Fourier synthesis of electron density

The electron density p(j , y, z) of the whole molecule, existing in the form of a crystal, is a periodic function in three dimensions. For this reason we can write the 

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From the heavy-atom positions obtained by this procedure, the structure factors are calculated from Equation (18), and the signs or phases are assigned to the measured structure amplitudes. The result is a Fourier synthesis of electron density. Because the phases depend only on the positions of the heavy atoms, this first Fourier synthesis often fails to reveal the whole structure of the molecule instead, it yields only a partial. structure, but this is generally enough to lead to better phases for the next Fourier synthesis. The structure should be solved after two or three (at most six or seven) of these successive Fourier syntheses of electron den-... 

After further refinement cycles, hydrogen atoms in chemically meaningful positions were assigned to the largest maxima in the difference Fourier synthesis of electron density. This is a Fourier synthesis in which the Fourier coefficients are the differences between the observed and calculated structure factors. When the hydrogen coordinates had been fitted to the measured data, the isotropic temperature coefficients of the hydrogen atoms were also refined. The R was now 5.3 %. Finally, weighting the squared deviations in... 

The situation, in truth, is somewhat more involved than this explanation would suggest. The individual reflections of the diffraction pattern are the interference sum of the waves scattered by all of the atoms in the crystal in a particular direction and, therefore, are themselves waves. Being waves they have not only an amplitude, but also a unique phase angle associated with each of them. This too depends on the distribution of the atoms, their xj, yj, Zj. The phase angle is independent of the amplitude of the reflection, but most important, it is an essential part of the individual terms that contribute to the Fourier synthesis, the electron density equation. Unfortunately, the phase angle of areflection cannot be recorded, as we record the intensity. In fact we have no practical way (and rather few impractical ways either) to directly measure it at all. But, without the phase information, no Fourier summation can be computed. In the 1950s, however, it became possible, with persistence, skill, and patience (and luck), to recover this elusive phase information for... 

The first step in the Fourier synthesis of the caffeine-pyrogallol complex was to choose an initial model so an electron density projection could be calculated. From the previous results it was determined that the unit cell was tetragonal measuring 23.26 X 23.26 X 6.99 A., and that eight caffeine-pyrogallol complex moieties reside in the unit cell. This information alone does not give any indication of the way in which the molecules are packed inside the unit cell, but the symmetry operations of the space group can be used to eliminate many models. For example, one of the symmetry operators in the unit cell is a fourfold axis. 

Figure 2a. Difference Fourier synthesis of oxymyoglobin, showing the electron density for the bound oxygen. The Fourier synthesis was computed with Fobserved — Fcalculated] as coefficients the calculated structure amplitudes were derived from the positions of all the atoms except the two oxygens.

Electron-density map A contour representation of electron density in a crystal structure. Peaks appear at atomic positions. The map is computed by a Fourier synthesis, that is, the summation of waves of known amplitude, periodicity, and relative phase. The electron density is expressed in electrons per cubic A. 

When a diffraction grating, such as a crystal, interacts with X rays, the electron density that causes this diffraction can be described by a Fourier series, as discussed in Chapter 6. The diffraction experiment effects a Fourier analysis, breaking down the Fourier series (of the electron density) into its components, that is, the diffracted beams with amplitudes, F[hkl). The relative phases a(hkl) are, however, lost in the process in all usual diffraction experiments. This loss of the phase information needed for the computation of an electron-density map is referred to as the phase problem. The aim of X-ray diffraction studies is to reverse this process, that is, to find the true relative phase and hence the true three-dimensional electron density. This is done by a Fourier synthesis of the components, but it is now necessary to know both the actual amplitude F[hkl) and the relative phase, a[hkl), in order to calculate a correct electron-density map (see Figure 8.1). We must be able to reconstruct the electron-density distribution in a systematic way by approximating, as far as possible, a correct [but so far unknown) set of phases In this way the crystallographer, aided by a computer, acts as a lens for X rays. 

When the application of Eq. (11) to a least squares analysis of x-ray structure factors has been completed, it is usual to calculate a Fourier synthesis of the difference between observed and calculated structure factors. The map is constructed by computation of Eq. (9), but now IFhid I is replaced by Fhki - F/f /, where the phase of the calculated structure factor is assumed in the observed structure factor. In this case the series termination error is virtually too small to be observed. If the experimental errors are small and atomic parameters are accurate, the residual density map is a molecular bond density convoluted onto the motion of the nuclear frame. A molecular bond density is the difference between the true electron density and that of the isolated Hartree-Fock atoms placed at the mean nuclear positions. An extensive study of such residual density maps was reported in 1966.7 From published crystallographic data of that period, the authors showed that peaking of electron density in the aromatic C-C bonds of five organic molecular crystals was systematic. The random error in the electron density maps was reduced by averaging over chemically equivalent bonds. The atomic parameters from the model Eq. (11), however, will refine by least squares to minimize residual densities in the unit cell. 

