Concepts and Methods of Solid-State NMR ... - ACS Publications

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Concepts and Methods of Solid-State NMR Spectroscopy Applied to Biomembranes Trivikram R. Molugu,† Soohyun Lee,† and Michael F. Brown*,†,‡ †

Department of Chemistry & Biochemistry and ‡Department of Physics, University of Arizona, Tucson, Arizona 85721, United States S Supporting Information *

ABSTRACT: Concepts of solid-state NMR spectroscopy and applications to fluid membranes are reviewed in this paper. Membrane lipids with 2H-labeled acyl chains or polar head groups are studied using 2H NMR to yield knowledge of their atomistic structures in relation to equilibrium properties. This review demonstrates the principles and applications of solid-state NMR by unifying dipolar and quadrupolar interactions and highlights the unique features offered by solid-state 2H NMR with experimental illustrations. For randomly oriented multilamellar lipids or aligned membranes, solidstate 2H NMR enables direct measurement of residual quadrupolar couplings (RQCs) due to individual C−2H-labeled segments. The distribution of RQC values gives nearly complete profiles of the segmental order parameters S(i) CD as a function of acyl segment position (i). Alternatively, one can measure residual dipolar couplings (RDCs) for natural abundance lipid samples to obtain segmental SCH order parameters. A theoretical mean-torque model provides acyl-packing profiles representing the cumulative chain extension along the normal to the aqueous interface. Equilibrium structural properties of fluid bilayers and various thermodynamic quantities can then be calculated, which describe the interactions with cholesterol, detergents, peptides, and integral membrane proteins and formation of lipid rafts. One can also obtain direct information for membrane-bound peptides or proteins by measuring RDCs using magic-angle spinning (MAS) in combination with dipolar recoupling methods. Solid-state NMR methods have been extensively applied to characterize model membranes and membranebound peptides and proteins, giving unique information on their conformations, orientations, and interactions in the natural liquid-crystalline state. 3.4. Schrö dinger’s Equation Entails Calculation of Quadrupolar Energies and Transition Frequencies 3.5. Quadrupolar Couplings in Deuterium NMR Spectroscopy Are Related to Molecular Geometry 3.6. Random (Powder-Type) Samples Yield Principal Values of Coupling Tensor 3.7. Spectral Inequivalence in Solid-State Deuterium NMR Spectroscopy Is Based on Differences in Molecular Mobility 3.8. Spectral Line Shapes and Quadrupolar Frequencies Manifest Spherical or Cylindrical Symmetry 3.9. Solid-State NMR Spectra of Uniaxially Aligned Immobile Samples Reveal Principal Axes Orientations and Principal Values 4. Residual Quadrupolar Couplings Characterize Nanostructures of Membrane Liquid Crystals 4.1. Orientational Order Parameters of Biomolecules Are Obtained from Residual Quadrupolar Couplings

CONTENTS 1. Introduction 2. Solid-State NMR Spectroscopy of Biomembranes 2.1. Solid-State Deuterium NMR Spectroscopy Elucidates Equilibrium and Dynamical Properties of Membrane Lipids 2.2. Average Properties of Membrane Lipids Encompass a Range of Time and Length Scales 2.3. Deuterium NMR Allows Direct Observation of Couplings Relevant to Membrane Structure and Dynamics 3. Molecular Distributions of Lipids and Proteins Are Revealed by Solid-State Deuterium NMR Spectral Line Shapes 3.1. Equilibrium Properties of Membrane Constituents Are Obtained from NMR Spectral Line Shapes 3.2. Coupling Hamiltonians in Solid-State NMR Spectroscopy Are Formulated Using Irreducible Tensor Operators 3.3. Irreducible Tensors Are Transformed from the Principal Axis System to Laboratory Frame by Wigner Rotation Matrices

© 2017 American Chemical Society

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Received: September 10, 2016 Published: September 14, 2017 12087

DOI: 10.1021/acs.chemrev.6b00619 Chem. Rev. 2017, 117, 12087−12132

Chemical Reviews 4.2. Multiple Coordinate Transformations Describe a Hierarchy of Motions by the Closure Property of the Rotation Group 4.3. Spectral Frequencies of Aligned Samples with Axial Rotation about the Director Yield Residual Quadrupolar Couplings 4.4. Moments of Deuterium NMR Line Shapes Quantify Distribution of Orientational Order Parameters for Nonaligned Samples 5. Segmental Order Parameters Are Related to Energy Landscapes for Liquid-Crystalline Membrane Lipids 5.1. Hierarchy of Motions Governs Dynamical Structures of Membrane Lipid Bilayers 5.2. Residual Quadrupolar Couplings Characterize Motional Averaging in Liquid-Crystalline Assemblies 5.3. Phospholipids in the Liquid-Crystalline State Yield a Distribution of Residual Quadrupolar Couplings 5.4. Dynamical Structures of Fluid Membranes Are Manifested by Segmental Order Parameters 5.5. Segmental Order Parameters Show a SiteSpecific Profile as a Function of Acyl Group Position 5.6. Bilayers Containing Cholesterol Enable Testing of Theories for Dynamical Structures of Membrane Assemblies 6. Material Properties, Nanostructures, and Biological Functions of Phospholipids Are Emergent at the Atomistic Level 6.1. Mean-Torque Model Describes Segmental Order Profiles in Terms of Lipid Properties 6.2. Moments of Segmental Orientational Distribution Are Related to Bilayer Structural Parameters 6.3. Acyl Packing Profiles Quantify Chain Extension along the Surface Normal of Lipid Nanostructures 6.4. Influences of Acyl Length, Lipid Polar Headgroups, and Cholesterol on Structural Parameters of Lipid Bilayers 6.5. Mixing of Sphingolipids and Phospholipids with Cholesterol Involves Configurational Entropy 6.6. Solid-State Deuterium NMR Reveals Coexistence of Ordered Domains with the LiquidDisordered Phase in Lipid Mixtures 6.7. Bilayer Properties Are Affected Oppositely by Nonionic Detergents and Raft-Like Cholesterol Mixtures 6.8. Polyunsaturated Lipid Acyl Chains Lead to Shifting of the Bilayer Mass Distribution 6.9. Non-Lamellar Lipid Nanostructures Involve Membrane Curvature 6.10. Peptides and Integral Membrane Proteins Are Affected by the Structural Properties of Lipid Membranes 7. Residual Dipolar Couplings Provide Structures and Topology of Membrane Lipids and Membrane Proteins

Review

7.1. Rotational-Echo Double-Resonance (REDOR) Experiments Yield Distance Constraints 7.2. Polarization-Inversion Spin Exchange at the Magic Angle (PISEMA) Provides Transmembrane Helix Topology 7.3. Residual Dipolar Couplings Specify Orientation of Helices Aligned in Lipid Bilayers 8. Challenges and Future Directions Appendix: Irreducible Tensor Calculus As Applied to NMR Spectroscopy of Biomembranes Associated Content Supporting Information Author Information Corresponding Author ORCID Notes Biographies Acknowledgments References

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1. INTRODUCTION Solid-state nuclear magnetic resonance spectroscopy constitutes a powerful approach to investigating biomolecules in their natural functional states. Biomembranes often exist in a liquidcrystalline phase whose molecular properties at mesoscopic length and time scales play crucial roles in understanding their biological actions. For both the protein and the lipid constituents, solid-state NMR methods provide information about the structure and dynamics at an atomistic level in the liquidcrystalline state.1−3 Moreover, solid-state NMR of biomembranes is highly complementary to other approaches, including X-ray and neutron scattering, as well as vibrational and electronic spectroscopy. Structural information in solid-state NMR spectroscopy is generally obtained from static or motionally averaged coupling tensors due to dipolar, chemical shift, or quadrupolar interactions.2,5−6 The corresponding dynamical information is acquired from the tensor fluctuations, which depend on the mean-squared amplitudes and rates of the motions, and affect the line shapes and relaxation times.1,14−17 There are few if any other physical methods that can match these unique capabilities. For these reasons, solid-state NMR spectroscopy is becoming increasingly prevalent in material science, biophysics, and structural biology, and this situation is expected to continue well into the future. For biomembrane investigations, deuterium nuclear magnetic resonance (2H NMR) spectroscopy plays a pivotal role. It also serves to illustrate the general principles of solid-state NMR spectroscopy as applied to soft biomolecular systems, where both structure and dynamics come clearly into play. Notably solidstate 2H NMR methods can be applied equally well to membrane proteins and membrane lipids through introduction of chemical and/or biological labeling strategies. Because 2H NMR spectroscopy relies on isotopic substitution of 2H for 1H, it probes the structures of organic or biological molecules in a site-specific, nonperturbing fashion. Applications of this technology give an unprecedented view of membranes and biomolecules, for instance, with regard to the binding of ligands or pharmaceuticals to protein receptors and ion channels23 or protein interactions with the surrounding lipid bilayer.24−26 Membrane constituents are studied at an atomistically resolved level of individual C−2H bonds; the method provides knowledge of local structure that is highly synergistic to X-ray and electron crystallography

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Figure 1. Cellular membranes often exist in a liquid-crystalline phase and are implicated in key biological functions. (a) The hydrophobic effect leads to sequestering of the nonpolar acyl chains within the bilayer interior, whereas the polar head groups interact with water at the membrane surface. Reprinted with permission from ref 111. Copyright 2002 American Chemical Society. (b) Side view of the simulated SecYEβ transmembrane protein in the lipid environment. Lipids are colored light blue, with their head groups highlighted as red, blue, and green spheres. Adapted with permission from ref 461. Copyright 2008 Rockefeller University Press.

crystalline state, knowing the dynamical structure of fluid bilayers is significant for understanding cellular functions in detail. Consequently, our focus here is on the equilibrium properties of membrane lipids in the liquid-crystalline state. As soft liquidcrystalline materials, they have well-defined nanostructures,111 which we illustrate for a canonical phospholipid bilayer and a proteolipid membrane in Figure 1. Notably, molecular understanding of the interactions of peptides and proteins with lipids requires experimental knowledge of the structure of the bilayer, the transmembrane topology and the bound protein conformations, and the bilayer structural changes due to the lipid− protein interactions.112 Although lipid bilayer structures are often investigated by X-ray and neutron diffraction methods, the highly disordered nature of fluid bilayers precludes atomicresolution crystallographic images. Even so, a static picture is to some extent a misrepresentation, because it provides only a snapshot of one instant of time, whereas the constituent molecules are in fact highly mobile with many degrees of freedom. Moreover, there are emergent membrane properties that are not readily visualized by such van der Waals surfaces.113 This review first describes the basic principles of solid-state 2H NMR spectroscopy. Lipid interactions with cholesterol,22,114 detergents,115,116 and membrane-bound peptides and proteins26,112,117 are considered in terms of their influences on bilayer properties, and we describe how to acquire structural knowledge and investigate the forces affecting soft membrane lipids using a statistical mean-torque model. Solid-state NMR applications involving dipolar couplings give complementary information at a molecularly site-resolved level that are analogous to quadrupolar couplings, including bond orientations and distance information. Lastly, we highlight the potential of solidstate NMR spectroscopy in studies of model lipid mixtures and biological membranes (Figure 1).

approaches, and dynamical information is acquired that is unrivaled in its chemical detail and applicability to molecular simulations. Representative examples27−37 of biomolecular applications of solid-state 2H NMR spectroscopy include membrane proteins and peptides,23,30−37 DNA fibers,28 and DNA−lipid complexes relevant to gene therapy.38 In the case of membranes, both saturated and polyunsaturated lipids39−69 as well as lipid mixtures associated with raft-like microdomains in cellular membranes7,26,70 have been investigated, and the range of applicability continues to increase. Experimental methods of solid-state 2H NMR spectroscopy for investigating both the structural and the dynamical properties of liquid-crystalline systems are particularly well developed.15−17,71−80 Because a single electrical interaction is dominant, the spectral interpretation and relaxation analysis is correspondingly simplified, and enables more complex biomolecular systems to be investigated than would otherwise be possible. For membrane lipids, the residual quadrupolar couplings (RQCs) are large in comparison to motionally averaged dipolar couplings or chemical shifts. Various atomistically resolved sites can be identified and observed directly. However, perhaps what is most compelling about solid-state 2H NMR methodologies is that the observables are exquisitely sensitive to molecular motions at typical NMR magnetic field strengths as applied to soft nanomaterials.15,75,81−89 In solid-state 2 H NMR spectroscopy, the line shapes correspond to equilibrium properties at the molecular level, and dynamical properties are accessible from solid-state 2H NMR relaxation rates and associated spectral densities of motion. By combining the results of solid-state 2H NMR line shape studies with relaxation studies, one can arrive at a more comprehensive understanding than with either method alone. The functional dynamics of the constituents can thus be probed in a manner that is largely inaccessible using other biophysical techniques. In this review, we describe the concepts and applications of solid-state NMR spectroscopy to lipid membranes as well as membrane-embedded peptides and proteins. Our presentation is geared toward beginning students of NMR spectroscopy as well as specialists in the field. Previous detailed accounts of membrane lipids are available,41,62,67,77,90−97 and extensive applications to membrane proteins and peptides are insightfully covered elsewhere.9,98−110 Because biomembranes often are in a liquid-

2. SOLID-STATE NMR SPECTROSCOPY OF BIOMEMBRANES Applications of solid-state NMR methods for biomembranes generally fall into two broad categories. First, one can estimate distances between nuclear spins from dipolar coupling tensors due to spin−spin interactions,118−126 as in the case of unaligned (amorphous) samples. Some specific examples include dipolar recoupling measurements for rotating samples under magic12089


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angle spinning (MAS). Moreover, the relative orientations of pairs of nuclear spin interaction tensors can be determined to obtain molecular torsion angles directly.127−129 Alternatively, solid-state NMR of aligned membranes can be used to study the orientations of membrane constituents.29,34,35,102,103,130−145 Further examples include solid-state 2H NMR of aligned bilayers,29,34,35,146,147 aligned biopolymers as in the case of DNA fibers,28,148 and separated local field NMR, where one measures orientation-dependent 15N chemical shifts and corresponding 1H−15N dipolar couplings to provide very similar topological information, as in the case of PISEMA experiments.104,149−151 In this later type of application, one directly determines the angles of the nuclear spin interaction tensors with respect to the macroscopic alignment axis in the case of oriented samples, e.g., the normal (director) to the bilayer surface.29,34,35,103,137,138,145,152−161 For polypeptides that can be prepared by solid-phase synthesis, an additional very powerful approach is the combination of 2H and 15N NMR measurements from various labeled sites.141,143,162 In principle, analogous information can be obtained from rotating samples under magicangle spinning conditions.12,163−170 Application of the MAS technique allows acquiring 1H nuclear Overhauser spectra of lipid membranes to probe spatial proton−proton contacts for structural studies analogous to solution NMR spectroscopy.171−175 Such studies are implemented to explore the effect of unsaturation and cholesterol on the lateral organization of lipids and lipid−protein interactions for the case of transmembrane peptides.10,176−184

Figure 2. Biological membranes contain different classes of glycerophospholipids. Polar head groups vary in size, hydrogen bonding, and charge. Examples are shown for zwitterionic phosphocholine (PC) and phosphethanolamine (PE) head groups and for the anionic phosphoserine (PS) headgroup. Nonpolar acyl chains differ in length and degree of unsaturation, as illustrated by oleic acid (18:1ω-9) and docosahexaenoic acid (22:6ω-3).

2.1. Solid-State Deuterium NMR Spectroscopy Elucidates Equilibrium and Dynamical Properties of Membrane Lipids

head groups differ in their size, capacity for hydrogen bonding, and charge, whereas the nonpolar acyl chains vary in their length and degree and position of unsaturation. One should recall that phase equilibria of phosphatidylcholines in excess water include three regions with increasing temperature: a lamellar gel phase with tilted chains (Lβ′), an intermediate ripple phase (Pβ′), and a lamellar liquid-crystalline phase (Lα)197 also known as the liquiddisordered (ld) phase. Binary mixtures of phosphatidylcholines with cholesterol reveal an additional liquid-ordered (lo) phase, where the lipids have liquid-like rotational disorder as in the Lα phase, and the acyl groups possess configurational order as in the gel or solid-ordered (so) phase.200−204 Such cholesterolcontaining lipid phases have attracted considerable attention in terms of so-called lipid rafts.3,205−208 In addition, phosphatidylethanolamines form the reverse hexagonal (HII) phase at temperatures above the Lα phase.197,209,210 Types of lipid systems investigated by 2H NMR include multilamellar dispersions, membranes aligned on planar glass substrates, bilayers adsorbed to small polystyrene beads, unilamellar vesicles, and reverse hexagonal phases.7,53,77,79,189,197

The general principles of solid-state NMR spectroscopy as applicable to biomembranes are well illustrated by 2H NMR of membrane lipids. An essential aspect of 2H NMR spectroscopy is that one introduces site-specific 2H labels, corresponding to the individual C−2H bonds. In this way, we obtain atomistically resolved information for noncrystalline amorphous or liquidcrystalline systems. For such studies, one must have an NMR spectrometer equipped for high radiofrequency-power studies of solid-like materials,185,186 as well as access to isotopically labeled lipids2,7,41,62,79,95,96,187−189 or proteins.23,34,35,98,190−193 Because the coupling interactions in solid-state NMR are sensitive to orientation and/or distance, their values correspond to the average structure of the system of interest. On the other hand, the relaxation parameters manifest the molecular motions that are also accessible in NMR spectroscopy.1,7,15,194 In solid-state 2H NMR of biomolecular systems, one acquires both line shape data and relaxation times as a means of comprehensively describing the structural dynamics. We can then go on to introduce concepts from statistical mechanics to explain the solid-state NMR line shapes, and correspondingly, the relaxation data are interpreted using dynamical theories in terms of intra- and intermolecular interactions. For membrane lipids, the balance of attractive and repulsive forces originates from the polar headgroups and the nonpolar acyl chains, giving an arresting polymorphism.2,7,22,113,195−199 In general, phospholipid bilayers are classified as smectic-A, lyotropic liquid crystals, as illustrated in Figure 1. The hydrophobic effect leads to sequestering of the nonpolar acyl chains within the bilayer interior, whereas the polar head groups interact with water molecules at the membrane surface. Figure 2 includes a representative set of glycerophospholipids found in cellular membranes together with cholesterol. Here, the polar

2.2. Average Properties of Membrane Lipids Encompass a Range of Time and Length Scales

The structural properties of membrane lipids in the lamellar state depend on the molecular packing and include the polar headgroup conformation, hydration water, and acyl chain configurational distribution. Hence, we must think in terms of ensembles of large numbers of molecules as well as the average behavior of the system. At the same time, one should keep in mind the underlying microscopic interactions and potentials at the single-molecule level that give rise to the observed behavior. In this context, the forces governing the nanostructures of membrane lipids are also related to their motions at equilibrium in accord with the fluctuation−dissipation theorem.211,212 One 12090


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nuclear spin angular momentum onto the axis of quantization (the magnetic field direction), with eigenstates of |m⟩ = |0⟩, |±1⟩ as given by the Hamiltonian Ĥ Z. According to quantum mechanics, the transitions between adjacent spin-energy levels are allowed, which yields two single-quantum nuclear spin transitions as we depict in Figure 3. Degeneracy of the allowed

can then regard the various material or biological properties in terms of broad classes of motions, with characteristic meansquared amplitudes and a wide range of time scales. The motions in biomolecular systems including membrane lipids can generally be considered in terms of fast or local motions (those that modulate the static coupling tensors of interest, i.e., dipolar, quadrupolar couplings, and chemical shifts), as well as slower fluctuations that modulate the residual couplings left over from the fast or local motions (so-called order fluctuations). The relatively long NMR spectral time scale (for 2H NMR < 10−5 s) implies that the molecular interactions can be averaged over mesoscopic length scales (collective behavior), ranging from the molecular size up to the bulk material dimensions.15,213,214 As a result, the solid-state 2H NMR observables for biomembranes can establish a powerful connection of atomistic-level interactions to emergent bilayer properties that approach the bulk behavior.113 It follows that we can relate the microscopic observables from 2 H NMR spectroscopy to the nano- or microstructures of the membrane lipid assemblies. Structural quantities of interest correspond to the mean interfacial area ⟨A⟩ in the lamellar state, together with the average thickness ⟨L⟩ of the bilayer hydrocarbon region and the mean aqueous distance separating the lamellae.79,111,215−217 The interfacial area plays an important role in validating molecular dynamics simulations of lipid membranes.25,111,218 For hexagonal phases (HI or HII) of amphiphiles comprising extended nanotubes, the radius of curvature Rc of the cylinders is important for the material properties.195,219 Various nanostructures result from the balance of forces acting at the level of the polar head groups and hydrocarbon chains.195,197,198,220 As a rule, deformation of the membrane film away from the equilibrium state is characterized by four material constants: (i) the surface tension γ (which is zero for a membrane bilayer at equilibrium), (ii) the area expansion modulus KA or alternatively the lateral compressibility C ≡ 1/KA, (iii) the bending rigidity KC and associated Gaussian curvature modulus κ, and (iv) the monolayer spontaneous curvature H0. The above structural quantities are fundamental to understanding the forces governing the nano- and microstructures of assemblies of membrane lipids and amphiphiles.221

Figure 3. The 2H nucleus has a spin of I = 1, with three Zeeman energy levels due to projection of the nuclear spin angular momentum onto the magnetic field direction. (a) The Zeeman Hamiltonian Ĥ Z is perturbed by the quadrupolar Hamiltonian Ĥ Q giving an unequal spacing of the nuclear spin energy levels, indicated by |m⟩, where m = 0, ±1. (b) The residual quadrupolar coupling (RQC) is designated by ΔνQ; it represents the difference in frequencies (ν±Q) of the single quantum transitions due to interaction of the 2H nuclear quadrupole moment with the electric field gradient (EFG) of the C−2H bond.

transitions in 2H NMR spectroscopy is removed by the quadrupolar coupling Ĥ Q as described below. The perturbing Hamiltonian Ĥ Q is due to interaction of the quadrupole moment of the 2H nucleus with the electric field gradient (EFG) of the C−2H bond.226 (Note that an electric quadrupole interacts with an electric field gradient analogously to interaction of an electric dipole with an electric fieldit represents the next higher term in the multipole expansion for even parity.) As a result, for each inequivalent site two spectral branches are clearly observed. Now in solid-state 2H NMR spectroscopy the experimentally observed residual quadrupolar coupling (RQC) is given by the difference in the frequencies of the spectral lines, ΔνQ ≡ ν+Q − ν−Q, as shown by Figure 4. In the next section, we give a mathematical description for calculating the 2H NMR transition frequencies and spectral line shapes. First, we start with the perturbing Hamiltonian; next, Schrödinger’s equation is solved to obtain the energy levels; last, we introduce the spectroscopic selection rules to calculate the frequencies of the spectral lines. This approach

2.3. Deuterium NMR Allows Direct Observation of Couplings Relevant to Membrane Structure and Dynamics

Besides the Zeeman interaction of the nuclear spin with the external magnetic field, various perturbations due to magnetic interactions (dipolar coupling, chemical shift) and electric interactions (quadrupolar coupling) are possible.222−224 These couplings provide a wealth of information regarding both the structure and the dynamics of biomolecular systems. As noted above, the principal values and principal axis systems of the various coupling tensors yield structural knowledge. Generally, in the simplest case of liquids the isotropic motions completely average the coupling tensor to its trace. By contrast, the tensor fluctuations give rise to spectral transitions that are related to the structural dynamics of the membrane constituents. Deuterium NMR spectroscopy affords a particularly simple illustration of how the principles of magnetic resonance are applied to molecular solids, liquid crystals, and biomembranes.79,91,94 This is because a single coupling is very large the electric quadrupolar interaction dominates over the magnetic dipolar couplings of the 2H and 1H nuclei as well as the 2H chemical shifts.225 The 2H nucleus has a spin of I = 1, and hence, there are three Zeeman energy levels due to the projection of the

Figure 4. In solid-state 2H NMR spectroscopy random samples give principal values of the coupling tensor as discontinuities in the spectral line shape. (a) Schematic illustration of the instantaneous orientation of the carbon−deuterium bond within the main magnetic field B0 and (b) its resulting powder-pattern line shape: the theoretical Pake doublet is a result of all of the orientations of the powder sample. Note that the density of states that exist in each orientation determines the intensity distribution. An overlay of the quadrupolar transitions is shown superimposed on the experimental line shape. 12091


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gives us the final result for the quadrupolar frequencies (νQ±). Additionally, our development is immediately applicable to other second-rank tensors, for instance, the magnetic dipolar coupling and the chemical shift interaction.96,222,224

integral steps. All symbols have their conventional meanings, (2) where T̂ (2) m and Vm are second-rank irreducible spherical tensors, representing the nuclear spin angular momentum operators I and the coupling tensor V, respectively. According to elementary physics, the spin angular momentum is quantized in the main external magnetic field B0 (defined by the laboratory frame, with coordinate axes X, Y, Z). With a change of basis, the corresponding irreducible tensor operators are given by222

3. MOLECULAR DISTRIBUTIONS OF LIPIDS AND PROTEINS ARE REVEALED BY SOLID-STATE DEUTERIUM NMR SPECTRAL LINE SHAPES Measurement of the 2H NMR line shapes yields knowledge of the average structure through the principal values of the coupling tensor as well as the principal axis system. Still, if we only determine the static coupling tensor using 2H NMR spectroscopy, then the method mainly provides us with structural knowledge as in X-ray scattering.227 An important aspect of solidstate 2H NMR is that information is also obtained regarding molecular motions, encompassing a range of different time scales. Through the measurement of RQCs (or residual dipolar couplings, RDCs), we can elucidate the influences of motional averaging by comparison of the static and residual coupling tensors. We can thereby obtain knowledge of the geometry, as well as types of molecular motions and their amplitudes in various membrane systems of interest.

