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Structure Alignment in
Polynomial Time
Rachel Kolodny
Stanford University
Nati Linial
The Hebrew University of Jerusalem
Problem Statement
• 2 structures in R3
A={a1,a2,…,an}, B={b1,b2,…,bm}
• Find subsequences sa and sb
s.t the substructures
{as (1),as (2),…, as (l)},{bs (1),bs (2),…, bs (l)}
a
a
are similar
a
b
b
b
Motivation
• Structure is better conserved than
amino acid sequence
– Structure similarity can give hints to
common functionality/origin
• Allows automatic classification of
protein structure
Correspondence  Position
• Given a correspondence the rotation
and translation that minimize the
cRMS distance can be calculated
Kabsch, W. (1978).
Position  Correspondence
• Given a rotation and translation one
can calculate the alignment that
optimizes a (separable) score
– Using dynamic programming
– Essentially similar to sequence alignment
• Example score
20
 # gaps 10

2
icorrespondance 1  d ( Ai , Bi ) / 5
Score  cRMS
• We want to give “bonus points” for longer
correspondences
– e.g. corresponding ONE atom from each
structure has 0 cRMS
• Even better scores ?
– vary gap penalty depending on position in
structure
– Incorporate sequence information
Score  cRMS
A specific correspondence
Previous Work
Distance Matrices
Heuristics in rotation
and translation space
DALI [Holm and Sander 93]
CONGENEAL [Yee & Dill 93]
SSAP [Taylor & Orengo 89]
Nussinov-Wolfson [89,93]
Godzik [93]
STRUCTAL [Subibiah et al 93]
COMPARER [Sali & Blundell 90]
LOCK [Singh & Brutlag 97]
CE [Shindyalov & Bourne 98]
…
Taylor (??) [93]
Zu-Kang & Sipppl 96 (?)
…
*most data taken from Orengo 94
“…It can be proved that, for these reasons, finding an optimal
structural alignment between two protein structures is an NP
hard problem and thus there are no fast structural alignment
algorithms that are guaranteed to be optimal within any given
similarity measure…”
Adam Godzik
‘The structural alignment
between two proteins: Is there a unique
answer’ 1996
“There is no exact solution to the protein structure alignment
problem, only the best solution for the heuristics used in the
calculation.”
Shindyalov & Bourne
‘Protein Structure Alignment by
Incremental Combinatorial (CE) of
the Optimal Path’ 1998
Focus on Scoring Functions
Exponentially
many
Focus on Scoring Functions
Exponentially
many
All Maxima are interesting
Noisy data !!
Exponentially
many
Good scoring functions
• Each of the functions is well-behaved
– Satisfies Lipschitz condition
• Thus, the maximum over a finite set is
well-behaved
• In each dimension two points at distance 
have function values that vary by O(n)
• Need O(n) samples in every dimension
Sampling is Sufficient
Polynomial Algorithm
• Sample in rotation and translation
space
– compute best score (and alignment) for
each sample point
• Return maximum score
• Need O(n6n2) time and O(n2) space
Internal Distance Matrices
• Invariant to position
and rotation of
structures  can be
compared directly
• Find largest common
sub-matrices (LCM)
whose distances are
roughly the same
LCM is NP-complete
0
1
2
3
2
3
3
4
5
2
1
0
1
2
1
1
2
3
4
1
2
1
0
3
2
2
3
4
5
2
3
2
3
0
1
2
3
4
5
2
2
1
2
1
0
1
2
3
4
1
3
1
2
2
1
0
1
2
3
1
3
2
3
3
2
1
0
1
2
2
4
3
4
4
3
2
1
0
1
3
5
4
5
5
4
3
2
1
0
4
2
1
2
2
1
1
2
3
4
0
0
1
1
1
1
1
1
0
1
1
1
1
1
1
0
1
1
1
1
1
1
0
1
1
1
1
1
1
0
1
1
1
1
1
1
0
• Harder than MAXCLIQUE
• Matrices encode
distances that are
positive, symmetric and
obey triangle inequality
Example
1dme
28 amino acids
1jjd
51 amino acids
Best STRUCTAL score 149
Best score found by exhaustive search 197
Heuristic
• Consider only
translations that
positions an atom
from protein A on
an atom of protein
B
• O(m*n) instead of
O((n+m)3)