Multiple correlation time models

Experimentalists often rely on motional models, based on hydrodynamics, in order to interpret their liquid state spectra. MD simulations, can be considered as model-free in the sense that they do not assume the molecular motion to be in any specific regime. MD can be used to evaluate the motional models and even replace them. MD simulations can be used to calculate both the correlation times and the whole correlation functions. This is useful in those cases when correlation times cannot be deduced from measurements of other isotopes in the same molecule or when there is no method available at all. Correlation functions give information about intermolecular interactions and reveal cases when several motional modes are contributing to relaxation mechanisms at slightly different time scales. This can be observed as multiple decay rates. Time correlation functions from MD simulations can be Fourier transformed to power spectra if needed to provide line shapes and frequencies. 

Korb el al. proposed a model for dynamics of water molecules at protein interfaces, characterized by the occurrence of variable-strength water binding sites. They used extreme-value statistics of rare events, which led to a Pareto distribution of the reorientational correlation times and a power law in the Larmor frequency for spin-lattice relaxation in D2O at low magnetic fields. The method was applied to the analysis of multiple-field relaxation measurements on D2O in cross-linked protein systems (see section 3.4). The reorientational dynamics of interfacial water molecules next to surfaces of varying hydrophobicity was investigated by Stirnemann and co-workers. Making use of MD simulations and analytical models, they were able to explain non-monotonous variation of water reorientational dynamics with surface hydrophobicity. In a similar study, Laage and Thompson modelled reorientation dynamics of water confined in hydrophilic and hydrophobic nanopores. 

The advances in time resolved techniques have fostered a reexamination of theories of the rotational motions of molecules in liquids. Models considered include the anisotropic motion of unsymmetrical fluorophores the internal motions of probes relative to the overall movement with respect to their surroundings, the restricted motion of molecules within membranes (e.g., wobbling within a cone), and the segmental motion of synthetic macromolecules [8]. Analyses of these models point to experimental situations in which the anisotropy can show both multi-exponential and none-exponential decay. Current experimental techniques are capable in principle of distinguishing between these different models. It should be emphasized, however, that to extract a single average rotational correlation time demands the same precision of data and analysis as fluorescence decay experiments which exhibit dual exponential decays. Multiple or non-exponential anisotropy experiments are thus near the limits of present capabilities, and generally demand favourable combinations of fluorescence and rotational diffusion times [48]. 

A table of correlations between the variables from the instrumental set and variables from the sensory set may reveal some strong one-to-one relations. However, with a battery of sensory attributes on the one hand and a set of instrumental variables on the other hand it is better to adopt a multivariate approach, i.e. to look at many variables at the same time taking their intercorrelations into account. An intermediate approach is to develop separate multiple regression models for each sensory attribute as a linear function of the physical/chemical predictor variables. 

Figure 8.21 shows model functions both for ideal and realistic cases. The dotted curve demonstrates the case of the ideal and infinitely extended ID lattice. Flere every time the ghost is displaced by an integer multiple of the lattice constant (x/L = 1, 2, 3,...), the correlation returns to the ideal value 1. For the ID lattice not only x0, but also the valley depths... 

---