Distinguishing Bicontinuous Lipid Cubic Phases from Isotropic Membrane Morphologies Using ³¹P Solid-State NMR Spectroscopy

Abstract

Nonlamellar lipid membranes are frequently induced by proteins that fuse, bend, and cut membranes. Understanding the mechanism of action of these proteins requires the elucidation of the membrane morphologies that they induce. While hexagonal phases and lamellar phases are readily identified by their characteristic solid-state NMR lineshapes, bicontinuous lipid cubic phases are more difficult to discern, since the static NMR spectra of cubic-phase lipids consist of an isotropic ³¹P or ²H peak, indistinguishable from the spectra of isotropic membrane morphologies such as micelles and small vesicles. To date, small-angle X-ray scattering is the only method to identify bicontinuous lipid cubic phases. To explore unique NMR signatures of lipid cubic phases, we first describe the orientation distribution of lipid molecules in cubic phases and simulate the static ³¹P chemical shift lineshapes of oriented cubic-phase membranes in the limit of slow lateral diffusion. We then show that ³¹P T₂ relaxation times differ significantly between isotropic micelles and cubic-phase membranes: the latter exhibit two-orders-of magnitude shorter T₂ relaxation times. These differences are explained by the different timescales of lipid lateral diffusion on the cubic-phase surface versus the timescales of micelle tumbling. Using this relaxation NMR approach, we investigated a DOPE membrane containing the transmembrane domain (TMD) of a viral fusion protein. The static ³¹P spectrum of DOPE shows an isotropic peak, whose T₂ relaxation times correspond to that of a cubic phase. Thus, the viral fusion protein TMD induces negative Gaussian curvature, which is an intrinsic characteristic of cubic phases, to the DOPE membrane. This curvature induction has important implications to the mechanism of virus-cell fusion. This study establishes a simple NMR diagnostic probe of lipid cubic phases, which is expected to be useful for studying many protein-induced membrane remodeling phenomena in biology.

Introduction

Nonlamellar lipid membranes with high curvature are generated during many protein-mediated biological processes such as virus-cell fusion, virus budding, endocytosis, and pore formation by lytic and antimicrobial peptides^1–5. Characterizing the type of membrane curvatures is important for understanding the mechanism of action of these proteins. Because lipid membranes are inherently non-crystalline and dynamic, solid-state NMR (SSNMR) spectroscopy is a natural technique for their characterization.

Liquid-crystalline phases formed by surfactants and lipids have been well studied by a variety of techniques (see, e.g. ref.⁶). The lamellar phase and the hexagonal phases give rise to characteristic static ³¹P NMR spectral lineshapes that allow these phases to be identified readily. However, among nonlamellar membrane morphologies, several types give rise to an isotropic peak in the static NMR spectra. These morphologies include micelles, small unilamellar vesicles, and cubic phases^7–8. Bicontinuous lipid cubic phases have received increasing attention in recent years because of the importance of this class of topological structures for membrane fusion, membrane scission, virus budding and pore formation. Bicontinuous cubic phases are periodic repeats of minimal surfaces on which every point has negative Gaussian curvature and zero mean curvature^9–11. The Gaussian curvature is the product of two principal curvatures at a point, while the mean curvature is the average. Thus, every point on the surface of bicontinuous cubic phases has equal magnitudes of positive and negative principal curvatures. Based on crystallographic space groups and symmetries, three bicontinuous lipid cubic phases can be distinguished: Pn3m (also called double diamond or D surface), Ia3d (gyroid or G surface) and Im3m (primitive or P surface)^12–13. A lipid bilayer drapes onto the minimal surface, the two sides of which lie a continuous region of water. Bicontinuous cubic phases separate water into two non-intersecting channels. For the Im3m (P), Pn3m (D), and Ia3d (G) phases, water channels meet at 6-way (90°), 4-way (109.5°), and 3-way (120°) junctions, respectively (Fig. 1).

Figure 1.

Geometries of three common bicontinuous cubic phases and the calculated quasi-static ³¹P NMR lineshape for an oriented cubic-phase membrane. Red and green colors denote the two surfaces of the bilayer. (a) The primitive Im3m phase. (b) The double diamond Pn3m phase. For clarity, an extended view containing 8 unit cells are shown on the right at a different angle. (c) The gyroid Ia3d phase. Due to symmetry, a saddle area is shown in grey in (a) and (c) to indicate the minimum surface used for calculating the ³¹P NMR lineshape. (d) Simulated ³¹P CSA lineshape of the three cubic phases, assuming an oriented cubic-phase membrane whose z-axis of the unit cell is parallel to the magnetic field. All three cubic phases give the same spectrum.

