1. Pairwise Sequence Alignment (cont.)
(Lecture for CS397-CXZ Algorithms in Bioinformatics)
Feb. 6, 2004
ChengXiang Zhai
Department of Computer Science
University of Illinois, Urbana-Champaign

2. 4 Basic Questions in Pairwise Alignment
(Modeling evolution)
Q1: How should we define s?
Q2: How should we define A?
(Application-specific)
Model: scoring function s: A
X=x1,…,xn
X=x1,…,xn
Possible alignments of X and Y: A ={a1,…,ak}
Find the best alignment(s) …
S(a*)= 21
Y=y1,…,ym
Y=y1,…,ym
Q4: Is the alignment biologically
Meaningful or just the best
alignment of two unrelated
sequences?
Q3: How can we find a* quickly?
(Dynamic programming)
Q1 & Q4 are related!
(Models for scores)

3. The Rest of This Lecture
Q4: How to assess the significance of an alignment score?
Classic approach: extreme value distribution
Bayesian approach: model comparison
Q1: How to define the scoring function?
Define the substitution score s
Define the gap penalty function g

4. First, Q4: Assessing Score Signficance
In general, larger s  more significant. The question is how large should s be?
Factors to be considered:
Sequence length: longer sequences are expected to give higher scores
# sequences in the database: the score of the best alignment is expected to be higher for a larger DB
Evolution time: longer evolution causes more mismatches, making a lower score more significant
The Challenge is how to quantify all these…

5. Log-odds score of the alignment
Two Basic Approaches
The classical approach: Extreme value distribution
Assume a null (random) model for scores M0
P(Score > s|M0, x, y)=?
The Bayesian approach: Model comparison
Assume two models for (x,y): random M0; aligned: M1
P(M1|x,y)/P(M0|x,y)=?
prior
Log-odds score of the alignment

6. Extreme Value Distribution
EVD: The asymptotic distribution of the maximum MN of a series of N independent normal random variables is
In general, the maximum of a large number of separate scores follows this distribution
Example: the best local match score between two long sequences
constants
mode

7. EVD of the Best Score in Ungapped Local Alignment
The number of unrelated local matches with score higher than S is approximately Poisson distributed, with mean
The probability that there is a match of score greater than S is
K and  can be fit using randomly generated data
This gives a way to test statistical significance p(x>21)= 0.01 vs. p(x>21)=0.3
Parameters
Sequence lengths

8. Bayesian Model Comparison
Assumptions:
M is a model for related sequences
R is a model for unrelated sequences (random)
Ungapped alignment n=m
Alignment of each pair is independent
Score
S(x,y)
Prior
(Subjective!)
This partially addresses Q1: how to design the scoring function?

9. Q1: How to Estimate Probabilities?
General idea: Exploit sequences with known (“reliable”) alignments
Simplest method: Max. Likelihood estimator
Improved method: Consider evolution time (phylogenetic tree, to be covered later)

10. Dayhoff PAM Matrices
Estimate p(b|a,t,M) (Substitution probabilities) rather than p(ba|M)
Use sufficiently similar sequence pairs to estimate p(b|a,t=1,M)
Compute p(b|a, t+1,M) based on p(b|a,t,M)
Compute the score matrix (e.g., PAM 250)

11. BLOSUM Matrices
Limitation of PAM: short time substitutions are dominated by trivial changes in the Codon triplets
BLOSUM tries to improve the estimation of p(ab|M,t) by re-sampling the aligned, ungapped sequences regions (e.g., based on PAM)
Time t is now connected with a threshold of sequence similarity, leading to different variations (e.g., BLOSUM50 & BLOSUM62)

12. Estimating Gap Penalties
Again the basic idea is to exploit known alignments
Basic assumptions:
The gap-open score d is linear in log(t)
The gap-extend score e is constant
Example: (g)=A+B*log(t)+C*log(g)
In practice, people choose the gap costs empirically for given substitution scores.