Molecular motions correlated

The effect of correlated molecular motions can be observed even in the isotropic phase of nematic liquid crystals near the phase transition temperature. This phenomenon was first studied by Wong and Shen and Frost and Lalanne. They found both a pretransitional increase of the Kerr coefficient and a critical slowing-down in the relaxation process. 

Many simulations attempt to determine what motion of the polymer is possible. This can be done by modeling displacements of sections of the chain, Monte Carlo simulations, or reptation (a snakelike motion of the polymer chain as it threads past other chains). These motion studies ultimately attempt to determine a correlation between the molecular motion possible and the macroscopic flexibility, hardness, and so on. 

Molecular modeling is an indispensable tool in the determination of macromolecular structures from NMR data and in the interpretation of the data. Thus, state-of-the-art molecular dynamics simulations can reproduce relaxation data well [9,96] and supply a model of the motion in atomic detail. Qualitative aspects of correlated backbone motions can be understood from NMR structure ensembles [63]. Additional data, in particular residual dipolar couplings, improve the precision and accuracy of NMR structures qualitatively [12]. 

In Eq. (4-62) Wq is the Larmor precessional frequency, and Tc is the correlation time, a measure of the rate of molecular motion. The reciprocal of the correlation time is a frequency, and 1/Tc may receive additive contributions from several sources, in particular I/t, where t, is the rotational correlation time, t, is, approximately, the time taken for the molecule to rotate through one radian. Only a rigid molecule is characterized by a single correlation time, and the value of Tc for different atoms or groups in a complex molecule may provide interesting chemical information. 

Turning from chemical exchange to nuclear relaxation time measurements, the field of NMR offers many good examples of chemical information from T, measurements. Recall from Fig. 4-7 that Ti is reciprocally related to Tc, the correlation time, for high-frequency relaxation modes. For small- to medium-size molecules in the liquid phase, T, lies to the left side of the minimum in Fig. 4-7. A larger value of T, is, therefore, associated with a smaller Tc, hence, with a more rapid rate of molecular motion. It is possible to measure Ti for individual carbon atoms in a molecule, and such results provide detailed information on the local motion of atoms or groups of atoms. Levy and Nelson " have reviewed these observations. A few examples are shown here. T, values (in seconds) are noted for individual carbon atoms. 

In presence of molecular motion the NMR line shape will change. A particularly simple situation arises, if the motion is rapid on timescale defined by the inverse width of the spectrum in absence of motion 6 1. In this fast exchange limit, which in 2H NMR is reached for correlation times tc < 1CT7 s, the motion leads to a partially averaged quadrupole coupling and valuable information about the type of motion can directly be obtained from analysis of the resulting line shapes. The NMR frequency is then given by... 

Up to now it has been tacitly assumed that each molecular motion can be described by a single correlation time. On the other hand, it is well-known, e.g., from dielectric and mechanical relaxation studies as well as from photon correlation spectroscopy and NMR relaxation times that in polymers one often deals with a distribution of correlation times60 65), in particular in glassy systems. Although the phenomenon as such is well established, little is known about the nature of this distribution. In particular, most techniques employed in this area do not allow a distinction of a heterogeneous distribution, where spatially separed groups move with different time constants and a homogeneous distribution, where each monomer unit shows essentially the same non-exponential relaxation. Even worse, relaxation... 

This connection between the correlation time of perturbation and that of response is a very general result independent of a model of molecular motion. It is valid not only when a molecule is perturbed by a sequence of instantaneous collisions (as in a gas), but also when it is subjected to perturbations that are continuous in time (caused by the nearest... 

How can one hope to extract the contributions of the different normal modes from the relaxation behavior of the dynamic structure factor The capability of neutron scattering to directly observe molecular motions on their natural time and length scale enables the determination of the mode contributions to the relaxation of S(Q, t). Different relaxation modes influence the scattering function in different Q-ranges. Since the dynamic structure factor is not simply broken down into a sum or product of more contributions, the Q-dependence is not easy to represent. In order to make the effects more transparent, we consider the maximum possible contribution of a given mode p to the relaxation of the dynamic structure factor. This maximum contribution is reached when the correlator in Eq. (32) has fallen to zero. For simplicity, we retain all the other relaxation modes = 1 for s p. 

The polymer at the gel point is in a critical state [3], and the name critical gel [4] is appropriate for distinguishing polymers at the gel point from the various materials which commonly are called gels. The critical gel has no intrinsic size scale except for the size of its oligomeric building block, and molecular motions are correlated over large distances. The combination of liquid and solid... 

For folded proteins, relaxation data are commonly interpreted within the framework of the model-free formalism, in which the dynamics are described by an overall rotational correlation time rm, an internal correlation time xe, and an order parameter. S 2 describing the amplitude of the internal motions (Lipari and Szabo, 1982a,b). Model-free analysis is popular because it describes molecular motions in terms of a set of intuitive physical parameters. However, the underlying assumptions of model-free analysis—that the molecule tumbles with a single isotropic correlation time and that internal motions are very much faster than overall tumbling—are of questionable validity for unfolded or partly folded proteins. Nevertheless, qualitative insights into the dynamics of unfolded states can be obtained by model-free analysis (Alexandrescu and Shortle, 1994 Buck etal., 1996 Farrow etal., 1995a). An extension of the model-free analysis to incorporate a spectral density function that assumes a distribution of correlation times on the nanosecond time scale has recently been reported (Buevich et al., 2001 Buevich and Baum, 1999) and better fits the experimental 15N relaxation data for an unfolded protein than does the conventional model-free approach. 

In the case of extreme narrowing in which fast isotropic molecular motions dominate as in the solution state, the spectral density is written by a single correlation time,... 

The relevant contribute of relaxation measurements on the use of NMR spectroscopy in studying interactions can be argued by considering the relationship between relaxation rates and spectral density function being the latter related to the correlation time, which accounts for the molecular motion. Therefore, spin-lattice and spin-spin can be used to probe interactions between, in principle, every species bearing an active NMR nucleus. 

Solid state 2H NMR parameters are almost exclusively governed by the quadrupole interaction with the electric field gradient (EFG) tensor at the deuteron site.1 8 The EFG is entirely intramolecular in nature. Thus molecular order and mobility are monitored through the orientation of individual C-2H bond directions. Therefore, 2H NMR is a powerful technique for studying local molecular motions. It enables us to discriminate different types of motions and their correlation times over a wide frequency range. Dynamics of numerous polymers has been examined by solid state 2H NMR.1 3,7,9 Dynamic information on polypeptides by NMR is however limited,10 26 although the main-chain secondary structures of polypeptides in the solid have been extensively evaluated by 13C and 15N CP/MAS NMR.27,28... 

Given the character of the water-water interaction, particularly its strength, directionality and saturability, it is tempting to formulate a lattice model, or a cell model, of the liquid. In such models, local structure is the most important of the factors determining equilibrium properties. This structure appears when the molecular motion is defined relative to the vertices of a virtual lattice that spans the volume occupied by the liquid. In general, the translational motion of a molecule is either suppressed completely (static lattice model), or confined to the interior of a small region defined by repulsive interactions with surrounding molecules (cell model). Clearly, the nature of these models is such that they describe best those properties which are structure determined, and describe poorly those properties which, in some sense, depend on the breakdown of positional and orientational correlations between molecules. 

The limiting value of A0 is never achieved in practice, and partial depolarization can result from molecular motion. For a chromophore which moves with the motion of a rigid spherical macromolecule to which it is attached, the observed anisotropy will decay exponentially as a function of , the rotational correlation time, according to... 

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