Cholesterol Effects on the Physical Properties of Lipid Membranes Viewed by Solid-state NMR Spectroscopy

Abstract

In this chapter, we review the physical properties of lipid/cholesterol mixtures involving studies of model membranes using solid-state NMR spectroscopy. The approach allows one to quantify the average membrane structure, fluctuations, and elastic deformation upon cholesterol interaction. Emphasis is placed on understanding the membrane structural deformation and emergent fluctuations at an atomistic level. Lineshape measurements using solid-state NMR spectroscopy give equilibrium structural properties, while relaxation time measurements study the molecular dynamics over a wide timescale range. The equilibrium properties of glycerophospholipids, sphingolipids, and their binary and tertiary mixtures with cholesterol are accessible. Nonideal mixing of cholesterol with other lipids explains the occurrence of liquid-ordered domains. The entropic loss upon addition of cholesterol to sphingolipids is less than for glycerophospholipids, and may drive formation of lipid rafts. The functional dependence of ²H NMR spin–lattice relaxation (R _1Z) rates on segmental order parameters (S _CD) for lipid membranes is indicative of emergent viscoelastic properties. Addition of cholesterol shows stiffening of the bilayer relative to the pure lipids and this effect is diminished for lanosterol. Opposite influences of cholesterol and detergents on collective dynamics and elasticity at an atomistic scale can potentially affect lipid raft formation in cellular membranes.

1 Introduction

Cholesterol is an essential component of mammalian cells constituting up to about 50 mol% of the total lipids in the plasma membrane [1, 2]. It plays key roles in maintaining the membrane integrity and regulating cell permeability [3]. Cellular functions are modulated by cholesterol in two different ways: either indirectly by affecting membrane properties due to lipid-cholesterol interactions [4,5,6,7] or directly by cholesterol-protein interactions [8,9,10]. As a steroid molecule, cholesterol has four fused rings and a hydrocarbon chain, which are hydrophobic, and a polar hydroxyl group making it overall amphiphilic as for other lipids (Fig. 1). Due to its chemical composition, most of the steroidal core structure fits into the lipid bilayer, with the fused rings in close proximity and interacting with the hydrocarbon chains of the lipids, while the polar hydroxyl group orients the molecule at the aqueous interface. When cholesterol is present in the liquid-ordered (l _o) phase, the lipid hydrocarbon chains become more ordered relative to the cholesterol-free (liquid-disordered, l _d) state. The lipid chain ordering induces alterations in the structural, dynamical [12], and physical properties of the lipid membrane, and thus can affect cellular signaling. The balance of attractive and repulsive forces for the membrane lipids is the consequence of both the polar headgroups and the nonpolar moieties, and yields a substantial polymorphism involving both lamellar and nonlamellar phases [13,14,15,16,17,18,19,20]. Notably, the structural and dynamical properties of biomembranes are mediated by the lipid composition and interactions with the proteins, water, cholesterol, and surfactants [7, 21,22,23,24,25,26,27,28,29,30,31,32,33]. Membrane remodeling requires mesoscopic elastic deformations of the lipids [30] that can play a central role in biological functioning with regard to lipid-protein interactions [6, 34, 35], domain formation, and various nano- and microstructures implicated in key cellular functions [34, 36,37,38]. Because the composition of lipids and cholesterol in cell membranes influences the lipid organization and membrane properties, it is not surprising that disorders in cellular cholesterol levels are implicated in various diseases [39, 40].

Fig. 1

Selected glycerophospholipid chemical structures, cholesterol, and lanosterol . The polar head groups vary in size, hydrogen bonding, and charge. Examples are shown for zwitterionic phosphocholine (PC) and phosphoethanolamine (PE) head groups, and for the anionic phosphoserine (PS) head group. Nonpolar acyl chains differ in length and degree of unsaturation, as illustrated by oleic acid (18:1ω-9) and docosahexaenoic acid (22:6ω-3). Cholesterol differs in the absence of methylation at the α-face relative to its biological precursor lanosterol. Figure adapted from [11]

Many of the molecular species of lipids and proteins in membranes do not mix ideally [19, 20, 41,42,43,44,45,46,47]—cholesterol is one such component that is unequally distributed in cellular membranes, allowing for the presence of liquid-ordered raft-like domains [19, 20, 32, 42, 46,47,48,49,50,51,52]. Such raft-like domains have attracted considerable attention as platforms for signaling regulation in cellular biology and pharmacology [53,54,55,56,57,58,59,60]. The concept that biomembranes are homogeneous two-dimensional fluids with randomly distributed proteins (the fluid mosaic model ) is challenged by the hypothesis that cellular membranes may contain such areas of lateral segregation [20, 55, 61,62,63,64,65,66,67,68,69,70,71,72,73]. By analysis of the liquid-ordered (l _o) phase [74, 75] and its possible connection with rafts in biological membranes [76, 77], the effect of cholesterol on domain formation in liquid-crystalline bilayers has been of great interest for membrane biophysicists [78]. For instance, raft-like domains are believed to occur in lipid systems with coexisting liquid-disordered (l _d) and liquid-ordered (l _o) phases . The l _d phase in these systems typically contains highly unsaturated lipids with a low phase-transition temperature, while the l _o phase predominantly consists of saturated glycerophospholipids or a sphingolipid component and cholesterol [2, 77]. Moreover, various proteins are endowed with the ability to interact with cholesterol via specific sensing mechanisms. The cholesterol recognition/interaction amino acid consensus (CRAC ) sequence motif [79,80,81,82] is an example of such a recognition element. In some cases, they include cationic clusters that allow interactions with phosphatidylinositol 4,5-bisphosphate (PIP₂ ) in a cholesterol-dependent manner. Notably, such CRAC domains are found in the Rhodopsin (Family A) G-protein-coupled receptors (GPCRs ) [83], and moreover posttranslational lipid modifications [84, 85] can promote sequestration into cholesterol-rich regions or microdomains.

Improving our understanding of complex lipid mixtures is an important aspect for current research in pharmacology, biophysics, and biochemistry as well as in cellular biology . Various experimental methods have been used to study lipid–cholesterol interactions, including electron spin resonance (ESR) [86,87,88,89,90,91,92,93,94], Raman [95,96,97], Fourier transform infrared (FT-IR) [98, 99], and fluorescence spectroscopy [100,101,102]; atomic force microscopy (AFM ) [103,104,105]; multidimensional nuclear magnetic resonance (NMR) spectroscopy [70, 73, 106, 107]; solid-state NMR spectroscopy [65, 101, 108,109,110,111,112,113,114,115,116,117,118,119]; and X-ray [72, 120, 121] and neutron diffraction methods [122,123,124]. Still, a thorough understanding of the physical basis for these observations in relation to the intricate lipid compositions of many biological membranes remains a conundrum [20, 51, 71, 125, 126]. This chapter covers recent developments in understanding lipid–cholesterol interactions in model membrane systems and implications for cellular function as seen by solid-state nuclear magnetic resonance (NMR) spectroscopy. First, we give a brief introduction to solid-state NMR methods for nonexperts to appreciate the results as applied to lipid-cholesterol systems. Next, we explain how solid-state NMR technology is applied for obtaining membrane structural and dynamical properties. We then discuss the sterol interactions with phospholipids in model membranes, including the role of configurational entropy in lipid raft formation. Emphasis is placed on how the average material properties emerge from the atomistic level interactions in lipid bilayers, as investigated by combining NMR spectroscopy with relaxation methods [11, 111, 112, 127].

