Scoring Matrices

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Scoring Matrices
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(No Transcript)
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Limitations to Needleman-Wunsch

• The problem with Needleman-Wunsch is the amount
  of processor memory resources it requires.
• Because of this, it is not favored for practical
  use, despite the guarantee of an optimal
  alignment.

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What is the problem?

• There are about 10 88 possible alignments for
  two sequences with 300 nucleotides long( There
  are only about 10 80 elementary particles in the
  universe.
• It is not possible to solve the alignment
  problem with brute force.
• Therefore, we need some smart methods (or
  algorithms to overcome this problem

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Limitations to Needleman-Wunsch

• The other difficulty is that the concept of
  global alignment is not used in pairwise sequence
  comparison searches.

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Global Alignment vs. Local Alignment
Global
Needleman-Wunsch Method
Local
Dot Plots Smith-Waterman FastA BLAST
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Global alignment The global alignment optimizes
the alignment over the full length of the
sequences. LGPSTKDFGKISESREFDN LNQLERSFGKINMRLE-D
A

• Local Alignment
• ---------------FGKI-----------
• ---------------FGKI-----------
• In local alignment ,stretches with the highest
  density of matches are given the highest
  priority.
• The alignment tends to stop at the ends of
  regions of identity or strong similarity.

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Purpose of Smith Waterman Algorithm

• Smith-Waterman dynamic programming algorithm,
  finds the most similar subsequences of two
  sequences, that has been generally recognized as
  the most sensitive sequence.
• The search sequences in protein and DNA databases
  searches for similarity to the query sequence by
  using Smith-Waterman algorithm as the core
  sequence comparison method.

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Smith-Waterman searches

• A more sensitive brute force approach to
  searching
• much slower than BLAST or FASTA
• uses dynamic programming
• SSEARCH is a GCG program for Smith-Waterman
  searches

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Differences
Needleman- Wunsch
Smith - Waterman

• Local alignments
• Residue alignment score may be positive or
  negative
• Requires a gap penalty to work effectively
• Score can increase, decrease or stay level
  between two cells of a pathway.

• Global alignments
• Requires alignments score for a pair of residues
  to be gt0
• No gap penalty required

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Scoring Matrix/Substitution Matrix

• To score quality of an alignment
• Contains scores for pairs of residues (amino
  acids or nucleic acids) in a sequence alignment
• For protein/protein comparisons a 20 x 20
  matrix of similarity scores where identical amino
  acids and those of similar character (e.g. Ile,
  Leu) give higher scores compared to those of
  different character (e.g. Ile, Asp).
• Symmetric, so often only half is shown.

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Substitution Matrices

• Not all amino acids are equal
• Some are more easily substituted than others
• Some mutations occur more often
• Some substitutions are kept more often
• Mutations tend to favor some substitutions
• Some amino acids have similar codons
• They are more likely to be changed from DNA
  mutation
• Selection tends to favor some substitutions
• Some amino acids have similar properties or
  structure
• They are more likely to be kept

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Substitution Matrix

• A substitution matrix describes the likelihood
  that two residue types would mutate to each other
  in evolutionary time.
• This is used to estimate how well two residues of
  given types would match if they were aligned in a
  sequence alignment.

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Substitution Matrix

• An amino acid substitution matrix is a
  symmetrical 2020 matrix, where each element
  contains the score for substituting a residue of
  type i with a residue of type j in a protein,
  where i and j are one of the 20 amino-acid
  residue types.
• Same residues should obviously have high scores,
  but if we have different residues in a position,
  how should that be scored?

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Scoring Matrices

• Scoring matrices tell how similar amino acids
  are.
• There are two main sets of scoring matrices PAM
  and BLOSUM.
• PAM is based on evolutionary distances
• BLOSUM is based on structure/function similarities

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Substitution Matrix Scoring

• The same residues in a position give the score
  value 1, and different residues give 0.
• The same residues give a score 1, similar
  residues (for example Tyr/Phe, or Ile/Leu) give
  0.5, and all others 0.
• One may calculate, using well established
  sequence alignments, the frequencies
  (probabilities) that a particular residue in a
  position is exchanged for another.

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Similarity Searching

• It is easy to score if an amino acid is identical
  to another (the score is 1 if identical and 0 if
  not). However, it is not easy to give a score
  for amino acids that are somewhat similar.
• Should they get a 0 (non-identical) or a 1
  (identical) or something in between?

