Solid-State 2H NMR Studies of Water-Mediated Lipid Membrane Deformation

Trivikram R. Molugu, Xiaolin Xu, Soohyun Lee, K. J. Mallikarjunaiah,and Michael F. Brown

ContentsIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Solid-State 2H NMR Provides Site-Specific Orientational Order Parameters of DeuteratedMembrane Lipids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Mean-Torque Model Allows Calculation of Structural Parameters for Lipid Membranesin the Liquid-Crystalline State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Solid-State 2H NMR Studies of Influences of Lipid Hydration Establish Correspondenceof Molecular Properties to Membrane Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Deuterium NMR Spectroscopy Reveals Emergence of Elasticity from Atomistic-LevelInteractions in Liquid-Crystalline Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Solid-State NMR Relaxation Uncovers Collective Excitations in Phospholipid MembranesDue to Quasielastic Director Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Hierarchical Energy Landscape Explains Spin Relaxation in Phospholipid MembranesDue to Collective and Noncollective Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Spin-Lattice Relaxation in Phospholipid Membranes Indicates Hydration-MediatedCollective Membrane Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

T.R. Molugu • S. LeeDepartment of Chemistry and Biochemistry, University of Arizona, Tucson, AZ, USAe-mail: mtvrao@email.arizona.edu; keivin7shlee@email.arizona.edu

X. XuDepartment of Physics, University of Arizona, Tucson, AZ, USAe-mail: xlxu@email.arizona.edu

K.J. MallikarjunaiahDepartment of Physics, Indian Institute of Science, Bangalore, Indiae-mail: kjmarjun@gmail.com

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

# Springer International Publishing AG 2018G.A. Webb (ed.), Modern Magnetic Resonance,https://doi.org/10.1007/978-3-319-28275-6_143-1

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Transverse Quadrupolar Echo Decay Rates Reveal Hydration-Optimized UltraslowCollective Dynamics of Phospholipid Bilayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Conclusions and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

AbstractThe application of solid-state 2H nuclear magnetic resonance (NMR) spectros-copy gives a powerful approach for investigating hydration-mediated effects onlipid bilayer structure and dynamics. The extent to which lipid bilayers aredeformed by dehydration stress is inherent to understanding how lipid-proteininteractions affect biomembrane functioning. For liquid-crystalline membranes,the average structure is manifested by the segmental order parameters (SCD) of thelipids. Structural quantities, such as the area per lipid and volumetric bilayerthickness, are obtained by a mean-torque analysis of 2H NMR order parameters.Removal of water in the liquid-crystalline state gives a reduction of the mean areaper lipid, together with a corresponding increase in volumetric bilayer thickness.Measurements of order parameters versus osmotic pressure yield the elastic areacompressibility modulus and the corresponding bilayer thickness at an atomisticlevel. Furthermore, solid-state 2H NMR relaxation rates of lipid bilayers atvarying hydration levels afford new insights into the role of water in membranestructural dynamics and viscoelastic properties. Model-free interpretation of spin-

lattice (R1Z) and transverse (RQE2 ) relaxation rates suggests that collective chain

motions described as order-director fluctuations dominantly contribute to therelaxation. In a continuum picture, elastic deformations in such materials arecollective hydrodynamic phenomena with motional time scales spanning manydecades (picoseconds to seconds). The dynamic processes mainly affecting thespin-spin relaxation have characteristic time scales much longer than thosecontributing to spin-lattice relaxation. Such studies probe membrane interactionsinvolving collective bilayer undulations, order-director fluctuations, and lipidmolecular protrusions, giving a unique source of information aboutintermolecular forces pertinent to biomembrane structure and function.

KeywordsLipid bilayers � Liquid crystals �Membranes deformation �Membrane elasticity �Molecular dynamics � NMR relaxation � NMR spectroscopy � Order-directorfluctuations � Osmotic stress

Introduction

Solid-state 2H NMR spectroscopy is highly informative about molecular structureand dynamics at the atomistic level. Particularly its applications to liquid-crystallinematerials such as lipid membranes uncover its potential as a unique tool providingvaluable information. The amphiphilic nature of lipids makes it possible to form

2 T.R. Molugu et al.

different structures, e.g., micelles, bilayers and vesicles, and nonlamellar phases inthe presence of polar liquids like water. Such structures are basically liquid crystal-line in nature and are important for cellular function. The interactions of lipidsinclude a balance of attractive long-range van der Waals forces, together withshorter-range repulsive forces that govern their micro- or nanostructures. Theliquid-crystalline properties of lipid membranes are functions of multiple variables:temperature, lipid composition, hydration, cholesterol, detergents, and osmolytes,leading to a rich polymorphism of lyotropic phases, e.g., solid-ordered (gel), liquid-disordered (smectic liquid-crystalline), liquid-ordered, micellar, hexagonal (HI andHII), and cubic phases [1–3]. Here we discuss the water-mediated lipid propertiesand their biological consequences. Such investigations entail various biophysicalmethods, e.g., solid-state NMR spectroscopy [4–8], X-ray [9–11] and neutronscattering [12–15], and vibrational [3, 16–18] and electronic spectroscopy [19]that are complementary to each other. However, solid-state 2H NMR spectroscopyis contrasted with other methods by uniquely providing multiscale spatial andtemporal information for lipid membrane systems in the functional liquid-crystallineenvironment. Structural information in solid-state NMR is obtained from static ormotionally averaged coupling tensors due to dipolar, chemical shift, or quadrupolarinteractions [4, 8, 20]. Particularly in solid-state 2H NMR spectroscopy, the domi-nating deuterium quadrupolar moment interacts in an orientation-dependent mannerwith the electric field gradient of its bonding environment, and makes it simple toobtain atomistic information site specifically. For membranes with 2H–labeled lipids,this method provides a distribution of residual quadrupolar coupling values, whichgives profiles of the segmental order parameters (SCD) of the lipid hydrocarbonchains [8]. These order profiles in combination with a statistical mean-torque modelprovide acyl packing profiles, representing the cumulative chain extension along thenormal to the aqueous interface, in relation to bilayer structural parameters such asthe area per lipid and bilayer thickness [8, 21–24]. In complementary fashion,dynamical information is acquired from the interaction tensor fluctuations, whichdepend on the mean-squared amplitudes and rates of the motions that affect thelineshapes and relaxation times [4, 25–29].

