Asymmetric part of the unit cell

Symmetry operations, therefore, can be visualized by means of certain symmetry elements represented by various graphical objects. There are four so-called simple symmetry elements a point to visualize inversion, a line for rotation, a plane for reflection and the already mentioned translation is also a simple symmetry element, which can be visualized as a vector. Simple symmetry elements may be combined with one another producing complex symmetry elements that include roto-inversion axes, screw axes and glide planes. 

---

Note that only those symmetry elements which intersect the asjnnmetric part of the unit cell are independent, exactly in the same way as only those atoms that are found in the asymmetric part of the unit cell are independent (see Figure 1.6). Once the locations of independent atoms and symmetry elements in the unit cell are known, the whole crystal can be easily... 

This defines a set of equations for the mean field Hamiltonians HPF. These equations have to be solved self-consistently since the thermodynamic values within the angle brackets in (109) involve the mean field Hamiltonians // F. In principle, all // F can be different in practice, we impose symmetry relations. Therefore, we choose a unit cell, compatible with the symmetry of the lattice introduced in Section II,D, and we put Hpf equal to // F whenever P and P belong to the same sublattice. Moreover, we apply unit cell symmetry that relates the mean field Hamiltonians on different sublattices. By using the symmetry-adapted functions introduced in Section II,B, the latter symmetry can be imposed as follows. We select a set of molecules constituting the asymmetric part of the unit cell. Then we assign to all other molecules P Euler angles tip-through which the mean field. Hamiltonian of some molecule P in the asymmetric part has to be rotated in order to obtain HrF. As a result, we... 

If we substitute these transformation relations into Eq. (109), we observe that the latter equation involves only the mean field Hamiltonians of the molecules in the asymmetric part of the unit cell. 

Asymmetric unit Smallest part of the unit cell that, when operated on by the symmetry operations, produces the whole unit cell. 

The independent part of the unit cell (e.g. the upper right half of the unit cell separated by a dash-dotted line and shaded in Figure 1.6) is called the asymmetric unit. It is the only part of the unit cell, for which the specification of atomic positions and other atomic parameters are required. 

The presence of rotational or screw symmetry means that the unit cell has internal symmetry. Therefore, only part of the unit cell, known as the asymmetric unit, is needed to uniquely define the unit cell. (The asymmetric unit may also contain more than one molecule, related by movements — symmetry operations — that are not part of the crystal symmetry - noncrystallographic symmetry operators. This can be very important in determining the protein structure, as discussed in Section 9.03.9.3). 

In molecular crystals, the asymmetric unit of the unit cell may be composed of a part of the chemical unit, of the chemical unit itself, or even of more than one chemical unit. The chemical unit is understood as the single molecule or the formula unit of the substance. For the free molecule (or formula unit) we can define "chemically equivalent"sites within the molecule. 

Crystallographic symmetry simplifies matters enormously. For example, the unit cell shown in Figure 10.21 could have an inversion center right in the middle of the unit cell, in which case we only have to describe the contents of half of a unit cell, while the rest can be constructed from this reduced part by applying the symmetry operations. This fraction is called the asymmetric unit, and its importance stems from the fact that it is only this part of a crystal structure that has to be determined by the diffraction experiment. 

Another useful physical property of the crystal is its density, which can be used to determine several useful microscopic properties, including the protein molecular weight, the proteinlwater ratio in the crystal, and the number of protein molecules in each asymmetric unit (defined later). Molecular weights from crystal density are more accurate than those from electrophoresis or most other methods (except mass spectrometry) and are not affected by dissociation or aggregation of protein molecules. The proteinlwater ratio is used to clarify electron-density maps prior to interpretation (Chapter 7). If the unit cell is symmetric (Chapter 4), it can be subdivided into two or more equivalent parts called asymmetric units (the simplest unit cell contains, or in fact is, one asymmetric unit). For interpreting electron-density maps, it is helpful to know the number of protein molecules per asymmetric unit. 

Another experimental verification of theoretical data may be provided by the analysis of a range of variation of endocyclic torsion angles in crystal structures containing uracil and cytosine fragments. It was demonstrated [32] that range of variation of the N1-C2-N3-C4 torsion angle in uracil and the C6-N1-C2-N3 torsion angle in cytosine amounts to 12-13°. For example, it was found that the value of the C6-N1-C2-N3 in the crystal of 5-bromo-2 -deoxycytidine is —7.1°, —12.4°, and — 13.9°, respectively, for three molecules in asymmetrical part of unit cell [39],... 

For single-crystal diffraction, a good-quality single crystal of the sample of interest is required. From the angles and intensities of diffracted radiation, the structure of the crystal can be elucidated and the positions of the molecules in the unit cell can be determined. The result is often displayed graphically as the asymmetric unit, which is the smallest part of a crystal structure from which the complete structure can be obtained using space-group symmetry operations. 

Many protein crystals exist with more than one molecule per asymmetric unit. These molecules are sometimes related by noncrystallographic symmetry (pseudosymmetry), that is, additional symmetry (such as a twofold rotation axis) that is not part of the symmetry defined by the space group. This feature can be very useful in finding the molecular structure using rotation functions. Additionally, it is also possible for a compound to crystallize in different forms with different packing (polymorphism). If the molecular transforms of the components of the crystal are known, it is possible, by the methods described above, to determine their positions and orientations in the respective unit cells. 

The choice of unit cell shape and volume is arbitrary but there are preferred conventions. A unit cell containing one motif and its associated lattice is called primitive. Sometimes it is convenient, in order to realise orthogonal basis vectors, to choose a unit cell containing more than one motif, which is then the non-primitive or centred case. In both cases the motif itself can be built up of several identical component parts, known as asymmetric units, related by crystallographic symmetry internal to the unit cell. The asymmetric unit therefore represents the smallest volume in a crystal upon which the crystal s symmetry elements operate to generate the crystal. 

The MM energy of a crystal is usually calculated for the asymmetric unit of a unit cell that is supposed to be part of an infinite lattice thus, the cell is surrounded by an infinite number of identical cells. This convention has no technical implications for the calculation of the terms in the MM energy function that are restricted to atoms within the same molecule (bond-stretching, an-... 

---