Solid-State Deuterium NMR Spectroscopyof Membranes

Trivikram R. Molugu, Xiaolin Xu, Avigdor Leftin,Silvia Lope-Piedrafita, Gary V. Martinez, Horia I. Petrache,and Michael F. Brown

AbstractSolid-state deuterium (2H) NMR spectroscopy provides a unique tool for lipidmembrane investigations. Knowledge of the average structure is obtained fromsolid-state 2H NMR lineshapes through principal values of the static ormotionally averaged coupling tensors due to quadrupolar interactions. For ran-domly oriented multilamellar lipids or aligned membranes, this technique pro-vides residual quadrupolar couplings (RQC) of the individual C–2H labeledsegments. The RQC values are used to calculate the segmental order parametersSCD(i) for each segment position (i), which are related to the average membrane

T.R. Molugu (*)Department of Chemistry and Biochemistry, University of Arizona, Tucson, AZ, USAe-mail: mtvrao@email.arizona.edu

X. XuDepartment of Physics, University of Arizona, Tucson, AZ, USA

A. LeftinDepartment of Medical Physics, Memorial Sloan Kettering Cancer Center, New York, NY, USA

S. Lope-PiedrafitaServei de Ressonància Magnètica Nuclear and Centro de Investigación Biomédica en Red-Bioingeniería, Biomateriales y Nanomedicina (CIBER-BBN), Universitat Autònoma de Barcelona,Cerdanyola del Vallès, Spain

G.V. MartinezDepartment of Cancer Imaging and Metabolism, H. Lee Moffitt Cancer Center and ResearchInstitute, Tampa, FL, USA

H.I. PetracheDepartment of Physics, Indiana University-Purdue University, Indianapolis, IN, USA

M.F. BrownDepartment of Chemistry and Biochemistry, and Department of Physics, University of Arizona,Tucson, AZ, USAe-mail: mfbrown@u.arizona.edu

# Springer International Publishing AG 2017G.A. Webb (ed.), Modern Magnetic Resonance,https://doi.org/10.1007/978-3-319-28275-6_89-1

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properties. The corresponding dynamical information is acquired from the tensorfluctuations, which depend on the mean-squared amplitudes and rates of themotions. Fluctuations of the coupling Hamiltonian due to the various membranedynamics cause the relaxation. The 2H solid-state NMR relaxation methods facilitatestudying the hierarchical dynamics of liquid-crystalline membranes over wide lengthand timescales. Model-free interpretation of spin–lattice relaxation rates as a functionof segmental order parameters enables understanding the complex lipid dynamics.Notably, the square-law functional dependence of R1Z rates and order parametersindicates the collective segmental dynamics, which in turn explain the liquid-crystalline material properties of lipid bilayer. Using solid-state 2H NMR relaxation,the influences of the acyl length, polyunsaturation, lipid polar head groups, cosur-factants, water, and incorporation of sterols are accessible in terms of the bilayerviscoelastic properties. These methods have been extensively applied to characterizemodel membranes and membrane-bound peptides as well as proteins for obtainingunique information on their conformations, orientation, and interactions.

KeywordsCholesterol • Liquid crystals • Lipids • Membranes • Molecular dynamics •Membrane elasticity • NMR relaxation • NMR spectroscopy • Order-directorfluctuations

ContentsIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Equilibrium and Dynamical Properties of Membrane Lipids Are Studied by Solid-StateDeuterium NMR Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Deuterium NMR Spectroscopy Allows Direct Observation of Coupling Tensors Relatedto Molecular Structure and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Molecular Structures and Motions Are Revealed by Deuterium NMR Lineshapes . . . . . . . . . . . . 6Deuterium NMR Provides Order Parameters as Spectroscopic Observables Related toAverage Membrane Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Deuterium Spin-Lattice Relaxation Times Reveal Dynamical Properties of LipidMembranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Model-Free Analysis Suggests that Collective Membrane Motions Govern the Relaxation . . . 13Spectral Densities and Correlation Functions Are Derived for Simplified Models inClosed Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Deuterium NMR Relaxation Allows Detailed Comparison of the Structural and DynamicalProperties of Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Introduction

Solid-state NMR spectroscopy is widely applicable to the investigation of non-crystalline or amorphous materials, e.g., polymers, glasses, protein precipitates,and membrane proteins. Rather than being mainly an alternative to X-ray crystal-lography, however, solid-state NMR is unique among current analytical andspectroscopic methodologies in that it provides both structural and dynamical

2 T.R. Molugu et al.

information at an atomically resolved level. In solid-state NMR, the structuralinformation is obtained from the static or motionally averaged coupling tensorsdue to dipolar, chemical shift, or quadrupolar interactions [1–3]. Correspondingdynamical information is acquired from the tensor fluctuations, which depend onthe mean-squared amplitudes and rates of the motions, and govern the NMRlineshapes and relaxation times [4]. For these reasons, solid-state NMR is findingincreasing applicability in the chemistry of materials, structural biology, and geno-mics research, and this trend can be expected to continue well into the future.

One area of solid-state NMR spectroscopy that has proven fruitful with regard tothe investigation of membranes is 2H NMR spectroscopy. Previous detailed reviews of2H NMR as applied to membrane lipids are available [3, 5–11]. Solid-state 2H NMRhas been used to investigate raft-like lipid mixtures implicated in membrane signalingfunctions [12–17], and in addition 2H NMR studies of membrane proteins [4, 18–29]and DNA fibers [30] have also been conducted. The present chapter is focused on theliquid-crystalline state of membrane lipids as investigated from combined 2H NMRlineshape and relaxation studies. A related aspect entails the correspondence of 2HNMR studies to molecular dynamics simulations [4, 24, 31]. The salient aspects of 2HNMR are that it enables both membrane lipids and membrane proteins to be studied bysubstitution of 2H for 1H, the structural data are highly complementary to X-ray [11,32, 33] and neutron diffraction studies, and virtually unique information regarding thefunctional dynamics of membrane constituents can be acquired.