Solid-State 2H NMR Provides Site-Specific Orientational OrderParameters of Deuterated Membrane Lipids

The solid-state 2H NMR line shapes represent the residual quadrupolar couplings.The static quadrupolar coupling is motionally averaged by local motions, and theresidual tensor (preaveraged by faster motions) is further averaged by slowermolecular motions, or collective fluctuations of the macromolecular assembly. In2H NMR spectroscopy, the structural information is contained in the principal valuesof the electric field gradient coupling tensor, together with the orientation of itsprincipal axes relative to the external magnetic field. Thus, to account for theobserved couplings, one needs to transform the principal axis system (PAS) of thestatic tensor to the laboratory frame through rotation represented by the Euler angles

Solid-State 2H NMR Studies of Water-Mediated Lipid Membrane Deformation 3

ΩPL (see Fig. 1). This transformation includes several intermediate rotations: theC–2H bond with respect to an internal (segmental) frame (ΩPI), the internal frame tothe molecular frame (ΩIM), molecular frame to local symmetry axis n (local director)(ΩMN), from the local director frame to average director frame (ΩND), and finallyfrom the average director to the laboratory frame (ΩDL) [6, 8]. According to Fig. 1,for a uniaxially aligned lipid bilayer, applying the closure property of the rotationgroup allows the overall transformation from the principal axis system of thecoupling tensor to the laboratory frame, described by the ΩPL Euler angles, to betreated as the effect of at least two consecutive rotations. The first transformation,with ΩPD Euler angles, is due to the time-dependent rotation from the principal axissystem to the director frame (whose z-axis is the bilayer normal); and the second,with ΩDL Euler angles, describes the time-independent rotation from the directorframe to the laboratory system:

D2ð Þ00 ΩPLð Þ ¼

Xn

D2ð Þ0n ΩPDð ÞD 2ð Þ

n0 ΩDLð Þ (1)

Because of the cylindrical symmetry about the director, all the Wigner rotationmatrix elements containing the index n are averaged to zero. (Note that Euler’sformula states that e�iϕ = cos ϕ � i sin ϕ, where ϕ = nαDL and all angles αDL areequally probable.)

The above analysis then leads us to the result that

Fig. 1 Depiction of coordinate-frame transformations of the coupling tensor for a lipid bilayer with2H-labeled acyl segments. The overall rotation from the principal axis system (PAS) to thelaboratory frame (defined by the magnetic field B0) is given by the ΩPL Euler angles. Using thegeometry of the system, the transformation is decomposed into several intermediate rotations shownby Ω with subscripts describing multiple reference frames. The z-axes of the various coordinateframes are indicated. Symbols are as follows: principal axis frame (P), internal or intermediatesegmental frame (I), molecular interaction frame (M), local director (N), average director frame (D),and laboratory frame (L) (Figure adapted with permission from Ref. [8])

4 T.R. Molugu et al.

D2ð Þ00 ΩPLð Þ

D E¼ D

2ð Þ00 ΩPDð Þ

D ED

2ð Þ00 ΩDLð Þ (2)

where the C–2H segmental order parameter SCD is defined as [30]

SCD � D2ð Þ00 ΩPDð Þ

D E¼ P2 cos βPDð Þh i (3a)

¼ 1

23 cos2 βPD � 1� �

(3b)

In the above formulation, D2ð Þ00 ΩPDð Þ is a Wigner rotation matrix element,

ΩPD = (0, βPD, 0) are the Euler angles denoting the orientation of the principalaxis system with regard to the director frame, P2(x) is the second-order Legendrepolynomial where x � cos βPD, and βPD is the time-dependent angle of the C–2Hbond axis to the director. The angular brackets represent an average over all themotions whose rates exceed the inverse anisotropy of the static quadrupolar coupling(correlation time< 10�5 s). The outcome of the motional averaging is thus to replacethe static quadrupolar coupling with a residual (or motionally averaged) quadrupolarcoupling. The static coupling is scaled by the segmental order parameter SCD to givethe residual coupling, so that the observed quadrupolar splitting reads

ΔνQ ¼ 3

2χQSCD

3 cos2 βDL � 1

2

� �(4)

Here βDL is the angle between the bilayer normal (director) and the main externalmagnetic field direction. Notably, the segmental order parameters correspond to theaverage structure of the membrane bilayer – they are related to the equilibriumproperties in terms of the configurational statistics of the molecules [21]. A repre-sentative solid-state 2H NMR spectrum for a multilamellar dispersion of DMPC-d54lipids with perdeuterated acyl groups is shown in Fig. 2. The sharp peaks of thepowder-type 2H NMR spectrum (thin line) of the randomly oriented bilayers orig-inate from the bilayer components whose membrane normal is perpendicular to themain magnetic field. The de-Paked spectrum (thick line) corresponds to the parallelcomponents of the membrane normal, and gives a twofold increase and sign reversalof the splittings. The residual quadrupolar couplings are designated by Δν ið Þ

Q and

yield the order parameters S ið ÞCD of the C–2H bonds directly.

Mean-Torque Model Allows Calculation of Structural Parametersfor Lipid Membranes in the Liquid-Crystalline State

To interpret the absolute |SCD| order parameters in terms of structural quantities,models have been introduced at various levels of detail [21, 31]. Because ofthe inherent structural complexity of membranes, usually these models are confined

Solid-State 2H NMR Studies of Water-Mediated Lipid Membrane Deformation 5

to simplified statistical treatments of lipid conformations. In early work, a simplediamond-lattice model was applied to describe the configurational statistics of thepolymethylene acyl chains of lipid bilayers in the fluid state [32–35]. Even so, thisapproach does not treat the influences of the acyl chain positional distribution incalculating the bilayer structural dimensions, and it overestimates the area per lipid atthe aqueous interface [36]. An alternative is a mean-torque potential model [21, 37,38], where statistical mechanical precepts are used to relate the measured orderparameters to the corresponding nanostructure of the lipid assembly.

