Rigid-body motions time scales

In the previous section, we have shown that switching the picture from the nearly integrable Hamiltonian to the Hamiltonian with internal structures may make it possible to solve several controversial issues listed in Section IV. In this section we shall examine the validity of an alternative scenario by reconsidering the analyses done in MD simulations of liquid water. As mentioned in Section III, since a water molecule is modeled by a rigid rotor, and has both translational and rotational degrees of freedom. So, the equation of motion involves the Euler equation for the rigid body, coupled with ordinary Hamiltonian equations describing the translational motions. The precise Hamiltonian is therefore different from that of the Hamiltonian in Eq. (1), but they are common in that the systems have internal structures, and the separation of the time scale between subsystems appears if system parameters are appropriately set. 

Spectral densities are calculated within the framework of the theoretical model for the dynamical evolution of the system. In the SRLS approach a two-body Smoluchowski equation describes the time evolution of the density probability of two relaxation processes (at different time scales) coupled by an interaction potential. In the application of this model to the description of protein dynamics, the two relaxing processes are interpreted as the slow global tumbling of the whole protein and the relatively fast local motion of the spin probe, the local motion of the N- H bond in our case. Both processes are described as rigid rotators the motion of which is coupled by a potential correlating their reorientation, and it is interpreted as providing the local ordering that the molecule imposes on the probe. 

Early work on NMR of polymers in dilute solution was reviewed by Heatley(29). It was already clear in that early review that relaxation times of dilute polymers were independent of polymer molecular weight, at least for molecular weights above a few to ten thousand, and were nearly independent of polymer concentration for concentrations up to 100-150 g/1 or so. A revealing exception to this rule was provided by polymers plausibly expected to rotate as nearly rigid bodies, for which Ti continued to depend on M up to much larger M. From these observations, it was plausibly inferred that local chain motions are primarily responsible for the observed relaxation times. Dependences of Ti on solvent temperature and viscosity were concluded to scale linearly with solvent viscosity, at least in most systems, a matter treated in more detail below. Heatley also considers correlations between Ti and chain structure. 

The decay constants for overall motion (or tumbling) can be obtained from hydrodynamic estimates of the diffusion tensor. While this is a complicated topic, especially for hydrated, semiflexible molecules, there are useful exact analytic formulas for spheres and ellipsoids, approximate analytic formulas for cylinders, and numerical methods for more complicated shapes. Using these approaches, the rigid body second-rank correlation times (i.e., T2) for medium-sized proteins can be estimated to be in the 5-10 ns range. Hence, internal motions are typically well separated in time-scale from tumbling for most proteins, and can be assumed uncoupled. For smaller molecules, tumbling can be faster than isomerization (e.g., butane ), or the two can be highly coupled (octane ). 

Because molecules studied by NMR are not rigid isolated bodies, there will always be some effects due to the surrounding molecules and also due to small and large amplitude motions within the molecule. It may be necessary to correct for these effects. Alternatively, they can provide subtle information about the environment of a molecule as well as the dynamics of the intramolecular motional averaging that takes place in the NMR time scale. 

---