Equilibrium and Dynamical Properties of Membrane Lipids AreStudied by Solid-State Deuterium NMR Spectroscopy

Phospholipid bilayers are classified as smectic A lyotropic liquid crystals, and anillustration of the liquid-crystalline lamellar phase is shown in Fig. 1. The hydro-phobic effect leads to a sequestering of the nonpolar acyl chains within the bilayerinterior, whereas the polar head groups interact with water at the membranesurface. The nanostructure of a membrane lipid aggregate is the result of a delicate

Fig. 1 Nanostructure of alipid bilayer in the fluid,liquid-crystalline (Lα) phase.Reprinted with permissionfrom Ref. [65] (Copyright2002 American ChemicalSociety)

Solid-State Deuterium NMR Spectroscopy of Membranes 3

balance of forces acting at the level of the polar head groups and the hydrocarbonregions of the membrane [34]. Representative glycerophospholipids are depicted inFig. 2, in which the polar head groups differ in their size, capacity for hydrogenbonding, and charge, whereas the nonpolar acyl chains vary in their length anddegree of unsaturation. The phase equilibria of phosphatidylcholines in excesswater include three regions as temperature increases: a lamellar gel phase withtilted chains (Lβ0), an intermediate ripple phase (Pβ

0), and a lamellar liquid-crystalline phase (Lα). Other types of phospholipid nanostructures are also possi-ble; for instance, unsaturated phosphatidylethanolamines form the reverse hexag-onal (HII) phase, and cubic phases exist. Moreover, when cholesterol is present thelipid mixtures can form condensed complexes, microdomains, or undergo phaseseparation [35–38], which may be associated with rafts and caveolae in cellularmembranes (Figs. 1 and 2).

An important feature of 2H NMR spectroscopy is that one introduces site-specific 2H labels, typically corresponding to the individual C–2H bonds, and inthis way one obtains atomically resolved information. In liquid-crystalline mem-branes, the residual quadrupolar couplings correspond to the segmental orderparameters of the flexible molecules – they can be directly measured as experimen-tal observables. Moreover, the nuclear spin relaxation rates can be measured, e.g.,the relaxation of Zeeman order (R1Z) or quadrupolar order (R1Q), which depends onthe molecular mobility. By combining 2H NMR order parameter measurementswith relaxation studies, one can probe the structural fluctuations of fluid membranelipids that give rise to averaging of the coupling tensors in solid-state NMRspectroscopy.

Fig. 2 (a) Chemical structures of representative glycerophospholipids. The polar head groups varyin their size, capacity for hydrogen bonding, and charge. (b) Representative examples are indicatedfor the zwitterionic head groups phosphocholine (PC) and phosphoethanolamine (PE), and theanionic head group phosphoserine (PS). (c) The nonpolar acyl chains vary in their length and degreeand position of unsaturation. (d) The sterol components of biomembranes include cholesterol,which is formed from its biological precursor lanosterol

4 T.R. Molugu et al.

Deuterium NMR Spectroscopy Allows Direct Observationof Coupling Tensors Related to Molecular Structure and Dynamics

Besides the Zeeman interaction of the nuclear spin with the external magnetic field,additional perturbations are due to magnetic interactions (dipolar coupling, chemicalshift) and electric interactions (quadrupolar coupling). These couplings provide awealth of information regarding both the structure and dynamics of biomolecularsystems. Generally speaking, the principal values and principal axis systems (PAS)of the various coupling tensors yield structural knowledge, whereas their fluctuationsgive rise to spectral transitions, and are related to the dynamics of the system ofinterest.

Deuterium NMR spectroscopy is particularly valuable as an illustration of theprinciples of solid-state NMR as applied to molecular solids, liquid crystals, andbiomembranes [3, 9, 39]. This is because a single coupling is very large – the electricquadrupolar interaction dominates over the magnetic dipolar couplings of the 2H and1H nuclei, as well as the 2H chemical shifts. The 2H nucleus has a spin of I = 1, andhence there are three Zeeman energy levels corresponding to the projection of thenuclear spin angular momentum, with eigenstates |m〉 = |0〉, |�1〉 given by theHamiltonian HZ, see Fig. 3, part (a). According to quantum mechanics, transitionsbetween the adjacent spin energy levels are allowed, giving two single-quantumnuclear spin transitions. In 2H NMR the degeneracy is removed due to the couplingof the quadrupole moment of the 2H nucleus with the electric field gradient (EFG) ofthe C–2H bond, as given by the Hamiltonian HQ. (An electric quadrupole interactswith an electric field gradient analogously to the interaction of an electric dipole withan electric field.) This is illustrated in Fig. 3, part (b), together with a representative2H NMR spectrum of a solid polymer, PMMA-d8, as shown in part (c) which will bediscussed subsequently (Fig. 3).

A general prescription for calculating the 2H NMR transition frequencies andspectral lineshapes is the following. First one starts with the perturbing Hamiltonian,next Schrödinger’s equation is solved to obtain the energy levels, and lastly oneintroduces the spectroscopic selection rules to calculate the frequencies of thespectral lines. This gives as a final result for the quadrupolar frequencies (ν�Q) that

ν�Q ¼ � 3

4χQ D

2ð Þ00 ΩPLð Þ � ηQffiffiffi

6p D

2ð Þ-20 ΩPLð Þ þ D

2ð Þ20 ΩPLð Þ

h i� �: (1)

Here χQ � e2qQ/h is the static quadrupolar coupling constant, ηQ is thecorresponding asymmetry parameter of the EFG tensor, and ΩPL � (αPL, βPL, γPL)are the Euler angles relating the PAS of the EFG tensor (P) and the laboratory frame(L). The experimentally observed 2H NMR quadrupolar splitting (Fig. 3) is thengiven by the difference in the frequencies of the spectral lines,ΔνQ � νþQ � ν�Q. Oneshould note that the development is also applicable to other second-rank tensors, for

Solid-State Deuterium NMR Spectroscopy of Membranes 5

instance, the magnetic dipolar interaction, electric quadrupolar interaction, and thechemical shift [1, 2, 39].