2 Solid-State NMR Spectroscopy of Lipid Membranes

Solid-state NMR spectroscopy [29, 128,129,130] offers a versatile and noninvasive method for studying the molecular organization of lipids within membranes. In particular, isotopic substitution of ²H for ¹H constitutes a minimal structural perturbation [131]. Interpretation of the ²H NMR spectra is relatively straightforward, due to the intramolecular nature of the quadrupolar interaction that dominates the lineshape. There are a number of reviews that give a comprehensive treatment of ²H NMR spectral analysis [30, 128, 129, 131,132,133,134,135]. An essential feature of ²H NMR spectroscopy is that one introduces site-specific ²H-labels, corresponding to the individual C–²H bonds. In this way, we obtain atomistically resolved information for noncrystalline amorphous or liquid-crystalline systems. Because the coupling interactions in solid-state NMR are sensitive to orientation and/or distance, their values correspond to the average structure of the system of interest. On the other hand, molecular motions are manifested by the relaxation parameters that are also accessible in NMR spectroscopy. A unique feature is that in solid-state ²H NMR of biomolecular systems, we acquire both lineshape data [118] and relaxation times [136] for investigating structural dynamics. Measurement of the ²H NMR lineshapes yields knowledge of the average structure through the principal values of the coupling tensor as well as the principal axis system. Yet, if we only determine the coupling tensors, then the method mainly provides us with structural knowledge as in X-ray crystallography. An important feature of solid-state ²H NMR spectroscopy is that information is also obtained regarding the molecular motions, encompassing a range of different timescales. Through the combined measurement of residual quadrupolar couplings (RQCs ) and relaxation rates, we thereby obtain knowledge of the geometry as well as investigate the multiscale molecular motions and their amplitudes in the membrane systems of interest. An analogous approach is introduced in the case of solid-state ¹³C NMR spectroscopy of membrane lipids and biomembranes at natural isotopic abundance.

2.1 Deuterium Solid-State NMR Spectroscopy

Notably, ²H NMR spectroscopy gives us a particularly simple illustration of the principles of magnetic resonance as applied to molecular solids, liquid crystals, and biomembranes. This is because the very large electric quadrupolar interaction dominates over the magnetic dipolar couplings of the ²H and ¹H nuclei as well as the ²H chemical shifts [134, 136]. The ²H nucleus has a spin of I = 1, and hence there are three Zeeman energy levels due to projecting the nuclear spin angular momentum onto the magnetic field direction. The three eigenstates |m〉 = |0〉 and |±1〉 are given by the Hamiltonian ${\widehat{H}}_Z$ for interaction of the nuclear magnetic moment with the static magnetic field. We learn in quantum mechanics that the transitions between adjacent spin energy levels are allowed, which yields the two single-quantum nuclear spin transitions. Moreover, the degeneracy of the allowed transitions in ²H NMR is removed by the quadrupolar coupling. Here, the perturbing Hamiltonian ${\widehat{H}}_{\mathrm{Q}}$ is due to interaction of the quadrupole moment of the ²H nucleus with the electric field gradient (EFG ) of the C–²H bond. It follows that for each inequivalent site, two spectral branches are observed in the experimental spectrum.

In solid-state ²H NMR spectroscopy, the experimentally observed quadrupolar coupling is given by the difference in the frequencies $\Delta {\nu}_{\mathrm{Q}}^{\pm}\equiv \Delta {\nu}_{\mathrm{Q}}^{+}-\Delta {\nu}_{\mathrm{Q}}^{-}$ of the spectral lines due to the perturbing Hamiltonian. The result for the quadrupolar frequencies (${\nu}_{\mathrm{Q}}^{\pm }$) thus reads:

$$ {\nu}_{\mathrm{Q}}^{\pm }=\pm \frac{3}{4}{\chi}_{\mathrm{Q}}\left\{{D}_{00}^{(2)}\left({\Omega}_{\mathrm{PL}}\right)-\frac{\eta_{\mathrm{Q}}}{\sqrt{6}}\left[{D}_{-20}^{(2)}\left({\Omega}_{\mathrm{PL}}\right)+{D}_{20}^{(2)}\left({\Omega}_{\mathrm{PL}}\right)\right]\right\}. $$

(1)

In the above formula, χ _Q ≡ e ² qQ/h is the static quadrupolar coupling constant, η _Q corresponds to the asymmetry parameter of the EFG tensor, ${D}_{00}^{(2)}\left({\Omega}_{\mathrm{PL}}\right)$ is a Wigner rotation matrix element, and Ω_PL ≡ (α _PL, β _PL, γ _PN) are the Euler angles [137] relating the principal axis system (PAS ) of the EFG tensor (P) to the laboratory frame (L) [29, 128, 136]. Furthermore, it turns out that the static EFG tensor of the C–²H bond is nearly axially symmetric (η _Q ≈ 0), which leads us to the simpler result:

$$ {\nu}_{\mathrm{Q}}^{\pm }=\pm \frac{3}{4}{\chi}_{\mathrm{Q}}\kern0.1em {D}_{00}^{(2)}\left({\Omega}_{\mathrm{PL}}\right). $$

(2)

The experimental quadrupolar splitting is thus given by:

$$ \Delta \kern0.1em {\nu}_{\mathrm{Q}}=\frac{3}{2}{\chi}_{\mathrm{Q}}\kern0.1em {D}_{00}^{(2)}\left({\Omega}_{\mathrm{PL}}\right). $$

(3)

Now in liquid-crystalline membranes, the motions of the constituent molecules are often cylindrically symmetric about the bilayer normal, an axis known as the director. The overall rotation of the principal axis system of the coupling tensor to the laboratory frame, described by the Ω_PL Euler angles , can thus be represented by the effect of two consecutive rotations. First, the Euler angles Ω_PD(t) represent the (time-dependent) rotation from the principal axis frame to the director frame, and second the Euler angles Ω_DL represent the (static) rotation from the director frame to laboratory frame. Using the closure property of the rotation group [136], and considering the cylindrical symmetry about the director, we can then expand Eq. (3), which now reads

$$ \Delta \kern0.1em {\nu}_{\mathrm{Q}}=\frac{3}{2}{\chi}_{\mathrm{Q}}\kern0.1em \left\langle {D}_{00}^{(2)}\left({\Omega}_{\mathrm{PD}}\right)\right\rangle {D}_{00}^{(2)}\left({\Omega}_{\mathrm{DL}}\right) $$

(4a)

$$ =\frac{3}{2}{\chi}_{\mathrm{Q}}\frac{1}{2}\left\langle 3{\cos}^2{\beta}_{\mathrm{PD}}-1\right\rangle \frac{1}{2}\left(3{\cos}^2\kern0.1em {\beta}_{\mathrm{DL}}-1\right). $$

(4b)

