Local Squaring Functions for Non-spherical Templates by alllona


									Local Squaring Functions for
  Non-spherical Templates

          Charles W. Carter, Jr.
           Jeffrey Roach
NSF ITR Site Visit, 14 November 2002
 Local Squaring Functions
        O(r,y) = ∫v|r(X)ro(x-y) - ro(x-y)|2dx3

FT    template)
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                                                               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) - 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.
            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
   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

Star Trek:
The Movie
                             Contemporary sense-
Aminoacyl-tRNA synthetases   antisense related genes

                              Class I aaRS
 Class I:                     F1 ATPase
 Rossmann fold                Nucleotide biosynthesis

                              Class II aaRS
  Class II:                   Actin
  Antiparallel b              ????

                             Carter and Duax, (2002)
                             Mol. Cell., 10:705-708

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