
• T floor (H / k)
• Sequence letters are assumed to be iid. The
  probability that
• a match occurs between a homologous region k-word
  and the query is
• p Mk
• no match occurs is
• q (1 p)
• at least one homologous region word matches the
  query is
• Phit 1 minus no matches 1 q T

119
BLAT Statistics for k Selection(Exact Match)

• The probability that two k words match by chance
  is
• r (1/A)k
• The number of
• query words is qw Q k 1
• database words is dw G/k
• The number of k words that are expected to match
  by chance is
• F qw dw r

120
BLAT Statistics for k Selection

• With increasing k,
• the probability of a valid hit, Phit, goes down
  because it becomes harder to find two words that
  match due to errors or mutations.
• the number of false hits, F, goes down because
  probability of random word matching decreases as
  words grow.
• For example, in DNA, with M 0.95, H 100, and
  Q 500, if
• k 11, then Phit 0.999 and F 32 512
• k 14, then Phit 0.991 and F 399
• By decreasing sensitivity slightly, the number of
  false hits can be reduced by almost a factor of
  100.

121
BLAT indexing

• Each k-word (nonoverlapping) in the database is
  converted to a number in base four
• T C A G T T A
• 3 1 0 2 3 3 04
• List I points to a list L, the sorted locations
  of the k-word in the database sequences.

A list, I, of all possible numbers is
maintained. For DNA, when k 7, this list has
47 or 16,384 entries.
I
3102330
3102331
L
100
730
600
350
930
870
122
Deficiencies in the BLAT Program

• Near exact matching with proteins requires an
  index which is too large to fit in memory, so a
  less efficient hashing scheme is used instead
• The alignment method is not standard optimal
  alignment and
• for DNA does not work well below 90 sequence
  identity
• for proteins does not work well with indels in
  one of the sequences

123
Topic 5 Outline

• Searching for Repetitive Motifs and Patterns
• Pattern Detection vs Alignment
• Short word methods
• TRF

124
Pattern Detection

• In pattern detection problems
• there is no query sequence
• we look for a repetitive pattern or motif in one
  long sequence or several sequences
• broad characteristics of the target pattern are
  specified in advance

125
Short Word Methods

• In short word methods, small matching words are
  detected. A cluster of short words indicates a
  potential pattern. A typical applications is the
  search for tandem repeats in DNA sequences.

Word spacings may differ.
126
Short Word Methods

• Short word methods are well suited to DNA
  sequences because
• the alphabet is small so exact matching words are
  common even in homologous regions that have
  experienced significant mutation.
• they can handle insertions and deletions that are
  common in DNA
• They are less well suited to protein sequences
  because
• large amino acid alphabet and frequent
  substitution make short matching words uncommon.

127
Tandem Repeats
A tandem repeat is any pattern of nucleotides
that has been duplicated so that it appears
several times in succession. For example, the
sequence fragment below contains a tandem repeat
of the trinucleotide CGT
tcgctggtcatacgtcgtcgtcgtcgttacaaacgtcttccgt
128
Approximate Tandem Repeats
More typically, the tandem copies are only
approximate due to mutations. Here is an
alignment of copies from a tandem repeat in C.
elegans.
Shown are the copies and a
consensus pattern
129
Tandem Repeats Associated with Human Disease

• Trinucleotide diseases caused by expansion of a
  trinucleotide repeat
• Fragile-X mental retardation
• Myotonic dystrophy
• Huntingtons disease
• Friedreichs ataxia
• Multilocus diseases linked in some cases to
  unstable or uncommon minisatellites
• Epilepsy
• Diabetes
• Ovarian cancer

130
Tandem Repeats Function and Usefulness

• Tandem repeats
• are involved in gene regulation and often
  contain putative transcription factor binding
  sites.
• exhibit copy number polymorphism, making them
  valuable genomic markers.

131
Tandem Repeats Finder (TRF)

• TRF finds tandem repeats in genomic DNA sequences
  using short word matches (k-tuple matches). It
  assumes
• repeats have on average gt80 sequence identity
• insertions and deletions occur on average in lt
  10 of the pattern positions
• These are average values for program parameter
  settings. Repeats with lower sequence identity
  and higher indel frequency are also found.