(2)lab

T0̂

(2)lab

T±̂ 1

(2)lab

T±̂ 2

How can one study the equilibrium and dynamical properties of membrane constituents in the liquid-crystalline phase? In NMR spectroscopy, the observable quantities are due to magnetic or electrical interactions of the nuclei, which depend on their orientation and/or relative spatial position. The nuclear spin couplings are formulated by second-rank tensors, which are most usefully represented in terms of a spherical basis, as opposed to the more familiar rectangular Cartesian coordinates (see Appendix). Here, we mainly emphasize 2H NMR spectroscopy as an illustration of the application of NMR methods to liquidcrystalline systems and molecular solids.12,28,29,78,81,84−86,100−103,148,228 Because a single electric quadrupolar interaction is dominant, the influences of magnetic dipolar interactions and chemical shielding are minimal. Moreover, the quadrupolar interaction is intramolecular, so that the effects of intermolecular interactions can be largely bypassed.

m =−2

1 ̂ ̂ I±S± 2

(2b)

(2c)

3 δλ 2

(3a)

V ±(2)PAS =0 1

(3b)

1 V ±(2)PAS = − δληλ 2 2

(3c)

3.3. Irreducible Tensors Are Transformed from the Principal Axis System to Laboratory Frame by Wigner Rotation Matrices

The reader should note that the above irreducible tensors are expressed in different reference frames; the spin operators I ̂ and Ŝ correspond to the laboratory coordinate system defined by the main magnetic field, whereas the quadrupolar and dipolar couplings involve the molecule-fixed frame. In order to express the Hamiltonian in a single frame, e.g., that of the laboratory, the transformation properties of irreducible tensors under rotations are utilized.230 The irreducible components of the coupling tensor in the laboratory frame and the principal axis system are related by (see Appendix)

(2)lab

( −1)m T̂ −m V m(2)lab

=

(2a)

In the above formulas δλ ≡ Vzz denotes the largest Cartesian principal component and the asymmetry parameter ηλ ≡ (Vyy − Vxx)/Vzz characterizes the deviation of the Cartesian principal values from axial symmetry, where ηλ ∈ [0, 1]. Notably, the coupling parameters for the electric quadrupolar interaction and the magnetic dipolar interaction are the following: quadrupolar coupling (I = 1), CQ = eQ/2ℏ, where Q is the electric quadrupole moment, e is the elementary charge, δQ = eq, and ηQ; dipolar coupling, CD = −γIγSℏμ0/2π, δD = ⟨r−3 IS ⟩, and ηD = 0. For the dipolar coupling, the inverse internuclear distance cubed is averaged over faster vibrational degrees of freedom, such that −3 −1/3 ⟨rIS ⟩ can be greater than the equilibrium internuclear distance,16,73,229 and analogous considerations apply to the quadrupolar coupling.89

Here, the case of spin I = 1 systems is considered: examples include the quadrupolar interaction of the 2H nucleus having I = 1 with the electric field gradient of its chemical bond, and also the direct 1H−1H and 13C−1H magnetic dipolar interactions involving spin I = 1/2 nuclei. The coupling of a spin I = 1 system is described by the contraction or scalar product of irreducible tensors of second rank, the first of which corresponds to the nuclear spin angular momentum operators and the second representing a specific coupling mechanism.222 The Hamiltonian operator then reads

∑

1 = ∓ (IẐ S±̂ + I±̂ SẐ ) 2

V 0(2)PAS =

3.2. Coupling Hamiltonians in Solid-State NMR Spectroscopy Are Formulated Using Irreducible Tensor Operators

2

1 (3IẐ SẐ − I·̂ Ŝ) 6

in which the raising and lowering operators are I±̂ ≡ IX̂ ± iIŶ and analogously for Ŝ±. Here, I ̂ indicates the angular momentum operators for the nuclear spin in units of ℏ, i.e., I ̂ = J/̂ ℏ, where J ̂ is the angular momentum. (Note that for the electric quadrupolar coupling or the homonuclear dipolar coupling I ̂ = Ŝ, whereas for the heteronuclear magnetic dipolar coupling I ̂ ≠ Ŝ.) By contrast, the irreducible components of the coupling tensor V(2) m are diagonal in a molecule-fixed principal axis system (PAS, with coordinate axes x, y, z)222

3.1. Equilibrium Properties of Membrane Constituents Are Obtained from NMR Spectral Line Shapes

Hλ̂ = Cλ T̂ ·V = Cλ

=

(1)

where the circumflex denotes an operator, Cλ is a constant, and λ refers to a specific interaction, e.g., the quadrupolar (Q) or dipolar (D) coupling. In the case of second-rank tensors, the summation over the projection index m runs from −2 to 2 in 12092


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2

V m(2)lab =

∑

IẐ |I , m⟩ = m|I , m⟩

(2) V s(2)PASDsm (Ω PL)

(4)

s =−2

From these relations, the following results for the three energy levels of an I = 1 system are then calculated, which is left as an exercise for the reader

Here, ΩPL = (αPL, βPL, γPL) are the Euler angles which describe transformation230 of the irreducible components from the principal axis system (P) to the laboratory frame (L). The convention of Brink and Satchler230 is used for the Wigner −im′α (j) rotation matrices, D(j) dm′m(β)e−imγ, where (m′, m) m′m(Ω) = e are generalized projection indices, j is the angular momentum quantum number, Ω ≡ (α, β, γ) are the Euler angles, and d(j) m′m(β) indicates the reduced rotation matrix elements. In discussing the NMR line shapes, we mainly consider the deuterium (2H) nucleus (I = 1), which is illustrative of the principles in general. The coupling due to the direct magnetic dipolar interaction as well as the anisotropic chemical shift can be similarly considered. Expressed within the laboratory frame, according to eqs 1−4, the quadrupolar Hamiltonian takes on the form 3 δQ CQ 2

Ĥ Q = −

⎧

(2) ∑ (−1)m T̂ −(2)lab m ⎨D0m (Ω PL) m

⎩

⎫ (2) [D−(2) 2m(Ω PL) + D2m (Ω PL)]⎬ ⎭ 6

ηQ

(5)

(6)

In what follows, let us label the spin eigenfunctions ψ by the total angular momentum I and its projection m onto the magnetic field axis, in units of ℏ, which in the Dirac bra-ket notation is denoted by |I, m⟩. The coupling energies due to the Zeeman and quadrupolar interactions in the |I, m⟩ basis are given to first order by the time-independent Schrödinger equation231

(9b)

⎛ e 2qQ ⎞⎧ (2) ⎟⎨D00 E−1 = γ ℏB0 + ⎜ (Ω PL) ⎝ 4 ⎠⎩ ηQ ⎫ (2) − [D−(2) 20(Ω PL) + D20 (Ω PL)]⎬ ⎭ 6

(9c)

(10)

3.5. Quadrupolar Couplings in Deuterium NMR Spectroscopy Are Related to Molecular Geometry

According to the above development, substituting the Wigner rotation matrix elements (see Appendix) in eq 10 then yields (2) [D−(2) 3/2 sin 2 β cos(2α), where Ω are 20(Ω) + D20 (Ω)] = 230,232 In this way, we then obtain for the generalized Euler angles. quadrupolar splitting ΔνQ that

(7)

where ℏĤ = ℏĤ Z + ℏĤ Q and ℏĤ Z = −γℏB0IẐ . The eigenvalues of the angular momentum operators I2̂ and IẐ correspond to the following equations231 2

⎛ −e 2qQ ⎞⎧ (2) ⎟⎨D00 E0 = ⎜ (Ω PL) ⎝ 2 ⎠⎩ ηQ ⎫ (2) − [D−(2) 20(Ω PL) + D20 (Ω PL)]⎬ ⎭ 6

In the above equation χQ ≡ δQCQ/π = e2qQ/h is the static quadrupolar coupling constant, η Q corresponds to the asymmetry parameter of the EFG tensor, D(2) 00 (ΩPL) is a Wigner rotation matrix element, P2(cos βPL) is the second-order Legendre polynomial, and ΩPL ≡ (αPL, βPL, γPL) are the Euler angles79,232 relating the principal axis system (PAS) of the EFG tensor (P) and the laboratory frame (L). For the case of the nuclear dipolar interaction, an analogous development holds where χD ≡ δDCD/2π = (−μ0γIγSℏ/4π2)⟨r−3⟩ is the dipolar coupling constant and r refers to internuclear distance.

3.4. Schrö dinger’s Equation Entails Calculation of Quadrupolar Energies and Transition Frequencies

I ̂ |I , m⟩ = I(I + 1)|I , m⟩

(9a)

ηQ 3 ⎧ (2) νQ± = ± χQ ⎨D00 (Ω PL) − [D−(2) 20(Ω PL) 4 ⎩ 6 ⎫ (2) + D20 (Ω PL)]⎬ ⎭

The quadrupolar frequencies are thus related to the eigenvalues of the secular part of the quadrupolar Hamiltonian, which commutes with Ĥ Z and causes first-order energy level shifts involving the unperturbed basis functions.

ℏĤ |I , m⟩ = Em|I , m⟩

⎛ e 2qQ ⎞⎧ (2) ⎟⎨D00 E+1 = −γ ℏB0 + ⎜ (Ω PL) ⎩ 4 ⎝ ⎠ ηQ ⎫ (2) − [D−(2) 20(Ω PL) + D20 (Ω PL)]⎬ ⎭ 6

Here, the first term is the Zeeman energy, Em = −γℏB0m and the second term is the shift of the energy levels to first order, because of the electric quadrupolar interaction of the 2H nucleus with the electric field gradient of its chemical bond. Lastly, the energy gaps of the two allowed single-quantum (Δm = ±1) transitions between the stationary states of the 2H nucleus are given by E−1 − E0 = hν+ and E0 − E+1 = hν− in terms of the Bohr frequency condition. Here, the transition frequencies are ν± = ν±Q − ν0, where ν0 = −γB0/2π is the Larmor frequency and ν±Q are the quadrupolar frequencies. These relations then led us to the results for the first-order quadrupolar frequencies

which is in dimensions of angular frequency. The expectation values are calculated by making the conventional high-field approximation in which the shifting of the energy levels due to the coupling interaction Ĥ Q is small in relation to the much larger Zeeman interaction, given by ℏĤ Z = −γℏB0IẐ . We thus restrict ourselves to the secular (m = 0) term in eq 4, also referred to as truncating the Hamiltonian. This is equivalent to the conventional first-order perturbation theory employed in elementary quantum mechanics, in which the smaller perturbing Hamiltonian commutes with the unperturbed Hamiltonian. It follows that the truncated internal Hamiltonian in dimensions of energy becomes e 2qQ 2 2 ⎧ (2) (3IẐ − I ̂ )⎨D00 (Ω PL) ℏĤ Q = ⎩ 4 ηQ ⎫ (2) [D−(2) − 20(Ω PL) + D20 (Ω PL)]⎬ ⎭ 6

(8b)

ΔνQ ≡ νQ+ − νQ−

(8a) 12093

(11a) DOI: 10.1021/acs.chemrev.6b00619 Chem. Rev. 2017, 117, 12087−12132

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ηQ ⎤ 3 ⎡1 sin 2 βPL cos(2αPL)⎥ = χQ ⎢ (3 cos2 βPL − 1) − ⎦ 2 ⎣2 2

residual tensor are obtained from the rotation pattern according to eq 10. The reader should note that when motion is present, the various rotation angles become time dependent, as discussed below.

(11b)

Furthermore, it turns out that the static EFG tensor of the C−2H bond is nearly axially symmetric (ηQ ≈ 0), which then leads us to the simpler result 3 (2) νQ± = ± χQ D00 (Ω PL) 4

3.6. Random (Powder-Type) Samples Yield Principal Values of Coupling Tensor

The above analysis teaches us how the orientation of a single crystal within the magnetic field can generate a rotation pattern for the quadrupolar splittings. On the other hand, still we often do not have an oriented sample, but rather a polycrystalline sample with a random or spherical distribution of the various C−2H bond orientations. This would be the case if the above single crystal was ground up into a powder. A spherically averaged distribution is then obtained, giving a powder (or powder-type) spectrum, from which we can read off the principal values of the coupling tensor from the discontinuities.222 Yet a drawback is that the orientation of the principal axis system (PAS) within the crystal frame is lost, since the spectral discontinuities correspond to the laboratory axes system. For biomolecules, randomly oriented samples are often studied, giving rise to what are called powder or powder-type spectra. A random dispersion of membranes in water involves a spherical distribution (Figure 5a), which as mentioned above is analogous to grinding a single crystal to form a powder, with a random distribution of crystallite axes. In such powder-type spectra, the three principal axes of the coupling tensor, i.e., static or residual, have an arbitrary orientation relative to the external magnetic field. Notably, the Euler angles are ΩXL where X = P for a static tensor (see above) and X = I for a residual tensor due to motional preaveraging versus an internal (I) frame. A distribution of resonance intensity S(ν) is thus present on account of the orientation dependence of the quadrupolar frequencies ν±Q due to the single-quantum |1,0⟩ → |1,−1⟩ and |1,+1⟩ → |1,0⟩ transitions. The axially symmetric powder-type spectrum for a spin I = 1 system is known as a Pake doublet, and the process of deconvoluting such a powder pattern to obtain the subspectrum due to a particular orientation is called de-Pakeing, as introduced by Myer Bloom and co-workers.233 In addition a cylindrical distribution can be considered (Figure 5b) as further discussed below.

(12)

The quadrupolar splitting thus reads 3 (2) χ D00 (Ω PL) 2 Q 2 3 ⎛ 3 cos βPL − 1 ⎞ ⎟⎟ = χQ ⎜⎜ 2 ⎝ 2 ⎠

ΔνQ =

(13a) (13b)

The above formulas describe the dependence of the quadrupolar splitting on the (Euler) angles that rotate the coupling tensor from its principal axes system to the laboratory frame, as defined by the main magnetic field. For the sake of illustration, let us now imagine a static oriented sample, e.g., a single crystal in the absence of molecular motions. To gain a simple physical feel for the above results, we consider the case that ηQ ≈ 0. Take an imaginary single crystal and begin with an alignment of the magnetic field along one of the principal axes of the coupling tensor, i.e., the z axis, which corresponds to the largest principal value of the electric field gradient. The crystal can be rotated in three dimensions with respect to the laboratory frame, giving discontinuities in the NMR spectrum corresponding to the main external magnetic field aligned along each of the three principal axes of the coupling tensor.222 For example, in the case of an aliphatic C−2H bond, the largest principal value is along the bond axis. According to eqs 13a and 13b, the quadrupolar splitting then takes its maximum value of ΔνQ = (3/2)χQ, where χQ = 170 kHz for an aliphatic C−2H bond. Rotating the single crystal about the z axis of the electric field gradient (associated with the orientation of the C−2H bonds within the crystal axes system) by the Euler angle αPL (≡ ϕ in spherical polar coordinates) does nothing, because the tensor is axially symmetric (ηQ ≈ 0). Even so, for a nonaxially symmetric tensor the behavior is more complicated, as described by eqs 11a and 11b. Next, the single crystal is rotated about the new y′ axis by the Euler angle βPL (≡ θ in spherical polar coordinates). The quadrupolar splitting now becomes progressively smaller and reaches zero at the so-called “magic angle” where βPL = cos−1(1/ √3) = 54.7°, beyond which the sign of the splitting changes. When βPL = 90°, the magnetic field is now aligned within the x−y plane of the axially symmetric coupling tensor, such that ΔνQ = −(3/4)χQ. Rotation about the third Euler angle (γPL), i.e., about the z′′ axis, is again inconsequential, because of the cylindrical symmetry around the main magnetic field direction. The outcome of these rotations is described by eq 12 for an axially symmetric tensor (ηQ = 0) and more generally by eq 10 for the case of a nonaxially symmetric tensor (ηQ ≠ 0). In addition, liquid-crystalline phospholipids deposited on a planar substrate are analogousthey can be thought of as a 1-D (liquid) crystal, where rapid axial averaging of the lipids occurs about the director axis (the membrane normal). In effect, we have a residual or effective coupling tensor that is preaveraged by the motions, but otherwise, the transformations under rotations are identical. In either case, the principal axes and principal values of the static or

3.7. Spectral Inequivalence in Solid-State Deuterium NMR Spectroscopy Is Based on Differences in Molecular Mobility

A particular advantage of NMR spectroscopy is that information is gained about both the average structural properties and the molecular motions. For example, in fluid bilayer membranes, the motions of the constituent molecules are cylindrically symmetric about the bilayer normal, an axis known as the director. The transformation of the various coordinate frames under rotations is most readily handled using irreducible tensor calculus.232 Treatment of the effects of rotations is facilitated by application of a simple principle from group theory, known as closure (see Appendix). In effect, this principle states that any overall rotation is equivalent to the result of a sequence of rotations involving various intermediate coordinate frames. Let us now take into account the effects of fluctuations of the coupling tensor on the 2H NMR line shapes. The occurrence of motion leads to averaging of the static coupling tensor to yield a residual coupling tensor, whose principal values depend on both the static coupling parameters as well as the type of motion.234 When motion occurs on a time scale comparable to or less than the inverse static splitting, then eq 13a for the case of an axially symmetric static EFG tensor (η = 0) is modified, leading to 12094


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3 (2) χ ⟨D00 (Ω PL)⟩ (14) 2 Q In the above formula, the brackets indicate an average over the possible tensor orientations with respect to the laboratory frame sampled on the NMR time scale (ergodic principle of statistical mechanics). The Wigner rotation matrix element D(2) 00 (Ω) is equal to the second Legendre polynomial P2(cos β) = (1/2) (3cos2 β − 1) where β is the colatitude. At this juncture it may be helpful to discuss the following simple examples as test cases. First, we consider deuterated hexamethylbenzene (HMB-d18), which has six C2H3 groups each undergoing rapid 3-fold rotation. An experimental 2H NMR spectrum of a randomly oriented, powder-type sample of HMBd18 is shown in Figure 6a. The peak splitting of the perpendicular ΔνQ =

Figure 6. Spectral inequivalence in solid-state 2H NMR spectroscopy is based on differences in molecular mobility. Representative solid-state 2 H NMR spectra of unoriented powder samples of (a) hexamethylebenzene (HMB-d18) and (b) deuterated poly(methyl methacrylate) (PMMA-d8), i.e., plexiglass. Note that the C2H2 groups are distinguishable from C2H3 groups, which undergo rapid rotation about their 3-fold axes on the NMR time scale (cf. the text). In (b) both contributions are clearly evident.

Figure 5. Spherical and cylindrical distributions are employed in calculating NMR line shapes of membrane constituents. (a) Spherical distribution in which the orientation of the main magnetic field B0 relative to the frame for the coupling tensor V (either static or residual; see text) is described by spherical polar angles (θ, ϕ); all angles are possible. Such a distribution is applicable to aqueous dispersions of membrane lipids and biomembranes having the bilayer normal (director) randomly oriented with respect to the main magnetic field. (b) Cylindrical distribution in which the z axes of the individual coupling tensors fall on the rim of a cone making a semiangle βXD with respect to the axis of cylindrical symmetry, where n0 in turn makes an angle βDL with respect to the main magnetic field B0 and is randomly oriented. Due to the uniaxial immobilized distribution, not all angles are possible. Such a semirandom cylindrical distribution is applicable to membrane proteins and other membrane constituents with an absence of axial motion about the bilayer normal.

