Positive membrane tension in the stretched plasma membrane of cells and in the stretched lipid bilayer of vesicles has been well analyzed quantitatively, whereas there is limited quantitative information on negative membrane tension in compressed plasma membranes and lipid bilayers. Here, we examined negative membrane tension quantitatively. First, we developed a theory to describe negative membrane tension by analyzing the free energy of lipid bilayers to obtain a theoretical equation for negative membrane tension. This allowed us to obtain an equation describing the negative membrane tension (σosm) for giant unilamellar vesicles (GUVs) in hypertonic solutions due to negative osmotic pressure (Π). Then, we experimentally estimated the negative membrane tension for GUVs in hypertonic solutions by measuring the rate constant (kr) of rupture of the GUVs induced by the constant tension (σex) due to an external force as a function of σex. We found that larger σex values were required to induce the rupture of GUVs under negative Π compared with GUVs in isotonic solution and quantitatively determined the negative membrane tension induced by Π (σosm) by the difference between these σex values. At small negative Π, the experimental values of negative σosm agree with their theoretical values within experimental error, but as negative Π increases, the deviation increases. Negative tension increased the stability of GUVs because higher tensions were required for GUV rupture, and the rate constant of antimicrobial peptide magainin 2-induced pore formation decreased.

The application of external forces to cells and lipid vesicles to increase the area of the plasma membrane and lipid bilayers (i.e., stretching) increases the counterbalancing membrane tension (σ) that tends to decrease the membrane area. Here, we define the direction of σ compressing the membrane as positive (i.e., positive membrane tension). A positive σ is produced and controlled quantitatively by several techniques, such as the micropipette aspiration method,1,2 the tether force method using optical tweezers,3,4 and other approaches.5–8 The quantitative theories underlying these approaches are well established using the Laplace law9 and other principles.10 Thus, the effects of positive σ on the structure and function of cells and lipid vesicles have been extensively investigated, and a positive σ is known to play important roles in the function of membrane proteins11–14 and pore formation in biomembranes and lipid bilayers.15–19 In contrast, when external forces applied to cells and lipid vesicles decrease the area of these membranes (i.e., compression), the direction of the resulting tension to expand the membrane is opposite to that of a positive σ, and thus, it is called negative σ.

Negative σ may play important roles in the function of membrane proteins and physical property of biomembranes as well as positive σ. For example, negative σ may affect the insertion of membrane proteins and lipids into various biomembranes inside cells and the activities of membrane proteins and peptides/proteins that interact with biomembranes. The interaction of cytoskeleton and curvature-sensing proteins with plasma membrane, exocytosis, and caveolae assembly may be affected by negative σ as well as positive σ.20–22 It is reported that positive σ increases the fluidity of lipid bilayers and diffusion coefficient of lipids/proteins,23–25 and thus, negative σ may change such physical properties of lipid bilayers/biomembranes. The pore formation in lipid bilayers/biomembranes induced by external forces and peptides/proteins may be affected by negative σ as well as positive σ.15–19 Since it is difficult to produce negative σ in lipid bilayers, the effect of negative σ on the function of proteins has been examined using lipid monolayers in the water–air interface.26–28 In these monolayers, as external force applied to them increases, the area per lipid in the monolayers decreases, which induces a negative σ (or a positive surface pressure in the monolayers). When proteins are inserted into lipid monolayers under their constant surface area condition, the positive surface pressure increases (i.e., the negative σ increases or |σ| increases).26 As the negative σ (controlled by an external force) increases, the activities of proteins (such as phospholipase A2 and phospholipase C) decrease.27,28 However, the disadvantage of the monolayer experiment is clear because most functions and physical properties of biomembranes are determined by lipid bilayers. On the other hand, using the asymmetric number distribution of lipids in two monolayers of a lipid bilayer, it is possible to create negative σ in one monolayer, which has a larger number of lipids, whereas the other monolayer (which has a smaller number of lipids) has positive σ.29,30 Such asymmetric tension affects activities of the mechanosensitive ion channel31 and antimicrobial peptide (AMP),18 although these effects are due to the positive σ in one monolayer. However, for lipid bilayers/biomembranes, the lack of a quantitative theory and experimental methodologies to estimate negative σ in these membranes makes it difficult to investigate the effect of negative σ on the activities of membrane proteins and peptides/proteins that interact with biomembranes and physical properties of lipid bilayers/biomembranes.

Here, we report a quantitative theory of negative σ for giant unilamellar vesicles (GUVs) in hypertonic solutions [i.e., under negative osmotic pressure (Π)]. We also experimentally estimated the Π-induced negative σ (σosm) for GUVs comprising dioleoylphosphatidylglycerol (DOPG) and dioleoylphosphatidylcholine (DOPC) (4/6 molar ratio) [hereafter PG/PC (4/6)]. For this purpose, we measured the rate constant (kr) of the constant tension-induced rupture of GUVs under Π. The total membrane tension (σt) due to the tension induced by an external force (σex) and σosm determines kr, and thus, the value of σosm is determined as the shift amount of the curve (kr vs σex) under Π from the curve for the isotonic solution.32,33 We found that at small negative Π, the experimental values of negative σosm agree with their theoretical values, but as negative Π increases, the deviation between these values of σosm increases. To explain the Π dependence of σosm, we experimentally estimated the excess area of GUV membranes. Finally, as an example of the effect of negative σ on the function of peptides/proteins, we investigated AMP-induced pore formation in PG/PC-GUVs under negative Π. Most AMPs induce cell membrane damage (e.g., pore formation) to kill bacterial cells.34–39 Here, we selected magainin 2 (Mag) as AMP because Mag-induced pore formation in lipid bilayers16,18,40,41 and its dependence on positive σ16,19,42 have been thoroughly examined.

