Fourier syntheses

Formally, a Fourier synthesis is a reconstruction of a repetitive function as a superposition of sine or cosine waves. Long-wavelength waves account for the general features of the structure, and the details are gradually filled in by incorporating shorter-wavelength waves. 

Braggs law is a very primitive approach to the interpretation of X-ray diffraction data. The huge amount of data obtained from a modern diffractometer has a much richer content than the separation of lattice planes, for in principle it contains information about the locations of individual atoms and the distribution of electron density throughout a unit cell. 

Example 11.5 ) Calculating an electron density by Fourier synthesis 

The determination of the three-dimensional structure of molecules is a key step in the rational design of therapeutic agents that bind specifically to receptor sites on proteins and nucleic acids Case study 11.2). Consider the (hOO) planes of a crystal of an organic molecule regarded as a candidate for a drug. In an X-ray analysis the structure factors were found as follows  

Construct a plot of the electron density projected on to the x-axis of the unit cell. 

The intensity of scattering from a crystal (assumed to be large and devoid of any lattice defects) is nonzero only when the scattering vector s coincides with the reciprocal lattice rkkl given by-Equation (3.2). The square root of the intensity Ihki observed at s = rkkl (and properly normalized) provides the absolute magnitude of the structure factor F(s) at rkkl, that is, 

Cmc2 A c-centered orthorhombic lattice with mirror plane 

P2 I a Primitive monoclinic unit cell with twofold screw axis 

The term asymmetric unit is the unit on which the space group operations act to produce the entire crystal structure (an infinitely extending pattern). This is the simplest unit that can be selected. The contents of the asymmetric unit combined with the relationships between positions of atoms listed for the space group are needed to derive a picture of the entire atomic contents of the crystal. 

Once we have determined the symmetry, the space group, and the unit-cell dimensions of the crystal, we can focus our attention on the intensities that describe the positions of the atoms. These calculations make use of the Fourier transforms as an essential techniques. 

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The goal of URT is to obtain reflectivity images from back-scattered measurements. This consists in a Fourier synthesis problem, and the first task is to correctly cover the frequency space of the "object" r. Let for simplicity the dimension of the physical space be 2. 

Perutz, M.F., et al. Structure of haemoglobin. A three-dimensional Fourier synthesis at 5.5 A resolution, obtained by x-ray analysis. Nature 185 416-422, 1960. 

Blake CC, Koeniz DF, Mair GA, North AC, Phillips DC, Sarma VR. Structure of hen egg-white lysozyme. A three-dimensional Fourier synthesis at 2 Angstrom resolution. Nature 1965 206 757-61. 

G. "Three-dimensional Fourier Synthesis of Horse Oxyhemoglobin at 2.8 A Resolution (II) The Atomic Model". Nature (1968), 219, 131. 

Because of the limitation intrinsic to the adoption of an explicit parametrised density model, many crystallographers have been dreaming of disposing of such models altogether. The thermally-smeared charge density in the crystal can of course be obtained without an explicit density model, by Fourier summation of the (phased) structure factor amplitudes, but the resulting map is affected by the experimental noise, and by all series-termination artefacts that are intrinsic to Fourier synthesis of an incomplete, finite-resolution set of coefficients. 

A second major source of computational difficulties associated with uniform prior-prejudice distributions is connected with the extremely fine sampling grids that are needed to avoid aliasing effects in the numerical Fourier synthesis of the modulating factor in (8). To predict the dependence of aliasing effects upon the prior prejudice, we need to examine more closely the way the MaxEnt distribution of scatterers is actually synthesised from the values of the Lagrange multipliers X. 

We wish to obtain an image of the scattering elements in three dimensions (the electron density). To do this, we perform a 3-D Fourier synthesis (summation). Fourier series are used because they can be applied to a regular periodic function crystals are regular periodic distributions of atoms. The Fourier synthesis is given in O Eq. 22.2 ... 

Although neither structure has yet been solved at atomic resolution Perutz has recentty built an atomic model of horse oxyhaemoglobin 41). This model is based on the Fourier sjmthesis of horse oxyhaemoglobin at 5.5 A resolution, the Fourier synthesis of myoglobin at atomic resolution,... 

