Translation function phased

Protocol 7.6 Full 6D search with the phased translation function ... 

Bentley, G. (1997) Phased translation function. Method Enzymol. 276, 611-619. 

Cowtan, K. (1998). Modified phased translation functions and their application to molecntlar-fragment location. Acta Crystallogr. D 54,750-756. 

Vagin, A. A. and Isupoy M. N. (2001). Spherically averaged phased translation function and its apphcation to the search for molecules and fragments in electron-density maps. Acta Crystallogr. D 57,1451-1456. 

Because ALBP is related to several proteins of known structure, molecular replacement is an attractive option for phasing. The choice of a phasing model is simple here just pick the one with the amino-acid sequence most similar to ALBP, which is myelin P2 protein. Solution of rotation and translation functions refers to the search for orientation and position of the phasing model (P2) in the unit cell of ALBP. The subsequent paper provides more details. 

It is possible, as shown by Rossmann and Blow (1962), to search for redundancies in Patterson space that correspond to the multiple copies of molecular transforms. Rossmann and Blow show, however, that the Patterson map does not need to be computed and used in any graphical sense, but that an equivalent search process can be carried out directly in diffraction or reciprocal space. Using such a search procedure, called a rotation function, they showed that noncrystallographic relationships, both proper and improper rotations, could be deduced in many cases directly from the X-ray intensity data alone, and in the complete absence of phase information. Translational relationships (only after rotations have been established) can also be deduced by a similar approach. Rotation functions and translation functions constitute what we call molecular replacement procedures. Ultimately the spatial relationships among multiple molecules in an asymmetric unit can be defined by their application. 

The low-resolution phases initially required may be obtained in a variety of ways, but frequently these depend on other imaging techniques outside of X-ray crystallography. These may include transmission electron microscopy, cryo-electron microscopy, or atomic force microscopy. Low-resolution phases are even more often obtained by placing the known structure of a closely related virus, or complex, in the correct disposition in the unit cell (determined by rotation and translation functions) and using its low resolution calculated phases. 

When a suitable model of the unknown crystal structure is available, it can be used to solve the phase problem. Examples are the use of the stracture of human thrombin to solve the structure of bovine thrombin, the use of a known antibody fragment to solve the stracture of an unknown antibody, or the use of the stracture of an enzyme to solve the stracture of an inhibitor complex of the same enzyme in a different crystal form. The model is oriented and positioned in the unit cell of the unknown crystal with the use of rotation and translation functions, and the oriented model is subsequently used to calculate phases and an electron-density map. 

If there is one protein molecule per asymmetric unit, but it has some local symmetry such as a twofold axis of rotation relating two identical subunits, then use can be made of this information in order to solve the structure. For example, the orientation of the twofold rotation axis can probably be determined from the rotation function. In a similar way, the translation function can lead to information on the positions of the two related subunits. If the twofold axis can be located, then the electron density for the two symmetry related portions of the molecule can be averaged. An envelope is drawn in the electron-density map that essentially defines the edge of each molecule. The electron density in the two independent molecules is then averaged and the solvent region is flattened. By Fourier transformation a better set of phases is obtained for a new electron-density map, which may be more readily interpreted (63). 

The symmetry of the structure we are looking for is imposed on the field 0(r) by building up the field inside a unit cubic cell of a smaller polyhedron, replicating it by reflections, translations, and rotations. Such a procedure not only guarantees that the field has the required symmetry but also enables substantial reduction of independent variables 0/ the function F (f)ij k )- For example, structures having the symmetry of the simple cubic phase are built of quadrirectangular tetrahedron replicated by reflection. The faces of the tetrahedron lie in the planes of mirror symmetry. The volume of the tetrahedron is 1 /48 of the unit cell volume. 

Short-time Brownian motion was simulated and compared with experiments [108]. The structural evolution and dynamics [109] and the translational and bond-orientational order [110] were simulated with Brownian dynamics (BD) for dense binary colloidal mixtures. The short-time dynamics was investigated through the velocity autocorrelation function [111] and an algebraic decay of velocity fluctuation in a confined liquid was found [112]. Dissipative particle dynamics [113] is an attempt to bridge the gap between atomistic and mesoscopic simulation. Colloidal adsorption was simulated with BD [114]. The hydrodynamic forces, usually friction forces, are found to be able to enhance the self-diffusion of colloidal particles [115]. A novel MC approach to the dynamics of fluids was proposed in Ref. 116. Spinodal decomposition [117] in binary fluids was simulated. BD simulations for hard spherocylinders in the isotropic [118] and in the nematic phase [119] were done. A two-site Yukawa system [120] was studied with... 

For smectic phases the defining characteristic is their layer structure with its one dimensional translational order parallel to the layer normal. At the single molecule level this order is completely defined by the singlet translational distribution function, p(z), which gives the probability of finding a molecule with its centre of mass at a distance, z, from the centre of one of the layers irrespective of its orientation [19]. Just as we have seen for the orientational order it is more convenient to characterise the translational order in terms of translational order parameters t which are the averages of the Chebychev polynomials, T (cos 2nzld)-, for example... 

One of the primary features of the Gay-Berne potential is the presence of anisotropic attractive forces which should allow the observation of thermally driven phase transitions and this has proved to be the case. Thus using the parametrisation proposed by Gay and Berne, Adams et al. [9] showed that GB(3.0, 5.0, 2, 1) exhibits both nematic and isotropic phases on varying the temperature at constant density. This was chosen to be close to the transitional density for hard ellipsoids with the same ellipticity indeed it is generally the case that to observe a nematic-isotropic transition for Gay-Berne mesogens the density should be set in this way. The long range orientational order of the phase was established from the non-zero values of the orientational correlation coefficient, G2(r), at large separations and the translational disorder was apparent from the radial distribution function. 

Similar behaviour is found for the singlet translational distribution function, p z), in the two smectic phases. According to the McMillan theory... 

The important information about the properties of smectic layers can be obtained from the relative intensities of the (OOn) Bragg peaks. The electron density profile along the layer normal is described by a spatial distribution function p(z). The function p(z) may be represented as a convolution of the molecular form factor F(z) and the molecular centre of mass distribution f(z) across the layers [43]. The function F(z) may be calculated on the basis of a certain model for layer organization [37, 48]. The distribution function f(z) is usually expanded into a Fourier series f(z) = cos(nqoz), where the coefficients = (cos(nqoz)) are the de Gennes-McMillan translational order parameters of the smectic A phase. According to the convolution theorem, the intensities of the (OOn) reflections from the smectic layers are simply proportional to the square of the translational order parameters t ... 

Starting with the partition function of translation, consider a particle of mass m moving in one dimension x over a line of length I with velocity v. Its momentum Px = mVx and its kinetic energy = Pxllm. The coordinates available for the particle X, px in phase space can be divided into small cells each of size h, which is Planck s constant. Since the division is so incredibly small we can replace the sum with integration over phase space in x and Px, and so calculate the partition function. By normalizing with the size of the cell h the expression becomes... 

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