1. Computational Genomics Lecture #2b
Scoring functions and DNA and AAs
Multiple sequence alignment
Background Readings: Chapters 2.5, 2.7 in the text book, Biological Sequence Analysis, Durbin et al., 2001.
Chapters , in Introduction to Computational Molecular Biology, Setubal and Meidanis, 1997.
 Chapter 15 in Gusfield’s book.
Much of this class has been edited from Nir Friedman’s lecture which is available at Changes made by Dan Geiger, then Shlomo Moran, and finally Benny Chor.

2. Scoring Functions, Reminder
So far, we discussed dynamic programming algorithms for
global alignment
local alignment
All of these assumed a scoring function:
that determines the value of perfect matches,
substitutions, insertions, and deletions.

3. Where does the scoring function come from ?
We have defined an additive scoring function by specifying a function ( ,  ) such that
(x,y) is the score of replacing x by y
(x,-) is the score of deleting x
(-,x) is the score of inserting x
But how do we come up with the “correct” score ?
Answer: By encoding experience of what are similar sequences for the task at hand. Similarity depends on time, evolution trends, and sequence types.

4. Why probability setting is appropriate to define and interpret a scoring function ?
Similarity is probabilistic in nature because biological changes like mutation, recombination, and selection, are random events.
We could answer questions such as:
How probable it is for two sequences to be similar?
Is the similarity found significant or spurious?
How to change a similarity score when, say, mutation rate of a specific area on the chromosome becomes known ?

5. A Probabilistic Model
For starters, will focus on alignment without indels.
For now, we assume each position (nucleotide /amino-acid) is independent of other positions.
We consider two options:
M: the sequences are Matched (related)
R: the sequences are Random (unrelated)

6. Unrelated Sequences
Our random model R of unrelated sequences is simple
Each position is sampled independently from a distribution over the alphabet 
We assume there is a distribution p() that describes the probability of letters in such positions.
Then:

7. Related Sequences
We assume that each pair of aligned positions (s[i],t[i]) evolved from a common ancestor
Let q(a,b) be a distribution over pairs of letters.
q(a,b) is the probability that some ancestral letter evolved into this particular pair of letters.
Compare to:

8. Odd-Ratio Test for Alignment
If Q > 1, then the two strings s and t are more likely to
be related (M) than unrelated (R).
If Q < 1, then the two strings s and t are more likely to
be unrelated (R) than related (M).

9. Log Odd-Ratio Test for Alignment
Taking logarithm of Q yields
Score(s[i],t[i])
If log Q > 0, then s and t are more likely to be related.
If log Q < 0, then they are more likely to be unrelated.
How can we relate this quantity to a score function ?

10. Probabilistic Interpretation of Scores
We define the scoring function via
Then, the score of an alignment is the log-ratio between the two models:
Score > 0  Model is more likely
Score < 0  Random is more likely

11. Modeling Assumptions
It is important to note that this interpretation depends on our simplified modeling assumption!!
For example, if we assume that the letter in each position depends on the letter in the preceding position, then the likelihood ratio will have a different form.
If we assume, for proteins, some joint distribution of letters that are nearby in 3D space after protein folding, then likelihood ratio will again be different.

12. Estimating Probabilities
Suppose we are given a long string s[1..n] of letters from 
We want to estimate the distribution q(·) that generated the sequence
How should we go about this?
We build on the theory of parameter estimation in statistics using either maximum likelihood estimation or the Bayesian approach (later on).

13. Estimating q()
Suppose we are given a long string s[1..n] of letters from 
s can be the concatenation of all sequences in our database
We want to estimate the distribution q()
That is, q is defined per single letters
Likelihood function:

14. Estimating q() (cont.)
How do we define q? Intuitively
Likelihood function:
ML parameters
(Maximum Likelihood)

15. Estimating p(·,·) Intuition:
Find a pair of aligned sequences s[1..n], t[1..n],
Estimate probability of pairs:
The sequences s and t can be the concatenation of many aligned pairs from the database
Number of times a is
aligned with b in (s,t)

16. Problems in Estimating p(·,·)
How do we find pairs of aligned sequences?
How far is the ancestor ?
earlier divergence  low sequence similarity
recent divergence  high sequence similarity
Does one letter mutate to the other or are they both mutations of a common ancestor having yet another residue/nucleotide acid ?

