Direct rotation function

Briinger and colleagues developed a direct rotation function, which is just a correlation coefficient between Eq s Ejy,Q (Omega), the normal-... 

DeLano, W. L. and Brunger, A. T. (1995) The direct rotation function Patterson correlation search applied to molecular replacement. Acta Crystallogr. D 51, 740-748. 

Figure 9.23 s-Polarized second-harmonic signal detected in transmitted direction as function of the azimuthal rotation angle. Twofold pattern clearly indicates C2 symmetry of sample. 

Figure 8.4 Rotation function results P2 into crystalline ALBP. (a)Plot of the 101 best solutions to the rotation function, each peak numbered in the horizontal direction (abscissa). The correlation between the Patterson s of the probe molecule and the measured ALBP X-ray results are shown in the vertical direction (ordinate) and are given in arbitrary units. (b) Description of the rotation studies after Patterson correlation refinement. The peak numbers plotted in both panels (a)and (b) are the same. Reprinted with permission from Z. Xu et al. (1992) Biochemistry 31,3484-3492. Copyright 1992 American Chemical Society.

FIGURE 8.21 (cont d). (c) The rotation function for crystalline insulin. Peaks (highlighted by arrows) correctly indicate the direction of the local twofold symmetry axes. These twofold axes were also indicated in the Patterson function in (b). The actual crystal structure, in the same orientation as in (a) to (c), is shown in (d), with the unit-cell axes a and b, and the local twofold axes indicated by arrows. (From Ref, 65. Courtesy the authors and Academic Press.)... 

Fig. 11 Image plot of the experimental (a) and calculated (b) X-band EPR spectrum at 1.8 K as a function of the magnetic field direction rotated from the (001) by 0 to the (110) direction...

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 initial vector v must also be evaluated. Instead of computing directly the vector representation of the given rotational function (i.e., the spherical harmonic in ftj, which is an element of the uncoupled basis set), one can evaluate the matrix representation of the function, which we call M, and then multiply it by the vector representation of which we call Vq, whose calculation is relatively easy utilizing the coupled basis set (see Appendix C.2). That is, let... 

In Eq. (IV.56h) we have chosen the space fixed F-axis to point in the direction of the virtual electric field Ets- The eigenvalue problem which corresponds to the above simplified Hamiltonian may be solved by a perturbation treatment. For this purpose the corresponding Hamiltonian matrix is calculated in a basis of functions built from products between electronic functions cz,..., aNe,bxfg,CNe) and rotational functions iptoi ,0,x)- As electronic functions we will use the eigenfunctions of Eq. (IV.56a), i.e.,... 

Equation (5.161) is very similar to that used by Bagchi et al. in their recent study [76]. However, since their expression is written in terms of the longitudinal ion-dipole direct correlation function and the orientational intermediate scattering function of the solvent in place of Cux k) and F fi k,t) in our formula, its application is limited to the calculation of the dielectric friction. As we have clarified in Sec. 5.3, both the translational and rotational motions of solvent molecules manifest themselves in Fxfi k, t), and Eq. (5.161) can be applied to the calculation of the friction coefficient which comprises the hydrodynamic as well as dielectric contributions. Thus Eq. (5.161) can be regarded as a more general microscopic expression for the friction coefficient. 

For rotational fine structure in electronic band spectra, the P- and R-branch line positions are still given by Eqs. 4.76, except that Vq now becomes Tg -I- G v ) — G"(y"). The important physical difference here is that B and B" are often grossly different in transitions between different electronic states. For example, B and B are 0.029 and 0.037 cm S respectively, for the <- transition in Ij (Fig. 4.7). The rotational line positions in the P and R branches are no longer even approximately equally spaced. For that matter, Vp and Vr both vary as (B — B")J for large J—since they are then dominated by terms quadratic in J—and hence they run in the same direction as functions of J. In contrast, Vp and Vr run in opposite directions as functions of J for small J, where the linear terms dominate. Hence, either the P or the R branch turns around as a function of J (depending on whether B < B" or B > B") at the bandhead as shown in Fig. 4.27. The approximate value of J at which one of these branches turns around can be found by differentiating the... 

T is a rotational angle, which determines the spatial orientation of the adiabatic electronic functions v / and )/ . In triatomic molecules, this orientation follows directly from symmetry considerations. So, for example, in a II state one of the elecbonic wave functions has its maximum in the molecular plane and the other one is perpendicular to it. If a treatment of the R-T effect is carried out employing the space-fixed coordinate system, the angle t appearing in Eqs. (53)... 

The catalytic subunit then catalyzes the direct transfer of the 7-phosphate of ATP (visible as small beads at the end of ATP) to its peptide substrate. Catalysis takes place in the cleft between the two domains. Mutual orientation and position of these two lobes can be classified as either closed or open, for a review of the structures and function see e.g. [36]. The presented structure shows a closed conformation. Both the apoenzyme and the binary complex of the porcine C-subunit with di-iodinated inhibitor peptide represent the crystal structure in an open conformation [37] resulting from an overall rotation of the small lobe relative to the large lobe. 

Field variables identified by their magnitude and two associated directions are called second-order tensors (by analogy a scalar is said to be a zero-order tensor and a vector is a first-order tensor). An important example of a second-order tensor is the physical function stress which is a surface force identified by magnitude, direction and orientation of the surface upon which it is acting. Using a mathematical approach a second-order Cartesian tensor is defined as an entity having nine components T/j, i, j = 1, 2, 3, in the Cartesian coordinate system of ol23 which on rotation of the system to ol 2 3 become... 

---