"Local Squaring Functions for Non-spherical Templates"
Local Squaring Functions for Non-spherical Templates Charles W. Carter, Jr. For Jeffrey Roach NSF ITR Site Visit, 14 November 2002 Local Squaring Functions O(r,y) = ∫v|r(X)ro(x-y) - ro(x-y)|2dx3 r0(x-y) (spherical FT template) QuickTime™ and a GIF decompressor are need ed to see this picture. r(x) Observed density Oh = (fh(2))/|V| ∑kFkFh-k - 2Fh(fh(3))* + d(h)K Roach, J.M. and Carter, C.W., Jr. (2002) Acta Cryst A58:215-220 Using Local Squaring Functions Fourier coefficients, Oh, facilitate simultaneous evaluation of O(r,x) throughout the unit cell: O(r,x) = 1/V ∑h Oh exp(2pih·x) O(r,x) enables comprehensive density modification within the molecular envelope => phase refinement Phase improvements average 20-30o. Map correlation coefficients improve by 15-20%. Probability distributions built from O(r,x) using different atom types => automatic map interpretation. Beyond Spherical Atomic Templates Surprising problem: cannot resolve C and O atoms in C=O, even at 0.9Å resolution!!! Spherical templates overcome the limitation on atom types, but not limited resolution. Extending the utility of LSFs requires using non- spherical templates. Local Squaring Functions for Non-spherical Templates Must model orientation and translation: O(r,q,y) = ∫v|r(X)ro(x-y) - ro(x-y)|2dx3 + K(q Orientation, q, parameterized by quaternions of unit magnitude. As a classical group: special unitary group of dimension 2 (SU2). Irreducible unitary representations give rotational Fourier series. Fragments with symmetry, eg., C=O, require theory of homogeneous spaces. Initial Application Three backbone molecular fragments Carbonyl, C. C T M Trans-peptide link, T. Main chain residue, M. Others possible… Sampling of orientations Carbonyl- 65 points evenly placed 3D sphere Others- 97 points evenly placed 4D sphere Use LSF to locate and orient fragments Conotoxin, 17 amino acids 2.0 A Multiple fragments give redundant information on possible location and possible orientation 9 residues had two or more fragments 4 residues had only one fragment 3 residues: no template placed on carbonyl carbon; however, templates for neighboring residues overlap Some Problems Model incorrect, but density reasonable Can be overcome with good data analysis Larger Applications: TrpRS Objective: Phase refinement and automated initial model building Better searching of more orientations and more fragments Distributed implementation Bezier/NURBS interpolation on sphere and on cartographic projection Data analysis reducing LSFs and electron- density to initial model Orientation Data Structure 400 orientations sampled uniformly on 4D sphere 2 level 20-ary tree 200 unique orientations Directory structure mimics orientation tree via symbolic links 3 level 20-ary tree has 2851 unique orientations V’Ger Star Trek: The Movie Contemporary sense- Aminoacyl-tRNA synthetases antisense related genes Class I aaRS Dehydrogenases Myosin Class I: F1 ATPase Rossmann fold Nucleotide biosynthesis G-proteins ???? Class II aaRS HSP70 Class II: Actin Antiparallel b ???? Carter and Duax, (2002) Mol. Cell., 10:705-708