Geometrical Membrane Curvature as an Allosteric Regulator of Membrane Protein Structure and Function

Abstract

Transmembrane proteins are embedded in cellular membranes of varied lipid composition and geometrical curvature. Here, we studied for the first time the allosteric effect of geometrical membrane curvature on transmembrane protein structure and function. We used single-channel optical analysis of the prototypic transmembrane β-barrel α-hemolysin (α-HL) reconstituted on immobilized single small unilamellar liposomes of different diameter and therefore curvature. Our data demonstrate that physiologically abundant geometrical membrane curvatures can enforce a dramatic allosteric regulation (1000-fold inhibition) of α-HL permeability. High membrane curvatures (1/diameter ∼1/40 nm⁻¹) compressed the effective pore diameter of α-HL from 14.2 ± 0.8 Å to 11.4 ± 0.6 Å. This reduction in effective pore area (∼40%) when combined with the area compressibility of α-HL revealed an effective membrane tension of ∼50 mN/m and a curvature-imposed protein deformation energy of ∼7 k_BT. Such substantial energies have been shown to conformationally activate, or unfold, β-barrel and α-helical transmembrane proteins, suggesting that membrane curvature could likely regulate allosterically the structure and function of transmembrane proteins in general.

Introduction

Transmembrane proteins are embedded in cellular membranes that locally differ in their lipid composition and their shape, or geometrical curvature (henceforth curvature). The term curvature describes here the geometrical shape of a membrane that has been imposed by an external force, e.g., by a protein coat. Thus, curvature describes a nonequilibrium state and is distinct from the intrinsic spontaneous curvature of a membrane, which favors equilibrium shapes and is zero for membranes of symmetric leaflet composition.

The lipid composition of cellular membranes has been long established as a critical effector of the structure and function of transmembrane proteins via both indirect physical effects and direct chemical interactions; and in that sense the lipid bilayer acts as an allosteric regulator (1–6). However, there is currently no evidence to demonstrate whether, under constant lipid composition, changes in membrane curvature that occur, e.g., during endo- or exocytosis, is missing can play an equally important regulatory role for transmembrane proteins (7). Most studies to date have instead focused on the ability of membrane curvature to regulate the localization of membrane-binding domains (8–11) and lipids (12). We therefore investigated here the direct allosteric effect of membrane curvature on the structure and function of a transmembrane protein, the prototypic β-barrel bacterial exotoxin α-hemolysin (α-HL) (13,14), in liposomes with symmetric leaflet lipid composition of zero spontaneous curvature. The wealth of knowledge available on α-HL (13,14) allowed us to handle α-HL as a membrane-force probe to experimentally quantify for the first time to our knowledge membrane force and energy as a function of membrane curvature.

In vivo, membranes of different curvature typically differ also in their lipid and protein composition (8,10). To deconvolve the regulatory effects of biochemical composition from curvature, we performed experiments in vitro, at a fixed lipid composition and over a range of physiologically important membrane curvatures (1/500–1/50 nm⁻¹). α-HL is one of the best characterized members of a family of transmembrane β-barrel proteins found in the mitochondria and chloroplasts of eukaryotic cells and the outer membranes of Gram-negative bacteria (14,15). We used this prototypical β-barrel to quantify the magnitude of intrabilayer forces exerted by membrane curvature on an imbedded protein, and thus evaluate whether such ubiquitous membrane mechanical effects are likely to regulate allosterically the structure and function of transmembrane proteins in general. By the very nature of α-HL, membrane insertion is unidirectional because of the bulky charged extracellular globular domain on each monomer that cannot pass through the membrane (14,16).

The most powerful method to measure the permeability of pore- and channel-forming proteins at the single-molecule level is through electrical recordings of ion conductance (17). However, neither patch-clamp technique (17) nor black lipid membrane techniques (16) can perform experiments with membranes bent to curvatures of 1/50–1/100 nm⁻¹ (18), which abound in eukaryotic membranes (e.g., in endocytotic buds, caveolae, synaptic or trafficking vesicles, curved regions of the golgi or the endoplasmic reticulum, etc. (19,20)).

To probe the effects of physiologically occurring membrane curvatures on the permeability of α-HL, we performed optical single-channel recordings (21) in conjunction with a high-throughput membrane curvature assay (22–24) (see Fig. 1 and Materials and Methods).

Figure 1.

