2. OUTLINE Scoring Matrices Probability of matching runs Quality of a database match

3. Scoring Matrices Two alternative models for differences in DNA / protein sequences – Random: All sequences are random selections of given pool of residues. – Nonrandom: Sequences are related, Evolutionary process.

4. Scoring Matrices – Random: p a : fraction of amino acid a in the pool (probbility of occurance of the amino acid). – Nonrandom: q a,b : the probability of finding particular residus a and b aligned

5. Scoring Matrices These two models can be compared: – q a,b / p a p b  odds ration – If q a,b > p a p b  nonrondom model is more likely to produce the alignement of these residues.

6. Scoring Matrices However, we need a single model – Assume: each position in an alignment will be regarded as independent. – Odds ratio of alignment:

7. Scoring Matrices log – odds ratio: Negative value: probability of the two residues aligned is greater in the random model than nonrandom model.

8. Scoring Matrices EXAMPLE: If M occurs in the sequences with 0.01 frequency and L occurs with 0.1 frequency. By random pairing, you expect 0.001 amino acid pairs to be M-L. If the observed frequency of M-L is actually 0.003, score of matching M-L will be log 2 (3)=1.585

9. Probability of matching runs Statistical significance measures: – p-value: the probability that at least one sequence will produce the same score by chance – E-value: expected number of sequences that will produce same or better score by chance

10. Probability of matching runs Analysis of coin tosses : – “H” indicates a head – p probability of head (p = 0,5) – Probability of 5 heads in a run: 0,5 5 =0,031 – The expected number of times that 5H occurs in above 14 coin tosses: 10x0,031 = 0,31

11. Probability of matching runs Analysis of coin tosses : – The expected number of a length l run of heads in n tosses: – Expected length R of the longest match in n tosses: (Erdös-Rényi)

12. Probability of matching runs Analysis of coin tosses : – Example: N = 20  R = log 2 (20) = 4,3 (in 20 coin tosses we expect 4,3 runs of heads, once )

13. Probability of matching runs DNA / protein sequences: Probability of an individual match p = 1 / 20 = 0,05

14. Probability of matching runs Expected number of matches: 8x6x0.05 = 2,4

15. Probability of matching runs Expected number of two successive matches: 8x6x0,05x0,05 = 0,12

16. Probability of matching runs Expected number of length l matches: Expected longest match of two sequences of length m and n: where p is the probability of occurance of a single residue.

17. Probability of matching runs Expected number of length l matches: Expected longest match of two sequences of length m and n: where p is the probability of occurance of a single residue.

18. Probability of matching runs Example: – DNA seq: m = 32, n = 32 R = log 4 (32x32) = 5 – Amino acid seq: m = 100, n = 80 R = log 20 (100x80) = 3

19. Probability of matching runs Under even the simplest random models and scoring systems, very little is known about the random distribution of optimal global alignment scores Statistics for the scores of local alignments, unlike those of global alignments, are well understood.

20. Probability of matching runs The optimal ungapped local alignment score follows the Gumble Extreme value distribution. Because we always choose the best-scoring alignments the distribution will be Gumble Extreme value distribution. Probability of obtaining an alignment of score S greater than a value x:

21. Quality of a database match How good is an alignment ? How believable the results of a database search ?

22. Quality of a database match The alignment reports are selected according to the alignment score. We need to know: – Whether the score is greater than we would expect from the alignment of the sequences with a random sequence.

23. Quality of a database match Statistical significance measures: – p-value: the probability that at least one sequence will produce the same score by chance – E-value: expected number of sequences that will produce same or better score by chance

24. Quality of a database match Score  Significance of the score. – By applying the Gumble Extreme value distribution, it is possible to estimate the probability of two random sequences aligned with a score greater than or equal to the alignment score. – E- value, p – value.

25. Quality of a database match E-value depends on: – The sequence length, – The number of sequences in the database, – Alignement score.

26. Quality of a database match A good E-value: – The smaller the E-value the better the alignment, – The threshold value generally is set to 0,01 or 0,001.

27. Quality of a database match

30. References M. Zvelebil, J. O. Baum, “Understanding Bioinformatics”, 2008, Garland Science Andreas D. Baxevanis, B.F. Francis Ouellette, “Bioinformatics: A practical guide to the analysis of genes and proteins”, 2001, Wiley.