3)(−1/2) = −21.6 kHz. (Applying closure the averaging by the composite motion can be written as the product of averaging due to the individual motions.) Amazingly, this theoretically predicted value coincides almost perfectly to the observed experimental value (21 kHz). A second example involves solid deuterated plexiglass (poly(methyl methacrylate), PMMA-d8), in which both static C2H2 groups as well as rapidly rotating C2H3 groups are present. Figure 6b shows an experimental solid-state 2H NMR spectrum of a randomly oriented, powder-type sample of deuterated plexiglass. Here, the outer quadrupolar splitting (127 kHz) of the powder pattern is due to the C2H2 groups of the PMMA-d8. Notably, motion is essentially absent on the 2H NMR time scale, and hence, quadrupolar splittings due to the static coupling tensor are observed. The experimental 2H NMR quadrupolar splitting corresponds to the θ = 90° orientation (ΔνQ⊥), leading to (−3χQ/4) = −127.5 kHz for immobile methylene groups. Weaker shoulders are also evident, corresponding to the θ = 0° orientation with a splitting of 3χQ/2 = 255 kHz. The central component (42 kHz) in Figure 6b is due to the methyl groups, which are rapidly rotating in the solid state. Three-fold rotation about the methyl axes averages the static coupling tensor to a residual or effective coupling tensor, which is axially symmetric eff (ηeff Q = 0). The largest principal value (χQ ) is reduced by a factor of −1/3 due to the tetrahedral geometry. Hence, for the θ = 90° component of the quadrupolar spectrum, the C2H3 splitting is (−3χQ/4)(−1/3) = 42.5 kHz, with shoulders due to θ = 0° giving a splitting of (−χQ/2) = −85.0 kHz, in good agreement with the experimental spectrum in Figure 6b. Thus, we can determine the

component (θ = 90°) of the experimentally observed spectrum (21 kHz) is much less than the magnitude of the theoretically predicted static quadrupolar coupling [(−3χQ/4) = −127.5 kHz], which indicates that there are molecular motions and that the resultant quadrupolar coupling is motionally averaged. To account for the experimental value, we need to introduce various possible types of rotational motions (Figure 7) and apply the closure property of the Wigner rotational matrix elements (see Appendix). Initially, we consider the 6-fold rotation of the benzene ring, i.e., the axis of rotation is perpendicular to the plane of the benzene ring (θ = 90°) as shown in Figure 7a. Then from the second-rank Legendre polynomial, this angle reduces the splitting by D(2) 00 (0°, 90°, 0°) = −1/2. Further, if we consider the rotation of methyl groups on a tetrahedron (θ = 109.5°) as shown in Figure 7b, the static splitting reduces by a factor of D(2) 00 (0°, 109.5°, 0°) = −1/3. None of the motions by themselves explains the experimentally observed spectral splitting value. Nevertheless, if we treat the composite motion as due to both 6fold benzene ring rotation as well as methyl rotation, as shown in Figure 7c, then the principal value is reduced by (−3χQ/4)(−1/ 12095


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Figure 7. Molecular motions reduce the spectral width according to the value of the second-order Legendre polynomial for the bond orientation to the rotation axis. Motionally averaged spectra are shown for two simple types of motion: (a) rotation (6-fold) of the benzene ring and (b) rotation (3-fold) of the methyl group. If both motions are present (c) then the reduction of the spectral width is given by the product of the corresponding Legendre polynomials. Adapted with permission from ref 186. Copyright 2013 American Chemical Society.

Figure 8. Solid-state 2H NMR spectra of amantadine-d15 in DMPC bilayers as a function of temperature and amantadine/peptide molar ratio. (a) Absence of M2 channel; calculated spectrum for 303 K reproduces the 1:3 frequency ratio and 4:1 intensity ratio of the two splittings. (b) Amantadine/peptide molar ratio of 1:4. The sum spectrum reproduces the 303 K spectrum by 1:9 combinations of the lipid-bound 303 K spectrum and peptide-bound 283 K spectrum (not shown). (c) Amantadine/peptide molar ratio of 4:4. The sum spectrum uses a 1:3 combination of the M2-bound spectrum (II) and lipid-bound spectrum (I). (d) Amantadine orientation in the M2 channel. (e) One of the two possible amantadine orientations in the lipid bilayer. Adapted with permission from ref 235. Copyright 2010 Nature Publishing Group.

coupling parameters and hence the type of motion directly from the experimental 2H NMR spectrum.222 Lastly, an example of the application of solid-state 2H NMR to membrane proteins in lipid bilayers is provided by the work of Hong and co-workers235 (Figure 8). Solid-state 2H NMR spectra for the perdeuterated antiviral compound amantadine in DMPC bilayers are shown for different drug to peptide ratios. For amantadine bound to an influenza virus M2 proton channel peptide, solid-state 2H NMR spectroscopy gives exquisite details about its orientation and dynamics in DMPC bilayers. Amantadine is a rigid amphiphile with a polar amine and a hydrophobic adamantane moiety centered on a 3-fold axis, ZM. Three axial C−2H bonds are parallel to ZM, while 12 equatorial C−2H bonds are at angles of 70° or 110° (θM) to ZM. The amantadine molecules partition strongly into protein-free DMPC vesicles and exhibit 2H quadrupolar splittings of 36 and 123 kHz due to θ = 90° with a 4:1 intensity ratio at low temperature (T = 243 K) (Figure 8a). These splittings reveal fast anisotropic rotation of the molecule around ZM, which scales the couplings from the rigid limit value of 127.5 kHz by P2(cos θM), giving 42.5 kHz for the 12 equatorial bonds and 127.5 kHz for the three axial bonds. Wobbling of the ZM axis by 6° accounts for the additional motional averaging. At various stoichiometric ratios of drug to peptide (Figure 8b and 8c), the amantadine 2H NMR spectrum at 243 K does not show any additional signatures due to M2 channel. The couplings remain unchanged as temperature is increased (from 243 to 303 K) across the membrane phase transition, indicating sequestration of the drug from the lipids. The constant scaling factor (0.93) compared to pure rotation around ZM indicates that amantadine rotates rapidly around the local bilayer normal (n) in a slightly tilted orientation (∼13°) between ZM and n. However, at higher temperature (303 K) the

couplings decrease (18 and 58 kHz), while maintaining the same 1:3 frequency ratio and 4:1 intensity ratio. The decrease in quadrupolar splitting with ∼0.5 scaling factor indicates that amantadine rotates rapidly around n in addition to its own axis, with ZM tilted from n by 37° or 80° (Figure 8d). An isotropic peak grows at high temperature, indicating a small fraction (∼12% at 303 K) of amantadine either near 54.7° from the membrane normal or undergoing large-angle tumbling in the channel. Finally, a weak 18-kHz splitting is seen at 303 K that matches the lipid-only coupling at this temperature. The spectrum is consistent with a 9:1 combination of the M2bound spectrum at 283 K without the 18-kHz splitting plus the 303 K lipid-bound spectrum, indicating that 10% of the drug partitions into the bilayer at 303 K (Figure 8e). The orientational constraints plus additional distance restraints indicate that the M2 proton channel has a single high-affinity site for amantadine, with at least a 40-fold greater affinity for the channel versus an additional peripheral site. At this point, we can conclude from the above examples that solid-state 2H NMR spectroscopy is highly informative about mobility and orientation in both molecular solids and biomolecular systems. 12096


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Table 1. Quadrupolar Frequencies and Discontinuities in Powder-Type 2H NMR Spectraa,b,c

a See text for definition of the various Euler angles. bThe above expressions correspond to a generalized electric field gradient (EFG) tensor. Either a eff static EFG tensor or a residual EFG tensor preaveraged by faster motions is considered. The coupling parameters are χ ≡ χQ or χeff Q and η ≡ ηQ or ηQ . Generalized Euler angles are denoted by ΩXD ≡ ΩPD or ΩID. The correspondence between the averaged Wigner rotation matrix elements and the c (2) (2) 2 order parameters Sii is ⟨D(2) 00 (ΩXD)⟩ = S33 and ⟨D−20(ΩXD)⟩ + ⟨D20 (ΩXD)⟩ = 3/2 ⟨sin βXD cos (2αXD)⟩ = 2/3 (S11 − S22). Note that the definition of Häberlen222 is used for the asymmetry parameter η, rather than the definition of Abragam.247

3.8. Spectral Line Shapes and Quadrupolar Frequencies Manifest Spherical or Cylindrical Symmetry

aggregate (analogous to the crystal frame) is clearly lost in such powder-type samples. On the other hand, the principal values correspond to discontinuities in the line shape distribution function, and can be read off the NMR spectra directly.222 Here, we discuss both spherical and cylindrical distributions of the coupling tensor, as introduced in Figure 5. For a complete description of the Pake doublet line shape for the powder type samples (spherical symmetry) (as in case (i) of Table 1), please refer to the SI. A spherical distribution is applicable to a random dispersion of bilayer membranes, to randomly dispersed lipid hexagonal phase aggregates in water, or to membrane proteins undergoing axially symmetric rotation about the bilayer normal, such as rhodopsin.29 In this situation, let (θ, ϕ) ≡ (βXL, αXL) which correspond to the spherical polar coordinates in the laboratory frame, defined by the main magnetic field, relative to the principal axis system of the coupling tensor shown in Figure

The foregoing sections have described how the quadrupolar frequencies ν±Q are obtained as a function of orientation for some simple motional cases. One can also consider various orientational distributions, in which the motions are not sufficiently rapid to produce averaging of the frequencies due to the principal values of the coupling tensor, i.e., essentially static distributions are considered. In such powder-type spectra, the principal values of the coupling tensor (either static or residual) correspond to characteristic features or discontinuities (singularities or edges) (see Supporting Information, SI). For a spherical distribution, these represent the frequencies of the NMR transitions for the main magnetic field aligned along one of the three orthogonal principal axes of the coupling tensor. Information regarding the orientations of the principal axes relative to the frame of the 12097


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5a. This might correspond to an axially symmetric static coupling tensor (ηλ= 0) or alternatively to an asymmetric residual tensor (ηeff λ ≠ 0), which in turn is modulated by axially symmetric motion about the director or cylinder axis. One should note that in the latter case, the axial symmetry about the director is imposed on the final averaged tensor. The quadrupolar (2) frequencies are then given by ν±Q ∝ ± D(2) 00 (θ), where D00 (θ) = 2 (1/2)(3 cos θ − 1) as in eq 13a. The characteristic scaling of eq 12 accounts for the axially symmetric, powder-pattern line shape. Analogously, a cylindrical distribution as shown in Figure 5b represents immobilized membrane proteins having uniaxial symmetry about the bilayer normal, where the latter is in turn randomly oriented. Alternatively hexagonal phase (HI or HII) aggregates with the cylinder axes oriented randomly between glass plates (or DNA fibers or fibrous proteins for that matter) can be considered.

(1) D00 (0, θ ̃, 0) =

cos θ ̃ = cos θB cos θ″ − sin θB sin θ″cos(ϕ + ϕ″)

(17a)

cos θ″ = cos θ′cos θ − sin θ′ sin θ cos ϕ′

(17b)

Notably, the angles ΩNL ≡ (ϕ″, θ″, 0) describe the overall rotation of the local frame to the laboratory frame, where θ″ is the tilt of the local normal, ϕ″ is the phase factor of the two combined transformations, and θ is the tilt angle of the alignment axis to the main B0 magnetic field. At this point, we are now in a position to choose a Monte Carlo approach to simulate the line shape. In this numerical procedure, the angles ϕ′, θ′, and ϕ describe the orientational and uniaxial distributions, which are random variables, while θ and θB are fixed parameters. By generating random values for the ϕ′, θ′, and ϕ variables and choosing θ and θB we can calculate θ″ using eq 17b. Substituting this value in eq 17a allows us to determine cos θ̃, which in turn is used to calculate the corresponding quadrupolar splitting using eq 13a. Alternatively, the line shape can be calculated in closed mathematical form. For this case we need to consider mapping the probability distribution for the overall bond angle θ̃ into the frequency space of the NMR spectrum. For an axially symmetric coupling tensor, the 1-dimensional NMR spectrum is given in terms of the reduced frequencies ξ± of the two I = 1 spectral branches 1 (2) ̃ ξ± = ±D00 (θ ) = ± (3 cos2 θ ̃ − 1) 2

(18a)

corresponding to |cos θ |̃ =

1 ± 2ξ± 3

(18b)

Conservation of probability in the angular or frequency domains in terms of reduced frequency p(ξ±) then yields ̃ θ̃ p(ξ±)dξ± = p(θ )d

(19)

Effectively, we need to map the 3-D geometrical distribution into the 1-D distribution in the spectral domain. The mathematical prescription is well known and involves what is called the Jacobian (determinant). Notably, the Jacobian matrix defines a linear mapping of the function between the two domains, and it gives us the factor by which the function is expanded or shrunk, e.g., the volume. The general problem of relating the distribution in the two spaces entails introduction of the Jacobian of all of the first derivatives of the function. For a 1D NMR spectrum, in terms of reduced frequency ξ± the Jacobian matrix corresponds to a single geometrical dimension. As a result, closure (cf. Appendix) can be introduced to collapse the relevant transformations to a domain of the single variable in the case of 1D NMR spectroscopy. Effectively, we need to transform the differential element dξ± (frequency domain) corresponding to the experimental solid-state 2H NMR spectrum (what we measure) into the differential element dθ̃ in the geometrical space of the aligned sample, e.g., as described by the static uniaxial distribution (see Figure 9). In the present case, the Jacobian is just the 1 × 1 determinant J = |dθ̃/dξ±|. The probability densities in the two domains are thus related by

2

∑ ∑

(16)

Inserting the matrix elements and simplifying leads us to (socalled poor man’s closure)

It is noteworthy that rather different, uniaxial powder-pattern line shapes are found for certain membrane proteins, such as rhodopsin,35 bacteriorhodopsin,146,154,156 and also for nucleic acid fibers.28 When axial averaging is absent, the 2H NMR line shape corresponds to a two-dimensional powder distribution in the plane perpendicular to the alignment axis.28,29,32 This situation is distinct from one-dimensional liquid-crystalline aligned bilayers,236 or a full three-dimensional powder as discussed above. For a static uniaxial distribution (as seen in (ii) of Table 1), the 2H NMR line shape analysis allows one to extract information about the molecular structure, as well as the alignment disorder of the sample in terms of the appropriate distribution functions.23,28,29,32 In the case of 2H NMR spectroscopy, the transition frequencies ν+Q and ν−Q are given by eq 10. If we assume an axially symmetric coupling tensor then the asymmetry parameter ηQ = 0, and the overall rotational transformation from the principal axis system (PAS) to the laboratory frame may be represented by a series of intermediate frames. Using closure (cf. Appendix), the elements of the Wigner rotation matrix are expressed by a sequence of frame transformations as 2

(1) D0(1) m (0, θB , ϕ)Dm0 (0, θ″ , 0)

m =−1

3.9. Solid-State NMR Spectra of Uniaxially Aligned Immobile Samples Reveal Principal Axes Orientations and Principal Values

(2) D00 (Ω PL) =

∑

(2) (2) D0(2) m ′(Ω PN)Dm ′ m(Ω ND)Dm0 (Ω DL)

m ′=−2 m =−2

(15)

Here, the first set of Euler angles ΩPN = (0, θB, ϕ) characterizes the orientation of the PAS of the coupling tensor (P) relative to the local membrane normal n, where θB is the bond orientation and ϕ is the random azimuthal rotation about n (local membrane frame represented by N). The second transformation ΩND = (0, θ′, ϕ′) treats the disorder of the local membrane normal versus the average membrane normal n0 (director frame represented by D) in terms of θ′ as well as ϕ′. The last transformation describes the tilt of the average membrane normal n0 to the laboratory frame (L). To calculate the quadrupolar splitting from eq 10 we need to know cos θ̃, which is the rank-1 Wigner rotation matrix element, ̃ ̃ D(1) 00 (0, θ, 0). Here, θ is the overall orientation of the PAS with respect to the laboratory frame (C−2H bond orientation to B0). Using the transformation properties of the Wigner rotation ̃ matrices we can express D(1) 00 (0, θ, 0) in the following way 12098


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dθ ̃ |p(ξ±)| = |p(θ )̃ | dξ±

(20)

Due to the even-rank coupling, only the absolute value can be determined. Conservation of the probability in the angular and frequency domains is thus described by eq 20 as discussed above. Considering the case of a spherical probability distribution, we then immediately obtain the well-known result of a Pake doublet: |p(ξ±)| = (1/2) 3(1 ± 2ξ±) . In addition, we are now fully equipped to consider more complicated semirandom distributions, e.g., the case of a static uniaxial distribution as applied either to aligned membrane proteins154,156,237 or to aligned fibers of polymers such as DNA.28,148 For the case of a static uniaxial distribution, we are able to calculate the line shape function p(ξ±) in mathematical closed form as indicated below |p(ξ±)| =

|p(θ )̃ | |sin θ |̃ 3(1 ± 2ξ±)

(21)

When alignment disorder is present (mosaic spread), as in the case of biological samples, the distribution of the colatitude p(θ′) is assumed to be Gaussian p(θ′) =

⎛ −θ′2 ⎞ 1 exp⎜ 2 ⎟ σ 2π ⎝ 2σ ⎠

(22)

where σ is the standard deviation about the mean of ⟨θ′⟩ = 0. Integration of p(θ̃) with respect to cos θ″ then yields two physical solutions, which involve complete elliptic integrals of the first kind.29 After a lengthy calculation, the general expression for the NMR line shape is (i) if α > γ > δ > β or γ > α > β > δ |p(ξ±)| ∝

1 |cos θ |̃

∫0

π

1 ⎛ x ⎞ ⎛ −θ′2 ⎞ K ⎜ ⎟exp⎜ ⎟sin θ′dθ′ y ⎝ y ⎠ ⎝ 2σ 2 ⎠

(23a)

(ii) if γ > α > δ > β or α > γ > β > δ |p(ξ±)| ∝

1 |cos θ |̃

∫0

π

1 ⎛⎜ y ⎞⎟ ⎛ −θ′2 ⎞ K exp⎜ ⎟sin θ′dθ′ x ⎝ x ⎠ ⎝ 2σ 2 ⎠

(23b)

where x ≡ (γ − δ)(α − β) and y ≡ (α − δ)(γ − β) in which the cosines of the sum and difference angles are defined as follows: α ≡ cos(θ̃ − θB); β ≡ cos(θ̃ + θB); γ = cos (θ − θ′); and δ = cos (θ + θ′). Here, the kernel K(k) = F(π/2, k) represents a complete elliptic integral of the first kind in the normal

2

Figure 9. Solid-state H NMR spectra of uniaxially aligned immobile samples reveal principal axes orientations and principal values. (a) Retinal all-trans conformer occurs in the active form of bacteriorhodopsin (bRall‑t) (light-adapted state). (b) Illustration of the retinal chromophore of bacteriorhodopsin in membranes as investigated by solid-state 2H NMR. Geometry of the tilt experiments is presented for a static uniaxial distribution of aligned bilayers. For a given methyl group θB is the angle of the C−C2H3 bond axis to the local membrane normal n, with static rotational symmetry given by the azimuthal angle ϕ. Alignment disorder is described by the angle θ′ of n relative to the average membrane normal n0 and is likewise uniaxially distributed as characterized by ϕ′. Tilt angle θ is from n0 to the main magnetic field B0 about which there is cylindrical symmetry. Lastly, θ″ and ϕ″ are the angles for the overall n to B0 transformation. Experimental solid-state 2H NMR tilt series (c) and spectral simulations (d) are shown for bacteriorhodopsin with (1R)-1-C2H3-labeled retinal at −50 °C. Data in the left panel represent 250 000 acquisitions each and include their deviation from the fit, which is shown in the right panel. Angles between the membrane normal and the main magnetic field are shown adjacent to the spectra. Simulations take into account the spectral distortions due to the finite pulse width. Adapted with permission from ref 34. Copyright 2007 Elsevier.

