Model-free treatments

In the previous discussion, the electron-nucleus spin system was assumed to be rigidly held within a molecule isotropically rotating in solution. If the molecule cannot be treated as a rigid sphere, its motion is in general anisotropic, and three or five different reorientational correlation times have to be considered 79). Furthermore, it was calculated that free rotation of water protons about the metal ion-oxygen bond decreases the proton relaxation time in aqua ions of about 20% 79). A general treatment for considering the presence of internal motions faster than the reorientational correlation time of the whole molecule is the Lipari Szabo model free treatment 80). Relaxation is calculated as the sum of two terms 8J), of the type... 

Ubiquitin is a small (76 amino acids) extremely stable protein containing a broad collection of secondary structure elements including parallel and antiparallel beta strands assembled into a mixed beta sheet, alpha and 3io helices and a variety of turns (Vijay-Kumar et al., 1987 Di Stefano Wand, 1987). In previous work, we have examined the fast main chain dynamics of ubiquitin by use of 15n NMR relaxation methods (Schneider et al., 1992). These data were analyzed in terms of the so-called model free treatment of Lipari and Szabo (1982a,b). The amplitudes of motion of the backbone amide N-H vectors of the packed regions of the protein are generally highly restricted and show no apparent correlation with secondary stmcture context but do show a strong... 

Given knowledge of the covalent geometry and fundamental constants underlying relaxation, the spectral densities, J(co), remain to be defined. Here, we adopt the so-called model free treatment due to Lipari and Szabo (1982a,b) which seeks to encapsulate the unique motional character expressed in observed relaxation behavior by defining the spectral density in terms of two local parameters ... 

A large number of structural models describing the random motions of the C—H internuclear vector have already been proposed for the calculations of G (t) or ,( ). In this section, the equations for J (( )) are shown for some representative structural models as well as for different levels of model-free treatments. 

This equation is expressed as a linear combination of Lorentzian contributions from the respective random motions, if the fourth term is assumed to be negligibly small. This indicates that Equation (3.35) is a model-free equation for three types of independently superposed random motions, which is in good accord with the results of the model-free treatments described in the following section, even though a specific structural model, shown in Fig. 3.6, was used for the derivation of Equation (3.35). 

The first-order model-free treatment. The time fluctuation of the C—H internuclear vector should be described in terms of the superposition of several independent random motions, and the autocorrelation function G<,(t)... 

Here, p is the number of random motions and are the correlation times of the respective motions. Ay seem to be the weights of the respective motions with EAy = 1, but are simply assumed to be adjustable parameters without giving any explicit physical meaning in the first-order model-free treatment [9, 13]. King and coworkers [14, 15] also derived Equation (3.36) in a more general fashion. 

To further develop this model-free treatment, we generalize Equation... 

This equation corresponds to Equation (3.35) for the 3t model as well as Equation (3.36) for p = 3 for the first-order model-free treatment. In contrast, Clore et al. [18, 19] have empirically derived the total correlation function similar to Equation (3.40) but Equation (3.40) is expressed more explicitly in terms of the respective order parameters. 

Figures 3.9 and 3.10 show the temperature dependencies of Ti and NOE of the CH2 (rrr) of the same PMMA solution and the results (solid and broken curves) simulated by the second-order model-free treatment with p = 3 [17]. Here, the Arrhenius equation was assumed for the respective correlation times tj = tio exp(AEi/RT) and ta/ = ta,o exp(AEA,/RT). In this case the simulated results with p = 3 are also in good accord with the experimental results, indicating the validity of the model-free treatment. Similar analyses of the temperature dependencies of the Tj were successfully performed for the rubbery components of the solid polyesters with different methylene sequences [20, 21]. These results are also well analyzed by the second-order model-free treatment with p = 3. There are a large number of the publications of the temperature dependencies of Ti and NOE analyzed by different models of molecular motions for polymers in the dis-...

Fig. 3.9. Temperature dependencies of NTi of the CH2 (rrr) carbon for PMMA/CDCI3 solution. The curves are the results simulated by the second-order model-free treatment.

In this contribution we will deal with electron-electron correlation in solids and how to learn about these by means of inelastic X-ray scattering both in the regime of small and large momentum transfer. We will compare the predictions of simple models (free electron gas, jellium model) and more sophisticated ones (calculations using the self-energy influenced spectral weight function) to experimental results. In a last step, lattice effects will be included in the theoretical treatment. 

FRP leads to the formation of statistical copolymers, where the arrangement of monomers within the chains is dictated purely by kinetic factors. The most common treatment of free-radical copolymerization kinetics assumes that radical reactivity depends only on the identity of the terminal unit on the growing chain. The assumption provides a good representation of polymer composition and sequence distribution, but not necessarily polymerization rate, as discussed later. This terminal model is widely used to model free-radical copolymerization according to the set of mechanisms in Scheme 3.11. 

Szabo proposed an interesting model-free formula for the time-resolved anisotropy in a macroscopically isotropic system [112]. He expressed r(f) as the autocorrelation function of orientations of the emission dipole moment at time t and absorption dipole moment at time t = 0 in a form suitable for general treatment of various systems, and particularly those with possible internal rotation ... 

Tlierc are two major sources of error associated with the calculation of free energies fi computer simulations. Errors may arise from inaccuracies in the Hamiltonian, be it potential model chosen or its implementation (the treatment of long-range forces, e j lie second source of error arises from an insufficient sampling of phase space. 

It is possible to go beyond the SASA/PB approximation and develop better approximations to current implicit solvent representations with sophisticated statistical mechanical models based on distribution functions or integral equations (see Section V.A). An alternative intermediate approach consists in including a small number of explicit solvent molecules near the solute while the influence of the remain bulk solvent molecules is taken into account implicitly (see Section V.B). On the other hand, in some cases it is necessary to use a treatment that is markedly simpler than SASA/PB to carry out extensive conformational searches. In such situations, it possible to use empirical models that describe the entire solvation free energy on the basis of the SASA (see Section V.C). An even simpler class of approximations consists in using infonnation-based potentials constructed to mimic and reproduce the statistical trends observed in macromolecular structures (see Section V.D). Although the microscopic basis of these approximations is not yet formally linked to a statistical mechanical formulation of implicit solvent, full SASA models and empirical information-based potentials may be very effective for particular problems. 

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