Optical recording of single-channel permeability as a function of membrane curvature. Nanoscale liposomes tethered via biotin-neutravidin on a passivated surface at a density dilute enough to be imaged with confocal fluorescence microscopy on an individual basis. A fluorescently labeled lipid (DHPE-Atto633) homogenously incorporated in the membrane provided accurate quantification of liposome diameter and therefore membrane curvature (Fig. 2). The broad range of randomly occurring liposome diameters (40–500 nm) allowed us to characterize the effect of membrane curvature in a high-throughput manner. Insertion of α-hemolysin (α-HL) into liposomes caused passive transport of a water-soluble fluorophore (Alexa488) trapped in their lumen, allowing us to record the permeability of single α-HL pores as a function of membrane curvature.

To experimentally probe the solute transport activity of α-HL (see Fig. 3 A), we prepared liposomes of diameters 40–500 nm (see Fig. 3) with fluorophore-labeled lipids (DHPE-Atto633) in their membrane loaded with a water-soluble fluorophore (Alexa488) in their lumen (25). The liposomes were tethered on a biofunctional surface at dilute densities (see Fig. 3 B). We used confocal fluorescence microscopy (see Materials and Methods) to image arrays of thousands of single, surface-tethered, intact (25), and nondeformed (24), nanoscale proteoliposomes of different diameters (D) and membrane curvatures (1/D) (22,23).

Figure 3.

Membrane curvature changes significantly the pore radius and permeability of the transmembrane protein α-HL as a consequence of membrane tension. (A) Schematic illustration of the hypothesis to scale. (B) Zoom of representative fluorescence images showing POPC/POPG (9:1 molar ratio) liposomes labeled fluorescently in their membrane and in their lumen. Scale bars, 5 μm. (C) Cropped micrographs of two representative time sequences show transport events after single pore formation in two liposomes with diameters of 146 nm (blue) and 88 nm (red). Temporal resolution 147 ms. (D) Fitting the normalized raw lumen intensity traces from the two liposomes in C allowed accurate quantification of the exponential transport rates, 0.488 ± 0.016 s⁻¹ and 0.094 ± 0.004 s⁻¹, respectively. Bleaching of Alexa488 was negligible on the timescale of transport. Complete transport demonstrates that the liposomes are unilamellar. (E) Increased membrane curvature reduces the pore permeability of α-HL by four orders of magnitude. Here, single-pore permeability data (N = 218) are fitted with a power function (pink line) for clarity, where the thickness of the fit reflects its uncertainty. (F) Increasing membrane curvature reduces the pore radius (black crosses) of α-HL due to the increasing tension in the membrane. Pore radii were binned using weighted averaging. The two independent calculations of bilayer tension using either the change in pore radius (red crosses) or the change in lipid packing (dashed pink line), are in good agreement. Pore radii and tension data are calculated using a cylinder model of the α-HL pore. Error bars represent the mean ± SD.

Insertion of a heptameric α-HL pore in any of the immobilized liposomes induced passive transport of Alexa488 down its concentration gradient (21) and a concomitant reduction of fluorescence intensity over time (see Fig. 3, C and D). To ensure that we were measuring transport events originating from single α-HL pores, we diluted the concentration of α-HL to the point where we observed rare stochastic pore formation on single vesicles. To exclude rare traces showing sequential stepwise insertion of two α-HL pores/liposome, we selected only transport events displaying monoexponential decays according to theory (Eq. 2). Furthermore, to exclude transport from multilamellar liposomes, traces were accepted only if they reached background intensity, which indicated fully emptied liposomes. The transport rate of Alexa488 (k) was then used to quantify the permeability (P) of a single α-HL pore, according to P = kV (Eq. 3), where V is the liposome volume (21). As seen from the raw intensity traces (see Fig. 3 D), bleaching of Alexa488 was negligible on the timescale of the transport events.

