Amino Acid Scoring Matrices

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Amino Acid Scoring Matrices

• Jason Davis

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Overview

• Protein synthesis/evolution
• Computational sequence alignment
• Smith-Waterman Algorithm
• BLAST
• Amino Acid Scoring Matrices
• PAM Point Accepted Mutations
• BLOSUM BLOck SUbstitution Matrix
• mPAM
• Metric Conversions

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Proteins

• 3-dimensional stuctures
• Composed of amino acids chained together
• Can be represented as a 2-dimensional sequence
• 20 different amino acids exist
• Usually 100-1500 amino acids long
• Have many different shapes and functions
• Function depends on both 3d shape and aa sequence

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Protein Synthesis

• DNA strand composed of 4 different base pairs
• A, T, C, G
• 20 amino acids 3 base pairs needed to encode
  each amino acid
• Degenerate coding

Protein
Signalling
Transcription/Translation
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Protein Evolution

• Protein families
• Set of homologous proteins
• Same function, different composition
• Similar structure
• Identifying families
• Pairwise sequence alignment
• Multiple sequence alignment
• NP-hard
• Other approaches
• Structural, experimental

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Pairwise Sequence Alignment

• Input
• 2 sequences p, q of lengths m,n
• 20x20 Amino Acid Substitution Matrix
• Insertion (gap) cost
• Global Alignment
• Find optimal set of insertions such that the
  resulting alignment (length lt mn) is optimal
  w.r.t. amino acid substitution matrix
• Difficult, less useful
• Local Alignment
• Find significant hotspot in the alignment

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Sequence Alignment Algorithms

• Dynamic Programming Approaches
• Global and Local variations
• Provably Optimal
• O(nm) space and time
• banded heuristics can reduce the state space
• FSA extensions allow varying penalties for gap
  openings and gap extensions
• Heuristics Approaches
• Blast, Fasta
• Sublinear time look for statistical
  significance in small local alignments between
  sequences

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Substitution Matrices - PAM

• Dayhoff, Schwartz, Orcutt (1978)
• Step 1 extrapolate mutation probabilites from 1
  step in evolutionary time
• Pick a set of protein families (71)
• Restrict proteins in each family to sequences
  with similarity above a certain threshold (gt85)
• Build a phylogenetic tree for each family
• Extrapolate frequencies Aab that amino acids a, b
  evolved from same amino acid
• Aab and Aba assumed to be the same
• Convert frequencies to probabilities
• p(ab) Bab Aab/?cAac

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Substitution Matrices PAM (2)

• Step 2 Infer greater evolutionary times
• Dayhoff defined a PAM1 matrix to have 1 expected
  substitutions
• For each row, scale off-diagonals and adjust
  diagonals to keep the matrix row stochastic
• To infer larger evolutionary times, we can view
  formed matrix C as a 20-state Markov Chain
• Cn is the result of performing n-steps in the
  Markov Process

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Substitution Matrices PAM (3)

• Create odds ratio of
• 1) the event that 2 amino acids i,j, evolved from
  the same ancestor, x
• fi observed frequency of amino acid i
• p(i,j have same ancestor) ?xfx Prx?i
  Prx?j ?xfx (CN)ix (CN)jx
  ?x (CN)ix fx (CN)jx ?x (CN)ix fj
  (CN)xj fj (C2N)ij
• 2) the event that the 2 amino acids align at
  random
• p(independent alignment of i,j) fi fj
• Final log odds ratio
• Dij averagelog((CN)ij / fi), log(CN)ji / fj))
• The log allows for an additive model
• Final numbers are rounded to nearest integer

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PAM250

• Different values on the diagonal correspond do
  mutability potential

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BLOSUM

• Henikoff Henikoff, 1992
• Uses aligned, ungapped blocks within protein
  families that have similarity greater than some
  level L
• qa ?bAab / ?c,d Acd
• pab Aab / ?c,d Acd
• S(a,b) log(pab / qaqb)
• Final entries are rounded
• Blosum62 (L62), Blosum50 (L50)
• More direct approach, usually yields better
  results

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Log-Odds Similarity Matrix Properties

• Negative numbers needed for Smith-Waterman local
  alignment algorithm
• Nice probabilistic interpretation
• Amino acid substitutions assumed independent
• Attempts to metricize these matrices
• Taylor, Jones 93 used various algebraic
  manipulations to arrive at a metric matrix with
  minimal disortion
• Dij a Sij
• Larger values of a yielded better metrics at the
  cost of high dimensionality
• Constant Shift Embedding
• Linial, et. al. constructed a near metric over
  aligned segments of length 50
• D(u,v) S(u,u) S(v,v) 2S(u,v)
• 10-7 error rate

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mPAM

• Metric substitution model
• Measures the expected time per 250 mutations
  among 100 amino acids
• Same rate as PAM250
• Exponential distribution assumed f(t) 1 e-?t
• Given pairwise substitution rates p(a,b)
• Solve for ? f(1) 1-e- ? p(a,b)
• Expected time t of an event occuring in an
  exponential distribution is 1/ ?
• mPAM(a,b) round(1/ ?)
• Two values needed to be adjusted to form a metric
• Rounding error?

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mPAM (2)

• Sellers Theorem
• If a pairwise alignment is found using a metric,
  resulting alignment scores are also metrics
• Optimized for BLAST-like lookup
• Smaller alignments
• Difficult to compare with other similarity
  matrices
• Dynamic programming algorithms rely on negative
  values in the similarity matrix
• Probabilistic interpretation larger positive
  alignments are statistically significant

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mPAM Disadvantages

• d(x,x) 0
• This does not capture the relative mutability
  among different amino acids
• PAM/BLOSUM capture this with different positive
  values along the diagonal
• Do amino acids substitute according to an
  exponential distribution?
• Amino Acid Substitution may be inherently
  non-metric
• Comparison to BLOSUM?