Here, β _DL ≡ θ is the angle of the bilayer normal to the static external magnetic field. The segmental order parameter S _CD is given by:

$$ {S}_{\mathrm{CD}}=\frac{1}{2}\left\langle 3{\cos}^2{\beta}_{\mathrm{PD}}-1\right\rangle $$

(5)

where the angular brackets denote a time/ensemble average. It follows that

$$ \Delta \kern0.1em {\nu}_{\mathrm{Q}}=\frac{3}{2}{\chi}_{\mathrm{Q}}\kern0.1em {S}_{\mathrm{CD}}\kern0.1em {P}_2\left(\cos \kern0.3em {\beta}_{\mathrm{DL}}\right) $$

(6)

where P ₂(cos β _DL) ≡ (3cos² β _DL − 1)/2 is the second-order Legendre polynomial. The above expression describes the dependence of the quadrupolar splitting on the (Euler) angles that rotate the coupling tensor from its principal axes system to the laboratory frame, as defined by the main magnetic field. Figure 2 illustrates the effect of cholesterol on the solid-state deuterium NMR lineshape for 1,2-diperdeuteriomyristoyl-sn-glycero-3-phosphocholine (DMPC-d ₅₄) bilayers. A gradual increase in the quadrupolar splittings (Δν _Q) for the acyl segments is observed as cholesterol concentration is increased, which reflects increased orientational order of the acyl chain segments with respect to the bilayer normal. This finding explains the well-known condensing effect of cholesterol at the molecular level, involving a reduction of the area per phospholipid molecule at the aqueous interface, accompanied by an increase in the bilayer hydrocarbon thickness.

Fig. 2

Solid-state ²H NMR spectra demonstrate the reduction in conformational degrees of freedom for the acyl chain orientation due to cholesterol : (a) DMPC-d ₅₄ in the liquid-disordered (l _d)phase, and (b–d) DMPC-d54 containing various mole fractions of cholesterol in the liquid-ordered (l _o) phase. Data were acquired at a magnetic field strength of 11.7 T (76.8 MHz) at T = 44 °C. Powder-type spectra (green) of randomly oriented multilamellar dispersions were numerically inverted (de-Paked) to yield sub-spectra corresponding to the θ = 0° orientation (blue). Note that a distribution of residual quadrupolar couplings (RQCs ) corresponds to the various C²H₂ and C²H₃ groups with a progressive increase due to cholesterol. Data are taken from [112]

We can now introduce simplifying precepts from statistical mechanics to further explain the solid-state NMR lineshapes in terms of membrane structural dynamics. For detailed explanations and applications of the statistical mean-torque theory, the interested reader is referred to the literature [30, 65, 70, 73, 138]. The microscopic observables from ²H NMR spectroscopy can then be related to the nano- or microstructure of the membrane lipid assembly. Structural quantities of interest for the lamellar state correspond to the mean interfacial area 〈A〉, together with the average thickness D _C of the bilayer hydrocarbon region, and the mean aqueous distance separating the lamellae [28, 128, 139,140,141,142]. Clearly, the area per lipid molecule plays an important role in molecular dynamics (MD) simulations of lipid membranes [71, 124, 142,143,144,145,146,147,148]. The various nanostructures are the result of a balance of forces acting at the level of the polar head groups and hydrocarbon chains [5, 13, 16, 138, 149, 150]. Notably, the deformation of a membrane film away from the equilibrium state is characterized by four material constants: (1) the surface tension γ (which is zero for a membrane bilayer at equilibrium), (2) the area expansion modulus K _A or alternatively the lateral compressibility C _A ≡ 1/K _A, (3) the bending rigidity K _C, and (4) the monolayer spontaneous H ₀ curvature. The above structural quantities are fundamental to the forces governing the nano- and microstructures of assemblies of membrane lipids and amphiphiles. Representative applications of solid-state ²H NMR spectroscopy to lipid membranes include studies of the influences of cholesterol [151,152,153] as well as acyl chain polyunsaturation [36, 67, 143, 150, 154,155,156,157,158].

2.2 Temperature–Composition Phase Diagram of Lipid/Sterol Mixtures

Lipid-lipid interactions have been investigated for a variety of systems and are important determinants of membrane organization. The lipid composition of the outer leaflet of a typical mammalian plasma membrane mainly comprises sphingomyelin (SM) , cholesterol, and unsaturated phosphatidylcholine (PC) [59, 159]. Preferential association of SM and cholesterol has been observed in various membrane environments including enrichment in detergent-resistant membranes [53, 160]; formation of ordered phases in three-component giant unilamellar vesicles [64]; and the direct observation of small domains in the membranes of living cells [161]. Despite the research efforts by many groups into lipid-cholesterol interactions, however, important questions remain concerning the role of cholesterol and other sterols in forming the tightly packed liquid-ordered (l _o) phase. For instance, liquid-disordered (l _d) and (l _o)phases coexist in 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC )/sterol model membranes [75, 162]. Still, the two phases remain submicroscopic in size. Because their small size makes these dynamic domains invisible by most imaging techniques, it has been postulated that l _d/l _o phase coexistence is absent in DPPC/cholesterol model membranes. An alternative explanation for membrane heterogeneity invokes lipid compositional fluctuations due to proximity to a critical point [126, 163, 164]. Complicating matters further in biological membranes is the asymmetric distribution of lipid molecules between the inner and outer leaflets [165]. In this regard, theoretical descriptions of phospholipid/sterol interactions are extensive [74, 166,167,168,169,170]. The original PC/sterol condensed complex model [166] has been modified to predict phase coexistence in DPPC /cholesterol model membranes above the main transition temperature [171] as well as DPPC phase diagrams with ergosterol [172] and lanosterol [173]. Research in this area remains very active as molecular simulations improve, and experimental input parameters become more accurate [174].

Using solid-state ²H NMR spectroscopy, influences of cholesterol on both the lipid head groups [151] and the acyl chains [152] have been extensively investigated. Cholesterol is located beneath the polar head groups, where it can interact strongly with the acyl chains and act as a spacer molecule, as first suggested [151] in terms of an umbrella-like model. Influences of the acyl chain ordering and dynamics are further discussed below. Previous NMR spectra of DPPC gave the first detailed phase diagram of this cholesterol/lipid mixture [74, 75] and entailed acquiring solid-state ²H NMR spectra for temperatures from 25 to 70 °C and cholesterol concentrations from 0 to 30%. Recently, the binary cholesterol/DPPC phase diagram has been confirmed and extended with 1-palmitoyl-2-perdeuteriopalmitoyl-sn-glycero-3-phosphocholine (DPPC-d ₃₁)/cholesterol membranes [175]. Analogous experiments have been conducted for DPPC-d ₆₂ using ergosterol [162] and lanosterol [173] as well as N-perdeuteriopalmitoyl-d-erythro-sphingosylphosphorylcholine (PSM-d ₃₁) . These comparative studies allow insightful comparisons to be made for data obtained using palmitoyl chains with similar conformations. Partial phase diagrams for multilamellar dispersions of sn-2 palmitoyl chain-perdeuterated DPPC-d ₃₁/cholesterol and for N-linked palmitoyl chain-perdeuterated PSM-d ₃₁/cholesterol mixtures have been constructed solely from ²H NMR measurements. Examples of the phase diagrams for the DPPC-d ₃₁/cholesterol and PSM-d ₃₁/cholesterol multilamellar dispersions obtained from the ²H NMR spectral analysis for various temperatures and cholesterol compositions are shown in Fig. 3. The phase diagrams are very similar and exhibit solid-ordered (s _o) + (l _o) and l _d + l _o coexistence regions with a clear three-phase line separating them [75, 175]. Three regions of two-phase coexistence and a three-phase line are identified in DPPC -d ₃₁/cholesterol multilamellar dispersions, as well as in the PSM-d ₃₁/cholesterol multilamellar dispersions, thus confirming the basic features of the phase diagram [75]. Narrowing of the l _d + l _o coexistence region at high temperature implies a critical point that corresponds to a cholesterol concentration between ~25 and 30 mol%.

