1. Protein Structure Alignment Human Myoglobin pdb:2mm1 Human Hemoglobin alpha-chain pdb:1jebA Sequence id: 27% Structural id: 90% Another example: G-Proteins: 1c1y:A, 1kk1:A6-200 Sequence id: 18% Structural id: 72%

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

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

4. 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)

5. 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.

6. 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

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

8. 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. Not a good approach – sequence alignment can be incorrect.

9. 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 aligned point pairs and sort the hypotheses by this number.

10. For the highest ranking hypotheses improve the transformation by replacing it by the best RMSD transformation for all the matching pairs. Complexity : O(n 3 m 3 ) * O(nm). Applying 3D grid gives practically O(n 3 m 3 ) * O(n) If one exploits protein backbone geometry + 3D grid : O(nm) * O(n)

11. Structural Alignment Approaches 1.Generate a set of 3D transformations. 2.Compute 3D alignment for each transformation. Two interrelated problems: 3D transformation and point correspondence (matching, alignment) 1.Generate a set of 3D transformations. 2.Cluster similar transformations. 3.Compute 3D alignment for each cluster representative. Geometric Hashing: Combines transformation and correspondence detection in one scheme. Some methods:

12. 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

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

14. 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 )*O(n) Extend while: rmsd( F ij (k) ) < 

15. Sequence-order Independent Alignment P:Q:

16. 4-helix bundle 2cbl:A 1f4n:A 1b3q 1rhg:A

17. Sequence Order Independent Alignment

18. 2cbl:A 1f4n 1rhg:A 1b3q 51103113169 358547 73126 34 12 306355354305 171147 chain A chain B

19. E. A. NALEFSKI and J. J. FALKE The C2 domain calcium-binding motif: Structural and functional diversity Protein Sci 1996 5: 2375-2390 The C2 domain calcium-binding motif

20. TRAF-Immunoglobulin Ensemble - helices ; - strands  Ensemble: 8 proteins from 2 folds.  Core: sandwich of 6 strands  Runtime: 21 seconds E- strand

21. Rasmol – Molecular VisualizationRasmol SCOP - Structural Classification of ProteinsSCOP MultiProt - Protein Structural (pairwise/multiple) AlignmentMultiProt MASS – Secondary Structure Based (pairwise/multiple) AlignmentMASS Some Links