Bioinformatics Group Institute of Biotechnology University of Helsinki

1. Bioinformatics Group Institute of Biotechnology University of Helsinki Significance in protein analysis Swapan ‘Shop’ Mallick

2. Overview • The need for statistics • Example: BLOSUM • What do the scores mean? • How can you compare two scores? • Example: BLAST • Problems with BLAST • Review of Distributions • Distribution of random BLAST results • P-values and e-values • Statistics of BLAST • Summary and Conclusion • Exercise

3. The need for statistics • Statistics is very important for bioinformatics. • It is very easy to have a computer analyze the data and give you back a result. • Problem is to decide whether the answer the computer gives you is any good at all. • Questions: • How statistically significant is the answer? • What is the probability that this answer could have been obtained by random? What does this depend on?

4. Basics N n Sample Population

5. Basics N Descriptive statistics n Sample Population Probability

6. Example: BLOSUM • The BLOSUM matrix assigns a probability score for each residue pair in an alignment based on: • the frequency with which that pairing is known to occur within conserved blocks of related proteins. • Simple since size of population = size of sample • BLOSUM matrices are constructed from observations which lead to observed probabilities

7. BLOSUM substitution matrices • BLOSUM matrices are used in ‘log-odds’ form based on actually observed substitutions. • This is because: • Ease of use: ‘Scores’ can be just added (the raw probabilities would have to be multiplied) • Ease of interpretation: • S=0 : substitution is just as likely to occur as random • S<0 : substitution is more likely to occur randomly than observed • S>0 : substitution is less likely to occur randomly than observed

8. Substitution matrices Score of amino acid a with amino acid b Pab is the observed frequency that residues a and b are correlated because of homology Lambda is a scaling factor equal to 0.347, set so that the scores can be rounded off to sensible integers fafb is the expected frequency of seeing residues a and b paired together, which is just the product of the frequency of residue a multiplied by the frequency of residue b Source: Where did the BLOSUM62 alignment score matrix come from?Eddy S., Nat. Biotech. 22 Aug 2004

9. Substitution matrices Lambda is a scaling factor equal to 0.347, set so that the scores can be rounded off to sensible integers Pab is the observed frequency that residues a and b are correlated because of homology fafb is the expected frequency of seeing residues a and b paired together, which is just the product of the frequency of residue a multiplied by the frequency of residue b

11. i) S=0 : O/E ratio=1 ii) Compare S=5 and S=10. Ratio is based on exponential function iii) S=-10: O/E ratio = 0.031 ≈ 1/32. iv) Ratio of scores S1, S2 in terms of probabilities of observed/random =

12. i) S=0 : O/E ratio=1 ii) Compare S=5 and S=10. Ratio is based on exponential function iii) S=-10: O/E ratio = 0.031 ≈ 1/32. iv) Ratio of scores S1, S2 in terms of probabilities of observed/random = 32.1 5.7

13. i) S=0 : O/E ratio=1 ii) Compare S=5 and S=10. Ratio is based on exponential function iii) S=-10: O/E ratio = 0.031 ≈ 1/32. iv) Ratio of scores S1, S2 in terms of probabilities of observed/random = 32.1 5.7

14. i) S=0 : O/E ratio=1 ii) Compare S=5 and S=10. Ratio is based on exponential function iii) S=-10: O/E ratio = 0.031 ≈ 1/32. iv) Ratio of scores S1, S2 in terms of probabilities of observed/random = 32.1 5.7

15. Example: BLAST • Motivations • Exact algorithms are exhaustive but computationally expensive. • Exact algorithms are impractical for comparing a query sequence to millions of other sequences in a database (database scanning), • and so, database scanning requires heuristic alignment algorithm (at the cost of optimality).

16. ID (GI #, refseq #, DB-specific ID #) Click to access the record in GenBank Links Gene/sequence Definition Bit score – higher, better. Click to access the pairwise alignment Expect value – lower, better. It tells the possibility that this is a random hit Interpret BLAST results - Description

17. Problems with BLAST • Why do results change? • How can you compare results from different BLAST tools which may report different types of values? • How are results (eg evalue) affected by query • There are _many_ values reported in the output – what do they mean?

18. Example: Importance of Blast statistics • But, first a review.

19. Review • What is a distribution? • A plot showing the frequency of a given variable or observation.

20. Review • What is a distribution? • A plot showing the frequency of a given variable or observation.

21. Features of a Normal Distribution • Symmetric Distribution • Has an average or mean value at the centre • Has a characteristic width called the standard deviation (S.D. = σ) • Most common type of distribution known m = mean

22. Standard Deviations (Z-score)

23. Mean, Median & Mode Mode Median Mean

24. Mean, Median, Mode • In a Normal Distribution the mean, mode and median are all equal • In skewed distributions they are unequal • Mean - average value, affected by extreme values in the distribution • Median - the “middlemost” value, usually half way between the mode and the mean • Mode - most common value

25. Different Distributions Unimodal Bimodal

26. Other Distributions • Binomial Distribution • Poisson Distribution • Extreme Value Distribution

27. Binomial Distribution 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 P(x) = (p + q)n

28. m =0.1 m = 1 m = 2 m = 3 m = 10 Proportion of samples Poisson Distribution P(x) x

29. Review • What is a distribution? • A plot showing the frequency of a given variable or observation. • What is a null hypothesis? • A statistician’s way of characterizing “chance.” • Generally, a mathematical model of randomness with respect to a particular set of observations. • The purpose of most statistical tests is to determine whether the observed data can be explained by the null hypothesis.