Fig. 3

Partial phase diagrams of: (a) PSM-d ₃₁/cholesterol and (b) DPPC -d ₃₁/cholesterol membranes obtained from solid-state ²H NMR spectroscopy. Symbols denote (filled circles) the midpoint of the transition obtained from the de-Paked ²H NMR spectra as a function of temperature; the onset or end of the transition as obtained (open triangles) by inspection of the temperature-dependent spectra, (open squares) from spectral subtractions, (open diamonds) from first-moment M ₁ curves, or (filled diamonds) by inspection of the de-Paked spectra against cholesterol concentration; and lastly (filled triangles) the onset of the transition in the first-moment M ₁(T) curves. Data are for multilamellar dispersions having cholesterol concentrations of 25 or 28.1 mol% in (a) PSM-d ₃₁/cholesterol bilayers and 25 mol% in (b) DPPC -d ₃₁/cholesterol bilayers. Figure modified from [175]

Notably, macroscopic (micron-sized) coexistence of l _d and l _o phases is not observed with solid-state ²H NMR spectroscopy. Instead, spectral line broadening in the l _d + l _o coexistence region points to intermediate exchange of lipids between the two types of domains. The length scales associated with the domains are estimated to be 75–150 nm for DPPC-d ₃₁/cholesterol and PSM-d ₃₁/cholesterol model membrane mixtures [175]. Distances have been estimated between l _d/l _o domain interfaces in DPPC /cholesterol and PSM/cholesterol multilamellar dispersions to be ~80 nm. The sizes are on the order of the dimensions of putative rafts in cellular membranes [165]. Hence, if rafts exist in cell membranes, they will be strongly governed by lipid/lipid interactions. The high cholesterol concentration found in many plasma membranes suggests that their preferred physical organization is most likely similar to the l _o phase. With the availability of a detailed picture of the DPPC/cholesterol and PSM/cholesterol binary systems in hand, future ²H NMR studies of ternary raft-like systems [65, 113] can be then conducted to gain added insight into the behavior of lipid rafts in cellular plasma membranes.

2.3 Separated Local-Field ¹³C NMR Spectroscopy: ¹³C–¹H Dipolar Couplings Allow Calculations of Lipid Bilayer Structure

Lipid membrane systems clearly can benefit from precise structural characterization using solid-state NMR methods. As explained in the previous section, solid-state ²H NMR spectroscopy is prominent among these methods. However, such applications are foreshadowed by the need for ²H-isotope labeling involving synthetic organic chemistry [154, 176]. By the introduction of dipolar-recoupling methods, it is possible to extend the approaches originally developed with regard to solid-state ²H NMR spectroscopy to other classes of biologically relevant lipids, as they occur in a membrane environment. In this regard, separated local-field (SLF) ¹³C NMR spectroscopy [70, 73, 115, 177, 178] at natural abundance expands the range of applications of solid-state NMR spectroscopy in membrane biophysics. For example, sphingolipids and other natural lipids may be investigated, together with their interactions with cholesterol in raft-like lipid mixtures and with membrane proteins. Additional applications of natural abundance ¹³C NMR methods to polyunsaturated lipid bilayers [178, 179] have been described. In these examples and others, we are interested in how the molecular properties of membrane lipids explain their biological functions within the broad context of structural biophysics [5, 6, 18, 180].

In what are called separated local-field (SLF) experiments, typically measurements of the direct ¹³C–¹H dipolar couplings are carried out involving liquid-crystalline systems, such as lipid bilayers at natural isotopic abundance. Through-space direct ¹³C–¹H dipolar interactions report on the orientations of the individual ¹³C–¹H bonds with respect to the bilayer normal, and are mathematically isomorphous to the C–²H bond order parameters, measured in solid-state ²H NMR spectroscopy. Various SLF methods include for example dipolar-recoupling methods [70, 73, 106, 115, 177, 181], switched-angle spinning, and off-magic-angle spinning experiments [182]. Segmental order parameters can be unambiguously determined by SLF methods, and hence it is a useful technique for lipid structural studies. In lipid systems, correspondingly the segmental order parameters are defined as:

$$ {S}_{\mathrm{CH}}=\frac{1}{2}\left\langle 3{\cos}^2{\beta}_{\mathrm{CH}}-1\right\rangle $$

(7)

where β _CH is the instantaneous angle between the ¹³C–¹H bond direction and the bilayer normal. Based on the geometrical considerations, the S _CH order parameters for a polymethylene chain are negative. Note that here we refer to the absolute order parameters |S _CH|, which are calculated from the relation:

$$ \left|{S}_{\mathrm{CH}}\right|=\frac{\left|\Delta \kern0.1em {\nu}_{\mathrm{D}}\right|}{\chi_{\mathrm{D}}\kern0.1em {\chi}_{\mathrm{P}}}. $$

(8)

In the above formula, ${\chi}_{\mathrm{D}}=\left(-{\mu}_0\kern0.1em {\gamma}_{\mathrm{C}}\kern0.1em {\gamma}_{\mathrm{H}}/4{\pi}^2\right)\left\langle {r}_{\mathrm{C}\mathrm{H}}^{-3}\right\rangle$ is the dipolar coupling constant (40.783 kHz corresponding to b _CH/2π = 20.392 kHz for an aliphatic ¹³C–¹H bond), χ _P = 0.393 is the pulse sequence scaling factor [73, 177], and Δν _D is the measured residual dipolar coupling (RDC) evaluated at the θ = 90° orientation of the lineshape (Pake powder pattern ). An illustration of the results obtained for mixtures of lipids and cholesterol using the dipolar recoupling with shape and scaling preservation (DROSS ) experiment is shown in Fig. 4 [73]. The solid-state NMR spectra show well-resolved ¹³C chemical shifts on the horizontal axis and the site-specific ¹H–¹³C dipolar couplings on the vertical axis. A representative INEPT ¹³C NMR spectrum showing highly resolved ¹³C chemical shift resonances for POPC lipid and cholesterol is provided in Fig. 5 [73], which is indicative of the applicability of separated local-field NMR experiments for obtaining the site-specific information about lipid bilayers and lipid mixtures.

