Spectral density molecular motion

Fig. 7.1.3 [Blii2] NMR-timescale of molecular motion and filter transfer functions of pulse sequences which can be utilized for selecting magnetization according to the timescale of molecular motion. The concept of transfer functions provides an approximative description of the filters. A more detailed description needs to take into account magnetic-field dependences and spectral densities of motion. The transfer functions shown for the saturation recovery and the stimulated-echo filter apply in the fast motion regime.

It should be noted that the JmL ( ) re quantities that are obtained from experiment without reference to any molecular dynamics model. When measured as a function of temperature and frequency, these spectral densities of motion provide the best test of motional models for liquid crystals. 

For an isolated spin-1 system, it is convenient to define sum and difference magnetizations [Eqs. (2.84)-(2.85)] in the J-B experiment. The decay of the difference (quadrupolar order) proceeds exponentially at a rate T q, while the sum (Zeeman order) recovers exponentially towards equilibrium at a different rate. The J-B experiment allows simulataneous determination of these rates from which Ji uJo) and J2 2ujo) can be separated. Table 5.1 briefly summarizes thermotropic liquid crystals in which spectral density measurements were reported. Figure 5.4 illustrates the temperature and frequency dependences of spectral densities of motion (in s by including the interaction strength Kq factor) for 5CB-di5. The result is fairly typical for rod-like thermotropic liquid crystals. The spectral densities increase with decreasing temperature in the nematic phase of 5CB. The frequency dependence of Ji uJo) and J2(2a o) indicate that molecular reorientation is likely not in the fast motion regime. However, the observed temperature dependence of the relaxation rates is opposite to what is expected for simple liquids. This must be due to the anisotropic properties (e.g., viscosity) of liquid crystals. 

The small step rotational diffusion model has been employed to extract rotational diffusion constants Dy and D from the measured deuterium spectral densities in liquid crystals [7.25, 7.27, 7.46, 7.49 - 7.53]. Both the single exponential correlation functions [Eq. (7.54)] and the multiexponential correlation functions [Eq. (7.60)] have been used to interpret spectral densities of motion. However, most deuterons in liquid crystal molecules are located in positions where they are rather insensitive to motion about the short molecular axis. Thus, there is a large uncertainty in determining D or Tq (t o) because of Dy > D and the rather small geometric factor [doo( )] for most deuterons in liquid crystal molecules. For 5CB, it is necessary to fix [7.52] the value of D using the known activation... 

The most difficult problem in any relaxation theory is the calculation of correlation functions or spectral densities of motion. It is often possible to determine the mean square spin interaction H t)) where H t) is a component of the spin Hamiltonian which fluctuates randomly in time owing to molecular motions. The time dependence of the correlation function - r)) can often be approximated... 

We call the correlation time it is equal to 1/6 Dj, where Dj is the rotational diffusion coefficient. The correlation time increases with increasing molecular size and with increasing solvent viscosity, equation Bl.13.11 and equation B 1.13.12 describe the rotational Brownian motion of a rigid sphere in a continuous and isotropic medium. With the Lorentzian spectral densities of equation B 1.13.12. it is simple to calculate the relevant transition probabilities. In this way, we can use e.g. equation B 1.13.5 to obtain for a carbon-13... 

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. 

In the first part to follow, the equations of motion of a soft solid are written in the harmonic approximation. The matrices that describe the potential, and hence the structure, of the material are then considered in a general way, and their properties under a normal mode transformation are discussed. The same treatment is given to the dissipation terms. The long wavelength end of the spectral density is of interest, and here it seems that detailed matrix calculations can be replaced by simple scaling arguments. This shows how the inertial term, usually absent in molecular problems, is magnified to become important in the continuum limit. 

While the assumption of an isotropic rotational motion is reasonable for low molecular weight chelates, macromolecules have anisotropic rotation involving internal motions. In the Lipari-Szabo approach, two kinds of motion are assumed to affect relaxation a rapid, local motion, which lies in the extreme narrowing limit and a slower, global motion (86,87). Provided they are statistically independent and the global motion is isotropic, the reduced spectral density function can be written as ... 

