23 - Brown Lab · solid-state deuterium nuclear magnetic resonance (NMR), can be used as complementary probes of structural fluc-tuations in lipid membranes. In the case of X-ray

From: Methods in Molecular Biology, vol. 400: Methods in Membrane LipidsEdited by: A. M. Dopico © Humana Press Inc., Totowa, NJ

341

23

X-Ray Scattering and Solid-State Deuterium Nuclear MagneticResonance Probes of Structural Fluctuations in Lipid Membranes

Horia I. Petrache and Michael F. Brown

SummaryMolecular fluctuations are a dominant feature of biomembranes. Cellular functions might rely on these prop-

erties in ways yet to be determined. This expectation is suggested by the fact that membrane deformation andrigidity, which govern molecular fluctuations, have been implicated in a number of cellular functions. However,fluctuations are more challenging to measure than average structures, which partially explain the small number ofdedicated studies. Here, it is shown that two accessible laboratory methods, small-angle X-ray scattering andsolid-state deuterium nuclear magnetic resonance (NMR), can be used as complementary probes of structural fluc-tuations in lipid membranes. In the case of X-ray scattering, membrane undulations give rise to logarithmicallyvarying positional correlations that generate scattering peaks with long (power-law) tails. In the case of 2H NMRspectroscopy, fluctuations in the magnetic-coupling energies resulting from molecular motions cause relaxationamong the various spin energy levels, and yield a powerful probe of orientational fluctuations of the lipid mole-cules. A unified interpretation of the combined scattering and 2H NMR data is provided by a liquid-crystallinemembrane deformation model. The importance of this approach is that it is possible to utilize a microscopic modelfor positional and orientational observables to calculate bulk material properties of liquid-crystalline systems.

Key Words: Bending rigidity; liquid crystals; membranes; molecular dynamics; order parameters; relaxation.

1. Introduction1.1. Molecular Fluctuations in Biology

Under physiological conditions, biological processes necessarily involve molecular rearrange-ment, fluctuations, and disorder. Most biological processes occur in solution and must obeylaws of either equilibrium or nonequilibrium thermodynamics. For example, the activation ofreceptor molecules by chemical or mechanical stimulants occurs against the inherent tendencyof fluid solutions to maximize their entropy (disorder). Whereas the average molecular struc-ture, be it that of a protein or of a lipid bilayer, is of immediate and obvious relevance, fluctua-tions around these averages can be at least equally important. Structural changes undergone bymembrane receptors are typically measured on the order of thermal (kBT) units.

By definition of thermal units, energies associated with spontaneous molecular fluctuationsare on the same energy scale. How biological systems use fluctuations is still an outstandingquestion, but there is only one way to find out: measure them. In this chapter, an overview oftwo complementary experimental methods is provided for studying the kinds of molecularfluctuations that are accessible through measurement of either positional or orientationalorder: small-angle X-ray scattering (1) and solid-state 2H nuclear magnetic resonance (NMR)spectroscopy (2).

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1.2. Fluctuation–Dissipation Theorem and Ergodicity

Scattering and spectroscopic methods are typically used to measure structural parameters,and also reveal that fluctuations around these averages are significant at physiologicaltemperatures. How important is measuring these fluctuations? According to the fluctuation–dissipation theorem, a direct relationship exists between spontaneous thermal fluctuationsand the macroscopic response of the system (here the membrane) to external perturbation.Typically, the mechanical (elastic) properties of the membrane material are measured by thestructural deformations upon external stress. This external stress can be imposed by osmoticpressure or mechanical pressure (3). As an attractive alternative, material properties can beobtained from measurements of spontaneous thermal fluctuations, a direct consequence ofthe fluctuation–dissipation theorem and ergodicity (4).

According to the ergodic hypothesis of statistical mechanics, the configurations sampledby a single molecule over a sufficiently long time correspond to those of the entire system atany given instant. This correspondence provides a link between 2H NMR and X-ray observ-ables. As mentioned above, it is expected that fluctuations and membrane deformation per seare involved in biological function (5), further motivating the development of techniquescapable of measuring thermal fluctuations of lipid membranes.