Materials and Methods

Materials

1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC), 1-palmitoyl-2-oleoyl-sn-glycero-3-phospho-1′-rac-glycerol sodium salt (POPG), and 1,2.dioeoyl-sn-glycero-3-phosphoethanolamine-N-(cap biotinyl) sodium salt (DOPE-cap-biotin) were purchased from Avanti Polar Lipids (Alabaster, AL). Neutravidin and Alexa488 hydrazide were purchased from Life Technologies (Naerum, Denmark). Atto633-1,2-dihexadecanoyl-sn-glycero-3-phosphoethanolamine (DHPE-Atto633) and Atto633-1,2.dioeoyl-sn-glycero-3-phosphoethanolamine (DOPE-Atto633) were purchased from Atto-Tec (Siegen, Germany). Poly(L-lysine) (20 kDa)-graft[3.5]-poly(ethylene glycol) (2 kDa) (PLL-PEG) and PLL-PEG-biotin (PLL-PEG (2)/PEG(3.4)-biotin (18%)) were purchased from Susos (Dübendorf, Switzerland). Dimethylsiloxane (DMSO) and α-HL from Staphylococcus aureus were all purchased from Sigma (Broendby, Denmark).

Preparation of liposomes and α-HL pore insertion

For single α-HL pore recordings, two kinds of liposomes were prepared. Liposomes referred to as POPC/POPG were composed of POPC/POPG/DHPE-Atto633/DOPE-cap-biotin (93.5:5:1:0.5). Liposomes were prepared as described previously (26). Briefly, lipids stored in chloroform were mixed in a chosen composition and dried under a nitrogen flow for 20 min followed by 40 min in vacuum. The created dried thin lipid film was rehydrated with 1 mM Alexa488 hydrazide in buffer A (100 mM NaCl and 10 mM HEPES, pH 7.4) (2 g/L), incubated at 37°C for 12 h to create a broad distribution of liposome diameters, and then freeze-thawed 20 times to maximize the percentage of unilamellar liposomes of all diameters to ∼95% (25,27). The 1 mM Alexa488 in the bulk was removed by dialysis, leaving 1 mM Alexa488 inside liposomes, which was expected to produce a negligible osmotic pressure difference, thus ensuring that there was no osmotic induced membrane stretching or additional surface tension. Liposomes referred to as DOPE-inserted (see Fig. 5) were surface-immobilized POPC/POPG liposomes on which DOPE-Atto633 (200 nM) was added 4 min before adding α-HL. This resulted in insertion of the DOPE-Atto633 only to the outer lipid monolayer. DOPE-Atto633 was dissolved in DMSO (99% v/v), ensuring a final concentration of DMSO of ≤0.5% in the microscope chamber, as described in Hatzakis et al. (23). In all cases, α-HL was solubilized in buffer A and added to the immobilized liposomes to a final concentration of 4.5 μM.

Figure 5.

Insertion effect of labeled DOPE in single liposomes on α-HL permeability. DOPE-Atto633 inserted exclusively into the outer membrane leaflet (asymmetric insertion) of the preformed and immobilized POPC/POPG liposomes of different membrane curvatures (N = 328). (A) As observed previously (23), the amount of labeled DOPE inserting into liposomes increased as a power function of membrane curvature by a factor of 10 at high curvatures and was quantified as the mol% lipid increase from the change in intensity of Atto633 before to after incubation. (B) Raw single α-HL pore-permeability measurements revealed that asymmetric DOPE insertion relaxed the tension on the single pore imposed by membrane curvature. A comparison of the slope (exponent) b = 1.94 ± 0.03 of the fit (blue line) to the slope b = 2.62 ± 0.03 for pores formed in POPC/POPG liposomes (Fig. 3E, red line) confirms a significant reduction of pore permeability by DOPE insertion. The uncertainty in permeability of the fit is represented by the line thickness, which is a propagation of a and b and their fit errors. (C) The relative x-fold increase in pore permeability as a function of DOPE insertion on a linear-log plot implies that DOPE insertion releases the membrane tension on the pore. The relative increase was calculated using the DOPE insertion data in A and the permeability fit parameters of B and POPC/POPG (Fig. 3E). Note the convoluted influence of DOPE insertion density and membrane curvature on pore permeability. Data in A and C were binned using weighted averaging and error is represented as the mean ± SD. To see this figure in color, go online.

Functionalizing surfaces and immobilization of liposomes

Surfaces used for α-HL pore recordings were prepared as described previously (26–28). Ultraclean glass slides (thickness 170 ± 10 μm) were plasma-etched for 2 min, mounted in the microscope chamber, and incubated for 30 min with a mixture of 1 g/L PLL-PEG and PLL-PEG-biotin (dissolved in 15 mM HEPES at pH 5.6) at a ratio of 1000:80. Unbound polymers were washed away with buffer A and the surfaces were incubated with 0.1 g/L neutravidin (dissolved in 15 mM HEPES) for 10 min followed by extensive washing with buffer A. The density of immobilized liposomes was controlled by recording liposome binding in real time after adding 4 μL of 0.05 g/L liposomes to a chamber with a total volume of 80 μL buffer. Unbound liposomes were removed by washing when the surface density reached ∼500 liposomes/51 × 51 μm² region.

