Kinematics Meets Crystallography: The Concept of a Motion Space1

2015 ◽  
Vol 15 (1) ◽  
Author(s):  
Gregory S. Chirikjian
Keyword(s):  
Rigid Body ◽  
Discrete Subgroup ◽  
Molecular Structures ◽  
New Drugs ◽  
Dimensional Structure ◽  
Biological Macromolecules ◽  
X Ray Diffraction ◽  
X Ray ◽  
Rigid Body Motions ◽  
Diffraction Patterns

In this paper, it is shown how rigid-body kinematics can be used to assist in determining the atomic structure of proteins and nucleic acids when using x-ray crystallography, which is a powerful method for structure determination. The importance of determining molecular structures for understanding biological processes and for the design of new drugs is well known. Phasing is a necessary step in determining the three-dimensional structure of molecules from x-ray diffraction patterns. A computational approach called molecular replacement (MR) is a well-established method for phasing of x-ray diffraction patterns for crystals composed of biological macromolecules. In MR, a search is performed over positions and orientations of a known biomolecular structure within a model of the crystallographic asymmetric unit, or, equivalently, multiple symmetry-related molecules in the crystallographic unit cell. Unlike the discrete space groups known to crystallographers and the continuous rigid-body motions known to kinematicians, the set of motions over which MR searches are performed does not form a group. Rather, it is a coset space of the group of continuous rigid-body motions, SE(3), with respect to the crystallographic space group of the crystal, which is a discrete subgroup of SE(3). Properties of these “motion spaces” (which are compact manifolds) are investigated here.

Kinematics Meets Crystallography: The Concept of a Motion Space

2014 ◽  
Author(s):  
Gregory S. Chirikjian
Keyword(s):  
Rigid Body ◽  
Molecular Structures ◽  
New Drugs ◽  
Dimensional Structure ◽  
Biological Macromolecules ◽  
Molecular Replacement ◽  
X Ray Diffraction ◽  
X Ray ◽  
Rigid Body Motions ◽  
Diffraction Patterns

In this paper, it is shown how rigid-body kinematics can be used to assist in determining the atomic structure of proteins and nucleic acids when using x-ray crystallography, which is a powerful method for structure determination. The importance of determining molecular structures for understanding biological processes and for the design of new drugs is well known. Phasing is a necessary step in determining the three-dimensional structure of molecules from x-ray diffraction patterns. A computational approach called molecular replacement (MR) is a well-established method for phasing of x-ray diffraction patterns for crystals composed of biological macromolecules. In MR, a search is performed over positions and orientations of a known biomolecular structure within a model of the crystallographic asymmetric unit, or, equivalently, multiple symmetry-related molecules in the crystallographic unit cell. Unlike the discrete space groups known to crystallographers and the continuous rigid-body motions known to kinematicians, the set of motions over which molecular replacement searches are performed does not form a group. Rather, it is a coset space of the group of continuous rigid-body motions, SE(3), with respect to the crystallographic space group of the crystal, which is a discrete sub-group of SE(3). Properties of these ‘motion spaces’ (which are compact manifolds) are investigated here.

scholarly journals Liquid-like and rigid-body motions in molecular-dynamics simulations of a crystalline protein

2019 ◽  
Author(s):  
David C. Wych ◽  
James S. Fraser ◽  
David L. Mobley ◽  
Michael E. Wall
Keyword(s):  
Molecular Dynamics ◽  
Rigid Body ◽  
Diffraction Data ◽  
Md Simulations ◽  
X Ray Diffraction ◽  
X Ray ◽  
Atom Pairs ◽  
Collective Motions ◽  
Rigid Body Motions ◽  
Insight Into

