k -Nearest-Neighbors Problem

cRMSD  cRMSD(c,c ’ ) is the minimized RMSD between the two sets of atom centers: min T [(1/n)  i=1, …,n ||a i (c) – T(a i (c’))|| 2 ] 1/2 where the minimization is over all possible rigid-body transform T

k -Nearest-Neighbors Complexity  O(N 2 (log k + L)) –N number of protein conformations to be compared –K number of nearest neighbors –L time to compare two conformations (cRMSD takes linear time).  Solution reduce L by reducing the number of centers to compare -> m- averaging

m-Averaged Approximation  Cut the backbone into fragments of m C  atoms  Replace each fragment by the centroid of the C  atoms

Evaluation: Test Sets [Lotan and Schwarzer, 2003]  FOLDTRAJ random partially unfolded structures -> good correlation with small m (few long segments)  Park-Levitt set [Park et al, 1997] compact native- like structures -> good correlation with large m (many short segments)  Use smaller m on unfolded proteins for greater time savings

Flexible m-averaging  ProteinA 47 residues  14 < r gyr < 24  6 < m < 12 r gyr

Results rgyrmk=100, %correctk=50, %correctk=10, %correct >= >= >= >=  Overhead for calculating and m-averaged structures and r gyration too high  Without averaging 28 sec and for all constant m’s 1 min  With flexible average 2 mins 20 sec  Easily fixed by precalculating r gyr and structures

Uses U F

Conclusions  Flexible m-averaging can save time (without sacrificing accuracy?)  Useful for quickly finding k nearest neighbors and building roadmaps  Precalculate m-averaged structures and r gyration for greater speed up