Molecular Structures and Motions Are Revealed by DeuteriumNMR Lineshapes

Measurement of the deuterium (2H) NMR lineshapes yields knowledge of theaverage structure through the principal values of the coupling tensor, as well as theprincipal axis system (PAS). For the sake of illustration, let us first consider a staticoriented sample, e.g., a single crystal in the absence of motions. The crystal can berotated with respect to the laboratory frame, giving discontinuities in the NMRspectrum, which correspond to the main external magnetic field aligned alongeach of the three principal axes of the coupling tensor. The case of an aligneddispersion of phospholipid bilayers deposited on a planar surface is exactly analo-gous. Here one has a residual or effective coupling tensor, which is pre-averaged bythe motions of the flexible lipid molecules in the Lα state, but otherwise thetransformation under rotations is identical. In either case, the principal axes andprincipal values of the static coupling tensor, or the residual tensor in the presence ofmotions, can be obtained from the rotation pattern according to Eq. 1. Even so, oftenone has a polycrystalline sample with a random or spherical distribution of thevarious C–2H bond orientations. A powder (or powder-type) spectrum is thenobtained, from which one can “read off” the principal values of the coupling tensordirectly from the spectral discontinuities [1, 3]. In this case, a drawback is that theorientation of the PAS of the coupling tensor within the crystal frame is unavailable,because the spectral discontinuities correspond to the laboratory system.

Fig. 3 (a) Energy levels in 2H NMR spectroscopy. The Zeeman Hamiltonian HZ is perturbed bythe quadrupolar Hamiltonian HQ giving an unequal spacing of the nuclear spin energy levels,indicated by |m〉 where m = 0, �1. (b) The quadrupolar splitting ΔνQ is the difference in thefrequencies (ν�Q) of the single-quantum transitions, and is due to the perturbing interaction of the 2Hnuclear quadrupole moment with the electric field gradient (EFG) of the C–2H bond. (c) Experi-mental 2H NMR spectrum of an unoriented powder sample of deuterated plexiglass, PMMA-d8.The contributions from the C2H2 groups differ from those of the C2H3 groups, which undergo rapidthree-fold rotation on the NMR time scale (cf. the text)

6 T.R. Molugu et al.

Returning to Fig. 3, an experimental 2H NMR spectrum of a randomly oriented,powder-type sample of deuterated plexiglass, PMMA-d8, is shown in part (c). Herethe outer splitting (�60 kHz) of the powder-pattern is due to the C2H2 groups ofPMMA-d8. For the C

2H2 groups, motion is essentially absent on the 2H NMR timescale, and the static coupling tensor is observed. The experimental 2H NMRsplitting (due to the large peaks) represents the θ = 90� orientation, for which(�3χQ/4) = �127.5 kHz in the case of immobile methylene groups. (Weakershoulders are also present, corresponding to the θ = 0� orientation with a splittingof 3χQ/2 = 255 kHz). On the other hand, the central component (�20 kHz) of theexperimental 2H NMR spectrum is due to the methyl groups, which are rapidlyrotating in the solid state. The three-fold rotation about the methyl axes means thatthe static coupling tensor is averaged to yield a residual coupling tensor, which isaxially symmetric (ηeffQ = 0), and whose largest principal value (χeffQ ) is correspond-ingly reduced by a factor of �1/3. Hence for the θ = 90� orientation, the C2H3

splitting is (�3χQ/4) (�1/3) = 42.5 kHz, in good agreement with the experimentalspectrum. (The weaker shoulders correspond to the θ = 0� orientation with asplitting of (3χQ/2) (�1/3) = �85.0 kHz.) According to this example, one canessentially “read off” the coupling parameters, and hence the types of motions,directly from the experimental 2H NMR spectrum [1, 40].

In passing, one can note that rather different, uniaxial powder-pattern line shapesare observed for certain membrane proteins, such as bacteriorhodopsin [18] orrhodopsin [20], and also for nucleic acid fibers [30]. From such 2H NMR lineshape investigations, one is able to extract information about the molecular structure,as well as the disorder of the sample in terms of the appropriate distributionfunctions [41].

Our next example involves the case of membrane lipid bilayers, where rapid axialaveraging occurs about the normal to the membrane film surface, referred to as thedirector axis. For membranes in the fluid state, the quadrupolar splittings are due to theorientational order parameters of the individual C–2H labeled groups, leading to aprofile as a function of acyl position. The segmental order parameter SCD describes theamplitudes of the angular excursions of the C–2H labeled groups, and is given by [8]:

SCD � D2ð Þ00 0, βPD, 0ð Þ

D E¼ P2 cos βPDð Þh i (2a)

¼ 1

23cos2 βPD � 1� �

: (2b)

In the above formula, D 2ð Þ00 ΩPDð Þ is a second-rank Wigner rotation matrix element,

P2(x) is the second Legendre polynomial where x � cos βPD, and βPD is the time-dependent angle between the C–2H bond axis and the director axis (perpendicular tothe surface of the membrane). The angular brackets denote an average over all themotions faster than the inverse of the anisotropy in the static quadrupolar coupling(< 10�5 s). The observed quadrupolar splitting then reads

Solid-State Deuterium NMR Spectroscopy of Membranes 7

ΔνQ ¼ 3

2χQSCD

3cos2 βDL � 1

2

� �(3)

where βDL is the angle between the bilayer normal (director) and the main externalmagnetic field direction. The segmental order parameters constitute experimentalobservables, and are related to the equilibrium properties and average structure of thesystem.