DOPC and DOPG were purchased from Avanti Polar Lipids, Inc. (Alabaster, AL). AlexaFluor 488 hydrazide (hereafter AF488) and 1,2-dioleoyl-sn-glycero-3-phosphoethanolamine-N-(7-nitro-2-1,3-benzoxadiazole-4-yl) (18:1 NBD-PE, hereafter NBD-PE) were purchased from Invitrogen (Carlsbad, CA, USA). Glass capillary (G-1) was purchased from Narishige (Tokyo, Japan). Bovine serum albumin (BSA) was purchased from FUJIFILM Wako Pure Chemical Co. (Osaka, Japan). Mag was synthesized by the solid-phase peptide synthesis base on the FastMoc method using an Initiator + Alstra peptide synthesizer (Biotage, Uppsala, Sweden).41,42 The crude Mag was purified using reverse phase HPLC, and its purified peptide was verified using mass spectroscopy.40–42 

Using the natural swelling method, 40 mol. %DOPG/60 mol. %DOPC-GUVs [hereafter PG/PC (4/6)-GUVs] were prepared in buffer (10 mM PIPES, pH 7.0, 150 mM NaCl, and 1 mM EGTA).33 A mixture of DOPG and DOPC in chloroform was prepared in a glass vial, and then, it was dried using N2 gas and then in a vacuum desiccator for more than 12 h.43 First, a dry PG/PC (4/6) film was pre-hydrated using 20 µl Milli-Q, and then, 1.0 ml of buffer containing 80 mM sucrose was added gently on it, which was incubated for 2 h at 37 °C, resulting in a GUV suspension. The GUV lumen contained 78 mM sucrose and 98% of the solutes in buffer, and thus, the osmolarity of the solution in the GUV lumen was 368 mOsm/l (i.e., Cin0 = 368 mOsm/l).33 The GUVs were partially purified by removing multilamellar vesicles and lipid aggregate using the centrifugation (14 000 g × 20 min at 20 °C) of the GUV suspension (producing a partially purified GUVs).

For the experiments of the interaction of Mag with single GUVs, GUVs were prepared in buffer containing 6.0 µM AF488 and 80 mM sucrose using the same method. To prepare purified GUVs, a GUV suspension was centrifuged using the above method, and then, using the membrane filtering method, a supernatant was purified to remove smaller GUVs, LUVs, and free AF488.44 For this purpose, a GUV suspension was filtered at a flow rate of 1 ml/min for 1 h through a Nuclepore membrane (a polycarbonate membrane containing pores with 10 µm diameter; Whatman, GE Healthcare, Ltd., Buckinghamshire, UK) in buffer containing 73 mM glucose (368 mOsm/l), and then, the unfiltered GUV suspension (which did not pass through the Nuclepore membrane) was obtained (producing a purified GUV suspension).

For the experiments of osmotic pressure-induced shape change of GUVs, DOPG/DOPC/NBD-PE (40/59/1)-GUVs were prepared in buffer containing 80 mM sucrose using the same method. The GUVs were partially purified by removing multilamellar vesicles and lipid aggregate using the centrifugation (14 000 g × 20 min at 20 °C) of the GUV suspension (producing a partially purified GUVs).

For this purpose, we used the method of the constant tension-induced rupture of PG/PC (4/6)-GUVs developed previously.45 In this method, a GUV is fixed at the edge of a micropipette with a diameter of ∼10 µm using an aspiration pressure, ΔPm. The membrane tension (σex) of the GUV produced by this aspiration can be described using ΔPm as follows:46,
σex=ΔPmdm41dmDV,
(1)
where dm is the internal diameter of the micropipette and Dv is the GUV diameter outside the micropipette. The glass surfaces of micropipettes and chambers were coated with BSA in the same solution as in the outside of the GUVs.33 The detailed experimental methods were described previously.16,45

To apply Π to the GUVs, a partially purified GUV suspension (i.e., Cin0 = 368 mOsm/l) was mixed with buffer containing various concentrations of glucose at a ratio of 1 to 14 and then transferred into a chamber pre-coated with BSA for observation using microscopy. For example, to apply Π due to ΔC0 = −11 mOsm/l to the GUVs, a partially purified GUV suspension was mixed with buffer containing 85 mM glucose at a ratio of 1 to 14 (i.e., after the mixing, the final osmolarity outside the GUV, Cout, became 379 mOsm/l). The GUVs were observed using a differential interference contrast (DIC) and phase-contrast microscope (IX-71, Olympus, Tokyo, Japan), which was connected to a CMOS camera (JCS-HR5UL, Olympus). The temperature of the chamber was controlled at 25 ± 1 °C using the thermoplate (Tokai Hit, Shizuoka, Japan).33,45 Osmolarities of buffer containing various concentrations of sucrose and glucose were calculated by the equation based on their experimental values.33 We waited for more than 5 min after the mixing (which reaches the shrinkage equilibrium of GUVs) because it is reported that the membrane permeability of water through a GUV membrane is high (e.g., 40−50 µm/s for lipids with a double bond, such as DOPC and DOPG), and thus, the volume of the GUV decreases rapidly to reach an equilibrium within 5 min.32,47,48 Then, we selected a target GUV and held it for 2 min at the tip of a micropipette (containing the buffer with the same osmolarity as that in the outside of the GUVs) using a small aspiration pressure (which corresponds to a σex of 0.50 mN/m). We then increased the membrane tension to a target one by increasing the aspiration pressure within 10 s. Under this condition, the GUV was observed until it was aspirated thoroughly into the micropipette.

