1. Extreme Elevation on a 2- Manifold Pankaj K. Agarwal Joint Work with H. Edelsbrunner, J. Harer, Y. Wang Duke University

2. Protein-Protein Docking

3. Sample Protein Complexes 1A221JLT3HLA

4. Docking Earlier methods: Lenhof, Nussinov and Wolfson (Tel Aviv ), Kuntz et al (UC San Diego), Singh and Brutlag (Stanford) Famous softwares: DOCK (Kuntz), AutoDock (Olson), Zdock (Wen) More Protein-protein docking: 3D-Dock, HEX, DOT, GRAMM, PPD, BIGGER

5. Docking: Geometric Approach Shape Complementarity Partial matching Rigid motion Extract features Locate feature points More global features? Matching Features Our goal: more than good feature points

6. Related Work Curvature Too local Connolly function [Connolly, 83] Ratio of inside/outside perimeters Atomic density [Mitchell et al., 01] FADE, PADRE

7. Our Results Elevation function Maxima of elevation Associated with feature Results being applied to docking Shape-matching algorithms Plugging into local-improvement algorithms

8. Elevation Function Elevation on Earth Fit an ellipsoid Choose the center to measure elevation For general manifolds No good choice for origin Function invariant under rigid motion

9. Persistence Critical points [Munkres, 84] Critical pairs, persistence [Edelsbrunner et al. 02] Critical pairs identify topological features Miss non-vertical features

10. Family of Height Functions Fix an arbitrary origin Given u  S 2, define h u : M  R, as h u (x) = H : M x S 2  R : a family of height functions H(x, u) = h u (x)

11. Height in All directions Each x  M critical in normal direction Pair (x,y) Persistence p(x) = p(y) = |h u (x) – h u (y)|

12. Elevation  Define E : M  R as E(x) = p(x) Interested in max of E Each x  M captures feature in its normal direction E(x) indicates size of feature However, E is not everywhere continuous

13. Surgery Reason: singular tangency Inflexion pt (birth-death pt) Double tangency (interchange) Blame the manifold! M’ : apply surgery on M Elevation function: E : M’  R

14. 2D Example

15. One Component

16. Pedal Surface Goal: help understand E Definition for P : M  R 3 Singularities Inflexion pt  cusp  Double tangency  xing

17. Relation Critical pts for h u  P  l(u) P = P(M) Singularities of P along l(u)

18. Dictionary of Singularities MHeightP inflexion ptbirth-death ptcusp double tangencyinterchangexing Jacobi pt2 birth-death ptsdovetail pt triple tangency3 interchangestriple pt bd-pt + interchangecusp intersection 2 birth-death ptscusp-cusp crossing 2 interchangesxing-xing crossing bd-pt + interchangecusp-xing crossing

19. Examples Examples of triple pt and cusp + intersection

20. Dictionary of Singularities MHeightP inflexion ptbirth-death ptcusp double tangencyinterchangexing Jacobi pt2 birth-death ptsdovetail pt triple tangency3 interchangestriple pt bd-pt + interchangecusp intersection 2 birth-death ptscusp-cusp crossing 2 interchangesxing-xing crossing bd-pt + interchangecusp-xing crossing

21. Another Example dovetail

22. Max of E An inflexion pt x cannot be a maximum

23. Dictionary of Singularities MHeightP inflexion ptbirth-death ptcusp double tangencyinterchangexing Jacobi pt2 birth-death ptsdovetail pt triple tangency3 interchangestriple pt bd-pt + interchangecusp intersection 2 birth-death ptscusp-cusp crossing 2 interchangesxing-xing crossing bd-pt + interchangecusp-xing crossing

24. Classification of Max for E Four types of max x No singularity involved: 1-legged : x regular, paired w/ regular pt Singularity involved: 2-legged : x regular, paired w/ double pt 3-legged : x regular, paired w/ triple pt 4-legged : x double pt, paired w/ double pt

25. New Manifold M´ Point x  M´ May correspond to multiple pts in M, and vice versa regular pt double pt : double tangency triple pt : triple tangency A max can be: k-legged, k = 1, 2, 3, or 4.

26. Illustration in M 3-legged max x

27. 2-Legged

28. 3-Legged

29. 4-Legged

30. Elevation: Example 1

31. Elevation: Example 2

32. Elevation vs. Atomic Density ElevationAtomic density

33. Summary Different feature information Positions (points) Shape (k-legs) Size (persistence) Direction (normal) Feature pairs: A max and (one of) its leg

34. Two-steps Coarse align via features Local improvement Input: Protein A, B Output: A set of potential alignments Docking

35. Conclusion Elevation function Relation w/ pedal function Classification of max of Elevation function Compute max for PL-case More efficient alg. to compute max Matching (docking)

36. Height Function Height function h : M  R Morse function No degenerate critical pts No two critical pts have same function value Critical points Capture topological features Pairs of critical pts Persistence Alg. (ELZ01) Pairing decided by order of heights x

37. Characterization of Max

38. PointMatch PointMatch Alg: Output: T point Take a pair from A, a pair from B Align two pairs, get transformation T Compute score between A and T(B) Rank transformations by score

39. PairMatch PairMatch Alg: Take a feature pair from each set Align two feature pairs, get transformation T Rank T ’s by their scores Output: T pair

40. Docking for 1BRS RankRMSD 11.516 22.313 335.947 421.598 521.599 617.995 734.582 834.774 945.726 1022.656

41. Docking Results pdb-idRankRMSD 1BRS11.516 1A2252.745 2PTC46.18 1MEE11.333 1CHO12.712 1JLT12.375 1CSE22.211 3SGB13.209 3HLA13.492