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

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={ai} i=1…n B={bj} j=1…m
Pairwise Correspondence:
(ak1,bt1) (ak2,bt2)… (akN,btN)
(1) Exact Matching: ||aki – bti||=0
(2) Bottleneck max ||aki – bti||
(3) RMSD (Root Mean Square Distance)
Sqrt( Σ||aki – bti||2/N)

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

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

7. A 3-D reference frame can be uniquely defined by the ordered vertices of a non-degenerate trianglep1 p2 p3

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. Complexity : O(n3m3 ) * O(nm) .
For the highest ranking hypotheses improve the transformation by replacing it by the best RMSD transformation for all the matching pairs.
Complexity : O(n3m3 ) * O(nm) .
Applying 3D grid gives practically O(n3m3) * O(n)
If one exploits protein backbone geometry + 3D grid :
O(nm) * O(n)

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

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

13. Accept Fij(k) if rmsd(Fij(k)) <e.
Complexity O(n3 n) * O(n) (assume n~m)
(For each Fij(k) we need compute its rmsd)
can be reduced to O(n3) * O(n)

14. Improvement : BLAST idea - detect short similar fragments, then extend as much as possible.
k+l-1
t+l-1 k t
i-1
i+1 i
j-1
j+1 j
ai ai ai+1
bj-1 bj bj+1
Extend while: rmsd(Fij(k)) <e.
Complexity: O(n2)*O(n)

15. Sequence-order Independent Alignment

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

17. Sequence Order Independent Alignment

18. Sequence Order Independent Alignment
2cbl:A
1f4n
1rhg:A
1b3q 51
103
113
169
chain A
chain B 3 58 54 7 73
126
171
147 34 12
chain A
chain B
306
355
354
305

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

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

21. Some Links Rasmol – Molecular Visualization
SCOP - Structural Classification of Proteins
FlexProt  (pairwise flexible alignment)
MultiProt  (multiple structural alignment)
MASS  (multiple structural alignment by secondary structures)
PatchDock  (molecule docking)
SiteEngine  (recognition of functional sites in protein structures)
3D-Jury Protein Structure Prediction