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. 

Since we deal with a periodic pattern, it is possible to apply a technique that was originally invented by the French physicist and mathematician Jean Baptiste Joseph Fourier (1768-1830). Fourier was the first who showed that every periodic process (or an object like in our case) can be described as the sum (a superposition) of an infinite number of individual periodic events (e.g. waves). This process is known as Fourier synthesis. The inverse process, the decomposition of the periodic event or object yields the individual components and is called Fourier analysis. How Fourier synthesis works in practice is shown in Figure 4. To keep the example most simple, we will first consider only the projection (a shadow image) of the black squares onto the horizontal a-axis in the beginning (Figure 3). 

In this demonstration of a Fourier series we will use only cosine waves to reproduce the shadow image of the black squares. The procedure itself is rather straightforward, we just need to know the proper values for the amplitude A and the index h for each wave. The index h determines the frequency, i.e. the number of full waves trains per unit cell along the a-axis, and the amplitude determines the intensity of the areas with high (black) potential. As outlined in Figure 4, the Fourier synthesis for the present case is the sum of the following terms ... 

Now let us assume that the black squares in the above example are the atoms in a real crystal structure and we want to locate these atoms by help of a Fourier synthesis. As we have seen in the previous paragraph we then need to know... 

Even without having the stmcture factor phases, e.g. from electron microscopy images, it is possible to get some insight into the atomic architecture of a crystal. A simple but powerful method to get this information was introduced hy A.L. Patterson about 70 years ago. Following Patterson the Fourier synthesis is carried out using the squared stmcture factor amplitudes Fha which are equal to the measured intensities for the reflections with index hkl. Moreover, all phase values must be set to zero, which leads to the following (auto-correlation) function ... 

In contrast to Fourier synthesis, which yields with electron diffraction data high electrostatic potential at the positions of the atoms, the maps obtained from Patterson synthesis show peaks at the tips of vectors. The length of each vector (drawn from the origin of the Patterson map) corresponds always to the distances between pairs of atoms and the direction each vector points... 

It is emphasized that the final result is the structure map of the examined crystal rather than a pseudo structure map. This is because the difftaction intensities have been pushed towards the corresponding kinematical values during the calculation of partial structure factor in each cycle of the correction. In addition, in the final step, structure refinement by Fourier synthesis modifies the peak heights towards the true values to some extent. It is obvious that all the missing structure information due to the CTF zero transfer is mended after phase extension. The amplitudes are provided by the electron diffraction data, and the phases are derived from the phase extension. As a result, the resolution of the structure analysis by this method is determined by the electron diffraction resolution limit. 

An example of this procedure is shown in Fig. 1. This example shows the build-up of the 2D potential of Ti2S projected along the short c axis, but the principle is the same for creating a 3D potential. The potential is a continuous function in real space and can be described in a map (Fig. 1). On the other hand, the structure factors are discrete points in reciprocal space and can be represented by a list of amplitudes and phases (Table 1). In this Fourier synthesis we have used the structure factors calculated from the refined coordinates of Ti2S °. 

If the Fourier synthesis is carried out by adding in the strong reflections first, we will see how fast the Fourier series converges to the projected potential. The positive potential contribution from the reflection is shown in white, whereas the negative potential contribution is shown in black. Most of the atoms are located in the white regions of each cosine wave, but the exact atomic positions will not become evident until a sufficiently large number of structure factors have been added up. 

Figure 1 Fourier synthesis of the projected potential map of Xi2S along the c-axis. Amplitudes and phases of the structure factors are calculated from the refined atomic coordinates of Ti2S and listed in Table 1. The space group of Xi2S is Pnnm and unit cell parameters a= 11.35, fc=14.05 and c=3.32 A.

As can be seen from Eq. 4, a Fourier synthesis requires phase angles as input, thus it cannot be used to locate heavy-atom positions in a derivative if no phase information exists. However, it can be used to determine such positions in a derivative, if phases are already available from one or more other derivatives. As in the case of a difference Patterson, the Fourier s)mthesis here also employs difference coefficients. They are of the form ... 

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