The following theoretical equation can fit to the experimental results of the rate constant (kr) for the constant tension (σt)-induced GUV rupture as a function of σt for PG/PC (4/6)-GUVs,49,
kr=Dr3kTσt+BexpπΓ2kBTσt+B,
(2)
where Γ is the line tension at the rim of pre-pore in the membrane, Dr is the diffusion coefficient for a particle in r-phase space, which represents the fluctuation of the radius of a pre-pore in the membrane, and B is a parameter due to the electrostatic interactions arising from the surface charges in the membrane (2.6 mN/m).49 

We used the same method described in Sec. II C for application of negative osmotic pressure to GUVs. DOPG/DOPC/NBD-PE (40/59/1)-GUVs under isotonic condition and negative osmotic pressure were observed using a confocal laser scanning microscope (CLSM; FV1000-D, Olympus) with a 60× objective at 25 ± 1 °C; the temperature was controlled by a stage thermocontrol system (Thermoplate, Tokai Hit, Shizuoka, Japan).18 

We used the same method described in Sec. II C for application of membrane tension to GUVs, observation of GUVs, and application of negative osmotic pressure. After reaching an equilibrium under negative osmotic pressure, we selected a target GUV and applied a membrane tension of 0.50 mN/m to the GUV held at the tip of a micropipette (containing the buffer with the same osmolarity as that in the outside of the GUVs) using a small aspiration pressure for 2 min. Then, we measured the length of various parameters required for the estimation of the excess area of GUVs. Using several equations in Sec. S.2., we calculated the fraction of excess area among the total area of a GUV membrane (Fex).

We investigated the interaction of Mag with GUVs in buffer using the single GUV method.42,50,51 To apply osmotic pressure to PG/PC (4/6)-GUVs, the method described in Sec. II C was used. A suspension of PG/PC (4/6)-GUVs containing 6.0 µM AF488 was purified according to Sec. II B, then mixed with buffer with a higher osmolarity at a ratio of 1 to 14, and then transferred into a chamber pre-coated with BSA. Under a fluorescent phase-contrast microscope (IX-73, Olympus, Tokyo, Japan) with a digital CMOS camera (ORCA-Flash 4.0 V3, Hamamatsu Photonics K.K., Hamamatsu, Japan), the interaction of Mag with single GUVs was observed at 25 ± 1 °C using a stage thermocontrol system (Thermoplate, Tokai Hit, Shizuoka, Japan).41,42 For the interaction, a Mag solution was added continuously to the neighborhood of a GUV through a micropipette with 20-μm-diameter using a positive micropipette pressure of 30 Pa, whose tip was located at a distance of 70 µm from the GUV. It was reported that under this condition, the Mag concentration near the GUV was 78% of that in the micropipette.16 To prevent the photobleaching of fluorescence of AF488, we used a neutral density (ND)-6 filter and a ND-25 filter to decrease the incident light intensity. The fluorescent images of GUVs were analyzed using HCImage software (Hamamatsu Photonics K.K.).41,42 The detailed procedure and analysis methods of the single GUV method were described in detail previously.16,42

Generally, the relationship between positive σ for the GUV membrane and the pressure difference between the inside and outside of the GUV, ΔP, can be described using the Laplace law for interface by replacing the surface tension with the membrane tension.52 However, since surface tension is always positive, it is difficult to apply the Laplace law to negative σ. Thus, here, we determine the relationship between σ and ΔP by analyzing the free energy of the lipid bilayer, F. The following equation provides the F of electrically neutral lipid bilayers, Fneu, which comprises an attractive interaction and a repulsive interaction between neighboring lipids,53 and its validity has been demonstrated for describing various phenomena in lipid bilayers,54 
Fneu=2Nγa+Ja.
(3)
Here, N is the number of lipids in a GUV monolayer, a is the area per lipid in the lipid bilayer, γ is the interfacial energy per unit area at the interface between water and a hydrocarbon chain, and J is a parameter describing the repulsion between electrically neutral lipids. In a stretched bilayer, the membrane tension, σneu, arises as follows:
σneu=FneuA=2γJa2,
(4)
where A (=Na) is the GUV membrane area.
A charged lipid bilayer requires that the effect of electrostatic interaction be included. If we adopt the low electric potential approximation (i.e., ψ0 < kBT/e, where ψ0 is the surface potential of the bilayer, kB is the Boltzmann constant, and e is the elementary charge), σ due to the electrostatic interaction (σel) can be expressed as54–56,
σel=X2e2εwε0κ1a2=2Jela2,
(5)
where X is the molar fraction of charged lipids in the membrane, εw is the relative permittivity of water, ε0 is the permittivity of free space, 1/κ is the Debye length, and Jel is a parameter describing the electrostatic repulsion between charged lipid molecules (=X2e2/2εwε0κ). The total membrane tension for the charged lipid bilayer, σ, can be expressed as
σ=σneu+σel=2γQa2,
(6)
where Q = J + Jel. At mechanical equilibrium where σ = 0, the total free energy of the charged lipid bilayer, F, has a minimum at the optimal area, afr (=(Q/γ)1/2). Thus, Q = γ afr2, and if a > afr, σ > 0. F can be expressed as
F=2Nγa+Qa=2Nγa+afr2a=2γA+Afr2A.
(7)
Figure 1 shows F as a function of A. The slope of the curve at A (>Afr (=Nafr)) is positive, indicating that σ > 0. If the radius of the GUV, r, increases by dr (>0), the area of the GUV membrane increases by Nda (=8πrdr). During this change, F increases by dF,
dF=2Nγa+da+Qa+daγaQa=2NγQa2da=σdA.
(8)
During this process, the volume of the GUV increases, and thus, the total work (δW) done by the pressure inside and outside of the GUV and the change in membrane area can be described as9,
δW=dFPinPoutdV,
(9)
where Pin and Pout are the pressure inside and outside of the GUV, respectively, V is the GUV volume, and dV = 4πr2dr since dr is small. For mechanical equilibrium, δW = 0. Using Eqs. (8) and (9), the following equation is obtained:
ΔP=PinPout=2σr.
(10)
Under this condition, σ > 0, and thus, Pin > Pout. Equation (10) is identical to the Laplace law if σ in Eq. (10) is replaced with the surface tension between water and air.9 
FIG. 1.

