1. Computational Biology Lecture #6: Matching and Alignment
Bud Mishra
Professor of Computer Science and Mathematics
10 ¦ 22 ¦ 2001
12/1/2018
©Bud Mishra, 2001

2. Inexact Matching ©Bud Mishra, 2001 Example: Edit Distance Problem:
Edit distance between two biological sequences
May correspond to:
Evolutionary Distance
Functional Distance
Structural Distance
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©Bud Mishra, 2001

3. Edit Distance ©Bud Mishra, 2001
Simplest distance function corresponds to:
EDIT DISTANCE
Atomic Edit Functions:
Insertion AATCGG a AATACGG
Deletion AATACGG a AATCGG
Substitution AATCGG a AATAGG
A composite edit function
' Function Composition of Atomic Edit Functions
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©Bud Mishra, 2001

4. Cost of a Composite Edit Function
(Based on the cost or distance for Atomic Edit Functions)
Given: Two strings S1 and S2
Distance(S1, S2) = min { cost(E) | E(S1) = S2}
where E = composite edit function mapping S1 to S2.
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©Bud Mishra, 2001

5. Some Properties of Distance Function
Assume:
(8 e= Atomic Edit Function) cost(e) = cost(e-1)
Distance(S1, S2) = Distance(S2, S1) Symmetric
Distance(S1, S1) = 0
Distance(S1, S2) + Distance(S2, S3) 5 Distance(S1, S3)
Triangle Inequality
Simplest Cost Function:
Each atomic edit function is of unit cost.
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©Bud Mishra, 2001

6. Edit Operations ©Bud Mishra, 2001
I: Insertion of a character into the first string S1
D: Deletion of a character from the first string S1
R: Replacement (or Substitution) of a character in the first string S1 with a character in the second string S2
M: Matching (Identity)
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7. Edit Transcript ©Bud Mishra, 2001 Example: GATT A CA | | | | |
The complete edit function is described by an “edit transcript”
EDIT TRANSCRIPT
= s 2 {D, M, R, I}*
Example (in left):
Edit transcript = DDMIRIMMII
Edit Distance = =7
GATT A CA
| | | | |
TTCGGCATT D 1 2 MM I 3 R 4 5 6 7
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©Bud Mishra, 2001

8. Levenshtein Distance or Edit Distance
Edit Distance between two strings S1 and S2 is defined as the minimum number of atomic edit operations – insertions, deletions (indels), and substitutions – needed to transform the first string into the second
Optimal Transcript = An edit transcript corresponding to the minimum number of atomic edit operations of unit cost.
EDP
The Edit Distance Problem
is to compute
the edit distance between two given strings, along with
an optimal edit transcript that describes the transformation (alignment)
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©Bud Mishra, 2001

9. Dynamic Programming Calculation of Edit Distance
Define:
D(i,j) ´ Min number of atomic edit operations needed to transform the first i characters of S1 into the first j characters of S2
´ EditDistance(S1[1..i], S2[1..j])
|S1| = n |S2| = m
Distance(S1, S2) = D(n,m)
Dynamic Programming: 3 components:
Recurrence Relation
Tabular Computation
Traceback
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©Bud Mishra, 2001

10. Recurrence ©Bud Mishra, 2001 D(i-1,j-1) D(i-1,j) D(i,j-1) D(i,j)
Base Relation:
D(0,0) = 0
EditDistance(l,l) = 0
Recurrence Relations:
In 1 coordinate:
D(i,0) = D(i-1,0)+1 (S1[i] deleted)
D(0,j) = D(0,j-1)+1 (S2[j] inserted)
EditDistance(S1[1..i], l) =i (i deletions)
EditDistance(l, S2[1..j]) =j (j insertions)
In both coordiates:
D(i,j) = min {D(i-1,j) +1, (S1[i] deleted)
{D(i,j-1)+1, (S2[j] inserted)
{D(i-1,j-1)+ D(i,j)} (substn or match)
D(i,j) = 1, if S1[i] ¹ S2[j]; 0 otherwise.
D(i-1,j-1)
D(i-1,j)
D(i,j-1)
D(i,j)
D(i,j) +1 +1
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©Bud Mishra, 2001

11. Efficient Tabular Computation of Edit Distance
Recursive Implementation a 2O(n+m)-time computation
Bottom-up computation
(n+1)£ (m+1) distinct values for D(i,j) to be computed
Dynamic Programming Table of size (n+1)£ (m+1)
String S_1 corresponds to the rows (Vertical Axis)
String S_2 corresponds to the columns (Horizontal Axis)
Fill out D(i,0) Ã First Column
Fill out D(0,j) Ã First Row
Fill out rows D(i,j) Ã Left-to-Right (increasing i)
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©Bud Mishra, 2001