Confocal laser scanning microscopy

Images were acquired on a Leica TCS-SP5 inverted confocal microscope setup using an HCX PL APO 100× magnification, 1.4 NA oil immersion objective (Leica, Wetzlar, Germany). For the single α-HL pore recordings, the lumen dye (Alexa488) and the membrane label (DHPE-Atto633) were excited with 488 nm and 633 nm laser lines, respectively. Images of membrane label intensities were acquired in the range 645–750 nm using photomultiplier tube detection and acoustic optical beam splitter technology. Membrane images were acquired at a resolution of 1024 × 1024 pixels, each pixel corresponding to 50.5 × 50.5 nm², and a bit depth of 16, with a line-scan speed of 100 Hz in the bidirectional mode, pinhole 1.5, and four line averages.

Lumen dye intensities were selected with a bandpass filter ET 525/50 (Chroma Technology, Brattleboro, VT) and detected with an Avalanche photodiode (Perkin Elmer, Waltham, MA). Lumen images for transport traces were acquired with a temporal resolution of 147 ms/image at a resolution of 256 × 256 pixels, each pixel corresponding to 202 × 202 nm², and a bit depth of 8, with a line-scan speed of 1000 Hz in the bidirectional mode, pinhole 1.5, and four line averages.

Analysis of single liposome diameter and α-HL pore transport events

We extracted membrane and lumen intensities from fluorescence images using software written in Igor Pro Ver. 5.01 (Wavemetrics, Lake Oswego, OR). The software identified liposomes as membrane label intensities (Atto633) above a user-defined threshold. Each of the identified liposome intensities was integrated inside a scalable region of interest twice as large as the liposome and subsequently background-subtracted. Diameters of imaged fluorescent liposomes (Fig. 2) were determined to a precision of ∼7% as described previously in Kunding et al. (27). In brief, the integrated intensity of a single liposome (I_M) is proportional to the number of fluorescently labeled lipids in the liposome membrane and thereby to the liposome surface area (A_ves). The diameter (D) with 7% error, δD, is consequently related to I_M by the proportionality factor C_cal according to Eq. 1:

$$I_M\propto A_{ves}=\pi D^2\Rightarrow D=C_{cal}\sqrt{I_M},\delta D=0.07D$$ (1)

C_cal was quantified by the use of a calibration sample where the liposomes were extruded 20 times through two 50 nm polycarbonate filters (Millipore, Billerica, MA) that produced a narrow size distribution. The calibration sample was first examined by dynamic light scattering (ALV-5000 Correlator equipped with a 633 nm laser line) followed by confocal microscopy using identical imagining conditions as for a normal sample. The mean integrated liposome intensity of the imaged calibration sample was correlated to the mean radius obtained by dynamic light scattering. Once the calibration factor was known, all integrated liposome intensities were converted to physical size.

Figure 2.

Size calibration of single liposomes. (A) The liposome diameter, D, is related to the intensity of the fluorescently labeled membrane lipids, I_M, through a calibration factor, C_cal, according to Eq. 1. (B) Distribution of POPC/POPG liposome diameters obtained in the α-HL experiments (Fig. 3). The number of liposomes/condition that exhibited monoexponential transport was typically >200, and covered a diameter range of 40–500 nm. To see this figure in color, go online.

The amount of intercalated DOPE-Atto633 (mol%) in the outer lipid membrane leaflet of individual liposomes was quantified as the percentage change of integrated intensity before and after the intercalation using the initial intensity from DHPE-Atto633 of 1.0 mol% as a reference (see Fig. 5 A).

Lumen intensities (Alexa488) of liposomes were colocalized with their membrane intensities (Atto633) and region of interest integrated in sequential frames providing a time trace of each transport event. To ensure that we monitored transport events originating from perfect pore formation of single α-HL pores, we selected pores that 1), showed monoexponential decays, and 2), reached zero (background) intensity according to permeability theory (Eq. 2). It is important to note that transport to zero intensity also ensured that we were investigating unilamellar vesicles. Of a total of 861 POPC/POPG liposomes, ∼46% (396) showed transport within the timeframe of the experiment, of which ∼55% (218) satisfied the above two criteria and were included in the analysis.