π /2

dx / 1 − k 2 sin 2 x . The trigonometric form K (k) ≡ ∫ 0 reader can also verify that the above results yield the appropriate limits for large and small alignment disorder. In the limit of very small mosaic spread, the probability distribution p(ξ±) reads as ⎧ ⎡ ⎛ 1 ± 2ξ± ⎞1/2 ⎤ ⎪⎛ 1 ± 2ξ ⎞ ± ⎢ ⎨ |p(ξ±)| ∝ ⎜ ⎟ −cos(θ − θB) + ⎜ ⎟ ⎥ ⎪⎝ ⎝ 3 ⎠ ⎥⎦ 3 ⎠⎢⎣ ⎩ ⎡ ⎛ 1 ± 2ξ± ⎞1/2 ⎤⎫ ⎪ × ⎢cos(θ + θB) − ⎜ ⎟ ⎥⎬ ⎪ ⎝ ⎢⎣ 3 ⎠ ⎥⎦⎭

−1/2

(24)

We also note that eq 24 predicts three weak singularities (infinite intensity, yet integrable), amounting to reduced frequencies of ξ± = ∓1/2, due to θ = βXL = ±90° ; ξ± = ±(1/ 12099


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2)⌊3 cos2 (βDL − βXD) − 1⌋, for θ = θmin = βDL − βXD; and last ξ± = ±(1/2)⌊3 cos2 (βDL + βXD) − 1⌋, corresponding to θ = θmax = βDL + βXD. Because the various tensors are distributed with their z axes on the rim of a cone (see Figure 5), the singularities do not correspond to a single orientation of the magnetic field, as in the case of a spherical or random distribution. Inspection of Table 1 indicates that the first discontinuity (ν±Q)⊥ is independent of orientation, and corresponds to the main magnetic field perpendicular to the z axis of the coupling tensor, which is present for orientations where θmax ≥ 90°. Similarly, (ν±Q)θmin corresponds to the magnetic field aligned along the minimum value of θ for a given orientation βDL, and (ν±Q)θmax corresponds to the maximum value (see Figure 5). We can apply this method to hydrophobic peptides and membrane proteins incorporated within lipid bilayers oriented on planar supports. Such an application has been reported for bacteriorhodopsin having the retinylidene chromophore 2H labeled in aligned membrane bilayers.29,34 A representative series of 2H NMR spectra as a function of bilayer orientation for (1R)-1-C2H3-labeled bacteriorhodopsin is shown in Figure 9. Additional applications of the uniaxial line shape theory to aligned fibers of DNA in the A form and B form have been described by Nevzorov et al.28 The results for the quadrupolar frequencies corresponding to the discontinuities (singularities or edges) in the powder-type 2H NMR spectra are summarized in Table 1 for various simplified motional cases. The first two cases (i) and (ii) correspond to the absence of motion, and the results for the solid-state 2H NMR line shapes are derived above. When motion occurs, as in cases (iii) and (iv), the quadrupolar frequencies are averaged to ⟨ν±Q⟩, yielding a reduction in the quadrupolar splitting ΔνQ of the two spectral branches. As discussed in further detail below, the presence of axial rotation about a preferred director is assumed, and so the quadrupolar frequencies transform as ±D(2) 00 (ΩDL) or (2) ±D00 (ΩCL), respectively. It follows that axially symmetric powder-type 2H NMR spectra are obtained, in which the spectral discontinuities ⟨ν±Q⟩⊥ and ⟨ν±Q⟩∥ are appropriately scaled.

motions bring a time dependence into the interaction tensor as expressed with respect to the laboratory system. One can then divide the transformation from the PAS to the laboratory frame into a time-dependent part and a time-independent part. For liquid-crystalline systems, often the motions are uniaxial about a preferred director axis (see above). The time dependence is then entirely expressed with respect to the director frame, whereas the static transformation from the director frame to the laboratory axis system is analogous to the case of a single crystal, as discussed above. Analytically, this can be achieved by using the closure property of the rotational transformations (cf. Appendix). Initially the PAS can be rotated to the alignment frame, viz., the director axis n0, and then the director frame is further rotated to the laboratory frame. The first transformation carries the time dependence of the fluctuating quadrupolar interactions, i.e., the electric field gradient, whereas the transformation from the director to the laboratory frame is time independent. For uniaxial liquid crystals, the average of the time-dependent part of the rotational transformation is called the order parameter, as further discussed below. As a specific example, in the case of acyl chain-deuterated lipid bilayers, the RQCs measured with solid-state 2H NMR spectroscopy correspond to the orientational fluctuations of C−2H bond vectors (PAS), which are time dependent. In this case, the average of the time-dependent transformation gives the orientational order parameters SCD of the individual C−2Hlabeled groups, which reveal a profile as a function of acyl position. For an axially symmetric, residual coupling interaction, the segmental order parameter79 of the C−2H labeled group is defined by SCD = ⟨P2(cos βPD)⟩ (described in more detail in the next section), where P2 represents the second-order Legendre polynomial, and βPD(t) is the time-dependent angle between the C−2H bond axis and the director (viz., the membrane normal). The angular brackets denote an average over all motions that are faster than the inverse anisotropy of the static quadrupolar coupling, (9χQ/8)−1 ≈ 5 × 10−6 s. The segmental order parameters quantify the average structure of the system, and manifest the equilibrium properties in terms of the configurational statistics of the molecules.220 Even so, the reader should recall (see above) that although the static coupling tensor for a C−2H bond is axially symmetric, because of the effects of motional averaging the residual coupling tensor can be biaxial, e.g., as in the case of low-temperature lipid phases.234

4. RESIDUAL QUADRUPOLAR COUPLINGS CHARACTERIZE NANOSTRUCTURES OF MEMBRANE LIQUID CRYSTALS Returning back to the case of membranes in the liquid-crystalline state, the reader should recall that measurements of randomly oriented samples give only the principal values of the coupling tensor. On the other hand, studies of oriented samples are needed to identity the principal axes orientations. Such measurements of liquid-crystalline phospholipid bilayers41,187 establish that the unique axis of the residual EFG tensor due to the motional averaging is the normal to the bilayer surface, known as the director n0. Indeed, such cylindrical symmetry of the bilayer properties due to axial averaging about the director is a characteristic signature of the phase and is pertinent to the collective behavior. Motions of the various segments of the flexible lipid molecules about the average director are involved, so that a broad distribution of the correlation times is expected.

4.2. Multiple Coordinate Transformations Describe a Hierarchy of Motions by the Closure Property of the Rotation Group

Let us next introduce a more general formulation that is capable of dealing with the various possible types of motions in a more unified way. In the liquid-crystalline state, a hierarchy of motions of the molecules occurs within the bilayer, as manifested by the frequency dependence of the 1H, 2H, and 13C nuclear relaxation rates.17,238−244 Evidently, one can expect contributions from segmental motions as well as molecular motions and possibly collective fluctuations of the bilayer itself. The general types of motional models that can be considered are illustrated by Figure 10. For segmental motions, the C−2H bond vector reorientation is described by the angles ΩPD(t) between the principal axis (P) of the coupling tensor and the director (D). Notably, the remaining transformations are collapsed. Analogously, molecular motions are considered in terms of the angles ΩMD(t) between the molecular frame (M) and the director (D). Finally, collective motions are formulated in terms of fluctuations of an

4.1. Orientational Order Parameters of Biomolecules Are Obtained from Residual Quadrupolar Couplings

For either oriented or powder-type solids, the observed principal values correspond to the orientation of the principal axis system with respect to the laboratory frame., i.e., it assumes a complete rotation of the PAS to the laboratory frame. For oriented crystals, this gives the principal values as well as the principal axis system. For partially ordered systems (e.g., liquid crystals), the molecular 12100


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components transform under rotation of the director axis like the single-crystal example described above. Moreover, the case of a lipid hexagonal phase (HI or HII) can be considered through a second application of closure.219 There is now another rotation of coordinates, which can be imagined to involve curling of a hypothetical lipid monolayer to form cylinders, which are hexagonally packed in the lipid assembly. Applying closure twice we thus obtain (2) D00 (Ω PL) =

∑ ∑ D0(2)n (ΩPD)Dnn(2)′ (ΩDC)Dn(2)′ 0(ΩCL) n

n′

(29)

Assuming there is axial symmetry about both the normal to the surface of the cylinder, corresponding to the director axis of a planar bilayer, and also the long axis of the cylinder leads us to

Figure 10. Multiple coordinate transformations are treated using the closure property of the rotation group. Illustration of coordinate frame transformations of the coupling tensor in a lipid bilayer with 2H-labeled acyl segments. The overall rotation from the principal axis system (PAS) to the laboratory frame (defined by the magnetic field B0) is given by the ΩPL Euler angles. Using the geometry of the system the transformation is decomposed into various intermediate rotations shown by Ω with subscripts describing multiple reference frames. The z axes of the various coordinate frames are indicated. Symbols are defined as follows: P, principal axis frame; I, internal or intermediate segmental frame; M, molecular interaction frame; N, local director frame; D, bilayer director frame; and L, laboratory frame.

(2) (2) (2) (2) ⟨D00 (Ω PL)⟩ = ⟨D00 (Ω PD)⟩D00 (Ω DC)D00 (ΩCL)

In general, for a lipid hexagonal phase (normal HI or reverse HII) one can assume that βDC = 90° in which case D(2) 00 (ΩDC) = −1/2. Substitution into eq 27 then gives the residual quadrupolar splitting, which reads ⎛ 3 cos2 β − 1 ⎞ 3 CL ⎟⎟ ΔνQ = − χQ SCD⎜⎜ 4 2 ⎝ ⎠

∑ D0(2)n (ΩPD)Dn(2)0 (ΩDL) n

(25)

Because of the cylindrical symmetry of the motions about the director, all Wigner rotation matrix elements containing the index n are averaged to zero. (The reader should recall that Euler’s formula states that e±iϕ = cos ϕ ± i sin ϕ, where in the present case ϕ ≡ nαDL and all angles αDL are equally probable.) This gives us the result that (2) ⟨D00 (Ω PL)⟩

=

(2) (2) ⟨D00 (Ω PD)⟩D00 (Ω DL)

(31)

Here, SCD is the order parameter relative to the normal to the cylinder surface, and is analogous to that in the lamellar phase.219 Assuming that SCD is the same, eq 31 shows that the quadrupolar splitting ΔνQ is reversed in sign, and reduced in absolute magnitude by a factor of one-half versus the lamellar phase, see eq 27. Note that for solid-state 2H NMR spectroscopy, only the absolute value of the splitting |ΔνQ| is measured. However, in the case of I = 1/2 nuclei such as 31P, this reversal in sign and reduction by one-half is seen in the chemical shift anisotropy Δσ directly.219 A similar formalism can be applied to the direct homonuclear dipolar interaction for the case of 1H NMR and the direct heteronuclear 13C−1H dipolar interaction for 13C NMR spectroscopy. As we discuss in the Appendix, one can use the closure property of the rotation group74,230 to express the Wigner rotation matrix elements for the overall transformation D(2) sm (ΩPL) in terms of the various intermediate frames (Figure 10). Given such a general formulation, one can write that245

instantaneous director (N) relative to the average director (D) and are represented by the ΩND(t) angles. Now for the case of a lipid bilayer, the overall rotation from the principal axis system of the coupling tensor to the laboratory frame, described by the ΩPL Euler angles, amounts to the effect of at least two consecutive rotations. The first, with ΩPD Euler angles, is due to the time-dependent rotation from the principal axis system to the director frame (whose z axis is the bilayer normal), while the second, with Euler angles ΩDL, represents the time-independent rotation from the director to the laboratory frame (2) D00 (Ω PL) =

(30)

(2) Dsm (Ω PL ; t ) =

∑ ∑ ∑ ∑ Dsr(2)(ΩPI; t )Drq(2)(Ω IM ; t ) r

q

p

n

(2) (2) × Dqp (Ω MN ; t )D(2) pn (Ω ND; t )Dnm (Ω DL)

(26)

(32)

Lastly, substituting eq 26 into eq 14 leads us to the final expression for the quadrupolar splitting

where all summations run from −2 to 2 in integral steps. In eq 32, the Euler angles ΩPI specify the transformation of the irreducible components from the principal axis system (P) of the static coupling tensor to the internal (I) motional frame, the angles ΩIM correspond to rotation from the internal frame to the molecular (M) axis system, the angles ΩMN indicate transformation from the molecular frame to the instantaneous director n(t), an axis of local cylindrical symmetry, ΩND correspond to rotation from the instantaneous director to the average n0 director, that is, the macroscopic bilayer normal, and last the angles ΩDL describe the fixed transformation from the average director frame (D) to the laboratory frame (L) of the static external magnetic field. In the ultimate transformation, the orientation of the magnetic field within the frame of the bilayer is denoted by the spherical polar

⎛ 3 cos2 β − 1 ⎞ 3 DL ⎟⎟ ΔνQ = χQ SCD⎜⎜ 2 2 ⎝ ⎠

(27)

Here, the C− H segmental order parameter SCD is defined as91 1 (2) SCD = ⟨D00 (Ω PD)⟩ = ⟨3 cos2 βPD − 1⟩ (28) 2 The effect of the motional averaging is thus to replace the static quadrupolar coupling with a residual (or motionally averaged) quadrupolar coupling. The largest principal value is projected onto the director (z axis), and the residual transverse 2

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Figure 11. Solid-state NMR spectroscopy reveals phospholipid membrane dynamics and structure over a range of time scales. The energy landscape of phospholipid mobility is characterized by a hierarchy of dynamic processes including segmental fluctuations, molecular diffusion, and viscoelastic membrane deformation. Orientational fluctuations correspond to geometry of interactions via Euler angles Ω and by correlation times τc of the motions. (a) Principal axis system of 13C−1H or C−2H bonds fluctuates due to motions of the internal segmental frame (I) with respect to the membrane director axis (D). (b) Diffusive phospholipid motions are described by anisotropic reorientation of the molecule-fixed frame (M) with respect to the membrane director axis (D). (c) The liquid-crystalline bilayer lends itself to propagation of thermally excited quasi-periodic fluctuations in membrane curvature expressed by motion of the local membrane normal (N) relative to the membrane director axis (D). Appropriate range of time scales of various complementary biophysical methods is indicated at the bottom of the figure. Figure adapted with permission from ref 7. Copyright 2011 Elsevier.

angles (ϕ, θ) ≡ (αDL, βDL). The reader should note that the Euler angles describing the various rotations in Figure 10 can be either time dependent or fixed; by contrast, the ΩDL transformation is stationary. By induction, the expansion in eq 32 can be further generalized to include an arbitrary number of coordinate transformations, or closure enables those transformations not specified in a particular motional model to be collapsed.74,245 Notably, for the hexagonal phases (HI or HII), closure can be used to further expand the transformation involving the Euler angles ΩDL into an additional summation Σn′ due to the rotation from the normal to the cylinder surface (D) to the cylinder long axis (C) with Euler angles ΩDC, followed by rotation from the cylinder axis to the magnetic field using ΩCL Euler angles for the transformation to the laboratory (L) frame. The reader should also note the following: As originally put forth by Brown and Söderman,234 the static coupling tensor can be preaveraged by rapid fluctuations to yield a residual coupling tensor that transforms in an analogous manner. Furthermore, even if the static tensor is axially symmetric (ηQ = 0), the residual or effective tensor can be nonaxially symmetric (ηeff Q ≠ 0). The possibility of such preaveraging accounts for the introduction of an internal coordinate frame in Figure 10. Within this frame, the residual coupling tensor is diagonal in a Cartesian basis, and the PAS of the static tensor is related to the residual tensor as designated by the ΩPI Euler angles. The residual coupling parameters are then given by234,245 χλeff ≡ ⟨χλ ⟩ ≡

(2) χλ ⟨D00 (Ω PI)⟩

where λ = Q in the case of the quadrupolar interaction. One can thus consider in a unified way various motional models, in which a static tensor is averaged by local motions, or alternatively a residual tensor (preaveraged by faster motions) is further averaged by slower molecular motions or collective fluctuations of the macromolecular assembly. The usefulness of this isomorphism becomes particularly evident regarding the treatment of motional models for NMR relaxation.194,236 4.3. Spectral Frequencies of Aligned Samples with Axial Rotation about the Director Yield Residual Quadrupolar Couplings

We shall now come back to the example of solid-state 2H NMR spectroscopy involving I = 1 nuclei. Using closure, eq 32, some general expressions can be derived for various limiting motional cases, which are pertinent to assemblies of amphiphiles in aqueous dispersions, such as membrane bilayers.79,91,94,96,97 To generalize the results to either a static or a residual coupling tensor, we slightly modify our notation and introduce the eff following definitions: χ ≡ χQ or χeff Q ; η ≡ ηQ or ηQ ; and ΩXD ≡ ΩPD or ΩID. A summary of expressions for the angularly dependent quadrupolar frequencies ν±Q, derived analytically in closed form, are included in Table 1. The following cases are included as discussed above. (i) A spherical distribution in which the principal axis system of the coupling tensor is immobile, representing cases where motion is largely absent. (ii) A cylindrical distribution with an immobile principal axis system of the coupling tensor, corresponding to membrane proteins with no axial rotation about the director axis. Additionally the effects of motion are considered by the following cases. (iii) A spherical distribution having axial rotation of the coupling tensor about the director axis, representing membrane bilayers in the fluid, lamellar phase. (iv) A spherical distribution in which there is axial rotation of the coupling tensor about the director axis, plus additional rotation about a cylinder axis, as in the case of hexagonal phases (HI or HII) of lipids or surfactants in water. Note that in Table 1, the third (iii) and fourth (iv) of the above motional cases involve axial rotation about the director on the 2H NMR time scale. The averaged values of the Wigner rotation matrix elements thus refer to a single-coordinate system, viz., that of the director frame. In such cases, the averaged Wigner rotation

(33a) (33b)

and ηλeff = − =−

6 ⟨D0(2) ± 2 (Ω PI)⟩ (2) ⟨D00 (Ω PI)⟩ 2 3 ⟨sin βPI cos(2γPI)⟩ (2) 2 ⟨D00 (Ω PI)⟩

(34a) (34b)

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matrix elements are related to the order matrix S introduced by Saupe246 and applied extensively by Joachim Seelig91 to lipid systems, having the diagonal elements Sii = (1/2)⟨3 cos2 βi − 1⟩, where i = x, y, or z. Here, the angles βi are for the direction cosines between the Cartesian axes of the coupling tensor and the director. Because of the orthonormality of the direction cosines, it follows that trace S ≡ ΣiSii = 0, and thus, only two order parameters are independent. The correspondence between the averaged Wigner rotation matrix elements in Table 1 and the order parameters Sii in a Cartesian basis is summarized as (2) ⟨D00 (Ω XD)⟩ = S33

⟨D−(2) 20(Ω XD)⟩ = =

+

Mk =

∞

∫0 S(ν)dν

(36)

where only one-half of the symmetric spectrum is considered, thereby allowing consideration of both the even and the odd moments. This leads to the following expressions for the first and second moments, which are given by94,97,216 π χ ⟨|SCD|⟩ M1 = (37a) 3 Q

(35a)

M2 =

(2) ⟨D20 (Ω XD)⟩

3 ⟨sin 2 βXD cos(2αXD)⟩ 2 2 (S11 − S22) 3

∫0 ν kS(ν)dν

9π 2 2 χ ⟨|SCD|2 ⟩ 20 Q

(37b)

In addition, the width of the distribution of order parameters about the mean can be included in terms of the parameter Δ2, defined as Δ2 =

(35b)

⟨|SCD|2 ⟩ − ⟨|SCD|⟩2 ⟨|SCD|⟩2

=

20 M 2 −1 27 M12

(38)

Here, Δ 2 is the fractional mean-squared width, which corresponds to the variance of the distribution of order parameters. The above formulas establish how the moments of the 2H NMR line shapes manifest the distribution of quadrupolar splittings, i.e., the order parameters. It is also possible to calculate the moments corresponding to the various motional models considered here. The distribution of quadrupolar splittings is characterized by the 2H NMR spectral moments and related to the equilibrium or average properties of the membrane lipids.

In the above formulas, we use the definition of Ulrich Häberlen222 for the asymmetry parameter η rather than the convention of Abragam.247 Perhaps at this point, some brief mention should also be made of the rates of the fluctuations that produce the motional averaging, i.e., the so-called the 2H NMR time scale. As remarked above, one of the significant advantages of NMR spectroscopy is that information is obtained about both the average (equilibrium) and the dynamical properties of the membrane constituents.248,249 For averaging to occur, the motions must be faster than the inverse anisotropy of the static coupling interaction.250,251 In 2H NMR spectroscopy, the two spectral branches are due to the single-quantum transitions, and the static anisotropy is 9χQ/8 = 191 kHz, corresponding to time scales less than about 10−5 s. Considering the static coupling, the slowest motion that could affect the averaging process must be less than the order of 10−5 s, which gives an upper limit for the time scale. Even so, for complex systems like lipid membranes, one must interpret the observed line shapes with regard to a hierarchy of motional time scales, which affects the motional averaging (Figure 11). Relatively fast motions of restricted amplitude can average the static coupling, leaving residual couplings that are further averaged by the next slower motions in the hierarchy. Successively averaging the coupling interaction in this way manifests the hierarchical energy landscape of the motions, whereby the residual anisotropy for the interaction is progressively diminished on down to the final observed value. According to this picture, for smaller anisotropies, significantly slower motions can affect the averaging of the inhomogeneously broadened NMR line shapes.

5. SEGMENTAL ORDER PARAMETERS ARE RELATED TO ENERGY LANDSCAPES FOR LIQUID-CRYSTALLINE MEMBRANE LIPIDS With the above development in hand we can now ask the following: What can be learned about the structural properties of membrane lipids in the liquid-disordered state from solid-state 2 H NMR spectroscopy? In liquid-crystalline membranes, comparison of the experimental RQCs to the static coupling constant (see above) reveals that the molecules are considerably disordered, e.g., due to entanglement of the acyl chains. The RQCs manifest a hierarchy of motions that contribute to the dynamical roughness of the energy landscape of the bilayer. In principle, this disorder can be due to trans−gauche rotational isomerizations of the acyl chains, together with molecular motions and whole-bilayer collective excitations as described by the equipartition principle. 5.1. Hierarchy of Motions Governs Dynamical Structures of Membrane Lipid Bilayers

Knowledge of the dynamics of the constituent molecules is clearly essential with regard to their physicochemical properties. The energy landscape principle suggests that a complex hierarchy of motions exists in biomolecular systems,15,22,77 including both membrane lipids and membrane proteins. We thus consider (i) segmental motions due to isomerizations of the flexible molecules,14 (ii) slower rotations of the highly entangled lipids,15,111 and (iii) collective deformations spanning a broad range which influence the entire assembly.15,252,253 The above motions may be coupled to lateral self-diffusion of the molecules within the film or membrane bilayer. Such motional processes cover a wide range of time scales as depicted in Figure 11. For biomembranes, the equilibrium and dynamical bilayer properties may influence their characteristic functions via lipid−protein

4.4. Moments of Deuterium NMR Line Shapes Quantify Distribution of Orientational Order Parameters for Nonaligned Samples

Another useful approach in the case of powder-type spectra involves calculation of the moments of the 2H NMR line shapes, e.g., for lipids with perdeuterated acyl chains as described by Bloom et al.77 The moments provide a general means of characterizing a distribution function, in the present case corresponding to the various quadrupolar splittings, or alternatively the segmental order parameters. It follows that the theory of distributions can be applied to further interpret the NMR line shapes. The kth spectral moment is defined as 12103


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Table 2. Summary of Quadrupolar Frequencies for Models Involving Axial Rotation about Directora

a

See text for definition of the various Euler angles. bThe expression corresponds to a generalized EFG tensor. Either a static EFG tensor or a residual eff EFG tensor preaveraged by faster motions is considered. The coupling parameters are χ = χQ or χeff Q and η = χQ or ηQ . Generalized Euler angles are denoted by ΩXM ≡ ΩPM or ΩIM.

interactions.112,117,198,221,254−258 One is thus challenged to identify the predominant motional contributions within the various regimes and to characterize their energetic parameters. Here, we describe how 2H NMR spectroscopy yields directly the model-free observables, i.e., the segmental order parameter profiles, which are related to the bilayer average structural properties. We then go on to formulate how the atomistically resolved 2H NMR quadrupolar couplings and derived segmental order parameters correspond to structural properties of the membrane bilayer, including the balance of forces that govern self-assembly of the amphiphiles.

The orientation of the internal frame relative to the diffusion axis system can also be bypassed,245 as in case (iii) of Table 1. What is more, according to the results in Table 2, the averaged quadrupolar frequencies ⟨ν±Q⟩ for the segmental, molecular, and collective motional models transform as D(2) 00 (ΩDL), thus yielding axially symmetric solid-state 2H NMR spectra. Clearly, line shape studies cannot be used to distinguish among such models involving rotational symmetry about a preferred director; rather, it is desirable to carry out NMR relaxation studies.15 Magnetic relaxation in lyotropic liquid crystals occurs due to a hierarchy of dynamic processes that include fast segmental motions due to rotational isomerizations of the flexible surfactant or lipid molecules, slower effective rotations of the entangled molecules, and finally collective bilayer deformations, which span a broad frequency range (Figure 11). By combining various relaxation mechanisms, for example, longitudinal (R1Z and R1Q), CPMG ), and relaxation in the rotating frame transverse (RQE 2 , R2 (R1ρ), one can probe the motional time scales over a wide range.2,7 In all of these relaxation processes, the equilibrium magnetization is perturbed by external radiofrequency pulses, and the magnetization equilibration is observed as a function of time to calculate the relaxation rates. In principle, this approach offers a powerful means of dissecting the types of motions that lead to averaging of the coupling tensors in NMR spectroscopy, including the amplitudes and time scales of the fluctuations.