17. Scoring Matrices Deal with DNA first (simpler) then AA (not too bad either)

18. What is it & why ? Let alphabet contain N letters N x N matrix
N = 4 and 20 for nucleotides and amino acids
N x N matrix
(i,j) shows the relationship between i-th and j-th letters.
Positive number if letter i is likely to mutate into letter j
Negative otherwise
Magnitude shows the degree of proximity
Symmetric

19. Scoring Matrices for DNA1 -3 A C G T 1 A C G T 1 -5 -1
Transitions &
transversions
identity
BLAST

20. The BLOSUM45 Matrix A R N D C Q E G H I L K M F P S T W Y V

21. Scoring Matrices for Amino Acids
Chemical similarities
Non-polar, Hydrophobic (G, A, V, L, I, M, F, W, P)
Polar, Hydrophilic (S, T, C, Y, N, Q)
Electrically charged (D, E, K, R, H)
Requires expert knowledge
Genetic code: Nucleotide substitutions
E: GAA, GAG
D: GAU, GAC
F: UUU, UUC
Actual substitutions
PAM
BLOSUM

22. Scoring Matrices: Actual Substitutions
Manually align proteins
Look for amino acid substitutions
Entry ~ log (freq(observed) / freq(expected))
Log-odds matrices

23. BLOSUM BLOcks Substitution Matrices
Henikoff & Henikoff, 1992
Next slides taken from lecture notes by Tamer Kahveci,
CISE Department, University of Florida
( fall2004/lectures/03-CAP5510-Fall04.ppt

24. BLOSUM Matrix
Begin with a set of protein sequences and obtain aligned blocks.
~2000 blocks from 500 families of related proteins
A block is the ungapped alignment of a highly conserved region of a family of proteins.
MOTIF program is used to find blocks
Substitutions in these blocks are used to compute BLOSUM matrix
block 1
block 2
block 3
WWYIR CASILRKIYIYGPV GVSRLRTAYGGRKNRG
WFYVR … CASILRHLYHRSPA … GVGSITKIYGGRKRNG
WYYVR AAAVARHIYLRKTV GVGRLRKVHGSTKNRG
WYFIR AASICRHLYIRSPA GIGSFEKIYGGRRRRG

25. Constructing the Matrix
Count the frequency of occurrence of each amino acid.
This gives the background distribution pa
Count the number of times amino acid a is aligned with amino acid b: fab
A block of width w and depth s contributes ws(s-1)/2 pairs.
Denote by np the total number of pairs.
Compute the occurrence probability of each pair: qab = fab/ np
Compute the expected probability of occurrence of each pair
eab = 2papb, if a ≠ b
papb otherwise
Compute twice (?) the log likelihood ratios, normalize, and round to nearest integer.
2* log2 qab / eab i
j >= i
a≠b

26. Computation of BLOSUM-X
The amount of similarity in blocks has a great effect on the BLOSUM score. BLOSUM-X is generated by taking only blocks with %X identity.
For example, a BLOSUM62 matrix is calculated from protein blocks with 62% identity.
So BLOSUM80 represents closer sequences (more recent divergence) than BLOSUM62.
On the web, Blast uses BLOSUM80, BLOSUM62 (the default), or BLOSUM45. a b

27. BLOSUM 62 Matrix Check scores for M I L V -small hydrophobic N D E Q
-acid, hydrophilic
H R K
-basic
F Y W
-aromatic
S T P A G
-small hydrophilic C
-sulphydryl

