Abstract
In aqueous systems with immobilized macromolecules, including biological tissues, 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 previously developed a rigorous EMOR relaxation theory for dipole-coupled two-spin and three-spin systems. Here, we extend the stochastic Liouville theory to four-spin systems and use these exact results as a guide for constructing an approximate multi-spin theory, valid for spin systems of arbitrary size. This so-called generalized stochastic Redfield equation (GSRE) theory includes the effects of longitudinal-transverse cross-mode relaxation, which gives rise to an inverted step in the relaxation dispersion profile, and coherent spin mode transfer among solid-like spins, which may be regarded as generalized spin diffusion. The GSRE theory is compared to an existing theory, based on the extended Solomon equations, which does not incorporate these phenomena. Relaxation dispersion profiles are computed from the GSRE theory for systems of up to 16 protons, taken from protein crystal structures. These profiles span the range from the motional narrowing limit, where the coherent mode transfer plays a major role, to the ultra-slow motion limit, where the zero-field rate is closely related to the strong-collision limit of the dipolar relaxation rate. Although a quantitative analysis of experimental data is beyond the scope of this work, it is clear from the magnitude of the predicted relaxation rate and the shape of the relaxation dispersion profile that the dipolar EMOR mechanism is the principal cause of water-1H low-field longitudinal relaxation in aqueous systems of immobilized macromolecules, including soft biological tissues. The relaxation theory developed here therefore provides a basis for molecular-level interpretation of endogenous soft-tissue image contrast obtained by the emerging low-field magnetic resonance imaging techniques.
Highlights
During the past two decades, a rigorous molecular theory has been developed for nuclear magnetic relaxation induced by exchange-modulated electric quadrupole1–3 or magnetic dipole4–7 couplings in aqueous systems with immobilized macromolecules
We make the leap to multi-spin systems, comprising one or two labile spins exchanging with an isotropic bulk phase and dipole-coupled to an arbitrary number of nonlabile spins in a solid-like environment
To better understand the rich relaxation behavior exhibited by the dipolar exchange-mediated orientational randomization (EMOR) model, we have developed a perturbation theory, based on the stochastic Redfield equation
Summary
During the past two decades, a rigorous molecular theory has been developed for nuclear magnetic relaxation induced by exchange-modulated electric quadrupole or magnetic dipole couplings in aqueous systems with immobilized macromolecules. We have developed a non-perturbative relaxation theory, based on the stochastic Liouville equation (SLE), and valid without restrictions on exchange rate, dipole couplings, and magnetic field strength.. We turn to the more computationally efficient and physically transparent SRE theory, extending its validity beyond the RMN regime by certain physically inspired, but essentially ad hoc, modifications These modifications were calibrated against the exact SLE solution of the EMOR model for four-spin systems, which is presented here. Unlike the GSRE and SLE theories, the ESE theory does not take into account longitudinal-transverse cross-mode relaxation or coherent transfer of magnetization (and higher spin modes) induced by the static dipole couplings. The results presented in this paper are based on a substantial amount of analytical and numerical work, described in more detail in the 14 appendices of the supplementary material
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