Scoring matrices

1
Scoring matrices

• Identity
• PAM
• BLOSUM

2
Scoring Matrices Types

• Identity matrix exact matches receive one score
  and non-exat matches a different score (say 1 and
  0, or 6 and 1 for local alignment.).
• Mutation data matrix a scoring matrix compiled
  based on observation of protein point mutation
  (PAM, BLOSUM).
• Physical properties matrix amino acids with
  with similar properties (e.G. hydrophobicity )
  receive high score.
• Genetic code matrix amino acids are scored
  based on similarities in the coding triple
  (codons).

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Substitution Matrix

• Amino acids substitute easily for another due to
  similar physicochemical properties
• Isoleucine for Valine (both small, hydrophobic)
• Serine for Threonine (both polar)
• Such changes conservative
• Thus, need a way to increase sensitivity of the
  alignment algorithm
• Solution substitution matrix
• Therefore, we need a range of values that depend
  on the nature of sequences being compared
• Identical amino acids gt Conservative
  substitutions gt Nonconservative substitutions

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Choice of scoring matrix is dictated by the
alignment goals

• Two proteins are homologous if (and only if) they
  are evolutionarily related (have a common
  ancestor)
• Homologous proteins are likely to have related
  functions (and have the same fold)
• Scoring matrices must in some way model our
  understanding of protein evolution.
• Based on the result of the search we have to be
  able to decide if the discovered sequence
  similarity could happen by chance or is a
  signature of likely homology.

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BLOSUM

• Block a short contiguous interval of multiple
  aligned sequences.
• BLOCKS data base of 3 000 blocks of highly
  conserved sequences representing hundreds of
  protein groups.
• Http//www.Blocks.fhcrc.Org/.
• BLOCKS ? substitutions frequency ? log odds
  score.
• Within each block cluster sequences within
  certain similarity threshold (80 similarity
  yields BLOSUM80) and have such cluster be
  represented by one sequence or average the
  contribution.
• BLOSUM62 most similar to PAM250 (believed to be
  better).

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BLOSUM METHOD
Data Base of blocks
Data base
?
?
7
Methods

• Deriving a frequency table from a data base of
  blocks.

Frequency table consisting of all possible amino
acid pairs in a column

• 9A 1S there are 87136 AA pairs
• 9 AS or SA pairs
• no SS pairs

For a block width of w and a depth of S, it
contribute WS(S-1)/2 ?1.10.(10-1)/245
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METHODS

• The result of this counting is a frequency table
  listing the number of time each of the
  20191210 different amino acid pairs occurs
  among the blocks.
• The table is used to calculate a matrix
  representing odds ratio between these observed
  frequency and those calculated by chance.

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METHODS

• Observed probability qij

?
fAA 36, fAS 9 qAA 36/45 0.8 qAS 9/45
0.2
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Methods

• Expected probability eij

?
pA 36 (9/2)/45 0.9 pS 00 (9/2) /45
0.1

• for ij ? eij pi.pj
• eAA pA.pA 0.9 x 0.9 0.81
• for i?j ? eij pi.pj pi.pj 2 pi.pj
• eAS pA.pS pA.pS 2 pA.pS 2 (0.9 x
  0.1) 0.18

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Methods

• The odds ratio

• An odds ratio matrix is calculated where each
  entry is qij/eij
• The logarithm of odds ratio (Lod) in bit unit
• Sij log2qij/eij
• A Lod is then calculated as score
• If the observed frequency is
• as the expected, then Sij 0
• if less than expected Sij lt 0
• if more than expected Sij gt 0

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METHODS

• Clustering segment within blocks

• Sequences are clustered within blocks, and each
  cluster is weighted. This is done by specifying a
  clustering percentage in which sequence segments
  that are identical for at least that percentage
  of amino acids are grouped together.
• The lod matrix derived from a database of blocks
  in which sequences that are identical at ? 80 of
  aligned residues are clustered is referred to as
  BLOSUM 80, and so forth.

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The Dayhoff Matrix (PAM)

• Developed by Margaret Dayhoff, 1978.
• Counted likelihood of all possible substitutions
  in closely related proteins.
• Derived mutability matrix Mi,j
• Probability that Ai mutates to Aj in one
  evolutionary unit, PAM.
• Multiplying M by itself extrapolate to higher
  evolutionary orders (Mk).

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PAM units

• Log-odds approach Scores proportional to the log
  of the ratio of target frequencies to background
  frequencies
• PAM Point Accepted Mutation /Percent Accepted
  Mutation
• Two sequences S and T are defined to be one PAM
  unit diverged if a series of accepted point
  mutation (and no insertion/deletion) can convert
  S to T with an average of one mutation per 100
  res.
• Point accepted mutation mutation of one residue
  accepted by evolution.

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PAM units

• Problem 1 given two sequences you cannot tell
  their PAM distance in the strict sense of the
  above definition since one residue could mutate
  more than once
• BUT If you take sequences that are closely
  related then problem above is unlikely to occur.
• Problem 2 A change could happen by
  deletion/insertion

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PAM Matrices - Summary

• There is a sequence of PAM matrices
• PAMn attempts to provide proper scoring for
  sequences that diverged n PAM units.
• PAMn matrix is obtained from PAM1 assuming Markov
  model of protein evolution where transition
  probabilities in 1 PAM step are given by PAM1.
• PAMn PAM1 n
• PAM1 is constructed based on highly similar
  sequences (believed to be apart at most few PAM
  units) so that Problems1 2 are unlikely to
  occur.)