12. The Algorithm ©Bud Mishra, 2001 Time complexity = O(nm)
for i=0 to n do
D(i,0) Ã i;
for j=0 to m do
D(0,j) Ã j;
for i=1 to n do
for j=1 to m do
D(i,j) Ã min[ D(i-1,j)+1,
D(i,j-1)+1,
D(i-1,j-1) + D(i,j)]
Time complexity = O(nm)
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13. Example ©Bud Mishra, 2001 S2 ! S1 ! A T C G Solutions with minimal
edit distance =4
(sol.1) T A-GGC
AATCGG
(Edit Script=
RMIRMR)
(sol.2) - -TAGGC
AATCGG -
(Edit Script =
IIMRMMD) A T C G 1 2 3 4 5 6
S1 !
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14. Trace Back ©Bud Mishra, 2001 Extracting Optimal Edit Transcript:
Set a pointer from:
Cell(i,j) ! Cell(i,j-1), if D(i,j) = D(i,j-1)+1
Horizontal Edge ) I, Insertion
Cell(i,j) ! Cell(i-1,j), if D(i,j) = D(i-1,j)+1
Vertical Edge ) D, Deletion
Cell(i,j) ! Cell(i-1,j-1), if D(i,j) = D(i-1,j-1)+D(i,j)
Diagonal Edge ) R, Substitution, if D(i,j)=1
M. Match, if D(i,j) = 0.
Optimal Edit Transcript can be computed in O(n+m) additional time.
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15. GAPS: The Scoring Model
Basic operations:
Sequencing Errors or Evolutionary processes of Mutations and Selections
Substitution: Changes one base to another.
Gaps: Insertions or Deletions:
Adds or removes a base.
Total Score Assigned to an Alignment=
Sum of terms for each aligned pair of bases plus terms for each gap.
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©Bud Mishra, 2001

16. Total Score of an Alignment with Gaps
Total Score Assigned to an Alignment
Corresponds to log of the
Relative likelihood that the two sequences are related compared to being unrelated.
Assumptions:
Mutations or Sequencing Errors at different sites in a sequence occur independently.
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17. Substitution Matrices
Notation: x and y ´ Pairs of sequences,
|x| = n and |y| = m.
x, y 2 (A+G+C+T)*
xi = ith symbol in x
yj = jth symbol in y
Random Model, R:
P(x,y|R) = Õ qxi Õ qyj
qa = probability that the letter “a” occurs independently at a given site.
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18. Random Model vs. Alternative Model
Alternative Model, M:
P(x,y | M) = Õ pxi, yj
pab = Probability that the letters “a” and “b” have each been derived independently from some common letter.
Log-Odds Ratio (LOD):
s(a,b) = ln (pab/qa qb)
P(x,y|M)/P(x,y|R) = Õ (pxi yi/ qxi qyi)
= Õ exp[s(xi,yi)] = exp[å s(xi, yi)]
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19. Score ©Bud Mishra, 2001 Score = ln [ P(x,y|M)/P(x,y|R) ]
= å s(xi, yi) = s(x,y)
Score Matrix or Substitution Matrix: A T C G 2 -1
Blossum 50
PAM
(Point Accepted Mutation)
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20. 1-PAM Matrix ©Bud Mishra, 2001
Let M be a probability transition matrix. Mab = Pr(a , b),
a, b = chracters
pa = Pr(“a” occurs in a string)
fab = The number of times the mutation a , b was observed to occur.
fa = åa ¹ b fab & f = å fa.
K = 1-PAM Evolutionary distance
“The amount of evolution that will change 1 in K characters on average.”
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©Bud Mishra, 2001

21. Maa = 1 – ma, Mab = fab /(Kf pa) = (fab/fa) ma
1-PAM Matrix
ma = fa /(Kf pa),
Maa = 1 – ma, Mab = fab /(Kf pa) = (fab/fa) ma
a-PAM Matrix = Ma
M* = lima à 1 Ma
Scorea(a,b) = 10 log10 Maab/pb
Sequence comparison with 40 PAM, 120 PAM & 250 PAM score functions…
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©Bud Mishra, 2001

22. Gaps: ©Bud Mishra, 2001 g = length of a gap,
exponentially distributed » Exp(l)
f(g) = l e-l g
P(g) = f(g) Õ qxi
ln P(g) = -l g + ln l + sum ln qxi
= -d – (g-1) e (Affine Score Model)
d = Gap-open Penalty
e = Gap-Extension Penalty
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©Bud Mishra, 2001