Now the expectation (mean) value of any physical observable (A(t)) = Yv Ap(t) can be calculated using Eq. (22) for the auto-correlation case (/ = /). For instance, A can be one of the relaxation observables for a spin system. Thus, the relaxation rate can be written as a linear combination of irreducible spectral densities and the coefficients of expansion are obtained by evaluating the double commutators for a specific spin-lattice interaction X in the auto-correlation case. In working out Gm x) [e.g., Eq. (21)], one can use successive transformations from the PAS to the (X, Y, Z) frame, and the closure property of the rotation group to rewrite D2mG(Qp ) so as to include the effects of local segmental, molecular, and/or collective motions for molecules in LC. The calculated irreducible spectral densities contain, therefore, all the frequency and orientational information pertaining to the studied molecular system. 

Equations (1-3) are widely used for protein dynamics analysis from relaxation measurements. The primary goals here are (A) to measure the spectral densities J(co) and, most important, (B) to translate them into an adequate picture of protein dynamics. The latter goal requires adequate theoretical models of motion that could be obtained from comparison with molecular dynamics simulations (see for example Ref. [23]). However, accurate analysis of experimental data is an essential prerequisite for such a comparison. 

This approach yields spectral densities. Although it does not require assumptions about the correlation function and therefore is not subjected to the limitations intrinsic to the model-free approach, obtaining information about protein dynamics by this method is no more straightforward, because it involves a similar problem of the physical (protein-relevant) interpretation of the information encoded in the form of SD, and is complicated by the lack of separation of overall and local motions. To characterize protein dynamics in terms of more palpable parameters, the spectral densities will then have to be analyzed in terms of model-free parameters or specific motional models derived e.g. from molecular dynamics simulations. The SD method can be extremely helpful in situations when no assumption about correlation function of the overall motion can be made (e.g. protein interaction and association, anisotropic overall motion, etc. see e.g. Ref. [39] or, for the determination of the 15N CSA tensor from relaxation data, Ref. [27]). 

Up to this point only overall motion of the molecule has been considered, but often there is internal motion, in addition to overall molecular tumbling, which needs to be considered to obtain a correct expression for the spectral density function. Here we apply the model-free approach to treat internal motion where the unique information is specified by a generalized order parameter S, which is a measure of the spatial restriction of internal motion, and the effective correlation time re, which is a measure of the rate of internal motion [7, 8], The model-free approach only holds if internal motion is an order of magnitude (<0.3 ns) faster than overall reorientation and can therefore be separated from overall molecular tumbling. The spectral density has the following simple expression in the model-free formalism ... 

As outlined above, the normalized spectral density is especially simple when a single correlation time is involved (for instance, when the molecular reorientation can be approximated by the motion of a sphere) ... 

By assuming an Arrhenius type temperature relation for both the diffusional jumps and r, we can use the asymptotic behavior of /(to) and T, as a function of temperature to determine the activation energy of motion (an example is given in the next section). We furthermore note that the interpretation of an NMR experiment in terms of diffusional motion requires the assumption of a defined microscopic model of atomic motion (migration) in order to obtain the correct relationships between the ensemble average of the molecular motion of the nuclear magnetic dipoles and both the spectral density and the spin-lattice relaxation time Tt. There are other relaxation times, such as the spin-spin relaxation time T2, which describes the... 

Both Ti and T2 relaxations of water protons are mainly due to fluctuating dipole-dipole interactions between intra- and inter-molecular protons [62]. The fluctuating magnetic noise from all the magnetic moments in the sample (these moments are collectively tamed the lattice) includes a specific range of frequency components which depends on the rate of molecular motion. The molecular motion is usually represented by the correlation time, xc, i.e., the average lifetime staying in a certain state. A reciprocal of the correlation time corresponds to the relative frequency (or rate) of the molecular motion. The distribution of the motional frequencies is known as the spectral density function. 

The persistence of the fluctuating local fields before being averaged out by molecular motion, and hence their effectiveness in causing relaxation, is described by a time-correlation function (TCF). Because the TCF embodies all the information about mechanisms and rates of motion, obtaining this function is the crucial point for a quantitative interpretation of relaxation data. As will be seen later, the spectral-density and time-correlation functions are Fourier-transform pairs, interrelating motional frequencies (spectral density, frequency domain) and motional rates (TCF, time domain). 

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