Bicontinuous lipid cubic phases commonly exhibit unit-cell dimensions of 10–20 nm, which are not much larger than the dimensions of micelles. Thus, lipid lateral diffusion on cubic-phase surfaces causes fast molecular reorientation. Due to the symmetry of these phases, this fast reorientation averages second-rank nuclear-spin interaction tensors to their isotropic values⁷. Thus, although bicontinuous cubic phases are structurally non-isotropic, their static solid-state NMR spectra exhibit an “isotropic” peak, indistinguishable from the spectra of truly isotropic membrane morphologies such as micelles and small vesicles. Here the term “isotropic peak” refers to a peak at the isotropic NMR frequency, while “isotropic” morphologies refer to the three-dimensional structures of micelles and small vesicles.

Although isotropic phases and cubic phases cannot be distinguished by static NMR lineshapes, they may be resolved by NMR relaxation times. In contrast to NMR lineshapes, whose averaging depends only on the lower bound of motional rates, relaxation times are sensitive to motions on a range of timescales. In lipid membranes, many molecular motions exist to drive nuclear-spin relaxation. These include segmental torsional motions, whole-body uniaxial rotational diffusion around the long molecular axis, wobble of the molecular axis in a cone, tumbling of vesicles or micelles, and lipid lateral diffusion on the membrane surface^14–15. The torsional motions, uniaxial rotation and wobble typically occur on the picosecond to nanosecond timescales for hydrated membranes^16–17. Tumbling of nanometer-sized vesicles and micelles occurs on the microsecond timescale. Lateral diffusion depends on the radius of curvature and the diffusion coefficient. For nanometer-sized vesicles or micelles, lateral diffusion occurs on the tens of microsecond timescale, whereas for ~100 nm or larger vesicles, lateral diffusion occurs on the millisecond timescale. Thus, tumbling and lateral diffusion are much slower than torsional motions and rotational diffusion.

While a large body of literature exists on using ³¹P, ²H, ¹³C, and ¹⁴N relaxation NMR to investigate lipid motions in lamellar membranes (see, e.g. ref.^18–19), the application of relaxation NMR for studying nonlamellar membrane morphologies is more scarce. ²H and ¹⁴N relaxation NMR has been used to study surfactant motions in various liquid-crystalline phases^20–22. ¹³C T₂ relaxation has been used to measure lateral diffusion coefficients of lipids in sonicated and extruded small vesicles²³. ³¹P and ²H relaxation NMR has been used to compare the curvatures of hexagonal and lamellar phases²⁴ and motions in spherical supported vesicles versus multilamellar vesicles²⁵. Halle and coworkers presented a theory of nuclear-spin relaxation in bicontinuous cubic-phase liquid crystals that suggested the possibility of extracting time correlation functions for different cubic phases²⁶. However, to our knowledge, no experimental demonstration of relaxation NMR for distinguishing cubic phases from isotropic phases has been reported.

In this work, we show that ³¹P T₂ relaxation times readily distinguish lipid isotropic phases and bicontinuous cubic phases. We use LMPC as a representative micelle and monoolein/POPC as a representative cubic-phase membrane. From their ³¹P T₁ and T₂ relaxation times, we extract correlation times of fast and slow motions, and compare them between the micelle and the cubic-phase samples. Based on these model-compound data, we investigate the ³¹P relaxation times of a DOPE membrane containing the transmembrane domain (TMD) of a viral fusion protein. This viral fusion TMD causes an isotropic ³¹P peak to the DOPE membrane. We show that the ³¹P T₂ relaxation times of this TMD-bound DOPE membrane are diagnostic of a cubic phase, which has significant implications to the mechanism of virus-cell fusion. We also examine the temperature dependences of the ³¹P relaxation times for all three membranes to obtain activation energies of fast and slow motions. In addition, we provide a mathematical description of the orientation distribution of lipids in cubic phases and show that, in the limit of slow lateral diffusion, an oriented cubic-phase membrane has unique NMR lineshapes.

