Effect of monolayer spontaneous curvature on constant tension-induced pore formation in lipid bilayers

Various external forces induce tension in the plasma membranes of cells and the lipid bilayers of vesicles, which plays important roles in their stability and the functions of membrane proteins and membrane active peptides [e.g., antimicrobial peptides (AMPs)].^1–6 As the membrane tension (σ) increases, the probability of pore formation in the membranes is enhanced, which induces cell and vesicle rupture under some conditions. When pores are formed in membranes, the pore rims increase the free energy of the membranes due to the structural instability of the pore rim. The extra free energy per unit length of the pore rim is the line tension Γ, which decreases the pore size. Therefore, the stability of a pore is determined by σ and Γ.^7–10 Other factors, such as a high electric field^11–13 and osmotic pressure,^14,15 also induce pore formation and vesicle rupture.

Micrometer-sized pores (i.e., micropores) in giant unilamellar vesicles (GUVs) produced by various external forces can be observed using optical microscopy.^{7,12,13,16,17} The membrane tension disappears immediately after pore formation, so the micropore closes rapidly.^8,18 Analysis of the time course of its closure provides Γ for a micropore.^{7,12,13,16,17} However, the processes and the mechanisms underlying micropore formation have not been sufficiently revealed. The application of membrane tension to a GUV using the negative pressure of a micropipette suddenly aspirates the GUV into the micropipette due to membrane tension-induced nanopore formation.^9,19–21 Under this condition, the membrane tension remains after nanopore formation, which increases the pore size and eventually induces GUV rupture. This method is useful for elucidating the kinetics and mechanism of tension-induced pore formation because it enables accurate control of the applied tension. It is considered that the thermal fluctuation of the lipid density in lipid bilayers transiently induces nanometer-sized defects with a lower lipid density (i.e., pre-pores), which close rapidly due to instability.^9,19–21 Analysis by dynamic tension spectroscopy^9,19 and the constant-tension method^20,21 allows us to determine the Γ for a pre-pore. The size of a pre-pore is much smaller than that of a micropore; therefore, the structure and stability of pre-pores may be different from those of micropores. Tension-induced pre-pores and nanopores also play an important role in the interaction of AMPs and cell-penetrating peptides (CPPs) with lipid bilayers.^5,6,22–24 Therefore, it is vital to elucidate the stability of pre-pores. However, the structure and stability of pre-pores are less well understood than those of micropores.

The effect of lipid composition on the rate of tension-induced pore formation and Γ for pre-pores is less well understood. The rate constant for constant tension-induced pore formation increases with the molar fraction of negatively charged lipids, i.e., the surface charge density of lipid bilayers, which indicates that the electrostatic interactions elevate the rate of pore formation.²¹ With increasing number of double bonds in the hydrocarbon chains of lipids, Γ for the pre-pores decreases.⁹ Shorter lipid hydrocarbon chains have a smaller Γ for pre-pores,⁹ which is evidenced by the much larger rate constant of constant tension-induced pore formation in lipid membranes composed of shorter hydrocarbon chains.²⁵ 