In what follows, we review the applicability of the mean-torque model inconjunction with the solid-state 2H NMR-derived segmental order parameters tointerpret the membrane structure. The main aim is to connect the lipid structuralparameters to hydrocarbon chain segmental order parameters. For disaturated lipids,a plateau in the 2H NMR-derived segmental order parameter profile is seen near tothe head group. At a certain depth of the bilayer, the effects of chain terminationsbecome significant [39, 40]. Acyl chains on adjacent molecules become moredisordered beyond this point, to keep the packing at hydrocarbon density[41]. Another finding from 2H NMR order profiles is that the plateau region showsa strong chain length dependence, whereas the chain extension profiles show thenonplateau regions are practically independent of length [21]. Changes in structuralparameters due to osmotic stress, or head group and acyl chain composition, are

Fig. 2 Representative solid-state 2H NMR powder-type spectrum (thin line) and de-Paked spec-trum (thick line) for multilamellar dispersion of DMPC-d54 lipids with perdeuterated acyl groups.The sharp peaks of the powder-type 2H NMR spectrum of randomly oriented bilayers correspond tothe membrane normal perpendicular to the main magnetic field (B0). The de-Paked spectrumcorresponds to the membrane normal parallel to B0 and gives a twofold increase and sign reversal

of the splittings. The residual quadrupolar couplings are designated by Δν ið ÞQ and yield the order

parameters S ið ÞCD of the C–2H bonds directly, where i = 2 to 14 is the acyl chain segment index

6 T.R. Molugu et al.

related to the balance of forces that underlie membrane assembly and lipid-proteininteractions [42–44].

Now at the molecular scale, on average each lipid in the membrane occupies aspace that corresponds to the volume and length of the hydrocarbon chainsaccording to:

DC ¼ 2VC

Ah i (5)

In the above formula,DC is the volumetric thickness of the hydrocarbon layer and VC

is the total volume of an individual acyl chain. For Eq. 5 the chain volume VC isgiven by densitometry measurements [45] and is conserved (i.e., constant). Thevolumetric thickness DC and the mean area 〈A〉 are thus inversely related, meaningthat the bilayer core has approximately the density of liquid hydrocarbon [41]. Evenso, the volumetric thickness DC is not the same as the mean projected acyl length[46]. Because of end effects of the acyl chains, the mean travel away from theaqueous interface is less than the distance to the bilayer midplane, as required for theassumption of constant volume to apply [21]. The chain volume at temperature T isfound from the methylene volume VCH2

by using the expression VCH2¼ Vo

CH2þ

αCH2T � 273:15ð Þ , where αCH2

is the isobaric thermal expansion coefficient formethylene groups [21]. It has also been determined that the volume of a methylgroup VCH3

� 2VCH2and that VCH � VCH2

=1:31 for the methine volume [45, 47].When the lipid composition is mixed, neighboring interactions between the

molecules can lead to a change in the average cross-sectional area per lipid. Toavoid considering chain upturns [21], for evaluating the average area per lipid,instead of the complete hydrocarbon chain one can consider the more ordered acylsegments near the head group region. The largest order parameters are due to theplateau region of the 2H NMR order profiles, where it is logical to assume that thesegmental cross-sectional area and projected length are inversely correlated[46]. Hence, it follows that the average cross-sectional area per lipid reads [21]

Ah i ¼ 4VCH2

1

D

� �(6)

where VCH2is the methylene volume [45] and D is the instantaneous travel of an

individual segment along the bilayer normal. The above expression can be furtherreformulated as

Ah i ¼ 4VCH2

DM

q (7)

where DM= 2.54 Å is the maximum projection onto the bilayer normal of the virtualbonds connecting every second carbon atom in the polymethylene chain (Fig. 3), and4VCH2

=DM is the lipid cross-sectional area of the extended all-trans conformation

Solid-State 2H NMR Studies of Water-Mediated Lipid Membrane Deformation 7

[21]. The area factor q is defined as 〈1/ cos β〉 where β is the angle that the virtualbond connecting the two Ci–1 and Ci+1 carbon atoms makes with the normal to thelipid bilayer surface.

Following this thinking, in Eq. 7 the shape of a statistical segment is approxi-mated by a geometrical prism with constant hydrocarbon volume [4]. For Euclideangeometry, the effective acyl segment length is averaged over the motions, whereasthe segmental volume clearly is not. As the amplitudes of the fluctuations increase,the area per lipid does likewise, whereas the volume per segment remains nearlyconstant. A Taylor series expansion about the all-trans reference value then allowsthe area factor q to be expressed by q � 3 � 3〈cos βi〉 + 〈cos2 βi〉 within a har-monic (second order) approximation [21]. The second moment 〈cos2 βi〉 can beacquired directly from the absolute order parameters j S ið Þ

CD j of an individual acylsegment (index i) by

cos2 βi� � ¼ 1� 4S

ið ÞCD

3(8)

To interpret the jSCDj segmental order parameters in terms of structural quantities,various models have been put forth [21]. Because of the inherent complexity ofmembrane structure, these models involve simplified statistical treatments of lipid

Fig. 3 Schematic representation of membrane structure calculations using solid-state 2H and13C–1H NMR spectral data by applying a simple mean-torque model. For a polymethylene chain,a three-carbon segment (virtual bond) is defined from carbon Ci�1 to Ci+1 having length DM = 2.54Ǻ and projection 〈Di〉 onto the molecular (M) axis. The orientation of the z-axis of the intermediate(I) frame to the molecular frame is shown. Using volumetric data, the hydrocarbon thicknessDC andcross-sectional area 〈AC〉 are obtained together with the area per lipid 〈A〉 (see text) (Figure adaptedwith permission from Ref. [6])

8 T.R. Molugu et al.

conformations. Even so, to calculate the mean acyl projections along the bilayernormal, the first moment 〈cos βi〉 of the acyl segment distribution is needed[46]. Calculation of the first moment 〈cos βi〉 , assuming a given value of the secondmoment 〈cos2 βi〉, is further explained below.