Deuterium NMR Provides Order Parameters as SpectroscopicObservables Related to Average Membrane Properties

The 2H NMR spectra of phospholipids in water allow the quadrupolar couplings ofthe C–2H labeled segments to be observed directly, e.g., as in the case of the acylchains [5, 7, 32, 42, 43] and head groups [44, 45]. What can be learned about themicro- and nanostructures of membrane lipids from 2H NMR spectroscopy? First,the 2H NMR spectra of multilamellar dispersions of randomly oriented phospho-lipids reveal weak singularities arising from the θ = 90� orientation of the bilayernormal (director axis) relative to the main magnetic field, whereas the weakershoulders are due to θ = 0�. A characteristic profile of residual quadrupolarcouplings is evident for phospholipids having either specifically deuterated [5, 46]or perdeuterated [6, 7, 32, 34, 47–50] acyl chains in the liquid-crystalline (Lα) phase.The largest values correspond to the segments close to the lipid polar head groups,with a progressive reduction along the lipid acyl chains [8]. Comparison of theresidual quadrupolar splittings to the static coupling constant (see above) indicatesthat the acyl groups are considerably disordered in the liquid-crystalline state.

In Fig. 4 representative 2H NMR spectra are shown for mixtures of arepresentative acyl chain perdeuterated phospholipid, 1,2-diperdeuteriomyristoyl-sn-glycero-3-phosphocholine, abbreviated DMPC-d54, at T = 44 �C containingcholesterol. For acyl chain perdeuterated phospholipids, one can deconvolute orde-Pake the powder-type spectra of random mutilamellar dispersions to obtain morehighly resolved subspectra corresponding to the θ = 0� orientation [6, 51, 52]. Uponde-Pakeing, the majority of the acyl C2H2 and C2H3 groups give resolvable signals,and it can be seen that a progressive increase in the splittings occurs with increasingmole fraction of cholesterol. Interaction with the rigid sterol frame leads to asubstantial reduction of the number of degrees of freedom of the flexible phospho-lipids, as evinced by the larger residual quadrupolar splittings, Fig. 4.

Now according to Eq. 3 the observed residual quadrupolar coupling ΔνQ isdirectly related to the segmental order parameter SCD. The residual quadrupolarcouplings vary substantially, giving a profile of the absolute segmental orderparametersS ið Þ

CD as a function of chain position (index i). This inequivalence arises fromthe effects of the bilayer packing on the trans-gauche isomerizations of the acylgroups. Figure 5 shows the splittings can be expressed in terms of the segmentalorder parameters S ið Þ

CD, which are plotted as a function of the acyl position (index i) forDMPC-d54 alone and DMPC-d54 in the presence of cholesterol (1:1). Knowing the

8 T.R. Molugu et al.

assignments from studies of specifically deuterated systems [53], essentially completeorder profiles can be obtained. This disorder manifests trans-gauche rotational isom-erizations of the acyl groups, together with the effects of molecular motions or wholebilayer collective motions.

Referring to Fig. 5, the plateau in the order profiles can be explained in terms ofpreferred configurations of the acyl chains parallel to the membrane normal. ForDMPC-d54 in the presence of cholesterol (1:1), the segmental order parametersapproach the limiting value of SCD = �1/2 as expected for an all-trans rotatingpolymethylene chain. The additional disorder can be due to internal degrees offreedom of the acyl chains, e.g., rotational isomerizations, molecular motions, orthermal disturbances of the bilayer lipids. In the absence of cholesterol, the smaller

absolute S ið ÞCD values for DMPC-d54 manifest additional degrees of freedom. Assum-

ing the disorder of the DMPC-d54 bilayer is due mainly to rotational isomerism, theacyl chains fall somewhere between a limiting crankshaft model with kink/jogconfigurations having 〈SCD〉 = �1/3 or �1/4, or only kinks (gauche�-trans-gauche�) with 〈SCD〉 = –1/4, and the classical oil-drop model having SCD = 0.Lastly, for DMPC-d54 both in the presence and absence of cholesterol, the increaseddisorder towards the chain ends must involve further acyl configurations. Within thehydrocarbon core, the chains are more disordered to occupy the free volume due tochain terminations, approaching the classical “oil-drop” limit for very long acylchain lengths. Note that the presence of an order profile suggests that variations inthe degree of chain entanglement are likely as a function of depth in the bilayer(Fig. 5).

Now in the lamellar state of membrane lipids, the structural properties include theaverage thickness 〈L〉 of the bilayer hydrocarbon region, together with the meaninterfacial area 〈A〉 [3, 8, 34, 37, 38, 54], which plays a key role in molecular

Fig. 4 Representative solid-state 2H NMR spectra for (a) DMPC-d54 in the Lα phase and (b)–(d)DMPC-d54 containing increasing mole fractions of cholesterol in the liquid-ordered (lo) phase. Thesamples contained 50 wt% 1H2O at 44 �C and the data were acquired at 76.8 MHz (magnetic fieldstrength of 11.7 T). Powder-type spectra (light blue) of randomly oriented mutilamellar dispersionswere numerically inverted (de-Paked) (dark red). Note that a distribution of residual quadrupolarcouplings is evident (Figure modified from Ref. [99] with permission)

Solid-State Deuterium NMR Spectroscopy of Membranes 9

dynamics simulations of lipid bilayers and membranes [55, 56]. Starting with theexperimental S ið Þ

CD order profiles, one can calculate the bilayer structural parametersusing a mean-torque model for the acyl chain distributions [3, 34, 37, 50]. Byinterpreting the 2H NMR spectra in terms of the distribution functions for thesegment orientations, one is able to gain insight into the intermolecular interactionsgoverning the nanostructures of various membrane lipids. Very briefly, the mean-torque model relates the experimental order parameters to the moments 〈cos β〉 and〈cos2 β〉, where β is the angle between the vector connecting the two neighboringcarbons of the ith segment and the bilayer normal [34]. For a disaturated phospho-lipid such as DMPC-d54, the mean area per molecule at the aqueous interface isgiven by 〈A〉 = 4VCH2

q/DM, where q � 〈1/cos β 〉 � 3 – 3〈cos β〉 + 〈cos2 β〉. HereVCH2

is the volume per methylene group, and DM is the length of the vectorconnecting the two neighboring carbons of the ith methylene segment (2.54 Å).The volumetric thickness of the bilayer is then given by DC = 2VC/〈A〉 where VC isthe hydrocarbon chain volume. Chain packing profiles can also be obtained, whichdescribe the accumulated segmental projections along the membrane normal, pro-ceeding from the midpoint of the bilayer towards the aqueous interface [34].

