scholarly journals The accuracy of atomic co-ordinates derived from Fourier series in X-ray crystallography. V

1948 ◽  
Vol 193 (1034) ◽  
pp. 305-310 ◽  
Keyword(s):  
Fourier Series ◽  
Closed Form ◽  
Experimental Error ◽  
Fourier Coefficients ◽  
Probable Error ◽  
X Ray ◽  
X Ray Crystallography ◽  
Experimental Errors

In the first paper of this series, the effect of experimental errors in the Fourier coefficients, upon the derived atomic co-ordinates was investigated. The assumption was made that the probable errors were independent of the magnitudes of their parent Bragg reflexions. It has been suggested that a more accurate assumption would be to take the probable error of any coefficient as being proportional to the magnitude of that coefficient. The present paper develops the theory on this basis, and a solution in closed form is obtained. The question of how the experimental error in the co-ordinates varies with the position of the atom in relation to its neighbours is also investigated, and it is shown that the variation is much smaller than that previously derived for finite summation errors.

The Reaction of Some N-(Nitrophenyl)azoles With Alkali: Preparation of the Corresponding Azoxybenzenes. X-Ray Structure of 2,2′-Bis(1″,2″,4″-triazol-1″-yl)azoxybenzene

1993 ◽  
Vol 46 (4) ◽  
pp. 417 ◽  
Author(s):  
MF Mackay ◽  
GJ Trantino ◽  
JFK Wilshire
Keyword(s):  
Solid State ◽  
Benzene Ring ◽  
Dihedral Angle ◽  
Experimental Error ◽  
Potassium Hydroxide ◽  
Potassium Hydroxide Solution ◽  
Triazole Ring ◽  
X Ray ◽  
X Ray Crystallography ◽  
Weak Electron

The reactions of some representative N-( nitrophenyl )azoles with boiling aqueous ethanolic potassium hydroxide solution gave the corresponding bis ( azolyl ) azoxybenzenes . It is deduced that, in these reactions, the N-attached azolyl groups concerned are acting as weak electron-withdrawing groups. The structure of 2,2′-bis(1″,2″,4″-triazol-1″-yl) azoxybenzene was determined in the solid state by X-ray crystallography. The monoclinic crystals belong to the space group P21/c with a 8.815(1), b 7.863(1), c 11.836(1) Ǻ, β 109.96(1)° and Z 2. The structure was refined to an R index of 0.041 for 1172 observed terms. The midpoint of the exocyclic N=N bond lies on an inversion centre so that the azoxy oxygen is statistically distributed between two sites. The benzene ring atoms are coplanar to within experimental error, as are the triazole ring atoms, and the dihedral angle between the perpendiculars to the two rings is 35.3(3)°.

Field experiments

1924 ◽  
Vol 14 (3) ◽  
pp. 407-412
Author(s):  
Reginald Arthur Berry ◽  
Daniel Grant O'Brien
Keyword(s):  
Experimental Error ◽  
Field Experiments ◽  
Probable Error ◽  
New Varieties ◽  
Experimental Errors ◽  
The Difference ◽  
Cent Error

Single plot trials owing to the magnitude of the experimental errors, are practically useless as a test for comparing the yields of grain of one variety of oat with that of another. At best the results are only applicable to the particular experiment in question. For the old Scotch varieties of oat the probable error on trials of this kind amounts to about 18 per cent, of the yield of grain and to about 20 per cent, for the new varieties. Adopting single plot trials an 18 per cent, error means that when determining the superiority in the yield of grain of one variety over that of another and when the difference is not likely to be more than about 5 per cent, it would be necessary to have 214 centres with no duplication of plots at any centre in order to endeavour to obtain a conclusive result from a single year's trials. When the difference is likely to be about 10 per cent, it would be necessary to have 53 centres.Data are supplied showing that the experimental error for different crops also for the grain and straw of the same crop is not the same. It is greatest for the grain and lowest for the hay crop.

scholarly journals The accuracy of bond-length estimations obtained by double Fourier series methods from X-ray data

1947 ◽  
Vol 190 (1022) ◽  
pp. 329-341 ◽  
Keyword(s):  
Fourier Series ◽  
Complex Structure ◽  
Maximum Error ◽  
Double Fourier Series ◽  
Systematic Errors ◽  
Probable Error ◽  
Organic Crystals ◽  
X Ray ◽  
Wide Range ◽  
Hypothetical Structure

