The physical basis of model-free analysis of NMR relaxation data from proteins and complex fluids

NMR relaxation experiments have provided a wealth of information about molecular motions in macromolecules and ordered fluids. Even though a rigorous theory of spin relaxation is available, the complexity of the investigated systems often makes the interpretation of limited datasets challenging and ambiguous. To allow physically meaningful information to be extracted from the data without commitment to detailed dynamical models, several versions of a model-free (MF) approach to data analysis have been developed. During the past 2 decades, the MF approach has been used in the vast majority of all NMR relaxation studies of internal motions in proteins and other macromolecules, and it has also played an important role in studies of colloidal systems. Although the MF approach has been almost universally adopted, substantial disagreement remains about its physical foundations and range of validity. It is our aim here to clarify these issues. To this end, we first present rigorous derivations of the three well-known MF formulas for the time correlation function relevant for isotropic solutions. These derivations are more general than the original ones, thereby substantially extending the range of validity of the MF approach. We point out several common misconceptions and explain the physical significance of the approximations involved. In particular, we discuss symmetry requirements and the dynamical decoupling approximation that plays a key role in the MF approach. We also derive a new MF formula, applicable to anisotropic fluids and solids, including microcrystalline protein samples. The so-called slowly relaxing local structure (SRLS) model has been advanced as an alternative to the MF approach that does not require dynamical decoupling of internal and global motions. To resolve the existing controversy about the relative merits of the SRLS model and the MF approach, we formulate and solve a planar version of the SRLS model. The analytical solution of this model reveals the unphysical consequences of the symmetrical two-body Smoluchowski equation as applied to protein dynamics, thus refuting the widely held belief that the SRLS model is more accurate than the MF approach. The different results obtained by analyzing data with these two approaches therefore do not indicate the importance of dynamical coupling between internal and global motions. Finally, we explore the two principal mechanisms of dynamical coupling in proteins: torque-mediated and friction-mediated coupling. We argue by way of specific analytically solvable models that torque-mediated coupling (which the SRLS model attempts to capture) is unimportant because the relatively slow internal motions that might couple to the global motion tend to be intermittent (jumplike) in character, whereas friction-mediated coupling (which neither the SRLS model nor the MF approach incorporates) may be important for proteins with unstructured parts or flexibly connected domains.