Patterson electron density

The isomorphous replacement method requires attachment of heavy atoms to protein molecules in the crystal. In this method, atoms of high atomic number are attached to the protein, and the coordinates of these heavy atoms in the unit cell are determined. The X-ray diffraction pattern of both the native protein and its heavy atom derivative(s) are determined. Application of the so-called Patterson function determines the heavy atom coordinates. Following the refinement of heavy atom parameters, the calculation of protein phase angles proceeds. In the final step the electron density of the protein is calculated. 

Calculations of the Patterson function may be carried out in exactly the same way as those of electron densities. Bragg s optical method may also be used indeed, in general it may be applied more readily to the formation of vector maps, since (the signs of the jF2 coefficients being all positive) the question of phase adjustment does not arise. The optical method has been shown to give a correct vector map for the 6 projection of haemoglobin. ... 

As I described earlier, this entails extracting the relatively simple diffraction signature of the heavy atom from the far more complicated diffraction pattern of the heavy-atom derivative, and then solving a simpler "structure," that of one heavy atom (or a few) in the unit cell of the protein. The most powerful tool in determining the heavy-atom coordinates is a Fourier series called the Pattersonfunction P(u,v,w), a variation on the Fourier series used to compute p(x,y,z) from structure factors. The coordinates (u,v,w) locate a point in a Patterson map, in the same way that coordinates (x,y,z) locate a point in an electron-density map. The Patterson function or Patterson synthesis is a Fourier series without phases. The amplitude of each term is the square of one structure factor, which is proportional to the measured reflection intensity. Thus we can construct this series from intensity measurements, even though we have no phase information. Here is the Patterson function in general form... 

Because the Patterson function contains no phases, it can be computed from any raw set of crystallographic data, but what does it tell us A contour map of p(x,y,z) displays areas of high density (peaks) at the locations of atoms. In contrast, a Patterson map, which is a contour map of P(u,v,w), displays peaks at locations corresponding to vectors between atoms. (This is a strange idea at first, but the following example will make it clearer.) Of course, there are more vectors between atoms than there are atoms, so a Patterson map is more complicated than an electron-density map. But if the structure is simple, like that of one or a few heavy atoms in the unit cell, the Patterson map may be simple enough to allow us to locate the atom(s). You can see now that the... 

The output is the integrated intensity value for that particular reflection. FHKL - This is the structure factor calculating program. The input is a list of hkl s and intensity values. The output consists of E values and phase angles to be used as input to the electron density program. ELECDEN - Calculates the electron density and contours the E-map on a Tektronix 4662 digital plotter. PATTERSON - Used to calculate three-dimensional Patterson maps. 

The phase problem can be solved, that is, phases of the scattered waves determined, either by Patterson function or by direct methods. The Patterson function P is a self-convolution of the electron density p, and its magnitude at a point u, v, w can be obtained by multiplying p (x, y, z) hy p (x + u, y + V, z + w) and summing these products for every point of the unit cell. In practice, it is calculated as... 

The Patterson or direct method solution will give a number of electron density peaks which can be identified as atoms of certain types. This is still a very crade model of the stracture, which should be optimized by the least squares (LS) refinement in the following way. Spherically symmetrical Hartree-Fock atoms are placed at the positions of the peaks and the coordinates (Section 2.2.2) and displacement parameters (Section 2.2.3) of these atoms are altered so as to minimize the function... 

The conunonly used methods for solving the phase problem required for stmcture solutions are the direct methods and Patterson maps. Direct methods use relationships between phases such as triplets (0 = 4>h + 4>k + -h-k The probabihty of 0 >= 0 increases with the magnitude of the product of the normalized stmcture factors of the three reflections involved. Once these triplets associated with high certainty are identified based on diffraction intensities, they are used to assign new phases based on a set of known phases. Since the number of phase relationships is large the problem is over determined. Another approach is based on the Sayre equation, which is derived based on the relationship between the electron density and its square ... 

