Abstract
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. We have embarked on a systematic program to develop, from the stochastic Liouville equation, a general and rigorous theory that can describe relaxation by the dipolar EMOR mechanism over the full range of exchange rates, dipole coupling strengths, and Larmor frequencies. Here, we present a general theoretical framework applicable to spin systems of arbitrary size with symmetric or asymmetric exchange. So far, the dipolar EMOR theory is only available for a two-spin system with symmetric exchange. Asymmetric exchange, when the spin system is fragmented by the exchange, introduces new and unexpected phenomena. Notably, the anisotropic dipole couplings of non-exchanging spins break the axial symmetry in spin Liouville space, thereby opening up new relaxation channels in the locally anisotropic sites, including longitudinal-transverse cross relaxation. Such cross-mode relaxation operates only at low fields; at higher fields it becomes nonsecular, leading to an unusual inverted relaxation dispersion that splits the extreme-narrowing regime into two sub-regimes. The general dipolar EMOR theory is illustrated here by a detailed analysis of the asymmetric two-spin case, for which we present relaxation dispersion profiles over a wide range of conditions as well as analytical results for integral relaxation rates and time-dependent spin modes in the zero-field and motional-narrowing regimes. The general theoretical framework presented here will enable a quantitative analysis of frequency-dependent water-proton longitudinal relaxation in model systems with immobilized macromolecules and, ultimately, will provide a rigorous link between relaxation-based magnetic resonance image contrast and molecular parameters.
Highlights
Soft-tissue contrast in clinical magnetic resonance imaging derives largely from spatial variations in the relaxation behavior of water protons
The I SP–in the same two-spin (I S) case might refer to the two protons (I S) of a water molecule temporarily trapped in a protein cavity, where the water protons are dipole-coupled to a nearby aliphatic proton (P)
We focus on the asymmetric I S–I case, highlighting differences compared to the symmetric I S–I S case
Summary
Soft-tissue contrast in clinical magnetic resonance imaging derives largely from spatial variations in the relaxation behavior of water protons. The lack of theoretical underpinning is a limitation in biophysical studies of, for example, water-protein interactions and intermittent protein dynamics by field-cycling measurements of the water 1H magnetic relaxation dispersion (MRD) in protein gels. Such data have been interpreted with semi-phenomenological models involving questionable assumptions about the relaxation-inducing motions.. Since the non-exchanging spins are not isotropically averaged, the longitudinal and transverse magnetizations are dynamically coupled in the anisotropic sites Such cross-mode relaxation, distinct from the cross-spin relaxation familiar from the Solomon equations, gives rise to an inverted relaxation dispersion at low field. Lengthy derivations and tables are relegated to six appendices.
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