Electron-density maps

2002 ◽  
Author(s):  
David Blow
Keyword(s):  
Fourier Transform ◽  
Electron Density ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Unit Cell ◽  
Density Maps ◽  
Wave Cycle ◽  
Density Map ◽  
The Fourier Transform ◽  
Electron Density Map

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.

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

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.

Phase improvement and extension

2013 ◽  
Author(s):  
Carmelo Giacovazzo
Keyword(s):  
Ab Initio ◽  
Electron Density ◽  
Model Building ◽  
Special Treatment ◽  
Fourier Synthesis ◽  
Density Maps ◽  
Phase Extension ◽  
Density Map ◽  
The Fourier Transform ◽  
Electron Density Map

The descriptions of the various types of Fourier synthesis (observed, difference, hybrid) and of their properties, given in Chapter 7, suggest that electron density maps are not only a tool for depicting the distribution of the electrons in the target structure, but also a source of information which may be continuously exploited during the phasing process, no matter whether ab initio or non-ab initio methods were used for deriving the initial model. Here, we will describe two important techniques based on the properties of electron density maps. (i) The recursive approach for phase extension and refinement called EDM (electron density modification). Such techniques have dramatically improved the efficiency of phasing procedures, which usually end with a limited percentage of phased reflections and non-negligible phase errors. EDM techniques allow us to extend phase assignment and to improve phase quality. The author is firmly convinced that practical solution of the phase problem for structures with Nasym up to 200 atoms in the asymmetric unit may be jointly ascribed to direct methods and to EDM techniques. (ii) The AMB (automated model building) procedures; these may be considered to be partly EDM techniques and they are used for automatic building of molecular models from electron density maps. Essentially, we will refer to proteins; the procedures used for small to medium-sized molecules have already been described in Section 6.3.5. Two new ab initio phasing approaches, charge flipping and VLD, essentially based on the properties of the Fourier transform, belong to the EDM category, and since they require a special treatment, they will be described later in Chapter 9. Phase extension and refinement may be performed in reciprocal and in direct space. We described the former in Section 6.3.6; here, we are just interested in direct space procedures, the so-called EDM (electron density modification) techniques. Such procedures are based on the following hypothesis: a poor electron density map, ρ, may be modified by a suitable function, f , to obtain a new map, say ρmod, which better approximates the true map: . . . ρmod (r) = f [ρ(r)]. (8.1) . . . If function f is chosen properly, more accurate phases can be obtained by Fourier inversion of ρmod, which may in turn be used to calculate a new electron density map.

scholarly journals Machine learning to estimate the local quality of protein crystal structures

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Ikuko Miyaguchi ◽  
Miwa Sato ◽  
Akiko Kashima ◽  
Hiroyuki Nakagawa ◽  
Yuichi Kokabu ◽  
...  
Keyword(s):  
Electron Density ◽  
Protein Structures ◽  
Three Dimensional ◽  
Protein Crystal ◽  
Low Resolution ◽  
X Ray Crystallography ◽  
Density Maps ◽  
Local Quality ◽  
Density Map ◽  
Electron Density Map

AbstractLow-resolution electron density maps can pose a major obstacle in the determination and use of protein structures. Herein, we describe a novel method, called quality assessment based on an electron density map (QAEmap), which evaluates local protein structures determined by X-ray crystallography and could be applied to correct structural errors using low-resolution maps. QAEmap uses a three-dimensional deep convolutional neural network with electron density maps and their corresponding coordinates as input and predicts the correlation between the local structure and putative high-resolution experimental electron density map. This correlation could be used as a metric to modify the structure. Further, we propose that this method may be applied to evaluate ligand binding, which can be difficult to determine at low resolution.

A Comparison of Two Algorithms for Electron-Density Map Improvement by Introduction of Atomicity: Skeletonization, and Map Sorting Followed by Refinement

1998 ◽  
Vol 54 (1) ◽  
pp. 81-85 ◽  
Author(s):  
F. M. D. Vellieux
Keyword(s):  
Electron Density ◽  
Initial Density ◽  
Acta Cryst ◽  
Density Maps ◽  
Resolution Electron ◽  
Crystallographic Symmetry ◽  
Density Map ◽  
Ornithine Transcarbamoylase ◽  
Electron Density Map ◽  
Pseudo Atom

