Overall Tumbling and 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]  

Here I is a (dimensionless) angular momentum operator. A non-spherical molecule will tumble more rapidly about some directions than about others, causing the diffusion constant Drot to become a tensor  

Here Q represents the Euler angles that specify the orientation of the macromolecule. 

In principle, molecular dynamics simulations should have a lot to say about these questions, since they provide a detailed (albeit approximate) description of macromolecular structure and dynamics. One can learn some information about global motion by extrapolations from even short simulations [39], but the longer time scales now available, which can be many times the mean rotational tumbling time, are expected to yield more reliable information. However, many popular water models (such as TIP3P) predict self-diffusion constants (and, presumably, viscosities) that are far from experiment (see Table 8.1), so that one would not 

For isotropic diffusional motion, solutions to Eq. (2) are easily computed for any value of I, and the correlation functions are single exponentials [32]  

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