Nuclear magnetic relaxation induced by exchange-mediated orientational randomization: Longitudinal relaxation dispersion for a dipole-coupled spin-1/2 pair

Abstract

In complex biological or colloidal samples, magnetic relaxation dispersion (MRD) experiments using the field-cycling technique can characterize molecular motions on time scales ranging from nanoseconds to microseconds, provided that a rigorous theory of nuclear spin relaxation is available. In gels, cross-linked proteins, and biological tissues, where an immobilized macromolecular component coexists with a mobile solvent phase, nuclear spins residing in solvent (or cosolvent) species relax predominantly via exchange-mediated orientational randomization (EMOR) of anisotropic nuclear (electric quadrupole or magnetic dipole) couplings. The physical or chemical exchange processes that dominate the MRD typically occur on a time scale of microseconds or longer, where the conventional perturbation theory of spin relaxation breaks down. There is thus a need for a more general relaxation theory. Such a theory, based on the stochastic Liouville equation (SLE) for the EMOR mechanism, is available for a single quadrupolar spin I = 1. Here, we present the corresponding theory for a dipole-coupled spin-1/2 pair. To our knowledge, this is the first treatment of dipolar MRD outside the motional-narrowing regime. Based on an analytical solution of the spatial part of the SLE, we show how the integral longitudinal relaxation rate can be computed efficiently. Both like and unlike spins, with selective or non-selective excitation, are treated. For the experimentally important dilute regime, where only a small fraction of the spin pairs are immobilized, we obtain simple analytical expressions for the auto-relaxation and cross-relaxation rates which generalize the well-known Solomon equations. These generalized results will be useful in biophysical studies, e.g

Nuclear magnetic relaxation by the dipolar EMOR mechanism: General theory with applications to two-spin systems

“…To illustrate the general theory, we present explicit results for the asymmetric two-spin case, which is contrasted with the previously treated symmetric two-spin case. 20 These results are directly applicable to, for example, a macromolecular hydroxyl proton in chemical exchange with water protons (asymmetric case) or to an internal water molecule in physical exchange with bulk water (symmetric case). This paper is organized as follows.…”

“…This is the case, for example, for chemical exchange of labile macromolecular protons with bulk water and for physical exchange of trapped (internal) water molecules with bulk water. 20,23 We can then ignore all dipole couplings among the labile spins in state B. If so desired, the small and frequencyindependent relaxation contribution from fast modulation of dipole couplings in state B can be added to the final expression for the overall relaxation rate.…”

“…19 As compared to quadrupolar relaxation, which only involves single spins, dipolar relaxation is theoretically more challenging. The EMOR SLE theory for dipolar relaxation of a homonuclear spin pair exchanging as a unit 20 is isomorphic with the corresponding theory for quadrupolar relaxation of a single spin-1, 11 but for heteronuclear spins, multispin (>2) systems and/or fragmentation of the spin system by exchange, qualitatively new phenomena appear in the dipolar relaxation. In a previous report, 20 hereafter referred to as Paper I, we developed the EMOR SLE theory for a (homonuclear or heteronuclear) spin pair that exchanges as an intact unit, a situation that we now refer to as symmetric exchange.…”

“…The EMOR SLE theory for dipolar relaxation of a homonuclear spin pair exchanging as a unit 20 is isomorphic with the corresponding theory for quadrupolar relaxation of a single spin-1, 11 but for heteronuclear spins, multispin (>2) systems and/or fragmentation of the spin system by exchange, qualitatively new phenomena appear in the dipolar relaxation. In a previous report, 20 hereafter referred to as Paper I, we developed the EMOR SLE theory for a (homonuclear or heteronuclear) spin pair that exchanges as an intact unit, a situation that we now refer to as symmetric exchange. Contrary to our earlier expectations, 20 the case of asymmetric exchange, where only one of the two dipole-coupled spins undergoes exchange, differs fundamentally from the symmetric case.…”

“…In a previous report, 20 hereafter referred to as Paper I, we developed the EMOR SLE theory for a (homonuclear or heteronuclear) spin pair that exchanges as an intact unit, a situation that we now refer to as symmetric exchange. Contrary to our earlier expectations, 20 the case of asymmetric exchange, where only one of the two dipole-coupled spins undergoes exchange, differs fundamentally from the symmetric case. In particular, since the non-exchanging spins are not isotropically averaged, the longitudinal and transverse magnetizations are dynamically coupled in the anisotropic sites.…”