28. PAM vs. BLOSUM Equivalent PAM and BLOSSUM matrices: PAM100 = Blosum90
BLOSUM62 is the default matrix to use.

29. And Now Ladies and Gentlemen Boys and Girls the holy grail Multiple Sequence Alignment

30. Multiple Sequence Alignment
S1=AGGTC
Possible alignment A - T G C
S2=GTTCG
S3=TGAAC
Possible alignment A G - T C

31. Multiple Sequence Alignment
Aligning more than two sequences.
Definition: Given strings S1, S2, …,Sk a multiple (global) alignment map them to strings S’1, S’2, …,S’k that may contain blanks, where:
|S’1|= |S’2|=…= |S’k|
The removal of spaces from S’i leaves Si

32. Multiple alignments
We use a matrix to represent the alignment of k sequences, K=(x1,...,xk). We assume no columns consists solely of blanks.
The common scoring functions give a score to each column, and set: score(K)= ∑i score(column(i)) x1 x2 x3 x4 M Q _ I L R - K P V
For k=10, a scoring function has 2k -1 > 1000 entries to specify. The scoring function is symmetric - the order of arguments need not matter: score(I,_,I,V) = score(_,I,I,V).

33. SUM OF PAIRS M Q _ I L R - K P V
A common scoring function is SP – sum of scores of the projected pairwise alignments: SPscore(K)=∑i<j score(xi,xj). M Q _ I L R - K P V
Note that we need to specify the score(-,-) because a column may have several blanks (as long as not all entries are blanks).
In order for this score to be written as ∑i score(column(i)),
we set score(-,-) = 0. Why ?
Because these entries appear in the sum of columns but not
in the sum of projected pairwise alignments (lines).

34. SUM OF PAIRS M Q _ I L R - K P V
Definition: The sum-of-pairs (SP) value for a multiple global alignment A of k strings is the sum of the values of all projected pairwise alignments induced by A where the pairwise alignment function score(xi,xj) is additive. M Q _ I L R - K P V

35. Example Consider the following alignment: a c - c d b - 3 3 +4
- c - a d b d
= 12
a - b c d a d
Using the edit distance and for ,
this alignment has a SP value of

36. Multiple Sequence Alignment
Given k strings of length n, there is a natural generalization of the dynamic programming algorithm that finds an alignment that maximizes
SP-score(K) = ∑i<j score(xi,xj).
Instead of a 2-dimensional table, we now have a k-dimensional table to fill.
For each vector i =(i1,..,ik), compute an optimal multiple alignment for the k prefix sequences x1(1,..,i1),...,xk(1,..,ik).
The adjacent entries are those that differ in their index by one or zero. Each entry depends on 2k-1 adjacent entries.

37. The idea via K=2 V[i,j] V[i+1,j] V[i,j+1] V[i+1,j+1]
Recall the notation:
and the following recurrence for V:
V[i,j]
V[i+1,j]
V[i,j+1]
V[i+1,j+1]
Note that the new cell index (i+1,j+1) differs from previous indices by one of 2k-1 non-zero binary vectors (1,1), (1,0), (0,1).

38. Multiple Sequence Alignment
Given k strings of length n, there is a generalization of the dynamic programming algorithm that finds an optimal SP alignment.
Computational Cost:
Instead of a 2-dimensional table we now have a k-dimensional table to fill.
Each dimension’s size is n+1. Each entry depends on 2k-1 adjacent entries.
Number of evaluations of scoring function : O(2knk)

39. Complexity of the DP approach
Number of cells nk.
Number of adjacent cells O(2k).
Computation of SP score for each column(i,b) is o(k2)
Total run time is O(k22knk) which is totally unacceptable !
Maybe one can do better?

40. But MSA is Intractable
Not much hope for a polynomial algorithm because the problem has been shown to be NP complete (proof is quite
Tricky and recent. Some previous proofs were bogus).
Look at Isaac Elias presentation of NP completeness proof.
Need heuristic or approximation to reduce time.

