Pattern-recognition-based detection of planar objects in three-dimensional electron-density maps

When everything has been done to make the phases as good as possible, the time has come to examine the image of the structure in the form of an electron-density map. The electron-density map is the Fourier transform of the structure factors (with their phases). If the resolution and phases are good enough, the electron-density map may be interpreted in terms of atomic positions. In practice, it may be necessary to alternate between study of the electron-density map and the procedures mentioned in Chapter 10, which may allow improvements to be made to it. Electron-density maps contain a great deal of information, which is not easy to grasp. Considerable technical effort has gone into methods of presenting the electron density to the observer in the clearest possible way. The Fourier transform is calculated as a set of electron-density values at every point of a three-dimensional grid labelled with fractional coordinates x, y, z. These coordinates each go from 0 to 1 in order to cover the whole unit cell. To present the electron density as a smoothly varying function, values have to be calculated at intervals that are much smaller than the nominal resolution of the map. Say, for example, there is a protein unit cell 50 Å on a side, at a routine resolution of 2Å. This means that some of the waves included in the calculation of the electron density go through a complete wave cycle in 2 Å. As a rule of thumb, to represent this properly, the spacing of the points on the grid for calculation must be less than one-third of the resolution. In our example, this spacing might be 0.6 Å. To cover the whole of the 50 Å unit cell, about 80 values of x are needed; and the same number of values of y and z. The electron density therefore needs to be calculated on an array of 80×80×80 points, which is over half a million values. Although our world is three-dimensional, our retinas are two-dimensional, and we are good at looking at pictures and diagrams in two dimensions.

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The phase problem and electron-density maps

Crystal Structure Analysis ◽

10.1093/oso/9780199576340.003.0015 ◽

2010 ◽

Author(s):

Jenny Pickworth Glusker ◽

Kenneth N. Trueblood

Keyword(s):

Crystal Structure ◽

Fourier Series ◽

Diffraction Pattern ◽

Electron Density ◽

Bragg Reflection ◽

Three Dimensional ◽

Density Maps ◽

Structure Factors ◽

Unit Cells ◽

Phase Angles

In order to obtain an image of the material that has scattered X rays and given a diffraction pattern, which is the aim of these studies, one must perform a three-dimensional Fourier summation. The theorem of Jean Baptiste Joseph Fourier, a French mathematician and physicist, states that a continuous, periodic function can be represented by the summation of cosine and sine terms (Fourier, 1822). Such a set of terms, described as a Fourier series, can be used in diffraction analysis because the electron density in a crystal is a periodic distribution of scattering matter formed by the regular packing of approximately identical unit cells. The Fourier series that is used provides an equation that describes the electron density in the crystal under study. Each atom contains electrons; the higher its atomic number the greater the number of electrons in its nucleus, and therefore the higher its peak in an electrondensity map.We showed in Chapter 5 how a structure factor amplitude, |F (hkl)|, the measurable quantity in the X-ray diffraction pattern, can be determined if the arrangement of atoms in the crystal structure is known (Sommerfeld, 1921). Now we will show how we can calculate the electron density in a crystal structure if data on the structure factors, including their relative phase angles, are available. The Fourier series is described as a “synthesis” when it involves structure amplitudes and relative phases and builds up a picture of the electron density in the crystal. By contrast, a “Fourier analysis” leads to the components that make up this series. The term “relative” is used here because the phase of a Bragg reflection is described relative to that of an imaginary wave diffracted in the same direction at a chosen origin of the unit cell.

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Three-Dimensional Electron Density from Tomographic Analysis of LASCO-C2 Images of the K-Corona Total Brightness

Solar Physics ◽

10.1007/s11207-010-9557-9 ◽

2010 ◽

Vol 265 (1-2) ◽

pp. 19-30 ◽

Cited By ~ 22

Author(s):

Richard A. Frazin ◽

Philippe Lamy ◽

Antoine Llebaria ◽

Alberto M. Vásquez

Keyword(s):

Electron Density ◽

Three Dimensional ◽

Dimensional Electron ◽

Tomographic Analysis

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Three-Dimensional Electron Density Mapping of Shape-Controlled Nanoparticle by Focused Hard X-ray Diffraction Microscopy

Nano Letters ◽

10.1021/nl100891n ◽

2010 ◽

Vol 10 (5) ◽

pp. 1922-1926 ◽

Cited By ~ 51

Author(s):

Yukio Takahashi ◽

Nobuyuki Zettsu ◽

Yoshinori Nishino ◽

Ryosuke Tsutsumi ◽

Eiichiro Matsubara ◽

...

Keyword(s):

Electron Density ◽

Three Dimensional ◽

Dimensional Electron ◽

X Ray Diffraction ◽

X Ray ◽

Density Mapping ◽

Shape Controlled ◽

Diffraction Microscopy

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Interactive visualization of electron density slices

Journal of Applied Crystallography ◽

10.1107/s0021889805008289 ◽

2005 ◽

Vol 38 (3) ◽

pp. 563-565 ◽

Cited By ~ 3

Author(s):

Filipe R. N. C. Maia ◽

Abraham Szöke ◽

Warren DeLano ◽

David van der Spoel

Keyword(s):

High Resolution ◽

Quantum Chemistry ◽

Electron Density ◽

Three Dimensional ◽

Structure Model ◽

Dimensional Model ◽

Three Dimensional Model ◽

Molecular Visualization ◽

Density Maps ◽

Quantum Chemistry Calculations

A new tool has been developed to aid in the visualization of electron density in crystals or from quantum chemistry calculations. It displays the fine details of the electron density on a plane and the three-dimensional model of the molecule at the same time. The program enables the user to examine the details of weak or irregular features. Such features frequently occur in low-resolution maps, where they determine the correct tracing of a protein backbone. In high-resolution maps, solvent regions are difficult or impossible to observe using isosurfaces. The tool has been integrated into an existing molecular visualization package (PyMol) making it possible to observe and interact both with a structure model and the electron density slices freely, simultaneously and independently. This visualization model fills a gap in the visualization methods available to crystallographers and others who work with electron density maps.

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Building brain network in electron density map determined by micro-CT

Acta Crystallographica Section A Foundations and Advances ◽

10.1107/s2053273314082473 ◽

2014 ◽

Vol 70 (a1) ◽

pp. C1752-C1752

Author(s):

Rino Saiga ◽

Susumu Takekoshi ◽

Naoya Nakamura ◽

Akihisa Takeuchi ◽

Kentaro Uesugi ◽

...

Keyword(s):

Density Distribution ◽

Electron Density ◽

Electron Density Distribution ◽

Model Building ◽

Three Dimensional ◽

Brain Network ◽

X Ray ◽

Density Maps ◽

Density Map ◽

Electron Density Map

In macromolecular crystallography, an electron density distribution is traced to build a model of the target molecule. We applied this method to model building for electron density maps of a brain network. Human cerebral tissue was stained with heavy atoms [1]. The sample was then analyzed at the BL20XU beamline of SPring-8 to obtain a three-dimensional map of X-ray attenuation coefficients representing the electron density distribution. Skeletonized wire models were built by placing and connecting nodes in the map [2], as shown in the figure below. The model-building procedures were similar to those reported for crystallographic analyses of macromolecular structures, while the neuronal network was automatically traced by using a Sobel filter. Neuronal circuits were then analytically resolved from the skeletonized models. We suggest that X-ray microtomography along with model building in the electron density map has potential as a method for understanding three-dimensional microstructures relevant to biological functions.

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