AbstractTo gain insight into crystalline protein dynamics, we performed molecular-dynamics (MD) simulations of a periodic 2×2×2 supercell of staphylococcal nuclease. We used the resulting MD trajectories to simulate X-ray diffraction and to study collective motions. The agreement of simulated X-ray diffraction with the data is comparable to previous MD simulation studies. We studied collective motions by analyzing statistically the covariance of alpha-carbon position displacements. The covariance decreases exponentially with the distance between atoms, which is consistent with a liquid-like motions (LLM) model, in which the protein behaves like a soft material. To gain finer insight into the collective motions, we examined the covariance behavior within a protein molecule (intra-protein) and between different protein molecules (inter-protein). The inter-protein atom pairs, which dominate the overall statistics, exhibit LLM behavior; however, the intra-protein pairs exhibit behavior that is consistent with a superposition of LLM and rigid-body motions (RBM). Our results indicate that LLM behavior of global dynamics is present in MD simulations of a protein crystal. They also show that RBM behavior is detectable in the simulations but that it is subsumed by the LLM behavior. Finally the results provide clues about how correlated motions of atom pairs both within and across proteins might manifest in diffraction data. Overall our findings increase our understanding of the connection between molecular motions and diffraction data, and therefore advance efforts to extract information about functionally important motions from crystallography experiments.

scholarly journals X-ray diffraction from nonuniformly stretched helical molecules

2016 ◽  
Vol 49 (3) ◽  
pp. 784-797 ◽  
Author(s):  
Momcilo Prodanovic ◽  
Thomas C. Irving ◽  
Srboljub M. Mijailovich
Keyword(s):  
Local Strain ◽  
Molecular Structures ◽  
X Ray Diffraction ◽  
Protein Assemblies ◽  
Helical Structures ◽  
X Ray ◽  
Molecular Systems ◽  
Multi Scale ◽  
Fibrous Protein ◽  
Diffraction Patterns

The fibrous proteins in living cells are exposed to mechanical forces interacting with other subcellular structures. X-ray fiber diffraction is often used to assess deformation and movement of these proteins, but the analysis has been limited to the theory for fibrous molecular systems that exhibit helical symmetry. However, this approach cannot adequately interpret X-ray data from fibrous protein assemblies where the local strain varies along the fiber length owing to interactions of its molecular constituents with their binding partners. To resolve this problem a theoretical formulism has been developed for predicting the diffraction from individual helical molecular structures nonuniformly strained along their lengths. This represents a critical first step towards modeling complex dynamical systems consisting of multiple helical structures using spatially explicit, multi-scale Monte Carlo simulations where predictions are compared with experimental data in a `forward' process to iteratively generate ever more realistic models. Here the effects of nonuniform strains and the helix length on the resulting magnitude and phase of diffraction patterns are quantitatively assessed. Examples of the predicted diffraction patterns of nonuniformly deformed double-stranded DNA and actin filaments in contracting muscle are presented to demonstrate the feasibly of this theoretical approach.

The biological crystallography without crystals

2017 ◽  
Vol 12 (1) ◽  
pp. 55-72 ◽  
Author(s):  
В.Ю. Лунин ◽  
V.Y. Lunin
Keyword(s):  
Three Dimensional ◽  
Scattering Data ◽  
Dimensional Structure ◽  
Biological Macromolecules ◽  
X Ray Diffraction ◽  
X Ray ◽  
Biological Objects ◽  
Standard Problem ◽  
Problems And Solutions

The main obstacle to the determination of the atomic structure of a biological macromolecule by X-ray structural analysis is the need to obtain a crystal of the object under study. This need is due to the complexity of the experimental registration of scattering from a separate molecule. However, it is not always possible to get crystals of biological objects. The development of experimental techniques, in particular the emergence of the X-ray free-electron lasers, allows to approach the practical solution of the problem of registration of the scattering from an isolated particle and thereby to obtain information about the three-dimensional structure of non-crystalline biological objects by X-ray diffraction methods. Sampling of experimental scattering data makes the task of the structure determination of a single particle equivalent to the standard problem of biological crystallography, which allows to extend the biological crystallography techniques to the study of isolated biological particles (individual cells, organelles, molecular machines and, in the future, biological macromolecules). This article is devoted to the state of the art in this area, problems and solutions.

