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Alignment of Flexible Molecular Structures Motivation Proteins are flexible. One would like to align proteins modulo the flexibility. Hinge and shear protein domain motions (Gerstein, Lesk , Chotia). Conformational flexibility in drugs. Problem definition Flexible Geometric Hashing Exploit the fact that neighboring parts share the joint - accumulate mutual information at the joint. Achieve complexity of the same order of magnitude as in rigid alignment. Flexible protein alignment without prior hinge knowledge FlexProt - algorithm detects automatically flexibility regions, exploits amino acid sequence order. Motivation Geometric Representation 3-D Curve {vi}, i=1…n Experimental Results Experimental Results FlexProt Algorithm Input: two protein molecules A and B, each being represented by the sequence of the 3-D coordinates of its Ca atoms. Task: largest flexible alignment by decomposing the two molecules into a minimal number of rigid fragment pairs having similar 3-D structure. FlexProt Main Steps Detection of Congruent Rigid Fragment Pairs Joining Rigid Fragment Pairs Clustering (removing ins/dels) Rigid Structural Comparison Structural Similarity Matrix Congruent Rigid Fragment Pair Detection of Congruent Rigid Fragment Pairs k+l-1 k i-1 t i+1 i j-1 j t+l-1 j+1 vi-1 vi vi+1 wj-1 wj wj+1 Fragkt(l) = vk wt … … vi ... vk+l-1 wj … wt+l-1 RMSD (Fragkt(l) ) < e RMSD Computation Vi …... Vi+l P= Wj ...… Wj+l Vk …... Vk+m Q= Wt ...… Wt+m P UQ RMSD( P ) RMSD( P U Q ) in O(1) time RMSD( Q ) NOT O( |P|+|Q| ) FlexProt Main Steps Detection of Congruent Rigid Fragment Pairs Joining Rigid Fragment Pairs Clustering (removing ins/dels) Rigid Structural Comparison How to Join Rigid Fragment Pairs ? Graph Representation Graph Node Graph Edge Graph Representation •The fragments are in ascending order. •The gaps (ins/dels) are limited. •Allow some overlapping. a b + Size of the rigid fragment pair (node b) - Gaps (ins/dels) Penalties - Overlapping Graph Representation W_t • DAG (directed acyclic graph) Optimal Solution ? W_t •“All Shortest Paths” O(|E|*|V|+|V|2) (for DAG) •“Single-source shortest paths” O(|E|+|V|) FlexProt Main Steps Detection of Congruent Rigid Fragment Pairs Joining Rigid Fragment Pairs Clustering (removing ins/dels) Rigid Structural Comparison Clustering (removing ins/dels) T1 T2 If joining two fragment pairs gives small RMSD (T1 ~ T2) then put them into one cluster. FlexProt Main Steps Detection of Congruent Rigid Fragment Pairs Joining Rigid Fragment Pairs Clustering (removing ins/dels) Rigid Structural Comparison Correspondence Problem Molecular Surface Representation Applications to docking Motivation Prediction of biomolecular recognition. Detection of drug binding ‘cavities’. Molecular Graphics. 1. Solvent Accessible Surface – SAS 2. Connolly Surface Connolly’s MS algorithm A ‘water’ probe ball (1.4-1.8 A diameter) is rolled over the van der Waals surface. Smoothes the surface and bridges narrow ‘inaccessible’ crevices. Connolly’s MS algorithm - cont. Convex, concave and saddle patches according to the no. of contact points between the surface atoms and the probe ball. Outputs points+normals according to the required sampling density (e.g. 10 pts/A2). Example - the surface of crambin Critical points based on Connolly rep. (Lin, Wolfson, Nussinov) Define a single point+normal for each patch. Convex-caps, concave-pits, saddle belt. Critical point definition Connolly => Shou Lin Solid Angle local extrema hole knob Chymotrypsin surface colored by solid angle (yellow-convex, blue-concave)