Scattering amplitude, isomorphous replacement

Once a suitable crystal is obtained and the X-ray diffraction data are collected, the calculation of the electron density map from the data has to overcome a hurdle inherent to X-ray analysis. The X-rays scattered by the electrons in the protein crystal are defined by their amplitudes and phases, but only the amplitude can be calculated from the intensity of the diffraction spot. Different methods have been developed in order to obtain the phase information. Two approaches, commonly applied in protein crystallography, should be mentioned here. In case the structure of a homologous protein or of a major component in a protein complex is already known, the phases can be obtained by molecular replacement. The other possibility requires further experimentation, since crystals and diffraction data of heavy atom derivatives of the native crystals are also needed. Heavy atoms may be introduced by covalent attachment to cystein residues of the protein prior to crystallization, by soaking of heavy metal salts into the crystal, or by incorporation of heavy atoms in amino acids (e.g., Se-methionine) prior to bacterial synthesis of the recombinant protein. Determination of the phases corresponding to the strongly scattering heavy atoms allows successive determination of all phases. This method is called isomorphous replacement. 

In order to exploit the heavy atom method with crystals of conventional molecules, or to utilize the isomorphous replacement method or anomalous dispersion technique for macro-molecular structure determination, it is necessary to identify the positions, the x, y, z coordinates of the heavy atoms, or anomalously scattering substituents in the crystallographic unit cell. Only in this way can their contribution to the diffraction pattern of the crystal be calculated and employed to generate phase information. Heavy atom coordinates cannot be obtained by biochemical or physical means, but they can be deduced by a rather enigmatic procedure from the observed structure amplitudes, from differences between native and derivative structure amplitudes, or in the case of anomalous scattering, from differences between Friedel mates. 

For the determination of the electron density map of a molecule the amplitudes and phases of the waves scattered by a single crystal are required for a number of Bragg reflections. Common X-ray techniques yield the product (Ah vh) (Ah 1waves scattered by the electrons of the molecule into the Bragg reflection H. Thus, the scattering amplitude Ah is obtained, but the phase information is lost. The solution of this phase problem for protein structure determination is based on Perutz and Kendrew s isomorphous replacement method (108-111). In this procedure Bragg reflections have to be measured at least three times, first on a crystal of native molecules, and then on two crystals, in which reference scatterers (for example Hg atoms) have been substituted at well-defined positions. From the difference of the measured intensities one can calculate the relative phases without ambiguity. 

If two centrosymmetric crystal structures are isomorphous, the arrangement of atoms is the same in both and only one atom (sometimes more than one) has a different atomic number in the two structures. The differences in the intensities for the Bragg reflections, therefore, result only from the differences in the scattering powers of the two atoms, M, and M2, that can replace each other. The contribution to the structure factors made by the rest of the structure, F/j, is the same for both crystal structures. If the structure amplitudes are F and F2 for a given Bragg reflection in the two structures, then the calculated difference is illustrated by the use of vectors as ... 

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