Structure Alignment in Polynomial Time

1
Structure Alignment in Polynomial Time

• Rachel Kolodny
• Stanford University
• Nati Linial
• The Hebrew University of Jerusalem

2
Problem Statement

• 2 structures in R3Aa1,a2,,an,
  Bb1,b2,,bm
• Find subsequences sa and sb s.t the
  substructuresasa(1),asa(2),,
  asa(l),bsb(1),bsb(2),, bsb(l) are similar

3
Motivation

• Structure is better conserved than amino acid
  sequence
• Structure similarity can give hints to common
  functionality/origin
• Allows automatic classification of protein
  structure

4
Correspondence ? Position

• Given a correspondence the rotation and
  translation that minimize the cRMS distance can
  be calculated
• Kabsch, W. (1978).

5
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

6
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

7
Score ?? cRMS
A specific correspondence
8
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
9

• 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

10
Focus on Scoring Functions
11
Focus on Scoring Functions
12
All Maxima are interesting
Noisy data !!
13
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

14
Sampling is Sufficient
15
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

16
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

17
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

• Harder than MAX-CLIQUE
• Matrices encode distances that are positive,
  symmetric and obey triangle inequality

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
18
Example
Best STRUCTAL score 149 Best score found by
exhaustive search 197
19
Heuristic

• Consider only translations that positions an atom
  from protein A on an atom of protein B
• O(mn) instead of O((nm)3)