The three-dimensional crystal

The general problem of the three-dimensional crystal consists in solving the following system of equations  

Here Aatom is the number of atoms per unit cell, N ew is the number of unit cells considered (very large), i and j labels include the x, y, z components of different atoms and q is a mass weighted coordinate, after Eq. (4.5). The interaction cannot depend on the absolute positions of the atoms but their relative positions. Again a major simplification to this equation comes from the application of the Bloch theorem and cyclic boundary conditions [8]. We shall represent the positions of all atoms in the solid by reference to a chosen unit cell, the zeroth cell (superscript 0), then for any k  

The summation over the 3 iVatomA ceii atomic coordinates is written as a double summation, for the atoms in the unit cell with an index i and over r eventually extended all over space. Eq. (4.50) can now be written  

The force constants now have an explicit dependence on the vector k. The periodicity of the crystal simplified the problem and we have a set of 3iVatom X 3iVatom equations for every value of k  

The process of finding the Ifequencies and the amplitudes of the displacement is mathematically the same as in the molecular case ( 4.2.2), with the exception of the k dependence. Therefore we must calculate the fi equencies and amplitudes across the whole range of k values. Normalising the vibrational amplitude, after Eq (4.20), we have  

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Surface crystallography is simply the two-dimensional analogue of bulk crystallography, in which we consider the structure of the different planes on which the atoms in the three-dimensional crystal reside. We limit ourselves to the most often encoun-... 

In the second view of the three-dimensional crystals the projected image normal to the plane of the lamellae (Fig. 7) shows ordered arrays of 40-50-A-diameter particles, that represent the cytoplasmic domains of ATPase molecules [156]. The crystals... 

In rhodopsin, EPR studies have demonstrated a clear helical periodicity in most of intracellular loop-3, except for a couple of residues in the middle (indicated in Figure 2.5). This would suggest that TM-V and TM-VI extend way into the cytosol and that only a very short loop connects these two helical extensions. However, in the three-dimensional crystals, most of intracellular loop-3 is a rather unstructured loop. Thus, in this case, it is likely that the EPR studies tell us something about the solution structure of the receptor, which may not be clear in the x-ray structure. 

Lyotropic lamellar (La) liquid crystals (LC), in a form of vesicle or planar membrane, are important for membrane research to elucidate both functional and structural aspects of membrane proteins. Membrane proteins so far investigated are receptors, substrate carriers, energy-transducting proteins, channels, and ion-motivated ATPases [1-11], The L liquid crystals have also been proved useful in the two-dimensional crystallization of membrane proteins[12, 13], in the fabrication of protein micro-arrays[14], and biomolecular devices[15]. Usefulness of an inverted cubic LC in the three-dimensional crystallization of membrane proteins has also been recognized[16]. 

The three-dimensional crystal structure of Pigment Violet 23 (117) elucidates molecular staples whereby the stables of the single molecular planes are arranged in an almost perpendicular orientation to each other. Interactions of the -ir-orbitals as well as van der Waals forces are responsible for the cohesion of the structure. The degree of overlapping is especially high in the dioxazine part of the molecule [3],... 

The three-dimensional crystal can be treated by a straightforward generalization of the method outlined above (6). A simple cubic lattice is defined by three integers (wii, rrii, m3), which take the values 0, 1,. . . , N. A free (100) surface is defined by the plane mi = 0, and the Coulomb integral is changed from a to a for all atoms in this plane. The wave functions are assumed to vanish on the other five surfaces of a cube. The wave function coefficients are given by... 

Divne, C., Stahlberg.J., Reinikainen, T., Ruohonen, L., Pettersson, G., KnowlesJ., et al. (1994) The three-dimensional crystal stractme of the catalytic core of cellobiohydrolase I from Trichoderma reesei. Science, 265, 524-528. 

Unlike the two-dimensional arrays in these examples, a crystal is a three-dimensional array of objects. If we rotate the crystal in the X-ray beam, a different cross section of objects will lie perpendicular to the beam, and we will see a different diffraction pattern. In fact, just as the two-dimensional arrays of objects we have discussed are cross sections of objects in the three-dimensional crystal, each two-dimensional array of reflections (each diffraction pattern recorded on film) is a cross section of a three-dimensional lattice of reflections. Figure 2.11 shows a hypothetical three-dimensional diffraction pattern, with the reflections that would be produced by all possible orientations of a crystal in the X-ray beam. 