The free energy of a lipid bilayer, F, and membrane tension, σ. (a) The F is plotted as a function of its total area, A (=Na), according to Eq. (7) (y axis: arbitrary unit). At mechanical equilibrium, F has a minimum, Fmin, at A = Afr (=Nafr) (i.e., the optimal area). (b) An enlarged figure of panel A near Afr. (c) A schematic drawing showing the relationship between F and σ. The slope of the curve at A1 (>Afr) is positive, indicating that the membrane tension at A1, σ1, is positive. In contrast, the slope of the curve at A2 (<Afr) is negative, indicating that the membrane tension at A2, σ2, is negative.

FIG. 1.

The free energy of a lipid bilayer, F, and membrane tension, σ. (a) The F is plotted as a function of its total area, A (=Na), according to Eq. (7) (y axis: arbitrary unit). At mechanical equilibrium, F has a minimum, Fmin, at A = Afr (=Nafr) (i.e., the optimal area). (b) An enlarged figure of panel A near Afr. (c) A schematic drawing showing the relationship between F and σ. The slope of the curve at A1 (>Afr) is positive, indicating that the membrane tension at A1, σ1, is positive. In contrast, the slope of the curve at A2 (<Afr) is negative, indicating that the membrane tension at A2, σ2, is negative.

Close modal

Next, we consider a spherical GUV with a compressed lipid bilayer where a < afr. σ for this GUV can be expressed by Eq. (6). Since Q = γ afr2 and a < afr, σ < 0. The slope of F at A (<Afr) is negative (Fig. 1), indicating that σ < 0. As the radius r of the GUV increases by dr (>0), Eqs. (8) and (9) hold, and thus, Eq. (10) also holds. Under this condition, σ < 0, and thus, Pin < Pout. Therefore, Eq. (10) also describes the pressure difference between the inside and the outside of the GUV with a negative σ.

Since the general form of repulsive interaction in Eq. (3) is Jn/an,54 we also obtained the theory for n = 2, which is almost the same as for n = 1 (see Sec. S.1. in the supplementary material).

When a GUV is transferred into a hypertonic solution, a negative osmotic pressure (Π) against the GUV is produced to induce an efflux of water. As a result, the GUV is shrunken (i.e., the volume of the GUV decreases), but this volume change is reversible because if the GUV is transferred to an isotonic solution, its volume returns to its initial value. This reversible volume change was demonstrated by an experimental result.47,48 The decrease in volume of the GUV results in the compression of the GUV membrane, inducing a negative membrane tension. Using the theory developed previously,32,33 we obtained the equation of the membrane tension of a GUV under a negative Π as follows. In an isotonic solution, a GUV has a radius of r0, its surface area and initial volume are A0 and V0, respectively, and the osmolarity of a buffer containing sucrose inside the GUV is Cin0. Here, we use a unit of osmolarity (mOsm/L), which is an effective molar concentration of all solutes (mmol/L or mM), identical with mol/m3. When a GUV is transferred into a hypertonic solution where the osmolarity of a buffer containing glucose is Cout (>Cin0) [i.e., ΔC0 = (Cin0Cout) (<0)], a negative Π against the GUV is produced to decrease its volume because Π = RTΔC0, where R is the gas constant and T is the absolute temperature, and thus, Π < 0. When the radius of the GUV decreases by Δr (<0) to reach its shrinkage equilibrium (i.e., its radius is r0 + Δr, and its surface area and its volume are A0 + ΔA and V0 + ΔV, respectively), ΔPr) (<0) can be expressed by the membrane tension due to Π, σosm (<0) by Eq. (11),
ΔP=PinPout=2σosmr0+Δr.
(11)
Under a negative osmotic pressure, as the volume of the GUV decreases, the negative tension of the GUV increases. At equilibrium, ΔP = Πeq = RTΔCeq, where ΔCeq = (CineqCout) (<0) (here, the superscript “eq” represents the physical values at equilibrium). Therefore, we can obtain σosm from ΔCeq as follows:
σosm=RTr0+Δr2ΔCeqRTr0+Δr2Cin01+3Δr/r0Cout,
(12)
where Cineq = Cin0V0/(V0 + ΔV), and we used an approximation for ΔV/V0 ≈ 3(Δr/r0) because the GUV is spherical and Δreq/r0 ≪ 1.32 On the other hand, σosm can be described by the fractional area change, δ (=ΔA/A0 ≈ 2(Δr/r0) (<0)), and the elastic modulus of the bilayer of the GUV, Kbil,
σosm=Kbilδ2KbilΔrr0.
(13)
By combining Eqs. (12) and (13), the following equation is obtained by neglecting (Δr/r0)2:32,
Δrr0=RTΔC0/22Kbilr0+3RTCout2RTΔC02.
(14)
In buffer containing a physiological ion concentration (150 mM NaCl), the summation of the first term and the third term in the denominator is much smaller than the second term, and therefore, Eq. (14) converts to the following equation:33 
Δrr0=ΔC03Cout.
(15)
By combining Eqs. (13) and (15), the following equation is obtained:
σosm=2KbilΔC03Cout.
(16)
Since ΔC0 < 0, σosm < 0, indicating a negative tension.