The accuracy obtainable by two-dimensional Fourier series methods has been studied experimentally for a hypothetical structure containing twelve atoms in general positions. An f curve appropriate to average hydrocarbon structures has been employed, and the results should be applicable to a wide range of organic crystals. Errors due to incompleteness in the Fourier series, and to random and systematic errors in the F values have been investigated. It is concluded that in careful determinations with extremely small and symmetric crystal specimens and with reliable photometric or very careful visual estimates of intensities, carried to the limit of copper radiation, the maximum error in a complex structure should not exceed 0⋅03 A, and the probable error will be about 0⋅015 A for atom separations of between 1⋅0 and 1⋅4 A. If the series is incomplete, or if systematic errors are present (such as might be due to absorption in a crystal specimen of irregular shape or to instrumental faults), then these limits will be exceeded.

scholarly journals Improvement of cryo-EM maps by density modification

2019 ◽  
Author(s):  
Thomas C. Terwilliger ◽  
Steven J. Ludtke ◽  
Randy J. Read ◽  
Paul D. Adams ◽  
Pavel V. Afonine
Keyword(s):  
Fourier Coefficients ◽  
X Ray ◽  
X Ray Crystallography ◽  
Modification Process ◽  
Sequence File ◽  
Reference Map ◽  
Modification Procedure ◽  
Model Correlation ◽  
Source Of Information ◽  
Distinct Approach

AbstractA density modification procedure for improving maps produced by single-particle electron cryo-microscopy is presented. The theoretical basis of the method is identical to that of maximum-likelihood density modification, previously used to improve maps from macromolecular X-ray crystallography. Two key differences from applications in crystallography are that the errors in Fourier coefficients are largely in the phases in crystallography but in both phases and amplitudes in electron cryo-microscopy, and that half-maps with independent errors are available in electron cryo-microscopy. These differences lead to a distinct approach for combination of information from starting maps with information obtained in the density modification process. The applicability of density modification theory to electron cryo-microscopy was evaluated using half-maps for apoferritin at a resolution of 3.1 Å and a matched 1.8 Å reference map. Error estimates for the map obtained by density modification were found to closely agree with true errors as estimated by comparison with the reference map. The density modification procedure was applied to a set of 104 datasets where half-maps, a full map and a model all had been deposited. The procedure improved map-model correlation and increased the visibility of details in the maps. The procedure requires two unmasked half-maps and a sequence file or other source of information on the volume of the macromolecule that has been imaged.

The crystal and molecular structure of ovalene a quantitative X-ray investigation

1953 ◽  
Vol 220 (1141) ◽  
pp. 157-170 ◽  
Keyword(s):  
Molecular Structure ◽  
Crystal Structure ◽  
Fourier Series ◽  
Crystal And Molecular Structure ◽  
Double Fourier Series ◽  
Probable Error ◽  
X Ray ◽  
Carbon Carbon Bonds ◽  
Theory And Experiment ◽  
Good Agreement

The crystal structure of ovalene (octabenzonaphthalene) has been determined in detail. The space group is P 2 1 / a , and the unit cell contains two planar, centro-symmetric molecules inclined at about 43° to (010), with a distance of 3.45Ǻ between the molecular planes. The structure has been refined by double Fourier series methods. The twelve chemically distinct carbon-carbon bonds in the molecule vary in length from 1⋅355 to 1⋅438Ǻ (probable error after correction for finite series effects about ±0⋅015Ǻ). These variations in bond length can be predicted by various theoretical treatments, and good agreement between theory and experiment is obtained. The extreme shortness of the bonds in ‘exposed’ positions on the periphery of the molecule, and the longer bonds in the interior, are special features of the results.

scholarly journals The accuracy of atomic co-ordinates derived from Fourier series in X-ray structure analysis. III

1947 ◽  
Vol 190 (1023) ◽  
pp. 482-489 ◽  
Keyword(s):  
Fourier Series ◽  
Power Law ◽  
Three Dimensional ◽  
Thermal Motion ◽  
One Dimensional ◽  
X Ray ◽  
Order Of Magnitude ◽  
Quantitative Examination ◽  
Experimental Errors ◽  
Dimensional Series