Materials and Methods

Membrane sample preparation

Three membrane samples were prepared: 1-myristoyl-2-hydroxy-sn-glycero-3-phosphocholine (LMPC), 1-monoolein (MO)/1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) (17: 3), and 1,2-dioleoyl-sn-glycero-3-phosphoethanolamine (DOPE) membrane containing the TMD of the parainfluenza virus 5 (PIV5) F protein. The peptide : lipid molar ratio of the TMD/DOPE sample is 1 : 15. LMPC was dissolved in water to a concentration of 400 mM and transferred into a 4 mm magic-angle-spinning (MAS) rotor. MO and POPC were codissolved in chloroform, while the PIV5 TMD and DOPE were dissolved in TFE and chloroform, respectively. The organic solvents were removed by nitrogen gas and the mixtures were lyophilized. The dry powders were suspended in 4 mL pH 7.5 HEPES buffer (10 mM HEPES-NaOH, 1 mM EDTA, and 1 mM NaN₃). The MO/POPC mixture was incubated at 4°C overnight, while the TMD/DOPE mixture was dialyzed against 1 L buffer for one day. The samples were centrifuged at 4°C and 55,000 rpm for 3 hours to obtain membrane pellets, which were equilibrated to ~55% hydration for MO/POPC and ~30% for TMD/DOPE before transfer into MAS rotors.

Solid-state NMR experiments

Static ³¹P NMR experiments were carried out on a Bruker DSX-400 MHz (9.4 T) spectrometer operating at Larmor frequencies of 400.49 MHz for ¹H and 162.12 MHz for ³¹P. ³¹P T₂ relaxation times were measured between 273 K and 295 K using a Hahn echo sequence (90° - τ – 180° - τ) under 30 kHz ¹H decoupling and echo delays (τ) of 50 µs - 30 ms. To extract T₂ values, echo intensities as a function of 2τ were fit to a single-exponential function A·e^−2τ/T₂, where A is close to 1. ³¹P T₁ relaxation times were measured using the inversion recovery sequence (180° - τ − 90°). The intensities was fit to a single-exponential function B·(1 − C·e^−τ/T₁), where B = 1 and C = 2 in the ideal case but are slightly adjustable in the fitting.

Results

Anisotropic NMR lineshapes of cubic-phase membranes

In static solid-state NMR spectra, cubic-phase membranes exhibit a narrow peak at the isotropic frequency because of fast lateral diffusion over the highly curved surface of the membrane. This isotropic spectrum is identical to that of micelles and small vesicles, even though the molecular orientational distribution in the cubic phase is non-isotropic. To elucidate this orientation distribution, we simulate the quasi-static ³¹P NMR lineshapes of oriented cubic-phase membranes in the absence of lateral diffusion. If the cubic phases are randomly oriented, then the NMR lineshape reverts to the powder lineshape of unoriented bilayers.

In liquid-crystalline lipid membranes, the three principal values of the rigid-limit ³¹P chemical shielding tensor, σ_xx, σ_yy and σ_zz, are averaged by the uniaxial rotation of the phospholipids to two components, σ_‖ and σ_⊥, where σ_‖ is the component parallel to the uniaxial rotational axis, which is the local bilayer normal, while σ_⊥ is the component perpendicular to it. The asymmetry parameter of the chemical shielding tensor, η ≡ (σ_yy − σ_xx)/(σ_zz − σ_iso), is averaged to 0, η̅ = 0, while the averaged anisotropy parameter, $\overline{\delta }\equiv \frac{2}{3}({\sigma }_{\Vert }-{\sigma }_{\perp })$, is ~30 ppm for most phospholipids.

The angle β between the magnetic field B₀ and the local bilayer normal gives the orientation-dependent ³¹P chemical shift frequency:

$${\omega }_{cs}(\beta )={\omega }_{\mathit{\text{iso}}}+\frac{1}{2}\overline{\delta }(3{\text{cos}}^2\beta -1)$$ (1)

where ω_iso is the trace of the chemical shift tensor. The Euler rotations that relate the laboratory frame to the bilayer frame are most conveniently considered through a coordinate system fixed to the cubic-phase crystal frame. When B₀ is coincident with the z-axis of this crystal frame (Fig. 1a–c), β is simply the polar angle of the local bilayer normal in the crystal frame. Thus, based on the distribution of β on the cubic-phase surface, we can calculate the ³¹P spectral lineshape. The lineshapes of other B₀ orientations relative to the crystal frame can be similarly calculated, as we show in the Supporting Information.