23. Multiple Sequence Alignment
Defn:Given strings S1, S2, …Sk a multiple (global) alignment maps them to strings S’1, S’2, …, S’k (by inserting chosen spaces) such that
|S’1| = |S’2| = L = |S’k|, and
Removal of spaces from S’i contracts it to Si, for 1 5 i 5 k.
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©Bud Mishra, 2001

24. Value of a Multiple Global Alignment
The sum of pairs (SP) value for a multiple global alignment A of k strings is the sum of the values of all Ck,2 pairwise alignments induced by A.
Given: Two strings S1 and S2. The expanded strings S’1 and S’2 correspond to a pairwise alignment.
d(x, y) = distance between two characters x and y
= 1, if x ¹ y and 0, if x = y.
d(x,-) = d(-,y) = 1.
Distance(S’1, S’2) = åi=1l d(S’1[i], S’2[i]),
where l = |S’1|=|S’2|.
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©Bud Mishra, 2001

25. Optimal Global Alignment
An optimal SP(global) alignment of strings S1, S2, …, Sk is an alignment that has a minimum possible sum-of-pairs value for these strings among all possible multiple sequence alignments.
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©Bud Mishra, 2001

26. Generalization of DP ©Bud Mishra, 2001
Assume |S1| = |S2| = L = |Sk| = n.
The generalized k-dimensional DP table has (n+1)k entries.
Each entry depends on 2k – 1 adjacent entries.
D(i1, 0, …, 0) = i1
D(0, i2,…, 0) = i2 M
D(0,0,…, ik) = ik
D(i1, i2, …ik) = min; ¹ S µ {1..k} [
D[…, ij-1,…]j2 S
+ ål¹ m 2 S d(il, im) + |S| £ (n-|S|) ]
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27. Complexity ©Bud Mishra, 2001
The time and space complexity of the generalized DP solution of the multiple alignment problem is = O((2n)k)
Theorem: The optimal SP alignment problem is NP-complete.
In the worst-case, one cannot expect to do much better unless P=NP.
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28. P-Time Heuristics ©Bud Mishra, 2001
A Polynomial Time Approximate Algorithm for Multiple String Alignment:
Assumption about the distance function:
Triangle Inequality:
8chars, x ,y, z d(x,z) 5 d(x,y) + d(y,z)
8char, x d(x, x) = 0
D(S1. S2)
, Value of the min. global alignment of S1 & S2.
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29. Algorithm ©Bud Mishra, 2001 Input: T = {S1, S2, …, Sk}
Step 1: Find S1 2 CT that minimizes
åS 2 T n {S1} D(S1, S)
Time Complexity = O(k2 n2)
Ck,2 DP each taking O(n2) time
Call the remaining strings S2, …, Sk
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30. = O(k2 n2) + åi=1k O(i n2) = O(k2 n2)
The ith Step
Step i: Assume S1, …Si-1 have been aligned as S’1, …, S’i-1
Add Si: Run DP to align S’1 & Si a S”i and S’i
Adjust S’2, …, S’i-1 by adding spaces where spaces were added in S’1
S1, S2, …, Si )aligned S’1, S’2, …, S’i.
Length(S’1) in step i 5 i n.
DP(S’1, Si) takes O(i n2) time
Total time Complexity
= O(k2 n2) + åi=1k O(i n2) = O(k2 n2)
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©Bud Mishra, 2001

31. Competitiveness ©Bud Mishra, 2001
M = Alignment induced by the algorithm
d(i,j) = Distance M induces on pair Si, Sj
M* = Optimal alignment
2 SP(M) = åi=1k åj=1, j ¹ ik d(i,j)
5 åi=1k åj=1, j ¹ ik d(i,1) + d(1,j)
(Triangle Inequality)
= åi=1k åj=1, j ¹ ik d(1,i) + åi=1k åj=1, j ¹ ik d(1,j)
(Symmetry)
= åi=2k (k-1) d(1,i) +åj=2k (k-1) d(1,j)
= 2(k-1) åi=2k d(1,i)
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©Bud Mishra, 2001

32. Competitiveness ©Bud Mishra, 2001 SP(M) 5 2 (1 – 1/k) SP(M*)
2SP(M*) = åi=1k åj=1, j¹ ik D(Si, Sj)
= åj=2k D(S1, Sj) + åj=1, j ¹ 2k D(S2, Sj)
+ L + åj=1k-1 D(Sk, Sj)
= k åi=2k d(1,i)
SP(M*) = (k/2) åi=2k d(1,i)
¸ (k/2) [SP(M)/(k-1)]
SP(M) 5 2 (1 – 1/k) SP(M*)
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©Bud Mishra, 2001