# Local Squaring Functions for Non-spherical Templates by alllona

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```									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) - ro(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) - 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

 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
   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

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