Formalism, model-free

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. 

Recent progress in protein dynamics studies by NMR was greatly facilitated by the invention of the model-free formalism [28, 32]. In this approach, the local dynamics of a protein are characterized by an order parameter, S, measuring the amplitude of local motion on a scale from 0 to 1, and the correlation time of the motion, T oc. The model-free expression for the correlation function of local motion reads... 

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 ... 

The backbone dynamics of 4-oxalocrotonate tautomerase, a 41-kDa homo-hexamer with 62 residues per subunit, and its complex with a substrate analogue have been analyzed by the model-free formalism.60 Binding of the analogue freezes the motion of some of the backbone NH vectors in the active site, leading to a loss of entropy (Chapter 2). 

The form of the free energy functional G appearing in the Polarizable Continuum Model is discussed in refs [35-37], Recently, Mennucci and Cammi have extended their integral equation formalism model for medium effects on shielding to the NMR shielding tensor for solutions in liquid crystals [38,39],... 

TABLE 1 Parameter sets commonly used in the model-free formalism... 

Unlike for globular proteins, internal motions of IDPs can hardly be interpreted within the framework of the Lipari-Szabo model-free formalism. IDPs are dominated by segmental motions with low or negligible cooperativity between... 

Meiler et al applied the model-free approach to the dynamic interpretation of residual dipolar couplings in globular proteins. Ishima et aO compared the methyl rotation axis order parameters derived from the model-free analyses of the and longitudinal and transverse relaxation rates measured in the same protein sample. Best et al reported the results of molecular dynamics simulations compared with NMR relaxation experiments for maltose and isomaltose. Using the model-free formalism they could estimate reliable order parameters. Baber et used an extended model-... 

Most of grammatical inference research has been focused on learning regular and context-free languages. Although these are the basic classes of the Chomsky hierarchy, it has been proved that even to learn these classes is already too hard under certain learning paradigms. Next, we review the main formal models proposed in this field and some of the main learnability results obtained. 

Keywords Finite automata Mildly context-sensitive languages Natural languages Formal linguistics Free-order languages Computational linguistics Formal models... 

This model can also be extended to further levels of librational motion, each with its own order parameter and correlation time, and has been underpinned more formally in the three-r case of two librational levels plus rotation [7]. It is clearly of a general nature. One may indeed choose to abandon relationship (4.23) and to re-interpret in (4.24) as a more general order parameter. One thereby achieves a model-free theory, which has advantages over alternative, more specific models for polymer motion (see chapter 6), when the motional details are not clear. 

In the extended model-free formalism, two internal motions on two separable time scales are taken into account [10] ... 

The hydration dependence studies of the internal protein dynamics of hen egg white lysozyme by and H NMR relaxation have been presented. The relaxation times were quantitatively analysed by the well-established correlation function formalism and model-free approach. The obtained data was described by a model based on three types of motion having correlation times around 10 , 10 and 10 s. The slowest process was shown to originate from correlated conformational transitions between different energy minima. The intermediate process was attributed to librations within one energy minimum, and the fastest one was identified as a fast rotation of methyl protons around the symmetry axis of methyl groups. A comparison of the dynamic behaviour of lysozyme and polylysine obtained from a previous study revealed that in the dry state both biopolymers are rigid on both fast and slow time scales. Upon hydration, lysozyme and polylysine showed a considerable enhancement of the internal mobility. The side chain fragments of polylysine were more mobile than those of lysozyme, whereas the backbone of lysozyme was found to be more mobile than that of polylysine. 

Specific models for internal motions can be used to interpret heteronuclear relaxation, such as restricted diffusion and site-jump models. However, model-free formal methods are preferable, at least for the initial analysis, since available experimental data generally are insufficient to completely characterize complex internal motions or to uniquely determine a specific motional model. The model-free approach of Lipari and Szabo for the analysis of relaxation data has been used for proteins and even for peptides. It attempts to reproduce relaxation rates by a weighted product of spectral density functions with different correlation times The weighting factors are identified as order parameters for the molecular rotational correlation time and optional further local correlation times r. The term (1-S ) would then be proportional to the amplitude of the corresponding internal motion. However, the Lipari-Szabo approach is based on the assumption that molecular and local correlation times are not coupled, i.e. they should be distinct enough (e.g. differing by at least a factor of 10 in time) to allow for this separation. However, in small molecules the rates of these different processes are of the same order of magnitude, and the requirements of the Lipari-Szabo approach may not be fulfilled. Molecular dynamics simulation provide a complementary approach for the interpretation of relaxation measurements. 

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. 

While one is free to think of CA as being nothing more than formal idealizations of partial differential equations, their real power lies in the fact that they represent a large class of exactly computable models since everything is fundamentally discrete, one need never worry about truncations or the slow aciminidatiou of round-off error. Therefore, any dynamical properties observed to be true for such models take on the full strength of theorems [toff77a]. 

The common disadvantage of both the free volume and configuration entropy models is their quasi-thermodynamic approach. The ion transport is better described on a microscopic level in terms of ion size, charge, and interactions with other ions and the host matrix. This makes a basis of the percolation theory, which describes formally the ion conductor as a random mixture of conductive islands (concentration c) interconnected by an essentially non-conductive matrix. (The mentioned formalism is applicable not only for ion conductors, but also for any insulator/conductor mixtures.)... 

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