1. Find the optimal alignment ? +

2. Optimal Alignment Find the highest number of atoms aligned with the lowest RMSD (Root Mean Squared Deviation) Find a balance between local regions with very good alignments and overall alignment

3. Geometric Matching task = Geometric Pattern Discovery

4. Structure Comparison Requirements 1.Which atom in structure A corresponds to what atom in structure B ? Answer: Sequence alignments THESESENTENCESALIGN--NICLEY ||| ||| || || ||||| |||||| THE—SEQ-EN-CE-ALIGNEDNICELY

5. Structure Comparison Requirements 2. What are the locations of atoms in the structures ? Answer: PDB-files (Dihedral angles, bond lengths …) Source: http://www.rcsb.org/pdb

6. Structure Comparison Requirements 3.Methods to superimpose structures Translation Rotation Answer: Translation and Rotation x 1, y 1, z 1 x 2, y 2, z 2 x 3, y 3, z 3 x 1 + d, y 1, z 1 x 2 + d, y 2, z 2 x 3 + d, y 3, z 3

7. Transformations Translation Translation and Rotation Rigid Motion (Euclidian Trans.) Translation, Rotation + Scaling

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

9. Distance Functions Two point sets: A={a i } i=1…n B={b j } j=1…m Pairwise Correspondence: (a k 1,b t 1 ) (a k 2,b t 2 )… (a k N,b t N ) (1) Exact Matching: ||a k i – b t i ||=0 (2) Bottleneck max ||a k i – b t i || (3) RMSD (Root Mean Square Distance) Sqrt( Σ||a k i – b t i || 2 /N)

10. 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 |T(p i ) - q i | 2 /n A closed form solution exists for this task. It can be computed in O(n) time.

11. RMSD Unit of RMSD => e.g. Ångstroms -identical structures => RMSD = “0” -similar structures => RMSD is small (1 – 3 Å) -distant structures => RMSD > 3 Å

12. Pitfalls of RMSD all atoms are treated equally (e.g. residues on the surface have a higher degree of freedom than those in the core) best alignment does not always mean minimal RMSD significance of RMSD is size dependent

13. Correspondence is Unknown find those rotations and translations of one of the point sets which produce “large” superimpositions of corresponding 3-D points. Given two configurations of points in the three dimensional space, T

14. 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.

15. A 3-D reference frame can be uniquely defined by the ordered vertices of a non- degenerate triangle p1p1 p2p2 p3p3

16. 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) ) < 

17. Protein zinc finger (4znf)

18. Superimposed 3znf and 4znf Lys30 30 CA atoms RMS = 0.70Å 248 atoms RMS = 1.42Å

19. Superimposed 3znf and 4znf backbones Lys30 30 CA atoms RMS = 0.70Å 248 atoms RMS = 1.42Å 30 CA atoms RMS = 0.70Å