Crystallization, superposition

Figure 5-5. a) Point defect potential in an ionic crystal superposition of the periodic lattice potential and the individual defect potential valley, b) Change of potential with time after a defect... 

The catalytic subunit of cAPK contains two domains connected by a peptide linker. ATP binds in a deep cleft between the two domains. Presently, crystal structures showed cAPK in three different conformations, (1) in a closed conformation in the ternary complex with ATP or other tight-binding ligands and a peptide inhibitor PKI(5-24), (2) in an intermediate conformation in the binary complex with adenosine, and (3) in an open conformation in the binary complex of mammalian cAPK with PKI(5-24). Fig.l shows a superposition of the three protein kinase configurations to visualize the type of conformational movement. 

Fig. 1. Superposition of three crystal structures of cAMP-dependent protein kinase that show the protein in a closed conformation (straight line), in an intermediate conformation (dashed line), and in an open conformation (broken line). The structures were superimposed on the large lobe. In three locations, arrows identify corresponding amino acid positions in the small lobe.

The isothermal curves of mechanical properties in Chap. 3 are actually master curves constructed on the basis of the principles described here. Note that the manipulations are formally similar to the superpositioning of isotherms for crystallization in Fig. 4.8b, except that the objective here is to connect rather than superimpose the segments. Figure 4.17 shows a set of stress relaxation moduli measured on polystyrene of molecular weight 1.83 X 10 . These moduli were measured over a relatively narrow range of readily accessible times and over the range of temperatures shown in Fig. 4.17. We shall leave as an assignment the construction of a master curve from these data (Problem 10). 

Figure 1. The charge-density difference (bonding charge density) between NiaX and the superposition of neutral Ni and X atomic charge densities on the (001) planes for (a) X = A1 and (b) X = Si. The solid (dotted) contours denote contours of increased (decreased) density as atoms are brought together to form the NiaX (X = Al, Si) crystal. Contours start from 4.0 X 10 e/(a.u.) cind increase successively by a factor of root 2.

Figure 9. Theoretical C,E curves (1, 2, 3) for single-crystal faces and (4) for a model polycrystalline surface calculated by the superposition of the C,E curves at E= const [Eq. (49)] with = 02 = = 1/3 1. (a) Faces with strong hydrophilicity and (b, c) faces with...

The resultant layer, with a total thickness of about 10 A, is electrically neutral. A crystal built up by the superposition of such layers would have the composition Si4AlaOi0(OH)2. It is very probable that the mineral pyrophillite, with this composition, has this structure. 

In contrast to single-crystal work, a fiber-diffraction pattern contains much fewer reflections going up to about 3 A resolution. This is a major drawback and it arises either as a result of accidental overlap of reflections that have the same / value and the same Bragg angle 0, or because of systematic superposition of hkl and its counterparts (-h-kl, h-kl, and -hkl, as in an orthorhombic system, for example). Sometimes, two or more adjacent reflections might be too close to separate analytically. Under such circumstances, these reflections have to be considered individually in structure-factor calculation and compounded properly for comparison with the observed composite reflection. Unobserved reflections that are too weak to see are assigned threshold values, based on the lowest measured intensities. Nevertheless, the number of available X-ray data is far fewer than the number of atomic coordinates in a repeat of the helix. Thus, X-ray data alone is inadequate to solve a fiber structure. 

Fig. 8. (a) Structure of the full-length Rieske protein from bovine heart mitochondrial bci complex. The catalytic domain is connected to the transmembrane helix by a flexible linker, (b) Superposition of the three positional states of the catalytic domain of the Rieske protein observed in different crystal forms. The ci state is shown in white, the intermediate state in gray, and the b state in black. Cytochrome b consists of eight transmembrane helices and contains two heme centers, heme and Sh-Cytochrome c i has a water-soluble catalytic domain containing heme c i and is anchored by a C-terminal transmembrane helix. The heme groups are shown as wireframes, the iron atoms as well as the Rieske cluster in the three states as space-filling representations. 

The crystal structures of PbTX-1 dimethyl acetal, PbTX-1, and dihydro PbTX-1 provide a total of four independent pictures of the same brevetoxin skeleton. It is rare that this quantity of structural data is available for a natural product of this size. A comparison of torsional angles shows that all four molecules have approximately the same conformations in all rings, except, of course, for the aldehyde side chain and the E-ring in one of the independent molecules of PbTX-1. Least squares superposition fits among the four molecules gave the following average distances ... 

Top DOS contributions of the different bands of a PtX - chain and their superposition to give the total density of states. Bottom COOP contributions of the different bands and their superposition to give the crystal orbital overlap population... 

At a certain stage in the refinement, the electron density map is interpreted using a model representation of the charge density distribution to extract the atomic coordinates. A commonly used scattering formalism is the independent-atom model (IAM), in which the total charge density in the crystal is approximated by the superposition... 

The idea is simple consider a polycrystalline material that is subjected to locally varying strain. Then every crystal is probing its local strain by small compression or expansion of the lattice constant. The superposition of all these dilated lattices makes the observable line profiles - and as a function of order their breadth has to increase linearly. According to Kochendorfer the polycrystalline material becomes inhomogeneous or heterogeneous . 

The treatments of Kochendorfer, Porod, and Warren-Averbach identify superposition with the mathematical operation of a convolution. While this is true for translational superposition, for dilational superposition it is a coarse approximation that is only valid for small polydispersity. In the latter case the convolution must be replaced by the Mellin convolution (Eq. (8.85), p. 168) governed by a dilation factor distribution and the structure of the reference crystal, the structure of each observed crystal is generated by affine dilation of the reference crystal (Stribeck [2]). 

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