1. Protein Sequence Comparison Patrice Koehl http://koehllab.genomecenter.ucdavis.edu/teaching/ecs129/06/lecture-notes

2. Why do we want to align protein sequences? If two sequences align well, the corresponding proteins are homologous; they probably share the same structure and/or function Sequence Alignment is a Tool for organizing the protein sequence space detection of homologous proteins build evolutionary history

3. Alignment Methods Rigorous algorithms - Needleman-Wunsch (global) - Smith-Waterman (local) Rapid heuristics - FASTA - BLAST

4. What is sequence alignment? Given two sequences of letters and a scoring scheme for evaluating letter matching, find the optimal pairing of letters from one sequence to the other. Different alignments: Favors identity Favors similarity ACCTAGGC ACCTAGGC AC-T-GG ACT-GG Gaps

5. Aligning Biological Sequences Aligning DNA: 4 letter alphabet ACGTTGGC AC-T-GG Aligning protein: 20 letter alphabet MCYTSWGC MC-T-WG

6. Computing Cost The computational complexity of aligning two sequences when gaps are allowed anywhere is exponential in the length of the sequences being aligned. Computer science offers a solution for reducing the running time: Dynamic Programming

7. Dynamic Programming (DP) Concept A problem with overlapping sub-problems and optimal sub-structures can be solved using the following algorithm: (1) break the problem into smaller sub-problems (2)solve these problems optimally using this 3-step procedure recursively (3)use these optimal solutions to construct an optimal solution to the original problem

8. DP and Sequence Alignment Key idea: The score of the optimal alignment that ends at a given pair of positions in the sequences is the score of the best alignment previous to these positions plus the score of aligning these two positions.

9. Test all alignments that can lead to i aligned with j i j ? DP and Sequence Alignment j i ?

10. Find alignment with best [previous score + score(i,j)] i j ? DP and Sequence Alignment j i Best alignment that ends at (i,j)

11. Implementing the DP algorithm for sequences 1)Build a NxM alignment matrix A such that A(i,j) is the optimal score for alignments up to the pair (i,j) 2) Find the best score in A 3) Track back through the matrix to get the optimal alignment of S1 and S2. Aligning 2 sequence S1 and S2 of lengths N and M:

12. Example 1 Sequence 1: ATGCTGC Sequence 2: AGCC Score(i,j) = 10 if i=j, 0 otherwise no gap penalty

13. Example 1 ATGCTGC A10000000 G0 C0 C0 1) Initialize

14. Example 1 ATGCTGC A10000000 G0 C0 C0 2) Propagate

15. Example 1 ATGCTGC A10000000 G0 20 C0 C0 2) Propagate

16. Example 1 ATGCTGC A10000000 G0 2010 2010 C0 C0 2) Propagate

17. Example 1 ATGCTGC A10000000 G0 2010 2010 C0 3020 30 C010 30 2) Propagate

18. Example 1 ATGCTGC A10000000 G0 2010 2010 C0 3020 30 C010 30 40 3) Trace Back ATGCTGC AXGCXXC Alignment: Score: 40

19. Mathematical Formulation W k : penalty for a gap of size k Global alignment (Needleman-Wunsch):

20. Complexity 1) The computing time required to fill in the alignment matrix is O(NM(N+M)), where N and M are the lengths of the 2 sequences 2) This can be reduced to O(NM) by storing the best score for each row and column. True if gap penalty is linear !

21. Example 2 Alignments: High Score: 30 AATGC A10 000 G0 2010 G0 20 C010 30 AATGC AG GC AATGC A GGC AATGC AGGC AATG C A GGC AATG C A GGC

22. Example 2 with Gap Alignments: High Score: 28 AATGC A108-2 G 108188 G-28101816 C-2881028 AATGC AG GC AATGC A GGC AATGC AGGC Gap cost: -2

23. Complexity (2) 1) The traceback routine can be quite costly in computing time if all possible optimal paths are required, since there may be many branches. 2) Usually, an arbitrary choice is made about which branch to follow. Then computing time is O(max(N,M)) By simply following pointers.

24. The Scoring Scheme Scores are usually stored in a “weight” matrix or “substitution” matrix or “matching” matrix. Defining the “proper” matrix is still an active area of research Usually, start from known, reliable alignment. Compute f i, the frequency of occurrence of residue type i, and q ij, the probability that residue types i and j are aligned; score is computed as:

25. CSTPAGNDEQHRKMILVFYW C9 -30 -4-3 -2 S41 101000 0 -2 -3 T141 101000 0 -2 -3 P 17 -2 -2 -2-3 -2-4-3-4 A01 40 -2 -2 -2 -3 G 01-206 -2 -3-4 0-3 -2 N-310-2 0610000-2-3 -2-4 D-301-21620 -2-3 -4-3 -4 E 00 -20252001 -3 -2-3 Q 00 -200250110-3-2 -3-2 H-30-2 110080-2-3 -22-2 R-3 -2-20 01052-3-2-3 -2-3 K 00 -2011 25 -3-2-3 -2-3 M -2-3-2-3-20 512-20 I -2 -3-4-3 14210-3 L-2 -3-4-3-4-3-2-3-2 22430-2 V-2 0-3 -2 -3 -21314 -3 F-2 -4-2-3 -3 000631 Y-2 -3-2-3-2-3-22-2 372 W-2-3 -4-3-2-4 -3-2 -3 -3-2-31211 Example of a Scoring matrix

26. Gap penalty Most common model: W N = G 0 + N * G 1 W N : gap penalty for a gap of size N G 0 : cost of opening a gap G 1 : cost of extending the gap by one N: size of the gap

27. Global versus Local Alignment Global alignment finds best arrangement that maximizes total score Local alignment identifies highest scoring subsequences, sometimes at the expense of the overall score Local alignment algorithm is just a variation of the global alignment algorithm!