The mean-torque model assumes that the orientational order for each chainsegment versus the local director n is described by an orientational potential U(β)(potential of mean torque). Combining the effects of thermal fluctuations, bothsegmental and molecular conformations assume a continuous distribution. Theprobability of finding a statistical segment with a virtual bond orientationβ (� βCH or βCD) at a given instant is given by the Boltzmann distribution,

f βð Þ ¼ 1

Zexp �U βð Þ

kBT

� �(9)

wherein the chain segmental index i is suppressed for clarity. Here the partitionfunction is:

Z ¼ðπ0

exp �U βð ÞkBT

� �sin βdβ (10)

Assuming a first-order mean-torque model [21], we define U(β) = U1(cos β), whereU1 is the first moment of the function U(β) expanded in Legendre polynomials. Theangular-dependent quantities are then integrated together with the distribution func-tion, yielding the following coupled equations:

cos βh i ¼ U1

kBT

� ��1

� cothU1

kBT

� �(11a)

cos2 β� � ¼ 1þ 2

U1

kBT

� ��2

� 2U1

kBT

� ��1

cothU1

kBT

� �(11b)

The analytical solution for 〈cos β〉 is found by introducing the approximation coth(�U1/kBT) � 1, which for an individual segment (index i) then gives the relation

cos βih i ¼ 1

21þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�8S

ið ÞCD � 1

3

s0@

1A (12)

Here the reader should note that for the all-trans conformation of the lipids, 〈cos βi〉= 〈cos2 βi〉 = 1 and hence the area factor q = 1. It follows that Eqs. 11a and 11bgive 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.

Solid-State 2H NMR Studies of Water-Mediated Lipid Membrane Deformation 9

Solid-State 2H NMR Studies of Influences of Lipid HydrationEstablish Correspondence of Molecular Properties to MembraneInteractions

We are now able to address how the structural parameters from solid-state 2H NMRspectroscopy can aid our understanding of the forces governing membrane organi-zation, remodeling, and deformation. Because solid-state 2H NMR spectroscopyyields atomistic insights for liquid-crystalline phospholipids, it can transform ourcomprehension of how the material properties emerge from intermolecular forces[48–51]. The atom-specific 2H NMR approach combined with perturbation bydehydration or osmotic pressure gives a direct avenue for relating molecular prop-erties to the thermodynamics of membrane interactions [36, 52, 53]. Central to thisstrategy is the idea of balancing the chemical potential of the multilamellar lipidphase with the stress due to an external force leading to removal of water. Defor-mation of the membrane lipid phase occurs by reduction of the water volume at theaqueous interface, thereby reducing the area per lipid with a concomitant increase involumetric bilayer thickness. In the case of osmotic pressure, the lipid phase isseparated from the polymer solution by either a semipermeable membrane or avirtual (imaginary) Gibbs dividing surface that bisects the system into thermody-namically distinct lipid and osmolyte phases [49, 54, 55]. Because of an unfavorableloss of entropy, the stressing polymer is excluded from the multilamellar lipid phase.In this way, the polymer solution does reversible work on the lipid phase byremoving water. For the osmolyte phase, the additional pressure increases thechemical potential of the water, which then becomes equal to the solvent chemicalpotential in the lipid phase. Deformation of the lipid phase occurs due to changingthe water volume, with temperature and pressure held constant.

For the lipid phase, the free energy (F) changes with the water volume under theconstant osmotic pressure. Denoting F as the Helmholtz free energy per lipid mole-cule, and nW as the moles of associated waters per lipid, yields dF = μWdnW as thedifferential. Conservation of energy then implies that the reversible work μWdnW doneon the lipid phase is equal but opposite to the work done by the osmolyte phase.Substituting μW ¼ ΠVW for the osmolyte phase yields dF = �ΠdVW in which casethe water volume per lipid is: VW ¼ VnW ¼ υWNW where υW ¼ VW=NA is the(partial) molecular volume of water, NA is the Avogadro number, and NW is thenumber of waters per lipid molecule. Often, it is assumed that the partial molar volumeVW is approximately equal to the water molar volumeV

�W and that it remains constant.

The directly measured removal of water corresponds to the reversible work done onthe lipid phase by changing the bilayer separation, plus any structural deformation ofthe bilayer. The removal of water is performed either osmotically or gravimetrically.By introducing the area per lipid 〈A〉 and the water thickness DW/2 as the latticevariables [21] (see Fig. 4), the total differential reads

dF ¼ @F

@ Ah i� �

DW=2

d Ah i þ @F

@DW=2

� �Ah idDW=2 (13)

10 T.R. Molugu et al.

The above formula says that for the lipid phase, the free energy depends only on thearea per lipid 〈A〉 and the interlamellar water spacing DW/2. We then write the watervolume in terms of the area per lipid molecule and the water spacing for a geomet-rical prism (see Fig. 4). As the result, we obtain VW = 〈A〉DW/2 leading to dF =� ΠDW/2d〈A〉 � Π〈A〉 dDW/2 where the osmotic pressure Π � constant.From the above total differential, the following thermodynamic relations are then

obtained [24, 56, 57]:

@F

@ Ah i� �

DW=2

¼ �ΠDW=2 ¼ τ (14a)

Fig. 4 Schematic representation of a lipid bilayer showing the structural measures from solid-state2H NMR spectroscopy and small-angle X-ray scattering (SAXS). The lamellar structure of thephospholipid membrane is depicted with the relevant structural quantities. Here the lamellar repeatspacing D = DW + DB is the sum of the interlamellar water distance DW = 2DW/2 and the bilayerthicknessDB= 2(DH +DC), whereDC is the hydrocarbon thickness per bilayer leaflet, andDH is thehead group thickness. The average cross-sectional area per lipid 〈A〉 together with the number oflipids (nL) gives the overall surface area of the membrane. Equilibrium structural quantities and thechanges due to bilayer stress conditions give an experimental view of the forces underlying lipidinteractions within the membrane (Figure adapted with permission from Ref. [24])

Solid-State 2H NMR Studies of Water-Mediated Lipid Membrane Deformation 11

and

@F

@DW=2

� �Ah i

¼ �Π Ah i ¼ �FR (14b)

Note that the above results allow us to divide the effect of osmotic pressure into theinfluences of separation forces (FR), and those of lateral tension (τ). To obtain thearea compressibility of the surface film, we substitute the relationDW=2 ¼ VWnW= Ah i

Π into Eq. 14a, which gives us the result that:

τ ¼ � VWnWAh i

� �Π (15)

Here VW is the partial molar volume of water at the bilayer aqueous interface, and Πis the osmotic pressure. For a lipid surface film, the area compressibility is defined as[58]

CA � 1

KA

� 1

Ah i@ Ah i@τ

� �T

¼ �1

VWnW

� �@ Ah i@Π

� �T

(16)

in which KA is the area compressibility modulus. Upon integration over the appliedpressure range, we can then rewrite our expression for the cross-sectional area interms of osmotic pressure as

Ah i ¼ VWnWKA

� �T

Πþ Ah i0 (17)

where 〈A〉0 is the mean cross-sectional area per lipid [21] at null osmotic pressure(full hydration) and constant temperature T. As noted above, it is usually assumedthatVW ¼ VW implying the partial molar volume is approximately equal to the molarvolume of pure water.