Further work on the problems considered in the previous papers of this series has resulted in a more satisfactory treatment of finite summation errors in the three-dimensional diatomic case. The results are extended to the two- and one-dimensional series, and the interesting result emerges that finite summation errors are of the same order of magnitude whatever the dimensions of summation. Using the new results a more quantitative examination of the effects of real thermal motion becomes possible. It is shown that the relative accuracies of parameters in structures, the higher order reflexions from which are suppressed by thermal motion, follows a simple power law in the corresponding reciprocal spacings. These considerations lead to an examination of the artificial temperature factor method of securing convergence, and it is shown that this produces greater errors due to overlapping than those it is designed to eliminate. A method of correcting these distortions is suggested. Finally, the treatment of the effect of experimental errors is extended to two and one dimensions, and it is shown that the three-dimensional summation is least affected by experimental inaccuracy. The errors for three-, two- and one-dimensional summation, in a particular case, are calculated to be in the ratio 1: 3: 10.

scholarly journals Geometric Constraints for the Phase Problem in X-Ray Crystallography

10.29007/p2pj ◽  
2018 ◽  
Author(s):  
Corinna Heldt ◽  
Alexander Bockmayr
Keyword(s):  
Electron Density Distribution ◽  
Fourier Coefficients ◽  
Three Dimensional ◽  
Geometric Constraints ◽  
Dimensional Structure ◽  
Biological Macromolecules ◽  
Phase Problem ◽  
X Ray ◽  
X Ray Crystallography

X-ray crystallography is one of the main methods to establish the three-dimensional structure of biological macromolecules. In an X-ray experiment, one can measure only the magnitudes of the complex Fourier coefficients of the electron density distribution under study, but not their phases. The problem of recovering the lost phases is called the phase problem. Building on earlier work by Lunin/Urzhumtsev/Bockmayr, we extend their constraint-based approach to the phase problem by adding further 0-1 linear programming constraints. These constraints describe geometric properties of proteins and increase the quality of the solutions. The approach has been implemented using SCIP and CPLEX, first computational results are presented here.

scholarly journals The accuracy of atomic co-ordinates derived from Fourier series in X-ray structure analysis IV. The two-dimensional projection of oxalic acid

1947 ◽  
Vol 190 (1023) ◽  
pp. 490-496 ◽  
Keyword(s):  
Experimental Data ◽  
Fourier Series ◽  
Oxalic Acid ◽  
Recent Work ◽  
Two Dimensional ◽  
X Ray ◽  
Oxalic Acid Dihydrate ◽  
Theoretical Predictions ◽  
Experimental Errors ◽  
Extensive Experimental Data

The methods, developed in previous papers of this series, have been applied to an examina­tion of the errors in the atomic co-ordinates derived from the ( h , 0, l ) projection of oxalic acid dihydrate. It is shown that the experimental errors are of the order ±0.01 A, and that the finite sum­mation errors are slightly larger—in agreement with the theoretical predictions of the former papers. Recent work, using more extensive experimental data, is discussed, and it is concluded that,owing to the introduction of an artificial temperature factor, the results are unlikely to be of greater accuracy than the originals.

Difference Fourier Analysis of Glucose Embedded and Frozen Hydrated Purple Membrane

1982 ◽  
Vol 40 ◽  
pp. 74-75
Author(s):  
Jules S. Jaffe ◽  
Robert M. Glaeser
Keyword(s):  
Purple Membrane ◽  
Data Set ◽  
High Resolution Data ◽  
X Ray ◽  
X Ray Crystallography ◽  
Fourier Techniques ◽  
Versus Protein ◽  
The Difference ◽  
Difference Fourier ◽  
Ideal Method

Although difference Fourier techniques are standard in X-ray crystallography it has only been very recently that electron crystallographers have been able to take advantage of this method. We have combined a high resolution data set for frozen glucose embedded Purple Membrane (PM) with a data set collected from PM prepared in the frozen hydrated state in order to visualize any differences in structure due to the different methods of preparation. The increased contrast between protein-ice versus protein-glucose may prove to be an advantage of the frozen hydrated technique for visualizing those parts of bacteriorhodopsin that are embedded in glucose. In addition, surface groups of the protein may be disordered in glucose and ordered in the frozen state. The sensitivity of the difference Fourier technique to small changes in structure provides an ideal method for testing this hypothesis.

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).

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