Figure 8.1 Notation of a cutting plane by Miller indices. The three-dimensional crystal is described by the three-dimensional unit cell vectors di, 02, and 03. The indicated plane intersects the crystal axes at the coordinates (3,1,2). The inverse is (, j, ). The smallest possible multiplicator to obtain integers is 6. This leads to the Miller indices (263).

The three-dimensional crystal structure of the complex between the Fab fragment of 7G12 and /V-rnethyl mesoporphyrin was determined... 

The three-dimensional crystal structure was solved for the AZ-28 antibody Fab complexed with hapten (Ulrich et al., 1997 Ulrich and Schultz,... 

Even in a macroscopic crystal, each electron, being a fermion, must possess a unique set of quantum numbers apart from the "internal" set of quantum numbers within the atom, ion, or molecule. Assuming that there is translational periodicity in the three-dimensional crystal, we obtain... 

The Ewald series for the three-dimensional crystal can also be differentiated. The first derivative yields expressions for the Madelung electric field Fm (due to local charges). The second derivative yields the Madelung field gradient, or, equivalently, the internal or dipolar or Lorentz field FD (due to local dipoles) [68-71],This second derivative can also generates the dimensionless 3x3 Lorentz factor tensor L with its nine components Lv/t ... 

B) Arrangement of domains within the three-dimensional crystal structure of (Xiong et al, 2001). Each domain is color coded as in A. (C) The structure in (B) with an I domain added. (See Color Insert.)... 

Due to the high spatial resolution and predictive scattering modes, TEMs are often employed to determine the three-dimensional crystal structure of solid-state... 

Rummel G, Hardmeyer A, Widmer C, Chiu ML, Nollert P, Locher KP, Pedruzzi I, Landau EM, Rosenbusch JP. Lipidic cubic phases new matrices for the three-dimensional crystallization of membrane proteins. J. Struct. Biol. 1998 121 82-91. 

Wiener MC. Existing and emergent roles for surfactants in the three-dimensional crystallization of integral membrane proteins. Curr. Opin. Coll. Int. Sci. 2001 6 412-119. 

The optical treatment in Chapter 1 is concerned with the nature of the image of a simple two-dimensional grid formed by a perfect lens, in focus, using monochromatic light. However, to understand the nature of the lattice image formed in the electron microscope, we must take into account the thickness and the orientation of the three-dimensional crystal, the defect of focus and the aberrations of the objective lens (see Chapter 2), and the beam convergence, because all these factors influence the relative phases of the diffracted beams that are permitted to pass through the objective aperture. 

The determination of the three-dimensional crystal structure of the 22-kDa fragment of apoE in 1991 represented a major milestone in the studies of the structure and function of apoE. With this structure, it is now possible to understand and interpret much of what was known previously about the protein. In addition, identification of the structures of the apoE2 and apoE4 variants provides new insight into how apoE interacts with the LDL receptor and how the preference for different lipoprotein classes might be influenced by structure. These structures represent the beginning of the next level of understanding of how... 

Zhang R-G, Scott DL, Westbrook ML, et al. (1995a) The three-dimensional crystal structure of cholera toxin. In J. Mol. Biol. 251 563-573. 

Different relative orientations of the columns in the three-dimensional crystal lattices depend on crystal symmetry. So far, three types of column packing have been observed (Fig. 5b) ... 

The fact that a solid is a metal or a nonmetal will therefore depend on three factors (i) the separation of the orbital energies in the free atom (ii) the lattice spacing and (iii) the number of electrons provided by each atom. For a realistic description of the three-dimensional crystal, we must therefore extend our simple Hiickel theory5 in two respects. First, we must consider more than a single type of AOs (e.g. 2s, 2 p, 3d, ), and, second, we must consider more than an electron per atom. By increasing the... 

Pure dried tylophorine is a yellow powder, and crystallization from common solvents is difficult. Spectral data, including MS, H-NMR, C-IMMR and chiroptical (ORD and CD), of phenanthroindolizidine alkaloids have been well established [18, 42, 44-45]. The three-dimensional crystal structure of tylophorine (1) was first determined by Wang et al [46]. The structure of tylophorine B, as the benzene solvate, has conjoined phenanthrene and indolizidine moieties. The aromatic rings lie almost in the same plane with dihedral angles of 1.7° (A/B), 2.8° (B/C), 2.2° (A/C), and 7.3° (B/D). The E ring adopts an envelope conformation and makes a dihedral angle of 6.7(3)° with the D ring [46]. 

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