Here, we examine the effect of the spontaneous curvature (H₀) for lipid monolayers^26,27 on the rate constant for constant tension-induced GUV rupture or pore formation in GUV membranes and on Γ for pre-pores. Lipids with different H₀ values were incorporated into GUV membranes to change the overall H₀ value for the GUV monolayer. Dioleoylphosphatidylethanolamine (DOPE) and 18:1 lysophosphatidylcholine (18:1 LPC—hereafter, LPC) were used. DOPE forms a hexagonal II phase at room temperature, which indicates a negative H₀,^28,29 whereas LPC forms a micelle, which indicates a positive H₀.^30,31 The H₀ values for DOPE and LPC are −0.36³² and +0.26 nm⁻¹,²⁸ respectively. As lipids with an essentially zero H₀, electrically neutral dioleoylphosphatidylcholine (DOPC) (H₀ = −0.091 nm⁻¹³³) and negatively charged dioleoylphosphatidylglycerol (DOPG) (H₀ = −0.0067 nm⁻¹ in 150 mM NaCl³²) were used. The incorporation of lipids with different H₀ values into membranes to change the overall H₀ for the mixture lipid monolayer has been verified by experimental results [e.g., H₀ for DOPG/DOPE and DOPC/DOPE decreases (or ∣H₀∣ increases) with increasing DOPE concentration^32,34,35 and H₀ for LPC/DOPE increases with increasing LPC concentration up to 10 mol. %²⁸]. Here, GUVs composed of negatively charged lipid bilayers were used, which are similar to plasma membranes, in a buffer containing 150 mM NaCl, which corresponds to the physiological ion concentration for most organisms. To eliminate the effect of electrostatic interactions,²¹ the lipid composition ratio was adjusted so that the surface charge densities for various membranes were the same as those for DOPG/DOPC (4/6, molar ratio, here and elsewhere)-GUVs.²⁰ First, we examined the constant tension-induced rupture of DOPG/DOPE (4/6)-GUVs [with H₀ of −0.22 nm⁻¹ in 150 mM NaCl³² < H₀ of DOPG/DOPC (4/6)] and compared the results with those for DOPG/DOPC (4/6)-GUVs (−0.091 < H₀ < −0.0067 nm⁻¹). Next, we examined the constant tension-induced rupture of DOPG/DOPC/LPC (40/55/10)-GUVs [>H₀ of DOPG/DOPC (4/6)]. We can approximate the area of LPC as 50% of DOPC;^36,37 therefore, the DOPG/DOPC/LPC (40/55/10) bilayer has the same surface charge density as the DOPG/DOPC (4/6) bilayer. Based on these results, the effects of H₀ on k_r and Γ are discussed.

The supplementary material provides the materials and methods.

The constant tension-induced rupture of DOPG/DOPE (4/6)-GUVs in a buffer was investigated using a previously described method.^20,21 The effect of tension at 7.5 mN m⁻¹ was first examined. The appearance of the GUV did not change for some time after the application of this tension, but suddenly, the GUV was aspirated into the micropipette and disappeared after less than 1 s. This rapid aspiration of the GUV has been considered to be due to the tension-induced nanopore formation in the GUV membrane because the membrane tension after pore formation expands the pore radius, which results in GUV rupture, and then ΔP induces the rapid aspiration of the GUV.²⁰ Eighteen GUVs (n = 18) were examined under the same conditions, and the aspiration of each GUV occurred at different times. To analyze such a stochastic phenomenon, we used the fraction of intact GUVs among all examined GUVs, P_intact(t), where t is the time of constant tension application to the GUV.²⁰ Figure 1(a) (black square) shows that P_intact(t) decreased with time. The tension-induced rupture of GUVs is a two-state transition from the intact state to the ruptured state; therefore, the following theoretical equation can be obtained:²⁰ 

where k_r is the rate constant for the rupture. The GUV ruptures immediately after pore formation, so the onset times for rupture and pore formation are the same within the experimental error. Therefore, we can infer that the rate constant k_p for pore formation is the same as k_r. Figure 1(a) shows a fit of the experimental data for P_intact(t) using Eq. (1), where the best fit value of k_r was 4.6 × 10⁻³ s⁻¹. The same experiment was performed three times (N = 3), each time using 18–21 GUVs. The mean value±standard error (SE) of k_r for a tension of 7.5 mN m⁻¹ was (4.8 ± 1.7) × 10⁻³ s⁻¹. We also examined the effect of higher values of tension on the stability of GUVs. Figure 1(a) (white square) shows that P_intact(t) at 8.5 mN m⁻¹ decayed more rapidly and its time course was well fitted using Eq. (1) with k_r = 3.2 × 10⁻² s⁻¹. Figure 1(b) shows that k_r increased with increasing σ. A higher tension is required to induce the rupture of DOPG/DOPE (4/6)-GUVs compared with DOPG/DOPC (4/6)-GUVs, and the k_r values for DOPG/DOPE (4/6)-GUVs are smaller than those for DOPG/DOPC (4/6)-GUVs under the same tension [Fig. 1(b)].