A comparison has been made of two methods for electron-density map improvement by the introduction of atomicity, namely the iterative skeletonization procedure of the CCP4 program DM [Cowtan & Main (1993). Acta Cryst. D49, 148–157] and the pseudo-atom introduction followed by the refinement protocol in the program suite DEMON/ANGEL [Vellieux, Hunt, Roy & Read (1995). J. Appl. Cryst. 28, 347–351]. Tests carried out using the 3.0 Å resolution electron density resulting from iterative 12-fold non-crystallographic symmetry averaging and solvent flattening for the Pseudomonas aeruginosa ornithine transcarbamoylase [Villeret, Tricot, Stalon & Dideberg (1995). Proc. Natl Acad. Sci. USA, 92, 10762–10766] indicate that pseudo-atom introduction followed by refinement performs much better than iterative skeletonization: with the former method, a phase improvement of 15.3° is obtained with respect to the initial density modification phases. With iterative skeletonization a phase degradation of 0.4° is obtained. Consequently, the electron-density maps obtained using pseudo-atom phases or pseudo-atom phases combined with density-modification phases are much easier to interpret. These tests also show that for ornithine transcarbamoylase, where 12-fold non-crystallographic symmetry is present in the P1 crystals, G-function coupling leads to the simultaneous decrease of the conventional R factor and of the free R factor, a phenomenon which is not observed when non-crystallographic symmetry is absent from the crystal. The method is far less effective in such a case, and the results obtained suggest that the map sorting followed by refinement stage should be by-passed to obtain interpretable electron-density distributions.

Diffraction by crystals

2002 ◽  
Author(s):  
David Blow
Keyword(s):  
Fourier Transform ◽  
Three Dimensional ◽  
Incident Beam ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Dimensional Structure ◽  
Angle Of Incidence ◽  
X Ray ◽  
Max Von Laue ◽  
The Fourier Transform

In Chapter 4 many two-dimensional examples were shown, in which a diffraction pattern represents the Fourier transform of the scattering object. When a diffracting object is three-dimensional, a new effect arises. In diffraction by a repetitive object, rays are scattered in many directions. Each unit of the lattice scatters, but a diffracted beam arises only if the scattered rays from each unit are all in phase. Otherwise the scattering from one unit is cancelled out by another. In two dimensions, there is always a direction where the scattered rays are in phase for any order of diffraction (just as shown for a one-dimensional scatterer in Fig. 4.1). In three dimensions, it is only possible for all the points of a lattice to scatter in phase if the crystal is correctly oriented in the incident beam. The amplitudes and phases of all the scattered beams from a three-dimensional crystal still provide the Fourier transform of the three-dimensional structure. But when a crystal is at a particular angular orientation to the X-ray beam, the scattering of a monochromatic beam provides only a tiny sample of the total Fourier transform of its structure. In the next section, we are going to find what is needed to allow a diffracted beam to be generated. We shall follow a treatment invented by Lawrence Bragg in 1913. Max von Laue, who discovered X-ray diffraction in 1912, used a different scheme of analysis; and Paul Ewald introduced a new way of looking at it in 1921. These three methods are referred to as the Laue equations, Bragg’s law and the Ewald construction, and they give identical results. All three are described in many crystallographic text books. Bragg’s method is straightforward, understandable, and suffices for present needs. I had heard J.J. Thomson lecture about…X-rays as very short pulses of radiation. I worked out that such pulses…should be reflected at any angle of incidence by the sheets of atoms in the crystal as if these sheets were mirrors.…It remained to explain why certain of the atomic mirrors in the zinc blende [ZnS] crystal reflected more powerfully than others.

scholarly journals HermiteFit: fast-fitting atomic structures into a low-resolution density map using three-dimensional orthogonal Hermite functions

2014 ◽  
Vol 70 (8) ◽  
pp. 2069-2084 ◽  
Author(s):  
Georgy Derevyanko ◽  
Sergei Grudinin
Keyword(s):  
Electron Density ◽  
Cross Correlation ◽  
Three Dimensional ◽  
Hermite Functions ◽  
Low Resolution ◽  
Atomic Structures ◽  
Resolution Electron ◽  
Laplacian Filter ◽  
Density Map ◽  
Electron Density Map

