Experimental error density maps

The second approach is to use Fourier methods to calculate the electron density based on the model (using calculated Fs and phases, the vector Fc) and compare this with the electron density based on the observations (with calculated phases, the vector Fo). An electron-density map is calculated based on I To I — I. Pc I- This so-called difference map will give an accurate representation of where the errors are in the model compared with the experimental data. If an atom is located in the model where there is no experimental observation for it, then the difference map will show a negative density peak. Conversely, when there is no atom in the model where there should be, then a positive peak will be present. This map can be used to manually move, remove, or add atoms into the model. 

Rees [27] has calculated the effect on the deformation electron density maps of experimental random errors in centrosymmetric crystals ... 

Ideally, a map calculated with experimental coefficients Fo(hkl). and phases a hkl) should accurately reproduce the electron density of the crystal to the precision of the experiment. In practice, there arc experimental errors in the measured values of F hkl) and errors in the model, such as the assumption of spherical atoms in the calculation of the atomic scattering factors, and limits in the extent to which disorder and thermal motion of atoms can be modeled. 

Deformation density The difference between the electron density in a molecule, with all its distortions as a result of bonding, and the promolecule density, obtained by forming a molecule with spherical electron density around each atom (free atoms). This map contains effects caused both by the errors in the relative phases of Bragg reflections, experimental errors in the data, and inadequacies in the representations of the scattering factors of free atoms. 

Simulated annealing refinement is usually unable to correct very large errors in the atomic model or to correct for missing parts of the structure. The atomic model needs to be corrected by inspection of a difference Fourier map. In order to improve the quality and resolution of the difference map, the observed phases are often replaced or combined with calculated phases, as soon as an initial atomic model has been built. These combined electron density maps are then used to improve and to refine the atomic model. The inclusion of calculated phase information brings with it the danger of biasing the refinement process towards the current atomic model. This model bias can obscure the detection of errors in atomic models if sufficient experimental phase information is unavailable. In fact during the past decade several cases of incorrect or partly incorrect atomic models have been reported where model bias may have played a role [67]. 

Experimentally determined electron density maps are never perfect because of defects in the data and phases, and as a consequence even the best of models will have errors in atomic positions, errors in dihedral angles, improper rotomers for side chains, or unacceptable contacts between atoms or chemical groups. When the quality of an electron density map is marginal or poor, serious errors in the model may occur, or the model may be fundamentally wrong. Even when the electron density map is good, and the model for most practical purposes correct, errors remain. The initial model for a protein will, in general, have rms errors in atomic positions of 1A or more, and an R factor computed from the model (with the exception perhaps of models obtained from molecular replacement) on the order of 0.50, an often discouraging result. 

Inasmuch as the phases dominate the appearance of the electron density map, errors in the phases will make the map much more difficult to interpret. Unfortunately, the initial phases obtained from the methods outlined above always contain errors. In experimental methods, very small differences in amplitudes are measured and the resulting phases are the statistical best estimates This does not mean they are right they may not yield the most interpretable map. Even before constructing a model to interpret the density, various forms of density modification are used to improve the map. 

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. 

Fig. 26.10 Fq - Fc difference maps, (a) Detection of errors in the model. The side-chain of Aspi A in human thrombin was deiiberateiy moved to a wrong position and an F — Fc difference map was calcuiated. The modei used to caiculate Fc is indicated in thick solid iines, the correct position of the side-chain is indicated in thick broken lines. Negative contours (4a beiow mean) are drawn in thin broken lines and positive contours (4a above mean) are drawn in thin solid iines. Strong negative difference density is present around the wrongiy piaced side-chain while strong positive density is present at the position where the side-chain shouid be according to the experimental data. These maps are extremely useful to spot errors in the model, (b) Fo - Fc omit map (see text) contoured at 3.5a of an inhibitor bound to porcine pancreatic eiastase. Protein atoms, used for calcuiation of Fc are indicated in thin iines, the inhibitor which has been removed from the modei is drawn in thick lines. This Fo - Fc density map has been caiculated with a model which contains no information whatsoever about the inhibitor. The difference density shown is therefore entireiy due to the experimental data and can be used to verify the correctness of the placement of the inhibitor.

The direct experimental result of a crystallographic analysis is an electron-density map, and not the atomic model everybody looks at If errors occur in crystal structures, they most often occur at the level of the (subjective) interpretation of the electron-density maps by the crystallographer. A severe problem, especially at low resolution (lower than 3.0 A), is the so-called model bias. To calculate an electron-density map, one needs amplitudes and phases. The amplitudes are determined experimentally, but the phases cannot be measured directly. In later stages of refinement, they are calculated from the model, which means that if the model contains errors, the phases will contain the same errors. Since phases make up at least 50% of the information which is used to calculate the electron-density maps, wrong features may still have reasonable electron density because of these phase errors. 

The difference Ag(X) = p(X) - Pm(X) between the actual density and the pro-molecule density is known as the deformation density and can be interpreted as the electron density reorganization that occurs when a collection of independent, isolated, spherically symmetric atoms is combined to form a molecule in a crystal. Since Aq is only a very small fraction of total Q in the region of the atoms, it is very susceptible to experimental error in the X-ray measurements and to inadequacies in the model, namely errors in the assumed atomic positions, atomic scattering factors, and ADPs. In one approximation, a deformation density map is obtained by direct subtraction of the two densities. The density map obtained in this way is smeared by vibrational motion of the atoms, but its peaks and troughs can often be interpreted in terms of some model of chemical bonding, e.g., peaks between bonded atoms being identified with bonding density and so on. A difference density map for tetrafluoroterephthalodinitrile [27] is shown in Fig. 3. 

Figure 7. l -Alanine. Fit to noisy data. Calculation B. 10% experimental noise level. MaxEnt deformation density and error map in the COO- plane. Mag size, orientation and contouring levels as in Figure 2. (a) MaxEnt dynamic deformation density Ai up O3) Error map qm -... 

In crystals, the scattering densities are periodic and the Bragg amplitudes are the Fourier components of these periodic distributions. In principle, the scattering density p(r) is given by the inverse Fourier series of the experimental structure factors. Such a series implies an infinite sum on the Miller indices h, k, l. Actually, what is performed is a truncated sum, where the indices are limited to those reflections really measured, and where all the structure factors are noisy, as a result of the uncertainty of the measurement. Given these error bars and the limited set of measured reflections, there exist a very large number of maps compatible with the data. Among those, the truncated Fourier inversion procedure selects one of them the map whose Fourier coefficients are equal to zero for the unmeasured reflections and equal to the exact observed values otherwise. This is certainly an arbitrary choice. 

Owing to experimental limitations (crystal quahty, errors, finite resolution, and the lack of phase information), the direct mapping of the density via Fourier summation... 

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