Fig. 4

Site-specific ¹³C–¹H residual dipolar couplings (RDCs) are measured using two-dimensional separated local-field ¹³C NMR (DROSS ) spectrum for POPC/cholesterol (1:1) binary mixture at T = 30 °C. (a) Selected recoupled powder patterns, showing experimental (gray) and simulated (red) lineshapes. (b) Oblique view of aliphatic fingerprint region of the DROSS spectrum of binary system POPC/cholesterol. (c) The 2D plane of the spectrum shown in (b). The ¹³C isotropic chemical shift (δ) spectrum is on the horizontal axis (red). The peak separation of the Pake doublet yields the site-specific ¹³C–¹H dipolar couplings along the vertical axis. Figure modified from [73]

Fig. 5

Experimental INEPT ¹³C NMR spectrum of the unsaturated lipid 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) in the presence of 33% cholesterol (chol) at 30 °C. The INEPT experiment is used as part of the separated local-field experiment DROSS . Several highly resolved ¹³C chemical shifts for POPC and cholesterol show the applicability of separated local-field NMR experiments for obtaining site-specific information on the lipid/cholesterol mixtures. Data obtained from [73]

3 Molecular Distributions of Lipids are Obtained Using Solid-State ²H NMR Spectral Lineshapes

The solid-state ²H NMR spectrum for 1,2-diperdeuteriomyristoyl-sn-glycero-3-phosphocholine (DMPC-d ₅₄) at 30 °C shown in Fig. 2a is illustrative of applications to phospholipids with perdeuterated chains in the physiologically relevant, liquid-crystalline state [111, 127]. Because the sample consists of bilayers randomly oriented in the aqueous medium relative to the direction of the magnetic field, the spectrum is a powder-type pattern that is a superposition of signals from the lamellae at all orientations. At each angle, the signal consists of doublets from the chain methyl and methylene positions along the perdeuterated chains [183]. The resulting spectrum has well-defined edges at ~±15 kHz due to a roughly constant order for the acyl methylene segments nearest to the head group of the lipid molecule. Individual peaks within the spectrum arise from less ordered methylene groups in the lower portion of the chains, and the highly disordered terminal methyl group produces the central pair of peaks [128]. From the distribution of the RQCs , structural quantities such as mean area per lipid 〈A〉 and volumetric bilayer thickness D _C are readily derived by a mean-torque (MT) model, and can be related to the corresponding material constants or elastic moduli [28, 30, 138]. The significant increase in the observed RQCs upon addition of cholesterol to DMPC bilayers is also shown in a series of spectra from Fig. 2b–d.

3.1 Bilayers Containing Cholesterol Enable Testing of Theories for Dynamical Structures of Membrane Assemblies

In general, lipid bilayers containing cholesterol allow an excellent model for testing theories for the configurational ordering and structural dynamics of liquid-crystalline membranes [30]. As we can see in Fig. 6, for the DMPC-d ₅₄ bilayer, both in the absence and presence of cholesterol , a well-defined profile of the segmental order parameters ${S}_{\mathrm{CD}}^{(i)}$ versus the acyl segment position (i) is evident. An approximate order parameter plateau occurs over the middle part of the chains, followed by a progressive decrease, which manifests the end effects within the bilayer central hydrocarbon core (Fig. 6). In addition, due to the orientation of the glycerol backbone nearly perpendicular to the membrane surface, the sn-1 and sn-2 acyl chains are inequivalent [29, 129, 132]. The initial chain geometry leads to smaller order parameters for the beginning of the sn-2 chain [143, 184]. Beyond the first few segments, the order parameters of the sn-2 chain become larger than those of the sn-1 chain [154]. Smaller statistical fluctuations are associated with greater travel (flux) of the sn-2 chain to compensate for the initial position closer to the aqueous interface. The order profile clearly suggests that variations in the degree of acyl chain entanglement occur as a function of depth within the bilayer hydrocarbon region. As a result, it is unlikely that the phospholipids move individually within the bilayer, even in the presence of cholesterol [153, 185, 186].

Fig. 6

Comparison of theoretical and solid-state ²H NMR spectroscopy results for the configurational ordering and structural dynamics of liquid-crystalline membranes. The bilayer dimensions correspond to the interfacial membrane area per lipid 〈A〉 and volumetric thickness D _C. Structural parameters are calculated from the acyl chain volume V _C and moments 〈cos β〉 and 〈cos² β〉 obtained from the order parameter plateau, where β is the angle between the virtual bond connecting two neighboring carbons of the ith segment and bilayer normal [138]. Profiles of segmental order parameters ${S}_{\mathrm{CD}}^{(i)}$ as a function of acyl chain position (i) for DMPC-d ₅₄ and DMPC-d ₅₄/cholesterol (1:1) are shown at T = 44 °C [111]. Filled and open symbols refer to inequivalent sn-1 and sn-2 acyl chains, respectively. Reference order parameters are indicated for limiting cases of an oil-drop model with S _CD = 0, a crankshaft model having S _CD = −1/3, and an all-trans rotating chain with S _CD = −1/2. Data are taken from [153]

The approach of using a model membrane, comprising a lipid that forms a fluid bilayer together with a lipid known to form a more ordered bilayer, as well as cholesterol in varying amounts, is probably most accessible as a mimic of biomembranes with a vast number of components [1, 15]. Interestingly, cholesterol has two different functions in model membrane systems: on the one hand, it increases the hydrophobic mismatch of the lipids at low concentrations and thereby enhances phase separation. On the other hand, it functions as a mixing agent at high concentrations. Let us consider in greater detail the results in Fig. 6 for the DMPC-d ₅₄ bilayer, both in the absence and presence of cholesterol . To calibrate our intuition, at this point it might be helpful for readers to consider some simple motional models as heuristic guides or limiting cases. Referring again to Fig. 6, the first example is an all-trans rotating polymethylene chain, where Eq. (2) with β _PD = 90^o then yields S _CD = −1/2 as a reference value. Next, we can consider a crankshaft model involving a polymethylene chain that is saturated with kink configurations gauche ^± -trans-gauche ^∓, leading to S _CD = −1/3 for comparison. Last, the classical oil-drop model completely neglects tethering of the acyl chains to the aqueous interface, in which case the isotropic motion gives S _CD = 0 as a limit. One can then compare the experimental order profiles to the above limiting cases as benchmarks.

As we have already seen in Fig. 2, for the DMPC-d ₅₄/cholesterol (1:1) binary mixture, in the liquid-ordered (l _o) phase there is a dramatic increase in the RQCs versus the liquid-disordered (l _d) phase of DMPC-d ₅₄ alone. This is due to a substantial reduction of the degrees of freedom of the flexible phospholipids, coming from the van der Waals interactions with the rigid sterol frame. The corresponding plateau in the order profile, cf. Fig. 6, can be understood in terms of a relatively constant probability of the acyl chain configurations, resulting from their tethering via the polar head groups to the aqueous interface, together with their travel (flux) toward the bilayer interior. For the top part of the acyl chains, the segmental order parameters approach the limiting value of S _CD = −1/2 when cholesterol is present, as expected for an all-trans rotating polymethylene chain [112]. However, there is still an approximate plateau indicating entanglement of the chain ends. Note that in the absence of cholesterol, the additional acyl disorder can arise from internal degrees of freedom of the phospholipids, e.g., due to segmental isomerizations, molecular motions, or collective thermal excitations of the bilayer. These additional degrees of freedom lead to smaller absolute S _CD values for the DMPC-d ₅₄ bilayer. Provided that the disorder of the DMPC-d ₅₄ bilayer is due mainly to rotational isomerism, then the acyl chains fall somewhere between the limiting crankshaft model with S _CD = −1/3, and the classical oil-drop model for which S _CD = 0. For the DMPC-d ₅₄ bilayer, both in the presence and absence of cholesterol, the acyl chains are more disordered within the hydrocarbon core to fill in the free volume that would otherwise be present due to chain terminations, approaching the classical “oil-drop” limit only in the center of the bilayer.