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Computation representation

• Define
• fp(a) probabilities of occurrence for each
  amino acid a.
• f(a,b) the number of times the mutation a?b
  ( f(a,b) f(b,a) )
• f(a) b?f(a,b) ( b?a )
• m(a) mutability of amino acid a f(a) / fp(a)

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Computation representation ,cnd

• M(a,b) the probability of amino acid a changing
  to amino acid b
• M(a,b) Pr(a?b)
• Pr(a?b a changed)Pr(a changed)
• f(a,b) m(a) / f(a)
• (the conditional probability above is
  estimated as the ratio between the a?b mutations
  and the total number of mutations involving a )
• M(a,a) 1- m(a) unchange probability
• (the diagonal
  elements)

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Relatedness odds Matrix

• M(a,b) gives the probability that amino acid a
  will change to b in a related sequence in a
  interval
• f(b) is the chance of a random occurrence of
  amino acid b
• Score(a,b) 10logM(a,b)/f(b)
• (symmetric matrix)

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PAM

• Let us assume to AA (or nucleotides) i and j,
  with frequency fi and fj.
• P(random alignment of i and j)fi fj.

                      
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PAM
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Long Distance Evolution

• There is a different mutation probability matrix
  for each evolutionary interval. These can be
  derived from the one for 1 PAM by matrix
  multiplication.
• e.g.
• in 2 PAM units of evolution
• a?c?b (c can be anything including a or
  b)
• In general Mn is the transition probability
  matrix for a period of n units of evolution

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Estimation of Evolutionary Distance

• Different mutation probability matrix for each
  evolutionary interval measured in PAMs.
• Calculate the percentage of amino acids that will
  be observed to change on the average in the
  interval
• P 100(1 ?f(i)M(i,i))
• A PAM250 matrix usually represents two sequences
  which have about 20 identity

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Nucleotide PAM scoring matrices
Assuming equal probability for each mutation PAM1
would be A T G C A
.99 .0033 .0033 .0033 T .0033 .99 .0033
.0033 G .0033 .0033 .99 .0033C .0033
.0033 .0033 .99 Some models would score higher
transitions (purine into purine pirimidine into
pirimidine) that transversions A T
G C A .99 .0002 .0006
.0002 T .0002 .99 .0002 .0006 G .0006
.0002 .99 .0002C .0002 .0006 .0002 .99
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Discrimination of real local alignment from by
chance alignment

• Method Compute mutual information
• Sx Syp(x,y) log (p(x,y)/ p(x)p(y))
• Recall that score s(x,y) log (p(x,y)/ p(x)p(y))
• Thus we simply compute
• Sx1..20 Sy1,..20 p(x,y) s(x,y)
• Examples (in bits)
• PAM160 .7 PAM250 .36
• Higher mutual information ? better discrimination
  between true and by chance alignment.

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Problems with PAM

• Defining PAM 1 in terms of amino acid mutation
  rather than number of nucleotide changes.
• Some mutation may be rare and underrepresented in
  PAM1 (which is based on closely related proteins
  only).
• The mutation rate depends on the position of an
  amino-acid in the structure.
• Require construction phylogenic tree which in
  turn need scoring matrices for proper
  construction. (remains a problem for many other
  methods)

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Some more problems with PAM Matrices

• Derived from global alignments of closely related
  sequences.
• Matrices for greater evolutionary distances are
  extrapolated from those for lesser ones.
• The number with the matrix (PAM40, PAM100) refers
  to the evolutionary distance greater numbers are
  greater distances.
• Does not take into account different evolutionary
  rates between conserved and non-conserved
  regions.

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BLOSUM matrices

• BLOcks SUbstitution Matrix
• Amino acid substitution matrices from protein
  blocks

S. HENIKOFF and J. HENIKOFF Proc. Natl. Acad.
Sci.USA Vol.89, pp. 10915-10919, November
1992 Biochmistry
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Comparison to PAM

• The BLOSUN series derived from alignments in
  blocks is fundamentally different from the
  Dayhoff PAM series, which is derived from the
  estimation of mutation rates.
• Nevertheless, the BLOSUM series based on percent
  clustering of aligned segments in blocks, can be
  compared to the Dayhoff matrices based on percent
  accepted mutation (PAM) using the measure of
  average information per residue pair in bits
  units called relative entropy.

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Comparison between BLOSUM 62 and PAM 160

• The BLOSUM 62 is less tolerant to substitutions
  involving hydrophilic amino acids, while it is
  more tolerant to substitutions involving
  hydrophobic amino acids.
• For rare amino acids especially cysteine and
  tryptophane, BLOSUM 62 is typically more tolerant
  to mismatches than is PAM 160.

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PAM vs BLOSUM

• Dayhoff estimated mutation rates from
  substitutions observed in closely related
  proteins and extrapolated those rates to models
  distant relationships.
• In BLOSUM approach, frequencies were obtained
  directly from relationships represented in the
  block, regardless of evolutionary distance.
• The Dayhoff frequency table included 36 pairs in
  which no accepted point mutations.

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Differences Between the PAM and BLOSUM Approach

• In contrast, the pairs counted with BLOSUM,
  included no fewer than 2369 occurrences of any
  particular substitution.
• The BLOSUM matrices depend only on the identity
  and composition of groups protein in Prosite.
• Therefore, there is no expectation that these
  substitution matrices will change significantly
  in the future.

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PAM Versus BLOSUM

• PAM is based on an evolutionary model.
• BLOSUM is based on protein families.
• PAM is based on global alignment.
• BLOSUM is based on local alignment.