Coherent and stochastic averaging in solid-state NMR

A master equation for spin systems far from equilibrium

The quantum dynamics of spin systems is often treated by a differential equation known as the master equation, which describes the trajectories of spin observables such as magnetization components, spin state populations, and coherences between spin states. The master equation describes how a perturbed spin system returns to a state of thermal equilibrium with a finite-temperature environment. The conventional master equation, which has the form of an inhomogeneous differential equation, applies to cases where the spin system remains close to thermal equilibrium, which is well satisfied for a wide variety of magnetic resonance experiments conducted on thermally polarized spin systems at ordinary temperatures. However, the conventional inhomogeneous master equation may fail in the case of hyperpolarized spin systems, when the spin state populations deviate strongly from thermal equilibrium, and in general where there is a high degree of nuclear spin order. We highlight a simple case in which the inhomogeneous master equation clearly fails, and propose an alternative master equation based on Lindblad superoperators which avoids most of the deficiencies of previous proposals. We discuss the strengths and limitations of the various formulations of the master equation, in the context of spin systems which are far from thermal equilibrium. The method is applied to several problems in nuclear magnetic resonance and to spin-isomer conversion.

Protein dynamics in the solid-state from 2H NMR lineshape analysis. III. MOMD in the presence of Magic Angle Spinning

We report on a new approach to the analysis of dynamic NMR lineshapes from polycrystalline (i.e., macroscopically disordered) samples in the presence of Magic Angle Spinning (MAS). This is an application of the Stochastic Liouville Equation developed by Freed and co-workers for treating restricted (i.e., microscopically ordered) motions. The ²H nucleus in an internally-mobile C-CD₃ moiety serves as a prototype probe. The acronym is ²H/MOMD/MAS, where MOMD stands for "microscopic-order-macroscopic-disorder." The key elements describing internal motions - their type, the local spatial restrictions, and related features of local geometry - are treated in MOMD generally, within their rigorous three-dimensional tensorial requirements. Based on this representation a single physically well-defined model of local motion has the capability of reproducing experimental spectra. There exist other methods for analyzing dynamic ²H/MAS spectra which advocate simple motional modes. Yet, to reproduce satisfactorily the experimental lineshapes, one has either to use unusual parameter values, or combine several simple motional modes. The multi-simple-mode reasoning assumes independence of the constituent modes, features ambiguity as different simple modes may be used, renders inter-system comparison difficult as the overall models differ, and makes possible model-improvement only by adding yet another simple mode, i.e., changing the overall model. ²H/MOMD/MAS is free of such limitations and inherently provides a clear physical interpretation. These features are illustrated. The advantage of ²H/MOMD/MAS in dealing with sensitive but hardly investigated slow-motional lineshapes is demonstrated by applying it to actual experimental data. The results differ from those obtained previously with a two-site exchange scheme that yielded unusual parameters.

Protein Rotational Dynamics in Aligned Lipid Membranes Probed by Anisotropic T1ρ NMR Relaxation

A membrane-bound form of Pf1 coat protein reconstituted in magnetically aligned DMPC/DHPC bicelles was used as a molecular probe to quantify for the viscosity of the lipid membrane interior by measuring the uniaxial rotational diffusion coefficient of the protein. Orientationally dependent ¹⁵N NMR relaxation times in the rotating frame, or T_1ρ, were determined by fitting individually the decay of the resolved NMR peaks corresponding to the transmembrane helix of Pf1 coat protein as a function of the spin-lock time incorporated into the 2D SAMPI4 pulse sequence. The T_1ρ relaxation mechanism was modeled by uniaxial rotational diffusion on a cone, which yields a linear correlation with respect to the bond factor sin⁴θ_B, where θ_B is the angle that the NH bond forms with respect to the axis of rotation. Importantly, the bond factors can be independently measured from the dipolar couplings in the separated local-field SAMPI4 spectra. From this dependence, the value of the diffusion coefficient D_|| = 8.0 × 10⁵ s^-1 was inferred from linear regression of the experimental T_1ρ data even without any spectroscopic assignment. Alternatively, a close value of D_|| = 7.7 × 10⁵ s^-1 was obtained by fitting the T_1ρ relaxation data for the assigned NMR peaks of the transmembrane domain of Pf1 to a wavelike pattern as a function of residue number. The method illustrates the use of single-helix transmembrane peptides as molecular probes to assess the dynamic parameters of biological membranes by NMR relaxation in oriented lipid bilayers.