Rigid body motion

Finally, there is the thermal motion problan in X-ray crystallography. Molecules vibrate internally, and molecules in a aystal lattice also vibrate with what is referred to as the rigid-body motion. That is simply the whole molecule moving back and forth and undergoing rotational types of oscillation in the crystal lattice. (These are motions that correspond to translations and rotations in the isolated molecule that are converted into vibrations by the constraints imposed by the crystal lattice.) The molecule as a whole is quite heavy, so these vibrational frequencies tend to be quite low, and the vibrational amplitudes are large. The most serious part of this problem comes not from the translational motions but rather from the rotational oscillations of the molecule in the crystal lattice. The effect of these is to shorten the apparent bond lengths and compact the apparent size of the molecule. 

we find that if the X-ray measurements are made at liquid nitrogen temperatures, the total errors are reduced by about half, say to about 0.003-0.005 A on average. Unfortunately, although X-ray crystallography is largely done at liquid nitrogen temperatures currently, most of the structural information in the Cambridge Database was determined in earlier times and at room temperature. 

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Figure 10 Mean values of bond lengths and bond angles of pyrazole structures (a) Ehrlich (X-ray) (b) Berthou et at. (X-ray) (c) Rasmussen et at. (neutron, corrected for rigid body motion) (d) Rasmussen et at. (X-ray, 295 K) ...

The basic concepts of shock and particle velocities are well illustrated by an example first introduced by Duvall and Band (1968). Here we assume that a string of beads of diameter d, mass m, and spaced a fixed distance / apart on a smooth (frictionless) wire is impacted by a rigid, massive piston at velocity v. Each bead is assumed to undergo perfectly elastic, rigid-body motion upon impact with its neighbor. 

In another promising method, based on the effective Hamiltonian theory used in quantum chemistry [19], the protein is divided into blocks that comprise one or more residues. The Hessian is then projected into the subspace defined by the rigid-body motions of these blocks. The resulting low frequency modes are then perturbed by the higher... 

A dynamic transition in the internal motions of proteins is seen with increasing temperamre [22]. The basic elements of this transition are reproduced by MD simulation [23]. As the temperature is increased, a transition from harmonic to anharmonic motion is seen, evidenced by a rapid increase in the atomic mean-square displacements. Comparison of simulation with quasielastic neutron scattering experiment has led to an interpretation of the dynamics involved in terms of rigid-body motions of the side chain atoms, in a way analogous to that shown above for the X-ray diffuse scattering [24]. 

Fig. 2.8.6 (a) (i) I implementation of vertical cylindrical Couette cell using concentric glass tubes (ii) velocity image taken across a horizontal slice and (iii) velocity profile taken across the cell. Note that the marker fluid in the inner cylinder exhibits rigid body motion... 

Farrens DL, Altenbach C, Yang K, Hubbell WL, Khorana HG. Requirement of rigid-body motion of transmembrane helices for light activation of rhodopsin. Science 1996 274(5288) 768-770. 

The resulting description of the average rigid-body motion is in terms of six independently distributed instantaneous motions—three screw librations about nonintersecting axes (with screw pitches given by S11/Lll, etc.) and three translations. The parameter set consists of three libration and three translation amplitudes six angles of orientation for the principal axes of L and T six coordinates of axis displacement and three screw pitches, one of which has to be chosen arbitrarily again, for a total of 20 variables. 

Shomaker V, Tmeblood KN (1968) On the rigid-body motion of molecules in crystals. Acta Crystallogr B 24 63-76... 

Rigid-Body Motions and Least-Squares Fitting... 

The general numerical approach employs rigid body motions and least-squares fitting. Given two sets of points xt,i = 1,2,..., N and yui= 1,2,..., N (here xt and yt are vectors specifying atomic coordinates), find the best rigid motion of the points y h) such that the sum of the squares of the deviations E xt — Yt 2 is a minimum. 

Relative importance of rigid-helix and side-chain motions were studied by Corbin et al. [14]. The contribution of two types of rigid-body motions were approximated. In the first set of calculations helices of deoxymyoglobin were treated as rigid units while in the second set side-chains were considered to be... 

Domain A structurally independent region in a crystal or in a protein structure. It is described as a semiindependent unit that is important for folding stability, evolutionary recombination, or functional rigid-body motions. 

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