We constructed the geometries of the three bicontinuous cubic phases (Fig. 1a–c) following the mathematical protocols of Klinowski^27–29 and Finch^30–31. Detailed equations are given in the Supporting Information. For symmetry reasons, only 1/8 of the unit cells of the primitive (Im3m) and gyroid (Ia3d) surfaces need to be sampled to obtain the lineshapes of the full unit cell (Fig. S1–S3). The CSA spectrum for B₀ along the z-axis of the unit cell is a superposition of an isotropic peak with an η̅ = 0 powder lineshape, and is identical for the three cubic phases (Fig. 1d). Thus, the cubic-phase spectrum differs from the lamellar bilayer spectrum at the isotropic frequency. Since this isotropic frequency corresponds to bilayer normals that are oriented at the magic angle, 54.7°, from the magnetic field, to show the high population of this magic-angle orientation, we plot the β distribution for the three cubic phases (Fig. 2). Indeed, the surface area at 54.7° and 125.3° dominates that of any other angles, and the distribution function can be approximated by the sum of sinβ and additional intensity at the magic angle. The CSA lineshape when B₀ points along the x- and y-axes of the crystal frame is identical with Fig. 1d (simulation not shown), but other B₀ orientations have distinct lineshapes, some of which are given in the Supporting Information (Fig. S4).

Figure 2.

Distribution of the angle β between the bilayer normal and the z-axis of the cubic-phase unit cell. (a) One saddle of the primitive surface Im3m. (b) One unit of the double diamond surface Pn3m. (c) One saddle of the gyroid surface Ia3d. Green, magenta and blue areas denote β angles of 90°, 0° or 180°, and 54.7° or 125.3°, respectively. (d) Normalized surface area as a function of the β angle. The smallest surface areas occur at β = 0° and 180°, while β = 54.7° and 125.3° exhibit the largest surface area.

While these simulated static ³¹P CSA lineshapes are interesting, in practice it is challenging to suppress lipid lateral diffusion and produce oriented cubic-phase membranes. Thus, we next explore ³¹P relaxation NMR to identify cubic phases under the realistic situation of random orientation and in the presence of fast lateral diffusion.

³¹P relaxation times of micelles versus cubic-phase membranes

The static ³¹P spectra of LMPC, MO/POPC, and TMD/DOPE membranes are shown in Fig. 3a. All three samples exhibit a single ³¹P isotropic peak at ambient temperature. The full widths at half maximum, Δ*, are ~90 Hz for LMPC and MO/POPC and increase to ~485 Hz for DOPE. In contrast, the ³¹P T₂ relaxation decays of the three membranes, plotted on a logarithmic timescale (Fig. 3b), show a two-orders-of-magnitude difference among the three samples: the LMPC micelle has T₂ values of more than 100 ms between 275 and 295 K, while the TMD/DOPE membrane shows ³¹P T₂’s of 0.3–1.3 ms (Table 1). POPC in the cubic-phase MO/POPC membrane has intermediate T₂’s of several milliseconds. All three samples exhibit increasing T₂ with temperature. These T₂ values translate to homogeneous linewidths, Δ' ≡ 1/πT₂, of 1.5 Hz for LMPC, 30 Hz for MO/POPC, and 245 Hz for TMD/DOPE at ambient temperature. Simulated Lorentzian lineshapes for these homogeneous linewidths are shown in Fig. 3a to compare with the observed apparent linewidths. It can be seen that although the LMPC sample has the narrowest apparent linewidth, it is more inhomogeneously broadened (i.e. large Δ*/Δ’) than MO/POPC and TMD/DOPE, suggesting that the micelles have a large distribution of sizes, while the TMD/DOPE spectrum is the most homogeneously broadened.

Figure 3.

³¹P T₂ relaxation times of LMPC, MO/POPC, and TMD-bound DOPE membranes. (a) Static ³¹P spectra of the three samples at 295 K. The measured spectra with apparent linewidths Δ* are shown in black whereas simulated Lorentzian lineshapes with homogeneous linewidths Δ’ based on the ³¹P T₂ are shown in red. (b) ³¹P spin-echo intensities as a function of echo delay from 273 K to 303 K. LMPC micelles have ³¹P T₂ values of 104 – 207 ms, MO/POPC shows T₂’s of 3.7 – 10.7 ms, while the TMD/DOPE sample shows the shortest ³¹P T₂ values of 0.38 – 1.3 ms.

