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A 3-D reference frame can be uniquely defined by the ordered vertices of a non-degenerate triangle

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##### A 3-D reference frame can be uniquely defined by the ordered vertices of a non-degenerate triangle

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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**Pattern Matching:Naive 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.**Naive 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.)**Object Recognition Techniques**• Pose Clustering • Geometric Hashing (and more …)**Pose 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**Pose Clustering (2)**• Problem: How to compare 2 transformations? Solutions: • Straightforward • Based on transformed points**Geometric Hashing**• Inavriant geometric relations • Store in fast look-up table**Geometric 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.**Geometric 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.).**Geometric 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.**Complexity of Geometric Hashing**O(n4 + n4 * BinSize) ~ O(n5 ) (Naive alg. O(n7))**Advantages :**• 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**Structural Comparison Algorithms**• Ca backbone matching. • Secondary structure configuration matching. • Molecular surface matching. • Multiple Structure Alignment. • Flexible (Hinge - based) structural alignment.**Protein 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