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.
Assume transformation T is given
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. 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.

6. Largest Common Point Set (LCP) problem
Given e>0 and two point sets A and B find a transformation T and equally sized subsets A’ (a subset of A) and B’ (a subset of B) of maximal cardinality such that dist(A’,T(B’)) ≤ e.
Bottleneck metric: optimal solution in O(n32.5) C. Ambuhl et al. 2000
RMSD metric: open problem

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

8. 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.
How?
-> maximal bipartite matching
(bottleneck metric)

9. Complexity : O(n3m3 ) * O(nm √(m +n) ) .
Can we say something about the quality of the final solution?
YES!
If there is a LCP of size L with error e, then the alignment method detects a LCP of size >= L with error 8e. M.T. Goodrich et al

10. 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.

14. Sequence-order Independent Alignment

15. 4-helix bundle
2cbl:A
1f4n:A
1rhg:A
1b3q

16. Sequence Order Independent Alignment

17. 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

18. 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 :

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