The nuclear magnetic resonance relaxation data analysis in solids: General R1/R1ρ equations and the model-free approach

Abstract

The advantage of the solid state NMR for studying molecular dynamics is the capability to study slow motions without limitations: in the liquid state, if orienting media are not used, all anisotropic magnetic interactions are averaged out by fast overall Brownian tumbling of a molecule and thus investigation of slow internal conformational motions (e.g., of proteins) in solution can be conducted using only isotropic interactions. One of the main tools for obtaining amplitudes and correlation times of molecular motions in the μs time scale is measuring relaxation rate R(1)(ρ). Yet, there have been a couple of unresolved problems in the quantitative analysis of the relaxation rates. First, when the resonance offset of the spin-lock pulse is used, the spin-lock field can be oriented under an arbitrary angle in respect to B(0). Second, the spin-lock frequency can be comparable or even less than the magic angle spinning rate. Up to now, there have been no equations for R(1)(ρ) that would be applicable for any values of the spin-lock frequency, magic angle spinning rate and resonance offset of the spin-lock pulse. In this work such equations were derived for two most important relaxation mechanisms: heteronuclear dipolar coupling and chemical shift anisotropy. The validity of the equations was checked by numerical simulation of the R(1)(ρ) experiment using SPINEVOLUTION program. In addition to that, the applicability of the well-known model-free approach to the solid state NMR relaxation data analysis was considered. For the wobbling in a cone at 30° and 90° cone angles and two-site jump models, it has been demonstrated that the auto-correlation functions G(0)(t), G(1)(t), G(2)(t), corresponding to different spherical harmonics, for isotropic samples (powders, polycrystals, etc.) are practically the same regardless of the correlation time of motion. This means that the model-free approach which is widely used in liquids can be equally applied, at least assuming these two motional models, to the analysis of the solid state NMR relaxation data.

Conformational Dynamics of a Seven Transmembrane Helical Protein Anabaena Sensory Rhodopsin Probed by Solid-State NMR

“…7 Underscoring the effect of environment on the membrane protein conformation, ASR oligomerizes into stable trimers in both detergents and lipids, 55 but exists in a dimeric state in crystals. 56 While seven helices form a similarly organized rigid transmembrane core in ASR in both crystals 56 and lipids, 7 its extra-membranous loop regions are less structurally defined.…”

“…7 Underscoring the effect of environment on the membrane protein conformation, ASR oligomerizes into stable trimers in both detergents and lipids, 55 but exists in a dimeric state in crystals. 56 While seven helices form a similarly organized rigid transmembrane core in ASR in both crystals 56 and lipids, 7 its extra-membranous loop regions are less structurally defined. Reduced NMR cross peak intensities of the residues in the loops (discussed in the following) and elevated B-factors in the crystal structure 47 for residues at the proteinsolvent interface point at their potential flexibility.…”

“…In particular, SSNMR has been successfully used for studies of membrane proteins in the physiologically relevant environment of a lipid bilayer. [2][3][4][5][6][7] Multidimensional magic angle spinning (MAS) SSNMR has also been used to sitespecifically characterize dynamics in proteins. Cross peak intensities in SSNMR spectra usually inversely correlate with the mobility of the corresponding residues, and often serve as indicators of increased local mobility.…”

Optimized “detectors” for dynamics analysis in solid-state NMR

“…(22) In this case, we see that there is almost no advantage to fitting the R1 and R1ρ rate constants simultaneously, since the block diagonal matrix shows that r 1 (q ,S) and r 2 (q ,S) will be fit only to the R1 data, and …”

“…where ωr and ω1 are the MAS frequency and the spin-lock field-strength, in angular-frequency units 21,22 (for a homonuclear spin-pair see 23 ). Although other relaxation experiments are possible, only R1 and R1 ρ data will be used as examples in this study.…”