The free energy [U (r)] of a pre-pore in a lipid bilayer under a membrane tension σ can be described as^20,21,38

where r is the pre-pore radius, Γ is the line tension at the pre-pore rim, and U₀ is the nucleation free energy required to form a hydrophilic pre-pore.^38,39U₀ has been experimentally detected in DOPC-GUVs and DOPG/DOPC-GUVs as the activation energy which does not depend on tension,^38,39 although its presence has been considered based on MD simulations.^40,41 Here, we assumed that U₀ exists in the U(r) of a pre-pore in the lipid bilayers with other lipid compositions. B represents the electrostatic interaction due to the surface charge on the GUV membrane [here, B = 2.6 mN m⁻¹ because the surface charge density for GUVs and the ion concentration in buffer are the same as those for DOPG/DOPC (4/6)-GUVs in the same buffer²¹]. The maximum value of U(r) at r = r_C = Γ/(σ + B), U_a, is the energy barrier (or the activation energy) for pore formation,

Due to the thermal force, the pre-pores form in the lipid bilayer under the membrane tension and the pre-pore radius fluctuates. The difference between the nanopores and pre-pores is that water-soluble fluorescent probes cannot pass through pre-pores, whereas they can permeate through nanopores. If the pre-pore radius is less than the critical value r_C, the pre-pore closes, but if it reaches r_C, the nanopore formation occurs (Fig. 2). The Arrhenius equation for k_r can be obtained from Eq. (3) and is expressed as^38,39

where A is the frequency factor, k is the Boltzmann constant, and T is the absolute temperature. By rearranging Eq. (4), we obtain

where the intercept of the curve {ln k_r vs [1/(σ + B)]}, C, is equal to (ln A − U₀/kT) and its slope is equal to πΓ²/kT. The experimental data used in Fig. 1(b) are replotted in Fig. 1(c) (red circle), i.e., ln k_r vs (1/(σ + B)), which was fitted using Eq. (5) and the obtained best-fit values of Γ = 15.6 ± 0.7 pN and C = 11. To compare the Γ value for DOPG/DOPE (4/6)-GUVs with that for DOPG/DOPC (4/6)-GUVs, we analyzed the results for DOPG/DOPC (4/6)-GUVs using Eq. (5) [Fig. 1(c), blue square] and the obtained best-fit values of Γ = 10.7 ± 0.4 pN and C = 5.1. Therefore, the Γ for pre-pores in DOPG/DOPE (4/6)-GUVs is 1.5 times larger than that for DOPG/DOPC (4/6)-GUVs.

The constant tension-induced rupture of DOPG/DOPC/LPC (40/55/10)-GUVs (where the molar fraction of LPC is 0.095) in a buffer was next investigated. LPC has only one hydrocarbon chain, so it can exist in an aqueous solution at relatively high concentrations (less than its critical micelle concentration, 3.0 µM³¹). Therefore, when GUVs containing LPC are present in the buffer, the LPC from the lipid monolayers of the GUV transfers to the aqueous solution and the binding equilibrium between the lipid monolayer and its facing solution is held. The binding constant (K_B) for LPC is large (1.1 × 10⁵ M⁻¹ for a fluorescent analog of LPC⁴²) and the volume of the GUV lumen is small; therefore, the number of LPC molecules that transfer from the inner leaflet of a GUV to the GUV lumen is negligible;⁴² the molar fraction of LPC in the inner leaflet at equilibrium (X_in^eq) decreases from 0.095 to 0.094 and the LPC concentration in the GUV lumen (C_lumen) is 0.85 µM. However, the volume outside the GUVs is large; therefore, the number of LPC molecules that transfer from a GUV outer leaflet to the aqueous solution outside the GUV is larger. To prevent this transfer and retain the LPC concentration in the outer leaflet, the LPC in the aqueous solution was added outside the GUVs. The molar fraction of LPC in the outer leaflet at equilibrium is given by X_out^eq = K_BC_out, where C_out is the LPC concentration in the aqueous solution outside the GUVs; therefore, for DOPG/DOPC/LPC (40/55/10)-GUVs with X_out^eq = 0.0952, the C_out has a value of 0.87 µM. The LPC concentration in the buffer outside the GUVs was, thus, adjusted in the experiments to 0.87 µM.