Table 1.

³¹P T₂ relaxation times (ms) of LMPC, MO/POPC and TMD/DOPE membranes.

Membrane	273 K	278 K	285 K	290 K	295 K
LMPC	104 ± 4	117 ± 7	150 ± 18		207 ± 11
MO/POPC		3.7 ± 0.5	6.1 ± 0.2	8.2 ± 0.6	10.7 ± 0.5
TMD/DOPE	0.38 ± 0.02	0.50 ± 0.03	0.77 ± 0.04		1.3 ± 0.05

Compared to T₂, ³¹P T₁ relaxation times are more uniform among the three samples, about 1 s at ambient temperature (Table 2 and Fig. 4). However, LMPC and MO/POPC show decreasing T₁ with decreasing temperature while TMD/DOPE manifests the opposite trend. Thus, the nanosecond motions in the TMD/DOPE membrane occur on the slow side of the T₁ minimum while those in the MO/POPC and LMPC membranes occur on the fast side (see below).

Table 2.

³¹P T₁ relaxation times (s) of LMPC, MO/POPC and TMD/DOPE membranes.

Membrane	273 K	278 K	285 K	290 K	295 K
LMPC	0.81±0.01	0.82±0.01	0.88±0.01		1.06 ±0.03
MO/POPC		0.74 ± 0.02	0.81 ± 0.02	0.88±0.03	0.95±0.03
TMD/DOPE	1.13±0.02	1.09 ± 0.03	1.01 ± 0.02		0.99 ± 0.02

Figure 4.

³¹P T₁ relaxation times of LMPC, MO/POPC and TMD/DOPE membranes as a function of temperature. The T₁ values increase with temperature for LMPC and MO/POPC and decrease with temperature for TMD/DOPE.

We now consider the mechanisms of ³¹P T₁ and T₂ relaxations in lipid membranes in order to extract motional correlation times for the three membranes. The nuclear-spin interactions relevant for ³¹P relaxation are the ³¹P CSA and ³¹P-¹H dipole coupling. At the magnetic field of 9.4 Tesla used here, the CSA mechanism dominates, and the T₁ and T₂ relaxation rates can be expressed as^32–33

$$R_1^{CSA}=\frac{2}{15}{\omega }_P^2{\sigma }^2(1+\frac{{\eta }^2}{3})J({\omega }_P),$$ (2)

$$R_2^{CSA}=\frac{1}{15}{\omega }_P^2{\sigma }^2(1+\frac{{\eta }^2}{3})[J({\omega }_P)+\frac{4}{3}J(0)],$$ (3)

where ω_P is the ³¹P Larmor frequency, $\sigma \equiv \frac{3}{2}({\sigma }_{zz}-{\sigma }_{\mathit{\text{iso}}})$ is the rigid-limit chemical shift anisotropy, and J(ω) is the spectral density at frequency ω and is the Fourier transform of the correlation function of motion. If the motion is Markovian (e.g. diffusive or jump-like), then the correlation function is exponential with a time constant τ_c and J(ω) is Lorentzian³⁴:

$$J(\omega )=\frac{{\tau }_c}{1+{(\omega {\tau }_c)}^2}.$$ (4)

To simplify analysis, we separate lipid motions into two types: fast motions with a correlation time τ_f and slow overall motions with a correlation times τ_s. With this approximation, the spectral density function becomes^35–36

$$J(\omega ;{\tau }_s,{\tau }_f)=\frac{S^2{\tau }_s}{1+{(\omega {\tau }_s)}^2}+\frac{(1-S^2){\tau }_f}{1+{(\omega {\tau }_f)}^2},$$ (5)

where S is the order parameter of the fast motion.

The rigid-limit ³¹P CSA (σ) is about 160 ppm, η is about 0.57^{16, 37}, and the Larmor frequency in our experiments is ω_p = 2π × 162 × 10⁶ rad/s. The order parameters of the glycerol backbone and the beginning of the headgroup are ~0.2 based on measured C-H dipolar couplings and ³¹P and ¹³C CSAs in liquid-crystalline phosphocholine^38–39. Thus, from the measured T₁ and T₂ values, we can extract the two unknowns, τ_f and τ_s, by solving the simultaneous equations (2) and (3). In principle, from these quadratic equations more than one solution of τ_f and τ_s are possible. However, by removing unphysical values and using the fact that τ_f and τ_s should decrease with increasing temperature, we can obtain a unique set of correlation times at each temperature.

