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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. r 0 (x-y) (spherical template). FT. r (x) Observed density. O h = (f h (2) )/|V| ∑ k F k F h-k - 2F h (f h (3) )* + d (h)K.

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Local Squaring Functions for Non-spherical Templates

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  1. Local Squaring Functions for Non-spherical Templates Charles W. Carter, Jr. For Jeffrey Roach NSF ITR Site Visit, 14 November 2002

  2. r0(x-y) (spherical template) FT r(x) Observed density Oh = (fh(2))/|V| ∑kFkFh-k - 2Fh(fh(3))* + d(h)K Local Squaring Functions O(r,y) = ∫v|r(X)ro(x-y) - ro(x-y)|2dx3 Roach, J.M. and Carter, C.W., Jr. (2002) Acta CrystA58:215-220

  3. Using Local Squaring Functions • Fourier coefficients, Oh, facilitate simultaneous evaluation of O(r,x) throughout the unit cell: • O(r,x) = 1/V ∑hOh 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.

  4. 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.

  5. Local Squaring Functions for Non-spherical Templates • Must model orientation and translation: • O(r,q,y) = ∫v|r(X)ro(x-y) - ro(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.

  6. T M C Initial Application • Three backbone molecular fragments • Carbonyl, C. • 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

  7. Conotoxin, 17 amino acids2.0 A

  8. 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

  9. Some Problems • Model incorrect, but density reasonable • Can be overcome with good data analysis

  10. 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

  11. 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

  12. V’Ger Star Trek: The Movie

  13. Contemporary sense- antisense related genes Aminoacyl-tRNA synthetases Class I aaRS Dehydrogenases Myosin F1 ATPase Nucleotide biosynthesis G-proteins ???? Class I: Rossmann fold Class II aaRS HSP70 Actin ???? Class II: Antiparallel b Carter and Duax, (2002) Mol. Cell., 10:705-708

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