5.2. Residual Quadrupolar Couplings Characterize Motional Averaging in Liquid-Crystalline Assemblies

We are now able to treat more specifically the various motional models, which involve axial rotation about the director in the liquid-crystalline state, as in case (iii) of Table 1. With regard to Lα phase bilayers, three broad classes of motions can give rise to a reduction of the quadrupolar frequencies from their maximal values.15 Several points are noteworthy, among which are the following. First, in the case of segmental motions χ and η refer to the static coupling tensor, where ΩXD = ΩPD. The order parameter for the frame undergoing the motion is given by ⟨D(2) 00 (ΩPD)⟩ in the case of segmental motions. Second, in the case of molecular motions, there are two possibilities: for rigid molecules such as cholesterol, the static coupling tensor is modulated as described above, whereas for flexible phospholipids χeff and ηeff correspond to the residual coupling tensor with ΩXD = ΩID → ΩMD for the Euler angles. The order parameters are given (2) by ⟨D(2) 00 (ΩPD)⟩ for rigid molecules and by ⟨D00 (ΩMD)⟩ for flexible molecules. Finally, for collective motions χeff represents the residual coupling tensor, where ηeff = 0 and ΩXD = ΩID → ΩND, i.e., the internal frame (I) corresponds to an instantaneous director axis (N). The order parameter for the motional frame is given by ⟨D(2) 00 (ΩND)⟩ for collective motions. Additionally, closure can be used to include the orientation of the internal frame relative to the frame undergoing the motion, e.g., the segmental (P) or molecular (M) frame or the instantaneous director (N) in the case of collective motions. This approach is useful for analysis of relaxation due to rotational diffusion, and enables the isomorphism (equivalence) of the results for the segmental and molecular models to be easily seen.

5.3. Phospholipids in the Liquid-Crystalline State Yield a Distribution of Residual Quadrupolar Couplings

In solid-state 2H NMR spectra of phospholipid bilayers, the experimental RQCs of the C−2H-labeled segments are observed directly. Studies of phospholipids with specifically 2H-labeled headgroups41,91,259−263 have been used as sensors of electrical potential at the membrane surface.259,261,262 Moreover, solidstate 2H NMR studies have been conducted of phospholipids in which the acyl chains are specifically 2H labeled at the various methylene segments2,7,14,91,264 or in the case of saturated fatty acyl groups by uniform perdeuteration.22,94,265−272 For phospholipids with 2H-labeled saturated acyl chains, the RQCs vary substantially, corresponding to a profile of the segmental order (i) parameters |SCD | as a function of chain position (index 7,41,178,267,272 The largest values represent the segments close i). to the lipid polar head groups, with a progressive decrease along 12104


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the lipid acyl chains.74,91,94 Motionally induced nonequivalence arises from the tethering of the chains to the aqueous interface, together with the effects of the bilayer packing. Knowing the PAS for motional averaging as outlined above, studies of randomly oriented multilamellar lipid systems can be very fruitfully employed. One such application involves dePakeing, a procedure originated by Myer Bloom and his coworkers.233,265 By numerical inversion, one can deconvolute or de-Pake the powder-type spectra of random multilamellar dispersions to acquire more highly resolved subspectra, corresponding to the θ ≡ βDL = 0° orientation of the bilayer director n 0 relative to the main external magnetic field.94,233,268−270 A substantial increase in resolution is obtained, because the spectra are spread out to the fullest possible extent in accord with eq 27. Moreover, they represent a single bilayer orientation. Upon de-Pakeing, the 2H NMR spectrum is essentially the same as when a multilamellar sample is aligned with n0 parallel to the main magnetic field B0.233,265,268,273,274 This approach constitutes a valuable alternative to the investigation of macroscopically aligned samples (see below). It is particularly suited for phospholipids with perdeuterated acyl chains in the liquid crystalline state, where many of the individually C−2H-labeled segments are motionally inequivalent and can be atomistically resolved. As one example, in Figure 12 we show illustrative powder-type and de-Paked 2H NMR spectra for mixtures of an acyl chainperdeuterated phospholipid, 1,2-diperdeuteriomyristoyl-sn-glycero-3-phosphocholine, abbreviated as DMPC-d54, with cholesterol in the liquid-ordered phase. After de-Pakeing, the majority of the acyl C2H2 segments as well as the C2H3 groups give resolvable signals. Moreover, the RQCs become progressively greater with increasing mole fraction of cholesterol. In this way, 2 H NMR studies of acyl chain perdeuterated phospholipids58,60,66,68,77,94,216,266,275−283 yield similar information as in the case of specifically deuterated phospholipids, which provide the reference NMR spectral assignments.14,17,264,284,285 A much larger variety of systems can thus be investigated than would be possible if specific deuteration was necessary in each case.

Figure 12. Phospholipids in the liquid-crystalline state yield a distribution of residual quadrupolar couplings. Representative solidstate 2H NMR spectra: (a) DMPC-d54 in the ld phase and (b)−(d) DMPC-d54 containing various mole fractions of cholesterol in the liquidordered (lo) phase. Data were acquired at a magnetic field strength of 11.7 T (76.8 MHz) at T = 44 °C.212 Powder-type spectra of randomly oriented multilamellar dispersions were numerically inverted (dePaked) to yield subspectra corresponding to the θ = 0° orientation. Note that a distribution of RQCs is evident, corresponding to the various C2H2 and C2H3 groups, with a progressive increase due to incorporation of cholesterol. Adapted with permission from ref 286. Copyright 2004 American Chemical Society.

5.4. Dynamical Structures of Fluid Membranes Are Manifested by Segmental Order Parameters

specifically 2H-labeled reference lipids,91,264 essentially complete order profiles can be obtained.286 Evidently in NMR spectroscopy, the second-rank tensors associated with the quadrupolar, dipolar, or chemical shift interactions are related to the equilibrium or averaged properties of the system.41,79 The correspondence of the motionally averaged tensors to equilibrium properties gives a connection to the results of other biophysical methods, including small-angle X-ray scattering (SAXS) and small-angle neutron scattering (SANS),289−291 as well as Fourier-transform infrared (FTIR) studies.292−295 Let us confine our attention at present to the lipid constituents of the membrane bilayer. With regard to the acyl chain region of the bilayer, such equilibrium properties include the thickness, mean cross-sectional or interfacial area per molecule, isobaric coefficient of thermal expansion, and isothermal compressibility. Additional properties are related to the curvature free energy of a monolayer film, including the spontaneous curvature and bending rigidity.296−299 The latter quantities account for the existence of nonlamellar phases, e.g., hexagonal (HI and HII) and cubic phases.197,303−307 We have discussed elsewhere how the properties of nonlamellar forming lipids may be important with regard to the functions carried out by membrane proteins, such as visual rhodopsin.198

According to eq 27 the observed residual quadrupolar coupling ΔνQ is directly related to the segmental order parameter SCD of the corresponding C−2H bond. Upon de-Pakeing, the 2H NMR spectra yield well-resolved RQCs, whose intensities are related to the number of contributing groups. The intensities can be quantified by fitting the de-Paked 2H NMR spectra to Gaussian basis functions, as indicated in Figure 13. Beginning with the smallest RQCs, due to the C2H3 groups in the bilayer center, onedimensional 2H NMR spectra allow a partial walk among the C2H2 segments, ending up with the unresolved couplings from the top part of the acyl chain closest to the aqueous interface. The RQCs enable determination of the order parameters of the individual C−2H bonds by using eq 27. This is summarized in Figure 14, where the RQCs are expressed in terms of absolute segmental order parameters |S(i) CD|, which are plotted as a function of the acyl position (index i) for DMPC-d54 alone and for DMPC-d54 in the presence of cholesterol (1:1). Moreover, comparing Figure 12a to Figure 12d, we see that the RQCs increase appreciably upon addition of cholesterol, leading to greater spectral resolution. From the relative intensities of the various RQCs, and knowing the assignments from studies of 12105


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Figure 14. Bilayer dimensions are given by the interfacial membrane area per lipid ⟨A⟩ and the volumetric thickness DC (cf. the text). (a) Structural parameters ⟨A⟩ and DC are calculated from the acyl chain volume VC and moments ⟨cos β⟩ and ⟨cos2 β⟩, obtained from the order parameter plateau, where β is the angle between the virtual bond connecting two neighboring carbons of the ith segment and the bilayer normal.220 (b) Profiles of segmental order parameters |S(i) CD| as a function of acyl chain position212 (i) for DMPC-d54 and DMPC-d54/cholesterol (1:1) at T = 44 °C. Filled and open symbols refer to inequivalent sn-1 and sn-2 acyl chains, respectively. Reference order parameters are indicated for limiting cases of an oil drop model with SCD = 0, a crankshaft model with kink/jog configurations having ⟨SCD⟩ = −1/3 or only kinks with ⟨SCD⟩ = −1/4, and an all-trans rotating chain with SCD = −1/2. Adapted with permission from ref 135. Copyright 1999 American Institute of Physics.

Figure 13. Dynamic structures of fluid membranes are manifested by segmental order parameters. Example of fitting solid-state 2H NMR spectra (de-Paked) to Gaussian basis functions for membrane lipids in the liquid-crystalline state: (a) DMPC-d54 in the liquid-disordered (ld) phase and (b) lanosterol/DMPC-d54 in the liquid-ordered phase, and (c) cholesterol/DMPC-d54 (1:1) in the liquid-ordered (lo) phase, both at T = 44 °C.286 Adapted with permission from ref 286. Copyright 2004 American Chemical Society.

Now in discussing the relationship of NMR observables to equilibrium and dynamical properties of lipid bilayers, solid-state 2 H NMR spectroscopy provides us with a general approach.91,94,96 The electric quadrupolar interaction is by far the largest coupling, and dominates over the chemical shift or direct homo- or heteronuclear dipolar interactions. The importance of 2 H NMR is that one can study the properties of an isolated spin I = 1 system without the added complications of other interactions of comparable magnitude. In addition, as noted above, the quadrupolar interaction is isomorphous to the direct (throughspace) dipolar coupling, so that the formalism can be readily extended to 1H and 13C nuclei. For significant motional averaging to take place,267,308 the rate of the orientational fluctuations of the acyl segments must be greater than about 105 s−1. Notably, if all segments were equivalent then a single quadrupolar splitting would be observed. Yet this is clearly not the caserather a family of splittings is found, which indicate variations in the degree of motional averaging along the acyl chains (Figure 12). Despite the fact that the static quadrupolar coupling is everywhere the same, the residual quadrupolar couplings of the individual segments are noticeably inequivalent. It is truly remarkable that well-defined residual quadrupolar interactions are observed, corresponding to the site-specific S(i) CD order parameters of the various acyl chain segments (index i).91 The system is quite fluid, yet the lipid molecules still give rise to distinct quadrupolar splittings in the solid-state 2H NMR spectra. Thus, one obtains essential information at the atomistic level in the disordered, liquid-crystalline state of the biologically relevant lipid bilayers.

5.5. Segmental Order Parameters Show a Site-Specific Profile as a Function of Acyl Group Position

In solid-state 2H NMR spectroscopy of fluid bilayers, the profiles (2,i) of the second-rank order parameters S(i) as a function of CD ≡ S acyl chain position (index i) constitute one of the primary experimental observables for modeling the equilibrium bilayer properties. They include the average projected acyl length and mean cross-sectional area per molecule, as well as properties connected to the force balance in lamellar and nonlamellar phases of membrane lipids. Such order profiles were first obtained by Seelig and co-workers91 from studies of lipids with specifically deuterated groups, and the approach has since been extended to lipids having perdeuterated acyl chains. As an example, a representative plot of the order parameters versus the acyl chain position is shown for 1,2-diperdeuteriomyristoyl-snglycero-3-phosphocholine (DMPC-d54) in Figure 14. The order profile includes a plateau in the S(i) CD values over the initial part of the chains, beyond which there is a decrease in absolute magnitude as the chain terminus is approached. The plateau in the order profile is associated with tethering of the acyl chains to the aqueous interface, as suggested qualitatively by a simple crankshaft model with kink configurations (gauche±-transgauche∓) and jog (gauche±-trans-trans-trans-gauche∓) configurations. Beyond the plateau region, the individual segments become more disordered to fill up the free volume that would be present otherwise, because of the chain terminations. Hence, the 12106


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that the phospholipids move individually within the bilayer, even in the presence of cholesterolrather, the dynamics are inherently collective.239 At this point, one can consider the results in Figures 12 and 13 for the DMPC-d54 bilayer in greater detail, both in the absence and in the presence of cholesterol. To gauge our intuition, it may be helpful to consider various motional models as simple limiting cases. Referring again to Figure 14, the first example is an all-trans rotating polymethylene chain, corresponding to eq 28 with βPD = 90° yielding SCD = −1/2 as a reference value. Next, we can consider a crankshaft model involving an infinite polymethylene chain saturated with kink configurations (gauche±-transgauche∓) and jog (gauche±-trans-trans-trans-gauche∓) configurations, leading to ⟨SCD⟩ = −1/3. Alternatively, for only kink configurations the limiting crankshaft model yields ⟨SCD⟩ = −1/ 4. Last, the classical oil-drop model completely neglects tethering of the acyl chains to the aqueous interface, or equivalently one can average over a tetrahedral lattice, in which case the isotropic motion gives SCD = 0. One can then compare the experimental order profiles to the above limiting cases as benchmarks. We have already seen in Figure 12 that for the DMPC-d54/cholesterol (1:1) binary mixture there is an arresting increase in the absolute RQCs versus DMPC-d54 alone. This finding is due to a substantial decrease of the degrees of freedom of the flexible phospholipids, e.g., due to van der Waals interactions with the rigid sterol frame. The corresponding plateau in the order profile (cf. Figure 14) can be interpreted by a relatively constant probability of the acyl chain configurations, due to their tethering via the polar head groups to the aqueous interface. For the top part of the acyl chains, the segmental order parameters approach the limiting value of SCD = −1/2 when cholesterol is present, as expected for an all-trans rotating polymethylene chain. The additional acyl disorder can arise from internal degrees of freedom of the phospholipids, e.g., due to segmental isomerizations, molecular motions, or collective thermal excitations of the bilayer. These additional degrees of freedom lead to smaller absolute S(i) CD values for the DMPC-d54 molecules in the absence of cholesterol. Provided the disorder of the DMPC-d54 bilayer is due mainly to rotational isomerism, the acyl chains fall somewhere in between the limiting crankshaft model with ⟨SCD⟩ = −1/3 and the classical oil-drop model for which SCD = 0, e.g., the greater disorder can entail stretches of the chain running perpendicular to the bilayer director. For the DMPC-d54 bilayer, both in the presence and in the absence of cholesterol, the acyl chains are more disordered within the hydrocarbon core to fill-in the free volume that would otherwise be present because of chain terminations, approaching the classical “oil-drop” limit only in the center of the bilayer.

chains are highly entangled to maintain the density constant at nearly that of liquid hydrocarbon. In addition, analogous 2H NMR studies of phospholipids have been carried out for lipids in various physical states, including the normal hexagonal (HI) and reverse hexagonal (HII) phases.77,209,219,309 For the case of 13C NMR spectroscopy, SCH order parameters for the phospholipid acyl chains and polar head groups are obtained from the residual dipolar or chemical shift tensors relative to their static values, yielding good agreement with previous 2H NMR results.238,240 As considered above, one can expect that the order parameters from 2H NMR spectroscopy may include contributions from local segmental motions, e.g., dihedral angle isomerizations, together with slower molecular motions and collective disturbances of the bilayer. For the purpose of discussion, we can roughly separate the contributions into those having a segmental order parameter S(2,i) due to fast motions, which varies f with the segment position (index i) in the chain, and those due to additional slower motions with an order parameter S(2) s that is approximately independent of chain position. The former comes largely from rapid trans−gauche isomerizations of the acyl chain segments, plus librational motions within the various conformational states, whereas the latter may arise from molecular motions and/or collective excitations of the bilayer. As an initial approximation, let us consider that the above motions are due to statistically independent Markov processes, e.g., associated with their time-scale separation. As a result, a simple product approximation can be made, namely, that S(2, i) = Sf(2, i)Ss(2)

(39)

where the fast and slow motions are decoupled. One possibility is that the equilibrium bilayer thickness and mean interfacial area per molecule arise mainly from local isomerizations of the chains, whereby the influences of slower motions are included by the order parameter S(2) s and are relatively small. In this case, the observed order parameter S(2,i) mainly reflects the local order parameter S(2,i) due to trans−gauche rotational isomerizations of f the chains, which can be used to monitor changes in bilayer properties due to lipid interactions with cholesterol, peptides, or membrane proteins. 5.6. Bilayers Containing Cholesterol Enable Testing of Theories for Dynamical Structures of Membrane Assemblies

In general, bilayers containing cholesterol provide an excellent model for testing theories for the configurational ordering and structural dynamics of liquid-crystalline membranes.22 As we can see in Figure 14, for the DMPC-d54 bilayer both in the absence and in the presence of cholesterol, a well-developed profile of the segmental order parameters S(i) CD versus the acyl chain position (i) is seen. A plateau occurs over the middle part of the chains, followed by a progressive diminution, which manifests the end effects within the bilayer central hydrocarbon core. Even so, due to the orientation of the glycerol backbone approximately perpendicular to the membrane surface, the sn-1 and sn-2 acyl chains are inequivalent, as shown by Seelig et al.41 The initial chain geometry leads to smaller order parameters for the beginning of the sn-2 chain310 (cf. Figure 14). Beyond the first few segments, the order parameters of the sn-2 chain become larger than those of the sn-1 chain, where smaller statistical fluctuations are associated with greater “stretching” of the sn-2 chain to compensate for the initial position closer to the aqueous interface.266 The order profile suggests that variations in the degree of acyl chain entanglement occur as a function of depth within the bilayer hydrocarbon region. As a result, it is unlikely

6. MATERIAL PROPERTIES, NANOSTRUCTURES, AND BIOLOGICAL FUNCTIONS OF PHOSPHOLIPIDS ARE EMERGENT AT THE ATOMISTIC LEVEL We are now able to investigate in further depth how the observables from solid-state 2H NMR spectroscopy correspond to the equilibrium properties of the system. Knowledge of the bilayer dimensions is highly significant to molecular dynamics computer simulations,25,111,311−315 as well as studies of lipid− protein interactions.69,98,221,254,258,316−319 The reader should take note that solid-state 2H NMR spectroscopy typically does not directly yield knowledge of positional order or distance information, as in the case of X-ray 66,217,227,320−323 or neutron53,291,324 diffraction. Rather, for the case of 2H NMR of lipid bilayers, orientational information is obtained from the 12107


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order profiles, corresponding to the individual acyl segments within the bilayer hydrocarbon region. This means that we must reconstruct the positional order within the bilayer hydrocarbon region from the orientational ordering of the acyl groups using concepts from statistical physics.220,227 Following this approach, the experimentally measured order parameters for liquidcrystalline lipid bilayers in the Lα (or liquid-disordered, ld) phase can then be related to the hydrocarbon thickness and the mean interfacial area per lipid. In terms of statistical mechanics, filling of the space by the acyl chains within the bilayer hydrocarbon core can be investigated by means of acyl chain packing profiles.220 Analogously, for lipids in the reverse hexagonal (HII) phase, solid-state 2H NMR can be used to calculate the radius of curvature of the nanotubes, and thus identify the neutral plane where bending occurs.219,325 Studies of the lateral compressibility of phospholipid bilayers have also been carried out using solid-state 2H NMR methods56,113 (see Figure 15). 6.1. Mean-Torque Model Describes Segmental Order Profiles in Terms of Lipid Properties

For membrane lipids, the quadrupolar couplings are reduced from their rigid-lattice (static) values due to segmental and molecular motions, as well as collective fluctuations of the entire bilayer.15 The various motions are described by their meansquared amplitudes and (reduced) spectral densities, and may include cross-correlations due to their statistical dependence.1,15,194 The segmental order parameters (SCD) describe the composite motion in terms of the residual quadrupolar couplings, as compared to the static values.7 The order parameters are model-free spectroscopic observables that are statistical averages over all motions with correlation times up to the inverse of the rigid-lattice coupling.79 In analogy with liquid crystals, each segment is considered individually, because the spectroscopic observables are site specific.79,326 Correlations among the motions along the chains and between the chains are included in the site-specific order parameters as a function of segmental position.220,327 To interpret the |SCD| order parameters in terms of structural quantities, various models have been developed at different levels of complexity.91,113,220, Because of the inherent difficulty of more exactly treating membrane structure, most of these models are confined to simplified statistical treatments of the lipid conformations. In early work, a simple diamond-lattice model was used to describe the configurational statistics of the polymethylene acyl chains of lipid bilayers in the fluid state.41,216,267,328 Yet such a highly simplified approach does not consider how the acyl chain positional distribution governs the calculation of the bilayer structural dimensions, and thus, it overestimates the area per lipid at the aqueous interface.327 An alternative is to consider a mean-torque potential model,60,66,220 where statistical mechanical precepts are applied that relate the measured order quantities to the corresponding nanostructure of the lipid assembly. Here, we introduce the applicability of the mean-torque model in connection with the solid-state 2H NMR-derived segmental order parameters to interpret the membrane structure. The main objective is to connect the lipid structural parameters to hydrocarbon chain segmental order parameters. For disaturated lipids, a plateau in the 2H NMR-derived segmental order parameter profile is evident near to the headgroup, because the acyl chains are effectively tethered to the aqueous interface. Within the ensemble, the acyl groups with a greater number of

Figure 15. Cross-sectional area per lipid ⟨A⟩ as a function of applied pressure (osmotic, dehydration, or hydrostatic) obtained by meantorque analysis of solid-state 2H NMR spectral data. Results allow the energetics of bilayer deformation to be quantified at an atomistic level. (a) Elastic area compressibility modulus (KA) is calculated from the values of ⟨A⟩ versus osmotic (squares) or dehydration (circles) pressure at 30 °C. (Inset) Percentage of total work due to bilayer deformation obtained by applying osmotic pressure versus cross-sectional area per lipid ⟨A⟩ for DMPC-d54 in the liquid-crystalline phase at 30 °C. (b) Corresponding semi-logarithmic plots of ⟨A⟩ against osmotic (Π) or bulk (P) pressure. Note that the 2-D compressibility κ⊥ (≡ 1/K⊥) obtained from bulk hydrostatic pressure data (triangles) does not directly involve removal of water. Thus it differs from the 2-D compressibility CA (≡ 1/KA) obtained from osmotic or dehydration pressure data. In both cases it is proposed that bilayer deformation is due to removal of water from the interlamellar space. Data taken with permission from ref 113. Copyright 2015 Elsevier.

