Oblique lattice

The S-layer lattices of B. coagulans E38/vl and B. stearothermophilus PV72/p2 are composed of subunits with a molecular mass of 97,0(X), show oblique lattice symmetry... 

The above discussion refers to the loss of mirror symmetry on adsorption leading to chirality at the level of the individual molecule. It is also common for oblique lattices to be formed following molecular adsorption, hence global chirality, even... 

In general, for an oblique lattice in two dimensions with primitive vectors ai and ao, the total conductance G can be expanded into a two-dimensional Fourier series. 

Rotational Symmetry of 2D Lattices. Each of the five lattices has rotational symmetry about axes perpendicular to the plane of the lattice. For the oblique lattice and both the primitive and centered rectangular lattices these are twofold axes, but there are several types in each case. The standard symbol for a twofold rotation axis perpendicular to the plane of projection is . In the case of the square lattice there are fourfold as well as twofold axes. The symbol for a fourfold axis seen end-on is For the hexagonal lattice there are two-, three-, and sixfold axes the latter two are represented by a and , respectively. In Figure 11.4 are shown all of the rotation axes possessed by each lattice. 

Suppose we were to center the oblique lattice of Figure 11.3a. This does not in any way improve its symmetry. All we have is another, denser oblique lattice, which would be properly defined as having a smaller set of defining translation vectors and a unit cell with half the area, as shown in Figure 11.6a. 

Let us now develop systematically the 14 lattices shown in Figure 11.11. Clearly, if we impose no special requirements on the set of defining vectors (Figure 11.12a), namely, a b c, and a p y, we have the 3D analog of our 2D oblique lattice. It is called a triclinic lattice. As with the 2D oblique lattice, there is no unique.way to choose the vector set, but normally one would choose the three shortest vectors. Even if one angle happens to be 90° or two of the three vectors happen to be equal, the lattice is still triclinic because these special relations do not enhance its symmetry. The triclinic lattice has inversion centers as its only symmetry elements. Moreover, a triclinic lattice is necessarily primitive, since if any additional points were introduced at the center of the cell or at any of the face centers, we would have to redefine our vector set in order to include them in a true lattice. 

A 3D lattice can be built up by stacking 2D lattices. If a 2D lattice is defined by two translation vectors, t, and t2, we need to introduce a third translation vector, t3, that defines the stacking pattern. For example, if we stack a set of (identical) oblique lattices (defined by t, and t2) employing a vector t3 that is not orthogonal to the 2D lattice planes, we generate the triclinic lattice, while if we require t3 to be orthogonal to the 2D lattice planes and connect each plane with a point in the nearest neighboring plane we get the primitive monoclinic lattice. 

Of the seven three-dimensional primitive lattices, (a) which one has a unit cell where the a and b lattice vectors form a base that is an arbitrary parallelogram (Uke the unit cell of a two-dimensional oblique lattice), while the c lattice vector is perpendicular to the other two (b) What is the lattice if the a and b lattice vectors form a base that corresponds to the two-dimensional hexagonal unit cell and the c lattice vector is perpendicular to the other two ... 

In the case of SbsC, with the various truncated forms it was demonstrated that the protein part between amino acids 258 and 880 is necessary for self-assembly and the C-terminal 179 amino acids can be deleted without affecting the oblique lattice structure (Jarosch et al., 2001). 

Truncation analysis of SbsC revealed that the C-terminal 179 amino acids are not required for the formation of the oblique lattice structure. In the case of this smallest SslA derivative, the C-terminal 172 amino acids were deleted and the... 

Fig. 3.7 Sketch of the modulated smectic phases. For the sake of clarity, the sinusoidal modulations are drawn in an exaggerated way. a Shows the SmA phase, which is described with a centered rectangular lattice and b shows the SmC phase in which the mesogens are found on an oblique lattice...

The most fundamental columnar phase is the hexagonal phase. In this phase the columns pack into a highly symmetrical hexagonal arrangement. If the cross section of the columns deviates from a perfect circular shape, e.g. because the discs are tilted within the columns, a hexagonal arrangement is not possible. Thus, such columns typically form rectangular or oblique lattices to avoid this unfavorable situation. 

Figure 11 Schematic representation of the formation of mono- and double-layer assembly products as described with S-layer subunits isolated from G. stearothermophilus NRS 2004/3a. This S-layer shows oblique lattice symmetry with center-to-center spacings of the morphological units of 9.4 and 11.6 nm and a base angle of 78°. The oblique lattice symmetry allows us to unambiguously determine the orientation of the constituent monolayer sheets in double-layer self-assembly products. On the oblique monolayer sheet A the axes of the two types of small (70 and 100 nm diameter) monolayer cylinders are formed as indicated. One of the axes includes an angle of 24° to the short base vector of the oblique S-layer lattice. The second axis is perpendicular to the first. Both monolayer cylinders have an identical direction of curvature. Owing to differences in the charge distribution on both the S-layers, PCF is only bound to the inner surface of both types of monolayer cylinders. Five types of double-layer self-assembly products with back-to-back orientation of the inner surface of the constituent monolayers have been found. The superimposition of sheets A and B in the double-layer assembly products of type I is demonstrated and the angular displacement of sheet B with respect to A around point X for the assembly products of type II to V is indicated. (Modified after P. Messner, D. Pum, and U.B. Sleytr. J. Ultrastruct. Mol. Struct. Res. 97 73-88,1986. With permission.)...

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