The above 2H NMR approach has been applied to a homologous series of differentphosphatidylcholines, in which both or one of the acyl chains at the sn-1 and sn-2positions of the glycerol backbone has been 2H labeled by perdeuteration [34]. Themain effect of increasing the acyl chain length is an increase in the bilayer hydrocarbonthickness. The homologous series of phosphatidylcholines exhibits a universal chain

Fig. 5 Bilayer dimensions are obtained from solid-state 2H NMR spectroscopy. (a) Membranelipid area 〈A〉 at aqueous interface and volumetric bilayer thicknessDC in the liquid-crystalline state(cf. the text). (b) Profiles of segmental order parameters S ið Þ

CD as a function of acyl chain position (i)for (u, uu) DMPC-d54 and (l, °) DMPC-d54/cholesterol (1:1) mixture at 44 �C. Filled andopen symbols refer to the inequivalent sn-1 and sn-2 acyl chains, respectively. Reference values ofthe order parameters for the limiting cases of an oil-drop model (SCD = 0), a crankshaft model(〈SCD〉=�1/3 or�1/4), and an all-trans rotating chain (SCD =�1/2) are indicated for comparison(Figure adapted from Ref. [38] with permission)

10 T.R. Molugu et al.

packing profile differing from that of phosphatidylethanolamines. With increasing acyllength, the lipid area becomes smaller at a given temperature, manifesting strongervan der Waals attractions for longer lipid chains. Studies of the lateral compressibilityof phospholipid bilayers have also been carried out using 2H NMR methods[11, 42]. Comparative investigations have focused on a homologous series ofmixed-chain, saturated-polyunsaturated phosphatidylcholines [32, 47, 49, 50, 57].Highly polyunsaturated acyl chains yield significant disordering of the bilayer, increas-ing the cross-sectional area per lipid head group relative to disaturated lipid bilayers.The resulting spontaneous curvature (or equivalently the lateral pressure profile) mayaffect the conformational energetics of membrane proteins, such as rhodopsin. Lipidsin nonlamellar phases have also been investigated with 2H NMR, including thehexagonal (HI) and reverse hexagonal (HII) phases [58], which are implicated inprotein-mediated functions of biomembranes.

Deuterium Spin-Lattice Relaxation Times Reveal DynamicalProperties of Lipid Membranes

Turning next to the topic of nuclear spin relaxation, for membranes in the liquid-crystalline state, a complex hierarchy of motions is to be expected. How can NMRrelaxation help to disentangle the various types of motions that account for thestatistically averaged membrane properties? Generally speaking, one can separatethe motions into broad classes with characteristic mean-squared amplitudes and timescales, which may be related to the material or biological properties of the system ofinterest. Examples of fast segmental motions include trans-gauche isomerizations ofthe flexible phospholipids in the liquid-crystalline state. Slower motions may be dueto noncollective rotational diffusion of the lipid molecules or collective disturbancesand excitations of the bilayer itself [8, 59–63].

The mechanism of NMR relaxation originates from fluctuations of the couplingHamiltonian arising from the various possible motions of the lipid molecules withinthe bilayer. In applying 2H–NMR spectroscopy to membrane lipids, one is ofteninterested in the relaxation of the Zeeman order (R1Z) or the quadrupolar order (R1Q).Experimentally, the spin-lattice relaxation rates R1Z and R1Q are measured by a pulsesequence designed to perturb the magnetization away from the equilibrium value –effectively one can think of these as “magnetization jump” experiments. The subse-quent return to equilibrium can then be followed by “reading out” the remainingmagnetization as a transverse coherence, which is detected as an oscillating NMRsignal, and picked up by the radiofrequency coil of the spectrometer probe.

An example is provided in Fig. 6, which depicts an inversion-recovery measure-ment of the R1Z values for a random dispersion of DMPC-d54/cholesterol (1:1) in theliquid-ordered phase. The partially relaxed 2H NMR spectra in part (a) correspond toa superposition of Pake doublets, due to the various motionally inequivalent C–2Hbonds (vide supra). Following the inverting π pulse, the z-magnetization appearsnegative, and gradually recovers to equilibrium as the delay increases. In part (b) of

Solid-State Deuterium NMR Spectroscopy of Membranes 11

Fig. 6, the various partially relaxed 2H NMR spectra have been numerically inverted(de-Paked) to yield the spectra corresponding to the θ = 0� orientation. Theincreased resolution allows a more accurate determination of the R1Z rates. It canbe seen that the magnetization recovery becomes progressively faster as the residualquadrupolar splittings increase. This behavior points to the influences of orderfluctuations, in which the pre-averaged or residual coupling tensor remaining fromthe local segmental motions is modulated by slower membrane motions, e.g., due tomolecular fluctuations or whole bilayer disturbances (vide infra).

According to time-dependent perturbation theory, the relaxation is due to orien-tational fluctuations of the individual C–2H bonds, which induce transitions betweenthe various energy levels of the 2H nuclear spin system [4]. The observable relax-ation rates are related to the spectral densities of the various motions within thelaboratory frame. According to theory, the R1Z and R1Q rates are given to secondorder by:

R1Z ¼ 1

T1Z

¼ 3

4π2χ2Q J1 ωDð Þ þ 4J2 2ωDð Þ½ � (4)

R1Q ¼ 1

T1Q

¼ 9

4π2χ2QJ1 ωDð Þ: (5)

In these formulas R1Z is the spin-lattice (longitudinal) relaxation rate, where T1Z is thecorresponding spin-lattice relaxation time, and R1Q is the spin-lattice relaxation rate forthe decay of quadrupolar order, in which T1Q is the quadrupolar order relaxation time.