1.3. Molecular Motions in Biomembranes

As shown in Fig. 1, large amplitudes of molecular motions within biomembranes give riseto broad spatial distributions (6,7). A hierarchy of molecular motions exists, from local uncor-related fluctuations to collective concerted motions (1,8). The collective molecular motionsare related to material (elastic) parameters describing physical quantities, such as compress-ibilities and bending rigidities (9). Because lipid bilayers combine properties of solids andfluids, the appropriate theoretical framework is the physics of liquid crystals (10). A key issueis the length scale at which the material properties begin to emerge (the so-called mesoscopicregime). Is it more fruitful to consider an all-atom molecular dynamics simulation, as in Fig. 1,

342 Petrache and Brown

Fig. 1. Molecular fluctuations can be simulated by computer modeling. (Left) A snapshot is shownof a DMPC bilayer containing a glycophorin A helix dimer. (Right) Spatial distributions are indicatedfor various molecular components sampled over a 5-ns trajectory (see ref. 6).

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and focus on extension to larger length scales? Or rather is it more appropriate to begin withthe macroscopic bilayer, and extrapolate down to the mesoscale approaching the bilayer andeven molecular dimensions? This question will be addressed later.

2. Fluctuations and Correlations2.1. Liquid-Crystalline Membranes are Described by Characteristic Intermolecular Correlation Functions

In multilamellar lipid suspensions, molecular correlations extend beyond individual mem-branes. Because of the van der Waals attraction between membranes, lipids form multilamellarvesicles. The average spacing between lamellae is measurable by X-ray scattering (Fig. 2).The scattering rings shown in Fig. 2 are characteristic of unoriented multilayers, for whichscattering domains are uniformly distributed relative to an incoming X-ray beam. Scatteringrings appear at angles satisfying the Bragg law:

(1)

where D is the interlamellar repeat spacing, λ is the X-ray wavelength, and h is an integerindex. Although multilamellar vesicles are aqueous suspensions, this scattering is called “powderscattering” by analogy with the random orientation obtained from powder crystallites (as in

2D hsinθ λ=

X-Ray and 2H NMR of Lipid Fluctuations 343

Fig. 2. Self-assembled lipid multilayers diffract incident X-rays to generate characteristic Braggrings. Diffraction angles θ satisfy the relationship 2Dsin θ = hλ, where D is the interlamellar repeatspacing, λ is the X-ray wavelength, and h is an integer index. (In X-ray scattering θ is defined relativeto the detector and is the glancing angle to the lamellar surface.) Unlike a typical crystalline material,fully hydrated DMPC at 35°C produces only two Bragg rings. Higher order rings (h > 2) vanishbecause of membrane thermal fluctuations.

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the Debye–Scherrer method). However, the lipid multilayers are imperfect crystals. As shownin Fig. 3 lipid multilayers can be regarded as a one-dimensional (1D) array of fluid mem-branes (8,9). However, because of thermally driven deformations (Fig. 3), correlations alongthe membrane stack decay rather quickly (11–13).

The Caillé-de Gennes model considers the local displacement of membrane patches asshown in Fig. 4A to calculate how intermembrane correlations behave along the membranestack (i.e., the variation of pairwise relative displacement from nearest neighbors to themiddle of the stack). Because periodic boundaries are used in the calculation, the system issymmetric about the middle of the stack (k = N/2). By letting N go to infinity, the asymptoticlimit of interneighbor displacement fluctuations shown in Fig. 4B can be determined.Membranes further apart are progressively less correlated; however, this decoupling divergesonly logarithmically.

2.2. Intermembrane Correlations are Measured by X-Ray Diffraction

The kind of collective motions considered previously generate X-ray scattering peaks withlong tails, as shown in Fig. 5. The numerical exponent of the power-law decay (η1) is in fact a(dis)order parameter that describes the collective (smectic) disorder of the membrane stack. Thelarger the value the more flexible (more disordered) the membrane. A simple relationship existsbetween this measurable parameter and two material parameters that describe membrane elastic-ity, namely, the bending rigidity (KC) and the stacking compression parameter (B). KC measuresthe energy needed to bend membranes locally, whereas B measures the harmonic term of inter-bilayer interactions (for mathematical relationships, see Subheading 4). It is possible then to useX-ray diffraction to determine the effect of lipid composition on membrane elasticity (11–14).

2.3. Lipid Dynamics are Probed by Deuterium NMR Spectroscopy

Correlated lipid motions are also measured by solid-state 2H NMR spectroscopy of deuterium-labeled acyl chains (15–17). However, as opposed to X-ray scattering, which measures positionalcorrelations between lipid domains, NMR measures orientational correlations relative to the


Fig. 3. Lipids exhibit anisotropic properties. (Left) A stack of lipid membranes in the fluid statecan be described as a 1D array of 2D fluids. Thermal undulations lead to loss of positional correla-tions of the stacked membranes. (Right) At shorter length scales, the finite thickness of each mem-brane involves additional splay, twist, and bend deformations. The order/disorder properties aredescribed by positional and/or orientational correlation functions. (Figure on right by T. Huber.)

