Macromolecular crystals symmetry

This might seem of scant use in protein crystallography, since we have no centric space groups. Crystals of biological macromolecules, as previously pointed out, cannot possess inversion symmetry. Sets of centric reflections frequently do occur in the diffraction patterns of macromolecular crystals, however, because certain projections of most unit cells contain a center of symmetry. The correlate of a centric projection, or centric plane in real space, is a plane of centric reflections in reciprocal space. A simple example is a monoclinic unit cell of space group P2. The two asymmetric units have the same hand, as they are related by pure rotation, and for every atom in one at xj, yj, Zj there is an equivalent atom in the other at —Xj, yj, —zj. If we project the contents of the unit cell on to a plane perpendicular to the y axis, namely the xz plane, by setting y = 0 for all atoms, however, then in that... 

For macromolecular crystals, the symmetry of the diffraction pattern (the Laue symmetry) must be generated by Friedel s law, plus the rotational components of symmetry axes present in the crystal. Once the rotational elements have been identified, it is necessary to deduce whether they are pure rotational operators or some sort of screw axes. For dyads,... 

It may not be obvious how we would locate the x, y, z coordinates of the heavy atom in the unit cell. Indeed it is sometimes not a simple matter to find those coordinates, but as for the heavy atom method described above, it can be achieved using Patterson methods (described in Chapter 9). As we will see later, Patterson maps were used for many years to deduce the positions of heavy atoms in small molecule crystals, and with only some modest modification they can be used to locate heavy atoms substituted into macromolecular crystals as well. Another point. It is not necessary to have only a single heavy atom in the unit cell. In fact, because of symmetry, there will almost always be several. This, however, is not a major concern. Because of the structure factor equation, even if there are many heavy atoms, we can still calculate Juki, the amplitude and phase of the ensemble. This provides just as good a reference wave as a single atom. The only complication may lie in finding the positions of multiple heavy atoms, as this becomes increasingly difficult as their number increases. 

As remarked in note (3) of Table 2.1 crystals of macromolecules cannot contain an inversion centre, a mirror or a glide plane. The diffraction pattern from a protein crystal can, however, contain an inversion centre (otherwise known as a centre of symmetry and a mirror plane. The symmetry symbols given here are for those symmetry elements seen therefore for macromolecular crystals and their diffraction patterns. 

Polarized analysis There is useful spectral information arising from the analysis of polarization of Raman scattered light. This, typically called as polarized analysis, provides an insight into molecular orientation, molecular shape, and vibrational symmetry. One can also calculate the depolarization ratio. Overall, this technique enables correlation between group theory, symmetry, Raman activity, and peaks in the corresponding Raman spectra. It has been applied successful for solving problems in synthetic chemistry understanding macromolecular orientation in crystal lattices, liquid crystals or polymer samples and in polymorph analysis. 

We therefore describe the basis of macromolecular crystallography and provide a summary of how to understand the results of a crystallographic experiment. We start with a mathematical description of what a crystal means in terms of symmetry this applies to all crystals, whether macromolecular or not. Later, we describe how protein crystals grow by using the hanging drop and sitting drop vapor diffusion methods this explains why protein crystals are so fragile and scatter X-rays very weakly. 

The combination of rotational and translational symmetry defines the space group of the crystal. It is shown that 235 space groups exist, but only 65 allow the handedness of the molecule to be preserved, and so only 65 can occur in macromolecular crystallography. The space groups are numbered, but are commonly referred to by their symbols, such as P212121. The most common in macromolecular crystallography are P212121, PI, P21 and C2. 

The exploitation of this radiation, particularly the brilliance and use of short A s, has made virus crystal data collection routine from difficult samples although it is at present necessary to use hundreds of crystals in the gathering of just one data set. Maybe the use of ultra-short wavelength beams ( =0.33 A) from a harmonic of an undulator could be harnessed to improve the lifetime of one such sample sufficiently to give a complete data set. Much larger macromolecular assemblies are currently under study, such as the ribosome, which possess little or no symmetry (unlike viruses) and are therefore more difficult to solve. 

The polarization of Raman scattered light also contains useful structural information. This property can be measured by using plane polarized light and a polarization analyzer. Spectra recorded with the analyzer set both parallel and perpendicular to the excitation plane can be used to calculate the depolarization ratio of each vibrational mode. This provides insight into molecular orientation and the symmetry of the vibrational modes, as well as information about molecular shape. It is often used to determine macromolecular orientation in crystal lattices, liquid crystals, or polymer samples. 

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