We estimated the membrane tension of PG/PC (4/6)-GUVs under a negative Π by examining the constant tension-induced rupture of single GUVs. The total membrane tension (σt) due to the tension induced by an external force (i.e., the aspiration pressure) (σex) and the tension induced by Π (σosm) (i.e., σt = σex + σosm) determines the rate constant (kr) for the constant tension-induced GUV rupture.32,33 We can therefore estimate the value of σosm as the shift amount of the curve (kr vs σex) for GUVs under Π from the curve for GUVs under isotonic condition.32,33

First, we examined the constant tension-induced rupture of GUVs under a negative Π due to ΔC0 = −11 mOsm/l. For this purpose, a GUV suspension whose osmolarity was 368 mOsm/l (i.e., Cin0 = 368 mOsm/l) was mixed with buffer containing 85 mM glucose at a ratio of 1 to 14 so that the osmolarity outside the GUV, Cout, after the mixing was 379 mOsm/l. First, we examined the effect on a GUV of a constant tension (σex) of 8.0 mN/m produced by aspiration pressure using a micropipette. After the application of the tension, the GUV was initially intact and apparently unchanged and then suddenly it was aspirated into the micropipette thoroughly. This phenomenon can be explained as follows: initially, nanopore formation occurs in the GUV membrane, then the pore size increases by the membrane tension due to the external force, resulting in GUV rupture, and finally, the GUV was aspirated into the micropipette owing to the aspiration pressure.45,57,58 We performed this experiment using 20 GUVs and observed that the aspiration time for each GUV differed, indicating stochastic rupture of the GUVs. To estimate the rate of this stochastic phenomenon, the fraction of intact GUVs among all examined GUVs, Pintact (t), can be used. Figure 2(a) shows that the time course of Pintact (t) fit well to the theoretical equation for the two-state transition from an intact GUV to a ruptured GUV as follows:45 
Pintactt=expkrt,
(17)
where kr is the rate constant of two-state transition or the rate constant of the GUV rupture. The best fitting provided a value of kr (1.3 × 10−2 s−1). Using the results of three independent experiments (N = 3), each time using 20 GUVs, a mean value ± standard error (SE) for the kr value was obtained: kr = (1.5 ± 0.3) × 10−2 s−1. Next, we examined the effect of different tensions on the GUVs under the same negative Π. Pintact (t) for 7.0 mN/m decreased more slowly than that for 8.0 mN/m [Fig. 2(a)], and analysis of the data provided a smaller value of kr. The values of kr for the rupture of PG/PC (4/6)-GUVs under a negative Π due to ΔC0 = −11 mOsm/l increased with increasing σex, and these values were much smaller than those for GUVs under isotonic condition at the same tension (σex) [Fig. 2(b)]. Thus, the curve (kr vs σex) under negative Π shifted toward the right (i.e., toward higher σex) compared to the curve obtained under isotonic condition [Fig. 2(b)].
FIG. 2.

Estimation of negative membrane tension in PG/PC (4/6)-GUVs under negative Π. (a) Effect of constant tension due to aspiration pressure, σex, on a GUV under Π due to ΔC0 = −11 mOsm/l. The change in the fraction of intact GUVs among all the examined GUVs, Pintact (t), over time. σex: (○) 7.0 and (□) 8.0 mN/m. In each experiment, 20 GUVs were examined. Red lines denote the best fit curves using Eq. (17). (b) Effect of σex on kr. (blue ▲) ΔC0 = −11 mOsm/L and (○) ΔC0 = 0 mOsm/l. Error bars show SEs. The black line corresponds to the best fit curve to Eq. (2) using Γ = 11.4 pN, B = 2.6 mN/m, and Dr = 165 nm2/s.33 The blue dashed-dotted line corresponds to Eq. (2) using σt = σex − 1.5 mN/m. (c) Experimental and theoretical values of negative membrane tension (σosm) for various values of Π (i.e., ΔC0). The mean values and SDs of the experimental values (blue ●) are shown (N = 3). A theoretical curve for σosm determined by Eq. (16) is shown by a red line, and its error bars are determined by the experimental errors of physical parameters used in Eq. (16).

FIG. 2.

Estimation of negative membrane tension in PG/PC (4/6)-GUVs under negative Π. (a) Effect of constant tension due to aspiration pressure, σex, on a GUV under Π due to ΔC0 = −11 mOsm/l. The change in the fraction of intact GUVs among all the examined GUVs, Pintact (t), over time. σex: (○) 7.0 and (□) 8.0 mN/m. In each experiment, 20 GUVs were examined. Red lines denote the best fit curves using Eq. (17). (b) Effect of σex on kr. (blue ▲) ΔC0 = −11 mOsm/L and (○) ΔC0 = 0 mOsm/l. Error bars show SEs. The black line corresponds to the best fit curve to Eq. (2) using Γ = 11.4 pN, B = 2.6 mN/m, and Dr = 165 nm2/s.33 The blue dashed-dotted line corresponds to Eq. (2) using σt = σex − 1.5 mN/m. (c) Experimental and theoretical values of negative membrane tension (σosm) for various values of Π (i.e., ΔC0). The mean values and SDs of the experimental values (blue ●) are shown (N = 3). A theoretical curve for σosm determined by Eq. (16) is shown by a red line, and its error bars are determined by the experimental errors of physical parameters used in Eq. (16).

