Fourier series structure determination

Since the variation of any physical property in a three dimensional crystal is a periodic function of the three space coordinates, it can be expanded into a Fourier series and the determination of the structure is equivalent to the determination of the complex Fourier coefficients. The coefficients are indexed with the vectors of the reciprocal lattice (one-to-one relationship). In principle the expansion contains an infinite number of coefficients. However, the series is convergent and determination of more and more coefficients (corresponding to all reciprocal lattice points within a sphere, whose radius is given by the length of a reciprocal lattice vector) results in a determination of the stmcture with better and better spatial resolution. Both the amplitude and the phase of the complex number must be determined for any Fourier coefficient. The amplitudes are determined from diffraction... 

About 1915 W.H. Bragg suggested to use Fourier series to describe the arrangement of the atoms in a crystal [1]. The proposed technique was somewhat later extended by W. Duane [2] and W.H. Zachariasen was the first who used a two-dimensional Fourier map in 1929 for structure determination [3], Since then Fourier synthesis became a standard method in almost in every structure determination from diffraction data. 

In Bragg s way of looking at diffraction as reflection from sets of planes in the crystal, each set of parallel planes described here (as well as each additional set of planes interleaved between these sets) is treated as an independent diffractor and produces a single reflection. This model is useful for determining the geometry of data collection. Later, when I discuss structure determination, I will consider another model in which each atom or each small volume element of electron density is treated as an independent diffractor, represented by one term in a Fourier series that describes each reflection. Bragg s model tells us where to look for the data. The Fourier series model tells us what the data has to say about molecular structure. 

In words, the structure factor that describes reflection hkl is a Fourier series in which each term is the contribution of one atom, treated as a simple sphere of electron density. So the contribution of each atom j to Fhkl depends on (1) what element it is. which determines jf, the amplitude of the contribution, and (2) its position in the unit cell (Xj, yj, z-)> which establishes the phase of its contribution. 

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... 

To compare apo- and holo-forms of proteins after both structures have been determined independently, crystallographers often compute difference Fourier syntheses (Chapter 7, Section IV.B), in which each Fourier term contains the structure-factor difference FAc>/c(-—F 0. A contour map of this Fourier series is called a difference map, and it shows only the differences between the holo-and apo- forms. Like the FQ — Fc map, the FAoio—F map contains both positive and negative density. Positive density occurs where the electron density of the holo-form is greater than that of the apo-form, so the ligand shows up clearly in positive density. In addition, conformational differences between holo- and apo-forms result in positive density where holo-protein atoms occupy regions that are unoccupied in the apo-form, and negative... 

The Rietveld method is employed both to finalize the model of the crystal structure, when necessary, e.g. to locate a few missing atoms in the unit cell by coupling it with Fourier series calculations, and to confirm the crystal structure determination by refining positional and other relevant parameters of individual atoms together with profile variables. The fully refined... 

The following five examples of completing the model of the crystal structure and Rietveld refinement are based on five materials discussed in sections 6.13 to 6.17. Therefore, in all cases the initial structural models will be employed exactly as they were derived in Chapter 6. When needed, the models will be completed by employing Fourier series calculation(s) using phase angles obtained after the initial models have been improved by using Rietveld refinement and the individual structure factors re-determined from the observed powder data after the refinement. ... 

G.12 H. Lipson and W. Cochrane. The Crystalline State. Vol. Ill The Determination of Crystal Structures (London George Bell, 1953). Advanced structure analysis by means of space group theory and Fourier series. Experimental methods are not included i.e., the problem of structure analysis is covered from the point at which jFp values have been determined by experiment to the final solution. Contains many illustrative examples. 

The main problem is solving the Fourier series in Eq. 12 is again the indeterminany of the signs of i.e. the phase problem. An important aid to the correct assignment can be obtained from the continuous structure factors F determined from experiments with random, unstacked samples provided, of course, that the membrane structures are the same in both cases. This has been discussed in detail recently by Blaurock Many other methods have been described in the literature, most of them, however, have to rely on trial-and-error searches involving information from other sources... 

The intensity of the diffracted beams on the other hand is determined by the distribution of the atoms, i.e. their relative geometric arrangement within a unit cell. As already mentioned, the periodic arrangement of the electron density ep in the crystal is essential for the diffraction of X-rays. Also, according to Bragg, the latter can be described in a Fourier series as a function of the structure factors Fhki, which determine the intensity of the reflections ... 

This behavior implies that the Fourier series expansions for the order parameter and free energy for each possible structure can be approximated by including terms ofjust one wavenumber, q = q, but summed over different directions determined by the symmetry of the phase. The simplest phase is the lamellar one, for which this reduces to... 

Before the calculation of the interfacial properties can proceed, the DFT must be used to determine the equilibrium freezing properties of the system at the required temperature. The structure and thermodynamics of the coexisting liquid and crystal phases are needed to determine the boundary conditions of the density profiles on either side of the interface. The equilibrium freezing calculation may be summarized as follows. First, the periodic single-particle density of the crystal is parametrized so that the minimization of the free energy functional can be performed. One popular parametrization is a Fourier series ... 

Cullis, R., et al. (1961). The structure of haemoglobin, VIII. A three-dimensional Fourier synthesis at 5.5 A resolution determination of the phase angles. Proceed. Royal Soc. London Series A 265,15-38. 

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... 

While it is very easy, when one knows the structure of the crystal and the wavelength of the rays, to predict the diffraction pattern, it is quite another matter to deduce the crystal structure in all Its details from the observed pattern and the known wavelength. The first step is lo determine the spacing of the atomic planes from the Bragg equation, and hence the dimensions of the unit cell. Any special symmetry of the space group of the structure will be apparent from space group extinction. A Irial analysis may (hen solve the structure, or it may be necessary to measure the structure factors and try to find the phases or a Fourier synthesis. Various techniques can be used, such as the F2 series, the heavy atom, the isomorphous series, anomalous atomic scattering, expansion of the crystal and other methods. 

In recent years the application of electrospray ionization (ESI) mass spectrometry, quadrupole time-of-flight (QqTOF) mass spectrometry, and Fourier transform ion cyclotron resonance (FT-ICR) are used for further structural characterization of DOM (Kujawinski et al., 2002 Kim et al., 2003 Stenson et al., 2003 Koch et al., 2005 Tremblay et al., 2007 Reemtsma et al., 2008). MS/MS capabilities provide the screening for selected ions, and FT-ICR allows exact molecular formula determination for selected peaks. In addition, SEC can be coupled to ESI and FTICR-MS to study different DOM fractions. Homologous series of structures can be revealed, and many pairs of peaks differ by the exact masses of -H2, -O, or -CH2. Several thousand molecular formulas in the mass range of up to more than 600 Da can be identified and reproduced in element ratio plots (O/C versus H/C plots). Limitations of ESI used by SEC-MS are shown by These and Reemtsma (2003). 

Although it is possible to determine the complete electron density distribution using the Fourier transform of the observed structure factors, Eq. (1), the errors inherent in the structure factor amplitudes and, in the case of non-centrosymmetric structures, the errors in their phases introduce significant noise and bias into the result. Because of this, it has become normal practice to model the electron density by a series of pseudo-atoms consisting of a frozen, spherical core and an atom centered multipole expansion to represent the valence electron density [2,17]. 

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