Fig. 6 Example of partially relaxed 2H NMR spectra of DMPC-d54 mutilamellar dispersion inthe liquid-crystalline (Lα) state at 44 �C. Panel (a) shows the experimental 2H NMR spectra,and panel (b) the numerically inverted (de-Paked) 2H NMR spectra (θ = 0�) as a function ofthe variable delay between the inversion pulse and the spectral acquisition. Data were obtainedat 76.8 MHz (magnetic field strength of 11.7 T) using an inversion-recovery pulse sequence,(π)x–t1–(π/2)x–τ1–(π/2)�y–τ2–acquire, with the variable delay t1 ranging from 5 ms to 3 s(Figure adapted from Ref. [94] with permission of American Physical Society)

12 T.R. Molugu et al.

The symbols Jm(ω) denote the irreducible spectral densities of motion, where m = 1,2 and ωD is the deuteron Larmor (resonance) frequency (Fig. 6).

The spectral densities Jm(ω) express the power spectrum of the motions as afunction of frequency ω. They correspond to the fluctuations of the individualWigner rotation matrix elements, which transform the coupling (EFG) tensor fromits principal axis system to the laboratory frame, and are given by the Fourier-Laplace transform:

Jm ωð Þ ¼ Re

ð1

�1Gm tð Þe�iωtdt: (6)

The autocorrelation functions Gm(t) depend explicitly on time and characterize theC–2H bond fluctuations; they read

Gm tð Þ ¼ D2ð Þ0m ΩPL; 0ð ÞD 2ð Þ

0m

ΩPL; t

D E� D

2ð Þ0m ΩPLð Þ

D E��� ���2 (7)

where D2ð Þ0m ΩPLð Þ denotes a Wigner matrix element (rank-2). Here the symbol ΩPL

indicates the three Euler angles (αPL, βPL, γPL) which rotate the principal axis systemof the EFG coupling tensor to the laboratory frame [8]. Note that the temporal decayof the correlation function is due to the angular reorientations of the C–2H labeledmolecular segments, whereas the long-time values correspond to the average bilayerproperties.

Model-Free Analysis Suggests that Collective Membrane MotionsGovern the Relaxation

Let us next turn to consider those aspects of the relaxation that are independent of aspecific motional model. We would like to identify the major features of thedynamics, leaving the details for subsequent atomistic computer simulations [56,64, 65]. We naturally make recourse to relatively fast and relatively slow motions,which successively yield further averaging of the coupling interaction on down tothe final residual value. The segmental order parameters clearly depend on theamplitudes of the C–2H bond fluctuations, whereas the relaxation rates depend onboth the orientational amplitudes and the rates of the C–2H bond fluctuations. As aresult, the ordering and rate of motion must be distinguished in explaining therelaxation of lipid bilayers [66].

Recall that in 2H NMR spectroscopy of membranes, the observables essentiallycomprise the order parameter and the relaxation rate profiles – thus it is useful toinquire as to the existence and nature of their functional dependence [59]. Accordingto the foregoing development, Eqs. 4, 5, 6, and 7, the interaction strength (the matrixelement) is squared in the transition probabilities that govern the relaxation rates(Fermi’s Golden Rule). Notably, the fast or local motions modulate the static

Solid-State Deuterium NMR Spectroscopy of Membranes 13

coupling constant, in which case the relaxation profile along the chains is governedessentially by the motional rates [46]. But for slow motions, the residual couplingtensor varies along the chains due to the pre-averaging by the faster segmentalmotions, yielding a profile of the interaction strength. If the slow motions affectthe lipid molecules to nearly the same extent, e.g., due to molecular rotations orwhole-bilayer lipid excitations (called order-director fluctuations, ODF), then therelaxation rates depend on the local order parameter squared.

In Fig. 7 representative examples are depicted of such a square-law dependenceof the relaxation and order parameter functions from 2H NMR spectroscopy of fluidlipid bilayers. Influences of cholesterol on the physical properties of DMPC bilayersare compared to its metabolic precursor lanosterol [67] in Fig. 7, part (a). Lanosterolis methylated on both the α-face and the β-face (Fig. 2) of the sterol frame thatinteracts with the phospholipids whereas only the β-face of cholesterol is methyl-ated. Notably, the slope of square-law plot is greater for lanosterol than for choles-terol, consistent with the bilayer stiffness being less for lanosterol versus itsmetabolic precursor cholesterol [67–69]. The site-specific analysis of the solid-state NMR results based on atomistic observables matches the results for themacroscopic bilayer elasticity. Part (b) of Fig. 7 shows the corresponding square-law plots for the multilamellar dispersions of DMPC-d54/cholesterol having differentmolar ratios of the two lipid components DMPC-d54 and cholesterol. It can be seenthat a square law functional dependence of the R ið Þ

1Z rates versus the order parametersS

ið ÞCD along the entire acyl chain is indeed observed [59]. Moreover, as the cholesterol

mole fraction XC increases, the slopes of the plots are systematically reduced,paralleling the macroscopic stiffening action of cholesterol [7, 70, 71]. By contrastan increase of the slope of square-law plot for the case of the detergent C12E8 inFig. 7, part (b), indicates the increase in the elastic behavior. The more molecularlysmooth van der Waals surface of the α-face of cholesterol enables a large increase inbilayer rigidity and stabilizes the liquid-ordered phase to an even greater degree thanlanosterol. The progressive increase in the bilayer rigidity on going from lanosterolto cholesterol parallels the metabolic pathway of sterol biogenesis, and may berelated to the optimization or evolution of the biophysical properties of cholesterol[72–75] (Fig. 7).