132

• LOCUS HUMFMR1 3765 bp
  mRNA PRI 08-NOV-1994
• DEFINITION Human Fragile X mental
  retardation 1 FMR-1 gene, 3' end, clones
• 1 gacggaggcg
  cccgtgccag ggggcgtgcg gcagcgcggc ggcggcggcg
  gcggcggcgg
• 61 cggcggaggc ggcggcggcg gcggcggcgg
  cggcggaggc ggcggcggcg gcggcggcgg
• 121 cggcggctgg gcctcgagcg cccgcagccc
  acctctcggg ggcgggctcc cggcgctagc
• 181 agggctgaag agaagatgga ggagctggtg
  gtggaagtgc ggggctccaa tggcgctttc
• 241 tacaaggcat ttgtaaagga tgttcatgaa
  gattcaataa cagttgcatt tgaaaacaac
• 301 tggcagcctg ataggcagat tccatttcat
  gatgtcagat tcccacctcc tgtaggttat
• 361 aataaagata taaatgaaag tgatgaagtt
  gaggtgtatt ccagagcaaa tgaaaaagag
• 421 ccttgctgtt ggtggttagc taaagtgagg
  atgataaagg gtgagtttta tgtgatagaa
• 481 tatgcagcat gtgatgcaac ttacaatgaa
  attgtcacaa ttgaacgtct aagatctgtt
• 541 aatcccaaca aacctgccac aaaagatact
  ttccataaga tcaagctgga tgtgccagaa
• 601 gacttacggc aaatgtgtgc caaagaggcg
  gcacataagg attttaaaaa ggcagttggt
• 661 gccttttctg taacttatga tccagaaaat
  tatcagcttg tcattttgtc catcaatgaa
• 721 gtcacctcaa agcgagcaca tatgctgatt
  gacatgcact ttcggagtct gcgcactaag
• 781 ttgtctctga taatgagaaa tgaagaagct
  agtaagcagc tggagagttc aaggcagctt
• 841 gcctcgagat ttcatgaaca gtttatcgta
  agagaagatc tgatgggtct agctattggt
• 901 actcatggtg ctaatattca gcaagctaga
  aaagtacctg gggtcactgc tattgatcta
• 961 gatgaagata cctgcacatt tcatatttat
  ggagaggatc aggatgcagt gaaaaaagct

133

• LOCUS HUMFMR1 3765 bp
  mRNA PRI 08-NOV-1994
• DEFINITION Human Fragile X mental
  retardation 1 FMR-1 gene, 3' end, clones
• 1 gacggaggcg
  cccgtgccag ggggcgtgcg gcagcgcggc ggcggcggcg
  gcggcggcgg
• 61 cggcggaggc ggcggcggcg gcggcggcgg
  cggcggaggc ggcggcggcg gcggcggcgg
• 121 cggcggctgg gcctcgagcg cccgcagccc
  acctctcggg ggcgggctcc cggcgctagc
• 181 agggctgaag agaagatgga ggagctggtg
  gtggaagtgc ggggctccaa tggcgctttc
• 241 tacaaggcat ttgtaaagga tgttcatgaa
  gattcaataa cagttgcatt tgaaaacaac
• 301 tggcagcctg ataggcagat tccatttcat
  gatgtcagat tcccacctcc tgtaggttat
• 361 aataaagata taaatgaaag tgatgaagtt
  gaggtgtatt ccagagcaaa tgaaaaagag
• 421 ccttgctgtt ggtggttagc taaagtgagg
  atgataaagg gtgagtttta tgtgatagaa
• 481 tatgcagcat gtgatgcaac ttacaatgaa
  attgtcacaa ttgaacgtct aagatctgtt
• 541 aatcccaaca aacctgccac aaaagatact
  ttccataaga tcaagctgga tgtgccagaa
• 601 gacttacggc aaatgtgtgc caaagaggcg
  gcacataagg attttaaaaa ggcagttggt
• 661 gccttttctg taacttatga tccagaaaat
  tatcagcttg tcattttgtc catcaatgaa
• 721 gtcacctcaa agcgagcaca tatgctgatt
  gacatgcact ttcggagtct gcgcactaag
• 781 ttgtctctga taatgagaaa tgaagaagct
  agtaagcagc tggagagttc aaggcagctt
• 841 gcctcgagat ttcatgaaca gtttatcgta
  agagaagatc tgatgggtct agctattggt
• 901 actcatggtg ctaatattca gcaagctaga
  aaagtacctg gggtcactgc tattgatcta
• 961 gatgaagata cctgcacatt tcatatttat
  ggagaggatc aggatgcagt gaaaaaagct

