Protein%20Structure%20Alignment

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Protein Structure Alignment
Human Hemoglobin alpha-chain pdb1jebA
Human Myoglobin pdb2mm1
Another example G-Proteins 1c1yA,
1kk1A6-200 Sequence id 18 Structural id 72
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Transformations

• Translation
• Translation and Rotation
• Rigid Motion (Euclidian Trans.)
• Translation, Rotation Scaling

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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?
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Distance Functions

• Two point sets Aai i1n
• Bbj j1m
• Pairwise Correspondence
• (ak1,bt1) (ak2,bt2) (akN,btN)

(1) Exact Matching aki bti0
(2) Bottleneck max aki bti (3) RMSD
(Root Mean Square Distance) Sqrt(
Saki bti2/N)
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Correspondence is Unknown
Given two configurations of points in the
three dimensional space,
find those rotations and translations of one
of the point sets which produce large
superimpositions of corresponding 3-D
points.
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Largest Common Point Set (LCP) problem
Given egt0 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
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A 3-D reference frame can be uniquely defined by
the ordered vertices of a non-degenerate triangle
p1
p2
p3
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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.

-gt maximal bipartite matching (bottleneck
metric)
How?
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• Complexity O(n3m3 ) O(nm v(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 gt L
with error 8e. M.T. Goodrich et al. 1994.
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Superposition - best least squares(RMSD Root
Mean Square Deviation)
Given two sets of 3-D points Ppi, Qqi ,
i1,,n rmsd(P,Q) v S ipi - qi 2 /n Find a
3-D rigid transformation T such that rmsd(
T(P), Q ) minT v S iT(pi) - qi 2 /n
A closed form solution exists for this task. It
can be computed in O(n) time.
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• Sequence-order Independent Alignment

P
Q
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4-helix bundle
2cblA
1f4nA
1rhgA
1b3q
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Sequence Order Independent Alignment
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Sequence Order Independent Alignment
2cblA 1f4n 1rhgA 1b3q
51
103
113
169
chain A
chain B
3
58
54
7
73
126
34
12
171
147
chain A
chain B
306
355
354
305
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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 1996 5 2375-2390
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TRAF-Immunoglobulin Ensemble
E- strand

• Ensemble 8 proteins from 2 folds.
• Core sandwich of 6 strands
• Runtime 21 seconds

- helices - strands