Theory

Quantification of single α-HL transport rates

We calculated the permeability of the single α-HL pores formed in liposomes of different curvatures (Fig. 3 E, and see Fig. 5 B) according to the methods of Hemmler et al. (21). Briefly, those authors calculated and experimentally validated that the pore permeability (P), in units of volume/time, is the product of the apparent transport rate (k) of the solute (Eq. 2) and the volume confining the solute (V). In our case, V was calculated via the experimentally measured liposome size, described in Materials and Methods. For each single pore transport event, k was extracted with an error δk by fitting the kinetics of solute intensity reduction with an exponential function:

$$I\left(t\right)=I_0\exp \left(-kt\right)$$ (2)

The permeability of each single pore with error δP was then calculated as

$$P=kV,\delta P=\sqrt{{\left(V\delta k\right)}^2+{\left(k\delta V\right)}^2}$$ (3)

α-HL pore model and calculation of effective single pore radii

To calculate the effective pore radius (r) of α-HL (Fig. 3 F, black crosses) from the quantified permeability of the single pores shown in Fig. 3 E, we used an analytical description of the pore permeability (Eq. 4) developed in Deen (29) and Peters (30), which takes into account hindrance of solute diffusion in the pore by a transport probability factor, φ(r). Briefly, the permeability of a pore is described by its effective interior cross-sectional area (A), its pore length (L), and the diffusion constant (D_i) of the transported solute in bulk. In our case, the diffusion of the solute Alexa488 with radius a inside the α-HL pore was hindered as the r of the compressed pore approached a. The probability factor, φ(r), described the pore area excluded by the solute itself and in the limits r = a and r >> a took the values φ(r) = 0 and φ(r) = 1, respectively. To link P and r, we assumed the interior of the homoheptameric pore to be circular along the z axis with a pore radius r:

$$P=\frac{D_{i}A}{L}\phi \left(r\right)=\frac{D_{i}A}{L}{\left(1-\frac{a}{r}\right)}^2=\frac{D_i\pi r^2}{L}{\left(1-\frac{a}{r}\right)}^2$$ (4)

To calculate the r of α-HL (Fig. 3 F, black crosses, and Fig. 4) from the quantified P, we used L = 10 nm as measured in its crystal structure (13) and a = 0.56 nm of Alexa488 with D_i = 400 μm² s⁻¹ (31,32). Furthermore, we neglected the role of charges of Alexa488 and of the pore interior, because the preference of α-HL for conducting negative relative to positive charges is only 10% (33).

Figure 4.

Single α-HL pores in liposomes. (A) A distribution of calculated pore radii of single pores formed in POPC/POPG liposomes has a peak at radius ∼7 Å with a mean ± SD value of 7.4 ± 1.8 Å. (B) The percentage distribution of the pore radius errors of the single pore radii shows low errors ranging between 1 and 8%, comparable to the spread of ∼7% obtained by electrical pore conductance measurements (37).

Calculation of membrane tension using the pore deformation

To be able to determine the tension (γ) of a curved membrane, needed to impose the observed pore deformation (Fig. 3 F, black crosses), we first estimated the compressibility of the α-HL pore. Because in a lipid bilayer transmembrane proteins experience pressures only in the plane of the membrane, we could not use literature values of volume compressibility, but had to estimate the area compressibility, k_A. The fluctuation-dissipation relation from thermodynamics (Eq. 5) was applied to estimate the k_A of the transmembrane β-barrel of the α-HL pore by using the fluctuations of the cross-sectional area of the pore’s interior:

$$k_A=\frac{\langle A^2\rangle -{\langle A\rangle }^2}{\langle A\rangle }\frac{N_A}{RT}=-\frac{1}{\langle A\rangle }\frac{d\langle A\rangle }{d\gamma }$$ (5)

This relation has been previously used to calculate area compressibilities of various membranes from the areal fluctuations obtained in simulated lipid bilayers (34,35) and has been validated experimentally (36). For the α-HL pore, we estimated the average compressibility of the pore section around the amino acids T145, where the minimal pore radius is found. A circular shape is a good approximation of the interior cross-sectional area of the symmetric α-HL pore. In that case, k_A takes the form of Eq. 6, where it is seen from the last term that the area compressibility only depends on the radial fluctuations (σ_r) of the pore:

$$k_{A,p}=\frac{\langle A^2\rangle -{\langle A\rangle }^2}{\langle A\rangle }\frac{N_A}{RT}=\frac{\mathrm{var}\left(A\right)}{\langle A\rangle }\frac{N_A}{RT}=\frac{{\sigma }_A^2}{\langle A\rangle }\frac{N_A}{RT}=\frac{{\left(2\pi r{\sigma }_r\right)}^2}{\langle \pi r^2\rangle }\frac{N_A}{RT}=4\pi {\sigma }_r^2\frac{N_A}{RT}$$ (6)

