Sequence Entropy

1. Sequence Entropy Sequence Analysis

2. Significance of Alignment Positions • Observed occurrence of amino acids at some position in an alignment that deviates from expected may indicate some (functional) significance • What ‘deviates from expected’? • unlikely occurrences • What is unlikely? • only (relatively) few possibilities to obtain observed result

3. Counting… • Number of possibilities W for finding given numbers Nx of aminoacids types x: • Problem: evaluates to huge numbers even for modest numbers of sequences…

4. Sequence Entropy • Entropy: • Using Stirlings approximation:ln M! ≈ M(ln M-1) for M≫1

5. Shannon’s ‘Information Entropy’: • ‘A Mathematical Theory of Communication’, The Bell System Technical Journal, Vol. 27, 1948. “ Can we define a quantity which will measure, in some sense, how much information is ‘produced’ by such a process, or better, at what rate information is produced? ” • He was thinking about the Transmission of Information, i.e., from a Source through some Channel to a Destination.

6. Choice, Uncertainty and Entropy • A set of ‘events’ with probabilities p1, p2, …, pn • Is there a measure that indicates how much ‘choice’ is possible, given those probabilities? • If there is, it should be: • continuous for all pi • monotonic in n if all probabilities are equal • additive for ‘sub-events’

7. Additivity: 1/2 1/2 1/2 1/3 1/3 2/3 1/2 1/3 1/6 1/6 H(1/2,1/2) H(2/3,1/3) H(1/2,1/3,1/6) = H(1/2,1/2) + 1/2 H(2/3,1/3)

8. Solution: Entropy • the entropy of a set of probabilities pi • measures information, choice and uncertainty • zero only if only one pi is not zero • there is only one choice • maximal if all pi are equal • most ‘uncertain’ situation: all options are possible

9. Information Content • Shannon was thinking about the Transmission of Information, i.e., from a Source through some Channel to a Destination. • …but it applies equally well to any type of ‘message’ • We can use it to measure the level of conservation in columns in an alignment

10. Simple Example: Sequence Entropy p1 = p2 = ½ p1 = 0 p2 = 0 p2 = f(‘A’) p1 = f(‘L’)

11. Conditional Probability / Prior Knowledge • Suppose we observe two things (A and B), and we suspect a relation (A causes B) • The fundamental question is then:How likely is A when we know B? • (n.b., this is Bayes statistics…) • Or, what is the uncertainty (entropy) of A knowing B?

12. Conditional Entropy • Entropy of joint occurrence of value i for event x and value j for event y : • and • so that • i.e., the entropy of a joint event is less than or equal to the sum of the individual entropies • it is equal only if the events are independent

13. Co-occurrence in practice • Measure of ‘mutial information’, by relative entropy: • more often written like: • i.e., what is the entropy in x, giveny

14. Relative Entropy in Sequence Analysis • Many biological problems relate to questions like: “Why do these proteins do this, and those proteins not?” • or “Why do these patients get sick, and those not?” • The answer can be related to similarities and differences between sequences • Similarities (conservation) relate to functionally critical positions • Differences can explain functional differences

15. Comparing groups of Sequences • For each position i in an alignment, we calculate the relative entropy for group Avs.B from the frequencies p (observed probabilities) of all aminoacid types x, as follows:

16. 3.0 A p å i , x A p log i , x B p 2.5 x i , x B p å B i , x p log i , x A p 2.0 x i , x Entropy / Harmony 1.5 1.0 0.5 0.0 Simple Example: Relative Entropy ‘A/B’ A B

17. Relative Entropy • Captures similarities and differences • Is infinite for completely dissimilar positions • e.g., A vs. L • Not symmetrical: • Maybe not easy for selecting dissimilar positions

18. 3.0 2.5 2.0 Entropy / Harmony 1.5 B p å i , x B p log i , x AB p x i , x 1.0 A p å i , x A p log i , x AB p x i , x 0.5 0.0 Simple Example: Relative Entropy ‘A/AB’ A B

19. Relative Entropy (A/AB) • Also captures similarities and differences • Is no longer infinite for completely dissimilar positions • Only symmetrical for equal size groups • in practice, not symmetrical • Maybe still not too easy for selecting dissimilar positions

20. Measuring Overlapping Distributions • Weigh both groups equally; take pA+pB in stead of pAB : • Fixed interval [0,1], but not completely symmetrical

21. 3.0 2.5 2.0 A B p p å å i i , , x x A B p p log log Entropy / Harmony 1.5 i i , , x x + + A A B B p p p p x x i i , , x x i i , , x x 1.0 0.5 0.0 Entropy vs. Sequence Harmony: Example A B

22. Sequence Harmony • Introduce symmetry by averaging: • May seem a trivial choice, but: Pirovano, Feenstra & Heringa. “Sequence Comparison by Sequence Harmony Identifies Subtype Specific Functional Sites”, Nucleic Acids Res., in press (2006).