134
LOCUS RATIGCA 4461 bp DNA ROD
18-APR-1994 DEFINITION Rat Ig germline
epsilon H-chain gene C-region, 3' end.
2881 cgccccaagt aggcttcatc atgctctttg
gtttagcaat agcccaaagc aagctatgca 2941
tccatctcag gcccagaggg atgaggagac cagaatcaag
acatacccac gcccatccca 3001 cgcccaacca
ccaaccacca gcacatcagg ttcacacacc tgagaccagt
ggctcccatc 3061 acacacacac acacacacac
acacacacac acacacacac acacacaagc ccgtacacat
3121 ccaccatatc cagagacaag tgtctgagtc tgagatacct
ctgaggatca ccaatggcag 3181 agtcggccag
cacctcagcc tccaggccaa tccttatact ttggcccact
gcaggccatg 3241 agagatggag gaggtggagg
cctgagctgt ggaaaaccag agacaggaag atggtctgta
3301 ctccaggcca atccttatac tttggcccac tgcaggccat
gagagatgga ggaggtggag 3361 gcctgagctg
tggaaaacca gagacaggaa gatggtctgt atggagagag
tagtaaacca 3421 gattataggg agactgaggc
aggagtagag ctcctacaag gccagtagtc taccttagag
3481 tcctataagt ctgggctggg agtccatgtg tcctgacttg
ctcctcagat atcacaacca 3541 agattcctgg
agccagagtg tgcatgcagg ccctagaaga aatgtggagc
ttagagccct 3601 tcctggaggg ccctgggcac
tctgaacaaa aggcaattct gtaggctgta tagaggcatc
3661 ctgtcagata cacacacaca tgcacacaca tacacacaca
gagacacaga cacacacaca 3721 tgcccacaca
catgcataca cacatgcaca cacatacaca cacagagaca
cagacacaca 3781 catgcccaca cacatgcata
cacacatgca tgcacacaca cacacacaca tacacataca
3841 cacacacaca cacaccccgc aggtagcctt catcatgctg
tctagcgata gccctgctga 3901 gggtgggaga
tactgggtca tggtgggcac cggagtagaa agagggaatg
agcagtcagg 3961 gtcaggggaa aaggacatct
gcctccaggg ctgaacagag acttggagca gtcccagagc
4021 aagtgggatg gggagctctg ccactccagt ttcaccagga
ctgcctgaga ccagtgaggg
135
LOCUS RATIGCA 4461 bp DNA ROD
18-APR-1994 DEFINITION Rat Ig germline
epsilon H-chain gene C-region, 3' end.
2881 cgccccaagt aggcttcatc atgctctttg
gtttagcaat agcccaaagc aagctatgca 2941
tccatctcag gcccagaggg atgaggagac cagaatcaag
acatacccac gcccatccca 3001 cgcccaacca
ccaaccacca gcacatcagg ttcacacacc tgagaccagt
ggctcccatc 3061 acacacacac acacacacac
acacacacac acacacacac acacacaagc ccgtacacat
3121 ccaccatatc cagagacaag tgtctgagtc tgagatacct
ctgaggatca ccaatggcag 3181 agtcggccag
cacctcagcc tccaggccaa tccttatact ttggcccact
gcaggccatg 3241 agagatggag gaggtggagg
cctgagctgt ggaaaaccag agacaggaag atggtctgta
3301 ctccaggcca atccttatac tttggcccac tgcaggccat
gagagatgga ggaggtggag 3361 gcctgagctg
tggaaaacca gagacaggaa gatggtctgt atggagagag
tagtaaacca 3421 gattataggg agactgaggc
aggagtagag ctcctacaag gccagtagtc taccttagag
3481 tcctataagt ctgggctggg agtccatgtg tcctgacttg
ctcctcagat atcacaacca 3541 agattcctgg
agccagagtg tgcatgcagg ccctagaaga aatgtggagc
ttagagccct 3601 tcctggaggg ccctgggcac
tctgaacaaa aggcaattct gtaggctgta tagaggcatc
3661 ctgtcagata cacacacaca tgcacacaca tacacacaca
gagacacaga cacacacaca 3721 tgcccacaca
catgcataca cacatgcaca cacatacaca cacagagaca
cagacacaca 3781 catgcccaca cacatgcata
cacacatgca tgcacacaca cacacacaca tacacataca
3841 cacacacaca cacaccccgc aggtagcctt catcatgctg
tctagcgata gccctgctga 3901 gggtgggaga
tactgggtca tggtgggcac cggagtagaa agagggaatg
agcagtcagg 3961 gtcaggggaa aaggacatct
gcctccaggg ctgaacagag acttggagca gtcccagagc
4021 aagtgggatg gggagctctg ccactccagt ttcaccagga
ctgcctgaga ccagtgaggg
136
Basic Assumption