41. Multiple Sequence Alignment – Approximation Algorithm
Now we will see an O(k2n2) multiple alignment algorithm for the SP-score that approximate
the optimal solution’s score by a factor of at most 2(1-1/k) < 2.

42. Star-score(K) = ∑j>0score(S1,Sj).
Star Alignments
Rather then summing up all pairwise alignments, select a fixed sequence S1 as a center, and set
Star-score(K) = ∑j>0score(S1,Sj).
The algorithm to find optimal alignment: at each step, add another sequence aligned with S1, keeping old gaps and possibly adding new ones.

43. Multiple Sequence Alignment – Approximation Algorithm
Polynomial time algorithm:
assumption: the function δ is a distance function:
(triangle inequality)
Let D(S,T) be the value of the minimum global alignment between S and T.

44. Multiple Sequence Alignment – Approximation Algorithm (cont.)
Polynomial time algorithm:
The input is a set Γ of k strings Si.
1. Find “center string” S1 that minimizes
2. Call the remaining strings S2, …,Sk.
3. Add a string to the multiple alignment that initially contains only S1 as follows:
Suppose S1, …,Si-1 are already aligned as S’1, …,S’i-1. Add Si by running dynamic programming algorithm on S’1 and Si to produce S’’1 and S’i.
Adjust S’2, …,S’i-1 by adding spaces to those columns where spaces were added to get S’’1 from S’1.
Replace S’1 by S’’1.

45. Multiple Sequence Alignment – Approximation Algorithm (cont.)
Time analysis:
Choosing S1 – running dynamic programming algorithm
times – O(k2n2)
When Si is added to the multiple alignment, the length of S1
is at most in, so the time to add all k strings is

46. Multiple Sequence Alignment – Approximation Algorithm (cont.)
Performance analysis:
M - The alignment produced by this algorithm.
d(i,j) - the distance M induces on the pair Si,Sj.
M* - optimal alignment.
For all i, d(1,i)=D(S1,Si)
(we performed optimal alignment between S’1 and Si and )

47. Multiple Sequence Alignment – Approximation Algorithm (cont.)
Performance
analysis:
Triangle inequality
Definition of S1

48. Multiple Sequence Alignment – Approximation Algorithm
Algorithm relies heavily on scoring function
being a distance. It produced an alignment
whose SP score is at most twice the minimum.
What if scoring function was similarity?
Can we get an efficient algorithm whose score
is half the maximum? Third of maximum? …
We dunno !

49. Tree Alignments
Assume that there is a tree T=(V,E) whose leaves are the sequences.
Associate a sequence in each internal node.
Tree-score(K) = ∑(i,j)Escore(xi,xj).
Finding the optimal assignment of sequences to the internal nodes is NP Hard.
We will meet again this problem in the study of
Phylogenetic trees (it is related to the parsimony problem).

50. Multiple Sequence Alignment Heuristics
Example - 4 sequences A, B, C, D. A. B D A C A B C D
Perform all 6 pair wise
alignments. Find scores.
Build a “similarity tree”.
distant
similar B.
Multiple alignment following the tree from A. B
Align most similar pairs allowing
gaps to optimize alignment. D A
Align the next most similar pair. C
Now, “align the alignments”, introducing gaps if
necessary to optimize alignment of (BD) with (AC).

51. (modified from Speed’s ppt presentation, see p. 81 in Kanehisa’s book)
The tree-based progressive method for multiple sequence alignment, used in practice (Clustal) (a) a tree (dendrogram) obtained by cluster analysis (b) pairwise alignment of 2 sequences’ alignments.
(a)
(b)
L W R D G R G A L Q
L W R G G R G A A Q
D W R - G R T A S G
DEHUG3
DEPGG3
DEBYG3
DEZYG3
DEBSGF
L R R - A R T A S A
L - R G A R A A A E
(modified from Speed’s ppt presentation, see p. 81 in Kanehisa’s book)

52. Visualization of Alignment Helps