Relaxation amplitude analysis

Figure 7. Relaxation amplitude analysis variation of AH with [Fe(CN)64 ]. AH = normal enthalpy of the relaxation mode observed (14). (a) At 23.5°C, plot according to linearized form of Equation 2. Total azurin 2-7 X 10 5 M. (b) At 6.5°C, plot according to Equation 2. Total azurin 2-11 X 10 5 M. The line drawn is the best fit to the experimental points with Kt = 1.0 X 102 M 1 and AH/ = —5.5 kcal/mole.

The amplitudes of chemical relaxation processes are determined by the equilibrium concentrations (and strictly speaking, associated activity coefficients) and by thermodynamic variables appropriate for the particular perturbation method used. Thus, for example, an analysis of the amplitudes of relaxation processes associated with temperature jump measurements can lead to determination of the equilibrium constants and enthalpies associated with the mechanism under study. As might be anticipated from our previous discussion, the relaxation amplitudes are determined by normal mode thermodynamic variables which are linear combinations of the thermodynamic variables associated with the individual steps in the mechanism. The formal analysis of relaxation amplitudes has been developed in considerable detail [2, 5,7],... 

Figure 4.59. Final data analysis and computation of values for Table 4.1. [A] Calculation of the amplitude-dependent relaxation time x. [B] Extrapolation of x to zero-modulation amplitude to eliminate the temperature dependence of r resulting from modulation.

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. 

The fitting of space relaxation data using Eq. (1) to this mechanistic scheme (space relaxation data are always isothermal, because transient temperature effects are not relevant for the amplitude change of a concentration disturbance this is just an advantage of wavefront analysis of reaction kinetics), reported in (3, 5, 12), supposing a Langmuir type chemisorption for (CO) and (I O) has confirmed that (see Figure 10, 11) ... 

The first area involves low amplitude (higher frequency) sound and is concerned with the physical effect of the medium on the wave and is commonly referred to as low power or high frequency ultrasound . Typically, low amplitude waves are used for analytical purposes to measure the velocity and absorption coefficient of the wave in a medium in the 2 to 10 MHz range. Information from such measurements can used in medical imaging, chemical analysis and the study of relaxation phenomena and this will be dealt with later. 

Fig. 1. Top Scheme of an inversion recovery experiment 5rielding the longitudinal relaxation time (inversion is achieved by mean of the (re) radiofrequency (rf) pulse, schematized by a filled vertical rectangle). Free induction decays (fid represented by a damped sine function) resulting from the (x/2) read pulse are subjected to a Fourier transform and lead to a series of spectra corresponding to the different t values (evolution period). Spectra are generally displayed with a shift between two consecutive values of t. The analysis of the amplitude evaluation of each peak from — Mq to Mq provides an accurate evaluation of T. Bottom the example concerns carbon-13 Tl of irans-crotonaldehyde with the following values (from left to right) 20.5 s, 19.8 s, 23.3 s, and 19.3 s.

The Debye Waller analysis of the S—B bonds gives A05 g2(HO) = 2.9 x 10 A. This value is lower than the pure Co value (3.6 x 10 A ). Due to the low density of the (110) face, one mi t have expected a large mean-square relative displacement. The measured small value reveals a stiffening of the force constant of the Co—Cu bond. This is consistent with the large eontraction of the Co—Cu interlayer distance (sell % see above). The stiffening in strongly relaxed surfaces has been observed before and overcompensate the effect of the reduced surface coordination in the perpendicular direction. Reversed surfaee anisotropy of the mean square relative atomic displacements has also been found on an other low-density surfaee C2 x 2 Cl/Cu(l 10) i.e. one half density of Cl vs. Cu(l 10) in plane density where the Cl atoms moves with amplitudes parallel to the surface eomparable with those of the Cu subtrate, but with a much reduced amplitude in the perpendicular direction... 

Kett et al. [1.162] studied Tg in freeze-dried formulations containing sucrose as a function of relative humidity and temperature during storage by TMDSC and ther-mogravimetric analysis. Craig et al. [1.163] found it helpful to asses the relaxation behavior of freeze-dried amorphous lactose by MTDSC. Relaxation times were calculated from measurements of Tg, and the magnitude of the relaxation endotherm. Scannnig was performed at 2°C/min with a modulation amplitude of 0.3 °C and a period of 60 s. 

Analysis of XH T2 relaxation as a function of temperature yields information on molecular motions. Side-groups and local chain motions (secondary relaxations) cause a change in T2 below the Tg. The value of T2 increases with the amplitude and the frequency of molecular motions. The glass transition that occurs in the time scale in the NMR T2 relaxation... 

The derived barriers for several thyroid hormones are in the range 36-38 kJ mol-1, which corresponds to large-amplitude ring flips on the milli- to microsecond time scale. From a combination of the relaxation data and the dynamic lineshape analysis data, it was possible to propose a unified model that accounts for both the fast and slow internal motions, as summarized in Fig. 13. 

As mentioned earlier, regions of disorder in the spatial ensemble of calculated NMR structures can, in principle, be due to internal motions but can also reflect a relative lack of NOEs in such regions. A recent analysis83 suggests that ill-defined regions in structural ensembles often do reflect slow, large-amplitude motions, and so it is not always necessary to resort to relaxation time measurements if only a general idea of molecular motions is required for a particular protein. Even if relaxation measurements are done, it is often not necessary to undertake and extensive analysis to derive correlation... 

The dielectric spectroscopy study of conductive samples is very complicated because of the need to take into account the effect of dc-conductivity. The dc-conductivity c>o contributes, in the frequency domain, to the imaginary part of the complex dielectric permittivity in the form of additional function a0/(so ). The presence of dc-conductivity makes it difficult to analyze relaxation processes especially when the contribution of the conductivity is much greater than the amplitude of the process. The correct calculation of the dc-conductivity is important in terms of the subsequent analysis of the dielectric data. Its evaluation... 

The dielectric relaxation properties in nano-PS with different thicknesses have been investigated recently over a wide range of frequency and temperature [160,161]. The dielectric properties of the PS samples were measured in the 20-Hz to 1-MHz frequency range and in the 173 to 493 K temperature interval. For all the dielectric measurements, the amplitude of a sinusoidal ac-voltage source was maintained 1 V so that the average electric field across our sample was of the order of hundreds of volts per centimeter depending on the sample thickness. It was verified that the response was linear with respect to the ac-voltage amplitude such that a linear response analysis could be utilized for our sample. 

---