• Mutated, adjacent copies of a pattern will
  contain runs of exact matches.

T C C A C G G A G A T A T T T A
T A T A C G T C G A G A C T T A
137
Basic Assumption

• Mutated, adjacent copies of a pattern will
  contain runs of exact matches.

T C C A C G G A G A T A T T T A
T A T A C G T C G A G A C T T A
Runs of matches are identified using k-tuple
matches.
138
Using k-tuple matches

• For purposes of program efficiency, fixed size
  k-tuples are used.

139
k-tuple matches
In pattern matching, a k-tuple is a window of
length k which contains text characters. Two
windows which contain the same text, form a
k-tuple match GAACGTTAGGTAACTGCAT CCTAGTTATACG
TTAAC
140
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141
Modeling Tandem Repeats

• Multiple k-tuple matches suggest the occurrence
  of a tandem repeat. The appropriate number of
  matches depends on how a tandem repeat is
  defined. For example, are the following two
  aligned sequence fragments two copies of the same
  underlying pattern?
• TCGGCATCAGTCTATGG
• TCAA-TG-GTGT-TGG

142
A Stochastic Model

• TRFs stochastic model is based on the
  probability of character matching and the
  frequency of insertion and deletion (indels)
  between aligned adjacent copies.
• CCACAACC-CGTCAGGCAAGT
• CTGCACCATCGTCTGGGAAGT
• HTTHHTHTTHHHHTHHTHHHH
• Note that the alignment has been converted into a
  Bernoulli (coin-toss) sequence.

143
Model Parameters

• PM the expected frequency of a match
• PI the expected frequency of an indel
• The parameters are applied to Bernoulli sequences
  to establish criteria for detecting the repeats.

144
Number of matches to indicate a repeat

• Sum of heads is the minimum required number of
  matches
• for a repeat with period n
• match probability p
• tuple size k
• Unless k 1, not all matches will be detected.
• For example
• HHTHHHHTHTHHHTTHHHT

k H seen
1 13
2 12
3 10
4 4
145
Sum of heads

• Suppose a random Bernoulli sequence has length
  100 and the expected number of heads is 75 (PM
  0.75). If we count the number of heads, then 95
  of the time we expect to count at least 68 heads.
  
• If we count only heads that occur in runs of
  length 5 or more, then 95 of the time we expect
  to count at least 26 heads. This is the sum of
  heads criteria.

146
Other Criteria

• Apparent Size Used to distinguish tandem from
  non-tandem repeats.
• Waiting time Used to pick a suitable tuple
  size.
• Random walk Used to accommodate insertions and
  deletions.

147
(No Transcript)
148
The Tandem Repeats Database

• The Tandem Repeats Database (TRDB) is
• A public database of information on tandem
  repeats.
• A private workspace for extended research on
  tandem repeats.