D20.4 The phase problem arises with the analysis of data in X-ray diffraction when seeking to perform a Fourier synthesis of the electron density. In order to carry out the sum it is necessary to know the signs of the structure factors however, because diffraction intensities are proportional to the square of the structure factors, the intensities do not provide information on the sign. For non-centrosymmetric crystals, the structure factors may be complex, and the phase a in the expression F/m = F w e is indeterminate. The phase problem may be evaded by the use of a Patterson synthesis or tackled directly by using the so-called direct methods of phase allocation. 

The structures were first related together by comparison of the main-chain and j3-C atoms. The best fit determined by an iterative least-squares method gave a r.m.s. difference of 1.32 A. By further comparison of the Fourier syntheses and also of each structure with the Fourier synthesis of the other, Hoi, Drenth, and their co-workers estimate that 0.5 A differences result from errors in the electron densities, the interpretation, and the model building. Further errors of 0.8 A are estimated to result from the determination of the co-ordinates from the model. Thus differences of less than 1.3 A are not significant. 

The extent to which the Fourier synthesis of the electron density can be inteipreted depends on the resolution with which the data could be meas-... 

Special difficulties arise when the heavy-atom structure, taken separately, contains one more symmetry element (usually a center of symmetry) than the overall structure. The first Fourier synthesis of the electron density then shows two light-atom structures along with the heavy-atom structure. In order to calculate the second Fourier. synthesis, the heavy-atom structure is then used together with only that part of the light-atom structure that can be assigned with high confidence to just one of the two light-atom structures indicated in the Fourier synthesis. The additional symmetry element vanishes in the next Fourier synthesis, and the structure analysis proceeds in the usual way. 

Ultimately, the only way to decide whether a phase set is the right one is to look for a chemically meaningful interpretation of the Fourier synthesis of the electron density. But it would be rather tedious to check, say, 64 Fourier syntheses in this way (256 or more in trouble.some cases). An attempt is therefore made to find figures of merit [91] that will make one phase set more probable than the others, even before the Fourier synthesis is calculated. Such a figure of merit might refer to the internal consistency of a phase set. The determination of phases or signs by direct methods then involves the following steps ... 

At this point, the least-squares refinement could be regarded as complete because the largest change in a parameter was smaller than 10% of the corresponding standard deviation. In a difference Fourier synthesis of the electron density, the values for the 10 largest maxima were 0.12-0.16 elec-trons/10 m ... 

It can be shown, at the price of a short course in Fourier analysis, that the structure factors are the coefficients of the Fourier synthesis of the electron density in the cell. We will not give the derivation here, and the reader should be satisfied that the... 

D. C. Phillips, and V. C. Shore, Nature 190, 666 (1961)] were even more rewarding. In fact, Kendrew s group recently succeeded in deriving an almost complete amino acid sequence from the electron-density distribution of their 2 A Fourier synthesis of sperm whale myoglobin. It is very likely that the amino acid sequences of these two proteins will be completely determined in the near future. 

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 ... 

The limitations mentioned do not apply if the work can be done on a machine various types of mechanical and electrical analogue machines for Fourier synthesis have been described (see Lipson and Cochran, 1953), culminating in Pepinsky s XRAC (X-ray analogue computer), in which structure amplitudes and phases for a two -dimensional synthesis are put in on an array of dials, and the electron density map appears at once on a cathode ray tube (Pepinsky, 1952). An increasing proportion of Fourier syntheses (and indeed crystallographic calculations of all types) is done on electronic digital computers. 

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 desired electron-density function is a Fourier series in which term hkl has amplitude IFobsl, which equals (7/, /)1/2, the square root of the measured intensity Ihkl from the native data set. The phase ot hkl of the same term is calculated from heavy-atom, anomalous dispersion, or molecular replacement data, as described in Chapter 6. The term is weighted by the factor whU, which will be near 1.0 if ct hkl is among the most highly reliable phases, or smaller if the phase is questionable. This Fourier series is called an Fobs or Fo synthesis (and the map an Fo map) because the amplitude of each term hkl is iFobsl for reflection hkl. 

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