1. Model Database

2. Scene

3. Recognition Lamdan, Schwartz, Wolfson, “Geometric Hashing”,1988.

4. Geometric Matching task = Geometric Pattern Discovery

5. Inexact Alignment. Simple case – two closely related proteins with the same number of amino acids. T Question: how to measure alignment error?

6. Superposition - best least squares (RMSD – Root Mean Square Deviation) Given two sets of 3-D points : P={p i }, Q={q i }, i=1,…,n; rmsd(P,Q) = √  i |p i - q i | 2 /n Find a 3-D rigid transformation T * such that: rmsd( T * (P), Q ) = min T √  i |p i - q i | 2 /n A closed form solution exists for this task. It can be computed in O(n) time.

7. Structure Alignment (Straightforward Algorithm) For each pair of triplets, one from each molecule which define ‘almost’ congruent triangles compute the rigid transformation that superimposes them. Count the number of point pairs, which are ‘almost’ superimposed and sort the hypotheses by this number.

8. For the highest ranking hypotheses improve the transformation by replacing it by the best RMSD transformation for all the matching pairs. Complexity : assuming order of n points in both molecules - O(n 8 ). O(n 4 ) if one exploits protein backbone geometry.

9. Accuracy improvement during detection of 3D transformation. Instead of 3 points use more. How many? Align any possible pair of fragments - F ij (k) i j i+k-1 j+k-1

10. Accept F ij (k) if rmsd( F ij (k) ) <  Complexity O(n 3 n). (For each F ij (k) we need compute its rmsd) can be reduced to O(n 3 )

11. Improvement : BLAST idea - detect short similar fragments, then extend as much as possible. j i+1 j+1 i j-1 i-1 a i-1 a i a i+1 b j-1 b j b j+1 k t k+l-1 t+l-1 Complexity: O(n 2 ) Extend while: rmsd( F ij (k) ) < 

12. Protein Structural Alignment based on Geometric Hashing

13. Sequence Based Structure Alignment Run pairwise sequence alignment. Based on sequence correspondence compute 3D transformation (least square fit can be applied). Iteratively improve structural superposition.

14. Alignment of Flexible Molecular Structures

15. 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.

17. Motivation

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

19. Examples

20. Experimental Results

21. largest flexible alignment by decomposing the two molecules into a minimal number of rigid fragment pairs having similar 3-D structure.Task: largest flexible alignment by decomposing the two molecules into a minimal number of rigid fragment pairs having similar 3-D structure.