Residual density maps

Final residual indices of the refinement strategies are given in Table 2. On the residual density maps shown in Figure 1, the maxima and minima do not exceed 0.2 e A-3. 

When the model used for Fcalc is that obtained by least-squares refinement of the observed structure factors, and the phases of Fca,c are assigned to the observations, the map obtained with Eq. (5.9) is referred to as a residual density map. The residual density is a much-used tool in structure analysis. Its features are a measure for the shortcomings of the least-squares minimization, and the functions which constitute the least-squares model for the scattering density. 

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. 

The Fourier syntheses of various residual density maps based on x-ray and neutron diffraction measurements seem to indicate that present-day diffraction data have sufficient information to pursue quantitative charge density analysis. One route is by a least squares analysis of x-ray data with generalized x-ray-scattering factors. However, published applications of the method do not lend themselves to a critical evaluation of... 

Figure 8.3 The DNA-binding protein Cro from bacteriophage lambda contains 66 amino acid residues that fold into three a helices and three P strands, (a) A plot of the Ca positions of the first 62 residues of the polypeptide chain. The four C-terminal residues are not visible in the electron density map. (b) A schematic diagram of the subunit structure. a helices 2 and 3 that form the helix-turn-helix motif ate colored blue and red, respectively. The view is different from that in (a), [(a) Adapted from W.F. Anderson et al., Nature 290 754-758, 1981. (b) Adapted from D. Ohlendorf et al., /. Mol. Biol. 169 757-769, 1983.]...

The structure was refined by block-diagonal least squares in which carbon and oxygen atoms were modeled with isotropic and then anisotropic thermal parameters. Although many of the hydrogen atom positions were available from difference electron density maps, they were all placed in ideal locations. Final refinement with all hydrogen atoms fixed converged at crystallographic residuals of R=0.061 and R =0.075. 

The difference electron density map following the last cycle of least squares refinement did not show evidence for a simple disorder model to explain the anomalously high B for the hydroxyl oxygen. Attempts to refine residual peaks with partial oxygen occupancies did not significantly improve the agreement index. 

Case 2 An experimental density map for recombinant type III antifreeze protein from eel pout (AFP), which contains 66 residues and in its crystalline form is a member of the P212121 space group [20]. 

With data averaged in point group m, the first refinements were carried out to estimate the atomic coordinates and anisotropic thermal motion parameters IP s. We have started with the atomic coordinates and equivalent isotropic thermal parameters of Joswig et al. [14] determined by neutron diffraction at room temperature. The high order X-ray data (0.9 < s < 1.28A-1) were used in this case in order not to alter these parameters by the valence electron density contributing to low order structure factors. Hydrogen atoms of the water molecules were refined isotropically with all data and the distance O-H were kept fixed at 0.95 A until the end of the multipolar refinement. The inspection of the residual Fourier maps has revealed anharmonic thermal motion features around the Ca2+ cation. Therefore, the coefficients up to order 6 of the Gram-Charlier expansion [15] were refined for the calcium cation in the scolecite. 

Once a suitable crystal is obtained and the X-ray diffraction data are collected, the calculation of the electron density map from the data has to overcome a hurdle inherent to X-ray analysis. The X-rays scattered by the electrons in the protein crystal are defined by their amplitudes and phases, but only the amplitude can be calculated from the intensity of the diffraction spot. Different methods have been developed in order to obtain the phase information. Two approaches, commonly applied in protein crystallography, should be mentioned here. In case the structure of a homologous protein or of a major component in a protein complex is already known, the phases can be obtained by molecular replacement. The other possibility requires further experimentation, since crystals and diffraction data of heavy atom derivatives of the native crystals are also needed. Heavy atoms may be introduced by covalent attachment to cystein residues of the protein prior to crystallization, by soaking of heavy metal salts into the crystal, or by incorporation of heavy atoms in amino acids (e.g., Se-methionine) prior to bacterial synthesis of the recombinant protein. Determination of the phases corresponding to the strongly scattering heavy atoms allows successive determination of all phases. This method is called isomorphous replacement. 

Fig. 13. Stereo drawing of one contour level in the electron density map at 2 A resolution for the residue 54-68 helix in staphylococcal nuclease. Carbonyl groups point up, in the C-terminal direction of the chain the asterisk denotes a solvent peak bound to a carbonyl oxygen in the last turn. Side chains on the left (including a phenylalanine and a methionine) are in the hydrophobic interior, while those on the right (including an ordered lysine) are exposed to solvent.

One of the most intriguing recent examples of disordered structure is in tomato bushy stunt virus (Harrison et ah, 1978), where at least 33 N-terminal residues from subunit types A and B, and probably an additional 50 or 60 N-terminal residues from all three subunit types (as judged from the molecular weight), project into the central cavity of the virus particle and are completely invisible in the electron density map, as is the RNA inside. Neutron scattering (Chauvin et ah, 1978) shows an inner shell of protein separated from the main coat by a 30-A shell containing mainly RNA. The most likely presumption is that the N-terminal arms interact with the RNA, probably in a quite definite local conformation, but that they are flexibly hinged and can take up many different orientations relative to the 180 subunits forming the outer shell of the virus particle. The disorder of the arms is a necessary condition for their specific interaction with the RNA, which cannot pack with the icosahedral symmetry of the protein coat subunits. 

Interpretation of the electron density maps showed that the large subunit could not be modelled beyond His536 (Fig. 6.10), that is fifteen amino acids short of the 551 residues predicted by the nucleotide sequence (Table 6.2). At about the same time, the cleavage of this fifteen-residue stretch, which is performed by a specific protease, was reported to be an obligatory step for the maturation of the enzyme (Menon et al. 1993). It is also of interest to note that in all [NiFe] hydrogenase crystal structures this buried C-terminal histidine is ligated to a metal atom which is either a magnesium or an iron (see above). 

Determination of electron density maps for the u-quartz polymorph establishes that the charge transfer between silicon and oxygen is not complete and that a residual charge of +1.0 ( 0.1) electron units (e.u.) remains localized on silicon, whereas a charge of —0.5 ( 0.1) e.u. is localized on each oxygen atom. The interpretation of this fact in terms of the bond ionicity is not as univocal as it may appear at first glance. 

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