Spectral Densities and Correlation Functions Are Derivedfor Simplified Models in Closed Form

The next question one can ask is: What are the slow motions that yield the orderfluctuations and produce the square-law functional dependence? First, we consider asimple noncollective model as a description of the effective rotations of the flexiblephospholipids in the liquid-crystalline state [76, 77]. In the case of phospholipids,rapid motions of the flexible molecules yield a reduced moment of inertia tensor,whose principal axes and principal values correspond to the D⊥and D|| off-axial andaxial diffusion constants. Modulation of the average or residual coupling tensor by

14 T.R. Molugu et al.

rotations of the “average molecule” about its average principal axes would thenproduce the relaxation. For such a noncollective molecular model, the spectraldensities then read

Jmolm ωð Þ ¼ D

2ð Þ00 ΩPIð Þ

D E��� ���2

Xq

Xn

D2ð Þ0q ΩIMð Þ � ηeffQffiffiffi

6p D

2ð Þ�2q ΩIMð Þ þ D

2ð Þ2q ΩIMð Þ

h i����������2

D 2ð Þqn ΩMDð Þ

��� ���2�

� D 2ð Þqn ΩMDð Þ

D E��� ���2δq0δn0� �

j 2ð Þqn ΩMD;ωð Þ D 2ð Þ

nm ΩDLð Þ�� ��2(8)

where ηeffQ is the effective asymmetry parameter of the residual EFG tensor due toaveraging by the faster segmental motions. In Eq. 8 the Euler angles ΩPI, ΩIM, ΩMD,and ΩDL denote transformation from the principal axis system (P) of the EFG tensorto an intermediate frame (I ), from the intermediate frame to the molecular frame (M ),from the molecular system to the director frame (D), and lastly from the director tothe laboratory axes system (L ). The symbols j 2ð Þ

qn (ΩMD; ω) indicate Lorentzian

Fig. 7 Solid-state 2H NMR relaxation shows emergence of membrane elastic fluctuations and theirsuppression. Data were collected at 76.8 MHz (11.8 T). (a) Solid-state 2H NMR relaxation indicatesstriking differences in membrane elastic fluctuations from the binary mixtures of DMPC withlanosterol versus cholesterol. Square-law functional dependence of relaxation rate R

ið Þ1Z and order

parameter S ið ÞCD profiles for DMPC-d54/sterol mixtures is shown at 55 �C. Note the decrease in the

square-law slopes is consistent with a gradual reduction in bilayer elasticity on going fromlanosterol to cholesterol. (b) Dependence of R ið Þ

1Z rates on Sið Þ2CD for resolved 2H NMR splittings of

DMPC-d54 showing influences of cholesterol in the liquid-ordered (lo) phase at 44 �C. The presenceof cholesterol leads to a large decrease in the square-law slopes, corresponding to a progressivereduction in bilayer elasticity. An opposite increase is seen for bilayers containing C12E8 nonionicdetergent at 42 �C (Figure adapted from Ref. [67] and Ref. [94] with permission of AmericanChemical Society and American Physical Society)

Solid-State Deuterium NMR Spectroscopy of Membranes 15

reduced spectral densities, whose correlation times τqn depend on the principalvalues of the rotational diffusion tensor, viz. D⊥and D|| [78].

Note that in Eq. 8 the spectral densities are pre-multiplied by the quantityj〈D 2ð Þ

00 ΩPIð Þ〉j2 where ΩPI has to do with only the fast motions. This is merely thesquare of the fast order parameter S 2ð Þ

f and so a squared dependence on the orderparameter is predicted in agreement with the experimental finding (Fig. 7). However,a molecular rotational diffusion model alone does not describe the NMR relaxationin the MHz range for phospholipids in the liquid-crystalline state, because thedistribution of correlation times is not sufficiently wide [59]. Rather, a significantcontribution from slow motions such as vesicle tumbling is also needed, as obtainedfrom the 2H NMR spectral linewidth [77].

Yet another possibility is to model the relatively slow order fluctuations interms of collective thermal disturbances of the bilayer arising from the ensembleof interacting molecules [59]. The bilayer is approximated as a continuousmaterial, and the relaxation is assumed to manifest a distribution of modes, eachundergoing overdamped or viscous relaxation. Within this framework, the spectraldensities Jcolm ωð Þ due to collective excitations of the bilayer are given by [79]

Jcolm ωð Þ ¼ 5

2D

2ð Þ00 ΩPNð Þ

D E��� ���2Dω� 2�d=2ð Þ D2ð Þ�1m ΩDLð Þ

��� ���2 þ D2ð Þ1m ΩDLð Þ

��� ���2� �

(9)

where D is a viscoelastic constant that depends on the elasticity, viscosity, andtemperature of the bilayer, and d is the dimensionality of the bilayer thermalexcitations. Here the Euler angles ΩPN rotate the principal axis system of theC–2H bond to the local (instantaneous) director frame (N ), and the angles ΩDL

transform from the bilayer director (D) to the laboratory system (L ). According toEq. 9 the spectral densities J colm (ω) depend on the square of the observed orderparameter SCD, and the slope of the square-law plot is inversely related to thesoftness of the membrane.

For such collective bilayer excitations, one can distinguish between two possiblelimits, depending on the wavelengths of the fluctuations relative to the membranethickness and the interlamellar separation. Collective order fluctuations in 2-D aredescribed by a flexible surface model, e.g., in analogy with smectic liquid crystals,where the finite thickness of the membrane bilayer is neglected. In keeping withEq. 9, for d= 2 the spectral densities Jcolm ωð Þhave an ω�1 frequency dependence [63,80]. Alternatively, 3-D order fluctuations are described by a collective membranedeformation model, where the bilayer is modeled analogously to nematic liquidcrystals [59, 81–84]. Equation 9 then predicts an ω–1/2 dependence of the spectraldensities Jcolm ωð Þ (d= 3) [3, 8, 39, 85, 86]. Although a 3-D director fluctuation modeladequately describes the frequency dispersion of the 2H R1Z rates for DMPC vesicles[87], by itself it does not explain the orientation dependence of the 2H R1Z and R1Q

relaxation data [88].How can one simultaneously account for both the orientational anisotropy of the

relaxation and the frequency dependence of lipid bilayers in the fluid state? Here weare struck by the fact that the collective membrane deformation model explains the

16 T.R. Molugu et al.

frequency dependence, whereas the noncollective molecular model best explains theangular anisotropy. One is naturally led to consider a composite membrane defor-mation model, in which collective axial rotations of the flexible phospholipidmolecules are superimposed onto the collective bilayer excitations in the liquid-crystalline state [89]. For such a composite process, the laboratory-frame spectraldensities read

Jm ωð Þ ¼ Jmolm ωð Þ þ Jcolm ωð Þ þ Jmol�col

m ωð Þ (10)

in which the relatively small contribution from segmental motions is neglected, andthe spectral densities Jmol

m ωð Þ and Jcolm ωð Þ are given above, Eqs. 8 and 9. The cross-term Jmol-col

m ωð Þ is geometrical in origin [89], and cross-correlations between themolecular fluctuations and the collective fluctuations are neglected as a simplifyingapproximation. Currently available data indicate that such composite model canexplain the angular and frequency dependencies of the NMR relaxation for lipidbilayers in the fluid phase [89].