gauche rotamers extend a shorter distance away from the aqueous interface, and those with fewer gauche rotomers penetrate more deeply into the bilayer core. At a certain depth of the bilayer, the influence of the chain terminations becomes important.329,330 Acyl chains on adjacent molecules become more disordered beyond this point to maintain the packing at liquid hydrocarbon density.239 Another observation from 2H NMR order profiles is that the plateau region shows a strong chain length dependence, whereas nonplateau regions are practically independent of the acyl chain length.220 Changes in structural parameters due to osmotic stress or headgroup and acyl chain composition are related to the balance of forces that govern membrane assembly and lipid−protein interactions.69,77,254,316,331 12108


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volume per segment spanned in space remains nearly constant. A Taylor series expansion about the all-trans reference value allows the area factor q to be approximated by q ≈ 3 − 3⟨cos βi⟩ + ⟨cos2 βi⟩ up to second order.220 The second moment ⟨cos2 βi⟩ can be obtained directly from the absolute order parameters |S(i) CD| of a given acyl segment (index i) by

Clearly, at the molecular scale on average each lipid in the membrane occupies a space that is related to the volume and length of the hydrocarbon chains according to DC =

2VC ⟨A⟩

(40)

where DC is the volumetric chain thickness of the hydrocarbon layer and VC is the total volume of an individual acyl chain. In eq 40 it is assumed that the chain volume VC is given by the densitometry measurements of John Nagle and co-workers217,332 and is conserved (i.e., constant). Note that the volumetric thickness DC and the mean area per lipid ⟨A⟩ at the aqueous interface are inversely related by the assumption of constant volume, meaning that the bilayer core has approximately the density of liquid hydrocarbon.239 However, the volumetric thickness DC is not the same as the mean projected acyl length.216 Due to end effects of the acyl chains, the mean travel away from the aqueous interface is less than the distance to the bilayer midplane, as required for the well-established assumption of constant volume to apply.220 The chain volume at temperature T is found from the methylene volume VCH2 using the expression VCH2 = VoCH2 + αCH2(T − 273.15), where αCH2 is the isobaric thermal expansion coefficient for methylene groups.220 It is well established that the volume of a methyl group VCH3 ≈ 2VCH2 and that VCH ≈ VCH2/1.31 for the methine volume.332,333 When the membrane composition is mixed, interactions among the neighboring lipids can lead to a change in the average cross-sectional area per lipid. To avoid complications from chain up turns,220 for estimating the average area per lipid, instead of the entire hydrocarbon chain, one can consider relatively ordered acyl segments near the headgroup region. The largest order parameter corresponds to the plateau region of the 2H NMR order profiles, where it is plausible to assume that the segmental cross-sectional area and projected length are inversely correlated, as clearly set forth by Brown et al. in ref 216. Accordingly,216 the average cross-sectional area per lipid reads220 1 D

⟨A⟩ = 4VCH2

⟨cos2 βi ⟩ =

4VCH2 DM

q

(43)

However, to calculate the mean acyl projections along the bilayer normal, the first moment ⟨cos βi⟩ of the acyl segment distribution is needed.216 To interpret the |SCD| order parameters in terms of structural quantities, several models have been developed.220 Because of the inherent complexity of membrane structure, most of these models are confined to simplified statistical treatments of lipid conformations. Calculation of the first moment ⟨cos βi⟩ assuming a given value of ⟨cos2 βi⟩ is considered in greater detail below. To continue further, the mean-torque model assumes that the orientational order for each chain segment versus the local director n is described by an orientational potential U(β) (potential of mean torque). With the combined effects of thermal fluctuations of the segmental orientations, both segmental and molecular conformations assume a continuous distribution. The probability of finding a statistical segment with a virtual bond orientation β (≡ βIN) at a given instant is given by the Boltzmann distribution f (β ) =

⎛ U (β ) ⎞ 1 exp⎜ − ⎟ Z ⎝ kBT ⎠

(44)

where the chain index i is suppressed for clarity. Here, the partition function is Z=

∫0

π

⎛ U (β ) ⎞ exp⎜ − ⎟sin β dβ ⎝ kBT ⎠

(45)

Assuming a first-order mean-torque model, we define U(β) = U1cos β, where U1 is the first moment of the function U(β) in terms of Legendre polynomials. Then, the angular-dependent quantities are integrated together with the distribution function to give the following coupled equations 220

(41)

Here, VCH 2 is the methylene volume332 and D is the instantaneous travel of an individual segment along the bilayer normal. The above expression can be further recast as ⟨A⟩ =

(i) 1 − 4SCD 3

⎛ U ⎞−1 ⎛U ⎞ ⟨cos β⟩ = ⎜ 1 ⎟ − coth⎜ 1 ⎟ ⎝ kBT ⎠ ⎝ kBT ⎠

(42)

(46a)

⎛ U ⎞−2 ⎛ U ⎞−1 ⎛U ⎞ ⟨cos2 β⟩ = 1 + 2⎜ 1 ⎟ − 2⎜ 1 ⎟ coth⎜ 1 ⎟ ⎝ kBT ⎠ ⎝ kBT ⎠ ⎝ kBT ⎠

where DM = 2.54 Å is the maximum projection onto the bilayer normal of the virtual bonds connecting the carbon atoms in the polymethylene chain and 4VCH2/DM is the lipid cross-sectional area of the extended all-trans conformation.220 The area factor q is defined as ⟨1/cos β⟩, where β is the angle between the virtual bond connecting the two Ci−1 and Ci+1 carbon atoms and the normal to the lipid bilayer surface.

(46b)

An analytical solution for ⟨cos β⟩ can be obtained by using the approximation coth(−U1/kBT) ≈ 1, which for an individual segment (index i) yields the relation

6.2. Moments of Segmental Orientational Distribution Are Related to Bilayer Structural Parameters

⟨cos βi ⟩ =

According to the above development, in eq 42 the shape of a statistical segment is approximated by a geometrical prism with constant hydrocarbon volume.79 For Euclidean geometry, the effective acyl segment length is averaged over the motions whereas the segmental volume obviously is not. As the motional amplitudes increase, so does the area per lipid, whereas the

⎛ 1⎜ 1+ 2 ⎜⎝

⎞ (i) −8SCD −1⎟ ⎟ 3 ⎠

(47)

It should be noted that for the all-trans conformation of the lipids, ⟨cos βi⟩ = ⟨cos2 βi⟩ = 1 and hence q = 1 for the area factor. It follows that eqs 46a and 46b give rise to a limiting area of 4VCH2/DM and limiting monolayer thickness of nCDM/2, where nC is the number of carbon atoms in the hydrocarbon chain. 12109


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We can conclude at this juncture that the average thickness for the lipid bilayer ensemble includes the headgroup plus backbone thickness in the sense of a Gibbs dividing surface. The volumetric (Luzzati) bilayer full thickness is DB = 2VL/⟨A⟩, where the value of VL is the lipid volume332 and the volumetric head group thickness is DH = 4 Å for phosphatidylcholines and 6 Å for sphingomyelin. Alternatively, a steric membrane bilayer thickness can be calculated using the expression DB′ = 2DC + 2DH′, where DC is the volumetric hydrocarbon half thickness due to the acyl chains of the lipid, and DH′ is the hydrated headgroup plus the backbone distance. In the case of phospholipids DH is 9 Å, and for most sphingolipids it is 7 Å.334−336 Use of the above values of DH′ together with DC corresponds to the steric bilayer thickness DB′ as defined by Nagle et al.217 Lastly, it is important to show that the first-order mean-torque model neglects the effects of collective slow motions.239 The above treatment of a meantorque model was originally formulated in terms of the 2H NMRderived SCD values. We can, however, use the SCH order parameters obtained from separated-local field 13C NMR spectroscopy to extend the calculation of membrane structural parameters of membrane lipids.337 Knowledge of the bilayer dimensions is important for validating computer simulations of the molecular dynamics of lipid membranes.111,312−315,338−346 Recently, the mean-torque model has been successfully applied to investigate the effects of osmotic pressure on the structure of model membranes.113 The remarkable changes observed in the membrane structural parameters for the DMPCd54 membrane system when the osmotic pressure (Π) is varied are shown in Figure 15. Using the mean-torque model, the changes in the average area per lipid ⟨A⟩, bilayer thickness DB, and water spacing (DW) were measured. Reduction of interlamellar water from NW = 20 to NW = 1.5 leads to a change of the water spacing from DW = 20.1 to 1.8 Å, a substantial range. Figure 15a shows that boosting the osmotic pressure up to ∼200 atm (20 MPa) gives a substantial reduction of the area per lipid, with a corresponding gain of the volumetric bilayer thickness (DB). Such large bilayer deformations have significant implications for hydrophobic matching to transmembrane proteins. In Figure 15b the logarithm of the average area ⟨A⟩ is shown as a function of osmotic pressure. The elastic area compressibility modulus (KA) is calculated as 142 ± 30 mJ m−2 from the initial slope of the plot of average cross-sectional area against osmotic pressure. The measured value of KA is in close agreement with the values reported independently by Gawrisch et al. (136 ± 15 mJ m−2) and by Petrache et al. (108 ± 35 mJ m−2) using SAXS measurements.

Figure 16. Acyl packing profiles quantify chain extension along the surface normal of the lipid nanostructures. (a) Segmental projections onto bilayer normal starting from the terminal methyl group (i = nC) and ending at the C2 carbon (i = 2) at 65 °C. At a fixed temperature, all PC lipids follow the same curve and terminate at the corresponding chain length. Note the significantly different behavior of DPPE due to its different headgroup type and the DMPC/cholesterol (1:1) mixtures. (b) Solid line shows theoretical fits for ⟨A⟩ versus 1/nC for PC lipids (open circles). Dashed lines show the range of uncertainty for the extrapolation at ν = 0. This value for ⟨A⟩ is very close to the DPPE area at 69 °C (starred symbol), indicating the PE headgroup allows for maximum packing of fluid acyl chains. Adapted with permission from ref 220. Copyright 2000 Elsevier.

chain extension due to the individual segment projections is considered to begin at the terminal methyl end, and the full extent for each lipid corresponds to the mean projected acyl length. Comparison can then be made to the effects of the lipid composition, as well as the presence of additional components such as cholesterol, membrane-bound peptides, and integral membrane proteins.

6.3. Acyl Packing Profiles Quantify Chain Extension along the Surface Normal of Lipid Nanostructures

As a further application of the mean-torque potential model, the formalism can be extended to treat equilibrium properties of bilayers in the fluid state (liquid disordered or liquid ordered) in terms of chain packing profiles.60,220,347 One-dimensional acyl chain packing profiles effectively divide the space within the bilayer hydrocarbon core without explicitly considering the acyl chain configurational statistics. The acyl packing is treated in terms of the cumulative chain projection, starting from the bilayer center and extending along the normal toward the aqueous interface. An illustration is given by the packing profiles for various phospholipid systems as shown in Figure 16, which are discussed in greater detail below. Here, the mean relative carbon positions are plotted in reverse order, starting from the terminal methyl group (i = nC), whose position is set to zero. The

6.4. Influences of Acyl Length, Lipid Polar Headgroups, and Cholesterol on Structural Parameters of Lipid Bilayers

To provide another example, we consider a homologous series of phosphatidylcholines with perdeuterated saturated chains ranging in length from C12:0 to C18:0 in the fluid state.220 For each of the lipids, the RQCs and C−2H bond order parameters have been obtained from de-Paked solid-state 2H NMR spectra of randomly oriented multilamellar samples. The solid-state 2H NMR spectra reveal directly that the RQCs of the end segments remain nearly constant with increasing acyl length, corresponding to the central hydrocarbon core of the bilayer. In consequence, the segmental conformations in the central bilayer 12110


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lateral packing of phospholipids is more sensitive to the headgroup type than to the acyl chain length, as already noted above. Turning now to the influence of cholesterol, the solid-state 2H NMR spectra for cholesterol/DMPC-d54 bilayers in the liquidordered phase again fall into a distinct category. The general trend is similar to PE, but for bilayers containing cholesterol, the RQCs reveal an appreciably greater acyl chain ordering versus either PC or PE.22,337,350 A concomitant reduction of the mean aqueous interfacial area per lipid occurs, with little effect on the headgroup conformation because of the spacer (umbrella) effect of cholesterol.351−353 The acyl chain packing profiles show the chain extension follows the rank order cholesterol/PC > PE > PC (not shown; see ref 350). Further interpretation reveals that bilayers containing cholesterol in the liquid-ordered (lo) phase give the largest possible chain extension and the smallest interfacial area per lipid, while still maintaining liquid crystallinity. The acyl chain configurational degrees of freedom are more decoupled from the translational and rotational degrees of freedom of the phospholipids. At the molecular level, the reduction in chain entanglement yields a liquid-ordered phase.200 However, the 2H NMR order profiles (Figure 14) imply that entanglement of the chain ends may still occur to a limited degree. The macroscopic condensing effect of cholesterol on the bilayer interfacial area is thus clearly manifested at the atomistically resolved level.354

region are nearly the same in all cases. Nonetheless, the number of chain segments contributing to the largest RQC values becomes greater with increasing acyl carbon length, representing the plateau region of the order profile.220 Hence, the order parameter plateau grows in length as the number of acyl carbons increases, but otherwise, the configurational statistics of the acyl chains are similar. How can these observations be explained in molecular terms? According to the mean-torque potential model, the solid-state 2H NMR data correspond to the orientational probability distributions for the various acyl chain segments.220 The RQCs and derived order profiles show that the chain length projected onto the director axis becomes progressively greater with increasing number of acyl carbons. Evidently, the main effect of greater acyl length involves increasing volumetric thickness of the bilayer at a particular temperature. Even so, a small progressive condensation of the area per lipid at the aqueous interface occurs due to stronger van der Waals attractions for longer lipid chains.220 In addition, it is striking that acyl chain packing profiles (Figure 16) for the homologous series of phosphatidylcholines fall on a universal curve at the same absolute temperature. The number of groups contributing to the central core regionthe neutral volumeremains unaltered, whereas those in the most ordered region, closer to the aqueous interface, become greater as the number of acyl carbons increases. With increasing acyl length, more ordered states are sampled by longer acyl chains, keeping the order in the bilayer center almost invariant. In this sense, the acyl chain packing for the saturated PC series is almost invariant, as noted above. Consequently, the major effect on the bilayer thickness arises from the interactions of the polar head groups with water, which govern the interfacial area per lipid, whereas the acyl chains adjust their configurations to fill the available space and minimize the free volume. At this point, one should recognize that the solid-state 2H NMR spectra and derived RQCs of various phospholipids fall into different categories. Regarding the influence of the polar head groups,219,348,349 for the DPPE-d62 bilayer in the Lα phase the experimental RQCs are larger than those of the corresponding phosphatidylcholine, DPPC-d62, at the same absolute temperature.348 It follows that a reduction occurs in the degree of configurational chain disorder and entanglement for PE versus PC bilayers in the fluid state. Phosphoethanolamine head groups give a smaller area per lipid at the aqueous interface, and the acyl length projected along the bilayer normal is commensurately longer than for phosphocholine headgroups. As PE head groups are smaller than for PC (cf. Figure 2), and in addition they undergo intermolecular hydrogen bonding, this observation conforms with our biophysical intuition. Further quantitative analysis in terms of the mean-torque model and derived acyl packing profiles (Figure 16) informs the balance of forces at the level of the polar headgroups and the nonpolar acyl chains. Here, we can easily discern that the chain packing profile for PE differs appreciably from the universal profile discovered above for the homologous PC series. It is evident from Figure 16 that the presence of phosphoethanolamine head groups leads to greater extension of the PE acyl chains along the bilayer normal than in the case of PCs. A fit to the area per lipid using an empirical free energy function shows that the PE interfacial area gives a limiting value for the packing of fluid acyl chains.220 These findings suggest that the balance of attractive and repulsive forces involving the polar headgroups of the bilayer is dominant with regard to the lipid organization in the fluid state. Evidently the

6.5. Mixing of Sphingolipids and Phospholipids with Cholesterol Involves Configurational Entropy

Previous work involving solid-state 2H NMR spectroscopy351,355 has established an umbrella-like model for cholesterol− phospholipid interactions, where the cholesterol C3−OH group (see Figure 2) is situated beneath the phospholipid head groups. Hence, it acts as a spacer molecule as first proposed by Brown and Seelig.351 Thermodynamically, the cholesterol interactions with lipid membranes are driven by the hydrophobic effect plus van der Waals interactions between the acyl chains and the sterol ring system, giving a liquid-ordered (lo) phase beyond a threshold cholesterol concentration,200,356 as insightfully studied by Feigenson et al.357 A recent solid-state 2H NMR study of acyl chain-perdeuterated POPC, PSM, and their binary and ternary mixtures with cholesterol shows that the addition of cholesterol significantly increases the hydrocarbon thickness of the lipid bilayer. Such large volumetric thickness perturbations are characteristic of both the phospholipid and the sphingolipid lo phases.337,350 The average cross-sectional areas in the lo phase are reduced owing to the condensing effect of cholesterol.337,358 The cholesterol ordering effect on lipid bilayer stiffening is limited by the maximum acyl length. Additionally, membranes comprising egg yolk sphingomyelin (EYSM) and cholesterol may be affected by sphingosine backbone hydrogen bonding and packing, together with interfacial hydrogen bonding, involving the NH donor and cholesterol C3−OH acceptor. Such interactions are observed in molecular dynamics simulations,359,360 although supporting experimental evidence remains elusive.361−366 Another feature is that the sphingosine backbone possesses both an OH hydrogen-bond acceptor and a NH hydrogen-bond donor that may lead to interlipid hydrogen bonding and possibly superlattice formation in the liquid-disordered (ld) phase.367 In a related study using separated-local field 13C NMR spectroscopy, cholesterol-mediated structural perturbations were found to be clearly less pronounced for the sphingolipids than for phospholipids in the liquid-ordered (lo) phase.337 12111


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mixtures are relevant to understanding the molecular mechanisms of lipid raft formation in mammalian cellular plasma membranes. Further applications of solid-state 2H NMR spectroscopy by Hong and co-workers have critically examined the presence of domains in bacterial membranes containing zwitterionic 1palmitoyl-2-oleoyl-sn-glycero-3-phosphoethanolamine (POPE) and anionic 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphoglycerol (POPG) lipids.117 Acyl chain-perdeuterated POPE and POPG lipids have been studied as single components, in binary POPE/ POPG (3:1) membranes, and in membranes containing one of four cationic peptides: two antimicrobial peptides of the βhairpin family of protein-1 (PG-1) and two cell-penetrating peptides, Penetratin, and the human immunodeficiency virus (HIV) trans-activator of transcription (TAT) peptide. In principle, homogeneously mixed POPE/POPG membranes give the average of the quadrupolar couplings for the two lipids, whereas the presence of membrane domains enriched in one of the two lipids would be expected to show distinct 2H quadrupolar couplings, indicating variations of the chain disorder. Representative spectra for the case of PG-1 are shown in Figure 17.

Correspondingly, the increase in the hydrocarbon thickness and condensation of area per lipid for sphingomyelin is less than for PC bilayers. Such a remarkable difference indicates that sphingolipids are in a relatively ordered state in the singlecomponent membrane. The higher absolute acyl segmental order parameters in single-component bilayers at a given temperature for sphingolipids versus phospholipids indicates the propensity of self-association for the hydrophobic moieties of sphingomyelin.207,208 Notably, these observations suggest that upon adding cholesterol the entropic loss is less pronounced for sphingolipids than for phospholipids. A greater loss of conformational entropy for these two lipid entities may explain the selective enrichment of sphingomyelin in putative raft-like microdomains.337 Mixing of cholesterol is more favorable for sphingolipids compared to phosphatidylcholines, potentially driving the formation of lipid rafts in multicomponent biomembranes.368−370 In other words, like dissolves likeas we learn in our introductory chemistry courses. 6.6. Solid-State Deuterium NMR Reveals Coexistence of Ordered Domains with the Liquid-Disordered Phase in Lipid Mixtures

Cellular membranes are known to be very heterogeneous in their lipid and protein compositions with domains or clusters suggested to occur in the 10−100 nm size range due to collective interactions among the molecules.357,371−373 As a counterexample to liquid-disordered, liquid-ordered phase separations, the putative rafts in cellular plasma membranes could involve a miscibility critical point, with a distribution of cluster or domain sizes. A critical point manifests a particular combination of pressure, temperature, and composition in a mixed system, where the entropy penalty due to assembling or disassembling a cluster of molecules becomes small in relation to the available thermal energy. For model lipid mixtures containing cholesterol, at subcritical or near-critical temperatures, collective fluctuations are observed by fluorescence microscopy and NMR spectroscopy,206 with correlation lengths that can range from 10 nm to the micrometer range. Exchange of lipids between the various clusters leads to the compositional fluctuations that are attributed to the proximity to a critical point. An enhancement of the 2H NMR transverse relaxation rates occurs due to exchange modulation of the acyl chain ordering, with correlation times in the microsecond time scale.206 As insightfully shown by Gawrisch, Keller, and co-workers206,354,374 solid-state 2H NMR spectroscopy thus represents a powerful approach to investigating the raft-like lipid heterogeneity of model lipid bilayers and cellular membranes. Interestingly, solid-state 2H NMR spectroscopy of sitespecifically deuterated N-stearoylsphingomyelin (SSM) and its ternary mixtures with 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) and cholesterol has shown raft-like coexistence of liquid-ordered (lo) domains and the liquid-disordered (ld) phase.375 This research uncovers how the detailed lipid-chain dynamics are simultaneously obtained from coexisting lo and ld domains. In addition, the quadrupolar splitting profile of lipids in the lo phase in the ternary system is almost identical to that in the SSM−cholesterol binary mixture, suggesting the order profile of the binary system is essentially applicable to more complicated membrane systems in terms of the acyl chain ordering. It establishes how 2H NMR spectroscopy can be used to reveal the mole fractions of components of the ternary mixtures distributed in the lo and ld domains of the lipids. Such compositional distributions as well as lipid segmental order profiles in ternary

Figure 17. Effect of antimicrobial peptide PG-1 on POPE/POPG membranes at 293 and 287 K. Solid-state 2H NMR spectra without PG1 are shown in dashed lines and with PG-1 are shown in solid lines. (a) POPE-d31/POPG (3:1) membrane at 293 K where POPE is predominantly in the liquid-crystalline phase in the absence of peptide and acquires significant gel-phase content upon PG-1 binding. (b) POPE-d31/POPG (3:1) membrane at 287 K where POPE is predominantly in the gel phase in both cases. (c) POPE/POPG-d31 (3:1) spectrum at 293 K where POPG is in the gel phase without PG-1 but becomes mostly disordered in the presence of PG-1. (d) Spectra of POPE/POPG-d31 (3:1) at 287 K where POPG is predominantly in the gel phase in the absence of peptide and becomes mostly fluid upon PG-1 binding. Adapted with permission from ref 117. Copyright 2013 Elsevier.

