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 (σ_{osm}) 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 (*k*_{r}) of rupture of the GUVs induced by the constant tension (σ_{ex}) due to an external force as a function of σ_{ex}. We found that larger σ_{ex} 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 (σ_{osm}) by the difference between these σ_{ex} values. At small negative Π, the experimental values of negative σ_{osm} 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.

## I. INTRODUCTION

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 law^{9} 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 proteins^{11–14} 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 channel^{31} 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 (*σ*_{osm}) 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 (*k*_{r}) of the constant tension-induced rupture of GUVs under Π. The total membrane tension (*σ*_{t}) due to the tension induced by an external force (*σ*_{ex}) and *σ*_{osm} determines *k*_{r}, and thus, the value of *σ*_{osm} is determined as the shift amount of the curve (*k*_{r} vs *σ*_{ex}) under Π from the curve for the isotonic solution.^{32,33} We found that at small negative Π, the experimental values of negative σ_{osm} agree with their theoretical values, but as negative Π increases, the deviation between these values of σ_{osm} increases. To explain the Π dependence of σ_{osm}, 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 bilayers^{16,18,40,41} and its dependence on positive σ^{16,19,42} have been thoroughly examined.

## II. MATERIALS AND METHODS

### A. Materials

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}

### B. Preparation of GUVs

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 N_{2} 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 *µ*l 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., *C*_{in}^{0} = 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 *µ*M 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 *µ*m 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).

### C. The constant tension-induced rupture of GUVs under negative osmotic pressure

^{45}In this method, a GUV is fixed at the edge of a micropipette with a diameter of ∼10

*µ*m using an aspiration pressure, Δ

*P*

_{m}. The membrane tension (

*σ*

_{ex}) of the GUV produced by this aspiration can be described using Δ

*P*

_{m}as follows:

^{46}

^{,}

*d*

_{m}is the internal diameter of the micropipette and

*D*

_{v}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., *C*_{in}^{0} = 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 Δ*C*^{0} = −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, *C*_{out}, 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 *µ*m/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 *σ*_{ex} 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.

*k*

_{r}) for the constant tension (

*σ*

_{t})-induced GUV rupture as a function of

*σ*

_{t}for PG/PC (4/6)-GUVs,

^{49}

^{,}

*Γ*is the line tension at the rim of pre-pore in the membrane,

*D*

_{r}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}

### D. Observation of negative osmotic pressure-induced shape change of GUVs

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}

### E. Estimation of the excess area of GUVs under negative osmotic pressure

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 (*F*_{ex}).

### F. Interaction of Mag with single GUVs

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 *µ*M 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-*μ*m-diameter using a positive micropipette pressure of 30 Pa, whose tip was located at a distance of 70 *µ*m 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}

## III. THEORY

### A. Negative membrane tension in lipid bilayers

*P*, 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 Δ

*P*by analyzing the free energy of the lipid bilayer,

*F*. The following equation provides the

*F*of electrically neutral lipid bilayers,

*F*

^{neu}, 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}

*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, σ

^{neu}, arises as follows:

*A*(=

*Na*) is the GUV membrane area.

*ψ*

_{0}<

*k*

_{B}

*T*/

*e*, where

*ψ*

_{0}is the surface potential of the bilayer,

*k*

_{B}is the Boltzmann constant, and

*e*is the elementary charge),

*σ*due to the electrostatic interaction (

*σ*

^{el}) can be expressed as

^{54–56}

^{,}

*X*is the molar fraction of charged lipids in the membrane,

*ε*

_{w}is the relative permittivity of water,

*ε*

_{0}is the permittivity of free space, 1/

*κ*is the Debye length, and

*J*

^{el}is a parameter describing the electrostatic repulsion between charged lipid molecules (=

*X*

^{2}e

^{2}/2

*ε*

_{w}

*ε*

_{0}

*κ*). The total membrane tension for the charged lipid bilayer,

*σ*, can be expressed as

*Q*=

*J*+

*J*

^{el}. At mechanical equilibrium where

*σ*= 0, the total free energy of the charged lipid bilayer,

*F*, has a minimum at the optimal area,

*a*

_{fr}(=(

*Q*/γ)

^{1/2}). Thus,

*Q*= γ

*a*

_{fr}

^{2}, and if

*a*>

*a*

_{fr},

*σ*> 0.