In Fig. 8 the frequency dispersion of the R1Z rates is shown for vesicles of DMPCin the liquid-crystalline state, comprising a total of 12 different magnetic fieldstrengths. Note that the R1Z values depend on frequency over the entire range from2.5 to 95.3 MHz – there is no plateau value at either low frequency or highfrequency. One cannot identify a regime where a single type of effective motiondominates the relaxation. The noncollective molecular model does not fit the lowfrequency end of the relaxation dispersion, even if an extreme value of the diffusiontensor anisotropy (η � D||/D⊥) is assumed [79, 90]. Consequently, an additionalcontribution from the vesicle tumbling is needed to explain the lower frequency data[77]. The 2-D flexible surface model is characterized by an ω�1 dependence andgives a rather poor fit to the data; it probably is most appropriate for interpreting therelaxation at very low frequencies in terms of collective undulations [76]. By

Fig. 8 Experimental 2H R1Z

relaxation rates plotted as afunction of frequency forvesicles of DMPC deuteratedat the C3 acyl segment in theliquid crystalline state at30 �C. Data were acquired at12 different frequencies(magnetic field strengths) andare shown together withtheoretical fits to variousmotional models (cf. the text)(Figure adapted from Ref.[79] with permission ofAmerican Institute of Physics)

Solid-State Deuterium NMR Spectroscopy of Membranes 17

contrast, the 3-D collective membrane deformation model fits the frequency-dependent R1Z data within the MHz range by an ω–1/2 dependence. Moreover, onecan extend the effective frequency range by considering the relaxation of the 13Cnucleus, which has a larger magnetogyric ratio than 2H [91]. In the case of 13C NMRthe relaxation is due to the magnetic dipolar interaction of the nucleus with its directlybonded hydrogen. The combined frequency-dependent data are described by asimple relaxation law of the form R

ið Þ1Z = Aτf + B S

ðiÞ2CD ω�1=2 , where τf is the fast

correlation time and A and B are constants, which is characteristic of order-directorfluctions (ODF) [85] (Fig. 8).

It follows that NMR results for soft membrane bilayers, involving flexible lipidmolecules with many degrees of freedom, can be interpreted using fairly simpleconcepts inspired by the physics of materials. The small contribution from internalmotions of the acyl chains matches the frequency-independent relaxation rates ofliquid n-alkanes [85], and agrees with more recent molecular dynamics simulations.Thus the bilayer microviscosity, where a bulk viscosity cannot be measured, iscomparable to an n-paraffin such as n-hexadecane [85]. The reason why the relax-ation is governed by collective order fluctuations is that the local segmental motionsof the lipids are very fast (τf � 10 ps), with spectral densities extending to very highfrequencies. Basically the membrane lipids are tethered to the aqueous interface viatheir polar head groups, and the bilayer interior is essentially liquid hydrocarbon.The experimental findings imply that quasi-coherent order fluctuations are alreadypresent in lipid bilayers involving lengths on the order of� the bilayer thickness andeven less, i.e., the mesoscopic length scale, as illustrated in Fig. 9.

Deuterium NMR Relaxation Allows Detailed Comparisonof the Structural and Dynamical Properties of Membranes

Clearly, an illuminating new aspect of the work described above is the extension ofthe concept of membrane elasticity to relatively short distances. This work providesstriking evidence that membrane deformational fluctuations occur over a wide rangeof length- and time-scales, which depend on the bilayer lipid composition. Indeedthe current results give evidence for a connection between bilayer properties on the

Fig. 9 Examples of collective membrane deformations within a continuum elastic approximation.(a) Planar bilayer, (b) splay, (c) twist, and (d) bend deformations, together with axial rotations aboutthe local director [85, 86] (Figure adapted from Ref. [93] with permission of American PhysicalSociety)

18 T.R. Molugu et al.

mesoscopic scale, as studied by NMR relaxation, and macroscopic properties of thebulk material. According to the foregoing development, by combining the 2H NMRrelaxation data with the order parameters, we obtain a more comprehensive pictureof the biophysical properties of membrane lipids than would otherwise be the case.Using solid-state 2H NMR relaxation, the influences of the acyl length (bilayerthickness), polyunsaturation, lipid polar head groups (interfacial area per molecule),addition of a cosurfactant, and incorporation of sterols have been studied andinterpreted in terms of the bilayer viscoelastic properties [32, 67, 86, 92–94]. Therotational dynamics of the phospholipid and cholesterol components in binarymixtures have also been studied as a probe of their intermolecular interactions [90,95–98]. A scale of bilayer softness is evident, ranging from highly deformable,surfactant or polyunsaturated bilayer systems, to systems containing phosphatidyl-ethanolamine with an increased bilayer stiffness, and ultimately the very rigid yetfluid bilayers containing cholesterol [86]. The theoretical interpretation of the NMRrelaxation data correlates well with previous macroscopic studies of membrane-bending deformations. Future applications of 2H NMR spectroscopy include studiesof membrane lipid nanostructures and lipid-protein interactions, as well as investi-gations of the structure and dynamics of membrane proteins in relation to theircharacteristic biological modes of action.

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