Proteins model-free approach

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 absence of a correlation between the local dynamics and the overall rotational diffusion of the protein, as assumed in the model-free approach, the total correlation function that determines the 15N spin-relaxation properties (Eqs. (1-5)) can be deconvolved (Tfast, Tslow < Tc) ... 

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]). 

One of the most widely used tools to assess protein dynamics are different heteronuclear relaxation parameters. These are in intimate connection with internal dynamics on time scales ranging from picoseconds to milliseconds and there are many approaches to extract dynamical information from a wide range of relaxation data (for a thorough review see Ref. 1). Most commonly 15N relaxation is studied, but 13C and 2H relaxation are the prominent tools to characterize side-chain dynamics.70 Earliest applications utilized 15N Ti, T2 relaxation as well as heteronuclear H- N) NOE experiments to characterize N-H bond motions in the protein backbone.71 The vast majority of studies applied the so-called model-free approach to translate relaxation parameters into overall and internal mobility. Its name contrasts earlier methods where explicit motional models of the N-H vector were used, for example diffusion-in-a-cone or two- or three-site jump, etc. Unfortunately, we cannot obtain information about the actual type of motion of the bond. As reconciliation, the model-free approach yields motional parameters that can be interpreted in each of these motional models. There is a well-established protocol to determine the exact combination of parameters to invoke for each bond, starting from the simplest set to the most complex one until the one yielding satisfactory description is reached. The scheme, a manifestation of the principle of Occam s razor is shown in Table l.72... 

Clore GM, et al. Deviations from the simple two-parameter model-free approach to the interpretation of nitrogen-15 nuclear magnetic relaxation of proteins. J. Am. Chem. Soc. 1990 112 4989-4991. 

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-... 

Meiler J et al (2001) Model-free approach to the dynamic interpretation of residual dipolar couplings in globular proteins. J Am Chem Soc 123(25) 6098-6107... 

Idiyatullin D, Daragan VA, Mayo KH (2003) (NH)-N-15 backbone dynamics of protein GB1 comparison of order parameters and correlation times derived using various model-free approaches. J Phys Chem B 107 2602-2609... 

The most popular model-free approach [8,9] for analyzing NMR relaxation data on proteins is based on the assumption that slow, overall molecular tumbling (tm) is isotropic and independent, fast internal bond rotations (Te) with a generalized order parameter, S. Unique information about the internal libration motions of or —H bond vectors (i.e. S and Tg)... 

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. 

The model-free approach is not applicable for non-globular, highly asymmetric, and partially folded or fuUy unfolded proteins as they do not meet the criteria of separation of internal and overall motions. The reconstruction of spectral density components, J(cOj), from Ri, R2, and N- H NOE measurements is possible but more experiments are required (e.g., relaxation measurements at several Bo values). Nevertheless, lUPs can also be analyzed in terms of raw data as done recently, e.g., for the unstructured protease inhibitor calpastatin (Kiss et al. 2008). 

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. 

Several authors reported investigations of protein dynamics on the pico- and nano-second time scales. A standard approach to the protein backbone dynamics in this range is to measure N Ti, and NOE and to interpret the data using the Lipari-Szabo model-free approach, while or relaxation measurements provide information on the side-chain motions. The Lipari-Szabo order parameters can also be derived from MD simulations. Some authors compared the experimentally derived with the MD-derived counterparts. Smith and co-workers reported a study of this kind for the backbone of lysozyme from a bacteriophage and... 

Combined use of Eqs. (43)—(45) allows free drug concentrations to be predicted for each subcompartment. This approach to modeling free drug concentrations would make use of protein binding parameters (i.e., Bt, Kt) obtained from in vitro experiments. 

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... 

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