*F*can be expressed as

*F*as a function of

*A*. The slope of the curve at

*A*(>

*A*

_{fr}(=

*Na*

_{fr})) 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*(=8π

*rdr*). During this change,

*F*increases by

*dF*,

*δW*) done by the pressure inside and outside of the GUV and the change in membrane area can be described as

^{9}

^{,}

*P*

_{in}and

*P*

_{out}are the pressure inside and outside of the GUV, respectively,

*V*is the GUV volume, and

*dV*= 4π

*r*

^{2}

*dr*since

*dr*is small. For mechanical equilibrium,

*δW*= 0. Using Eqs. (8) and (9), the following equation is obtained:

*σ*> 0, and thus,

*P*

_{in}>

*P*

_{out}. Equation (10) is identical to the Laplace law if

*σ*in Eq. (10) is replaced with the surface tension between water and air.

^{9}

Next, we consider a spherical GUV with a compressed lipid bilayer where *a* < *a*_{fr}. *σ* for this GUV can be expressed by Eq. (6). Since *Q* = γ *a*_{fr}^{2} and *a* < *a*_{fr}, *σ* < 0. The slope of *F* at *A* (<*A*_{fr}) 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, *P*_{in} < *P*_{out}. 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 *J*_{n}/*a*^{n},^{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).

### B. Negative membrane tension of a GUV induced by a negative osmotic pressure

^{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

*r*

_{0}, its surface area and initial volume are

*A*

_{0}and

*V*

_{0}, respectively, and the osmolarity of a buffer containing sucrose inside the GUV is

*C*

_{in}

^{0}. 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/m

^{3}. When a GUV is transferred into a hypertonic solution where the osmolarity of a buffer containing glucose is

*C*

_{out}(>

*C*

_{in}

^{0}) [i.e., Δ

*C*

^{0}= (

*C*

_{in}

^{0}−

*C*

_{out}) (<0)], a negative Π against the GUV is produced to decrease its volume because Π =

*RT*Δ

*C*

^{0}, where

*R*is the gas constant and

*T*is the absolute temperature, and thus, Π < 0. When the radius of the GUV decreases by Δ

*r*(<0) to reach its shrinkage equilibrium (i.e., its radius is

*r*

_{0}+ Δ

*r*, and its surface area and its volume are

*A*

_{0}+ Δ

*A*and

*V*

_{0}+ Δ

*V*, respectively), Δ

*P*(Δ

*r*) (<0) can be expressed by the membrane tension due to Π,

*σ*

_{osm}(<0) by Eq. (11),

*P*= Π

^{eq}=

*RT*Δ

*C*

^{eq}, where Δ

*C*

^{eq}= (

*C*

_{in}

^{eq}−

*C*

_{out}) (<0) (here, the superscript “eq” represents the physical values at equilibrium). Therefore, we can obtain

*σ*

_{osm}from Δ

*C*

^{eq}as follows:

*C*

_{in}

^{eq}=

*C*

_{in}

^{0}

*V*

_{0}/(

*V*

_{0}+ Δ

*V*), and we used an approximation for Δ

*V*/

*V*

_{0}≈ 3(Δ

*r*/

*r*

_{0}) because the GUV is spherical and Δ

*r*

_{eq}/

*r*

_{0}≪ 1.

^{32}On the other hand,

*σ*

_{osm}can be described by the fractional area change,

*δ*(=Δ

*A*/

*A*

_{0}≈ 2(Δ

*r*/

*r*

_{0}) (<0)), and the elastic modulus of the bilayer of the GUV,

*K*

_{bil},

*r*/

*r*

_{0})

^{2}:

^{32}

^{,}

^{33}

*C*

^{0}< 0,

*σ*

_{osm}< 0, indicating a negative tension.

