Random walks reorientation

In terms of the average time required for random-walk reorientation through unit angle (one radian), which as we have noted is 1 /6D (Equation (2.25)), this may be expressed... 

The orientation of linear rotators in space is defined by a single vector directed along a molecular axis. The orientation of this vector and the angular momentum may be specified within the limits set by the uncertainty relation. In a rarefied gas angular momentum is well conserved at least during the free path. In a dense liquid it is a molecule s orientation that is kept fixed to a first approximation. Since collisions in dense gas and liquid change the direction and rate of rotation too often, the rotation turns into a process of small random walks of the molecular axis. Consequently, reorientation of molecules in a liquid may be considered as diffusion of the symmetry axis in angular space, as was first done by Debye [1],... 

Consider for a moment a rod-shaped particle of unit length. The orientation of the rod, u, can be specified by a unit vector u directed along its axis with spherical polar coordinates, D - id, random walk along the surface of the unit sphere. Debye [16] in 1929 developed a model for the reorientation process based on the assumption that collisions are so fiiequent that a particle can rotate throu only a very small angle before having another reorienting collision (i.e., small step diffusion). Debye began with the diffusion equation... 

Fig. 7.3.1. u(0) and u(/) are unit vectors representing the orientation angles of the symmetry axis of a cylindrically symmetric molecule at times 0 and t, respectively. The locus of all the possible vectors u(f) is the surface of a sphere of unit radius (a unit sphere). The reorientation of the molecule can be regarded as a trajectory on the surface of the unit sphere. A random walk trajectory gives rise to rotational diffusion. 

Overall rotational tumbling is regulated by frequent collisions with light water molecules. For a nearly rigid protein, this physical model should lead to diffusive rotational behavior, where the reorientation of a unit vector attached to the molecule undergoes a random walk on the surface a sphere. If c(n, t) is the probability density for finding the vector pointing direction n at time f, a spherical molecule should follow a simple diffusion equation [31,32] ... 

The standard model for diffusive motion in polymers is Brownian diffusion, which occurs as a series of infinitesimal reorientational steps. This model is most appropriate for intermediate-to-large sized spin probes and spin-labeled macromolecules, where the macromolecule is much larger than any solvent molecules. Because of this broad applicability, the Brownian diffusion model is the most widely used. This type of rotational diffusion is completely analogous to the one-dimensional random walk used to describe translational diffusion in standard physical chanistry texts, with the difference that the steps are described in terms of a small rotational step 59 that can occur in either the positive or negative direction. In three dimensions, rotations about each of three principal axes of the nitroxide must be taken into account. A diffusion constant may be defined for each of these rotations motions, in a way that is completely analogous to the definition of translational diffusion constant for the one-dimensional random walk. 

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