Model-Free Approaches

Lipari G and Szabo A 1982 Model-free approach to the interpretation of nuclear magnetic resonance relaxation in macromolecules 1. Theory and range of validity J. Am. Chem. Soc. 104 4546-59... 

Another way to describe deviations from the simple BPP spectral density is the so-called model-free approach of Lipari and Szabo [10]. This takes account of the reduction of the spectral density usually observed in NMR relaxation experiments. Although the model-free approach was first applied mainly to the interpretation of relaxation data of macromolecules, it is now also used for fast internal dynamics of small and middle-sized molecules. For very fast internal motions the spectral density is given by ... 

ANNs need supervised learning schemes and can so be applied for both classification and calibration. Because ANNs are nonlinear and model-free approaches, they are of special interest in calibration. 

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. 

Another powerful tool for examining this issue is the use of time-resolved fluorescence spectra, especially when combined with the technique of Time-Resolved Area Normalized Emission Spectra (TRANES) developed by Periasamy and coworkers [78-80]. In this method, separate decay curves are collected over a wide range of emission wavelengths and reconstructed into time-resolved spectra, which are then normalized to constant area. In this model-free approach, it is possible to deduce the nature of heterogeneity of the fluorescent species from the... 

The same result has been obtained by Lipari and Szabo in their model-free approach.158 The first term in the above equation accounts for the effect of local order on the isotropic rotation through the factor 1 — S2, and the second term is due to the slower fluctuations in the local ordered clusters. This model-free approach has become popular among workers in the area of lyotropics and biomembranes. 

In the extended model-free approach [29], the local dynamics are deconvolved into a fast and a slow motion rfagt < ttsiow, S2 = S2lowS ast) ... 

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) ... 

The model-free approach is essentially based on a parametrization of the spectral densities using a small number of fitting parameters, which then allows Eqs. (1-3) to be solved. The analysis of experimental data using this method will be discussed in a later section. 

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 ... 

If one likes to include molecular motional effects of the ligand when bound to the macromolecule, one can introduce an order parameter using the model-free approach, as has been applied for the interpretation of CCR rates by Carlomagno et al. [46]. 

The model-free approach applies every time there is evidence of local motions or when the value obtained for is smaller than the value expected from the Stokes-Einstein equation. 

In a recent work by Tolman,50 a refinement approach similar to the model free approach was described which extends the realm of applicability to cases in which RDC datasets acquired in five different alignment media are not available. The theoretical development proceeds from recognition that, in general, the RDC data in its entirety can be represented by a single... 

A model-free approach to analysis of DEER data in the absence of orientation selection was proposed based on shell factorization.22 The decay curves are simulated as the products of orientationally averaged thin shells of interacting electrons. The dipolar time-evolution data can be separated into a linear contribution and a non-linear contribution from background. The linear contribution can be converted to a radial distribution function for spin-spin interaction. 

In the following, three different approaches to estimation of the effect of the heat released by the reaction on the system dynamics are presented the first two are based on the results in [15, 51], while the third is one of the most interesting model-free approaches in the literature. In detail ... 

The model-free approach proposed in [27], in which both the heat transfer coefficient and the thermal power are estimated as unknown parameters. As for the previous one, this approach does not need either the knowledge of the reaction kinetics or the estimation of the concentrations. 

When an accurate model of the reaction kinetics is not available (e.g., due to the lack of reliable data for identification), the previously developed approach may be ineffective and model-free strategies for the estimation of the effect of the heat released by the reaction, aq, must be adopted. In detail, the approach in [27] can be considered, where aq is estimated, together with the heat transfer coefficient, via a suitably designed nonlinear observer [24], Other model-free approaches can be adopted, e.g., those based on the adoption of universal interpolators (neural networks, polynomials) for the direct online estimation of the heat [16] and purely neural approaches [11], Approaches based on the combination of neural and model-based paradigms [2] or on tendency models [25] can be considered as well. 

In the following, two model-free approaches based on adaptive observer are presented the first one is based on the results in [51], and the second one is the well-established observer proposed by [24] and applied to batch reactors in [27],... 

It can be argued that the differences between the compared schemes are mainly due to the different estimation accuracy of the quantity aq (Fig. 5.6). It can be seen that, after the initial transient phase in which the model-free observers present an inverse response, both the adaptive (model-based and model-free) approaches achieve very good estimates. As for the parameter estimate, since both the adaptive observers (0O) and the controller (0C) estimates converge to the true value of 0 (see Fig. 5.7), it is possible to argue that the persistency of excitation condition is fulfilled. 

It can be concluded that the exponential stability property confers to the adaptive model-based scheme a satisfactory degree of robustness. Therefore, even in the presence of large model uncertainties, its performance is comparable with or better than that of model-free approaches. 

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... 

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