First, we examined the 4.0 mN m⁻¹ tension-induced rupture of DOPG/DOPC/LPC (40/55/10)-GUVs using 20 GUVs (n = 20). Figure 3(a) (black square) shows that the time course of P_intact(t) could be well fitted using Eq. (1) and provided a k_r value of 2.4 × 10⁻³ s⁻¹. The mean value±SE of k_r for 4.0 mN m⁻¹ was (2.9 ± 0.6) × 10⁻³ s⁻¹ (N = 3). The effect of higher values of tension on the stability of GUVs was also examined. Figure 3(a) (white square) shows that P_intact(t) at 6.0 mN m⁻¹ decayed more rapidly and the time course could be well fitted using Eq. (1) with k_r = 1.3 × 10⁻² s⁻¹. The value of k_r increased with increasing membrane tension, as shown in Fig. 3(b). The values of k_r for DOPG/DOPC/LPC (40/55/10)-GUVs were larger than those for DOPG/DOPC (4/6)-GUVs for tensions ≤6.0 mN/m [Fig. 3(b)]. Figure 3(c) shows fits to the experimental data using Eq. (5), which provide the best fit values of Γ = 6.7 ± 0.2 pN and C = −0.65. The value of Γ for the DOPG/DOPC/LPC (40/55/10) bilayer is almost half that for the DOPG/DOPC (4/6) bilayer.

As a control experiment, the constant tension-induced rupture of DOPG/DOPC/LPC (40/55/10)-GUVs was examined in the absence of LPC in the aqueous solution. The mean value±SE of k_r for 5.0 mN m⁻¹ was (4.3 ± 0.7) × 10⁻³ s⁻¹ (N = 3), which is slightly smaller than that in the presence of LPC [(5.4 ± 0.8) × 10⁻³ s⁻¹] [Fig. 3(b)]. This can be explained by the decrease in the LPC concentration in the outer leaflet due to the transfer to the aqueous solution. Generally, the labeling of fluorescent probes to peptides/substances increases their K_B values to lipid bilayers,⁴³ and thus, the K_B for nonlabeled LPC may be smaller than that for a fluorescent analog of LPC. In this case, X_in^eq and X_out^eq would decrease and C_lumen and C_out would increase. However, the k_r value of DOPG/DOPC/LPC (40/55/10)-GUVs in the absence of LPC in the aqueous solution is much larger than that of DOPG/DOPC (40/60)-GUV and a little smaller than that in the presence of LPC in the buffer [Fig. 3(b)], demonstrating that the decrements of X_in^eq and X_out^eq are small.

If we use the activation energy without including U₀, we can obtain the same Γ values as those obtained above because Γ is determined by the slope in Eq. (5). We can also obtain the Γ values by the analysis using the mean first-passage time technique^20,21 (see the supplementary material).

According to Eq. (5), the k_r value is affected by the energy barrier determined by Γ and the intercept C determined by A and U₀. C(DOPG/DOPE) > C(DOPG/DOPC) > C(DOPG/DOPC/LPC), and thus, this order is opposite to the order of k_r. Therefore, k_r for these GUVs is mainly determined by Γ (i.e., k_r increases with decreasing Γ). Table I shows that Γ decreases with increasing H₀. Here, we discuss this dependence of Γ. The value of Γ is determined by the structure of pre-pores. There are several models for pre-pores (e.g., hydrophobic pre-pores, where the wall is composed of hydrocarbon chains, and hydrophilic pre-pores, where the walls are composed of lipid headgroups).^{11,40,41,44–49} Our recent result (the tension-enhanced increase in the rate of transbilayer movement of phospholipids without pore formation)⁵⁰ supports a hydrophilic pre-pore, such as a toroidal pre-pore [Fig. 4(a)], where the outer and inner monolayers bend and connect to each other at the rim and the surface of the wall is composed of hydrophilic headgroups [Fig. 4(b)]. If continuum mechanics is applied to obtain the free energy of the pre-pore rim, we can consider a sheet (or a large monolayer) connected to the outer and inner monolayers. At the pre-pore rim, this sheet bends significantly, and thus, the monolayer has a large, positive curvature in its side view, as shown in Fig. 4(b). The bending energy E_b that is required to bend the monolayer from H₀ to this large, positive curvature (H) is proportional to κ_m(H − H₀)², where κ_m is the monolayer bending modulus. However, in the top view shown in Fig. 4(c), the monolayer has a negative curvature, whose magnitude depends on the pre-pore radius. It has been reported that a water-soluble fluorescent probe (AF647) cannot pass through DOPG/DOPC (4/6)-GUVs in a buffer with a σ of 6 mN m⁻¹⁵⁰ and that the Stokes–Einstein radius for AF647 is 0.8–0.9 nm, based on that for a similar compound (0.88 nm).⁵¹ It is, thus, inferred that the maximum radius of pre-pores is smaller than 0.8 nm. Therefore, the negative curvature at the mid-surface of the lipid bilayer is large [Fig. 4(c)]and its average absolute value is larger than that of the positive curvature shown in Fig. 4(b). The toroidal pre-pore rim, thus, has a kind of a saddle surface [Fig. 4(a)]. The curvature elastic energy f_curv for the monolayer can generally be expressed based on the Helfrich model^52–55 as