Tables 3 lists the measured fast and slow correlation times for the three membranes at various temperatures. We found τ_f values of 0.27–0.43 ns for LMPC and MO/POPC but a 10-fold longer τ_f value of 3–4 ns for the TMD/DOPE sample. Thus, phosphocholines in the micelle and the cubic-phase monoolein undergo faster motions than DOPE lipids in complex with the fusion protein TMD. The longer correlation time of DOPE may result from intermolecular hydrogen bonding between phosphoethanolamine headgroups and from DOPE – TMD interactions.

Table 3.

Fast and slow correlation times of three membranes at different temperatures.

Membrane		273 K	278 K	285 K	290 K	295 K
LMPC	τf (ns)	0.38	0.37	0.34	-	0.27
	τs (µs)	0.077	0.067	0.050	-	0.035
MO/POPC	τf (ns)	-	0.43	0.38	0.34	0.31
	τs (µs)	-	2.5	1.5	1.2	0.89
TMD/DOPE	τf (ns)	3.9	3.7	3.4	-	3.3
	τs (µs)	25	19	12	-	7.3

For slow motions, the τ_s values differ by two orders of magnitude among the three membranes (Table 3). LMPC has τ_s values of 0.04–0.08 µs over the temperature range studied, while POPC in cubic-phase MO/POPC exhibits τ_s values of 0.9–2.5 µs. TMD-bound DOPE has the longest τ_s of 7.3–25 µs.

To understand the origin of these very different τ_s values, we estimate the correlation times of various slow motions in lipid membranes. Two main sources of slow motions are whole-body tumbling and lipid lateral diffusion, whose correlation times can be calculated as⁴⁰

$$\frac{1}{{\tau }_s}=\frac{1}{{\tau }_s^d}+\frac{1}{{\tau }_s^t}={\left(\frac{r_L^2}{6D_L}\right)}^{-1}+{\left(\frac{4\pi r_t^3{\eta }_W}{3kT}\right)}^{-1},$$ (6)

where ${\tau }_s^d$ is the correlation time for lipid lateral diffusion, ${\tau }_s^t$ is the correlation time for tumbling, D_L is the lateral diffusion coefficient, r_L is the radius of curvature for lateral diffusion, r_t is the radius of the tumbling vesicle, T is the absolute temperature, k is the Boltzmann constant, and η_W is the viscosity of the aqueous solution. For lipid cubic phases, r_L ≪ r_t because of the extended nature of the cubic-phase assembly, thus ${\tau }_s^d$ is much shorter than ${\tau }_s^t$, making lateral diffusion the determining factor for the overall τ_s. Using a typical D_L value of 10⁻⁸ cm²/s and a radius of 10 nm, we estimate a τ_s value of ~ 17 µs when lateral diffusion dominates the slow motion. For micelles, r_L = r_t. Using an η_W of 0.891·10⁻³ kg/m·s, a radius of 5 nm, and T of 295 K, we estimate a tumbling correlation time ${\tau }_s^t$ of 0.13 µs, which is much shorter than ${\tau }_s^d$ and thus dominates τ_s. Thus, micelles should have two-orders-of-magnitude shorter τ_s than lipid cubic phases, consistent with the measured τ_s differences between LMPC and MO/POPC. Therefore, the fact that TMD/DOPE exhibits even longer τ_s assigns this membrane to the cubic phase.

We can extract the activation energies E_a of the fast and slow motions from the measured temperature dependence of the correlation times. Assuming Arrhenius behavior, the correlation time depends on T as

$$\tau ={\tau }_0{\mathrm{e}}^{\frac{E_a}{RT}},$$ (7)

where R is the ideal gas constant. Fig. 5 plots ln(τ) as a function of 1000/T, the slope of which gives the activation energies. For fast motions, we found similar activation energies of 13–14 kJ/mol for LMPC and MO/POPC, while the TMD/DOPE membrane has a 3-fold lower activation energy of 5.2 kJ/mol. For slow motions, activation energies of 26–41 kJ/mol were found, with LMPC giving the smallest value while MO/POPC and TMD/DOPE exhibit larger and similar activation energies.

