scholarly journals Wilson statistics: derivation, generalization and applications to electron cryomicroscopy

2021 ◽  
Vol 77 (5) ◽  
Author(s):  
Amit Singer
Keyword(s):  
Bayesian Inference ◽  
Theoretical Foundation ◽  
Three Dimensional ◽  
Electron Cryomicroscopy ◽  
High Frequencies ◽  
X Ray ◽  
X Ray Crystallography ◽  
Bayesian Inference Framework ◽  
Higher Order Spectra ◽  
Mathematical Derivation

The power spectrum of proteins at high frequencies is remarkably well described by the flat Wilson statistics. Wilson statistics therefore plays a significant role in X-ray crystallography and more recently in electron cryomicroscopy (cryo-EM). Specifically, modern computational methods for three-dimensional map sharpening and atomic modelling of macromolecules by single-particle cryo-EM are based on Wilson statistics. Here the first rigorous mathematical derivation of Wilson statistics is provided. The derivation pinpoints the regime of validity of Wilson statistics in terms of the size of the macromolecule. Moreover, the analysis naturally leads to generalizations of the statistics to covariance and higher-order spectra. These in turn provide a theoretical foundation for assumptions underlying the widespread Bayesian inference framework for three-dimensional refinement and for explaining the limitations of autocorrelation-based methods in cryo-EM.

scholarly journals Wilson Statistics: Derivation, Generalization, and Applications to Electron Cryomicroscopy

2021 ◽  
Author(s):  
Amit Singer
Keyword(s):  
Bayesian Inference ◽  
Theoretical Foundation ◽  
Three Dimensional ◽  
Electron Cryomicroscopy ◽  
High Frequencies ◽  
X Ray ◽  
X Ray Crystallography ◽  
Bayesian Inference Framework ◽  
Higher Order Spectra ◽  
Mathematical Derivation

The power spectrum of proteins at high frequencies is remarkably well described by the flat Wilson statistics. Wilson statistics therefore plays a significant role in X-ray crystallography and more recently in electron cryomicroscopy (cryo-EM). Specifically, modern computational methods for three-dimensional map sharpening and atomic modelling of macromolecules by single particle cryo-EM are based on Wilson statistics. Here we provide the first rigorous mathematical derivation of Wilson statistics. The derivation pinpoints the regime of validity of Wilson statistics in terms of the size of the macromolecule. Moreover, the analysis naturally leads to generalizations of the statistics to covariance and higher order spectra. These in turn provide theoretical foundation for assumptions underlying the widespread Bayesian inference framework for three-dimensional refinement and for explaining the limitations of autocorrelation based methods in cryo-EM.

scholarly journals Structure of birnavirus-like particles determined by combined electron cryomicroscopy and X-ray crystallography

2005 ◽  
Vol 86 (8) ◽  
pp. 2339-2346 ◽  
Author(s):  
Joan Pous ◽  
Christophe Chevalier ◽  
Malika Ouldali ◽  
Jorge Navaza ◽  
Bernard Delmas ◽  
...  
Keyword(s):  
Three Dimensional ◽  
Infectious Pancreatic Necrosis Virus ◽  
Icosahedral Symmetry ◽  
Reconstruction Method ◽  
Rna Packaging ◽  
Electron Cryomicroscopy ◽  
X Ray ◽  
X Ray Crystallography ◽  
Bursal Disease ◽  
Pancreatic Necrosis Virus

