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# General expressions for R_{1}_{ρ} relaxation for N-site chemical exchange and the special case of linear chains.

### Author information

- 1
- Department of Biochemistry and Molecular Biophysics, Columbia University, 630 West 168th Street, New York, NY 10032, United States.
- 2
- Department of Molecular Genetics, Biochemistry and Microbiology, University of Cincinnati, Cincinnati, OH 45267, United States.
- 3
- Department of Biochemistry and Molecular Biophysics, Columbia University, 630 West 168th Street, New York, NY 10032, United States. Electronic address: agp6@columbia.edu.

### Abstract

Exploration of dynamic processes in proteins and nucleic acids by spin-locking NMR experiments has been facilitated by the development of theoretical expressions for the R_{1}_{ρ} relaxation rate constant covering a variety of kinetic situations. Herein, we present a generalized approximation to the chemical exchange, R_{ex}, component of R_{1}_{ρ} for arbitrary kinetic schemes, assuming the presence of a dominant major site population, derived from the negative reciprocal trace of the inverse Bloch-McConnell evolution matrix. This approximation is equivalent to first-order truncation of the characteristic polynomial derived from the Bloch-McConnell evolution matrix. For three- and four-site chemical exchange, the first-order approximations are sufficient to distinguish different kinetic schemes. We also introduce an approach to calculate R_{1}_{ρ} for linear N-site schemes, using the matrix determinant lemma to reduce the corresponding 3N×3N Bloch-McConnell evolution matrix to a 3×3 matrix. The first- and second order-expansions of the determinant of this 3×3 matrix are closely related to previously derived equations for two-site exchange. The second-order approximations for linear N-site schemes can be used to obtain more accurate approximations for non-linear N-site schemes, such as triangular three-site or star four-site topologies. The expressions presented herein provide powerful means for the estimation of R_{ex} contributions for both low (CEST-limit) and high (R_{1}_{ρ}-limit) radiofrequency field strengths, provided that the population of one state is dominant. The general nature of the new expressions allows for consideration of complex kinetic situations in the analysis of NMR spin relaxation data.

#### KEYWORDS:

Chemical exchange; N-site chemical exchange; NMR spectroscopy; Nuclear spin relaxation; Rotating-frame relaxation

- PMID:
- 27866072
- PMCID:
- PMC5199238
- DOI:
- 10.1016/j.jmr.2016.10.015