A multiple-common-lines method to determine the orientation of snapshot diffraction patterns from single particles

2014 ◽  
Vol 70 (4) ◽  
pp. 364-372 ◽  
Author(s):  
Liang Zhou ◽  
Tian-Yi Zhang ◽  
Zhong-Chuan Liu ◽  
Peng Liu ◽  
Yu-Hui Dong
Keyword(s):  
Three Dimensional ◽  
Dimensional Structure ◽  
X Ray Diffraction ◽  
Single Particles ◽  
X Ray ◽  
Three Dimensional Structure ◽  
Individual Object ◽  
Common Lines ◽  
Diffraction Imaging ◽  
Diffraction Patterns

With the development of X-ray free-electron lasers (XFELs), it is possible to determine the three-dimensional structures of noncrystalline objects with coherent X-ray diffraction imaging. In this diffract-and-destroy mode, many snapshot diffraction patterns are obtained from the identical objects which are presented one by one in random orientations to the XFEL beam. Determination of the orientation of an individual object is essential for reconstruction of a three-dimensional structure. Here a new method, called the multiple-common-lines method, has been proposed to determine the orientations of high- and low-signal snapshot diffraction patterns. The mean errors of recovered orientations (α, β, γ) of high- and low-signal patterns are about 0.14, 0.06, 0.12 and 0.77, 0.31, 0.60°, respectively; both sets of errors can meet the requirements of the reconstruction of a three-dimensional structure.

scholarly journals Determination of the Structure of Biological Macromolecular Particles Using X-Ray Lasers. Achievements and Prospects

2020 ◽  
Vol 15 (2) ◽  
pp. 195-234
Author(s):  
T.E. Petrova ◽  
V.Y. Lunin
Keyword(s):  
Single Molecule ◽  
Experimental Approach ◽  
Biological Macromolecules ◽  
X Ray Diffraction ◽  
X Ray ◽  
Current State ◽  
Fundamental Possibility ◽  
Diffraction Patterns ◽  
Further Development

X-ray diffraction analysis is the main experimental approach to determining the atomic structure of biological macromolecules and their complexes. The most serious limitation of its applicability, to date, is the need to prepare a sample of the object under study in the form of a single crystal, which is caused by the extremely low intensity of rays scattered by a single molecule. The commissioning of X-ray Free-Electron Lasers with their super-powerful (by many orders of magnitude exceeding the brightness of modern synchrotrons) and ultra-short (less than 100 fs) pulse is an experimental breakthrough that allows us to expect to obtain diffraction patterns from individual biological particles and then determine their structure. The first experimental results demonstrate the fundamental possibility of such an approach and are accompanied by the publication of a significant number of articles on various aspects of the development of the method. The purpose of this article is to discuss the current state of art in this area, evaluate the results achieved and discuss the prospects for further development of the method based on the analysis of publications in the world scientific literature of recent years and the experience of work carried out by the review authors and their colleagues.

Evaluation of Freezing Methods by Low Temperature X-Ray Diffraction

1977 ◽  
Vol 35 ◽  
pp. 330-333
Author(s):  
T. Gulik-Krzywicki ◽  
M.J. Costello
Keyword(s):  
Biological Materials ◽  
Molecular Organization ◽  
X Ray Diffraction ◽  
Glass Capillary ◽  
X Ray ◽  
Rate Dependent ◽  
Before And After ◽  
Freeze Etching ◽  
Diffraction Patterns ◽  
Set Up

Freeze-etching electron microscopy is currently one of the best methods for studying molecular organization of biological materials. Its application, however, is still limited by our imprecise knowledge about the perturbations of the original organization which may occur during quenching and fracturing of the samples and during the replication of fractured surfaces. Although it is well known that the preservation of the molecular organization of biological materials is critically dependent on the rate of freezing of the samples, little information is presently available concerning the nature and the extent of freezing-rate dependent perturbations of the original organizations. In order to obtain this information, we have developed a method based on the comparison of x-ray diffraction patterns of samples before and after freezing, prior to fracturing and replication.Our experimental set-up is shown in Fig. 1. The sample to be quenched is placed on its holder which is then mounted on a small metal holder (O) fixed on a glass capillary (p), whose position is controlled by a micromanipulator.