Isoleucine
Leucine
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Scoring Similarity

• 1) Can only score aligned sequences
• 2) DNA is usually scored as identical or not
• 3) Modified scoring for gaps - single vs.
  multiple base gaps (gap extension)
• 4) AAs have varying degrees of similarity
• a. of mutations to convert one to another
• b. chemical similarity
• c. observed mutation frequencies
• 5) PAM matrix calculated from observed mutations
  in protein families

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Dayhoff Matrix

• This was done originally be Margaret Dayhoff.
  Her matrices are called the PAM (Point Accepted
  Mutation) matrices, which describe the exchange
  frequencies after having accepted a given number
  of point mutations over the sequence.
• Typical values are PAM 120 (120 mutations per 100
  residues in a protein) and PAM 250.
• There are many other substitution matrices
  BLOSUM, Gonnet, etc.

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Dayhoff Matrix

• Derived from how often different amino acids
  replace other amino acids in evolution.
• Created from a dataset of closely similar protein
  sequences (less than 15 amino acid difference).
  These could be unambiguously aligned.
• A mutation probability matrix was derived where
  the entries reflect the probabilities of a
  mutational event.
• This matrix is called PAM 1. An evolutionary
  distance of 1 PAM (point accepted mutation) means
  there has been 1 point mutation per 100 residues

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Importance of Scoring Matrices

• Scoring matrices appear in all analyses involving
  sequence comparisons.
• The choice of matrix can strongly influence the
  outcome of the analysis.
• Scoring matrices implicitly represent a
  particular theory of relationships.
• Understanding theories underlying a given scoring
  matrix can aid in making proper choice.

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Scoring Matrix Conventions

• Scoring matrices are conventionally numbered with
  numeric indices corresponding to the rows and
  columns of the matrix.
• For example, M11 refers to the entry at the first
  row and the first column.
• In general, Mij refers to the entry at the ith
  row and the jth column.

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Scoring Matrices

• To use this for sequence alignment, we simply
  associate a numeric value to each letter in the
  alphabet of the sequence.
• For example, if the matrix is A,C,T,G then A
  1, C 2, etc. Thus, one would find the score for
  a match between A and C at M12.

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The Filled-in F matrix for global alignment of
xAAGT and YAGCGT(using BLOSUM50 substitution
matrix)
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Global alignment using BLOSUM50 substitution
matrix
alignment AAG _T AGCGT
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Amino Acid Scoring Matrices

• There are two major scoring matrices for amino
  acid sequence comparisons
• PAM-derived from sequences known to be closely
  related (Eg. Chimpanzee and human). Ranges from
  PAM1 to PAM500
• BLOSUM-derived from sequences not closely related
  (Eg. E. coli and human). Ranges from BLOSUM
  10-BLOSUM 100

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PAM250 Matrix
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The Point-Accepted-Mutation (PAM) model

• This model implies that amino acids (AA) mutate
  independently of each other with a probability
  which depends only on the AA.
• Since there are 20 AA, the transition
  probabilities are described by a 20X20-mutation
  matrix, denoted by M. A standard M defines a
  1-PAM change.
• Point Accepted Mutation (PAM) Distance A 1-PAM
  unit changes 1 of the amino acids on average
• where fi is the frequency of AAi, and Mii is the
  frequency of no change in amino acid i.

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The Point-Accepted-Mutation (PAM) model

• Started by Margaret Dayhoff, 1978
• A series of matrices describing the extent to
  which two amino acids have been interchanged in
  evolution
• PAM-1 was obtained by aligning very similar
  sequences. Other PAMs were obtained by
  extrapolation

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The Point-Accepted-Mutation (PAM) model of
evolution and the PAM scoring matrix
A 2-PAM unit is equivalent to two 1-PAM unit
evolution (or M2). A k-PAM unit is equivalent to
k 1-PAM unit evolution (or Mk). Example 1
CNGTTDQVDKIVKILNEGQIASTDVVEVVVSPPYVFLPVVKSQLRPEIQ
V
CNGTTDQVDKIVKIRNEGQIASTDVVEVVVSPPYVFLPVVKSQLRPE
IQV length 50 1 mismatch PAM distance 2
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The Point-Accepted-Mutation (PAM) model of
evolution and the PAM scoring matrix
Observed Sequence Difference
Evolutionary Distance In PAMs
1 5 10 20 40 50 60 70 80
1 5 11 23 56 80 112 159 246
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Assumptions in the PAM model
1. Replacement at any site depends only on the
amino acid at that site and the probability given
by the table (Markov model). 2. Sequences that
are being compared have average amino acid
composition.
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Steps to building the first PAM

• Aligned sequences that were at least 85
  identical.
• Reconstructed phylogenetic trees and inferred
  ancestral sequences. 71 trees containing 1,572 aa
  exchanges were used.
• Tallied aa replacements "accepted" by natural
  selection, in all pairwise comparisons (each Aij
  is the number of times amino acid j was replaced
  by amino acid i in all comparisons).

