The key to obtaining the model-free description of the dynamics of a macromolecule is the optimisation of the model-free and Brownian rotational diffusion parameters using the collected R 1 , R 2 and steady-state NOE relaxation data. The problem of optimising the chi-squared value is often assumed to be trivial, however, the long chain of dependencies required for its calculation complicates the model-free chi-squared space. Convolutions are induced by the Lorentzian form of the spectral density functions, the linear recombinations of certain spectral density values to obtain the relaxation rates, the calculation of the NOE using the ratio of two of these rates, and finally the quadratic form of the chi-squared equation itself. Two major topological features of the model-free space complicate optimisation. The first is a long, shallow valley which commences at infinite correlation times and gradually approaches the minimum. The most severe convolution occurs for motions on two timescales in which the minimum is often located at the end of a long, deep, curved tunnel or multidimensional valley through the space. A large number of optimisation algorithms will be investigated and their performance compared to determine which techniques are suitable for use in model-free analysis. Local optimisation algorithms will be shown to be sufficient for minimisation not only within the model-free space but also for the minimisation of the Brownian rotational diffusion tensor. In addition the performance of the programs Modelfree and Dasha are investigated. A number of model-free optimisation failures were identified: the inability to slide along the limits, the singular matrix failure of the Levenberg-Marquardt minimisation algorithm, the low precision of both programs, and a bug in Modelfree. Significantly, the singular matrix failure of the Levenberg-Marquardt algorithm occurs when internal correlation times are undefined and is greatly amplified in model-free analysis by both the grid search and constraint algorithms. The program relax (http://www.nmr-relax.com) is also presented as a new software package designed for the analysis of macromolecular dynamics through the use of NMR relaxation data and which alleviates all of the problems inherent within model-free analysis.

Finding the dynamics of an entire macromolecule is a complex problem as the model-free parameter values are intricately linked to the Brownian rotational diffusion of the molecule, mathematically through the autocorrelation function of the motion and statistically through model selection. The solution to this problem was formulated using set theory as an element of the universal set U-the union of all modelfree spaces (d'Auvergne EJ and Gooley PR (2007) Mol BioSyst 3(7), 483-494). The current procedure commonly used to find the universal solution is to initially estimate the diffusion tensor parameters, to optimise the model-free parameters of numerous models, and then to choose the best model via model selection. The global model is then optimised and the procedure repeated until convergence. In this paper a new methodology is presented which takes a different approach to this diffusion seeded model-free paradigm. Rather than starting with the diffusion tensor this iterative protocol begins by optimising the model-free parameters in the absence of any global model parameters, selecting between all the model-free models, and finally optimising the diffusion tensor. The new model-free optimisation protocol will be validated using synthetic data from Schurr JM et al. (28), 9673-9679 is reanalysed. The result is that the dynamics for certain secondary structural elements is very different from those originally reported.

Model-free analysis of NMR relaxation data, which is widely used for the study of protein dynamics, consists of the separation of the global rotational diffusion from internal motions relative to the diffusion frame and the description of these internal motions by amplitude and timescale. Five model-free models exist, each of which describes a different type of motion. Model-free analysis requires the selection of the model which best describes the dynamics of the NH bond. It will be demonstrated that the model selection technique currently used has two significant flaws, under-fitting, and not selecting a model when one ought to be selected. Under-fitting breaks the principle of parsimony causing bias in the final model-free results, visible as an overestimation of S2 and an underestimation of taue and Rex. As a consequence the protein falsely appears to be more rigid than it actually is. Model selection has been extensively developed in other fields. The techniques known as Akaike's Information Criteria (AIC), small sample size corrected AIC (AICc), Bayesian Information Criteria (BIC), bootstrap methods, and cross-validation will be compared to the currently used technique. To analyse the variety of techniques, synthetic noisy data covering all model-free motions was created. The data consists of two types of three-dimensional grid, the Rex grids covering single motions with chemical exchange [S2,taue,Rex], and the Double Motion grids covering two internal motions [S f 2,S s 2,tau s ]. The conclusion of the comparison is that for accurate model-free results, AIC model selection is essential. As the method neither under, nor over-fits, AIC is the best tool for applying Occam's razor and has the additional benefits of simplifying and speeding up model-free analysis.

Fibrosterol sulfate A is a polysulfated bis-steroid with an atypical side chain. Due to the flexibility of the linker, large scale motions that change dramatically the shape of the entire molecule are expected. Such motions pose major challenges to the structure elucidation and the correct determination of configuration. In this study, we will describe the determination of the relative configuration of fibrosterol sulfate A through an RDC-based multiple alignment tensor analysis complemented by molecular dynamics. For completeness, we applied also the single tensor approach which is unreliable due to the large scale motions and compare the results.

The dynamics of the glycosidic bond of lactose was studied by a paramagnetic tagging-based NMR technique, which allowed the collection of an unusually large series of NMR data for a single compound. By the use of distance- and orientation-dependent residual dipolar couplings and pseudocontact shifts, the simultaneous fitting of the probabilities of computed conformations and the orientation of the magnetic susceptibility tensor of a series of lanthanide complexes of lactose show that its glycosidic bond samples syn/syn, anti/syn and syn/anti ϕ/ψ regions of the conformational space in water. The analysis indicates a higher reliability of pseudocontact shift data as compared to residual dipolar couplings with the presently available weakly orienting paramagnetic tagging technique. The method presented herein allows for an improved understanding of the dynamic behaviour of oligosaccharides.