## IV. RESULTS AND DISCUSSION

### A. Effect of negative osmotic pressure on constant tension-induced rupture of GUVs

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 (*σ*_{t}) due to the tension induced by an external force (i.e., the aspiration pressure) (*σ*_{ex}) and the tension induced by Π (*σ*_{osm}) (i.e., *σ*_{t} = *σ*_{ex} + *σ*_{osm}) determines the rate constant (*k*_{r}) for the constant tension-induced GUV rupture.^{32,33} We can therefore estimate the value of *σ*_{osm} as the shift amount of the curve (*k*_{r} vs *σ*_{ex}) for GUVs under Π from the curve for GUVs under isotonic condition.^{32,33}

*C*

^{0}= −11 mOsm/l. For this purpose, a GUV suspension whose osmolarity was 368 mOsm/l (i.e.,

*C*

_{in}

^{0}= 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,

*C*

_{out}, after the mixing was 379 mOsm/l. First, we examined the effect on a GUV of a constant tension (

*σ*

_{ex}) 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,

*P*

_{intact}(

*t*), can be used. Figure 2(a) shows that the time course of

*P*

_{intact}(

*t*) fit well to the theoretical equation for the two-state transition from an intact GUV to a ruptured GUV as follows:

^{45}

*k*

_{r}is the rate constant of two-state transition or the rate constant of the GUV rupture. The best fitting provided a value of

*k*

_{r}(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

*k*

_{r}value was obtained:

*k*

_{r}= (1.5 ± 0.3) × 10

^{−2}s

^{−1}. Next, we examined the effect of different tensions on the GUVs under the same negative Π.

*P*

_{intact}(

*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

*k*

_{r}. The values of

*k*

_{r}for the rupture of PG/PC (4/6)-GUVs under a negative Π due to Δ

*C*

^{0}= −11 mOsm/l increased with increasing

*σ*

_{ex}, and these values were much smaller than those for GUVs under isotonic condition at the same tension (

*σ*

_{ex}) [Fig. 2(b)]. Thus, the curve (

*k*

_{r}vs

*σ*

_{ex}) under negative Π shifted toward the right (i.e., toward higher

*σ*

_{ex}) compared to the curve obtained under isotonic condition [Fig. 2(b)].

Since *σ*_{t} determines the value of *k*_{r}, we can conclude that the shift amount of the curve for *k*_{r} vs *σ*_{ex} of a GUV under Π toward the right from the curve for the isotonic condition equals the value of *σ*_{osm}.^{32,33} Hence, the value of *σ*_{osm} can be determined by the difference between the values of *σ*_{ex} for a GUV under Π and isotonic condition, both of which induce the same *k*_{r}, since *σ*_{osm} = *σ*_{t} − *σ*_{ex}, and *σ*_{ex} for a GUV under isotonic condition equals to *σ*_{t}. To determine the value of *σ*_{ex} inducing a specific value of *k*_{r} under isotonic condition accurately, we used Eq. (2),^{49} which fits well to the experimental values for isotonic conditions [Fig. 2(b)]. For example, *k*_{r} is 3.1 × 10^{−3} s^{−1} at *σ*_{ex} = 7.0 mN/m under a negative Π due to Δ*C*^{0} = −11 mOsm/l, and using Eq. (2), we obtained the *σ*_{ex} value inducing the same *k*_{r} value under isotonic condition as 5.6 mN/m. Thus, *σ*_{osm} = *σ*_{t} − *σ*_{ex} = 5.6 − 7.0 = −1.4 mN/m. Table I shows the *σ*_{osm} values for other *k*_{r} values. We obtained the mean values ± SDs of *σ*_{osm} by averaging three different *σ*_{ex} values: −1.5 ± 0.2 mN/m, which is the experimental value of *σ*_{osm} under this negative Π. Using *σ*_{t} = *σ*_{ex} −1.5 mN/m, the theoretical curve [Eq. (2)] of *k*_{r} vs *σ*_{t} was obtained, which fit to the experimental results [Fig. 2(b), dashed-dotted line].