The area compression modulus of the α-HL pore, K_A,p ≡ 1/k_A,p, is hereby derived, with its propagated error, as

$$K_{A,p}=\frac{RT}{4\pi {\sigma }_r^2{N}_A},\delta K_{A,p}=\frac{RT}{2\pi {\sigma }_r^2{N}_A}\left(\frac{\delta {\sigma }_r}{{\sigma }_r}\right)$$ (7)

We calculated an average pore compression modulus of K_A,p = 131.2 ± 26.4 mN/m using σ_r = 0.5 Å, as reported for T145 in a 50-ns-long all-atom simulation of a preequilibrated α-HL pore in a flat DPPC membrane (33). We assigned δσ_r = 10% error to σ_r. The fluctuation of the pore structure found in the simulation, σ_r = 0.5 Å, which corresponds to a fluctuation of ∼7% for the pore radius at T145 is in good agreement with the variation/fluctuation in NaCl conductance of 6.8% obtained by electrical conductance measurements (37).

To calculate the bilayer tension (Fig. 3 F, red crosses) needed for the observed deformation, we combined the estimated K_A,p value with our observation of the changed pore radius for different membrane curvatures (Fig. 3 F, black crosses). Integrating the standard relation between tension and areal change (Eq. 8) with the assumed pore circularity gives tension as a function of pore radius:

$$d\gamma =-K_{A,p}\left(\frac{dA}{A}\right)\Rightarrow \gamma =-K_{A,p}\ln \left(A/A_0\right)=-2K_{A,p}\ln \left(r/r_0\right)\delta \gamma =\sqrt{{\left(\gamma \frac{\delta K_{A,p}}{K_A,p}\right)}^2+{\left(2K_{A,p}\frac{\delta r}{r}\right)}^2}$$ (8)

This implies an increase in bilayer tension of γ = 53 ± 13 mN/m in a highly curved membrane (40 nm) relative to zero tension in a flat membrane (500 nm).

Theoretical calculation of tension in a liposome

We calculated the theoretical tension (γ_m) as a function of liposome diameter (Eq. 9), thus curvature (Fig. 3 F, dashed red line), using the expansion and compression of the outer and inner membrane monolayers (19,35,38) according to the methods of Malinin and Lentz (39), who derived the corresponding energy term. Briefly, for a liposome with a mean diameter D, both expansion and compression of the lipid areas in the outer (A_out) and inner (A_in) membrane leaflets, respectively, are derived by employing spheres of diameter D + 2h and D − 2h in the first term in (Eq. 9) and integrating over the neutral leaflet surfaces of the outer (A_out = π(D + 2h)²) and inner (A_in = π(D − 2h)²) membranes (39).

$$d{\gamma }_m=-K_{A,m}\left(\frac{dA}{A}\right)\Rightarrow {\gamma }_m=K_{A,m}\ln \left(\frac{A_{out}}{A_{in}}\right)=2K_{A,m}\ln \left(\frac{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.D+h}{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.D-h}\right)$$ (9)

Here, h = 1.8 nm is the distance from the membrane middle to the neutral surfaces of the leaflets where expansion/compression is decoupled from membrane bending. We used a membrane compression modulus for POPC K_A,m = 235 mN/m (36).

Calculation of membrane curvature deformation energy

The energy, E, corresponding to a deformation of a channel or a pore like α-HL, is the product of the change in pore area, ΔA, and the applied 2-dimensional pressure (membrane tension), γ_m, as

$$E={\gamma }_m\text{Δ}A,\delta E=\sqrt{{\left(\text{Δ}A\delta {\gamma }_m\right)}^2+{\left({\gamma }_m\delta \text{Δ}A\right)}^2}$$ (10)

Results and Discussion

Increasing membrane curvature inhibits single α-HL pore permeability

We were then able to plot the permeability from single α-HL transport events as a function of liposome diameter (Fig. 3 E). The α-HL pore is structurally extremely stable, which suggests that large forces would be required to induce detectable changes (16,40). Interestingly, the experiments revealed that membrane curvature reduced pore permeability by nearly four orders of magnitude over the investigated range of curvatures. This demonstrates a remarkably potent allosteric inhibitory effect of membrane curvature on the activity of α-HL.

