Title: A 3-D reference frame can be uniquely defined by the ordered vertices of a non-degenerate triangle
1A 3-D reference frame can be uniquely defined by
the ordered vertices of a non-degenerate triangle
p1
p2
p3
2Pattern MatchingNaive algorithm
- For each pair of triplets, one from each molecule
which define almost congruent triangles compute
the rigid motion that superimposes them. - Count the number of point pairs, which are
almost superimposed and sort the hypotheses by
this number.
3Naive algorithm (continued )
- For the highest ranking hypotheses improve the
transformation by replacing it by the best RMSD
transformation for all the matching pairs. - Complexity assuming order of n points in both
molecules - O(n7) . - (O(n3) if one exploits protein backbone
geometry.)
4Object Recognition Techniques
- Pose Clustering
- Geometric Hashing
- (and more )
5Pose Clustering
- Clustering of transformations.
- Match each triplet from the first object with
triplet from the second object. - A triplet defines 3D transformation. Store it
using appropriate data structure. - High scoring alignments will result in dense
clusters of transformations.
Time Complexity O(n3m3)Clustering
6Pose Clustering (2)
- Problem How to compare 2 transformations?
-
- Solutions
- Straightforward
- Based on transformed points
7Geometric Hashing
- Inavriant geometric relations
- Store in fast look-up table
8Geometric Hashing - Preprocessing
- Pick a reference frame satisfying pre-specified
constraints. - Compute the coordinates of all the other points
(in a pre-specified neighborhood) in this
reference frame. - Use each coordinate as an address to the hash
(look-up) table and record in that entry the
(ref. frame, shape sign.,point). - Repeat above steps for each reference frame.
9Geometric Hashing - Recognition 1
- For the target protein do
- Pick a reference frame satisfying pre-specified
constraints. - Compute the coordinates of all other points in
the current reference frame . - Use each coordinate to access the hash-table to
retrieve all the records (ref.fr., shape sign.,
pt.).
10Geometric Hashing - Recognition 2
- For records with matching shape sign. vote for
the (ref.fr.). - Compute the transformations of the high scoring
hypotheses. - Repeat the above steps for each ref.fr.
- Cluster similar transformation.
- Extend best matches.
11(No Transcript)
12Complexity of Geometric Hashing
O(n4 n4 BinSize) O(n5 )
(Naive alg. O(n7))
13Advantages
- Sequence order independent.
- Can match partial disconnected substructures.
- Pattern detection and recognition.
- Highly efficient.
- Can be applied to protein-protein interfaces,
surface motif detection, docking. - Database Object Recognition a trivial extension
to the method - Parallel Implementation straight forward
14Structural Comparison Algorithms
- Ca backbone matching.
- Secondary structure configuration matching.
- Molecular surface matching.
- Multiple Structure Alignment.
- Flexible (Hinge - based) structural alignment.
15Protein Structural Comparison
PDB files
Feature Extraction
Geometric Matching
Verification and Scoring
Rotation and Translation Possibilities
Least Square Analysis
Ca
Other Inputs
Geometric Hashing
Backbone
Secondary Structures
Transformation Clustering
Flexible Geometric Hashing
H-bonds
Sequence Dependent Weights