HermiteFit, a novel algorithm for fitting a protein structure into a low-resolution electron-density map, is presented. The algorithm accelerates the rotation of the Fourier image of the electron density by using three-dimensional orthogonal Hermite functions. As part of the new method, an algorithm for the rotation of the density in the Hermite basis and an algorithm for the conversion of the expansion coefficients into the Fourier basis are presented.HermiteFitwas implemented using the cross-correlation or the Laplacian-filtered cross-correlation as the fitting criterion. It is demonstrated that in the Hermite basis the Laplacian filter has a particularly simple form. To assess the quality of density encoding in the Hermite basis, an analytical way of computing the crystallographicRfactor is presented. Finally, the algorithm is validated using two examples and its efficiency is compared with two widely used fitting methods,ADP_EMandcoloresfrom theSituspackage.HermiteFitwill be made available at http://nano-d.inrialpes.fr/software/HermiteFit or upon request from the authors.

scholarly journals A moment invariant for evaluating the chirality of three-dimensional objects

2010 ◽  
Vol 8 (54) ◽  
pp. 144-151 ◽  
Author(s):  
Johan Hattne ◽  
Victor S. Lamzin
Keyword(s):  
Electron Density ◽  
Three Dimensional ◽  
Peptide Fragments ◽  
Moment Invariant ◽  
Three Dimensional Objects ◽  
Density Maps ◽  
Wide Range ◽  
Polypeptide Backbone ◽  
Dimensional Object ◽  
Electron Density Map

Chirality is an important feature of three-dimensional objects and a key concept in chemistry, biology and many other disciplines. However, it has been difficult to quantify, largely owing to computational complications. Here we present a general chirality measure, called the chiral invariant (CI), which is applicable to any three-dimensional object containing a large amount of data. The CI distinguishes the hand of the object and quantifies the degree of its handedness. It is invariant to the translation, rotation and scale of the object, and tolerant to a modest amount of noise in the experimental data. The invariant is expressed in terms of moments and can be computed in almost no time. Because of its universality and computational efficiency, the CI is suitable for a wide range of pattern-recognition problems. We demonstrate its applicability to molecular atomic models and their electron density maps. We show that the occurrence of the conformations of the macromolecular polypeptide backbone is related to the value of the CI of the constituting peptide fragments. We also illustrate how the CI can be used to assess the quality of a crystallographic electron density map.

scholarly journals Rapid chain tracing of polypeptide backbones in electron-density maps

2010 ◽  
Vol 66 (3) ◽  
pp. 285-294 ◽  
Author(s):  
Thomas C. Terwilliger
Keyword(s):  
Electron Density ◽  
High Electron ◽  
Visual Evaluation ◽  
Density Maps ◽  
Density Values ◽  
Speed Up ◽  
Density Map ◽  
Electron Density Map ◽  
And Storage

A method for the rapid tracing of polypeptide backbones has been developed. The method creates an approximate chain tracing that is useful for visual evaluation of whether a structure has been solved and for use in scoring the quality of electron-density maps. The essence of the method is to (i) sample candidate Cαpositions at spacings of approximately 0.6 Å along ridgelines of high electron density, (ii) list all possible nonapeptides that satisfy simple geometric and density criteria using these candidate Cαpositions, (iii) score the nonapeptides and choose the highest scoring ones, and (iv) find the longest chains that can be made by connecting nonamers. An indexing and storage scheme that allows a single calculation of most distances and density values is used to speed up the process. The method was applied to 42 density-modified electron-density maps at resolutions from 1.5 to 3.8 Å. A total of 21 428 residues in these maps were traced in 24 CPU min with an overall r.m.s.d. of 1.61 Å for Cαatoms compared with the known refined structures. The method appears to be suitable for rapid evaluation of electron-density map quality.

Automatic procedure for crystal-structure analysis with the heavy-atom technique*

1971 ◽  
Vol 134 (1-2) ◽  
Author(s):  
Vasantha Pattabhi ◽  
K. Venkatesan
Keyword(s):  
Electron Density ◽  
Least Squares Method ◽  
Heavy Atom ◽  
Site Occupancy ◽  
Crystal Structure Analysis ◽  
Density Maps ◽  
Structure Factors ◽  
Subsequent Calculation ◽  
Density Map ◽  
Electron Density Map

AbstractThis paper deals with a possible method of determining molecular structure directly, using x-ray data. The peaks chosen from the electron-density map calculated using the signs obtained from the contribution of the heavy atom are assigned site-occupancy values which depend upon the peak heights. The occupancy parameter is refined by the least-squares method and the peaks for which the value of the occupancy parameters increases are taken as real atoms in the subsequent calculation of structure factors and electron-density maps. The method has been tested in two hypothetical cases and in a real case.

Sign in / Sign up

Export Citation Format

Share Document