3.2 Order Parameter Profiles of Binary Mixtures of Cholesterol with Phospholipids and Sphingolipids: Implications for Rafts in Cellular Membranes

In addition to glycerophospholipids , most eukaryotic cells contain sphingolipids and sterols as additional classes of lipids (30–40 mol% cholesterol and 10–20 mol% sphingomyelin). Notably, plasma membranes of animal cells are enriched in cholesterol, which is metabolically derived from lanosterol by removal of the methyl groups from the α-face of the molecule, raising the question of their evolution and function in the organization of the bilayer. Despite extensive research, whether the well-known ordering effect of cholesterol for sphingomyelin and other saturated glycerophospholipids leads to lateral phase segregation and microdomains remains under discussion, as the situation in vivo is far too complex to be exactly determined. However, the structural properties and phase-transition temperatures of sphingomyelins near body temperature (37 °C) suggest that they may play an important role in the formation of specialized domains in membranes, such as lipid rafts [47, 55, 69, 187].

Various studies have been reported on the comparison of order parameter profiles for sphingolipids and phospholipids, and their binary and tertiary mixtures with cholesterol [65, 73, 113, 152, 153]. Results have been obtained from solid-state ²H NMR lineshapes of both POPC-d ₃₁ and PSM-d ₃₁ in a complementary way in the corresponding ternary systems. Addition of cholesterol at 20 mol% has been shown to have a major impact on the spectra of PSM-d ₃₁ in ternary mixtures with POPC (1:1). The range of phase coexistence is increased by the addition of POPC, thereby resisting the formation of the l _d phase from the l _o phase. By contrast, addition of PSM shows an increased ordering effect on the palmitoyl chain in POPC relative to the PSM-free POPC-d ₃₁/cholesterol mixture. Analysis of solid-state ²H NMR data using a first-order mean-torque model [65, 138] has uncovered interesting insights into the effect of cholesterol on the lateral organization of lipid-cholesterol mixtures. As described by Bartels et al. [65], the addition of cholesterol initially drives the phase separation (l _d–l _o) by inducing greater lateral order in sphingomyelin than in POPC lipids. However, at 20 mol% cholesterol, discrete components due to POPC-d ₃₁ and PSM-d ₃₁ phase separations are observed. At physiological temperature, the lipids in the ternary mixture (1:1 PSM/POPC with 20 mol% cholesterol) show distinct structural parameters (bilayer thickness and area per lipid ). Structural parameters of POPC are highly temperature dependent, whereas sphingomyelin shows resistance to thermally driven structural deformation, and stronger affinity of cholesterol that can drive the formation of membrane domains. In mixtures with high amounts of cholesterol (33 mol%), saturation of the ordering effect for PSM seems to facilitate ideal mixing of the components, and hence similar results are observed. These observations naturally bring the intuition that the phase separation is driven by hydrophobic mismatch of the acyl chains of the various lipids, e.g., it is assumed the thickness difference is induced by unequal sterol partitioning into the two phases.

Another study using solid-state ¹³C NMR spectroscopy [73] has shown that the cholesterol-mediated structural perturbations are less pronounced for egg-yolk sphingomyelin (EYSM) than for POPC. In Fig. 7, the order parameter profiles derived using separated local-field (SLF) ¹³C NMR spectroscopy for EYSM and POPC bilayers are shown. The influence of cholesterol for the l _o phase of POPC and EYSM is clearly distinguishable. The higher-order parameter values for both the lipid bilayers at various carbon positions are an arresting indication of the l _o phase [6, 65, 92, 106, 111,112,113, 188,190,190], due to interaction with cholesterol [192,193,193]. In addition, the nonequivalence of the segments of the sn-1 and sn-2 chains [132] is clearly reflected in the case of the POPC bilayers (Fig. 6a, b). Monounsaturation of the oleoyl chain at the C9 and C10 sites renders the two vinyl ¹³C–¹H positions orientationally nonequivalent, both to each other and to the other saturated chain segments. The terminal methyl groups of the acyl chains exhibit very small residual dipolar couplings (RDC), because of the reorientation and three-fold symmetry of the methyl ¹³C–¹H bonds. The largest couplings of these sites are observed at the C3 position, which may participate in interfacial exchange-type C3–OH hydrogen bonding, and/or C3–OH acceptor and NH donor hydrogen bonding. The large value is suggestive of a glycerol backbone conformation that is stabilized through lipid packing assisted by hydrogen bonding in the l _d phase.

Fig. 7

Segmental order parameter S _CH profiles from solid-state ¹³C NMR spectroscopy (separated local-field) indicate lipid-specific loss of conformational disorder due to cholesterol. Absolute order profiles are plotted for: (a) the sn-2 oleoyl chain of POPC, (b) the sn-1 palmitoyl chain of POPC, (c) the sn-2 sphingosine chain of EYSM, and (d) the sn-1 palmitoyl chain of EYSM. Circles represent pure lipids and squares represent lipid mixtures with cholesterol (1:1). For POPC, data are shown at two temperatures, T = 28 °C (gray-filled symbols) and T = 48 °C (solid symbols), and for EYSM at T = 48 °C. Note that upon adding cholesterol the absolute S _CH order parameters increase more for POPC than in EYSM. Figure adapted from [73]

Most striking, upon addition of 50 wt% cholesterol, the increment in segmental order parameters is ∆|S _CH| ≈ 0.25 for POPC and ≈ 0.12 for EYSM, as indicated in Fig. 7 (comparisons are for the maximum absolute |S _CH| values due to the plateau region of the |S _CD| profiles). Correspondingly, the increment in the hydrocarbon thickness and condensation of the area per lipid for EYSM is lower than for POPC. Such a remarkable difference indicates that EYSM is in a relatively ordered state in the single-component membrane. The higher acyl segmental order parameters in single-component bilayers at a given temperature, for EYSM relative to POPC, indicate the high propensity of self-association for the hydrophobic moieties of sphingomyelin lipids [55, 69]. Notably, these observations suggest that upon adding cholesterol, the entropic loss is less pronounced for EYSM than for POPC, as discussed in [29]. Mixing of cholesterol is more favorable for sphingolipids compared to phosphatidylcholines, potentially driving the formation of lipid rafts in multicomponent biomembranes [194, 195]. In other words , like dissolves like, as we learn in our introductory chemistry courses.

