An average-magnetization analysis of R 1ρ relaxation outside of the fast exchange limit

Approximate expressions for the NMR spin relaxation rate constant in the rotating frame of reference R 1ρ are derived for two-site chemical exchange by consideration of the evolution of the average density operator using the stochastic Liouville equation. R 1ρ is obtained as a linearized approximation to the largest (least negative) eigenvalue of the matrix describing the evolution of the average density operator in the long-term limit. The expressions obtained are more accurate than existing expressions when exchange is not fast and the populations of the exchanging sites are close to equal. The new expressions for R 1ρ facilitate the interpretation of chemical exchange phenomena in proteins and other biological macromolecules.