A complete derivation of the Karger equations for analyzing NMR diffusion measurements of exchanging systems

The use of NMR methods to quantitatively determine association/dissociation constants in biological and chemical systems (eg, drug binding/protein-ligand interactions) is long established. Among these, NMR diffusion-based techniques are increasingly prominent. In this article, the Karger equations, used for analyzing NMR diffusion-based measurements of exchange, are derived from first principles and then solved in detail for the case of a two-site freely diffusing system under simplifying assumptions that the chemical shift and the relaxation in both sites are the same. Care is taken to clarify the connections between the experimental variables and the parameters of the Karger equations.