Group screw rotation

The unit cell considered here is a primitive (P) unit cell that is, each unit cell has one lattice point. Nonprimitive cells contain two or more lattice points per unit cell. If the unit cell is centered in the (010) planes, this cell becomes a B unit cell for the (100) planes, an A cell for the (001) planes a C cell. Body-centered unit cells are designated I, and face-centered cells are called F. Regular packing of molecules into a crystal lattice often leads to symmetry relationships between the molecules. Common symmetry operations are two- or three-fold screw (rotation) axes, mirror planes, inversion centers (centers of symmetry), and rotation followed by inversion. There are 230 different ways to combine allowed symmetry operations in a crystal leading to 230 space groups.12 Not all of these are allowed for protein crystals because of amino acid asymmetry (only L-amino acids are found in proteins). Only those space groups without symmetry (triclinic) or with rotation or screw axes are allowed. However, mirror lines and inversion centers may occur in protein structures along an axis. 

Crystal symmetries that entail centering translations and/or those symmetry operations that have translational components (screw rotations and glides) cause certain sets of X-ray reflections to be absent from the diffraction pattern. Such absences are called systematic absences. A general explanation of why this happens would take more space and require use of more diffraction theory than is possible here. Thus, after giving only one heuristic demonstration of how a systematic absence can arise, we shall go directly to a discussion of how such absences enable us to take a giant step toward specifying the space group. 

The strips may be divided into unit cells or eventually reduced unit cells (see Fig. 1), when the space group of the strip contains operations involving glide-reflections or screw-rotations, i.e. a combination of an improper reflection or two-fold rotation followed by a non-primitive translation of half a unit cell, which by themselves do not leave the lattice invariant. Each cell will contain w sites, w x L = 2N. 

A few pairwise interfaces in the range 1300-1900 are nevertheless observed in crystals. Almost all result from the presence of twofold and other point-group symmetry elements, which are relatively uncommon in crystals of monomeric proteins. Their occurrence suggests that the formation of dimers or other small oligomers in solution precedes crystallization under the conditions where these particular crystals are obtained at protein concentrations typically in the range 10 —10 M. The large majority of the crystal contacts are associated with lattice translations and screw rotations not found in oligomeric proteins. Their size distribution resembles that of the transient interfaces created by the random collision of two small proteins simulated in the computer... 

The group of rotating positive displacement pumps comprises a lot of different versions. When setting a borderline for high-pressure usabUity at 10 MPa, only the gear pumps (70 MPa), the screw pumps (32 MPa), and the progressing cavity pumps (15 MPa) remain. The basic conveying characteristics of these pumps can be... 

Table 14.5 Combinations of symmetry operations in organic crystals A, twofold rotation M, mirror reflection G, glide reflection S, twofold screw rotation 1, inversion tbrongb a point Ct, centering translation. The labels preceding each space group symbol are as follows C, cluster, R, row, L, layer and 3D, fuU three-dimensional structure. When several possihihties are given for an arrangement, they depend on the relative orientation of the symmetry operations...

If the space group contains screw axes or glide planes, the Patterson fiinction can be particularly revealing. Suppose, for example, that parallel to the c axis of the crystal there is a 2 screw axis, one that combines a 180° rotation with... 

Notice that the symmetry operations of each point group by continued repetition always bring us back to the point from which we started. Considering, however, a space crystalline pattern, additional symmetry operations can be observed. These involve translation and therefore do not occur in point groups (or crystal classes). These additional operations are glide planes which correspond to a simultaneous reflection and translation and screw axis involving simultaneous rotation and translation. With subsequent application of these operations we do not obtain the point from which we started but another, equivalent, point of the lattice. The symbols used for such operations are exemplified as follows ... 

The products of the type are the components of an additional tensor, S, called the screw tensor, as the coupling between translations and rotations describes a screw-type motion. Unlike the tensors T and L, S is unsymmetrical, since is different from . The terms involving elements of S may be grouped as (for Ul2)... 

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