Fourier series Patterson

Patterson A L 1934 A Fourier series method for for the determination of the components of interatomic distances in crystals Phys. Rev. 46 372-6... 

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... 

In words, the difference Patterson function is a Fourier series of simple sine and cosine terms. (Remember that the exponential term is shorthand for these trigonometric functions.) Each term in the series is derived from one reflection hkl in both the native and derivative data sets, and the amplitude of each term is (IFHpI — IFpl)2, which is the amplitude contribution of the heavy atom to structure factor FHp. Each term has three frequencies h in the u-direction, k in the v-direction, and l in the w-direction. Phases of the structure factors are not included at this point, they are unknown. 

It is true that in those years there was at least one method that might be called a more direct method. This was the method that Patterson had developed by introducing his famous Patterson function, which was simply the Fourier series with coefficients that were the intensities of the scattered X-rays. With this method, one was able to deduce information about the interatomic vectors, and, if the structure was not too complicated, one could then calculate the actual arrangement of the atoms. Crystallographers were surely getting structures in this way. 

Lipson. H., and Beevers, C. A. An improved numerical method of two-dimensional Fourier synthesis for crystals. Proc. Phys. Soc. 48, 772-780 (1936). Patterson, A. L., and Tunell, G. A method for the summation of the Fourier series used in the X-ray analysis of crystal structures. Amer. Mineralogist 27. 655-679 (1942). 

The resultant function, unfortunately, does not reveal the distribution of atoms in the unit cell directly but it represents the distribution of interatomic vectors, all of which begin in a common point - the origin of the unit cell. Thus, Puvw is often called the function of interatomic vectors and it is also known as the Patterson function of the F -Fourier series. The corresponding vector density distribution in the unit cell is known as the Patterson map. 

A.L. Patterson, A Fourier series representation of the average distribution of the scattering power in crystals, Phys. Rev. 45, 763 (1934), A.L. Patterson, A Fourier series method for the determination of the components of the interatomic distances in crystals, Phys. Rev. 46, 372(1934). 

Based on the three-dimensional fimction proposed by Patterson in 1934, a new Fourier series that could be calculated directly from the measured intensities. This function is defined as the self-convolution of the electron density, p r), and has the same periodicity as the electron density ... 

The uranyl heavy atom parameters were then used to calculate Fj and Fj two sets of phases for the protein were calculated - one for each possible heavy atom enantiomorph - by the method of single isomorphous replacement with the inclusion of anomalous scattering data. The value of E" was initially calculated from the r.m.s. error in the least squares refinement for the centric zone, and E" was made equal to E/3. The phases, were used to compute difference Fouriers with coefficients (Fp - Fp exp.iap for the other derivatives. The phases calculated from uranyl positions gave a clear indication of the heavy atom positions of other derivatives which agreed well with those positions determined from the Patterson functions. The enantiomorphic set of uranyl positions gave no clear indication of the heavy atom positions. Thus, the correct enantio-morphs and relative origins for derivatives were established. We then carried out a series of phase refinement cycles (16,17). 

In 1935 A. L. Patterson showed that by applying Fourier analysis using the phaseless quantities a series of peaks is obtained (called a Patterson map), each one corresponding to an interatomic vector if a peak exists in the map with coordinates u,v,w, then the unit cells contain atoms at x, y, z, and Xj, y2, Z2, such that... 

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