Transcript Local Squaring Functions for Non-spherical

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
FT
r0(x-y)
(spherical
template)
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are needed 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
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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
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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
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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 nonspherical templates.
Local Squaring Functions for
Non-spherical Templates
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Must model orientation and translation:
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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
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Three backbone molecular fragments
Carbonyl, C.
 Trans-peptide link, T.
 Main chain residue, M.
 Others possible…
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C
T
M
Sampling of orientations
Carbonyl- 65 points evenly placed 3D sphere
 Others- 97 points evenly placed 4D sphere
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Use LSF to locate and orient fragments
Conotoxin, 17 amino acids
2.0 A
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Multiple fragments give redundant
information on possible location and
possible orientation
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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
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Model incorrect, but density reasonable
Can be overcome with good data analysis
Larger Applications: TrpRS
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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 electrondensity to initial model
Orientation Data Structure
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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
Aminoacyl-tRNA synthetases
Class I:
Rossmann fold
Class II:
Antiparallel b
Contemporary senseantisense related genes
Class I aaRS
Dehydrogenases
Myosin
F1 ATPase
Nucleotide biosynthesis
G-proteins
????
Class II aaRS
HSP70
Actin
????
Carter and Duax, (2002)
Mol. Cell., 10:705-708