4 Nuclear Spin Relaxation Reveals Multiscale Dynamics of Membrane Lipids

An important further aspect is that analysis of the nuclear spin relaxation rates yields experimental information about the molecular dynamics that is unobtainable with other biophysical methods [128]. The possible types of motions that occur in lyotropic liquid crystals are: (1) segmental motions due to rotational isomerizations of the flexible surfactant or lipid molecules; (2) slower effective rotations of the entangled molecules; and (3) collective deformations of the bilayer which span a broad range, and can influence interactions involving the assembly [29, 111, 112, 129, 196]. At the high frequencies, bond stretching and bending vibrations are most likely too fast to influence significantly the nuclear spin relaxation, but rather lead to a pre-averaging of the coupling tensor [196, 197]. Identifying the predominant contributions within the various motional regimes, and characterizing their energetic parameters, would reveal the atomistic interactions that lead to bulk material properties based on the current NMR technology [29]. For liquid-crystalline membranes , elastic deformations (modeled as splay, twist, and bend) within the hydrocarbon core are interpreted as collective lipid dynamics on the order of membrane dimensions [127]. It follows that NMR relaxation studies of lipid membranes in the l _o and l _d phases can strongly benefit our understanding of the atomistic lipid–cholesterol interactions, leading to changes in bulk membrane physical properties, with striking biological consequences.

4.1 Fluctuating Molecular Interactions Due to Various Types of Lipid Motions Within the Bilayer Cause the Nuclear Spin Relaxation

The process of NMR relaxation is due to fluctuations of the coupling Hamiltonian, on account of the various possible motions of the lipid molecules within the bilayer. According to time-dependent perturbation theory, these fluctuations give rise to transitions between the various adjacent energy levels [136]. In ²H NMR relaxometry of liquid-crystalline membrane lipids, one is often interested in the spin–lattice (R _1Z) relaxation rates. Experimental R _1Z relaxation rate measurements involve perturbation of the magnetization away from the equilibrium value, and then following the attainment of equilibrium by observing the magnetization recovery as a function of time. The observable relaxation rates are related to the spectral densities of motion in the laboratory frame by:

$$ {R}_{1Z}=\frac{3}{4}{\pi}^2\kern0.1em {\chi}_{\mathrm{Q}}^2\left[{J}_1\left({\omega}_0\right)+4{J}_2\left(2{\omega}_0\right)\right]. $$

(9)

In the above expression, R _1Z is the spin–lattice (longitudinal) relaxation rate, and J _m(ω ₀) denotes the irreducible spectral densities of motion, where m =1, 2, and ω ₀ is the deuteron Larmor frequency. The spectral densities J _m(ω ₀) describe the power spectrum of the motions as a function of frequency ω ₀ in terms of fluctuations of the Wigner rotation matrix elements for transformation of the coupling (EFG ) tensor from its principal axis system to the laboratory frame. They are the Fourier transform partners of the orientational correlation functions G _m(t) which depend on time , and characterize the C–²H bond fluctuations.

4.2 Generalized Model-Free Aspects of the Nuclear Spin Relaxation of Membrane Lipid Bilayers

Here, we give a brief introduction to the model-free interpretation of the relaxation rates for general readers, while dealing with actual lipid relaxation data. For a more complete description of generalized model-free (GMF) analysis, readers are referred to the review by Xu et al. [136]. Contributions to the nuclear spin relaxation from motions with different characteristic mean-squared amplitudes and timescales are possible in terms of a hierarchical energy landscape [185]. They include: (1) the static coupling tensor that is modulated by rapid local segmental motions, such as trans-gauche isomerizations of the hydrocarbon chains of the lipid or surfactant molecules; and (2) the residual coupling tensor (leftover from the fast motions) that is further modulated by slower motions. The slower motions in principle might include whole-molecule motions of the flexible phospholipids, or alternatively collective thermal excitations involving the various lipid molecules [146, 198]. Clearly, the segmental order parameters depend only on the amplitudes of the C–²H bond fluctuations. On the other hand, the relaxation rates depend on both the orientational amplitudes and the rates of the C–²H bond fluctuations. According to the GMF approach of relaxation rate analysis [136, 197, 199], a simple linear dependence of the R _1Z rates on the squared segmental order parameters (${S}_{\mathrm{CD}}^2$) (square-law) along the chain would result (Fig. 8). In the limit of short-wavelength excitations, on the order of the bilayer thickness and less, the spectral density reads [200]:

$$ {J}_m\left(\omega \right)=\frac{5}{2}{S}_{\mathrm{CD}}^2\kern0.1em D{\omega}^{-\left(2-d/2\right)}\left[{\left|{D}_{-1m}^{(2)}\left({\beta}_{\mathrm{DL}}\right)\right|}^2+{\left|{D}_{1m}^{(2)}\left({\beta}_{\mathrm{DL}}\right)\right|}^2\right]. $$

(10)

Fig. 8

Solid-state ²H NMR relaxation at 76.8 MHz (11.8 T) shows emergence of membrane elastic fluctuations and their suppression by sterols. Striking differences are uncovered in membrane elastic fluctuations from cholesterol versus lanosterol at the atomistic level. (a) Dependence of spin–lattice relaxation rates ${R}_{1\mathrm{Z}}^{(i)}$ on squared order parameters ${S}_{\mathrm{CD}}^{(i)}$ for resolved ²H NMR splittings of DMPC-d ₅₄ showing influences of cholesterol in the liquid-ordered (l _o) phase at T = 44 °C. The presence of cholesterol leads to a large decrease in the square-law slopes, corresponding to a progressive reduction in bilayer elasticity . An opposite increase is seen for bilayers containing C₁₂E₈ nonionic detergent at T = 42 °C. Data are taken from [129]. (b) Dependence of spin–lattice relaxation rates ${R}_{1\mathrm{Z}}^{(i)}$ on squared order parameters ${S}_{\mathrm{CD}}^{(i)}$ in liquid-ordered phase (l _o) for resolved ²H NMR splittings of DMPC-d ₅₄ showing influences of lanosterol and cholesterol at T = 55 °C. Note that the decrease in the square-law slopes is consistent with a gradual reduction in bilayer elasticity on going from lanosterol to cholesterol. Data are taken from [112]

Here, ω is the angular frequency, D is the viscoelastic constant, d is the dimensionality, and D ⁽²⁾ indicates the second-rank Wigner rotation matrix [137]. The irreducible spectral densities J _m(ω) depend on the square of the observed S _CD order parameters, and the slope of the square-law plot is inversely related to the softness of the membrane. For 3D quasielastic fluctuations, the viscoelastic constant is given by $D=3{k}_{\mathrm{B}}T\sqrt{\eta }/5\pi \sqrt{2\kern0.1em {K}^3}{S}_{\mathrm{s}}^2$, where a single elastic constant K is assumed, in which η is the corresponding viscosity coefficient, S _s is the order parameter for the relatively slow motions, and other symbols have their usual meanings. No distinction is made between splay, twist, and bend deformations. In addition to the bending modulus κ, the compression modulus K _B may come into play [141].