σ_{ex} (mN/m) with ΔC^{0}
. | k_{r} (s^{−1})
. | σ_{ex} (mN/m) under ΔC^{0} = 0 obtained using Eq. (S2)
. | σ_{osm} (mN/m) = σ_{ex} (ΔC^{0} = 0) − σ_{ex} (ΔC^{0})
. | Experimental σ_{osm} (mN/m)
. | Theoretical σ_{osm} (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 |

σ_{ex} (mN/m) with ΔC^{0}
. | k_{r} (s^{−1})
. | σ_{ex} (mN/m) under ΔC^{0} = 0 obtained using Eq. (S2)
. | σ_{osm} (mN/m) = σ_{ex} (ΔC^{0} = 0) − σ_{ex} (ΔC^{0})
. | Experimental σ_{osm} (mN/m)
. | Theoretical σ_{osm} (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 *σ*_{osm} can be obtained using Eq. (16). For Δ*C*^{0} = −11 mOsm/l, *C*_{out} = 379 mOsm/l and *K*_{bil} = 141 ± 5 mN/m,^{16} and thus, *σ*_{osm} was −2.7 mN/m. The error in the theoretical values of *σ*_{osm} is estimated from the experimental errors for Δ*C*^{0}, *C*_{out}, and *K*_{bil}. If we estimate the relative errors in *C*_{out} and *C*_{in}^{0} as 0.005 due to the preparation of the solutions, the error in the theoretical values of *σ*_{osm} 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 *σ*_{osm} is slightly smaller than its theoretical value.

Second, we examined the effect of a lower negative Π due to Δ*C*^{0} = −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 *C*_{out} = 374 mOsm/l. Figure S1 shows that the values of *k*_{r} for the GUVs under this Π were much smaller than that for GUVs under isotonic condition at the same *σ*_{ex} but similar to that obtained at Δ*C*^{0} = −11 mOsm/l. Using the same method, we estimated the experimental *σ*_{osm} 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 Δ*C*^{0} = −16 mOsm/l by mixing a GUV suspension under isotonic condition with buffer containing 90 mM glucose so that *C*_{out} = 384 mOsm/l (Fig. S1). The experimental *σ*_{osm} was estimated as −1.2 ± 0.3 mN/m (Table III). Figure 2(c) shows that the experimental and theoretical values of *σ*_{osm} under Δ*C*^{0} = −6 mOsm/l are almost the same, but the negative *σ*_{osm} did not significantly increase above Δ*C*^{0} = −6 mOsm/l, and thus, the deviation between the experimental and theoretical values of *σ*_{osm} increases.

σ_{ex} (mN/m) with ΔC^{0}
. | k_{r} (s^{−1})
. | σ_{ex} (mN/m) under ΔC^{0} = 0 obtained using Eq. (S2)
. | σ_{osm} (mN/m) = σ_{ex} (ΔC^{0} = 0) − σ_{ex} (ΔC^{0})
. | Experimental σ_{osm} (mN/m)
. | Theoretical σ_{osm} (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 |

σ_{ex} (mN/m) with ΔC^{0}
. | k_{r} (s^{−1})
. | σ_{ex} (mN/m) under ΔC^{0} = 0 obtained using Eq. (S2)
. | σ_{osm} (mN/m) = σ_{ex} (ΔC^{0} = 0) − σ_{ex} (ΔC^{0})
. | Experimental σ_{osm} (mN/m)
. | Theoretical σ_{osm} (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 |

σ_{ex} (mN/m) with ΔC^{0}
. | k_{r} (s^{−1})
. | σ_{ex} (mN/m) under ΔC^{0} = 0 obtained using Eq. (S2)
. | σ_{osm} (mN/m) = σ_{ex} (ΔC^{0} = 0) − σ_{ex} (ΔC^{0})
. | Experimental σ_{osm} (mN/m)
. | Theoretical σ_{osm} (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 |