where H is the mean curvature at the neutral surface of the monolayer, K is the Gaussian curvature at the neutral surface, κ_G is the Gaussian curvature modulus of the monolayer, and the integration extends over the monolayer’s neutral surface. Currently, there is no information on κ_m and κ_G for the lipid compositions used in this report. For electrically neutral lipid monolayers, κ_G ranges from −5 to −10 kT.^56–58 The pre-pore formation in a GUV transforms two closed monolayers to a closed merged monolayer (Fig. S3 in the supplementary material). According to the Gauss–Bonnet theorem, the integral of K of a closed surface can be calculated as follows:

where g is the genus.⁵⁹ A sphere composed of a closed monolayer with no handle has a genus g = 0, and thus, ∫KdA = 4π. For the two spherical closed monolayers before the pre-pore formation, ∫KdA = 8π, and for the closed merged monolayer after the pre-pore formation, ∫KdA = 4π. Therefore, the formation of a pre-pore induces a change in the integrated Gaussian curvature of −4π. This inference is similar to that used for the formation of a fusion pore.⁶⁰ As a result, the value of f_curv due to K (i.e., f_curv^G) is estimated as −4πκ_G⁶⁰ and, thus, 63 kT for κ_G = −5 kT. The Γ due to f_curv^G is estimated as f_curv^G/2πr = 10 kT/r = 82 pN for r = 0.5 nm, which is much larger than the experimental values of Γ, although f_curv^G may be related to the nucleation free energy because it does not depend on r.⁶¹ Moreover, the experimental results for the dependence of Γ on H₀ indicate that regions with a positive curvature [Fig. 4(b)] are dominant in the pre-pore rim, which is also in contradiction to the toroidal pre-pore structure [Fig. 4(a)]. These contradictions suggest that the pre-pore does not have a regular toroidal structure but an irregular structure with a hydrophilic surface, whose line tension is not determined by Eq. (6) but mainly by the bending energy of the regions with a positive curvature.

We examined the constant tension-induced rupture of GUVs composed of lipid compositions with different spontaneous curvatures of monolayers (H₀). The rate constant of the rupture of GUVs (k_r) due to pore formation increases with increasing H₀. The line tension (Γ) for pre-pores in GUV membranes, determined by the analysis of the tension dependence of k_r, decreases with increasing H₀, which results in an increase in k_r. This suggests that the bending energy of the regions with a positive curvature of pre-pores contributes to Γ.

See the supplementary material for the materials and methods, the analysis of the tension dependence of k_r using the mean first passage time (MFPT), and the illustration of the integrated Gaussian curvature of closed monolayers in a single GUV before and after pre-pore formation.

This work was supported in part by a Grant-in-Aid for Scientific Research (B) (Grant No. 19H03193) from the Japan Society for the Promotion of Science (JSPS) to M.Y.

The authors have no conflicts to disclose.

Kanta Tazawa: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Masahito Yamazaki: Conceptualization (equal); Data curation (lead); Formal analysis (equal); Funding acquisition (lead); Investigation (lead); Methodology (lead); Project administration (lead); Resources (lead); Software (lead); Supervision (lead); Validation (lead); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.