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See supplementary material at for tables of ISTO basis operators (Appendix A); Zeeman Liouvillian supermatrix elements (Appendix B); dipolar Liouvillian supermatrix elements (Appendix C); general ILRR expressions for all cases (Appendix D); the local relaxation supermatrix in the MN regime (Appendix E); specific ILRR expressions for the the MN regime (Appendix F); chemical shift effects in the MN regime (Appendix G); and the effect of the static dipole coupling for theISPI case in the MN regime (Appendix H).[Supplementary Material]
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In aqueous systems with immobilized macromolecules, including biological tissue, the longitudinal spin relaxation of water protons is primarily induced by exchange-mediated orientational randomization (EMOR) of intra- and intermolecular magnetic dipole-dipole couplings. Starting from the stochastic Liouville equation, we have developed a non-perturbative theory that can describe relaxation by the dipolar EMOR mechanism over the full range of exchange rates, dipole couplings, and Larmor frequencies. Here, we implement the general dipolar EMOR theory for a macromolecule-bound three-spin system, where one, two, or all three spins exchange with the bulk solution phase. In contrast to the previously studied two-spin system with a single dipole coupling, there are now three dipole couplings, so relaxation is affected by distinct correlations as well as by self-correlations. Moreover, relaxation can now couple the magnetizations with three-spin modes and, in the presence of a static dipole coupling, with two-spin modes. As a result of this complexity, three secondary dispersion steps with different physical origins can appear in the longitudinal relaxation dispersion profile, in addition to the primary dispersion step at the Larmor frequency matching the exchange rate. Furthermore, and in contrast to the two-spin system, longitudinal relaxation can be significantly affected by chemical shifts and by the odd-valued (“imaginary”) part of the spectral density function. We anticipate that the detailed studies of two-spin and three-spin systems that have now been completed will provide the foundation for developing an approximate multi-spin dipolar EMOR theory sufficiently accurate and computationally efficient to allow quantitative molecular-level interpretation of frequency-dependent water-proton longitudinal relaxation data from biophysical model systems and soft biological tissue.


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