σ_{ex} (mN/m) with ΔC^{0}
. | k_{r} (s^{−1})
. | σ_{ex} (mN/m) under ΔC^{0} = 0 obtained using Eq. (S2)
. | σ_{osm} (mN/m) = σ_{ex} (ΔC^{0} = 0) − σ_{ex} (ΔC^{0})
. | Experimental σ_{osm} (mN/m)
. | Theoretical σ_{osm} (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 (Δ*C*^{0} = 0 mOsm/l) and negative Π (Δ*C*^{0} = −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 Δ*C*^{0} = 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 *σ*_{osm} 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 (*A*_{tot}) is π*D*^{2}. However, generally GUV membranes have excess area (*A*_{ex}) due to their thermal undulation motion,^{46} and thus, *A*_{tot} = π*D*^{2} + *A*_{ex}. We experimentally determined the values of the fraction of *A*_{ex} among the total GUV area (*F*_{ex}) (Sec. S.2 in the supplementary material) and found that the increment of *F*_{ex} at small negative Π is small, but at larger negative Π, *F*_{ex} 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 σ_{osm} 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 (*A*_{ex}). 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 *σ*_{osm} (i.e., −1.4 mN/m, which is the mean value of *σ*_{osm} for Δ*C*^{0} = −6 and −11 mOsm/l). Thus, the maximum fractional area change, *δ*, is −0.01 since *δ* = *σ*_{osm}/*K*_{bil}, and thus, at *a* = 0.99*a*_{fr}, *σ*_{osm} = −1.4 mN/m [see Fig. 1(b)]. According to Eq. (7), the free energy of the lipid bilayer, *F*, at *a* = 0.99*a*_{fr} is larger than *F* at *a* = *a*_{fr} by 0.005% of *F* (*a*_{fr}) since *F* (0.99*a*_{fr})/*F* (*a*_{fr}) = 2.0001*a*_{fr}/2*a*_{fr} = 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* = *a*_{fr} to *a* = 0.99*a*_{fr} 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 *A*_{ex} 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 σ_{osm} no longer increases. This is the probable explanation on the experimental results that a negative σ_{osm} is produced in the membrane of GUVs under negative Π, but the negative σ_{osm} does not increase with increasing negative Π (and as a result, the deviation between the experimental and theoretical values of *σ*_{osm} increases).

### B. Effect of negative osmotic pressure on magainin 2-induced pore formation

*σ*

_{osm}on the rate constant (

*k*

_{p}) of Mag-induced nanopore formation. First, under Π due to Δ

*C*

^{0}= −6 mOsm/l, we investigated the interaction of 31

*µ*M Mag with PG/PC (4/6)-GUVs containing a fluorescent probe, AF488 (

*C*

_{in}

^{0}= 368 mOsm/l) in a hypertonic solution (

*C*

_{out}= 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

*k*

_{p}, we analyzed the time course of the fraction of intact GUVs with no leakage among all examined GUVs,

*P*

_{intact}(

*t*). Since nanopore formation is considered as a two-state transition from the intact state to the initial pore state,

*P*

_{intact}can be expressed as

^{40,42}

*t*

_{eq}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

*k*

_{p}value of 5.4 × 10

^{−3}s

^{−1}. The value of

*k*

_{p}for this condition was (6.2 ± 0.9) × 10

^{−3}s

^{−1}(

*N*= 3). We performed the same experiments under isotonic conditions and Δ

*C*

^{0}= −11 mOsm/l and obtained

*k*

_{p}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

*k*

_{p}value under Δ

*C*

^{0}= −6 mOsm/l is smaller than that under isotonic conditions but similar to that under Δ

*C*

^{0}= −11 mOsm/l.

The value of *k*_{p} 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 (*σ*_{osm}). If *σ*_{osm} > 0, the total tension increases, and thus, *k*_{p} increases.^{16,42} In contrast, if *σ*_{osm} < 0, the total tension decreases, and thus, *k*_{p} decreases. Therefore, the above results suggest that a negative Π induces a negative *σ*_{osm} in the GUV membrane, but the negative *σ*_{osm} due to Δ*C*^{0} = −11 mOsm/l is similar to that due to Δ*C*^{0} = −6 mOsm/l, indicating that the negative *σ*_{osm} values for these conditions are similar, which supports the results obtained by constant tension-induced GUV rupture.

## V. CONCLUSION

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 σ_{osm} for GUVs. At small negative Π, the experimental values of negative σ_{osm} 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.

## SUPPLEMENTARY MATERIAL

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

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**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).

## DATA AVAILABILITY

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

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