Close modal

Since σt determines the value of kr, we can conclude that the shift amount of the curve for kr vs σex of a GUV under Π toward the right from the curve for the isotonic condition equals the value of σosm.32,33 Hence, the value of σosm can be determined by the difference between the values of σex for a GUV under Π and isotonic condition, both of which induce the same kr, since σosm = σtσex, and σex for a GUV under isotonic condition equals to σt. To determine the value of σex inducing a specific value of kr under isotonic condition accurately, we used Eq. (2),49 which fits well to the experimental values for isotonic conditions [Fig. 2(b)]. For example, kr is 3.1 × 10−3 s−1 at σex = 7.0 mN/m under a negative Π due to ΔC0 = −11 mOsm/l, and using Eq. (2), we obtained the σex value inducing the same kr value under isotonic condition as 5.6 mN/m. Thus, σosm = σtσex = 5.6 − 7.0 = −1.4 mN/m. Table I shows the σosm values for other kr values. We obtained the mean values ± SDs of σosm by averaging three different σex values: −1.5 ± 0.2 mN/m, which is the experimental value of σosm under this negative Π. Using σt = σex −1.5 mN/m, the theoretical curve [Eq. (2)] of kr vs σt was obtained, which fit to the experimental results [Fig. 2(b), dashed-dotted line].

TABLE I.

Negative membrane tension (σosm) due to a negative Π for ΔC0 = −11 mOsm/l and Cout = 379 mOsm/l. Experimental values of σosm were determined by the difference between the constant tension in a GUV produced by aspiration pressure, σex, for a GUV under Π and isotonic condition, both of which induce the same kr. The theoretical values of σosm were obtained using Eq. (16).

σex (mN/m) with ΔC0kr (s−1)σex (mN/m) under ΔC0 = 0 obtained using Eq. (S2)σosm (mN/m) = σexC0 = 0) − σexC0)Experimental σosm (mN/m)Theoretical σosm (mN/m)
7.0 (3.1 ± 0.4) × 10−3 5.6 −1.4 −1.5 ± 0.2 −2.7 ± 1.0 
7.5 (4.6 ± 0.4) × 10−3 5.8 −1.7 
8.0 (1.5 ± 0.3) × 10−2 6.7 −1.3 
σex (mN/m) with ΔC0kr (s−1)σex (mN/m) under ΔC0 = 0 obtained using Eq. (S2)σosm (mN/m) = σexC0 = 0) − σexC0)Experimental σosm (mN/m)Theoretical σosm (mN/m)
7.0 (3.1 ± 0.4) × 10−3 5.6 −1.4 −1.5 ± 0.2 −2.7 ± 1.0 
7.5 (4.6 ± 0.4) × 10−3 5.8 −1.7 
8.0 (1.5 ± 0.3) × 10−2 6.7 −1.3 

The theoretical values of σosm can be obtained using Eq. (16). For ΔC0 = −11 mOsm/l, Cout = 379 mOsm/l and Kbil = 141 ± 5 mN/m,16 and thus, σosm was −2.7 mN/m. The error in the theoretical values of σosm is estimated from the experimental errors for ΔC0, Cout, and Kbil. If we estimate the relative errors in Cout and Cin0 as 0.005 due to the preparation of the solutions, the error in the theoretical values of σosm is 1.0 mN/m (i.e., −2.7 ± 1.0 mN/m). Based on the error analysis, we can conclude that the absolute experimental value of σosm is slightly smaller than its theoretical value.

Second, we examined the effect of a lower negative Π due to ΔC0 = −6 mOsm/l. For this purpose, a GUV suspension under isotonic condition was mixed with buffer containing 79 mM glucose at a ratio of 1 to 14 so that Cout = 374 mOsm/l. Figure S1 shows that the values of kr for the GUVs under this Π were much smaller than that for GUVs under isotonic condition at the same σex but similar to that obtained at ΔC0 = −11 mOsm/l. Using the same method, we estimated the experimental σosm as −1.3 ± 0.3 mN/m (Table II), which agrees with the theoretical value (−1.5 ± 1.0 mN/m) within experimental error. We also examined the effect of a higher negative Π due to ΔC0 = −16 mOsm/l by mixing a GUV suspension under isotonic condition with buffer containing 90 mM glucose so that Cout = 384 mOsm/l (Fig. S1). The experimental σosm was estimated as −1.2 ± 0.3 mN/m (Table III). Figure 2(c) shows that the experimental and theoretical values of σosm under ΔC0 = −6 mOsm/l are almost the same, but the negative σosm did not significantly increase above ΔC0 = −6 mOsm/l, and thus, the deviation between the experimental and theoretical values of σosm increases.

TABLE II.

Negative membrane tension (σosm) due to a negative Π for ΔC0 = −6 mOsm/l and Cout = 374 mOsm/l. Experimental values of σosm were determined by the difference between the constant tension in a GUV produced by aspiration pressure, σex, for a GUV under Π and isotonic condition, both of which induce the same kr. The theoretical values of σosm were obtained using Eq. (16).

σex (mN/m) with ΔC0kr (s−1)σex (mN/m) under ΔC0 = 0 obtained using Eq. (S2)σosm (mN/m) = σexC0 = 0) − σexC0)Experimental σosm (mN/m)Theoretical σosm (mN/m)
6.5 (2.3 ± 0.2) × 10−3 5.4 −1.1 −1.3 ± 0.3 −1.5 ± 1.0 
7.0 (4.4 ± 0.7) × 10−3 5.8 −1.2 
8.0 (1.0 ± 0.1) × 10−2 6.4 −1.6 
σex (mN/m) with ΔC0kr (s−1)σex (mN/m) under ΔC0 = 0 obtained using Eq. (S2)σosm (mN/m) = σexC0 = 0) − σexC0)Experimental σosm (mN/m)Theoretical σosm (mN/m)
6.5 (2.3 ± 0.2) × 10−3 5.4 −1.1 −1.3 ± 0.3 −1.5 ± 1.0 
7.0 (4.4 ± 0.7) × 10−3 5.8 −1.2 
8.0 (1.0 ± 0.1) × 10−2 6.4 −1.6 
TABLE III.

