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.

Beyond Spherical Atomic Beyond Spherical Atomic TemplatesTemplates

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

Larger Applications: Larger Applications: TrpRSTrpRS

Objective: Phase refinement and Objective: Phase refinement and automated initial model buildingautomated initial model building

Better searching of more Better searching of more orientations and more fragmentsorientations and more fragments

Distributed implementationDistributed implementation Bezier/NURBS interpolation on Bezier/NURBS interpolation on

sphere and on cartographic sphere and on cartographic projectionprojection

Data analysis reducing LSFs and Data analysis reducing LSFs and electron-density to initial modelelectron-density to initial model

Orientation Data Orientation Data StructureStructure

400 orientations 400 orientations sampled uniformly on sampled uniformly on 4D sphere 4D sphere

2 level 20-ary tree2 level 20-ary tree 200 unique orientations200 unique orientations Directory structure Directory structure

mimics orientation tree mimics orientation tree viavia symbolic links symbolic links

3 level 20-ary tree has 3 level 20-ary tree has 2851 unique 2851 unique orientationsorientations

Star Trek: Star Trek: The MovieThe Movie

V’Ger

Class I:Rossmann fold

Class II:Antiparallel

Aminoacyl-tRNA synthetasesContemporary sense- antisense related genes

Class I aaRSDehydrogenasesMyosinF1 ATPaseNucleotide biosynthesisG-proteins????

Class II aaRSHSP70Actin????

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