4.3 Nuclear Spin Relaxation of Lipid Membranes in the Liquid-Ordered Phases Reveals Atomistic Lipid–Cholesterol Interactions and Bulk Membrane Physical Properties

In the example presented here, a solid-state ²H NMR relaxation study of effect of cholesterol on lipid bilayers shows that a square-law functional dependence of the R _1Z rates versus the order parameters S _CD is evident along the entire acyl chain for the multilamellar dispersions of DMPC-d ₅₄/cholesterol bilayers (Fig. 8a) [111]. This dependence on the motional amplitudes signifies relatively slow bilayer motions that modulate the residual coupling tensors leftover from faster segmental motions (Fermi’s golden rule). Given a simple composite membrane deformation model [196, 201], the R _1Z rates are due to a broad spectrum of 3D collective bilayer excitations, with effective rotations of the lipids. Transverse ²H NMR spin relaxation studies also provide evidence for 2D collective motions of the membrane film, albeit at lower frequencies [203,204,205,205]. By contrast, local trans-gauche isomerizations along the chains modulate the same static NMR coupling tensor, and do not yield such a square-law. With regard to splay deformations, the so-called bending rigidity is κ ≈ Kt, where t = 2D _C is the bilayer thickness, giving a κ ^−3/2 dependence of the R _1Z rates [127]. Moreover, 3D director fluctuations (d = 3) yield a ω ^−1/2 frequency dispersion as a characteristic signature [185, 206]. In this case, the reduction in the square-law slope, cf. Fig. 8a, b, reflects an increase in κ and/or S _s due to short-range cholesterol–phospholipid interactions. One should also note that at the molecular level, a dynamical protrusion of cholesterol across the midplane, i.e., between the apposed monolayers, is suggested by quasielastic neutron scattering studies [27, 207]. Indeed, the square-law functional dependence as first discussed [185] is a model-free correlation among the experimental observables, and it does not rest on any specific molecular interpretation. For the longitudinal relaxation rates of liquid-crystalline lipid membranes , the above analysis is expected to be generally applicable.

4.4 The Progressive Increase in Bilayer Rigidity from Lanosterol to Cholesterol Parallels the Metabolic Pathway of Sterol Biogenesis

Influences of cholesterol on the physical properties of DMPC bilayers have also been compared to its metabolic precursor lanosterol in a related study [112] (Fig. 8b). Notably, cholesterol is a two-faced molecule—the α-face is smooth and the β-face is molecularly rough due to the methyl substituents. Even so, lanosterol is methylated on both the α-face and the β-face (Fig. 1), and it presents a more balanced countenance to the phospholipids [208]. In terms of biomolecular NMR spectroscopy, it has been observed that the slope of the square-law plot is greater for lanosterol than for cholesterol, consistent with the bilayer stiffness being less for lanosterol versus its metabolic product cholesterol [25, 112, 209]. Again, it is found that the site-specific analysis of the solid-state NMR results based on atomistic observables matches the results for the macroscopic bilayer elasticity [207, 210] (Fig. 9). The more molecularly smooth van der Waals surface of the α-face of cholesterol [92, 193, 211, 212] enables a large increase in bilayer rigidity and stabilizes the liquid-ordered phase to an even greater degree than lanosterol. Figure 9a illustrates the comparison of the bending rigidity (modulus) (κ) for the DMPC bilayers using deuterium NMR relaxometry and thermal shape fluctuation data. The progressive increase in the bilayer rigidity on going from lanosterol to cholesterol (Fig. 9b) parallels the metabolic pathway of sterol biogenesis [162, 173, 214,215,216,217,217], and may be related to the optimization or evolution of the biophysical properties of cholesterol. In a similar study, the effect of several sterols on the lipid order and bilayer rigidity has been investigated for macroscopically aligned bilayers of DMPC or DPPC using ²H NMR spectra and spin–lattice relaxation rates [25]. The bending modulus of the bilayers was calculated from plots of the relaxation rates versus the square of the order parameter. Clear differences were obtained in the efficiency of the sterols to increase the stiffness of the bilayers. These differences are correlated to the ability of the sterols to induce the liquid-ordered phase in binary as well as in ternary systems.

Fig. 9

Bending rigidity of lipid bilayers obtained from solid-state ²H NMR relaxation analysis. (a) Comparison of solid-state ²H NMR square-law slopes to bending rigidity κ obtained from thermal shape fluctuation data for lipid vesicles. Results for homologous series of PCs with acyl carbon lengths of n _C = 12, 14, and 16 in the liquid-disordered (l _d) phase at 55.4 MHz (squares); data at 76.8 MHz for DMPC/cholesterol mixtures with X _C = 0, 0.33, and 0.50 in the liquid-ordered (l _o) phase (triangles). The inset shows the values of κ estimated from the ²H NMR model for the homologous series of PCs. Bending rigidity estimates from solid-state ²H NMR are in good agreement with the values obtained from thermal shape fluctuation data. Figure adapted from [111]. (b) Comparison of bending rigidities of DMPC lipid bilayers obtained by solid-state ²H NMR relaxation at various compositions of DMPC and cholesterol (Chol/DMPC), and DMPC and lanosterol (Lan/DMPC) mixtures. Significant difference is seen in the influence of cholesterol versus lanosterol on the elastic properties of DMPC membrane bilayers. Data taken from [112]

5 Biophysical Conclusions and Outlook

Solid-state NMR methods offer excellent experimental techniques in medicine and biology. They uniquely probe the physical properties of lipid membranes and provide information complementary to other spectroscopic methods. Interestingly, owing to their physiological liquid-crystalline nature , membrane elastic deformations together with their multiscale molecular dynamics clearly fall in the solid-state ²H NMR time- and length scales. The membrane stiffening effect upon addition of sterols has been investigated at an atomistically resolved level, showing a direct correspondence with bulk elasticity. Phase separation in bilayers of ternary lipid mixtures occurs mainly due to dissimilar affinities for different cholesterol concentrations. It is likely that configurational entropy plays a major role in the formation of the lipid domains called rafts [73]. It has also been shown in model membranes [20, 46, 47, 218] that formation of such domains entails Gibbs free energies that are lower than those for protein-lipid interactions, which therefore may indicate a role for proteins in domain formation. Further detailed information regarding protein–lipid interactions can contribute to understanding the lateral organization of cellular plasma membranes, and how lipid rafts may be implicated in their functional mechanisms. Moreover, nuclear spin–lattice relaxation studies of fluid lipid bilayers manifest their quasielastic deformation on short length and timescales, on the order of the membrane thickness and less. Interpretation of the atom-specific NMR relaxation data is in broad agreement with molecular dynamics (MD) simulations of flexible surfactant films and lipid membranes [51, 142, 146, 198, 219]. Molecular simulations [219] show that local trans-gauche isomerizations are accompanied by concerted isomerizations about multiple bonds in the assembly of lipid acyl chains. The continuum model for NMR relaxation approximates the collective excitations in mathematical closed form, wherein the influences of cholesterol and the membrane thickness correspond to the bilayer bending energy. A key remaining question is how the bilayer softness as studied with NMR relaxation may be significant to lipid–protein interactions in fluid membranes [6, 221,222,222], where elastic curvature deformation may play an important role.