Deuterium NMR Spectroscopy Reveals Emergence of Elasticityfrom Atomistic-Level Interactions in Liquid-CrystallineMembranes

Next we can ask: how are the atomistic results of solid-state 2H NMR connected withthe membrane structure and underlying intermolecular forces? Let us now turn tohow 2H NMR spectroscopy allows us to investigate the possibility of membranedeformation due to osmotic stress [59–62]. In Fig. 5 we summarize the arrestingchanges in the solid-state 2H NMR spectra and derived C–2H bond order parameterprofiles obtained for model DMPC-d54 membranes [63] due to dehydration andosmotic stress. The de-Paked 2H NMR spectra are shown at the left of Fig. 5a forDMPC-d54 samples in the liquid-crystalline (liquid-disordered) state, where the

12 T.R. Molugu et al.

water-to-lipid mass ratio is varied gravimetrically. Upon water removal, the observedquadrupolar splittings continuously increase from 30 wt.% H2O (NW = 18) until3.1 wt.% H2O (NW = 1.5) is reached. Moreover, in Fig. 5b at the left, we see thatsimilarly dramatic changes are evident in the 2H NMR spectra of DMPC-d54 whenosmotic stress is introduced by controlling the water activity through exposure topolymer solutions containing polyethylene glycol of molar mass Mr = 1500(PEG-1500). For the de-Paked 2H NMR spectra corresponding to DMPC-d54 sam-ples with different PEG-1500 mass ratios, there is a striking increase of thequadrupolar splittings as the concentration of osmolyte increases, or equivalentlyas the osmotic pressure increases from 0% PEG-1500 (excess hydration) to 87.6%PEG-1500 (NW � 1.3). For either gravimetric dehydration or osmolyte addition, thespectral changes are due to varying the water activity of the samples. Thecorresponding order parameter profiles for DMPC-d54 obtained under conditionsof dehydration or osmotic stress are shown at the right in Fig. 5a, b. In the liquid-crystalline state, the lipids are effectively tethered to the aqueous interface throughtheir polar head groups. Among the various rotational isomeric states (e.g., trans,

Fig. 5 Experimental solid-state 2H NMR spectra for DMPC-d54 bilayer in the liquid-crystalline(liquid-disordered, ld) at 35 �C. Segmental order parameter profiles at 30 �C corresponding todifferent (a) gravimetric hydrations and (b) PEG-1500 osmolyte concentrations. Remarkablysimilar changes in order parameters suggest substantial alterations in membrane structural proper-ties (Figure adapted with permission from Ref. [63])

Solid-State 2H NMR Studies of Water-Mediated Lipid Membrane Deformation 13

gauche�, gaucheþ), correlations of the lipid chains promote their extension (travel)away from the aqueous interface. As the bilayer center is approached, there is aprogressive drop in segmental order due to the effective chain terminations; seeFig. 5a. Because the chain ends are statistically distributed, for the acyl groups withshorter projections along the bilayer normal, greater disorder of the surrounding acylchains is required to fill the voids and maintain the hydrocarbon density � constant[4, 39, 64].

Figure 6 shows the differences observed in the cross-sectional area per lipid forthe DMPC-d54 membrane system when the osmotic pressure is varied [24, 63]. Themean-torque model allows changes in the average cross-sectional area per lipid 〈A〉,bilayer thickness DB = 2DC + 2DH, and water spacing DW to be established[63]. Notably, Fig. 6 reveals that boosting the osmotic pressure up to �200 atm(20 MPa) gives an appreciable reduction of the area per lipid, accompanied by a gainof the volumetric bilayer thickness. According to 2H NMR spectroscopy, the cross-sectional area per lipid shrinks from 60.2 Å2 at full hydration (NW � 20) to 50.2 Å2

(NW � 1.5) for both gravimetric and osmolyte samples at 30 �C. Overall, the lipid

Fig. 6 Cross-sectional area per lipid 〈A〉 for DMPC-d54 bilayer as a function of applied pressure(osmotic, dehydration, or hydrostatic) obtained by mean-torque analysis of solid-state 2H NMRspectral data. Corresponding plots of 〈A〉 against osmotic (П) or bulk (P) pressure are shown. Theelastic area compressibility modulus (KA) is determined from the values of 〈A〉 versus osmotic(squares) or dehydration (circles) pressure at 30 �C. Inset: percentage of total work of bilayerdeformation due to applying osmotic pressure versus cross-sectional area per lipid 〈A〉 for DMPC-d54 in the liquid-crystalline phase at 30 �C. Note that the 2-D compressibility κ⊥ (�1/K⊥) obtainedfrom bulk hydrostatic pressure data (triangles) does not directly involve removal of water. Thus, itdiffers from the value of CA (�1/KA) obtained from osmotic or dehydration pressure data (Data takenwith permission from Ref. [24])

14 T.R. Molugu et al.

cross-sectional area deformation is ΔA = 10 Å2 and represents a 17% area contrac-tion. Correspondingly, the volumetric bilayer thickness DB expands from 43.6 Å(NW � 20) to 48.8 Å (NW � 1.5). The resulting bilayer thickness deformation isΔDB = 5.2 Å giving a 20% swelling of the hydrocarbon thickness (2DC). Lastly, theelastic area compressibility modulus (KA) is calculated as 142� 30 mJ m�2 from theinitial slope of the plot of average cross-sectional area against osmotic pressure(Fig. 6) in accord with Eq. (16). The measured value of KA is in close agreement withthe values reported independently by Koenig et al. [65] (136 � 15 mJ m�2) and byPetrache et al. [62] (108 � 35 mJ m�2), using SAXS and/or solid-state 2H NMRmeasurements. Bulk water has been found to play an important role in lipid-mediated GPCR activation [66] and other membrane protein functions [55,67–69]. These results show that the bilayer deformation due to the lipids alone caninfluence how osmotic stress affects membrane protein activity.