Effective pore radius changes with membrane curvature

Next, we converted the measured permeability values (Fig. 3 E) to absolute effective pore radii, adopting the widely used approximation of hindered solute diffusion inside a cylindrical pore ((29,30) and Eq. 4). The accurate measurements of permeability allowed us to calculate the individual effective pore radii with an average weighted standard deviation (SD) of 0.24 Å (Fig. 4). Uncertainties spanned 1–8%, which is comparable to the spread in potassium conductance of 7% obtained by high-resolution electrical recordings of α-HL (37).

For pores formed in liposomes of low curvature (500 nm), we calculated a mean radius (± SD) of 7.1 Å ± 0.4 Å. This was in good agreement with the average radius of 7 Å measured in the crystal structure (13), and with the experimentally measured effective pore radius of 9 Å (21), further supporting the reliability of our activity measurements and the quantitative data analysis based on the pore cylinder model. The effective pore radius of α-HL was continuously reduced by membrane curvature, down to a mean value of 5.7 Å ± 0.3 Å for the smallest measured liposome diameter of 40 nm (Fig. 3 F, black crosses).

Linking pore compression to effective membrane tension

We next calculated the membrane tension needed to induce the observed conformational change (Fig. 3 F, red crosses) using the direct relation between membrane tension, pore compressibility, and pore deformation (Eqs. 5–8). Because it is not possible to measure directly the area compressibility of α-HL, we estimated it from the amplitude in thermal fluctuations of the pore structure obtained from an atomic simulation of α-HL in an unperturbed planar membrane (Eq. 7). We found a tension of ∼50 mN/m in highly curved membranes that declined toward zero in flatter membranes (Fig. 3 F, red crosses). This finding is in agreement with the idea that an unperturbed planar membrane is tensionless, as the symmetric lateral pressure profiles of its two leaflets equalize, whereas membrane curvature enforces an asymmetric pressure profile, produced by the rearrangement of the lipid packing in the two membrane leaflets. This lipid packing rearrangement, occurring with increased curvature-enhanced lipid compression and pressure in the lipid headgroup region of the inner membrane leaflet, but also a remarkable pressure in the alkyl chain region of the outer monolayer, resulted in increased tension (41). These two regions of increased pressure on the α-HL pore imply a reasonable cylindrical pore model that is concentrically compressed upon increased membrane tension. Values of pore radius, membrane tension, and the related deformation energy are effective values and their variations across the membrane and the asymmetry of the membrane are not taken into account in these calculations. The pore model used neglects a possible dependence of pore area compressibility on membrane curvature that most likely would decrease with increased pore compression and therefore yield a higher membrane tension than the ∼50 mN/m cited above for highly curved membranes. Our findings suggest that activity and deformation of α-HL depended on the degree of membrane tension imposed by membrane curvature.

As a cross-validation, we performed an independent theoretical calculation of the tension present in a membrane due to curvature-induced changes in lipid packing of the outer and inner membrane leaflets. According to other reports in the literature (19,35,38,39), the change in lipid packing of a spherical liposome can be calculated as the expansion and compression of lipid area in the outer and inner layers (Eq. 9), assuming that deformations are elastic, and that the compression moduli of the pore and the membrane are constant with curvature. As seen in Fig. 3 F (dashed pink line), the theory was indeed in good quantitative agreement with the tension values calculated independently from pore deformation data (Fig. 3 F, red crosses).

Controlling pore permeability by changing membrane leaflet asymmetry

To further test the dependence of α-HL pore deformation on membrane tension, we performed a control experiment designed to modulate selectively the lipid composition of only the outer membrane leaflet and thereby the curvature-induced asymmetry of the lateral pressure profile (42). This was achieved by addition of a lipid at the exterior of the liposomes, after liposome formation and immobilization. We used a lipid that was labeled fluorescently at the headgroup (DOPE-Atto633) for two reasons: 1), to allow in situ quantification of the absolute amount of incorporation into the outer leaflet of each single liposome shown in Fig. 5 A, in agreement with Hatzakis et al. (23); and 2), to minimize flip-flop to the inner membrane leaflet during data acquisition, because Atto633 is a large zwitterionic and hydrophilic moiety. The short time delay from addition of DOPE-Atto633 to acquisition of α-HL events, on the order of a few minutes, further minimized possible flip-flop (see Materials and Methods).