23. 3.0 2.5 2.0 Entropy / Harmony 1.5 1.0 0.5 0.0 Entropy vs. Sequence Harmony: Example A B

24. Analyzing multiple groups • Relative Entropy and Sequence Harmony are defined to compare a set of groups (N=2) • problem for multiple groups (N>2) • A solution: Entropy-variability plots • variability is number of different aminoacid types in a certain alignment position • problem: variability always tends towards maximum (20) for larger number of sequences • Another solution: Two-entropy plots • Total family entropy vs. sum of sub-family entropy

25. Two-Entropies analysis of GPCRs • Goal: to identify the function of individual positions Ye, Lameijer, Beukers & IJzerman. “A Two-Entropies Analysis to Identify Functional Positions in the Transmembrane Region of Class A G Protein-Coupled Receptors”, Proteins 63, pp1018–1030 (2006)

26. G-protein Coupled Receptors (GPCRs) • Huge family of integral cell-membrane proteins • crucial in signal transduction • 70 subfamilies • 1935 Class A GPCRs • Three main regions: • extracellular side • transmembrane (TM) • cytoplasmic side

27. Two-entropies vs. Entropy-variability:upper vs. lower domain Red: upper domainBlue: lower domain

28. Two-entropies vs. Entropy-variability:relative solvent accessibility (RSA) Red: RSA < 15%Blue: RSA > 15%

29. Two-entropies vs. Entropy-variability:ligand binding site vs. other positions Subfamily specific binding Red: <4 Å from retinalBlue: >4 Å from retinal Subfamily specific binding Common activation mechanism Common activation mechanism

30. GPCR Two-entropies analysis Red: ligand binding Green: coupling & activation Blue: others

31. GPCR Functional Site Prediction: Method Comparison Finding known sites for ligand binding in Bovine Rhodopsin (left) and Aminergic Receptors (right)

32. 1 0 260 280 300 320 340 360 380 400 420 440 460 Smad-MH2 Alignment & Sequence Harmony • Walter Pirovano*, K. Anton Feenstra* and Jaap Heringa. “Sequence Comparison by Sequence Harmony Identifies Subtype Specific Functional Sites”, Nucleic Acids Res., in press (2006). • K. Anton Feenstra, Walter Pirovano and Jaap Heringa. “Sub-type Specific Sites for SMAD Receptor Binding Identified by Sequence Comparison using ‘Sequence Harmony’ ”. in: From Computational Biophysics to Systems Biology. pp. 73-78. Eds. U.H.E. Hansmann, J. Meinke, S. Mohanty and O. Zimmermann, Jülich, NIC Series, Vol. 34, 2006. • Elena Marchiori*, Walter Pirovano, Jaap Heringa and K. Anton Feenstra*. “A Feature Selection Algorithm for Detecting Subtype Specific Sites for Smad Receptor Binding”, Bio-ICMLA06, accepted (2006).

33. 262 270 280 290 300 310 AR BR Smads: Comparing two Groups

34. Smad-MH2 Alignment & Functionally Specific Sites • 29 known sites of functional specificity • based mostly on site-specific mutants and characterized on affinity for binding to BMPR-I vs. TBR-I receptor types

35. Finding Low-harmony sites in Smad-MH2 270 280 290 300

36. Finding Low-harmony sites in Smad-MH2 Pirovano, Feenstra & Heringa. “Sequence Comparison by Sequence Harmony Identifies Subtype Specific Functional Sites”, Nucleic Acids Res., in press (2006).www.few.vu.nl/~feenstra/articles/NAR 2006 Sequence Harmony.pdf

37. Smad-MH2: Low Harmony Patches

38. Smad-MH2: Functional Clusters R427 TbR-I/BMPR-I/ALK1/2 A323 receptor-binding M327 T430 V325 TbR-I/ALK1/2 Q284 TbR-I/BMPR-I A354 R410 V461 W368 P378 R462 C463 P360 ? Y366 Q407 FAST1, Mixer, SARA S460 R334 Q400 Q364 N381 R365 L440 ? T298 R337 F346 co-repressors A392 SARA/Mixer L297 retention & transcription factors Q309 N443 c-Ski/SnoN I341 S308 P295 F273 Q294 A272 SARA S269 T267

39. Conclusions Smad-MH2 • 40 Sites of Low Sequence Harmony in Smad-MH2 • different between the AR (TGF-b) and BR (BMP) sub-type Smads • Low Harmony sites in Smad-MH2 are functionally relevant • Other methods cannot select all known sites! • Functional Sites are Interaction Surfaces on Protein Surface: • Next: Analyze Interaction Partners in the Pathway • 14 Low Harmony Sites in Smad-MH2 of unknown function • 11 putative functions from structural considerations • promising candidates that determine TGF-b/BMP specificity • confirm (or rebuke) putative functions?

40. Sequence Harmony Webserver http://www.ibi.vu.nl/programs/seqharmwww1-b/

41. Sequence Harmony Webserver: Groups

42. Sequence Harmony Webserver: Reference

43. Sequence Harmony Webserver: Structure

44. SMAD Sequence Harmony: Raw Table

45. SMAD Sequence Harmony: Results

46. SMAD Sequence Harmony: Structure