Figure 5.

Activation energies of slow and fast motions in the three lipid membranes, extracted from the slope of the correlation times with respect to inverse temperature. (a) ln(τ_f) as a function of temperature. The activation energies range from 5.2 to 14 kJ/mol. (b) τ_s as a function of temperature. The activation energies range from 26 to 41 kJ/mol.

Discussion

These ³¹P relaxation data reveal that cubic-phase membranes can be distinguished from isotropic micelles by their different T₂ relaxation times or homogeneous linewidths. Although LMPC and MO/POPC have the same apparent ³¹P linewidths (Fig. 3a), the underlying homogeneous linewidths are dramatically different. The narrower homogeneous linewidth of the LMPC micelle is empirically consistent with the fluid nature of the micelle sample. The estimated correlation times for whole-body tumbling and lateral diffusion over a radius of 5–10 nm provide insights into the different orders of magnitudes of T₂ relaxation times. To understand τ_s and τ_f’s contribution to ³¹P T₂ and T₁ relaxation times, we consider the dependences of the spectral density function on correlation times. Setting τ_f to 0.1 – 10 ns and τ_s to 10 ns – 100 µs, the spectral density at the Larmor frequency, J(ω_p; τ_s, τ_f), depends on τ_s and τ_f as

$$\frac{\partial J({\omega }_p;{\tau }_s,{\tau }_f)}{\partial {\tau }_s}=S^2\frac{1-{({\omega }_P{\tau }_s)}^2}{{[1+{({\omega }_P{\tau }_s)}^2]}^2}\approx -\frac{S^2}{{({\omega }_P{\tau }_s)}^2},$$ (8)

$$\frac{\partial J({\omega }_p;{\tau }_s,{\tau }_f)}{\partial {\tau }_f}=(1-S^2)\frac{1-{({\omega }_P{\tau }_f)}^2}{{[1+{\left({\omega }_P{\tau }_f\right)}^2]}^2}.$$ (9)

The approximation in eqn. (8) results from the fact that (ω_Pτ_s)² ≫ 1. Since S is ~0.2, J(ω_p; τ_s, τ_f) has negligible dependence on τ_s, indicating that slow motion has little impact on T₁. In contrast, since ω_Pτ_f ~ 1, τ_f has a significant effect on J(ω_p; τ_s, τ_f). These dependences of J(ω_p; τ_s, τ_f) are plotted in Fig. 6a. Around the T₁ minimum τ_f = 1/ω_P, T₁ decreases with increasing τ_f when τ_f <1/ω_P but increases with increasing τ_f when τ_f >1/ω_P (Fig. 6b).

Figure 6.

Dependences of ³¹P T₁ and T₂ relaxation times on fast and slow correlation times τ_f and τ_s. (a) log(T₁) versus τ_f and τ_s. In the range of 10 ns < τ_s <10 µs, T₁ is insensitive to τ_s. (b) ³¹P T₁ relaxation time as a function of τ_f when τ_s = 1 µs. Under our experimental condition T₁ has a minimum of ~0.54 s, which is achieved when τ_f = 1/ω_P ≈ 1 ns. (c) ³¹P T₂ relaxation time as a function of τ_s while τ_f is 0.2, 1 and 4 ns. T₂ decreases with increasing τ_s and τ_f. The dependence of T₂ on τ_f diminishes with increasing τ_s. When τ_s > 1 µs, T₂ is independent of τ_f, and the relationship between T₂ and τ_s is approximately linear.

T₂ relaxation times depend on spectral densities at both the Larmor frequency and ω = 0. From Eqn. (5), J(0) = S²τ_s + (1 − S²)τ_f, indicating that J(0) is proportional to both τ_f and τ_s. Fig. 6c plots the calculated T₂ values as a function of τ_s for several τ_f values. When τ_s > 1 µs, T₂ is dominated by τ_s and mostly independent of τ_f. In this regime, R₂ can be simplified as

$$R_2^{CSA}\approx \frac{1}{15}{\omega }_P^2{\sigma }^2(1+\frac{{\eta }^2}{3})\left(\frac{4}{3}{S}^2{\tau }_s\right),$$ (10)