Birnaviruses possess a capsid with a single protein layer in contrast to most double-stranded RNA viruses infecting multicellular eukaryotes. Using freeze-drying and heavy metal shadowing, the capsids of two birnaviruses, infectious bursal disease virus (IBDV) and infectious pancreatic necrosis virus, as well as of an IBDV virus-like particle (VLP) are shown to follow the same T=13 laevo icosahedral geometry. The structure of the VLP was determined at a resolution of approximately 15 Å (1·5 nm) by a combination of electron cryomicroscopy and a recently developed three-dimensional reconstruction method, where the scattering density is expressed in terms of symmetry-adapted functions. This reconstruction methodology is well adapted to the icosahedral symmetry of viruses and only requires a small number of images to analyse. The atomic model of the external capsid protein, VP2, recently determined by X-ray crystallography, fits well into the VLP reconstruction and occupies all the electron densities present in the map. Thus, similarly to the IBDV virion, only VP2 forms the icosahedral layer of the VLP. The other components of both VLP and IBDV particles that play a crucial role in the capsid assembly, VP1, VP3 and the peptides arising from the processing of pVP2, do not follow the icosahedral symmetry, allowing them to be involved in other processes such as RNA packaging.

scholarly journals Congruent Docking of Dimeric Kinesin and ncd into Three-dimensional Electron Cryomicroscopy Maps of Microtubule–Motor ADP Complexes

1999 ◽  
Vol 10 (6) ◽  
pp. 2063-2074 ◽  
Author(s):  
Keiko Hirose ◽  
Jan Löwe ◽  
Maria Alonso ◽  
Robert A. Cross ◽  
Linda A. Amos
Keyword(s):  
Conformational Changes ◽  
Bound States ◽  
Three Dimensional ◽  
Coiled Coil ◽  
Direct Interaction ◽  
Electron Cryomicroscopy ◽  
X Ray ◽  
X Ray Crystallography ◽  
Microtubule Motor ◽  
Atomic Coordinates

We present a new map showing dimeric kinesin bound to microtubules in the presence of ADP that was obtained by electron cryomicroscopy and image reconstruction. The directly bound monomer (first head) shows a different conformation from one in the more tightly bound empty state. This change in the first head is amplified as a movement of the second (tethered) head, which tilts upward. The atomic coordinates of kinesin·ADP dock into our map so that the tethered head associates with the bound head as in the kinesin dimer structure seen by x-ray crystallography. The new docking orientation avoids problems associated with previous predictions; it puts residues implicated by proteolysis-protection and mutagenesis studies near the microtubule but does not lead to steric interference between the coiled-coil tail and the microtubule surface. The observed conformational changes in the tightly bound states would probably bring some important residues closer to tubulin. As expected from the homology with kinesin, the atomic coordinates of nonclaret disjunctional protein (ncd)·ADP dock in the same orientation into the attached head in a map of microtubules decorated with dimeric ncd·ADP. Our results support the idea that the observed direct interaction between the two heads is important at some stages of the mechanism by which kinesin moves processively along microtubules.

3-D reconstruction of molecular shape from microcrystal thin sections

1983 ◽  
Vol 41 ◽  
pp. 748-749
Author(s):  
S. Cusack ◽  
J.-C. Jésior
Keyword(s):  
Three Dimensional ◽  
Molecular Shape ◽  
Biological Molecules ◽  
Fourier Space ◽  
Single Particles ◽  
Thin Sections ◽  
Standard Methods ◽  
X Ray ◽  
X Ray Crystallography ◽  
Dimensional Reconstruction

Three-dimensional reconstruction techniques using electron microscopy have been principally developed for application to 2-D arrays (i.e. monolayers) of biological molecules and symmetrical single particles (e.g. helical viruses). However many biological molecules that crystallise form multilayered microcrystals which are unsuitable for study by either the standard methods of 3-D reconstruction or, because of their size, by X-ray crystallography. The grid sectioning technique enables a number of different projections of such microcrystals to be obtained in well defined directions (e.g. parallel to crystal axes) and poses the problem of how best these projections can be used to reconstruct the packing and shape of the molecules forming the microcrystal.Given sufficient projections there may be enough information to do a crystallographic reconstruction in Fourier space. We however have considered the situation where only a limited number of projections are available, as for example in the case of catalase platelets where three orthogonal and two diagonal projections have been obtained (Fig. 1).

