Alignment of Flexible Molecular Structures

1. Alignment of Flexible Molecular Structures

2. Motivation • Proteins are flexible. One would like to align proteins modulo the flexibility. • Hinge and shear protein domain motions (Gerstein, Lesk , Chotia). • Conformational flexibility in drugs.

3. Problem definition

4. Flexible Geometric Hashing • Exploit the fact that neighboring parts share the joint - accumulate mutual information at the joint. • Achieve complexity of the same order of magnitude as in rigid alignment.

5. Flexible protein alignment without prior hinge knowledge FlexProt - algorithm • detects automatically flexibility regions, • exploits amino acid sequence order.

6. Motivation

7. Geometric Representation 3-D Curve {vi}, i=1…n

8. Experimental Results

9. Experimental Results

10. FlexProt Algorithm • Input:two protein molecules A and B, each being represented by the sequence of the 3-D coordinates of its Caatoms. • Task:largest flexible alignment by decomposing the two molecules into a minimal number of rigid fragment pairs having similar 3-D structure.

12. FlexProt Main Steps Detection of Congruent Rigid Fragment Pairs Joining Rigid Fragment Pairs Rigid Structural Comparison Clustering (removing ins/dels)

13. Congruent Rigid Fragment Pair Structural Similarity Matrix

14. k+l-1 k t+l-1 t Fragkt(l) = vk…vi ...vk+l-1 wt…wj…wt+l-1 RMSD (Fragkt(l) ) < e Detection of Congruent Rigid Fragment Pairs i-1 i+1 i j-1 j+1 j vi-1vivi+1 wj-1 wjwj+1

15. RMSD( P ) RMSD( PUQ ) in O(1) time NOT O( |P|+|Q| ) RMSD( Q ) RMSD Computation Vi…...Vi+l Wj...…Wj+l Vk…...Vk+m Wt...…Wt+m P= Q= PU Q

16. FlexProt Main Steps Detection of Congruent Rigid Fragment Pairs Joining Rigid Fragment Pairs Rigid Structural Comparison Clustering (removing ins/dels)

17. How to Join Rigid Fragment Pairs ?

18. Graph Representation Graph Node Graph Edge

19. Graph Representation • The fragments are in ascending order. • The gaps (ins/dels) are limited. • Allow some overlapping. W a b +Size of the rigid fragment pair (node b) - Gaps (ins/dels) - Overlapping Penalties

20. W_k W_m W_n W_t W_i Graph Representation • DAG (directed acyclic graph)

21. Optimal Solution ? W_k W_m W_n W_t W_i • “All Shortest Paths” • O(|E|*|V|+|V|2) (for DAG) • “Single-source shortest paths” • O(|E|+|V|)

22. FlexProt Main Steps Detection of Congruent Rigid Fragment Pairs Joining Rigid Fragment Pairs Rigid Structural Comparison Clustering (removing ins/dels)

23. Clustering (removing ins/dels) T1 T2 If joining two fragment pairs gives small RMSD (T1 ~ T2) then put them into one cluster.

24. FlexProt Main Steps Detection of Congruent Rigid Fragment Pairs Joining Rigid Fragment Pairs Rigid Structural Comparison Clustering (removing ins/dels)

25. Correspondence Problem

26. Molecular Surface Representation Applications to docking

27. Motivation • Prediction of biomolecular recognition. • Detection of drug binding ‘cavities’. • Molecular Graphics.

28. 1. Solvent Accessible Surface – SAS2. Connolly Surface

29. Connolly’s MS algorithm • A ‘water’ probe ball (1.4-1.8 A diameter) is rolled over the van der Waals surface. • Smoothes the surface and bridges narrow ‘inaccessible’ crevices.

30. Connolly’s MS algorithm - cont. • Convex, concave and saddle patches according to the no. of contact points between the surface atoms and the probe ball. • Outputs points+normals according to the • required sampling density (e.g. 10 pts/A2).

31. Example - the surface of crambin

32. Critical points based on Connolly rep. (Lin, Wolfson, Nussinov) • Define a single point+normal for each patch. • Convex-caps, concave-pits, saddle - belt.

33. Critical point definition

34. Connolly => Shou Lin

35. Solid Angle local extrema hole knob

36. Chymotrypsin surface colored by solid angle (yellow-convex, blue-concave)