Negative membrane tension (σosm) due to a negative Π for ΔC0 = −16 mOsm/l and Cout = 384 mOsm/l. Experimental values of σosm were determined by the difference between the constant tension in a GUV produced by aspiration pressure, σex, for a GUV under Π and isotonic condition, both of which induce the same kr. The theoretical values of σosm were obtained using Eq. (16).

σex (mN/m) with ΔC0kr (s−1)σex (mN/m) under ΔC0 = 0 obtained using Eq. (S2)σosm (mN/m) = σexC0 = 0) − σexC0)Experimental σosm (mN/m)Theoretical σosm (mN/m)
6.5 (2.7 ± 0.4) × 10−3 5.5 −1.0 −1.2 ± 0.3 −3.9 ± 1.1 
7.0 (6.1 ± 0.4) × 10−3 6.0 −1.0 
8.0 (1.0 ± 0.1) × 10−2 6.4 −1.6 
σex (mN/m) with ΔC0kr (s−1)σex (mN/m) under ΔC0 = 0 obtained using Eq. (S2)σosm (mN/m) = σexC0 = 0) − σexC0)Experimental σosm (mN/m)Theoretical σosm (mN/m)
6.5 (2.7 ± 0.4) × 10−3 5.5 −1.0 −1.2 ± 0.3 −3.9 ± 1.1 
7.0 (6.1 ± 0.4) × 10−3 6.0 −1.0 
8.0 (1.0 ± 0.1) × 10−2 6.4 −1.6 

To examine the effect of negative Π on the shape of GUVs, we observed DOPG/DOPC/NBD-PE (40/59/1)-GUVs under isotonic condition (ΔC0 = 0 mOsm/l) and negative Π (ΔC0 = −16 mOsm/l) using confocal laser scanning microscopy (CLSM). Figure 3(a) shows the representative data. We did not find any large shape changes of GUVs nor tube formation in the GUV lumen under both conditions. A large undulation motion (with a micrometer scale) of the GUV membrane was not observed under both conditions. We observed 100 GUVs and 92 GUVs under ΔC0 = 0 and −16 mOsm/l, respectively, in three independent experiments, and obtained similar results. It is noted that using optical microscopy, such as CLSM, we cannot observe any nanometer-scale membrane undulation motions of GUV membranes. Next, we consider the mechanism underlying the Π dependence of negative σosm using the excess area of a GUV membrane. If a GUV is completely a sphere with a diameter of D, the total area of the GUV membrane (Atot) is πD2. However, generally GUV membranes have excess area (Aex) due to their thermal undulation motion,46 and thus, Atot = πD2 + Aex. We experimentally determined the values of the fraction of Aex among the total GUV area (Fex) (Sec. S.2 in the supplementary material) and found that the increment of Fex at small negative Π is small, but at larger negative Π, Fex greatly increases [Fig. 3(b)]. Based on this result, we can consider the following scenario. Lipid bilayers in the liquid-crystalline phase contain a lot of water molecules in their membrane interface and the thermal fluctuation of lipid density in the lipid bilayers is very large, and thus, the area per lipid decreases in some time from its optimal area and it increases in the other time. Thus, when a negative Π is applied to a GUV and then the GUV volume starts to decrease, the area per lipid rapidly decreases to respond this change in Π. This decrease in the area per lipid induces a negative σosm in the GUV membrane. If the negative Π is small and, thus, the decrease in the GUV volume is small, the decrease in area per lipid can respond to Π without a significant increase in excess area (Aex). The maximum decrease in area per lipid in the experiments shown in Fig. 2 can be calculated by the maximum of the experimental values of negative σosm (i.e., −1.4 mN/m, which is the mean value of σosm for ΔC0 = −6 and −11 mOsm/l). Thus, the maximum fractional area change, δ, is −0.01 since δ = σosm/Kbil, and thus, at a = 0.99afr, σosm = −1.4 mN/m [see Fig. 1(b)]. According to Eq. (7), the free energy of the lipid bilayer, F, at a = 0.99afr is larger than F at a = afr by 0.005% of F (afr) since F (0.99afr)/F (afr) = 2.0001afr/2afr = 1.00005. Although the value of increment in itself depends on the equation of the free energy of lipid bilayer, F [e.g., Eq. (S5) in the supplementary material], this result indicates that the free energy increment from a = afr to a = 0.99afr is very small. As the negative Π increases, the GUV volume decreases further. Under this situation, the area of the GUV membrane (i.e., the area per lipid) does not decrease, but the membrane bends locally to deviate from the spherical shape (i.e., the Aex increases greatly) because the further decrease in area greatly increases the free energy of the lipid bilayer, but the bending energy of the membrane is small.59 As a result, the negative σosm no longer increases. This is the probable explanation on the experimental results that a negative σosm is produced in the membrane of GUVs under negative Π, but the negative σosm does not increase with increasing negative Π (and as a result, the deviation between the experimental and theoretical values of σosm increases).

FIG. 3.

Effect of negative osmotic pressure on shape of GUVs and excess area of GUV membranes. (a) The time course of CLSM images of a DOPG/DOPC/NBD-PE (40/59/1)-GUV under isotonic condition (ΔC0 = 0 mOsm/l) (1) and under negative osmotic pressure (ΔC0 = −16 mOsm/l) (2). Bar, 10 µm. (b) The fraction of excess area (Fex) among the total area of a GUV membrane for GUVs under negative osmotic pressure. The values of Fex were calculated using Eq. (S15) or Eq. (S16) in the supplementary material. We performed this experiment using total 40 GUVs in two independent experiments, and the mean values and SEs of Fex are shown.