The determination of the atomic structure of a reconstruction requires the quantitative measurement of as many allowed reflections as possible. Given the structure factors, standard Fourier methods of crystallography, such as Patterson function or electron-density difference function, are used. The experimental Patterson function is the Fourier transform of the experimental intensities, which is directly the electron density-density autocorrelation function within the unit cell. Practically, a peak in the Patterson map means that the vector joining the origin to this peak is an interatomic vector of the atomic structure. Different techniques may be combined to analyse the Patterson map. On the basis of a set of interatomic vectors obtained from the Patterson map, a trial structure can be derived and model stracture factor amplitudes calculated and compared with experiment. This is in general followed by a least-squares minimisation of the difference between the calculated and measured structure factors. Of help in the process of structure determination may be the difference Fourier map, which is... 

Suitable small crystals of 2XeF,AsF, were obtained by sublimation under nitrogen (at atmos.) in sealed quartz AT-ray capillaries. A tablet measuring <01 mm. in any dimension was used for the intensity data. The crystals are monoclinic with unit-cell dimensions a = 16-443, b = 8-678, c = 20-888 A, )S = 90-13°, V = 2799 A -The space group is 12/a, and Z 12 Three-dimensional data, amounting to 1182 non-zero independent reflections, were obtained. Two xenon and one arsenic atoms were located with a three-dimensional Patterson map, and the remaining atomic p>ositions from subsequent electron-density maps. Full-matrix least-squares refinement led to a final conventional R-value of 0-066. 

Crystals of the compound of empirical formula FiiPtXe are orthorhombic with unit-cell dimensions a = 8-16, h = 16-81. c = 5-73 K, V = 785-4 A . The unit cell volume is consistent with Z = 4, since with 44 fluorine atoms in the unit cell the volume per fluorine atom has its usual value of 18 A. Successful refinement of the structure is proceeding in space group Pmnb (No. 62). Three-dimensional intensity data were collected with Mo-radiation on a G.E. spectrogoniometer equipped with a scintillation counter. For the subsequent structure analysis 565 observed reflexions were used. The platinum and xenon positions were determined from a three-dimensional Patterson map, and the fluorine atom positions from subsequent electron-density maps. Block diagonal least-squares refinement has led to an f -value of 0-15. Further refinements which take account of imaginary terms in the anomalous dispersion corrections are in progress. 

Determination of Structure. Analysis of a Patterson map indicated two sets of heavy atoms in genera) positions, a result incompatible with the preconceived opinion of the composition of the materia). For this reason the subsequent analysis was carried out using the diffraction data to establish the composition. The peaks for the heavy atoms were of appropriate relative height to correspond to Xe and As, and other peaks were found which corresponded to six F atoms around the As atom. Fourier maps phased with these eight atoms revealed the seventh fluorine atom. Another smaller peak was tested as a possible fluorine atom, but it was rejected by the least-squares refinement. A later electron density map, prior to the absorption correction and with R - = O.IO, showed no other peaks... 

The Patterson map, commonly designated P(uvw), is a Fourier synthesis that uses the indices, h,k,l, and the square of the structure factor amplitude, F(hkl), of each diffracted beam. It is usual to describe the Patterson map in vector space defined by u, v, and w, rather than x,y,z as used in electron-density maps. 

This Equation 8.12 has the same form as the equation for electron density (Equation 6.3, Chapter 6), but note that there is no phase angle in the expression The coefficients of the Patterson function are the observed intensities, after some geometric corrections involved with the data collection process are made. Because the Patterson function uses... 

Heavy-atom method Relative phases calculated for a heavy atom in a location determined from a Patterson map are used to calculate an approximate electron-density map. Further portions of the molecular structure may be identified in this map and used to calculate better relative phases, and therefore a more realistic electron-density map results. Several cycles of this process may be necessary in order to determine the entire crystal structure. 

Bragg following advice to do so from his father, W. H. Bragg/ The calculations involved were many and tedious, and therefore Henry Lipson, C. Arnold Beevers, A. Lindo Patterson, and George Tunell provided more convenient methods of computing the electron-density functions in the days before high-speed computers were available. Currently, the computation of an electron-density map is simple and fast because of the efficiency of available computers. 

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