Solid-State NMR Relaxation Uncovers Collective Excitationsin Phospholipid Membranes Due to Quasielastic DirectorFluctuations

Many biological membranes are in the liquid-crystalline phase with a hierarchy ofmotions of the molecules within the bilayer, as manifested by the frequency depen-dence of the 1H, 2H, and 13C nuclear relaxation rates [41, 70–76]. The energylandscape includes fast segmental motions at short length scales, individual lipidmolecular diffusion and collective bilayer fluctuations at intermediate length scales,and collective undulations of larger patches of the membrane at larger length scales(Fig. 7). In biology, the insertion and functioning of membrane proteins are thoughtto be heavily dependent on membrane dynamics and elastic properties. Here solid-state 2H NMR relaxation methods contribute noninvasive, powerful tools to inves-tigate phospholipid membrane dynamics [4, 25, 77–82]. The general type of energylandscape that can be considered is shown schematically in Fig. 7. For segmentalmotions, the C–2H bond vector reorientation is described by the angles ΩPD(t)between the principal axis system (PAS) of the coupling tensor and the director(D), where the remaining transformations are collapsed. Analogously, molecularmotions are considered in terms of the angles ΩMD(t) between the molecular frame(M) and the director. Lastly, collective motions are formulated in terms of fluctua-tions of an instantaneous director (N) relative to the average director (D) and arerepresented by the ΩND(t) angles. Clearly, water is an essential component ofbiomembranes and influences the membrane elasticity, yet how the above motionalprocesses are affected as function of hydration has not received adequate attention.Notably, a comprehensive knowledge of membrane dynamics as a function of thenumber of water molecules per lipid can help to shed light on the optimal membranehydration for their biological function.

Here we give an overview of recent longitudinal (R1Z) and transverse relaxationrate (RQE

2 ) studies of the model phospholipid 1,2-diperdeuteriomyristoyl-sn-glycero-

Solid-State 2H NMR Studies of Water-Mediated Lipid Membrane Deformation 15

3-phosphocholine (DMPC-d54) in the liquid-crystalline phase as a function ofhydration. In combination, the R1Z and RQE

2 rates can report on slow and fastdynamics over multiple decades; hence, one can investigate the emergence of slowcollective motions of the bilayer system related to bulk material properties. Theresults show that collective fluctuations of the lipid segments govern the spin-latticerelaxation in phospholipid membranes. At high hydration levels, such collectivemodes are caused by quasielastic shape fluctuations, together with effective axialrotations of the lipids. Furthermore, slower motional modes possibly involving thelayer undulations are detected in membrane bilayers at optimally high hydrationlevels using transverse quadrupolar relaxation methods.

Hierarchical Energy Landscape Explains Spin Relaxationin Phospholipid Membranes Due to Collective and NoncollectiveMotions

The theory of NMR relaxation considers the behavior of a spin probe in the presenceof a randomly fluctuating, time-dependent perturbation of the Zeeman Hamiltonian,

H tð Þ ¼ HZ þ Hλ tð Þ (18)

Fig. 7 Solid-state NMR spectroscopy uncovers phospholipid membrane dynamics and structureover multiple time scales. The energy landscape of phospholipid mobility characterizes a hierarchyof dynamic processes including segmental fluctuations, molecular diffusion, and viscoelasticmembrane deformation. Orientational fluctuations manifest the geometry of interactions via Eulerangles Ω and by correlation times τc of the motions. (a) Principal axis system of 13C–1H or C–2Hbonds fluctuates due to motions of internal segmental frame (I ) with respect to the membranedirector axis (D). (b) Diffusive phospholipid motions are described by anisotropic reorientation ofthe molecule-fixed frame (M) versus the lipid membrane director axis (D). (c) The liquid-crystallinebilayer involves the propagation of thermally excited quasi-periodic fluctuations in membranecurvature expressed by motion of the local membrane normal (N) relative to the membrane directoraxis (D). The range of time scales is indicated at the bottom (Figure adapted with permissionfrom Ref. [8])

16 T.R. Molugu et al.

Here the perturbing Hamiltonian is expressed as

Hλ ¼ ℏCλ

X2m¼�2

�1ð ÞmT 2ð Þlab�m V 2ð Þlab

m (19)

and corresponds to various tensorial interactions (quadrupolar coupling, dipolarcoupling, or chemical shift, indicated by λ) in an irreducible representation. For thecase of 2H NMR spectroscopy (λ � Q), the perturbing quadrupolar interactiondepends on the second-rank electric field gradient (EFG) coupling tensorV 2ð Þlabm ΩPLð Þ, where the Euler angles ΩPL(t) describe the time dependence of the

principal axis system (PAS) orientation relative to the main magnetic field. Theprincipal values of the EFG tensor are associated with a particular C–2H bond(segment), and we have to calculate the tensor principal components in the laboratoryframe by using the Wigner rotation matrix, leading to

V 2ð Þlabm ¼

X2n¼�2

V 2ð ÞPASn D 2ð Þ

nm ΩPLð Þ (20)

For the case of an axially symmetric static EFG coupling tensor with V2ð ÞPAS0 as the

only nonzero term, the above expression for the coupling tensor components in thelaboratory frame can be reduced to

V 2ð Þlabm ¼ V

2ð ÞPAS0 D

2ð Þ0m ΩPLð Þ (21)

Clearly, the fluctuations of the EFG tensor in a lipid bilayer relative to the laboratoryframe are embedded in the Euler angles ΩPL(t). In general these fluctuations are thecombination of fast segmental motions, noncollective molecular reorientations, andcollective excitations of the bilayer formulated as order-director fluctuations. Notably,the correlation functions of the EFG tensor fluctuations are then expressed in terms ofthe irreducible components using the second-rank Wigner rotation matrix D(2)(ΩPL) by

Gm τð Þ ¼ D2ð Þ0m ΩPL; tþ τð Þ � D

2ð Þ0m ΩPLð Þ

D Eh i�D

2ð Þ0m ΩPL; tð Þ � D

2ð Þ0m ΩPLð Þ

D Eh iD E:

(22)

In the above expression,m= 0,�1, or�2 is the projection index for the second-rankinteractions, and the elements of the Wigner rotation matrix are given elsewhere[4, 8, 83–85]. Here we use the Rose convention for the Wigner rotation matrices[83]. The spectral densities of motion Jm(ω) in the laboratory frame are defined byFourier transformation of the above correlation functions, and read

