Local Squaring Functions Local Squaring Functions for Non-spherical for Non-spherical Templates Templates Charles W. Carter, Jr. Charles W. Carter, Jr. For For Jeffrey Roach Jeffrey Roach NSF ITR Site Visit, 14 NSF ITR Site Visit, 14 November 2002 November 2002
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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. - PowerPoint PPT Presentation
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Local Squaring Functions Local Squaring Functions for Non-spherical for Non-spherical
TemplatesTemplatesCharles W. Carter, Jr.Charles W. Carter, Jr.
For For
Jeffrey RoachJeffrey Roach
NSF ITR Site Visit, 14 November NSF ITR Site Visit, 14 November 20022002
LocalLocal Squaring Squaring FunctionsFunctions
0(x-y)(sphericaltemplate)
(x)Observed density
O(,y) = ∫v|(X)o(x-y) - o(x-y)|2dx3
FT
Oh = (fh(2))/|V| ∑kFkFh-k - 2Fh(fh
(3))* + (h)K
Roach, J.M. and Carter, C.W., Jr. (2002) Acta Cryst A58:215-220
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Using Local Squaring FunctionsUsing Local Squaring Functions
Fourier coefficients, Fourier coefficients, OOhh, facilitate simultaneous , facilitate simultaneous evaluation of O(evaluation of O(,x) throughout the unit cell: ,x) throughout the unit cell: O(O(,x) = 1/V ,x) = 1/V ∑h OOh h expexp(2(2ih·x)ih·x)
O(O(,x) enables ,x) enables comprehensive density comprehensive density modificationmodification within the molecular envelope => within the molecular envelope => phase refinement phase refinement Phase improvements average 20-30Phase improvements average 20-30oo. . Map correlation coefficients improve by 15-20%. Map correlation coefficients improve by 15-20%.
Probability distributions built from O(Probability distributions built from O(,x) using ,x) using different atom types => automatic map different atom types => automatic map interpretation.interpretation.
Surprising problem: cannot resolve C and Surprising problem: cannot resolve C and O atoms in C=O, even at 0.9Å O atoms in C=O, even at 0.9Å resolution!!! resolution!!!
Spherical templates overcome the Spherical templates overcome the limitation on atom types, but not limited limitation on atom types, but not limited resolution.resolution.
Extending the utility of LSFs requires Extending the utility of LSFs requires using non-spherical templates.using non-spherical templates.
Local Squaring Local Squaring Functions for Non-Functions for Non-
spherical Templatesspherical Templates Must model Must model orientation and translation:orientation and translation:
O(,,y) = ∫v|(X)o(x-y) - o(x-y)|2dx3 + K(
Orientation, Orientation, , parameterized by , parameterized by quaternions of unit magnitude.quaternions of unit magnitude.
As a classical group: special unitary group As a classical group: special unitary group of dimension 2 (SU2).of dimension 2 (SU2).
Irreducible unitary representations give Irreducible unitary representations give rotational Fourier series.rotational Fourier series.
Fragments with symmetry, eg., C=O, Fragments with symmetry, eg., C=O, require theory of homogeneous spaces.require theory of homogeneous spaces.
Three backbone molecular fragmentsThree backbone molecular fragments Carbonyl, C.Carbonyl, C. Trans-peptide link, T.Trans-peptide link, T. Main chain residue, M.Main chain residue, M. Others possible…Others possible…
Sampling of orientationsSampling of orientations Carbonyl- 65 points evenly placed 3D sphereCarbonyl- 65 points evenly placed 3D sphere Others- 97 points evenly placed 4D sphereOthers- 97 points evenly placed 4D sphere
Use LSF to Use LSF to locate and orientlocate and orient fragments fragments
Initial ApplicationInitial Application
T MC
Conotoxin, 17 amino Conotoxin, 17 amino acidsacids2.0 A2.0 A
Multiple fragments give redundant Multiple fragments give redundant information on possible location information on possible location and possible orientationand possible orientation 9 residues had two or more fragments9 residues had two or more fragments 4 residues had only one fragment4 residues had only one fragment 3 residues: no template placed on 3 residues: no template placed on
carbonyl carbon; however, templates carbonyl carbon; however, templates for neighboring residues overlapfor neighboring residues overlap
Some ProblemsSome Problems
Model incorrect, but density reasonableModel incorrect, but density reasonable Can be overcome with good data analysisCan be overcome with good data analysis