Near physiological temperature (308 K), it is observed that no or only small coupling differences exist between POPE and POPG in the presence of any of the cationic peptides. However, near ambient temperature (293 K), at which gel and liquid-crystalline phases coexist in the POPE/POPG membrane, the peptides cause distinct effects on the quadrupolar splittings for the two lipids, indicating domain formation (see Figure 17). In addition, the antimicrobial peptide PG-1-bound lipid spectrum indicates that the peptide mixes unevenly with the different components of the lipid mixture. Upon PG-1 binding, the POPE-d31 lipids show 12112


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of the arachidonic acid (AA) and DHA polyunsaturated acyl chains of phospholipid bilayers have also been conducted.61,65,66 For highly polyunsaturated acyl chains, the solid-state 2H NMR data reveal significant disordering of the bilayers of mixedchain phospholipids. A greater cross-sectional area per lipid headgroup is evident versus the corresponding disaturated lipid bilayers.60,267 Application of the mean-torque potential model indicates that mixed-chain polyunsaturated lipids are characterized by acyl chain packing profiles that are shifted as compared to the homologous saturated series.60 Introduction of polyunsaturated ω-3 DHA chains translates the order profile along the saturated chains, allowing more disordered states to become accessible within the bilayer central region. According to Petrache et al.60 the center of mass of the polyunsaturated chains is located closer to the aqueous interface, whereas the saturated chains extend deeper into the bilayer hydrocarbon core.60 As a consequence, the area per lipid headgroup for polyunsaturated bilayers is increased versus the disaturated bilayers as first noted by Salmon et al.267 In addition, the resulting spontaneous curvature (or equivalently the lateral pressure profile) may affect the conformational energetics of membrane proteins such as visual rhodopsin.69,221,258,389 Packing differences have been found for bilayers of DHA versus arachidonic acid (AA) phospholipids, which may be connected with the different biological roles of ω-3 and ω-6 essential fatty acids.66

an ordering effect (Figure 17a and 17b) while POPG-d31 indicates a disordering (Figure 17c and 17d). The Gramnegative-selective PG-1 mutant, IB549, causes even larger differences in the POPE and POPG lipids.117 The POPE lipids partition into the ordered phase, whereas all POPG remains in the disordered phase. In comparison, the TAT peptide stiffens both POPE and POPG similarly in the binary membrane at ambient temperature, indicating that TAT does not cause dynamic heterogeneity, but rather interacts with the membrane by a different mechanism. Penetratin also increases POPE order but disorders POPG like the PG-1 peptide, indicating domain separation. These results provide insight into the extent of domain formation in bacterial membranes, and the possible peptide structural requirements for such phenomena. As a result, solid-state 2H NMR provides important knowledge of the dynamic heterogeneity of lipid membranes, in addition to the phase diagrams of ternary lipid mixtures containing cholesterol.206,374,376 6.7. Bilayer Properties Are Affected Oppositely by Nonionic Detergents and Raft-Like Cholesterol Mixtures

Detergents have received substantial attention in connection to bilayer interactions involving cholesterol, because of so-called rafts in mammalian cellular plasma membranes.377 Solid-state 2H NMR spectroscopy has been utilized to study interactions of phospholipids in the fluid state with various nonionic detergents,115,378−383 as well as mixtures of DPPC with lysoPC (lysophosphatidylcholine), where one or the other component has a perdeuterated acyl chain.216 Generally, introduction of nonionic detergents such as octaethylene glycol-mono-n-dodecyl ether (C12E8) leads to smaller RQCs and C−2H segmental order parameters for the DMPC-d54 bilayer, consistent with increased configurational freedom. Use of the mean-torque potential model reveals that the presence of a cosurfactant gives a reduction in the acyl length projected along the director axis and the associated volumetric thickness of the bilayer, together with an increase in the area per lipid at the aqueous interface. Interestingly, the effect of nonionic detergents such as C12E8 on bilayer properties is opposite to cholesterol, and parallels the influences observed at the macroscopic level. Recalling that like dissolves like (see above), this antipathy may be one of the factors associated with the occurrence of detergent-resistant fractions in cellular membranes.377,381−383

6.9. Non-Lamellar Lipid Nanostructures Involve Membrane Curvature

Phospholipids in nonlamellar phases have likewise been investigated with solid-state 2H NMR spectroscopy, including the hexagonal (H I ) and reverse hexagonal (H I I ) phases.55,209,219,309,325 For soft hexagonal phases comprising lipids in the fluid state, collective axial rotations of the lipids about the local surface normal, together with translational diffusion over the cylindrical surface, scales the lamellar phase RQCs by a factor of −1/2 as described by eq 31. Apart from the purely geometric scaling of the RQCs, however, a further reduction of the 2H NMR order parameters is evident due to curvature of the hexagonally packed nanotubes. By comparing the order profiles of the same phospholipid in the lamellar (Lα or ld) phase and in the HII phase, we can identify the neutral plane where the crosssectional area per lipid is constant, as well as estimate the radius of curvature of the cylinders.219 Future applications of solid-state 2 H NMR can address how material properties such as the spontaneous curvature (H0) and bending rigidity (KC) of biological lipid mixtures are affected by the polar headgroup and acyl chain compositions.197,198,256,390,391

6.8. Polyunsaturated Lipid Acyl Chains Lead to Shifting of the Bilayer Mass Distribution

Additional solid-state 2H NMR studies of model lipid membranes in the liquid-crystalline state involve a homologous series of mixed-chain, saturated, and polyunsaturated phosphatidylcholines.54,56,57,60,64,67,277,384−386 Such polyunsaturated lipids have long been implicated in cardiovascular disease, visual disorders, cancer, aging, and other pathological conditions. Here, the saturated sn-1 chain is perdeuterated and acts as an intrinsic probe of the polyunsaturated bilayer structure with site-specific resolution.60,65,263,266,277,281,387,388 As the length of the saturated sn-1 acyl group adjacent to docosahexaenoic acid (DHA) at the glycerol sn-2 position increases (cf. Figure 2), the RQCs and ordering of the initial acyl segments become greater, while those of the end segments are smaller. This is in marked contrast with the corresponding disaturated series (see above), where the RQCs of the end segments remain almost unchanged with increasing acyl length, thus revealing a universal chain-packing profile (Figure 16).220 Furthermore, solid-state 2H NMR studies

6.10. Peptides and Integral Membrane Proteins Are Affected by the Structural Properties of Lipid Membranes

Applications of solid-state 2 H NMR spectroscopy also encompass studies of phospholipids with perdeuterated acyl chains together with hydrophobic additives392 and peptides,393−396 including gramicidin A,397,398 LL-37, a human antimicrobial peptide,399 and a ras-derived peptide implicated in human cancers,178 as well as the integral membrane protein rhodopsin.25,400,401 The influences of membrane-bound peptides and proteins on the 2H NMR spectra and derived order profiles of the lipid bilayer illustrate the biological relevance of solid-state 2 H NMR studies of model lipid systems. For example, membranes containing a heptameter derived from the Cterminus of human N-ras protein with two hexadecyl modifications and additional peptides have been studied with 2 H NMR spectroscopy by Huster and co-workers.183,402−406 12113


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the dipolar coupling and obtain internuclear distances. Analysis of REDOR data involves the simple measurement of signal intensities and comparing the normalized dipolar dephased signal to a universal dipolar dephasing curve. The universal dipolar dephasing curve depends only on the dipolar coupling, and it is independent of all other NMR parameters such as chemical shift anisotropy and resonance offset. Ease of experimental implementation and straightforward data analysis are properties that make the REDOR experiment particularly useful for structural characterization of complex molecular systems. For the case of the heteronuclear dipolar interaction, we consider the spin pair I and S, so that the first-order Hamiltonian in the laboratory frame reads

Applying the mean-torque model, little influence of the ras peptide on the DMPC-d54 chain packing profile is evident, whereas for the ras-d66 peptide the longer hexadecyl chains are substantially less extended.406 Despite the fact that the ras-d66 chains each contain two additional methylene groups versus DMPC-d54, the projected chain length is almost the same in both cases. One is then led to a picture in which the bilayer volumetric thickness is governed by the host DMPC-d54 matrix, whereas the intercalated chains of the surface-bound ras-d66 peptide are less extended to match the bilayer dimensions. Systematic analysis of solid-state 2H NMR data also provides a basis for studies of lipid interactions with integral membrane proteins, such as visual rhodopsin, where its photochemical and signaling functions may depend on the presence of highly unsaturated lipid membranes.25,221,316,389

ĤD =

7. RESIDUAL DIPOLAR COUPLINGS PROVIDE STRUCTURES AND TOPOLOGY OF MEMBRANE LIPIDS AND MEMBRANE PROTEINS While solid-state 2H NMR uses quadrupolar couplings to inform structural aspects of biomolecules, related information can be obtained using dipolar couplings by separated-local field (SLF) NMR methods. Complementary SLF experiments provide the correlations of isotropic chemical shifts (13C or 15N) to the corresponding dipolar (13C−1H or 15N−1H) couplings. Such motionally averaged direct dipolar couplings are referred to as residual dipolar couplings (RDCs) and are analogous to the RDCs in solution NMR spectroscopy.407 Dipolar recoupling onaxis with shape and sign preservation (DROSS)22,337,408−410 as well as RPDLF411−413 and off-magic-angle spinning (OMAS) are among the best known examples for such experiments. These innovative methods characterize how the membrane parameters respond to the changes in the lipid environment, e.g., conformational changes of membrane-bound peptides or proteins, lipid composition, and hydration. In addition, such experimental techniques can be extended to obtain direct information for membrane-bound peptides or proteins by making use of partially averaged anisotropic dipolar interactions in liquid-crystalline systems (complementary to Figure 8). Rotational-echo double-resonance (REDOR) and polarizationinversion spin-exchange at the magic angle (PISEMA) are among the prominent experimental methods that are applicable to solidstate NMR spectroscopy of membrane proteins. Additional methods include 13C−15N correlation methods and techniques such as proton-driven spin diffusion and dipolar-assisted rotational resonance (DARR), which have the advantage that samples with multiple isotopic labels and even uniformly labeled proteins can be studied.

⎧ (2) δ DC D (3IẐ SẐ − I·̂ Ŝ)⎨D00 (Ω PL) ⎩ 2 ⎫ η (2) − D [D−(2) 20(Ω PL) + D20 (Ω PL)]⎬ ⎭ 6

(48)

The above Hamiltonian in angular frequency units corresponds to eq 6 for the quadrupolar case. It represents the secular part of the dipolar coupling, i.e., the part that commutes with the main Zeeman Hamiltonian and yields the first-order energy level shifts, but does not cause transitions between the nuclear spin states. (Note that the commutator [IẐ , IẐ ŜZ] = [IẐ , IẐ ]ŜZ + IẐ [IẐ , ŜZ] = 0.) Introducing the time dependence due to the magicangle spinning (MAS), and assuing axial symmetry (ηD = 0), the above Hamiltonian becomes (2) ĤD(t ) = δ DC DIẐ SẐ D00 (Ω PL ; t )

(49)

The axis of rotation (R) is introduced using the closure property of the rotation group (cf. Appendix) which gives (2) D00 (Ω PL ; t ) =

∑ D0(2)n (ΩPL ; t )Dn(2)0 (ΩRL) n

(50)

where ΩPR(t) = (αPR + ωRt, βPR, 0) and ωR is the rotor frequency. The dipolar frequencies of the two spectral branches for the coupled spin system then read χ (2) vD±(t ) = ± D D00 (Ω PL ; t ) (51) 2 Here, D(2) 00 is the Wigner rotation matrix element, ΩPL ≡ (αPL, βPL, γPL) defines the orientation of the PAS of the dipolar coupling tensor (P) to laboratory frame (L), and χD = (−μ0γIγSℏ/4π2)⟨r−3⟩ is the dipolar coupling constant. By observing the spin I, to measure the direct (through-space) dipolar coupling, the signal from spin S is subjected to dephasing with continuous rf pulses for a time period of nτR, where n is the number of pulses applied and τR is the rotor period. The signal intensity S with dephasing and S0 without dephasing are measured for the same duration. The normalized dephased signal (S− S0)/S0 is thus415−418,421−423

7.1. Rotational-Echo Double-Resonance (REDOR) Experiments Yield Distance Constraints

Rotational-echo double-resonance NMR is a high-resolution, solid-state NMR experiment for measuring the dipolar couplings between heteronuclear spin pairs.123,414−419 The internuclear distance dependence ⟨r−3⟩ of the dipolar coupling makes REDOR useful for structural characterization of solids, and it has become a valuable tool for characterizing a wide range of materials, including peptides and proteins,420 polymers, zeolites, guest−host systems, glasses, and more. In REDOR experiments, the NMR signal of the observed nucleus is attenuated when dipolar-dephasing radiofrequency pulses are applied to the nonobserved nucleus. The dependence of this signal reduction on the dipolar evolution time provides a direct way to determine

ΔS 1 =1− S0 4π

∫0

× sin β dβ

2π

dα

∫0

π

cos[nτR χD 2 sin(2β)sin α] (52)

By analyzing the REDOR-dephasing curves of ΔS/S0 as a function of the dephasing time nτR, the value of χD and hence the internuclear distance r can be determined for the case of dilute spin pairs, or small spin networks with known geometry. In 12114


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aligned proteins.151 The patterns observed in PISEMA spectra (Figure 18) for oriented, labeled peptides or proteins have been

situations where the observed spin is dephased by a group of spins without a known spatial arrangement, the distribution of dipolar couplings can still be characterized using the analytical solutions for eq 52.421,424 Thus, REDOR has grown since its invention into one of the most frequently employed solid-state NMR techniques for accurate measurement of internuclear distances. Numerous applications have been reported that make use of the REDOR technique.27,126,425−432 For example, in one such work the M2 proton channel from influenza virus was reconstituted into DMPC vesicles at pH 7.5 under fully tetrameric conditions.235 The peptide residues were uniformly 13 C labeled, and 13C chemical shifts were assigned from twodimensional correlation spectra. Moreover, amantadine was perdeuterated, thus enabling 13C REDOR distance measurements. With these distance restraints, and including additional solid-state 2H NMR orientational restraints, the high-resolution structure and conformation of the amantadine binding site of the M2 proton channel was obtained. The results indicate that amantadine physically occludes the M2 channel, thus paving the way for developing new antiviral drugs against influenza viruses. The study demonstrates the ability of solid-state NMR to elucidate small-molecule interactions with membrane proteins, and determine the high-resolution structures of their complexes. A similar approach has been applied to probe the membrane locations of the fusion peptide regions of the HIV gp41 and influenza virus hemagglutinin proteins, which both catalyze fusion of the viral and host cell membranes during initial infection of the cell.431 The HIV fusion peptide forms an intermolecular, antiparallel β-sheet, and the REDOR data support molecular major and minor populations. Furthermore, a significant fraction of the influenza fusion peptide molecules forms a tight hairpin with antiparallel N- and C-terminal αhelices, where the REDOR data indicate a single-peptide population with a deeply inserted N-terminal helix. The shared feature of deep insertion of the fusion peptide β- and α-secondary structures may be relevant for catalysis of fusion via the resultant local perturbation of the membrane bilayer. The REDOR strategy is generally used for accessing site-specific atomic-level information to yield parameters characterizing structure, dynamics, and metabolism in large, heterogeneous systems but not for 3-D structure determination. Therefore, REDOR complements other MAS5,8,180,433−442 and static NMR experiments4,21,35,103 by providing useful tools for structural biologists in answering specific questions regarding the structure and function of membrane proteins, and poorly soluble biomolecular complexes generally. Although elucidations of entire protein structures are typically outside the scope of REDOR, it presents a vital source of structural details. Used with labels placed in key positions, the solid-state NMR distance measurements can address and solve various interesting biological questions. In addition, as a heteronuclear method, REDOR has proven to be a particularly powerful tool for investigating the formation of intermolecular complexes in the course of biological processes.

Figure 18. Illustration of PISEMA experimental approach. (a) Transformation of the principal axis frame for each peptide plane to the helix axis frame allows for the tilt (τ) and rotation (ρ) of the helix to be calculated with respect to B0. The PISEMA pulse sequence correlates the amide 15N−1H dipolar coupling (blue) with the anisotropic 15N chemical shift tensor (σ11, σ22, σ33) (red). (b) For transmembrane proteins oriented in lipid bilayers the helical structures are manifested as characteristic PISA wheel patterns that emerge from PISEMA spectra. For membrane proteins and peptides with the director axis aligned parallel to B0, the orientation of each amide bond and therefore the tilt and rotation of the helix axis can be determined with respect to the lipid bilayer. Adapted with permission from ref 462. Copyright 2007 Elsevier. (c) Uniform oscillation of the anisotropic chemical shift and dipolar interactions is displayed with wave patterns having 3.6 residues per cycle. The membrane protein CrgA from Mycobacterium tuberculosis has two helices and Rv1861 has three helices. Representative data are shown here for each peptide. Adapted with permission from ref 463. Copyright 2013 American Chemical Society.

7.2. Polarization-Inversion Spin Exchange at the Magic Angle (PISEMA) Provides Transmembrane Helix Topology

termed polarity-index slant-angle (PISA) wheels by Marassi et al.102,137,149,443,444 These PISA wheels are useful for both assigning resonances and determining the average orientation of α-helices and β-sheets with respect to the applied external magnetic field (B0). Furthermore, PISEMA spectra of powder samples provide information on the average orientation of the molecules that undergo axial rotation without orienting the sample. Such capabilities, together with the high-resolution

In two-dimensional solid-state NMR spectroscopy, resonance patterns observed from transmembrane helices and β-sheets provide important structural details of the molecule in a membrane lipid bilayer. The polarization-inversion spin exchange at the magic angle (PISEMA) experiment correlates anisotropic dipolar and chemical shift interactions for labeled 12115


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Figure 19. Example of structure determination of the GPCR family protein CXCR1 using solid-state NMR spectroscopy. (a) Strip plots from threedimensional experiments taken at specific 15N and 13C chemical shifts for three representative regions of CXCR1: N-terminus (residues 31−35) (red), TM2 (residues 78−82) (blue), and ECL2 (residues 175−179) (green). (b) 13C-Detected 1H−15N separated-local field (SLF) spectra. (c) Dipolar wave plot of the experimentally measured 1H−15N dipolar coupling values as a function of residue number. Sinusoidal fits (cyan) to the data (4.1 kHz rmsd) highlight the transmembrane (TM1−TM7) and C-terminal (H8) helices. (d) Ensemble of the 10 lowest energy structures of CXCR1 aligned in the membrane (n denotes the bilayer normal). Adapted with permission from ref 169. Copyright 2012 Nature Publishing Group.

secondary structure of a protein, the value of ΔνD = ν+D − ν−D as a function of residue number demonstrates the periodic wavelike variations in the absence of chemical shift effects. Assuming the N−H bond vector coincides with the z axis of the principal axis system, the heteronuclear residual dipolar couplings are determined as a function of the residue number along the helix as

spectra, make the PISEMA experiment and its variants into powerful tools for obtaining structural information about membrane peptides and proteins, complementary to X-ray crystallography and solution NMR spectroscopy. By analogy to the case of solid-state 2H NMR of lipid membranes, in a PISEMA experiment the transformation of the principal axis frame for each peptide plane to the helix axis frame allows for calculation of tilt (τ) and rotation (ρ) angles of the helix with respect to B0 (Figure 18). For an experimental PISEMA spectrum, the PISA wheel is composed of a set of discrete resonances, each of which corresponds to an individual peptide plane. The center of the PISA wheel is governed uniquely by the helix tilt angle.445 It follows that the observation of helical wheels in the PISEMA spectra provides a mechanism to obtain detailed structural information without resonance assignments; moreover, they report on distortions in the α-helices such as kinks and bends. Many membrane proteins such as G-proteincoupled receptors (GPCRs) have multiple transmembrane helices,169 which preclude the resolution of discrete PISA wheels in uniformly 15N-labeled samples. Even so, such transmembrane helices can be studied by cleaving at the loops between helices with little loss of functional activity169,446 (Figure 19). This approach may provide an opportunity to label individual helices or pairs of helices while the remainder is unlabeled.