Jm ωð Þ ¼ Re

ð1�1

Gm τð Þe�iωτdτ (23)

Solid-State 2H NMR Studies of Water-Mediated Lipid Membrane Deformation 17

Based on the Redfield theory, upon evaluating the appropriate matrix elements,the various experimental relaxation rates correspond to the spectral densities ofmotion by

R1Z � 1

T1Z

¼ 3

4π2χ2Q J1 ω0ð Þ þ 4J2 2ω0ð Þ½ (24)

RQE2 � 1

TQE2

¼ 3

4π2χ2Q

3

2J0 0ð Þ þ 5

2J1 ω0ð Þ þ J2 2ω0ð Þ

� �(25)

Here T1Z stands for the spin-lattice relaxation time and TQE2 for the transverse

relaxation time, where R1Z and RQE2 are the corresponding relaxation rates, and

ω0 is the nuclear resonance (Larmor) frequency. Among the spectral densities,J0(0) is prominent for relatively slow motions, whereas all three spectral densitiesJ1(0), J1(ω0), and J2(2ω0) contribute for the case of relatively fast motions. With theabove development in mind, comparison of the R1Z and RQE

2 expressions immedi-ately tells us that for spin-lattice (longitudinal) relaxation, high-frequency relativelyfast motions are manifested, whereas for transverse relaxation the contribution fromslow motions would be more apparent. By conducting both types of relaxationmeasurements, a more complete picture of the hydration-mediated dynamics ispossible.

For example, in the case of spin-lattice relaxation, the experimentally observablerelaxation rates include contributions from relatively fast (f) and slow (s) motions, asdenoted by:

R1Z � 1

T1Z

¼ 1

Tf1Z

þ 1

Ts1Z

(26)

As this level of approximation, cross-correlations between motions of different timescales are neglected, which can be introduced as an additional refinement. The fastrelaxation rate is directly proportional to the correlation time τf in the limit that1/τf > ω0 leading to:

Rf1Z ¼ 1

Tf1Z

¼ A 1� S2f

τf � Aτf (27)

Here S2f is the squared order parameter characterizing the rapid lipid bilayerfluctuations (e.g., trans-gauche rotational isomerization) that modulate the orienta-tion of the PAS of the coupling tensor (e.g., the C–2H bond orientation) to theinstantaneous (local) director, which can be neglected considering that S2f 1 as aninitial approximation [25, 86].

In lipid bilayers, the relatively slow dynamics are characterized by two types offluctuations: collective and noncollective motions. The collective dynamics repre-sents the local director fluctuations, where the instantaneous director or local director

18 T.R. Molugu et al.

fluctuates with respect to the average director (membrane normal). These motionsare hydrodynamic modes, and encompass a continuum of quasielastic deformationsover a range of length scales, spanning the segmental dimensions on up to the fullbilayer size. Potentially, they correspond to the emergent bulk material properties.The spin-lattice relaxation rates due to three-dimensional or two-dimensional direc-tor fluctuations are characterized by a Larmor frequency dependence ofω�1=2

0 orω�10

respectively. The relatively slow noncollective motions have been discussed exten-sively, and correspond to molecular rotations about the principal axes of an averagedinertial tensor identified with the molecular system (parallel and perpendicular to thelong axis). The noncollective molecular rotations are characterized by Lorentzians(with a ω�2

0 frequency dependence in the long correlation time limit). The three-dimensional director fluctuations represent the material deformation (e.g., splay,twist, and bend) approaching the bilayer thickness and less, e.g., as a continuumapproximation for the collective motions of the flexible lipids. On the other hand, thetwo-dimensional director fluctuations are large-scale undulatory motions involvingthe bilayer surface, i.e., due to the interface of the lipids with water. Experimentally,the spin-lattice relaxation rates approximately follow the ω�1=2

0 trend expected fordominant three-dimensional collective motions in the MHz frequency regime. How-ever, the surface undulations in lipid bilayers correspond to collective lipid motionsslower than the time scales of the molecular motions, giving rise to a predicted ω�1

0

dependence.For the DMPC-d54 lipid bilayer, its hydrophobic tails constitute the saturated

polymethylene chains, which can be expected to lead to a stiffer bilayer as comparedwith unsaturated lipid bilayers. For such a system (composed of DMPC-d54 lipidmolecules), the two-dimensional bilayer undulations are expected to be small inamplitude or too slow to be measured by spin-lattice relaxation methods, whereasthree-dimensional collective motions are the major part. In this case, the contributionfrom collective slow motions to the spin-lattice relaxation rate is expressed as:

Rs1Z ¼ 1

Tf1Z

¼ S2f S2sBω

�1=20 (28)

where Sf and Ss are the order parameters for relatively fast and slow motions,respectively. Putting the fast and slow relaxation rates together then gives us theapproximate expression of spin-lattice relaxation,

R1Z � 1

T1Z

¼ Aτf þ BS2CDω�1=2 (29)

Notably, when the amplitude of the relative slow bilayer fluctuations is approxi-mately constant within the bilayer core, the slow relaxation rate will depend on S2fand hence on the square of the experimental order parameter along the acyl chains. Itshould also be emphasized that at this level, the analysis is relatively model free,apart from the assumption of a stationary Markov process, and the specific symmetrycorresponding to the molecular structure as well as the phase of interest. Therefore,

Solid-State 2H NMR Studies of Water-Mediated Lipid Membrane Deformation 19

no model-dependent assumptions are contained in the interpretation of the director-frame spectral densities. For example, the spectral densities within the frame of thedirector can be written in terms of mean-squared amplitudes and reduced spectraldensities for the bilayer fluctuations.

