Evolution and Scoring Rules Example Score = 5 x (# matches) + (-4) x (# mismatches) + + (-7) x (total length of all gaps) Example Score = 5 x (# matches) + (-4) x (# mismatches) + + (-5) x (# gap openings) + (-2) x (total length of all gaps)

Scoring Matrices

Scoring Rules vs. Scoring Matrices Nucleotide vs. Amino Acid Sequence The choice of a scoring rule can strongly influence the outcome of sequence analysis Scoring matrices implicitly represent a particular theory of evolution Elements of the matrices specify the similarity of one residue to another

DNA: A T G C 1:1 RNA: A U G C 3:1 Protein: 20 amino acids Transcription Translation Replication Translation - Protein Synthesis: Every 3 nucleotides (codon) are translated into one amino acid

Nucleotide sequence determines the amino acid sequence

Translation - Protein Synthesis 5’ -> 3’ : N-term -> C-term RNA Protein

Log Likelihoods used as Scoring Matrices: PAM - % Accepted Mutations: 1500 changes in 71 groups w/ > 85% similarity BLOSUM – Blocks Substitution Matrix: 2000 “blocks” from 500 families

Log Likelihoods used as Scoring Matrices: BLOSUM

Likelihood Ratio for Aligning a Single Pair of Residues Above: the probability that two residues are aligned by evolutionary descent Below: the probability that they are aligned by chance Pi, Pj are frequencies of residue i and j in all protein sequences (abundance)

Likelihood Ratio of Aligning Two Sequences

The alignment score of aligning two sequences is the log likelihood ratio of the alignment under two models Common ancestry By chance

PAM and BLOSUM matrices are all log likelihood matrices More specificly: An alignment that scores 6 means that the alignment by common ancestry is 2^(6/2)=8 times as likely as expected by chance.

BLOSUM matrices for Protein S. Henikoff and J. Henikoff (1992). “Amino acid substitution matrices from protein blocks”. PNAS 89: Training Data: ~2000 conserved blocks from BLOCKS database. Ungapped, aligned protein segments. Each block represents a conserved region of a protein family

Constructing BLOSUM Matrices of Specific Similarities Sets of sequences have widely varying similarity. Sequences with above a threshold similarity are clustered. If clustering threshold is 62%, final matrix is BLOSUM62

A toy example of constructing a BLOSUM matrix from 4 training sequences

Constructing a BLOSUM matr. 1. Counting mutations

Constructing a BLOSUM matr. 2. Tallying mutation frequencies

Constructing a BLOSUM matr. 3. Matrix of mutation probs.

4. Calculate abundance of each residue (Marginal prob)

5. Obtaining a BLOSUM matrix

Constructing the real BLOSUM62 Matrix

1.2.3.Mutation Frequency Table

4. Calculate Amino Acid Abundance

5. Obtaining BLOSUM62 Matrix

PAM Matrices (Point Accepted Mutations) Mutations accepted by natural selection

PAM Matrices Accepted Point Mutation Atlas of Protein Sequence and Structure, Suppl 3, 1978, M.O. Dayhoff. ed. National Biomedical Research Foundation, 1 Based on evolutionary principles

Constructing PAM Matrix: Training Data

PAM: Phylogenetic Tree

PAM: Accepted Point Mutation

Mutability

Total Mutation Rate is the total mutation rate of all amino acids

Normalize Total Mutation Rate

Mutation Probability Matrix Normalized Such that the Total Mutation Rate is 1%

Mutation Probability Matrix (transposed) M*10000

-- PAM1 mutation prob. matr. --PAM2 Mutation Probability Matrix? -- Mutations that happen in twice the evolution period of that for a PAM1

PAM Matrix: Assumptions

In two PAM1 periods: {A  R} = {A  A and A  R} or {A  N and N  R} or {A  D and D  R} or … or {A  V and V  R}

Entries in a PAM-2 Mut. Prob. Matr.

PAM-k Mutation Prob. Matrix

PAM-1 log likelihood matrix

PAM-k log likelihood matrix

PAM-250

PAM60—60%, PAM80—50%, PAM120—40% PAM-250 matrix provides a better scoring alignment than lower-numbered PAM matrices for proteins of 14-27% similarity

Sources of Error in PAM

Comparing Scoring Matrix PAM Based on extrapolation of a small evol. Period Track evolutionary origins Homologous seq.s during evolution BLOSUM Based on a range of evol. Periods Conserved blocks Find conserved domains

Choice of Scoring Matrix

Global Alignment with Affine Gaps Complex Dynamic Programming

Problem w/ Independent Gap Penalties The occurrence of x consecutive deletions/insertions is more likely than the occurrence of x isolated mutations We should penalize x long gap less than x times of the penalty for one gap

Affine Gap Penalty w2 is the penalty for each gap w1 is the _extra_ penalty for the 1 st gap

Scoring Rule not Additive! We need to know if the current gap is a new gap or the continuation of an existing gap Use three Dynamic Programming matrices to keep track of the previous step

S1 is the vertical sequence S2 is the horizontal sequence (From Diagonal) a(i,j): current position is a match (From Left) b(i,j): current position is a gap in S1 (From Above) c(i,j): current position is a gap in S2 Filling the next element in each matrix depends on the previous step, which is stored in the three matrices.

Last step a match a gap in S2 a gap in S1 new gap in S2 a continued gap in S2 a gap in S2 following a gap in S1

Decisions in Seq. Alignment Local or global alignment? Which program to use Type of scoring matrix Value of gap penalty

A ij *10

PAM-k log-likelihood matrix