Compact expressions for R-1 rho relaxation for N-site chemical exchange using Schur decomposition

摘要

The rotating-frame spin relaxation rate constant, R-1 rho, is a powerful probe of macromolecular chemical and conformational dynamics in relaxation dispersion, CEST, and DEST NMR experiments. The R-1 rho, relaxation rate constant is given by the absolute value of the largest (least negative) eigenvalue of the Bloch-McConnell evolution matrix; however, estimation of this eigenvalue require inversion of 3 N x 3 N dimensional matrices, in which N is the number of interconverting sites or states for a given nuclear spin in a molecule. The Schur complement is used to reduce the problem of calculating the characteristic polynomial of a 3 N x 3 N matrix to that of calculating the characteristic polynomial of a 3 x 3 matrix. The resulting expressions for N-site chemical exchange are more numerically tractable, because the largest matrix inversion also is of dimension 3 x 3. In addition, the simplifications offered by the Schurr complement conveniently illustrate the effects of fast or slow kinetic steps within an N-site kinetic topology. (C) 2020 Elsevier Inc. All rights reserved.