Scoring Matrices

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.

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

Limitations to Needleman-Wunsch

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

Global Alignment vs. Local Alignment

Global

LocalDot PlotsSmith-WatermanFastABLAST

Needleman-Wunsch Method

Global alignment:The global alignment optimizes the alignment over the full length of the sequences.

LGPSTKDFGKISESREFDNLNQLERSFGKINMRLE-DALocal 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.

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.

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

Differences

Needleman- Wunsch Smith - Waterman

• Global alignments

• Requires alignments score for a pair of residues to be >=0

• No gap penalty required

• 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.

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.

Substitution MatricesNot 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

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.

Substitution Matrix

An amino acid substitution matrix is a symmetrical 20*20 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?

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 distancesBLOSUM is based on structure/function similarities

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.

H2N CH C

CH2

OH

O

CH CH3

CH3

H2N CH C

CH

OH

O

CH3

CH2

CH3

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?

Leucine Isoleucine

Scoring Similarity1) Can only score aligned sequences2) DNA is usually scored as identical or not3) 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

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.

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

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.

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.

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.

The Filled-in F matrix for global alignment of x=AAGT and Y=AGCGT(using BLOSUM50

substitution matrix)

Y/X D A A G TD 0 -8 -16 -24 -32A -8 5 -3 -11 -19G -16 -3 5 5 -3C -24 -11 -3 2 4G -32 -19 -11 5 0T -40 -27 -19 -3 10

Global alignment using BLOSUM50 substitution

matrix

Y/X D A A G TD 0 -8 -16 -24 -32A -8 5 -3 -11 -19

G -16 -3 5 5 -3C -24 -11 -3 2 4G -32 -19 -11 5 0T -40 -27 -19 -3 10

alignment: AAG _T AGCGT

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

PAM250 Matrix

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.

The Point-Accepted-Mutation (PAM) model

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

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: CNGTTDQVDKIVKILNEGQIASTDVVEVVVSPPYVFLPVVKSQLRPEIQV

|||||||||||||| |||||||||||||||||||||||||||||||||||

CNGTTDQVDKIVKIRNEGQIASTDVVEVVVSPPYVFLPVVKSQLRPEIQV

length = 50 1 mismatch PAM distance = 2

The Point-Accepted-Mutation (PAM) model of evolution and the PAM scoring

matrix

Observed %Sequence Difference

Evolutionary DistanceIn PAMs

1510204050607080

1511235680112159246

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.

Steps to building the first PAM

1. Aligned sequences that were at least 85% identical.

2. Reconstructed phylogenetic trees and inferred ancestral sequences. 71 trees containing 1,572 aa exchanges were used.

3. 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).

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

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.

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 to250 PAM. (Mij

k = 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.

The bottom line on PAM

Frequency of alignmentFrequency of occurrence

The probability that two amino acids, i and j arealigned by evolutionary descent divided by the

probability that they are aligned by chance

BLOSUM Matrix (BLOcks SUbstitution Matrices)

Blocks Sum-created from BLOCKS databaseA series of matrices describing the extent to which two amino acids are interchangeable in conserved structures of proteinsThe number in the series represents the threshold percent similarity between sequences, for consideration for calculation

(For example, BLOSUM62 means 62% of the aa’s were similar)

BLOSUM Matrices

BLOSUM is built from distantly related sequences within conserved blocks whereas PAM is built from closely related sequencesBLOSUM 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)

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...BBCDADABCDA.A.BBCBBBBBCDABA.BCCAAAAACDAC.DCBCDBCCBADAB.DBBDCCAAACAA...BBCCC

Building BLOSUM Matrices1. 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.

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 35%

If you are comparing sequences that are very similar, useBLOSUM 80. Sequences that are more divergent (dissimilar)than 20% are given very low scores in this matrix.

Which Scoring Matrix to use?

PAM-1BLOSUM-100

Small evolutionary distanceHigh identity within short sequences

PAM-250BLOSUM-20

Large evolutionary distanceLow identity within long sequences

The PAM 250 Scoring Matrix

GCG Wisconsin Package GAPGAP 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.

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.

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