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externally applied magnetic field (2,18). Moreover, one can expect the fluctuations measuredwith 2H NMR to correspond with the quasi-3D limit over short wavelengths on the order of thebilayer thickness (19,20), as shown in Fig. 6. Coupling of the 2H-nuclear electric quadrupolemoment to the electric field gradient of the C–2H bond gives an orientation-dependent


Fig. 4. Caillé-de Gennes smectic liquid crystal model used to analyze experimental X-ray data. (A) Localdeformation of membrane n at point (x,y) is described by the field un(x,y). (B) Mean-square fluctuationsbetween membranes n and n + k increase logarithmically with k, as shown by the (normalized) relative dis-placement function Σ(k) on the right. Collective undulatory motions in liquid crystals generate X-ray scat-tering peaks with long tails, as compared with Gaussians in the case of crystals.

Fig. 5. (A,B) Multilamellar DMPC samples give sharp scattering peaks. (C) High-resolution X-rayscattering reveals long, power-law tails characteristic of smectic liquid crystals (1). The power-lawexponent depends on the (Caillé) order parameter η1, with a harmonic scaling for high-order scatteringpeaks, i.e., η2 = 4η1. (D) Theoretical scattering profiles in log–log plots show this scaling (1).

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perturbation of the magnetic Zeeman Hamiltonian . The two energy gaps shown in Fig. 6A then differ by a certain amount (∆νQ), which is measurable by 2H NMR. Because thedynamics of C–2H bonds vary along the acyl chain (owing to the bilayer depth), the measured∆νQ values differ. A profile of ∆νQ values as a function of segment index is thus generated (21).

Molecular motions affect the 2H NMR measurement in two ways. This is because there aretwo kinds of observables: order parameters, indicative of equilibrium properties, and relax-ation parameters, which characterize the molecular dynamics. As shown below (Subheading2.4), the relationship between these quantities tells us about the types of molecular motionswithin the bilayer. For fluid bilayers, the dynamics of the lipids may encompass significantcontributions from collective motions of the bilayer aggregate, in addition to the localmotions. The observed NMR relaxation rates are then determined by the composite or result-ant (effective) motions of the lipids. Relaxation spectra for 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC) are shown in Fig. 7. Within a continuum description of lipidbilayers, the contribution of collective motions to relaxation depends on material propertiesof lipid bilayers; for example, bilayer softness or membrane deformability over relatively

( ˆ )HZ( ˆ )HQ


Fig. 6. (A) Energy levels and resonance lines showing perturbation of 2H-nuclear Zeeman energy levels by quadrupolar interaction in solid-state 2H NMR spectroscopy. The Zeeman Hamiltonian is perturbed by the quadrupolar Hamiltonian giving an unequal spacing of the nuclear spinenergy levels, indicated by |m| where m = 0, ±1. The residual quadrupolar coupling (RQC) is des-ignated by ∆vQ; it represents the difference in frequencies of the single quantum transitions owing tointeraction of the 2H-nuclear quadrupole moment with the electric field gradient of the C–2H bond. (B)Illustration of the types of fluctuations in the lipids that give rise to NMR relaxation (see ref. 15 foran explanation of coordinate systems). (C) Representative 2H NMR spectra of unoriented powdersamples of polyunsaturated DHA phospholipid bilayers having perdeuterated sn-1 acyl chains.Powder-type spectra of randomly oriented mutilamellar dispersions (right) were numerically inverted(de-Paked) (left) to yield spectra corresponding to the θ = 0° orientation. (In NMR θ is definedwith respect to the normal to the lamellar surface, which differs from X-ray scattering.) Note that adistribution of RQCs is evident, corresponding to contributions from the various C2H2 and C2H3groups. The quadrupolar splittings differ for C–2H bonds along the lipid acyl chain because of variationsin orientational potential for the different C2H2 groups and C2H3 groups (21).

( )υQ±

HQ

HZ

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short length scales (8,22). According to this view, bulk bilayer properties may be emergenton length scales approaching the bilayer thickness, and even less.