30. Review • What is a distribution? • A plot showing the frequency of a given variable or observation. • What is a null hypothesis? • A statistician’s way of characterizing “chance.” • Generally, a mathematical model of randomness with respect to a particular set of observations. • The purpose of most statistical tests is to determine whether the observed data can be explained by the null hypothesis.

31. Review • Examples of null hypotheses: • Sequence comparison using shuffled sequences. • A normal distribution of log ratios from a microarray experiment. • LOD scores from genetic linkage analysis when the relevant loci are randomly sprinkled throughout the genome.

32. Empirical score distribution • The picture shows a distribution of scores from a real database search using BLAST. • This distribution contains scores from non-homologous and homologous pairs. High scores from homology.

33. Empirical null score distribution • This distribution is similar to the previous one, but generated using a randomized sequence database.

34. Review • What is a p-value?

35. Review • What is a p-value? • The probability of observing an effect as strong or stronger than you observed, given the null hypothesis. I.e., “How likely is this effect to occur by chance?” • Pr(x > S|null)

36. Review • What is the name of the distribution created by sequence similarity scores, and what does it look like? Extreme value distribution, or Gumbel distribution. It looks similar to a normal distribution, but it has a larger tail on the right.

37. Review • What is the name of the distribution created by sequence similarity scores, and what does it look like? • Extreme value distribution, or Gumbel distribution. • It looks similar to a normal distribution, but it has a larger tail on the right.

38. Statistics • BLAST (and also local i.e. Smith-Waterman and BLAT scores) between random, unrelated sequences follow the Gumbel Extreme Value Distribution (EVD) • Pr(s>S) = 1-exp(-Kmn e-lS) • This is the probability of randomly encountering a score greater than S. • S alignment score • m,n query sequence lengths, and length of database resp. • K, l parameters depending on scoring scheme and sequence composition • Bit score : S’ = lS – log(K) log(2)

39. BLAST output revisited S’SE n m K From: Expasy BLAST

40. Review • EVD for random blast • Upper tail behaviour: Pr( s > S ) ~ Kmn e-lS This is the EXPECT value = Evalue

41. Summary • Want to be able to compare scores in sequences of different compositions or different scoring schemes • Score: S = sum(match) – sum(gap costs)

42. Summary • Want to be able to compare scores in sequences of different compositions or different scoring schemes • Score: S = sum(match) – sum(gap costs) • Bit score • S’ = lS – log(K) log(2)

43. Score and bit score grow linearly with the length of the alignment Summary • Want to be able to compare scores in sequences of different compositions or different scoring schemes • Score: S = sum(match) – sum(gap costs) • Bit score • S’ = lS – log(K) log(2)

44. Score and bit score grow linearly with the length of the alignment Summary • Want to be able to compare scores in sequences of different compositions or different scoring schemes • Score: S = sum(match) – sum(gap costs) • Bit score • S’ = lS – log(K) log(2) • E-value of bit score • E = mn2-S’

45. Score and bit score grow linearly with the length of the alignment Summary E-Value shrinks really fast as bit score grows • Want to be able to compare scores in sequences of different compositions or different scoring schemes • Score: S = sum(match) – sum(gap costs) • Bit score • S’ = lS – log(K) log(2) • E-value of bit score • E = mn2-S’

46. Score and bit score grow linearly with the length of the alignment Summary E-Value shrinks really fast as bit score grows • Want to be able to compare scores in sequences of different compositions or different scoring schemes • Score: S = sum(match) – sum(gap costs) • Bit score • S’ = lS – log(K) log(2) • E-value of bit score • E = mn2-S’ E-Value grows linearly with the product of target and query sizes.

47. Score and bit score grow linearly with the length of the alignment Summary E-Value shrinks really fast as bit score grows • Want to be able to compare scores in sequences of different compositions or different scoring schemes • Score: S = sum(match) – sum(gap costs) • Bit score • S’ = lS – log(K) log(2) • E-value of bit score • E = mn2-S’ E-Value grows linearly with the product of target and query sizes. Doubling target set size and doubling query length have the same effect on e-value

48. Conclusion • You should now be able to compare BLAST results from different databases, converting values if they are reported differently (which happens frequently) • You should now know why BLAST results might change from one day to the next, even on the same server • You should understand also the dependance of query length on E-value. • Statistical rankings are reported for (almost) every database search tool. When making comparisons between databases, between sequences it is useful to know how the statistics are derived to know if comparisons are meaningful.

49. THE END

50. Supplemental Section