Spin-Lattice Relaxation in Phospholipid Membranes IndicatesHydration-Mediated Collective Membrane Dynamics

What is more, experimentally the R1Z relaxation data for the various segmentalpositions along the DMPC-d54 acyl chains do not vary strongly as a function ofnumber of water molecules per lipid, whereas the order parameters are stronglydependent on hydration (see Fig. 8). Most striking, the spin-lattice relaxation ratesplotted against squared segmental order parameters reveal the expected theoreticallinear functional behavior for all hydration levels investigated. It has been shownthat such a behavior is characteristic of collective dynamics in phospholipid mem-branes. Additional relaxation experiments have been conducted over a wide tem-perature range with varying hydration levels. A square-law functional dependence ofthe R1Z relaxation rates versus the segmental order parameters is observed for allhydration levels at all temperatures studied, implying a strong contribution to thespin-lattice relaxation from the collective membrane dynamics (Fig. 9). The slopesof the square-law plots show a weak temperature dependence at 50 wt% of water andare invariant at lower hydration. For high hydration level samples, excess watersoftens the bilayer membrane, resulting in a smaller value of the elastic constantK (in analogy to springs), which can explain the greater square-law slopes. Because

Fig. 8 Effect of hydration on the segmental longitudinal R1Z relaxation rates. (a) Square-lawrepresentation of R1Z profiles at different hydration levels at 30 �C for DMPC-d54 bilayers in theliquid-crystalline phase. (b) Square-law representation of R1Z profiles at different PEG-1500 levelsat 45 �C for DMPC-d54 bilayers in the liquid-crystalline phase. Such a functional relation indicatesorder fluctuations due to the collective 3-D director fluctuations or noncollective motions of flexiblelipid molecules. The slope of the square-law plot is sensitive to hydration showing that theviscoelastic properties of the membranes vary with the water content

20 T.R. Molugu et al.

the time scales measured by T1Z relaxation are nanoseconds, relatively slow dynam-ics are not accessible to this spin-lattice relaxation method. In addition, at lowerhydration levels, e.g., 10 wt.% water corresponding to 4.5 water molecules per lipid,the bilayer system is underhydrated. For such systems, the slower thermal fluctua-tions are suppressed due to lack of sufficient water molecules, and we end with thesame slope in the square-law plot. If we conclude that the temperature variation ofthe square-law slope manifests stabilization of the bilayer system, then the number ofwater molecules per lipid would most likely be the determining factor.

Transverse Quadrupolar Echo Decay Rates Reveal Hydration-Optimized Ultraslow Collective Dynamics of PhospholipidBilayers

By contrast, to study the slow dynamics not accessible from spin-lattice relaxation,one can employ transverse relaxation rate measurements, which are dominated bycontributions from relatively slow motions. The characteristic time scale of thefluctuations (up to milliseconds) is appreciably longer than the time scale (nanosec-onds) covered by the R1Z measurements (see above). Upon increasing the bilayerhydration, the water molecules mainly increase the area per lipid accompanied by adecrease in bilayer thickness, with potentially greater penetration into the bilayerinterfacial region. As a result, the transverse RQE

2 rates are dramatically enhancedwith increasing hydration levels at all temperatures (Fig. 10). Even so, fastercollective motions occurring deeper within the bilayer core can remain relatively

Fig. 9 Square-law representation of R1Z relaxation rates versus segmental order parameters (SCD)at various temperatures. (a) 10 wt.% of H2O and (b) 87 wt.% of PEG-1500 for DMPC-d54 bilayersin the liquid-crystalline phase. The slopes of square-law profiles are temperature independent. Thesquare-law functional dependence of R1Z and SCD order parameters indicates that in DMPC-d54multilamellar dispersions, the spin-lattice relaxation is dominantly contributed by the 3-D collectivedirector fluctuations at the Larmor frequencies studied. Consequently, the membrane hydrocarboninterior behaves analogously to a nematic-like liquid crystal

Solid-State 2H NMR Studies of Water-Mediated Lipid Membrane Deformation 21

unaffected. Because of the increased hydrated area per lipid, accompanied by greaterwater penetration into the interface, the membrane surface is modified, leading tobreaking of the square law in Fig. 10. The emergence of two-dimensional collectivebilayer undulations with longer wavelengths reduces the frequency dependencefrom ω�1=2

0 to a mixture of ω�1=20 and ω�1

0 , thus giving a diminution of the square-law slope as seen experimentally. We also note that the breaking of the square lawoccurs with higher order parameter values, which correspond to carbon positionscloser to the lipid head groups. This finding makes sense because these positions arecloser to the water molecules, and thus have strong correlations to the lipid polarhead groups.

Conclusions and Future Directions

Amphiphilic phospholipids form lyotropic liquid-crystalline membranes in the pres-ence of polar solvents like water. These membranes are deformable elastic materials,and solvent concentration plays an important role in their liquid-crystalline behavior.Hydrated lipids form liquid crystals exhibiting long-range order in their molecularorganization. A statistical mean-torque model characterizes membrane deformationson the order of the bilayer thickness, and combined with osmotic pressure measure-ments, the elastic moduli for membrane deformation can be determined. The corre-spondence to bulk material values shows the emergence of bulk material propertiesover mesoscopic distance scales approaching the bilayer thickness. The additionalapplications of 2H spin-lattice and spin-spin (transverse) relaxation methods allowone to begin to understand the connection between the atomistic structure and thebulk material properties. In terms of a continuum picture, elastic deformations in

Fig. 10 Square-law representation ofRQE2 profiles at different hydration levels at 45 �C for DMPC-d54

bilayers in the liquid-crystalline phase. At low hydration (3 wt.% of H2O) the R2QE rates follow a

functional square-law: however, at higher hydration the R2QE rates are enhanced and the square law

breaks down. This behavior may indicate the presence of slower dynamics possibly involving 2-D layerundulations at high hydration

22 T.R. Molugu et al.

such materials are manifested as director fluctuations, which are collective hydro-dynamic phenomena with motional time scales spanning many decades (picosec-onds to seconds). These fluctuations cause 2H NMR relaxation corresponding todynamic parameters of the mesogenic organization. Solid-state 2H longitudinal (R1Z)rates for acyl-chain-perdeuterated lipid multilamellar dispersions follow an empiricalsquare-law functional behavior on the segmental order parameters, showing theemergence of 3-D director fluctuations. The dynamic processes mainly affectingthe spin-spin relaxation have characteristic time scales much longer than thosecontributing to spin-lattice relaxation. Transverse nuclear spin relaxation experi-ments thus constitute an additional powerful tool for probing slow dynamic pro-cesses, such as fluctuations of the orientational director in partially ordered phases.EnhancedRQE

2 rates indicate additional contributions from slower dynamics, whereasbreaking of the square-law suggests the emergence of undulations as 2-D directorfluctuations in phospholipid membranes. The square-law confinement suggestsrestricted water penetration into the hydrophobic bilayer interior, in which the lipidhydration at the transition of 3-D to 2-D director fluctuations may have importantbiological consequences for biomembrane functioning.

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