Square-law plots of relaxation rates and order parameters constitute a model-free relation betweenthe structural and dynamical 2H NMR observables. The slope of this relationship is predicted the-oretically and found experimentally to be inversely related to the softness of the membranebilayer. When evaluated at the same amplitude of motion as in Fig. 8, 2H NMR relaxation parame-ters provide a naturally calibrated scale for comparison of lipid bilayers. Reference data are providedby benchmark DMPC and its mixtures with cholesterol and a nonionic detergent, whose


Fig. 7. Partially relaxed 2H NMR spectra of randomly oriented DMPC-d54 mutilamellar dispersion inthe liquid-crystalline (Lα) state at T = 44°C. (A) Experimental 2H NMR spectra. (B) Numerically inverted(de-Paked) 2H NMR spectra (θ = 0°). Data were obtained at 76.8 MHz (magnetic field strength of 11.7T )using an inversion-recovery pulse sequence, with the variable delay τ ranging from 5 ms to 3 s (22).

Fig. 8. Dependence of relaxation rate and order parameter profiles for various phospholipidsin the liquid-crystalline (Lα) state. Data for randomly oriented (powder-type) samples were acquired at76.8 MHz (magnetic field strength of 11.7 T ) and numerically inverted (de-Paked). (A) Logarithmicplots are shown, wherein data are compared with the slope of n = 2 predicted for order fluctuations.(B) Square-law functional dependence of corresponding relaxation and order profiles. The square-lawplots of NMR observables indicate differences in membrane rigidities, wherein a steeper slopereflects a softer, more deformable bilayer (23).

S i

CD

( )R i1Z( )

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viscoelastic properties are known. When a detergent is present as cosurfactant, an increase inslope is found (20). In contrast, a bilayer-stiffening agent such as cholesterol or lanosterolleads to a dramatic reduction in the square-law slope (22).

By this method, as an interesting example, significant differences have been found in theviscoelasticity of the ω-3 and ω-6 bilayers, a possibly biologically relevant property that dis-tinguishes between the two fatty acid types (23). One interesting possibility is that anincreased deformability of bilayers containing arachidonic (22:4, ω-6) lipids might berelated to the fatty acid selectivity of phospholipase A2 enzymes. It would be interesting tosee whether X-ray scattering measurements confirm these results.

2.4. Motional Models Describe Collective Excitations of Lipid Bilayers

By considering combinations of deformation modes as shown in Fig. 3, one can in princi-ple derive a number of motional models, ranging from molecular models with local correla-tions to full 3D models that consider both membrane displacement and thickness deformation(16). Continuum models represent bulk bilayer properties, where the length scale of the emer-gence of bulk properties may be more transparent than in molecular dynamics simulations. Theweakness is that molecular detail is relinquished. One can consider a membrane deformationmodel, which assumes two types of motions for the relaxation mechanism. The segmentalmotions are fast when compared with the resonance frequency, and the ordering repre-sents a small correction. In contrast, the slow motions are owing to collective fluctuationsof the lipids, which modulate the residual quadrupolar coupling (RQC) left over from thefast motions, and depend on the square of the local-order parameter. If relatively slowmotions are the major contribution to relaxation, an approximately linear dependence of the spin-lattice relaxation rate vs along the chain (index i) is expected (18), as observed for disaturated lipids (Fig. 8). See Subheading 4, for further details of the mathematicaldescription of NMR relaxation in terms of membrane elasticity.

Further insight into the types of bilayer fluctuations is obtained from the frequencydependence of the relaxation. Motional models can be distinguished based on the frequencydependence of the nuclear spin relaxation, which corresponds to the fluctuating part of theelectric quadrupolar coupling (2). Figure 9 presents spin-lattice relaxation rates for DMPCobtained at a series of different magnetic field strengths, each corresponding to a differentresonance frequency. The relaxation is distinctly nonexponential, with very long relaxationtails being observed in the time domain. In this regard, the data are quite analogous to thecase of X-ray scattering. It is plausible that the origin of long relaxation tails in both cases issimilar, namely that they originate from membrane motions owing to collective interactionsamong the bilayer lipids with each other and with water.

Basically, a model involving only molecular motions is doubtful as an explanation of therelaxation data. A 2D flexible surface model, which is shown by X-ray diffraction to be appli-cable to longer wavelength fluctuations, predicts a ω−1 frequency-dependence and fails toaccount for the 2H R1Z relaxation dispersion. However, such a frequency-dependence explainsthe relaxation dispersion obtained at appreciably lower frequencies in Carr–Purcell–Meiboom–Gillexperiments (24), which may correspond to longer wavelength undulations as seen in X-rayscattering experiments (see Subheading 2.2). For the higher frequency end of the relaxationdispersion (in the MHz regime), a 3D membrane deformation model predicts a ω−1/2 frequencylaw for the R1Z measurements, and explains the experimental data in Fig. 9.

