An average-magnetization analysis of R _1ρ relaxation outside of the fast exchange limit

Abstract

Approximate expressions for the NMR spin relaxation rate constant in the rotating frame of reference R I P are derived for two-site chemical exchange by consideration of the evolution of the average density operator using the stochastic Liouville equation. R I P 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 I P facilitate the interpretation of chemical exchange phenomena in proteins and other biological macromolecules.

Approximate Solutions of the Bloch–McConnell Equations for Two‐Site Chemical Exchange

“…[7,15] In the case of two-site exchange, the evolution of the average magnetization hMi(t) can be shown to obey Equation (17) in the Laplace domain, hMiðsÞ ¼ hKihMið0Þ ð17Þ…”

“…By following the approach of Trott and co-workers, [7] the Laplace transformation of hM ƒ + i(t) given by Equation (16) is used to solve the Bloch±McConnell equations.…”

“…The same approach has been used previously to analyze R 11 rotating-frame relaxation in the presence of a radiofrequency field. [7,15] The simpler case of R 2 relaxation in the absence of radiofrequency fields, discussed herein, illuminates the basis for the improved approximations obtained using this approach and provides insight into the definition of the chemical exchange regime.…”

“…[3] The analytical expressions that can be derived for two-site exchange in, for example, free-precession evolution, [5,6] R 11 relaxation, [7,8] and spinecho pulse trains [9,10] are extremely useful for interpreting experimental data and for obtaining insight into the phenomenology of more complex exchange processes.…”

Structure determination and dynamics of protein–RNA complexes by NMR spectroscopy

“…There have been significant advances in the experimental characterization of slow motions in proteins involving a variety of nuclei as probes [364,376,377]. In addition, new strategies for analysis of relaxation dispersion data have been reported [378,379]. More recently, the application of similar techniques to RNA have been reported [380].…”