Set theory formulation of the model-free problem and the diffusion seeded model-free paradigm

Model-free analysis of NMR relaxation data, which describes the motion of individual atoms, is a problem intricately linked to the Brownian rotational diffusion of the macromolecule. The diffusion tensor parameters strongly influence the optimisation of the various model-free models and the subsequent model selection between them. Finding the optimal model of the dynamics of the system among the numerous diffusion and model-free models is hence quite complex. Using set theory, the entirety of this global problem has been encapsulated by the universal set [Fraktur U], and its resolution mathematically formulated as the universal solution . Ever since the original Lipari and Szabo papers the model-free dynamics of a molecule has most often been solved by initially estimating the diffusion tensor. The model-free models which depend on the diffusion parameter values are then optimised and the best model is chosen to represent the dynamics of the residue. Finally, the global model of all diffusion and model-free parameters is optimised. These steps are repeated until convergence. For simplicity this approach to will be labelled the diffusion seeded model-free paradigm. Although this technique suffers from a number of problems many have been solved. All aspects of the diffusion seeded paradigm and its consequences, together with a few alternatives to the paradigm, will be reviewed through the use of set notation.