indicating that the T₂ relaxation time is dominated by and inversely proportional to τ_s (Fig. 6c). As a result, tumbling micelles with shorter τ_s values (< 0.1 µs) have longer T₂ relaxation times than cubic-phase membranes with longer τ_s values (1–30 µs) due to lateral diffusion. These significant τ_s differences between LMPC micelles and the MO/POPC cubic phase are not manifested in the ³¹P spectral lineshapes because both motions are faster than the averaged ³¹P CSA of ~5 kHz. However, at larger magnetic field strengths, the ³¹P CSA may become sufficiently large such that the cubic-phase ³¹P spectrum may no longer be averaged to an isotropic frequency. Higher magnetic fields may also facilitate alignment of the cubic-phase membrane, if the magnetic susceptibility anisotropy of the cubic-phase membrane can be made sufficiently large using dopants such as lanthanide ions, so that the calculated ³¹P lineshapes may become observable.

The measured activation energies give useful insights into the nature of various lipid motions. For slow motions, TMD/DOPE and MO/POPC exhibit an activation energy of ~40 kJ/mol whereas the LMPC micelle has a much smaller activation energy of 26 kJ/mol. Pulsed-field gradient NMR data have yielded the activation energies of lipid lateral diffusion in various membranes. For example, hydrated POPC⁴¹ and MO in cubic-phase MO/water mixtures⁴² were reported to have activation energies of ~30 kJ/mol. The good agreement between these literature values and our data for TMD/DOPE and MO/POPC supports the assignment of the slow motion in these samples to lateral diffusion. For fast motions, the activation energies are smaller: LMPC and MO/POPC gave a value of 13–14 kJ/mol while TMD/DOPE showed the smallest activation energy of 5.2 kJ/mol. The former is in excellent agreement with the values obtained from previous field-dependent ³¹P T₁ relaxation data, which indicated that most phospholipids have an activation energy of 13.2 ± 1.9 kJ/mol in the liquid-crystalline phase^43–44. This energy barrier was assigned to diffusive motions in a spatially rough potential energy landscape⁴⁴, and is associated with motions with correlation times of several nanoseconds. We found τ_f values of less than 1 ns for the LMPC micelle, which suggests that other motional processes may also be present. Single-field ³¹P relaxation times as measured here are not sufficient to separate multiple fast motions such as torsional motion, headgroup rotation, and lipid uniaxial rotation⁴⁵; however, such a detailed separation of motional mechanisms is not the focus of this study. Even with our simplifying assumption of only one fast motion, the good agreement between our activation energies for LMPC and MO/POPC and literature values suggests that the fast motion can be reasonably assigned to rotational diffusion combined with wobble of the molecular axis.

The ³¹P T₂ relaxation data indicate that the TMD of the fusion protein F of the parainfluenza virus 5 converts the DOPE membrane to a cubic phase. This is consistent with small-angle X-ray scattering (SAXS) data that showed the formation of an Ia3d cubic phase (unpublished data). Thus, the TMD of this viral fusion protein induces negative Gaussian curvature to the DOPE membrane, the type of curvature that is present in hemifusion intermediates and fusion pores⁴. The active participation of the TMD to viral fusion may not be restricted to the PIV5 fusion protein, but may occur in other viral fusion proteins as well. In addition to fusion proteins, the influenza M2 protein has also been shown to induce an isotropic peak in the ³¹P NMR spectra⁴⁶ and cubic phases in the SAXS spectra⁴⁷, and this curvature-inducing ability has been correlated with the membrane-scission function of the M2 protein⁴⁸. The current ³¹P relaxation NMR approach should be useful for further characterization of M2-induced membrane restructuring as well as for de novo determination of the membrane morphologies with an associated isotropic ³¹P peak as generated by other proteins.

In conclusion, the lipid membrane morphology can be identified by first measuring the static ³¹P NMR lineshapes. Lamellar and hexagonal phases exhibit unique anisotropic powder patterns while micelles and cubic phase exhibit an isotropic peak. If the latter is found, then non-spinning ³¹P T₂ relaxation times should be measured. If the T₂ is longer than ~100 ms at room temperature, then the membrane is in an isotropic phase, while T₂’s of less than ~10 ms indicate that the membrane is in a bicontinuous cubic phase. Temperature-dependent T₁ and T₂ relaxation times can be further measured to obtain more detailed information about the correlation times of lipid motion. The slow correlation time is especially distinct between the cubic phases (microseconds) and the micellar phase (10 – 100 ns).