ΔνD ≡ νD+ − νD− ⎧ (2) η = χD ⎨⟨D00 (Ω PL)⟩ − D [⟨D−(2) 20(Ω PL)⟩ ⎩ 6

}

(2) + ⟨D20 (Ω PL)⟩]

(53a)

(53b)

which corresponds to eq 11 and eq 26. Here it is assumed the director n0 is parallel to the main magnetic field B0 so that ΩPL → ΩPD enabling the helix angles (ρ, τ) to the bilayer normal to be determined (Figure 18a). For the general case of an arbitrary director orientation, the rotation matrix elements are expanded in terms of the intermediate director frame using closure (cf. Appendix). Analogously, the residual (motionally averaged) dipolar couplings in solution NMR spectra of weakly aligned samples give strikingly similar patterns of variations, evincing a convergence of solid-state and solution NMR methodologies in structural studies of proteins (Figure 19). It follows that the dipolar waves can be applied to determine the orientations of the transmembrane helices with respect to the membrane normal, together with the deviations from the ideal helical structure. One can thus conclude that solid-state NMR spectroscopy is highly applicable for investigating the partially averaged anisotropic interactions in a unified way, involving both dipolar and quadrupolar couplings together with the anisotropic chemical shifts. The spatial dependencies of the dipolar tensor and electric field gradient tensor are formally isomorphous, and thus, complementary information about molecular structures is obtained. In the case of solid-state 2H NMR, the quadrupolar couplings dominate the other spin interactions, making the interpretation of the spectrum comparatively simple. For the dipolar couplings and chemical shift anisotropy, they compete in

7.3. Residual Dipolar Couplings Specify Orientation of Helices Aligned in Lipid Bilayers

Notably, the projection of the secondary structure on a plane is directly related to the 2-D PISEMA spectrum of an aligned sample, as pointed out by Opella et al.443 and Cross et al.444 for the PISA wheels discussed in the previous section. Moreover, the dipolar coupling values (called a dipolar wave), measured from either weakly or completely aligned samples, can be correlated with its projection onto a single axis (Figures 18 and 19). Both PISA wheels and dipolar waves originate from the orientational dependence of the anisotropic spin interactions, and the periodicity inherent in the secondary structure elements of proteins. The magnitudes of the heteronuclear dipolar couplings ν±D are a function of a given ΩPL orientation. For a periodic 12116


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with perdeuterated acyl chains are investigated. For different atomistically resolved acyl segments, the SCD order parameters of the C−2H bonds are obtained, corresponding to the statistical amplitude of the orientational fluctuations with respect to the bilayer normal. The distribution of RQCs enables nearly complete order profiles to be directly determined as a function of the acyl segment index (i). Such 2H NMR studies allow one to investigate the equilibrium properties of membrane lipids in relation to the forces governing their nanostructures in the liquid-crystalline state. For fluid bilayers, the relevant structural properties include the mean area per lipid at the aqueous interface, together with the average acyl length projected along the bilayer normal, and the volumetric thickness of the hydrocarbon core. The area per lipid determined from 2H NMR spectroscopy is essential with regard to validating molecular dynamics (MD) computer simulations of lipid membranes. Additional equilibrium properties include the monolayer area compressibility and the curvature elastic modulus, which describe the deformation of the membrane film in response to stress. Membrane deformability is important with regard to lipid polymorphism, phase equilibria of multicomponent lipid systems (formation of rafts), and lipid−protein interactions. Applying a mean-torque potential model, the acyl chain packing profiles allow solid-state 2H NMR studies of different phospholipids to be visualized in terms of the accumulated chain extension, proceeding from the bilayer center toward the aqueous interface. For lipids in nonlamellar phases, the average structural properties can likewise be obtained, such as the radius of curvature of inverted hexagonal (HII) phase nanotubes. Analogous approaches are possible from measurements of dipolar couplings and chemical shifts, thereby expanding the range of applicability of multinuclear NMR methods to the study of biomembranes. As one of the premier spectroscopic techniques, solid-state 2H NMR enjoys a number of significant advantages. It is highly complementary to other NMR methods that are becoming increasingly prevalent in studies of biomembranes. The benefits in observing the 2H nucleus are several fold: first, the molecular geometry and single-spin interaction makes the spectral analysis straightforward, which uniquely facilitates probing site-specific structure (orientation) and dynamics; it provides unambiguous information on the types, time scales, and degree of anisotropy of the motions; last, the 2H NMR parameters are well understood theoretically and experimentally. An additional feature is that solid-state 2H NMR relaxation studies can probe molecular dynamics over a large range of time scales, covering many decades of frequency (103−108 Hz), which facilitates investigating the collective dynamics of soft, liquid-crystalline materials. With such unique capabilities, in future applications multinuclear solid-state NMR methods can be expected to provide a rich source of structural and dynamic information for both the lipid and protein components of biomembranes.

the spectral broadening, thus potentially complicating the interpretation. Yet with recent developments in recoupling methods to reintroduce the anisotropic interactions in a controlled way, it is possible to eliminate the spectral overlaps and thereby directly detect the dipolar couplings as an important adjunct to quadrupolar couplings. In addition, with the dramatic improvements in polarization transfer methods, such as 1H detection447−451 and dynamic nuclear polarization (DNP) as implemented by Robert Griffin et al.,452−458 the sensitivity and resolution of the solid-state NMR experiments are increasing drastically. Solid-state NMR spectroscopy thus offers the considerable potential of providing atomistic-level information in both structure and dynamics in complex biological systems.

8. CHALLENGES AND FUTURE DIRECTIONS Solid-state nuclear magnetic resonance has emerged as a powerful spectroscopic technique that is capable of providing atomic-scale information in complex biological membranes and protein systems. The new structural and dynamical knowledge contributes insights into how membrane lipids and membrane proteins carry out their detailed functions in living systems. Over the past two decades, solid-state NMR spectroscopy has evolved dramatically in terms of the range of applicability to complex biomolecular systems based on improvements in hardware and experimental and assignment protocols, thus enabling a comprehensive multinuclear approach. For future implementations, measurements of chemical shifts, dipolar couplings, and quadrupolar couplings will be key to an enhanced understanding of structure and function. In this context, the knowledge obtained from studies of well-defined single-spin systems as in 2 H NMR spectroscopy is highly complementary to studies of coupled spin systems. Multinuclear approaches provide more accurate formulations of biomolecular structural and dynamic properties than are possible using a single method by itself. Taken as a whole, these methodological and conceptual improvements provide for an enlarged scope of applicability of the solid-state NMR methods to biomembrane systems. Applications include studies of complex integral membrane proteins (GPCRs, ion channels, membrane pumps, and transporters) and amyloid fibrils in a natural-like environment. For the case of partially ordered systems, few of the methods can substitute for the unique features afforded by NMR spectroscopy. Furthermore, solid-state 2H NMR spectroscopy gives a striking and conceptually simple illustration of the applications of irreducible tensor calculus and their transformations under rotations as a gateway into more complex implementations. Various multinuclear approaches are highly complementary in the case of biomembranes, involving both lipids and protein constituents. For investigations of liquid-crystalline membranes and embedded membrane proteins, solid-state 2H NMR methods play an important role in conjunction with multinuclear approaches. In all of these applications, the emphasis is on a unified representation in terms of quadrupolar and dipolar coupling tensors that provides a rich source of structural information for partially ordered systems. For membrane constituents, solid-state 2H NMR spectroscopy yields information about the structure and dynamics at an atomistically resolved level in the disordered, liquid-crystalline state. Values of the residual quadrupolar couplings (RQCs) are measured directly using solid-state 2H NMR spectroscopy. Studies of membrane lipids are conducted where the acyl chains or the polar head groups are specifically 2H labeled, or phospholipids

APPENDIX: IRREDUCIBLE TENSOR CALCULUS AS APPLIED TO NMR SPECTROSCOPY OF BIOMEMBRANES Here, we briefly describe the application of irreducible tensor methods in NMR spectroscopy of membranes. The formalism is equally applicable to other branches of biomolecular spectroscopy, including fluorescence and infrared spectroscopy.14 Familiarity with irreducible spherical tensors and their transformations under rotations gives an essential toolkit for the application of angular momentum theory in NMR spectroscopy. 12117


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1 V ±(2)2 = − δη (A2c) 2 In these formulas, δ ≡ Vzz is the largest principal value and η ≡ (Vyy − Vxx)/Vzz is the asymmetry parameter. For a given coupling interaction (label λ), the above results correspond to eqs 3a−3c of the main text. The great utility of spherical tensor calculus in molecular spectroscopy is that the irreducible tensor operators transform under rotations in an amazingly simple manner. As a result, the treatment of multiple rotations is straightforward once a few elementary concepts have been learned. Rotations are described most generally in terms of three parameters, the most useful for our purposes being the three Euler angles α, β, and γ. Figure A1

In fact, the sooner one is able to change from a Cartesian (x, y, z) basis to an irreducible or spherical basis, the better off one is. The investment paid at the beginning is recovered many times over by the ease with which complicated problems can be handled in analytical closed form. Notably, this is in contrast to the use of direction cosines, which become quite cumbersome if more than a single transformation of coordinates is considered. As a general rule, the coupling Hamiltonians in NMR spectroscopy can be formulated by the scalar product of two irreducible tensors, one of which represents the spin angular momentum operators and the other the coupling of the nuclear spins, e.g., due to the electric quadrupolar or magnetic dipolar interactions, or the anisotropic chemical shift. First, we outline the construction of irreducible tensor operators from the more familiar vector operators within a Cartesian basis. Then we describe briefly the relation between irreducible and Cartesian tensors and how they transform under rotations. Lastly, we include some of their useful properties, including closure and the construction of rotation matrices of different rank. We state without proof the following results from group theory, which are covered in greater detail elsewhere.222,230,232 Let us initially consider two vector operators Â and B̂ in a Cartesian frame (basis), for instance, corresponding to the nuclear spin angular momentum. The irreducible (or spherical) components of rank j = 1 are formed from linear combinations of the Cartesian components as follows: T̂ 0(1) ≡ Â 0 = Â Z, (1) T±̂ 1 ≡ A±̂ 1 = ∓ 1/2 (AX̂ ± iAŶ ), and analogously for B̂ 0 and B̂ ±1. Taking the direct product of the above irreducible tensor operators, one can construct irreducible tensor operators T̂ (j) m of higher rank j, where the projection quantum number m runs from −j to +j in integral steps. This procedure is described in various standard texts on group theory. For the quadrupolar or dipolar couplings, one has j = 2, leading us to the result that (2)

T0̂ = (2)

T±̂ 1 =

1 (3Â 0 B0̂ − Â ·B̂ ) 6

(A1a)

1 (Â 0 B±̂ 1 + A±̂ 1B0̂ ) 2

(A1b)

(2) T±̂ 2 = A±̂ 1B±̂ 1

Figure A1. Illustration of Euler angles employed in the transformation of irreducible (spherical) tensors from an initial (x, y, z) coordinate frame to a final (x‴, y‴, z‴) system. The first rotation is about the initial z axis by the Euler angle α. The second rotation is about the new y′ axis by the Euler angle β. The third rotation is about the penultimate z″ axis by the Euler angle γ. Consideration of the Euler angles allows one to formulate explicitly various models for the rotational dynamics of membrane constituents, including both lipids and proteins.1

(A1c)

Substituting Â = I ̂ and B̂ = Ŝ we then obtain eqs 2a−2c of the main text in terms of the angular momentum operators. Furthermore, we need to consider the nature of the coupling interactions involving the angular momentum in NMR spectroscopy. These interactions are generally described by secondrank tensors V in terms of their principal values and principal axis systems, rather than by construction from vector operators. Second-rank Cartesian tensors can be decomposed most generally into a sum of irreducible tensors of ranks j = 0, 1, and 2 with respect to the full three-dimensional rotation group,232 that is, V = V(0) + V(1) + V(2). For the case of quadrupolar or dipolar interactions, it is the traceless symmetric part V(2) that is of interest. It follows that the irreducible components V(2) m of the coupling tensors are related to their Cartesian components Vii (i = x, y, z) within the principal axis system (PAS) by222 V 0(2) =

V ±(2)1 = 0

3 δ 2

indicates how a physical system within an initial coordinate frame is transformed via the three Euler angles α, β, and γ to a final coordinate frame.232 First, the system within the (x, y, z) frame is rotated about the z axis by the angle α to the (x′, y′, z′) frame. Next, the primed system is rotated about the new y′ axis by the angle β to the (x″, y″, z″) system. Finally, the double-primed system is rotated about the z″ axis by the angle γ to yield the system transformed to the (x‴, y‴, z‴) frame, where γ is the angle between the y‴ axis and the so-called line of nodes. The matrix which rotates the components of an irreducible tensor of rank j from the initial to the final coordinate frame is called the Wigner rotation matrix D(j), and the elements for the case of j = 2 are summarized in Table A1. Notably, the elements of the Wigner rotation matrix can be considered as generalized spherical harmonics, just as the spherical harmonics are a generalization of the Legendre polynomials. They are mathematical objects that describe the effect of rotating an axially symmetric system as a rigid body, and their dependence on the three Euler angles Ω ≡ (α, β, γ) reads

(A2a)

Dm(j)′ m(α , β , γ ) = e−im ′ αdm(j)′ m(β)e−imγ

(A2b) 12118

(A3) DOI: 10.1021/acs.chemrev.6b00619 Chem. Rev. 2017, 117, 12087−12132

Here, d(j) m′m(β) indicates the elements of the reduced Wigner rotation matrix. The Wigner rotation matrix elements have a number of helpful symmetry properties, including the following, ⎛ 1 + cos β ⎞2 −2iγ ⎟ e e−2iα⎜ ⎝ ⎠ 2

⎛ 1 + cos β ⎞ −2iγ ⎟e e−iα sin β ⎜ ⎝ ⎠ 2

Review

3 sin 2 βe−2iγ 8

⎛ cos β − 1 ⎞ −2iγ ⎟e − e+iα sin β ⎜ ⎝ ⎠ 2

⎛ cos β − 1 ⎞2 −2iγ ⎟ e e+2iα sin β ⎜ ⎝ ⎠ 2

2

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Dm(j)′ m(Ω) = (−1)m ′− m D−(jm) *′− m(Ω)

where the asterisk denotes complex conjugation. Setting m = 0 yields the correspondence to the more familiar spherical harmonics, where relabeling the indices yields ⎛ 4π ⎞1/2 (j) * β , 0) = ⎜ ⎟ Y m (β , α ) ⎝ 2j + 1 ⎠

⎛ 1 + cos β ⎞ −iγ ⎟e − e−2iα sin β ⎜ ⎝ ⎠ 2

(j) (j) D00 (Ω) = d00 (β) = Pj(cos β)

Table A2. Legendre Polynomials up to Fourth Order

⎛ cos β − 1 ⎞ +iγ ⎟e e−2iα sin β ⎜ ⎝ ⎠ 2

⎛ 1 − cos β ⎞2 +2iγ ⎟ e e−2iα⎜ ⎝ ⎠ 2

j

Legendre polynomial Pj(x)

0 1 2 3 4

1 x (1/2)(3x2 − 1) (1/2)(5x3 − 3x) (1/8)(35x4 − 30x2 + 3)

We are now able to indicate how the elements of an irreducible (spherical) tensor rotated by the Euler angles Ω ≡ (α, β, γ) correspond to the elements of the irreducible tensor in the original frame. Under such a unitary transformation, the irreducible tensor elements become (j)

Tn̂ =

(j) (j) Dpn (Ω)

∑ Tp̂

where the projection indices are n, p = −j, ..., j. In fact, the above transformation under rotation defines an irreducible tensor operator of rank j. Note that the elements of the transformed irreducible tensor operator T̂ n(j) correspond to a linear combination of the components of the same tensor operator (j) T̂ (j) p , where Dpn (Ω) indicates the elements of the Wigner rotation matrix. Next, we consider the effect of consecutive transformations of an irreducible tensor to a succession of different coordinate frames. Application of eq A7 then involves the Euler angles Ω1 and a second application gives (j)

Tm̂ =

(j) (j) Dnm(Ω 2)

∑ Tn̂

(A8a)

n

=

(j) (j) (j) Dpn (Ω1)Dnm (Ω 2 )

∑ ∑ Tp̂ n

(A8b)

p (j)

(j) (Ω 2)] ∑ Tp̂ [∑ D(pnj)(Ω1)Dnm

(A8c)

n

Of course, we can also consider the overall rotation from the initial coordinate frame to the final frame with Euler angles Ω3 which yields

2

1

0

−1

(A7)

p

p

−2

(A6)

where Pj(x) is a Legendre polynomial of rank j and x ≡ cos β. For convenience, the Legendre polynomials up to rank j = 4 are summarized in Table A2.

=

m′/m

(A5)

Here, the connection to the spherical polar angles is given by (θ, ϕ) = (β, α). Finally, taking m = 0 in eq A5 one obtains the more familiar Legendre polynomials given by

1 −2iα 3 e sin 2 β 2 8

⎛ cos β + 1 ⎞ −iγ ⎟e e−iα(2 cos β − 1)⎜ ⎝ ⎠ 2 3 sin β cos β 2 − e−iα

⎛ 1 − cos β ⎞ +iγ ⎟e e−iα(2 cos β + 1)⎜ ⎝ ⎠ 2 ⎛ cos β − 1 ⎞ +2iγ ⎟e e−iα sin β ⎜ ⎝ ⎠ 2

3 sin β cos βe−iγ 2

1 (3 cos2 β − 1) 2 3 sin β cos βe+iγ 2 −

3 sin 2 βe+2iγ 8

⎛ 1 − cos β ⎞ −iγ ⎟e e+iα(2 cos β + 1)⎜ ⎝ ⎠ 2 3 sin β cos β 2 e+iα ⎛ cos β + 1 ⎞ +iγ ⎟e e+iα(2 cos β − 1)⎜ ⎝ ⎠ 2

⎛ 1 + cos β ⎞ +2iγ ⎟e − e+iα sin β ⎜ ⎝ ⎠ 2

⎛ cos β − 1 ⎞ −iγ ⎟e − e+2iα sin β ⎜ ⎝ ⎠ 2 3 sin 2 β 8 e+2iα

⎛ 1 + cos β ⎞ +iγ ⎟e e+2iα sin β ⎜ ⎝ ⎠ 2 ⎛ 1 + cos β ⎞ +2iγ ⎟ e e+2iα⎜ ⎝ ⎠ 2

−2

2

−1

0

1

Dm(j0) (α ,

Table A1. Elements of the Second-Rank Wigner Rotation Matrix

(A4)

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Tm̂ =

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Biographies

(j) (j) Dpm(Ω3)

∑ Tp̂

(A9)

p

Trivikram R. Molugu received his Ph.D. degree in Condensed Matter Physics from the University of Hyderabad, India, in 2009. His doctoral work includes applications of fast field-cycling NMR (FFCNMR) relaxometry for probing the molecular dynamics of liquid crystals and proteins. In 2009 he received a research associate fellowship from the Council of Scientific and Industrial Research (CSIR), India, for pursuing research with the Biomolecular NMR Spectroscopy Group at the Center for Cellular and Molecular Biology (CCMB), India. He applied 3-D solution NMR methods to study protein structure and functional dynamics. In 2011 he joined the University of Arizona as a research associate and has been employing both high-resolution solid-state and solution NMR methods to investigate biomembranes. His research interests are NMR spectroscopy, relaxometry, and biological consequences of physicochemical properties of biomembranes.

Hence, we arrive at the closure property of the group of rotations j) D(pm (Ω3) =

(j) (Ω 2 ) ∑ D(pnj)(Ω1)Dnm n

(A10)

Note that the above formula is a generalization of the spherical harmonic addition theorem to the three Euler angles Ω ≡ (α, β, γ) involving the Wigner rotation matrices. It describes how an overall rotation through Euler angles Ω3 is produced by first rotating the system by the Euler angles Ω1 and then by the Euler angles Ω2. By induction, one can use closure to decompose the Wigner rotation matrix elements for any overall rotation into those for an arbitrary number of intermediate rotations. Alternatively, closure can be introduced to collapse the summations involving various intermediate rotations. Equation A10 indicates that the Wigner rotation matrix elements themselves transform as irreducible tensors. By forming their direct products, one can construct rotation matrices of higher rank j in the manner outlined above. In NMR spectroscopy, tensors up to second rank are combined,14 yielding Wigner rotation matrices up to rank j = 4. The products of the Wigner rotation matrix elements can be expanded in a Clebsch−Gordan series,232 giving for the squared moduli

Soohyun Lee is pursuing his Ph.D. degree in Physical Chemistry at the University of Arizona under the guidance of Prof. Michael F. Brown. He received his B.S. degree in Biochemistry/Chemistry and M.S. degree in Physical Chemistry from the University of California at San Diego (UCSD). At UCSD he worked on improving the protein folding simulation software with Dr. Peter Wolynes, and at the Scripps Research Institute he carried out quantum chemistry calculations to explain the changes in pKA for cytochrome c with Dr. Lou Noodleman. From the beginning he has had a strong motivation toward the understanding of biomembranes. Currently, he is engaged in investigating viral membrane proteins using high-resolution solid-state NMR methods.

⎛2 2 j ⎞⎟ |Dm(2)′ m(Ω)|2 = (−1)m − m ′ ∑ (2j + 1)⎜ ⎝ m′ − m′ 0 ⎠ j ⎛2 2 j ⎞ (j) ×⎜ ⎟D00 (Ω) ⎝m − m 0⎠

Michael F. Brown is Professor of Chemistry and Professor of Physics at the University of Arizona. He is also a member of the Program in Applied Mathematics and the Committee on Neuroscience. His undergraduate and Ph.D. degrees were earned at the University of California at Santa Cruz. He conducted postdoctoral research as an NIH fellowship recipient at the Biozentrum of the University of Basel in Switzerland and at the Max-Plank Institute for Molecular Physics in Heidelberg, Germany. At this time, he initiated solid-state NMR studies of membranes together with mentors Joachim Seelig and Ulrich Häberlen. His current research focuses on solid-state NMR of membrane proteins and lipids, nuclear spin relaxation of biomolecules, and lipid−protein interactions.

(A11)

Here, (m′, m) and Ω are generic projection indices and Euler angles, respectively, and j = 0, ..., 4 in integral steps, corresponding to the triangle condition in quantum mechanics. The mathematical objects in the large parentheses denote Wigner 3-j symbols, which are themselves related to the Clebsch−Gordan or vector coupling coefficients.230 The 3-j symbols have a number of particularly helpful symmetry properties, and extensive tables exist.459,460 This approach greatly facilitates evaluation of the mean-squared values of the rotation matrix elements and related quantities. The above formalism is directly applicable to studies of the equilibrium and dynamical properties of membrane constituents, including both lipids and proteins as described in the text.

ACKNOWLEDGMENTS Research from the laboratory of the authors is supported by the U.S. National Institutes of Health. During the initial preparation of this review M.F.B. was a fellow of the Japan Society for the Promotion of Science (JSPS) at Osaka University, and he thanks Prof. H. Akutsu and his group for scientific interactions at the Institute for Protein Research. We also gratefully acknowledge A. Struts and X. Xu for helpful discussions. Special thanks are due to our group members and collaborators for their many contributions to this research.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.chemrev.6b00619. Further description of spectral line shapes and quadrupolar frequencies for powder-type samples (PDF)

REFERENCES (1) Xu, X.; Struts, A. V.; Brown, M. F. Generalized Model-Free Analysis of Nuclear Spin Relaxation Experiments. eMagRes 2014, 3, 275−286. (2) Leftin, A.; Xu, X.; Brown, M. F. Phospholipid Bilayer Membranes: Deuterium and Carbon-13 NMR Spectroscopy. eMagRes 2014, 3, 199− 214. (3) Vogel, A.; Scheidt, H. A.; Baek, D. J.; Bittman, R.; Huster, D. Structure and Dynamics of the Aliphatic Cholesterol Side Chain in Membranes as Studied by 2H NMR Spectroscopy and Molecular Dynamics Simulation. Phys. Chem. Chem. Phys. 2016, 18, 3730−3738.

AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. ORCID

Michael F. Brown: 0000-0002-7693-3224 Notes

The authors declare no competing financial interest. 12120


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