Fourier difference

Moews, P.C., Kretsinger, R.H. Refinement of the structure of carp muscle calcium-binding parvalbumin by model building and difference Fourier analysis. 

X-Ray diffraction from single crystals is the most direct and powerful experimental tool available to determine molecular structures and intermolecular interactions at atomic resolution. Monochromatic CuKa radiation of wavelength (X) 1.5418 A is commonly used to collect the X-ray intensities diffracted by the electrons in the crystal. The structure amplitudes, whose squares are the intensities of the reflections, coupled with their appropriate phases, are the basic ingredients to locate atomic positions. Because phases cannot be experimentally recorded, the phase problem has to be resolved by one of the well-known techniques the heavy-atom method, the direct method, anomalous dispersion, and isomorphous replacement.1 Once approximate phases of some strong reflections are obtained, the electron-density maps computed by Fourier summation, which requires both amplitudes and phases, lead to a partial solution of the crystal structure. Phases based on this initial structure can be used to include previously omitted reflections so that in a couple of trials, the entire structure is traced at a high resolution. Difference Fourier maps at this stage are helpful to locate ions and solvent molecules. Subsequent refinement of the crystal structure by well-known least-squares methods ensures reliable atomic coordinates and thermal parameters. 

Conventional implementations of MaxEnt method for charge density studies do not allow easy access to deformation maps a possible approach involves running a MaxEnt calculation on a set of data computed from a superposition of spherical atoms, and subtracting this map from qME [44], Recourse to a two-channel formalism, that redistributes positive- and negative-density scatterers, fitting a set of difference Fourier coefficients, has also been made [18], but there is no consensus on what the definition of entropy should be in a two-channel situation [18, 36,41] moreover, the shapes and number of positive and negative scatterers may need to differ in a way which is difficult to specify. 

Several steps were needed to determine the structure of the core particle to higher resolution (Fig. Id). The X-ray phases of the low-resolution models were insufficient to extend the structure to higher resolution, since the resolution of the early models of the NCP was severely limited by disorder in the crystals. The disorder was presumed to derive from both the random sequences of the DNA and from heterogeneity of the histone proteins caused by variability in post-translational modification of the native proteins. One strategy for developing an atomic position model of the NCP was to develop a high-resolution structure of the histone core. This structure could then be used with molecular replacement techniques to determine the histone core within the NCP and subsequently identify the DNA in difference Fourier electron density maps. 

The refinement includes an anisotropic vibration for all the non-hydrogen atoms. Difference-Fourier syntheses phased by the phosphorus, nitrogen, sulfur, oxygen and carbon atoms revealed the hydrogen atoms in their expected positions. 

By far, the most common procedure for the determination of heavy-atom positions is the difference Patterson method it is often used in combination with the difference Fourier technique to locate sites in second and third derivatives. 

Difference Fourier techniques are most useful in locating sites in a multisite derivative, when a Patterson map is too complicated to be interpretable. The phases for such a Fourier must be calculated from the heavy-atom model of other derivatives in which a difference Patterson map was successfully interpreted, and should not be obtained from the derivative being tested, in order not to bias the phases. Also, difference Fourier techniques can be used to test the correctness of an already identified heavy-atom site by removing that site from the phasing model and seeing whether it will appear in... 

Although low-resolution difference Fourier maps for oxy-Hr and deoxy-Hr show little change in the protein structures, some of the iron center properties are significantly altered in deoxy-Hr. The differences provide a rationale for an oxygen-binding mechanism. The Mossbauer spectrum for deoxy-Hr has a single quadrupole doublet with an isomeric shift typical of high-spin ferrous iron (8 = 1.14 mm/sec AEq = 2.76 mm/sec) (Clark and Webb, 1981). As for met-Hr the two iron environments are similar, yet differ in coordination number for exam-... 

Fig. 9a and b. Difference Fourier maps calculated from Laue diffraction data showing maltoheptose bound in phosphorylase b. The Laue map shown in a is calculated with a subset of 9029 unique data at 2.5 A resolution. A positive contour at half maximal peak height is shown, b is an enlargement of a and shows 4 of the 7 sugar units, the 3 central units have the highest occupancies. Side chain movements produce the two extra lobes of density. (Figures courtesy of J. Hajdu)... 

The percarboxylic acid proton of 3-oxo-l,2-benzisothiazole-2(377)-peroxypropanoic acid 1,1-dioxide (51) (Pnma, 0—0 = 1.469, C—O—O—H = 180.0°) was located on the difference Fourier map . Hydrogen bonding in the peracid 51 (Figure 22) occurs from the peracid proton to the carbonyl O of the saccharin entity (O O = 2.618 A) to provide chains of peracid molecules that are stacked via additional C—H O contacts (not shown in Figure 22) in sheets along the b axis. 

Using time-resolved crystallographic experiments, molecular structure is eventually linked to kinetics in an elegant fashion. The experiments are of the pump-probe type. Preferentially, the reaction is initiated by an intense laser flash impinging on the crystal and the structure is probed a time delay. At, later by the x-ray pulse. Time-dependent data sets need to be measured at increasing time delays to probe the entire reaction. A time series of structure factor amplitudes, IF, , is obtained, where the measured amplitudes correspond to a vectorial sum of structure factors of all intermediate states, with time-dependent fractional occupancies of these states as coefficients in the summation. Difference electron densities are typically obtained from the time series of structure factor amplitudes using the difference Fourier approximation (Henderson and Moffatt 1971). Difference maps are correct representations of the electron density distribution. The linear relation to concentration of states is restored in these maps. To calculate difference maps, a data set is also collected in the dark as a reference. Structure factor amplitudes from the dark data set, IFqI, are subtracted from those of the time-dependent data sets, IF,I, to get difference structure factor amplitudes, AF,. Using phases from the known, precise reference model (i.e., the structure in the absence of the photoreaction, which may be determined from... 

Henderson, R., and Moffat, J. K. 1971. The difference Fourier technique in protein crystallography and their treatment. Acta Crystallogr. B 27 1414-20. 

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