Examination of Rabbit Fc Crystals

1968 ◽  
Vol 26 ◽  
pp. 140-141
Author(s):  
J. P. Robinson ◽  
P. G. Lenhert
Keyword(s):  
Molecular Weight ◽  
Flat Plate ◽  
Phosphate Buffer ◽  
Asymmetric Unit ◽  
X Ray Diffraction ◽  
X Ray ◽  
Neutral Ph ◽  
Small Crystals ◽  
Carbon Coated ◽  
Diffraction Patterns

Crystallographic studies of rabbit Fc using X-ray diffraction patterns were recently reported. The unit cell constants were reported to be a = 69. 2 A°, b = 73. 1 A°, c = 60. 6 A°, B = 104° 30', space group P21, monoclinic, volume of asymmetric unit V = 148, 000 A°3. The molecular weight of the fragment was determined to be 55, 000 ± 2000 which is in agreement with earlier determinations by other methods.Fc crystals were formed in water or dilute phosphate buffer at neutral pH. The resulting crystal was a flat plate as previously described. Preparations of small crystals were negatively stained by mixing the suspension with equal volumes of 2% silicotungstate at neutral pH. A drop of the mixture was placed on a carbon coated grid and allowed to stand for a few minutes. The excess liquid was removed and the grid was immediately put in the microscope.

Control of the phase composition of advanced calcium phosphates using an x-ray diffractometer with a curved position-sensitive detector

2020 ◽  
Vol 86 (6) ◽  
pp. 29-35
Author(s):  
V. P. Sirotinkin ◽  
O. V. Baranov ◽  
A. Yu. Fedotov ◽  
S. M. Barinov
Keyword(s):  
Phase Composition ◽  
Calcium Phosphates ◽  
Position Sensitive Detector ◽  
Sensitive Detector ◽  
X Ray Diffraction ◽  
X Ray ◽  
Impurity Phases ◽  
Position Sensitive ◽  
Diffraction Peaks ◽  
Diffraction Patterns

The results of studying the phase composition of advanced calcium phosphates Ca10(PO4)6(OH)2, β-Ca3(PO4)2, α-Ca3(PO4)2, CaHPO4 · 2H2O, Ca8(HPO4)2(PO4)4 · 5H2O using an x-ray diffractometer with a curved position-sensitive detector are presented. Optimal experimental conditions (angular positions of the x-ray tube and detector, size of the slits, exposure time) were determined with allowance for possible formation of the impurity phases during synthesis. The construction features of diffractometers with a position-sensitive detector affecting the profile characteristics of x-ray diffraction peaks are considered. The composition for calibration of the diffractometer (a mixture of sodium acetate and yttrium oxide) was determined. Theoretical x-ray diffraction patterns for corresponding calcium phosphates are constructed on the basis of the literature data. These x-ray diffraction patterns were used to determine the phase composition of the advanced calcium phosphates. The features of advanced calcium phosphates, which should be taken into account during the phase analysis, are indicated. The powder of high-temperature form of tricalcium phosphate strongly adsorbs water from the environment. A strong texture is observed on the x-ray diffraction spectra of dicalcium phosphate dihydrate. A rather specific x-ray diffraction pattern of octacalcium phosphate pentahydrate revealed the only one strong peak at small angles. In all cases, significant deviations are observed for the recorded angular positions and relative intensity of the diffraction peaks. The results of the study of experimentally obtained mixtures of calcium phosphate are presented. It is shown that the graphic comparison of experimental x-ray diffraction spectra and pre-recorded spectra of the reference calcium phosphates and possible impurity phases is the most effective method. In this case, there is no need for calibration. When using this method, the total time for analysis of one sample is no more than 10 min.

Infrared spectroscopic study of the YPO4-YCrO4 system

1985 ◽  
Vol 50 (10) ◽  
pp. 2139-2145
Author(s):  
Alexander Muck ◽  
Eva Šantavá ◽  
Bohumil Hájek
Keyword(s):  
Spectroscopic Study ◽  
Infrared Spectra ◽  
Point Of View ◽  
Mixed Crystals ◽  
Crystal Symmetry ◽  
Infrared Spectroscopic Study ◽  
X Ray Diffraction ◽  
X Ray ◽  
Infrared Spectroscopic ◽  
Diffraction Patterns

The infrared spectra and powder X-ray diffraction patterns of polycrystalline YPO4-YCrO4 samples are studied from the point of view of their crystal symmetry. Mixed crystals of the D4h19 symmetry are formed over the region of 0-30 mol.% YPO4 in YCrO4. The Td → D2d → D2 or C2v(GS eff) correlation is appropriate for both PO43- and CrO43- anions.

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