Single lattice unit

Highly simplified models of protein structure embedded into low coordination lattices have been used for tertiary structure prediction for almost 20 years [65, 66, 75]. For example, Covell and Jemigan [64] enumerated all possible conformations of five small proteins restricted to fee and bcc lattices. They found that the nativelike conformation always has an energy within 2% of the lowest energy. Virtually simultaneously. Hinds and Levitt [28] used a diamond lattice model where a single lattice unit represents several residues. While such a representation cannot reproduce the geometric details of helices or P-sheets, the topology of native folds could be recovered with moderate accuracy. 

At this point, we will comment on how this procedure generalizes to other polymers. The other case that was considered by us [28,30,32,175,176] was concerned with bisphenol-A-polycarbonate (BPA-PC) (cf. Fig. 5.1). While for PE we had a correspondence that five chemical repeat units correspond to one effective bond of the bond fluctuation model, for BPA-PC the mapping ratio was inverse - one chemical repeat unit was mapped onto three effective bonds One must consider, however, the very different sizes of the chemical repeat units while for PE this is a single CH2 group, in BPA-PC the repeat unit involves 12 C-C or C-0 bonds along the backbone, and the end-to-end distance of the repeat unit is of the order of 10 A. Thus in this case also one effective bond corresponds to a group of four successive covalent bonds along the backbone of the chain, and a lattice unit corresponds to about 2.03 A [175],... 

The attenuation of die dipole of the repeat unit owing to thermal oscillations was modeled by treating the dipole moment as a simple harmonic oscillator tied to the motion of the repeat unit and characterized by the excitation of a single lattice mode, the mode, which describes the in-phase rotation of the repeat unit as a whole about the chain axis. This mode was shown to capture accurately the oscillatory dynamics of the net dipole moment itself, by comparison with short molecular dynamics simulations. The average amplitude is determined from the frequency of this single mode, which comes directly out of the CLD calculation ... 

Fig. 3. (10 ) CTR or the a- Al203(0001)-(lxl) surface. Experimental (solid circles) and best-fit models for each possible termination single A1 layer (thick solid line), double A1 layer (dashed line) and oxygen terminated surfaces (dotted line). The logarithm of the structure factor is reported as a function of the out-of-plane momentum transfer in reciprocal lattice units of AI2O3.

Although the Flory-Huggins theory was derived from a lattice model in which units of polymer A and polymer B are-the same size (i.e., they each occupy a single lattice cell), the theory is readily generalized so that it can apply to realistic cases in which the volumes of monomers A and B are unequal ... 

FIGURE 3.15 The types of unit cells that form the basis for the allowable lattices of all crystals (known as the Bravais lattices). There are 15 unique lattices (see International Tables, Volume I, for further descriptions). All primitive (/ ) cells may be considered to contain a single lattice point (one-eighth of a point contributed by each of those at the corners of the cell), face-centered (C) and body-centered (/) cells contain two full points, and face-centered (F) cells contain four complete lattice points. 

This is the familiar formulation of Bragg s law for a three-dimensional point lattice. It says that the Fourier transform of a point lattice is absolutely discrete and periodic in diffraction space, and that we can predict when a nonzero diffraction intensity will appear for any family of planes hkl, and what the angle of incidence and reflection 0 must be in order for an intensity to appear. Bragg s law, notice, is completely independent of atoms, or molecules, or unit cell contents. The law is imposed by the periodicity of the crystal lattice, and it strictly governs where we may observe any nonzero intensity in diffraction space. It tells us when the resultant waves produced by the scattering of all of the atoms in the many individual unit cells, each represented by a single lattice point, are exactly in phase. 

What is seen for one dimension is quite the same for the two- or three-dimensional cases as well. Just as the resultant wave created by the interference of the scattered waves from all of the atoms in the molecules could be considered as arising from discrete lattice points, the same is true for a real crystal. We can consider the resultant waves produced by the scattering of all of the atoms in the unit cells to simply be emerging from a single lattice point common to each cell, as in Figure 5.10. Because the contents of the unit cell are continuous and nonperiodic, their transform, or resultant waves F-s will be nonzero for all s. Because the lattice points in a crystal are discrete and periodic, however, the waves from all lattice points will constructively interfere and be observable only in certain directions according to Bragg s law, that is, when s = h. 

Each of these terms is identical to the structure factor as described in Equation (14), except that the summation is over a subset of atoms. Fuc is summed only over atoms within a single bulk unit cell, Fsurf is summed over all near-surface atoms that might be displaced from their ideal bulk lattice positions (typically 2-3 layers deep into the crystal) plus any adsorbed layers attached to the surface, and Fwater describes the fluid structure above the interface, including any structuring of the fluid near the mineral surface. Equation (16a) can be rewritten to express the scattering intensity,... 

Traditionally, the thermodynamics of polymer mixtures was developed in terms of a lattice model, with each monomer unit of the polymer chains occupying a single lattice site. The free energy of mixing of polymers in solution can be described by the Flory-Huggins equation ... 

Let us illustrate this simple approach by a few examples, but we first add that numerical values for molar volumes (measured in cm /mol) are converted into the crystallographer s A /f.u. (f.u. = formula unit) by multiplying them by a factor of 1.661 and vice versa. According to the Biltz table, sodium chloride (NaCl) should have a molar volume of 6.5 cm /mol (Na" ") + 20 cm /mol (Cl ) = 26.5 cm /mol. In a given crystal structure, a single formula unit of NaCl will therefore occupy 44.0 A. In fact, the crystal structure of NaCl has a lattice parameter of 5.64 A and a unit cell volume of 179.4 A with four formula units, which yields an experimental 44.9 A per formula unit and an incremental error of only 2%. ... 

Using your largest spheres, construct a single simple cubic lattice unit cell. Using dots to represent centers of atoms, draw a diagram to represent your model. 

Construct a single body-centered cubic lattice unit cell. 

S-layers are composed of single protein or glycoprotein subunits which, after secretion, crystallize into two-dimensional lattices. The lattices can have different types of symmetries. Depending on the lattice type, one lattice unit consists of one, two, three, four, or six protein monomers rendering, therefore, oblique (pi, p2), trimeric (p3), square (p4), or hexagonal symmetry (p6) to the lattice (Figure 3.4). The lattice stracture is further characterized by the lattice constants a and b as well as by the base angle 7. The center to center distance of the units varies from 3.5 to 35 nm. 

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