FIG. 3.

Effect of negative osmotic pressure on shape of GUVs and excess area of GUV membranes. (a) The time course of CLSM images of a DOPG/DOPC/NBD-PE (40/59/1)-GUV under isotonic condition (ΔC0 = 0 mOsm/l) (1) and under negative osmotic pressure (ΔC0 = −16 mOsm/l) (2). Bar, 10 µm. (b) The fraction of excess area (Fex) among the total area of a GUV membrane for GUVs under negative osmotic pressure. The values of Fex were calculated using Eq. (S15) or Eq. (S16) in the supplementary material. We performed this experiment using total 40 GUVs in two independent experiments, and the mean values and SEs of Fex are shown.

Close modal
Next, we investigated the effect of negative σosm on the rate constant (kp) of Mag-induced nanopore formation. First, under Π due to ΔC0 = −6 mOsm/l, we investigated the interaction of 31 µM Mag with PG/PC (4/6)-GUVs containing a fluorescent probe, AF488 (Cin0 = 368 mOsm/l) in a hypertonic solution (Cout = 374 mOsm/l). A phase-contrast image of a GUV [Fig. 4(a)] before the interaction exhibited high contrast due to the differences in sucrose and glucose concentrations inside and outside the GUV. Fluorescent microscopy images of the GUV show that high fluorescence intensity due to AF488 remained constant inside the GUV for the first 100 s and then decreased rapidly to zero. Subsequent phase-contrast image showed spherical GUVs with decreased phase contrast. These results indicate that Mag-induced nanopore formation in the membrane started at 100 s.40–42 We repeated the same experiments using 16 GUVs and found that the onset time of nanopore formation differed. To estimate kp, we analyzed the time course of the fraction of intact GUVs with no leakage among all examined GUVs, Pintact(t). Since nanopore formation is considered as a two-state transition from the intact state to the initial pore state, Pintact can be expressed as40,42
Pintactt=expkptteq,
(18)
where teq is the time required to reach the binding equilibrium of Mag between aqueous solution and the GUV membrane. Equation (18) fits well to the result [Fig. 4(c)], providing a kp value of 5.4 × 10−3 s−1. The value of kp for this condition was (6.2 ± 0.9) × 10−3 s−1 (N = 3). We performed the same experiments under isotonic conditions and ΔC0 = −11 mOsm/l and obtained kp values of (1.5 ± 0.4) × 10−2 s−1 and (6.5 ± 1.1) × 10−3 s−1, respectively (N = 3−4). Hence, the kp value under ΔC0 = −6 mOsm/l is smaller than that under isotonic conditions but similar to that under ΔC0 = −11 mOsm/l.
FIG. 4.

Mag-induced nanopore formation in GUVs under negative Π. (a) Microscopic images of a PG/PC (4/6)-GUV under Π due to ΔC0 = −6 mOsm/l during the interaction with 31 µM Mag: (1) and (3) phase contrast images and (2) fluorescence images due to encapsulated AF488. The number below each image denotes the interaction time. Bar, 20 µm. (b) Change in the normalized fluorescence intensity, I(t)/I(0), of the GUV over time shown in panel A. (c) Change in the fraction of intact GUVs among all the examined single GUVs, Pintact, over time. A red line is the best fit curve to Eq. (18). (d) Effect of Π (ΔC0) on the rate constant of Mag-induced pore formation, kp. The mean values and SE values of kp are shown (N = 3–4).

FIG. 4.

Mag-induced nanopore formation in GUVs under negative Π. (a) Microscopic images of a PG/PC (4/6)-GUV under Π due to ΔC0 = −6 mOsm/l during the interaction with 31 µM Mag: (1) and (3) phase contrast images and (2) fluorescence images due to encapsulated AF488. The number below each image denotes the interaction time. Bar, 20 µm. (b) Change in the normalized fluorescence intensity, I(t)/I(0), of the GUV over time shown in panel A. (c) Change in the fraction of intact GUVs among all the examined single GUVs, Pintact, over time. A red line is the best fit curve to Eq. (18). (d) Effect of Π (ΔC0) on the rate constant of Mag-induced pore formation, kp. The mean values and SE values of kp are shown (N = 3–4).

Close modal

The value of kp increases as the stretching of the inner leaflet increases (i.e., the positive membrane tension increases).18,42 The total tension in the inner leaflet is the summation of the Mag-induced positive tension and the membrane tension due to Π (σosm). If σosm > 0, the total tension increases, and thus, kp increases.16,42 In contrast, if σosm < 0, the total tension decreases, and thus, kp decreases. Therefore, the above results suggest that a negative Π induces a negative σosm in the GUV membrane, but the negative σosm due to ΔC0 = −11 mOsm/l is similar to that due to ΔC0 = −6 mOsm/l, indicating that the negative σosm values for these conditions are similar, which supports the results obtained by constant tension-induced GUV rupture.

We have developed a quantitative theory of negative σ for GUVs under negative Π. The results of constant tension-induced rupture of GUVs allowed us to experimentally estimate the negative σosm for GUVs. At small negative Π, the experimental values of negative σosm agree with their theoretical values, but as negative Π increases, the deviation between these values increases. Negative tension increases the stability of GUVs because higher tension is required for GUV rupture, and the rate constant for Mag-induced pore formation decreases.

See the supplementary material for the theory using a different repulsive interaction between neighboring lipids and the excess area of a GUV.

The authors have no conflicts to disclose.

Marzuk Ahmed: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Md. Masum Billah: Methodology (equal); Writing – review & editing (equal). Yukihiro Tamba: Methodology (supporting). Masahito Yamazaki: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

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

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