28. Modifications for local alignment 1)The scoring matrix has negative values for mismatches 2)The minimum score for any (i,j) in the alignment matrix is 0. 3)The best score is found anywhere in the filled alignment matrix These 3 modifications cause the algorithm to search for matching sub-sequences which are not penalized by other regions (modif. 2), with minimal poor matches (modif 1), which can occur anywhere (modif 3).

29. Mathematical Formulation W k : penalty for a gap of size k Local alignment (Smith Waterman):

30. Global versus Local Alignment ACCTGS A100000 C021000 C013000 N000100 S000001 Match: +1; Mismatch: -2; Gap: -1 ACCTGS A1-3 C 21-2 C-313 N-3-2100 S-3-20 1 Global: ACCTGS ACC-NS Local: ACC

31. Heuristic methods O(NM) is too slow for database search Heuristic methods based on frequency of shared subsequences Usually look for ungapped small sequences FASTA, BLAST

32. FASTA Create hash table of short words of the query sequence (from 2 to 6 characters) Scan database and look for matches in the query hash table Extend good matches empirically Seq1Seq2Seq3Seq4Seq5Seq6Seq7…SeqN Word1 Word2 Word3 … WordP

33. http://www.ncbi.nlm.nih.gov/Education/BLASTinfo/information3.html Tutorial: BLAST 1)Break query sequence and database sequences into words 2)Search for matches (even not perfect) that scores at least T 3)Extend matches, and look for alignment that scores at least S

34. Summary Dynamic programming finds the optimal alignment between two sequences in a computing time proportional to NxM, where N and M are the sequence lengths Critical user choices are the scoring matrix, the gap penalties, and the algorithm (local or global)

35. Statistics of Sequence Alignment

36. Significance We have found that the score of the alignment between two sequences is S. Question: What is the “significance” of this score? Otherwise stated, what is the probability P that the alignment of two random sequences has a score at least equal to S ? P is the P-value, and is considered a measure of statistical significance. If P is small, the initial alignment is significant.

37. A given experiment may yield the event A or the event not(A) with probabilities p, and q=1-p, respectively. If the experiment is repeated N times and X is the number of time A is observed, then the probability that X takes the value k is given by: Basic Statistics: Binomial Distribution With the binomial coefficient: Properties: E(X)=Np Var(X)=Np(1-p) N=40; p=0.2

38. The Poisson distribution is a discrete distribution usually defined over a volume or time interval. Given a process with expected number of success  in the given interval, the probability of observing exactly X success is given by: Basic Statistics: Poisson Distribution =8 Properties: E(X)= Var(X)=

39. A normal distribution in a variable X with mean  and variance  2 is a statistical distribution with probability function: Basic Statistics: Normal Distribution N=40; p=0.2 The normal distribution is the limiting case of a binomial distribution P(n,N,p) with:

40. A extreme value distribution in a variable X is a statistical distribution with probability function: Basic Statistics: Extreme Value Distribution a=0; b=1 Note the presence of a long tail

41. BLAST Input

42. BLAST results

43. BLAST results (2)

44. Statistics of Protein Sequence Alignment Statistics of global alignment: Unfortunately, not much is known! Statistics based on Monte Carlo simulations (shuffle one sequence and recompute alignment to get a distribution of scores) Statistics of local alignment Well understood for ungapped alignment. Same theory probably apply to gapped-alignment

45. Statistics of Protein Sequence Alignment What is a local alignment ? “Pair of equal length segments, one from each sequence, whose scores can not be improved by extension or trimming. These are called high-scoring pairs, or HSP” http://www.people.virginia.edu/~wrp/cshl98/Altschul/Altschul-1.html

46. The E-value for a sequence alignment S The expected number of HSP with score at least S is given by: HSP scores follow an extreme value distribution, characterized by two parameters, K and. m, n : sequence lengths E : E-value

47. The Bit Score of a sequence alignment Raw scores have little meaning without knowledge of the scoring scheme used for the alignment, or equivalently of the parameters K and. Scores can be normalized according to: S’ is the bit score of the alignment. The E-value can be expressed as:

48. The P-value of a sequence alignment The number of random HSP with score greater of equal to S follows a Poisson distribution: (E: E-value) Then: Note: when E <<1, P ≈E

49. Database search, where database contains N S sequences corresponding to N R residues: 1)All sequences are a priori equally likely to be related to the query : 2)Longer sequences are more likely to be related to the query: BLAST reports E DB2 The database E-value for a sequence alignment

50. Summary Statistics on local sequence alignment are defined by: –Raw score –Bit score (normalized score) –E-value –P-value Statistics on database search: - database E-value