Alignments, Matrices

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Alignments, Matrices Markov Models

• Chris Bailey
• Bacterial Pathogenesis Genomics Unit
• cmb036_at_bham.ac.uk

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The Matrix

• When analysing nucleotide sequences
• Nucleotides match (GG)
• Or they dont (G?C)
• One nucleotide substitution is more relevant than
  another
• (Except in the light of what that nucleotide
  sequence ends up coding for)

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The Matrix

• But in proteins
• Some amino acids are readily substitutable (i.e.
  1 hydrophobic residue for another)
• And others are badly tolerated (i.e. 1
  hydrophobic residue for a charged residue)
• Assuming you want the 2º 3º structure to be the
  same

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Accommodating for Change

• So we adapt our scoring function (s)
• So that scores for matches and mismatches take
  account of amino acid substitutability
• We do this using protein substitution matrices

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Whats a matrix

• Lookup grid (20x20), with each row/column
  containing an amino acid

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Whats a Matrix

• Values in the matrix are scores of one amino acid
  vs another
• For example
• Big positive score amino acid 1 is readily
  substitutable for amino acid 2
• Big negative score amino acid 1 is not
  substitutable for amino acid 3

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Part of a Matrix
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How do we fill the table?

• Obvious what the values should represent (protein
  similarity)
• But how do we calculate the actual numbers
• Since we are looking for homology an evolutionary
  model is a good place to start

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Why look at evolution?

• Homology is a binary property
• Similarity ? Homology
• Homology indicates two proteins had a common
  ancestor

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The PAM Matrix

• PAM Point Accepted Mutation
• Construct 71 phylogenetic trees of protein
  families
• Observe amino acid substitutions on each branch
  of tree
• Also need probability of occurrence for each
  amino acid (pa)

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The PAM Matrix

• Using substitution data calculate fab the
  observed frequency of the mutation a ? b
• Also note that fab fba
• Using this information calculate fa, the total
  number of mutations in which a involved

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The PAM Matrix

• And also calculate f, the total occurences of
  amino acid substitutions
• From here we go on to calculate relative
  mutability

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The PAM Matrix

• Relative mutability Probability that a given
  amino acid will change in the evolutionary period
  of interest
• Now we calculate the matrix

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The PAM Matrix

• 20 x 20 Matrix where Mab is the probability of
  amino acid a changing into amino acid b
• Maa 1 ma
• Mab is more complicated requires conditional
  probability
• E.g. P(A and B) P(A)P(BA)

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The PAM Matrix

• In this case
• Or

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The PAM Matrix

• These equations allow us to calculate a PAM1
  matrix
• The number after PAM is the number of amino acid
  substitutions per 100 residues
• PAM40 40 substitutions per 100 residues
• PAM250 250 substitutions per 100 residues
• All matrices calculated by multiplication of PAM1
  matrix

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The PAM matrix

• The final scores in a PAM matrix are expressed as
  a lod (logarithm of odds) score
• Compare probability of mutation vs probability of
  random occurrence
• Gives odds ratio
• Scoring Matrix S is calculated by

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The full PAM 250 matrix
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The BLOSUM Matrix

• Derived using similar mathematical principals as
  a PAM matrix
• However substitution data is derived in a
  different manner

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The BLOSUM Matrix

• Substitution data comes from multiple alignments.
  
• Straightforward count of number of substitutions
  in the alignement
• Number after BLOSUM (e.g. 62) denotes the minimum
  level (as a age) of similarity between sequences
  within the alignments.

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BLAST

• The Basic Local Alignment Search Tool
• It is a Heuristic (i.e. it does not guarantee
  optimal results)
• Why

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BLAST

• Genbank currently stores about 1010 residues of
  protein data.
• Trying to form alignments against such a huge
  database is unfeasible (even with vast computing
  power)
• So we need to shortcut

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PAM Vs BLOSUM

• PAM100 ? BLOSUM90
• PAM120 ? BLOSUM80
• PAM160 ? BLOSUM60
• PAM200 ? BLOSUM52
• PAM250 ? BLOSUM45

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How BLAST Works

• 3 steps
• Compile a list of High Scoring words
• Search for hits hits give seeds
• Extend seeds

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Step 1 - Preprocessing

• BLAST creates a series of words, of length W
  where
• W is 2..4 for proteins
• W is gt10 for DNA
• These words are based on subsequences of the
  query (Q)

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Step 1 - Preprocessing

• Get each subsequence of Q, that is of length W
• E.g.
• LVNRKPVVP
• LVN
• VNR
• NRK
• RKP
• etc

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Step 1 - Preprocessing

• For each of these words, find similar words
• How does blast define similarity
• Uses scoring matrices to score 2 words
• E.g.
• W1 R K P
• W2 R R P
• 9 -1 7 Total 15

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Step 1 - Preprocessing

• Words are similar if their score is greater than
  a value T (T 12 usually)
• For RKP the following are examples of high
  scoring words
• QKP, KKP, RQP, REP, RRP, RKP

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Step 2 Looking for Hits

• Formatdb create a hash lookup table
• E.g. KKP ? 12054345, 23451635, 23452152
• Maps each word to an entry in the database of
  proteins
• Allows us to retrieve sequences which has a word
  match in constant time

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Step 3 Extending the matches

• Starting at the word match, extend the alignment
  in both directions
• Alignment scored using an adapted smith-waterman
  algorithm
• Alignment stopped once score has dropped below
  the specified threshold

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Multiple alignments

• The problem
• Pairwise alignment requires n2 time
• Multiple alignment requires nx where x is the
  number of sequences to align

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Progressive alignment

• Pairwise alignment of each combination of
  sequence pairs (requires xn2 time)
• Use alignment scores to produce dendogram using
  neighbour-joining method
• Align sequences sequentially, using relationships
  from tree

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Progressive alignment

• For 5 sequences align 1v2, 1v3 4v5
• Make a tree

1
3
4
5
2
1
3
4
5
2
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Progressive alignment
1
3
1.
4
2.
5
1
3
4
3.
5
1
3
4
4.
5
2
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Hidden Markov models

• Prediction tool
• Two related things X and Y
• Using information about X and Y to build model
• Predict Y using X or vice versa

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Markov Models

• Defines a series of states and the probability of
  moving to another state
• Eg. The weather

Sun
Rain
Cloud
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Markov Models

• Example probabilities

• This is a state transition matrix (A)

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Markov Models

• Initialise the system use a vector of initial
  probabilities (? vector)
• In this case the type of weather on day 1.

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Hidden Markov Models

• Use something observable to predict something
  hidden
• E.g. Barometric pressure and the weather (if you
  couldnt see outside)
• Every observable state is connected to every
  hidden state

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Hidden Markov Models
Observable
28 Hg
31 Hg
29 Hg
30 Hg
Hidden
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Hidden Markov Models

• Also need a probability matrix for connections
  between observable hidden states (confusion
  matrix) (B)

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Hidden Markov Models

• You can do many different things using HMMs
• Match a series of observation to a HMM
  (Evaluation)
• Determine the hidden sequence most likely to have
  generated a sequence of observations (Decoding)
• Determining the parameters (p, A and B) most
  likely to have generates a sequence of
  observations (Learning)

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Hidden Markov models

• There are a series of algorithms that you can use
  to
• Evaluate (Forward algorithm)
• Decode (Viterbi algorithm)
• Learn (Forward-Backward algorithm)

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Hidden Markov Models