Three-dimensional crystallization of membrane proteins

1988 ◽  
Vol 21 (4) ◽  
pp. 429-477 ◽  
Author(s):  
W. Kühlbrandt
Keyword(s):  
High Resolution ◽  
Membrane Proteins ◽  
Membrane Protein ◽  
Protein Complexes ◽  
Three Dimensional ◽  
Biological Macromolecules ◽  
X Ray ◽  
X Ray Crystallography ◽  
Membrane Protein Complexes ◽  
Essential Prerequisite

As recently as 10 years ago, the prospect of solving the structure of any membrane protein by X-ray crystallography seemed remote. Since then, the threedimensional (3-D) structures of two membrane protein complexes, the bacterial photosynthetic reaction centres of Rhodopseudomonas viridis (Deisenhofer et al. 1984, 1985) and of Rhodobacter sphaeroides (Allen et al. 1986, 1987 a, 6; Chang et al. 1986) have been determined at high resolution. This astonishing progress would not have been possible without the pioneering work of Michel and Garavito who first succeeded in growing 3-D crystals of the membrane proteins bacteriorhodopsin (Michel & Oesterhelt, 1980) and matrix porin (Garavito & Rosenbusch, 1980). X-ray crystallography is still the only routine method for determining the 3-D structures of biological macromolecules at high resolution and well-ordered 3-D crystals of sufficient size are the essential prerequisite.

scholarly journals Structure of a 14-3-3 Coordinated Hexamer of the Plant Plasma Membrane H+-ATPase by Combining X-Ray Crystallography and Electron Cryomicroscopy

2007 ◽  
Vol 25 (3) ◽  
pp. 427-440 ◽  
Author(s):  
Christian Ottmann ◽  
Sergio Marco ◽  
Nina Jaspert ◽  
Caroline Marcon ◽  
Nicolas Schauer ◽  
...  
Keyword(s):  
Plasma Membrane ◽  
Electron Cryomicroscopy ◽  
X Ray ◽  
X Ray Crystallography ◽  
Plant Plasma Membrane

scholarly journals Low-Zpolymer sample supports for fixed-target serial femtosecond X-ray crystallography

2015 ◽  
Vol 48 (4) ◽  
pp. 1072-1079 ◽  
Author(s):  
Geoffrey K. Feld ◽  
Michael Heymann ◽  
W. Henry Benner ◽  
Tommaso Pardini ◽  
Ching-Ju Tsai ◽  
...  
Keyword(s):  
Protective Antigen ◽  
Three Dimensional ◽  
Two Dimensional ◽  
X Ray Diffraction ◽  
Fixed Target ◽  
X Ray ◽  
X Ray Crystallography ◽  
Transmission Electron ◽  
Linac Coherent Light Source ◽  
Efficient Alternative

X-ray free-electron lasers (XFELs) offer a new avenue to the structural probing of complex materials, including biomolecules. Delivery of precious sample to the XFEL beam is a key consideration, as the sample of interest must be serially replaced after each destructive pulse. The fixed-target approach to sample delivery involves depositing samples on a thin-film support and subsequent serial introductionviaa translating stage. Some classes of biological materials, including two-dimensional protein crystals, must be introduced on fixed-target supports, as they require a flat surface to prevent sample wrinkling. A series of wafer and transmission electron microscopy (TEM)-style grid supports constructed of low-Zplastic have been custom-designed and produced. Aluminium TEM grid holders were engineered, capable of delivering up to 20 different conventional or plastic TEM grids using fixed-target stages available at the Linac Coherent Light Source (LCLS). As proof-of-principle, X-ray diffraction has been demonstrated from two-dimensional crystals of bacteriorhodopsin and three-dimensional crystals of anthrax toxin protective antigen mounted on these supports at the LCLS. The benefits and limitations of these low-Zfixed-target supports are discussed; it is the authors' belief that they represent a viable and efficient alternative to previously reported fixed-target supports for conducting diffraction studies with XFELs.

Sign in / Sign up

Export Citation Format

Share Document