Bond vector

The range of systems that have been studied by force field methods is extremely varied. Some force fields liave been developed to study just one atomic or molecular sp>ecies under a wider range of conditions. For example, the chlorine model of Rodger, Stone and TUdesley [Rodger et al 1988] can be used to study the solid, liquid and gaseous phases. This is an anisotropic site model, in which the interaction between a pair of sites on two molecules dep>ends not only upon the separation between the sites (as in an isotropic model such as the Lennard-Jones model) but also upon the orientation of the site-site vector with resp>ect to the bond vectors of the two molecules. The model includes an electrostatic component which contciins dipwle-dipole, dipole-quadrupole and quadrupole-quadrupole terms, and the van der Waals contribution is modelled using a Buckingham-like function. 

The vector 1 is the bond vector for bond i and T is the 3x3 matrix that transform coordinates in the reference frame for bond (i -F1) to those in the frame of bond i. In thi case the square end-to-end distance cam be calculated from ... 

The relationship between two bond vectors can be represmted using a distance, two angles and a torsion indicated (top). To derive the data for the database all possible pairs ofexocyclic vectors are considered and ur geometric parameters calculated. 

Figure 1 The principal sources of structural data are the NOEs, which give information on the spatial proximity d of protons coupling constants, which give information on dihedral angles < i and residual dipolar couplings, which give information on the relative orientation 0 of a bond vector with respect to the molecule (to the magnetic anisotropy tensor or an alignment tensor). Protons are shown as spheres. The dashed line indicates a coordinate system rigidly attached to the molecule.

We represent the four-atom problem in terms of diatom-diatom Jacobi coordinates R, the vector between the AB and CD centers of mass, and rj and r2, the AB and CD bond vectors. In a body-fixed coordinate system [19,20] with the z-axis chosen to R, only six coordinate variables need be considered, which we choose to be / , ri, and ra, the magnitudes of the Jacobi vectors, and the angles 01, 02, and (j). Here 0, denotes the usual polar angle of r, relative to the z-axis, and 4> is the difference between the azimuthal angles for ri and r2 (i.e., a torsion angle). 

The average square of the projection of a bond vector on the x-axis is not zero. It may be calculated in a similar manner as follows ... 

Reif, B., Steinhagen, H., Junker, B., Reggelin, M., Griesinger, C. Determination of the orientation of a distant bond vector in a molecular reference frame by cross-correlated relaxation of nuclear spins. 

Table 4.3. Conformations at the two bonds of length l to bead i in terms of the vectors i-1 —> i (rows) and i —> i+1 (columns) for bond vectors of length L for one relationship between the 2nnd lattice and its underlying diamond lattice... 

Although the underlying physics and mathematics used to convert relaxation rates into molecular motions are rather complex (Lipari and Szabo, 1982), the most important parameter obtained from such analyses, the order parameter. S 2, has a simple interpretation. In approximate terms, it corresponds to the fraction of motion experienced by a bond vector that arises from slow rotation as a rigid body of roughly the size of the macromolecule. Thus, in the interior of folded proteins, S2 for Hn bonds is always close to 1.0. In very flexible loops, on the other hand, it may drop as low as 0.6 because subnanosecond motions partially randomize the bond vector before it rotates as a rigid body. 

The most important feature of the information in dipolar couplings is that it is independent of distance. The data can be envisioned as reflecting the relative orientation of pairs of bond vectors, with the intervening distance having no effect (Meiler et al., 2000). Thus dipolar couplings can potentially provide a method for characterizing the structure of denatured proteins, provided that the denatured state ensemble of conformations retains several levels of nonrandomness. 

Here the bond vector b is along the same direction as lifl. The symbol < > represents an average. The vector r is an average,... 

Figure 3.67 Methyl tilt in CH3NH2, showing individual N-C-H angle deviations (solid lines) from the mean value (111.3 0.1°) and the overall tilt angle (dotted line) of the methyl symmetry axis with respect to the C—N bond vector. The anti C—H bond (circles) corresponds to Fig. 3.59(a) at (p = 0°.

The projections of a single side chain on the plane perpendicular to the helix axis are shown in Fig. 20(a), and the directions of a CK-2H bond are shown in Fig. 20(b). The C 2H bond vectors are distributed nearly symmetrically around the average direction. The same result was obtained for a Cf 2H bond. Therefore, jumps among multiple sites which are axially symmetric about the average direction are a good approximation for the large amplitude motion. 

Fig. 20. (a) Allowed side chain conformations, (b) Distribution of CK-2H bond vectors, (c) Three-site jump model as an approximation of multi-site jumps. Reproduced with permission from the Society of Polymer Science, Japan. 

When one implements an MC stochastic dynamics algorithm in this model (consisting of random-hopping moves of the monomers by one lattice constant in a randomly chosen lattice direction), the chosen set of bond vectors induces the preservation of chain connectivity as a consequence of excluded volume alone, which thus allows for efficient simulations. This class of moves... 

Figure 3 Sketch of the bond-fluctuation lattice model. The monomer units are represented by unit cubes on the simple cubic lattice connected by bonds of varying length. One example of each bond vector class is shown in the sketch.

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