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Steps to building PAM

• 4. Computed amino acid mutability, mj (the
  propensity of a given amino acid, j, to be
  replaced by any other amino acid)
• 5. Combined data from 3 4 to produce a
  Mutation Probability Matrix for one PAM of
  evolutionary distance, according to the following
  formula

Replacements
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Steps to building PAM
6. Take the log odds ratio to obtain each
score Sij log (Mij/fi) Where fi is the
normalized frequency of aai in the sequences
used. 7. Note must multiply the Mij/fi by a
factor of 10 prior to avoid fractions.
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Sources of error in PAM model
1. Many sequences depart from average aa
composition. 2. Rare replacements were observed
too infrequently to determine probabilities
accurately (for 36 aa pairs (out of 400 aa pairs)
no replacements were observed!). 3. Errors in 1
PAM are magnified when extrapolated to 250 PAM.
(Mijk k PAM) 4. The idea that each amino acid
is acting independently is an imperfect
representation of evolution. Actually, distantly
related sequences usually have islands (blocks)
of conserved residues implying that replacement
is not equally probable over entire sequence.
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The bottom line on PAM
Frequency of alignment
Frequency of occurrence
The probability that two amino acids, i and j
are aligned by evolutionary descent divided by
the probability that they are aligned by chance
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BLOSUM Matrix (BLOcks SUbstitution Matrices)

• Blocks Sum-created from BLOCKS database
• A series of matrices describing the extent to
  which two amino acids are interchangeable in
  conserved structures of proteins
• The number in the series represents the threshold
  percent similarity between sequences, for
  consideration for calculation
• (For example, BLOSUM62 means 62 of the aas were
  similar)

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BLOSUM Matrices

• BLOSUM is built from distantly related sequences
  within conserved blocks whereas PAM is built from
  closely related sequences
• BLOSUM is built from conserved blocks of aligned
  protein segments found in the BLOCKS database
  (the BLOCKS database is a secondary database that
  depends on the PROSITE Family database)

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BLOSUM Matrices (cont.1)

• Version 8.0 of the Blocks Database consists of
  2884 blocks based on 770 protein families
  documented in PROSITE. PROSITE supplies
  documentation for each family.

Hypothetical entry in red box in BLOCK record
AABCDA...BBCDA DABCDA.A.BBCBB BBBCDABA.BCCAA AAACD
AC.DCBCDB CCBADAB.DBBDCC AAACAA...BBCCC
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Building BLOSUM Matrices

• 1. To build the BLOSUM 62 matrix one must
  eliminate sequences that are identical in more
  than 62 of their amino acid sequences. This is
  done by either removing sequences from the Block
  or by finding a cluster of similar sequences and
  replacing it with a single representative
  sequence.
• 2. Next, the probability for a pair of amino
  acids to be in the same column is calculated. In
  the previous page this would be the probability
  of replacement of A with A, A with B, A with C,
  and B with C. This gives the value qij
• 3. Next, one calculates the probability that a
  certain amino acid frequency exists, fi.

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Building BLOSUM Matrices (cont.)

• 4. Finally, we calculate the log odds ratio si,j
  log2 (qij/fi). This value is entered into the
  matrix.
• Which BLOSUM to use?

BLOSUM Identity 80
80 62
62 (usually default value) 35
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If you are comparing sequences that are very
similar, use BLOSUM 80. Sequences that are more
divergent (dissimilar) than 20 are given very
low scores in this matrix.
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Which Scoring Matrix to use?

• PAM-1
• BLOSUM-100
• Small evolutionary distance
• High identity within short sequences

• PAM-250
• BLOSUM-20
• Large evolutionary distance
• Low identity within long sequences

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The PAM 250 Scoring Matrix
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GCG Wisconsin Package GAP

• GAP is the implementation of the Needleman-Wunsch
  algorithm in the GCG program package.
• The NW algorithm will present you with a single
  globally optimal alignment, not all possible
  optimal alignments - different alignments may
  exist that give the same score.
• GAP presents you with one member of the family of
  best alignments that align the full length of one
  sequence to the full length of a second sequence.
  
• There may be many members of this family, but no
  other member has a higher score.

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GCG Wisconsin Package GAP

• The primary use of a global alignment algorithm
  is when you really want the whole of two
  sequences to be aligned, without truncation.
• GAP could completely bypass a region of high
  local homology, if a better (or even just as
  good) path can be found in a different way.
• This is problematic if one short sequence is
  aligned against a longer one with internal
  repeats.
• If there is weak or unknown similarity between
  two sequences, a local alignment algorithm
  (BESTFIT) is the better choice.
• Use GAP only when you believe the similarity is
  over the whole length.

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Global Alignment vs. Local Alignment

• Global alignment is used when the overall gene
  sequence is similar to another sequence-often
  used in multiple sequence alignment.
• Clustal W algorithm
• Local alignment is used when only a small portion
  of one gene is similar to a small portion of
  another gene.
• BLAST
• FASTA
• Smith-Waterman algorithm