149
C. elegans Distribution of Repeat Pattern Size
Chr 1
150
Human Distribution of Pattern Size Chr 1
151
C. elegans Distribution of Repeat Location Chr 1
152
Human Distribution of Repeat Location Chr 1
153
Human Distribution of Repeat Location Chr 1 for
Patternsize gt 5
154
Clusters in 41 bp Repeats
155
Two identical repeats
156
Four repeats from a tandem repeat cluster
157
Topic 6 Outline

• Composition Alignment
• Sequence composition and composition match
• Composition alignment algorithm
• Composition match scoring functions
• Limiting the length of a composition match
• Growth of local composition alignment scores
• Biological examples

158
Sequence Composition

• Composition is a vector quantity describing the
  frequency of occurrence of each alphabet letter
  in a particular string. Let S be a string over
  S. Then,
• C(S)(fs1 , fs2 , fs3 , , fsS)
• is the composition of S, where fsi is the
  fraction of the characters in S that are si.
• Note that the order of letters is irrelevant as
  it has no effect on the composition.

159
Composition Example

• S ACTGTACCTGGCGCTATT
• C(S) ( 0.17, 0.28, 0.22, 0.33 )
• A C G
  T

160
Composition Match

• Two strings, S and T, have a composition match if
  their lengths are equal and C(S) C(T).
• For example, S and T below have a composition
  match
• S ACTGTACCTGGCGCTATT
• T AAACCCCCGGGGTTTTTT

161
Composition and Sequence Features

• Isochores Multi-megabase, specifically GC-rich
  or GC-poor. GC-rich isochores have greater gene
  density.
• CpG Islands Several hundred nucleotides, rich
  in the dinucleotide CG which is underrepresented
  in eukaryotic genomes. Methylation of the
  cystine in these dinucleotides affects gene
  expression.
• Protein binding regions Tens of nucleotides,
  dinucleotide composition contributes to DNA
  flexibility, allowing the helix to change shape
  during protein binding.

162
Composition Alignment Problem

• Given Two sequences, S of length m, and T of
  length n, over an alphabet S, and a scoring
  function cm(s, t) for the score of a composition
  match between substrings s and t.
• Find The best scoring alignment (global or
  local) of S with T such that the allowed scoring
  options include composition match between
  substrings of S and T as well as the standard
  options of single character match, single
  character mismatch, insertion and deletion.

163
Example of composition alignment

• S AACGTCTTTGAGCTC
• T AGCCTGACTGCCTA
• Alignment
• AACGTCTTTGAGCTC
• lt-gt lt---gt
• AGCCTGACT-GCCTA

164
Algorithm Analysis

• Given two sequences, S and T, the best alignment
  of the prefix strings
• S1, i s1 si
• T1, j t1 tj
• ends in one of four ways, mismatch, insertion,
  deletion, or composition match between suffixes
  of length l
• 1 l min(i, j, limit)
• i.e., between substrings Si l 1, i and Tj
  l 1, j

165
Time Complexity

• Computing the optimal composition alignment is
  done with dynamic programming and is similar to
  standard alignment, except for the composition
  match scoring option. The overall time
  complexity is
• O(nmZ)
• where Z is the time required per (i, j) pair to
  find the best length l for the composition match.

166
Computing length of the shortest composition
match

• Our goal here is to start with two strings, S and
  T, of equal length, and for each prefix pair S1,
  k, T1, k, find the length of the shortest
  suffixes that have a composition match. For
  example, let
• S AACGTCTTTGAGCT
• T AGCCTGACTGCCTA
• Then for k 6, the shortest suffixes which have
  a composition match have length 3
• S AACGTC
• T AGCCTG

167
Composition difference

• Composition difference is a vector quantity for
  two strings x and y
• CD(x, y) (cs1 , , csS)
• where csi is the difference between the number
  of times si occurs in x and in y.

168
Using composition difference
Key observation two identical composition
differences at prefix lengths k and g indicate a
composition match of length k g.
169
Sorting to find shortest composition matches
Sort on composition difference using stable sort.
Adjacent tuples with the same composition
difference identify shortest composition matches.
170
Time complexity for composition matches

• O(nmS) to find nm shortest composition match
  lengths for two strings of length n and m.
• In our work, S, is a small constant (4 for DNA,
  16 for dinucleotides). For larger alphabets, the
  method of Amir, Apostolico, Landau and Satta
  (2003) can be used.