S iCD( )

2R i

1Z( )


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3. ConclusionsAs complementary methods, X-ray scattering and 2H NMR of multilamellar lipid vesicles

reveal a unified view of the hierarchy of molecular motions within biomembranes. Spontaneousthermal fluctuations as well as intermolecular correlation functions can be measured andanalyzed in terms of material parameters relevant to membrane function. For example, anantagonistic effect of cholesterol and polyunsaturated lipids on membrane bending rigidity isfound by both X-ray scattering and NMR spectroscopy. A dual analysis by X-ray and NMRmethods explains the molecular origin of these effects. Continuum analysis of X-ray and 2HNMR data is complementary to all-atom molecular dynamics simulations and bothapproaches have much to offer. Although line shape analysis of either X-ray or NMR meas-urements are not yet common, considerations of motional models (possibly inspired bycomputer simulations) are promising approaches for a full description of biomembranefunction and interactions.

4. Notes4.1. Membrane Deformation and Elasticity–Positional Fluctuations

Scattering methods provide information about molecular positions as in the case of X-ray(1,25) and neutron (14,26,27) diffraction studies of membrane bilayers. To calculate posi-tional correlations owing to spontaneous membrane fluctuations, the Caillé-de Gennes modelconsiders the energies associated with 2D membrane deformations. In the fluid state, lipidmembranes can bend locally with an energy cost proportional to the square of the local mem-brane curvature and KC (28). Interactions between nearest-neighbor membranes are accountedfor by a harmonic compressibility term depending on the material parameter B and the relativemembrane displacement. The interaction per unit area between neighboring membranes of


Fig. 9. Frequency-dependence of relaxation for different motional models. Experimental 2H R1Zrelaxation rates are plotted as a function of frequency for vesicles of DMPC deuterated at the C3 acylsegment in the liquid crystalline state at T = 30°C. Data at 12 different resonance frequencies (magneticfield strengths) are shown, together with theoretical fits to various motional models that are describedin the text. Quasi-elastic lipid motions give long relaxation tails, and govern relaxation mechanismsmeasurable by solid-state 2H NMR spectroscopy. The results allow one to distinguish among various lim-iting relaxation laws (16).

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size L × L is represented as an integral of the deformation terms plus a bare interaction term,V(DW), where DW represents the interlamellar water spacing. This bare interaction term representsthe intermembrane interaction in the absence of fluctuations (1):

(2)

Using statistical mechanics, the mean-square fluctuation of the interbilayer separation canbe calculated in relation to the material parameters KC and B:

(3)

Moreover, correlations between neighboring membranes n and n + k, for k = 1, 2, 3, ... N canbe calculated to determine the scattering correlation function which is given by:

(4)

Here, Σ(k) (shown in Fig. 4B) represents the normalized mean-square fluctuations of the rel-ative displacement between membranes n and n + k, and the Caillé order parameter η1 is afunction of KC and B:

(5)

The Caillé parameter η1 containing information on Kc and B is obtained experimentally byfitting X-ray data using the correlation function in Eq. (4).

4.2. Membrane Deformation and Elasticity–Angular Fluctuations

Solid-state 2H NMR spectroscopy detects angular fluctuations of the 2H-labeled molecules andis complementary to scattering methods. 2H NMR data are obtained at an atomically specific leveland correspond to a membrane ensemble. Temporal fluctuations of the molecules are describedby a liquid-crystalline membrane-deformation model. The bilayer is approximated as a continu-ous material, and the relaxation is assumed to be owing to a distribution of overdamped modes.

In 2H NMR, membrane fluctuations are described by spectral densities, which are the Fouriertransform of correlation functions that depend on changes in the orientation of the C–2H bonds ofthe labeled lipid molecules. The symbols Jm(ω) denote the spectral densities of motion (m = 0, ±1,and ±2), which correspond to the different irreducible components of the second-rank quadrupo-lar-coupling tensor in the laboratory frame. For liquid-crystalline lipids, a composite model formembrane fluctuations is considered (16). Collective bilayer excitations are superimposed withcollective axial rotations of the flexible phospholipid molecules, giving for the spectral densities:

(6)

