Towards interpretation of intermolecular paramagnetic relaxation enhancement outside the fast exchange limit

In an exchanging system between major and minor species, the transverse paramagnetic relaxation enhancement rate observed on the resonances of the major species (Γ₂^app) is dependent upon the exchange regime between the species. Quantitative analysis of PRE data in such systems typically assumes that the overall exchange rate k_ex between the species is fast on the PRE time scale (k_ex ≫ Γ₂). Recently, we have characterized the kinetics of binding of the model protein ubiquitin to large (LUV) and small (SUV) unilamellar lipid-based nanoparticles or liposomes (Ceccon A, Tugarinov V, Bax A, Clore GM (2016). J Am Chem Soc 138:5789–5792). Building upon these results and taking advantage of a strong paramagnetic agent with an isotropic g-tensor, Gd³⁺, we were able to measure intermolecular methyl carbon and proton PREs between paramagnetically-tagged liposomes and ubiquitin. In the limit of fast exchange (k_ex ≫ Γ₂) the ratio of the apparent proton to carbon methyl PREs, (¹H_m–Γ₂^app)/(¹³C_m–Γ₂^app), is equal to the square of the ratio of the gyromagnetic ratios of the two nuclei, (γ_Η/γ_C)². However, outside the fast exchange regime, under intermediate exchange conditions (e.g. when Γ₂ is comparable in magnitude to k_ex) the (¹H_m–Γ₂^app)/(¹³C_m–Γ₂^app) ratio provides a reliable measure of the 'true' methyl PREs.