171
Composition match scoring functions

• Functions based on match length, k
• Function 1 cm(k) ck
• Function 2 cm(k) cv k
• where c is a constant.
• Functions based on substring composition
• Function 4 cm(C, B, k) ck H(C,B)
• where H is the relative entropy function, C is
  the composition of the matching substrings and B
  is a background composition.

172
Additive and subadditive scoring functions

• The functions based on length are additive or
  subadditive
• cm(i j) cm(i) cm(j)
• Lemma For additive or subadditive composition
  match scoring functions, any best scoring
  alignment is equivalent in score to an alignment
  which contains only shortest composition matches.
  
• Theorem Composition alignment with additive or
  subadditive match scoring functions and finite
  alphabet has time complexity O(nm).

173
The limit parameter

• Intuitively, allowing scrambled letters to match
  should increase the amount of matching between
  sequences. If too much matching occurs,
  alignments will not be meaningful.
• The limit parameter is an upper bound on the
  length l of the longest single composition match.
  

174
Limit and fraction of matching charactersrandom,
ungapped alignments
Sequence length limit 1 2 5 10
binary 50 62.5 75.6 82.4
DNA 25 30 37.5 44.2
Dinucleotide 6.2 6.5 7.1 7.5
175
Limit and fraction of matching charactersrandom,
ungapped alignments
Sequence length 100, all letters equal
probability p 0.25
limit 1 2 5 10
DNA 25 33.7 44.4 51
Sequence length 400, all letters equal
probability p 0.25
limit 1 2 10 50
dinucleotide 6.25 6.81 7.76 7.78
176
Growth of local alignment scoreFunction 1
177
Global score as a predictor of local parameter
suitability Function 1
178
Growth of local alignment score Function 2
179
Global score as a predictor of local parameter
suitability Function 2
180
Limit values for DNA

• Function 1 cm(k) ck Limit 3.
• Function 2 cm(k) cvk Limit 10.
• Function 4 cm(C, B, k) ck H(C, B)
• Limit 50.

181
Biological examples

• Composition alignment was tested on a set of 1796
  promoter sequences from the Eukaryotic Promoter
  Database. Each sequence is 600 nucleotides long,
  500 bases upstream and 100 downstream of the
  transcription initiation site.
• Two local alignment scores were produced using
  function 1, W using composition alignment and S
  using standard alignment. The examples shown have
  statistically significant W with W 3 S to
  exclude good standard alignments.

182
Example 1
Composition alignment and standard alignment of
two promoters. Standard alignment is not
statistically significant. Sequences are
characteristic of CpG islands.
Composition Alignment GCCCGCCCGCCGCGCTCCCGCCCGCC
GCTCTCCGTGGCCC-CGCCG-CGCTGCCGCCGCCGCCGCTGC lt-gt
ltgtltgtltgt ltgtltgt lt-gt ltgtltgt ltgtltgt
ltgt lt-gt CCGCGCCGCCGCCGTCCGCGCCGCCCCG-CCCT-TGG
CCCAGCCGCTCGCTCGGCTCCGCTCCCTGGC Standard
Alignment CGCCGCCGCCG CGCCGCCGCCG
183
Example 2
Composition alignment of two promoter sequences.
Composition changes at vertical line.
A C G T Left
(0.01, 0.61, 0.30, 0.08) Right (0.19, 0.16,
0.56, 0.09)
GCCCCGCGCCCCGCGCCCCGCGCCCCGCGCGCCTC-CGCCCGCCCCT-GC
TCCGGC---C-TTGCGCCTGC-GCACAGTGGGATGCGCGGGGAG lt-gtlt
gtltgt ltgt lt-gtltgt ltgt
lt-gt ltgtltgtlt-gt ltgtltgtltgtlt-gtlt-gt CCGCGC
GCCCCC-GCCCCCGCCCCGCCCCGGCCTCGGCCCCGGCCCTGGC-CCCGG
GGGCAGTCGCGCCTGTG-AACGGTGAGTGCGGGCAGGG
184
Final Slide

• Recognize that similarity among biological
  sequences most likely exists in ways which we can
  not today perceive and for which we have no
  detection tools. You are encouraged, therefore,
  to think broadly, beyond the embellishment or
  refinement of current methods, to new definitions
  of similarity and new problems of comparison.