Spectral densities and correspond to collective fluctuations and molecular rotations, respectively, and is a geometrical cross-term.Jm

mol col− ( )ωJm

mol ( )ωJmcol ( )ω

J J J Jm mmol

mcol

mmol col( ) ( ) ( ) ( )ω ω ω ω= + + −

η π σ π1 2 22

1= =22

B

CD

k T

D BK

G k iq u u q kh +k h2( ) exp exp ( )= − −n n( )[ ] ≅

1

22∆ == − 1exp ( )η h k2∑

σπ

2 2

2

1= − =+u uk T

Bn n 1

B

C

( )K

HNL

dx dy Ku

x

u

y

L L

= + +1 1

220 0

2

2

2

2

2

∫ ∫ ∂∂

∂∂

C

n n 11

2

2

1

B u u V Dn

N

n n+=

− +1 W( )

( )∑


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For collective fluctuations, the spectral densities Jmcol(ω) are given by:

(7)

Here, is a viscoelastic constant that includes the elastic

constant (K), viscosity (η), temperature (T), and the order parameter for collective slowmotions; d is the dimensionality of the bilayer thermal excitations. Euler angles ΩPD trans-form from the frame of the C–2H bond to the director frame (D); they are time-dependent andcorrespond to the fast order parameter . Angles ΩDL are fixed and transform from the

bilayer director to the laboratory axis (L). Formulas for the spectral densities for molecular

motions and the cross-term are given elsewhere (16).Moreover, for distances approaching the bilayer thickness, the dimensionality of the fluc-

tuations comes into play, for example, for 3D vs 2D membranes. According to the above for-mula, collective order fluctuations in 2D are described by a flexible-surface model, whereinthe finite thickness of the membrane bilayer is neglected (d = 2). This gives an ω−1 frequency-dependence, and effectively corresponds to the Caillé-de Gennes model. Alternatively, 3Dorder fluctuations are described by a collective membrane-deformation model (d = 3), whichpredicts an ω−1/2 frequency-dependence of the spectral densities , in analogy with aPincus-de Gennes liquid-crystal model. KC may also differ in the two limits because of a pos-sible length scale dependence. In either case, depends on the square of the observed

order parameter, . The slope of a square-law plot of against (where

i is the chain index) is inversely related to the softness of the membrane.

The C–2H bond segmental order parameters, , are obtained as experimental observ-ables in 2H NMR spectroscopy. They are evaluated from the RQCs (∆νQ) using the relation:

(8)

Here, χQ = e2qQ/h is the static quadrupolar coupling constant (=167 kHz), P2 is the secondLegendre polynomial, and θ is the angle between the bilayer normal and the static magneticfield. 2H NMR spectra in the Lα phase are numerically deconvoluted with the de-Pakeingalgorithm (29–31) to obtain subspectra corresponding to the θ = 0° orientation of the bilayernormal to the main magnetic field.

Finally, information on membrane motions is accessible through measurement of various2H NMR relaxation rates. To compute the relaxation curve, spin-lattice (Zeeman) relaxationtimes (T1Z) corresponding to each of the resolved splittings are obtained by fitting the data toa three-parameter decaying exponential function:

(9)

where R1Z = 1/T1Z is the spin-lattice relaxation rate and τ is a variable delay. R1Z is related to thespectral densities Jm(ω) of the membrane motions by:

(10)R J J1 1 2

3

44 2Z

2Q2

0 0= +π χ ω ω( ) ( )

S S S S R( ) ( ) ( ) ( ) expτ τ= − − −∞ ∞[ ] ( )0 1Z

∆υ χ θQ Q CD= 3

2 2S P (cos )

S iCD( )

S iCD( )

2R i

1Z( )S S SCD f

2s2= ( ) ( )

Jmcol ( )ω

Jmcol ( )ω

Jmmol col− ( )ωJm

mol ( )ω

( )Sf(2)

( )Ss(2)

D k T Scol B1 2

s2= (3 / 5) // / /( )η π21 2 3 2

2

K ( )

J D D Dmcol

PD col2 d 2

m2( )ω ω= − − /

−15

2 002

2( ) ( ) ( )( )Ω ΩΩ ΩDL m2

DL( ) ( )

( )2 2+ 1D


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where ω0 is the deuteron resonance (Larmor) frequency. For membrane lipids, the relaxationdata are described by a simple law of the form:

(11)

where τf is the correlation time for segmental fast motions of the lipids, and a and b are heldconstant. A model for collective bilayer fluctuations explains the combined angular and fre-quency dependencies of the NMR relaxation for lipid bilayers in the fluid phase (16).

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R a b Si i1

1 22

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23 - Brown Lab · solid-state deuterium nuclear magnetic resonance (NMR), can be used as complementary probes of structural fluc-tuations in lipid membranes. In the case of X-ray