Labeling of DOPE at the headgroup also converted it effectively to an inverted cone-shaped amphiphile. This shape, which energetically favors the outer monolayer, prevented its flip-flop to the inner monolayer (19) and favored the relaxation of the membrane tension upon incorporation. The latter is confirmed in Fig. 5 B, where insertion of labeled DOPE into the outer leaflet resulted indeed in a relaxation of the membrane and a reduction of the membrane-curvature effect on pore permeability by a factor of 5 for the smallest liposomes, thus validating the contributing effect of membrane tension on pore permeability and hence pore structure, when compared to POPC/POPG liposomes (Fig. 3 E). To further verify how the DOPE-Atto633 insertion directly affected the α-HL pore relative to pores in POPC/POPG liposomes, we calculated the relative increase in pore permeability (Fig. 5 C) using the DOPE insertion trend in Fig. 5 A and the permeability fit parameters of Figs. 5 B and 3 E. Hitherto, Fig. 5 C shows the x-fold increase in α-HL pore permeability upon DOPE-Atto633 insertion that additionally verifies the relaxation of the liposome membrane tension on the pore.

Effective membrane curvature deformation energy

Our data on increasing pore deformation with membrane curvature, and the independent theoretically calculated membrane tension, provide a measure of pore-deformation energy (Eq. 10). The quantified pore-deformation energy is zero for a flat membrane (Fig. 6) but increases with membrane curvature and reaches ∼7 k_BT (∼28.8 pN nm, ∼17.4 kJ mol⁻¹) in membranes of high curvature (1/40 nm⁻¹).

Figure 6.

Deformation energy created by membrane curvature. The change in the α-HL pore radius (black) with membrane curvature can be used to determine the pore-deformation energy (red) utilized by the membrane (Eq. 10). Error is represented as the mean ± SD. To see this figure in color, go online.

Introducing lipids of spontaneous curvature can modulate the strong dependence of energy and tension on membrane curvature as verified by adding DOPE-Atto633. In general, biological lipid membranes are not symmetric in composition, which introduces a nonzero spontaneous membrane curvature, and it is therefore the mismatch between geometrical and spontaneous curvature that defines the induced membrane tension and energy, as described by Helfrich elastic membrane theory (43).

Conclusion

It is well established that changes in the lipid composition of membranes can regulate the localization and structure of both membrane-associated and transmembrane proteins (2–5). In contrast, geometrical membrane curvature has attracted recent attention as a putative regulatory mechanism that can operate independently of the composition of membranes (8,44). A number of experiments have suggested that for a fixed membrane composition, curvature can regulate the spatial localization of many soluble proteins possessing a variety of different membrane-binding domains (8,9) (including BAR domains (45), amphipathic helices (10), and lipid anchors (23)), as well as lipid diffusion (46), sorting, and segregation (12). Our data expand significantly the potential implications of membrane curvature for biological function by demonstrating that it can also regulate allosterically the structure and function of a transmembrane protein.

Curvature deforms membranes asymmetrically, leading to an excited state that in vivo is stabilized by supramolecular protein assemblies (7,19). Our data show that the excited curved membrane of zero spontaneous curvature is under considerable tension (∼50 mN/m for curvatures of 1/40 nm⁻¹, Fig. 3 F), which can impose protein deformation energies (Fig. 6) of ∼7 k_BT (∼29 pN nm, ∼17 kJ mol⁻¹). Such substantial energies have been shown to conformationally activate, and in certain cases unfold, numerous β-barrel and α-helical transmembrane proteins (18,47,48), suggesting that membrane curvature could likely regulate allosterically the structure and function of transmembrane proteins in general, both in vivo and in vitro (e.g., in the highly curved lipid phases used for crystallization of transmembrane proteins). Because the magnitude of membrane lateral pressure and protein compressibility is not constant across the bilayer, simulations and further experimental studies would be necessary to understand precisely how membrane forces are transduced in the different parts of the protein body. Since membrane curvature can change extremely rapidly in living cells (e.g., over millisecond timescales during exocytosis), we speculate that it may enable fast dynamic regulation of transmembrane protein conformation and function.