Interaction and aggregation of lipid vesicles (DLVO theory versus modified DLVO theory)

Colloids and Surfaces B: Biointerfaces 14 (1999) 27 – 45 www.elsevier.nl/locate/colsurfb

Interaction and aggregation of lipid vesicles (DLVO theory versus modified DLVO theory) Shinpei Ohki a,*, Hiroyuki Ohshima b a

Department of Physiology and Biophysics, School of Medicine and Biomedical Sciences, State Uni6ersity of New York at Buffalo, 224 Cary Hall, Buffalo, NY, USA b Faculty of Pharmaceutical Sciences, Science Uni6ersity of Tokyo, Shinjuku-ku, Tokyo, Japan

Abstract Interaction and aggregation of acidic phospholipid (phosphatidylserine) vesicles were studied with variation of cation species and their concentrations in vesicle suspensions, and of vesicle sizes. Aggregation was determined by measuring turbidity of vesicle suspension. The experimental results of aggregation of vesicles induced by monovalent cations (Na + , K + , Cs + and TMA + ) were explained well in terms of the interaction energy of two interacting vesicles using the ordinary Derjaguin–Landau – Verwey – Overbeek (DLVO) theory for both small and large lipid vesicles. However, the experimental results of aggregation of vesicles induced by divalent cations (Ca2 + , Mg2 + and Ba2 + ) were not explained by the ordinary DLVO theory. In order to explain the experimental results of these vesicle aggregation phenomena, it was necessary to modify the theory by including hydration interaction energies which are due to hydrated water at membrane surfaces, and their magnitude and sign depend upon the nature (hydrophobicity) of the membrane surface. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Vessicle interaction; Aggregation; Lipid vesicles ; DLVO theory; Modified DLVO theory

1. Introduction Membrane fusion is an essential molecular event involved in many cellular processes, such as, exocytosis, endocytosis, intracellular vesicle transport, fertilization, viral infection, etc. [1–5]. The elucidation of molecular mechanism of membrane fusion would, therefore, contribute greatly to a better understanding of these cellular phenomena. Since biological membranes are com-

* Correponding author. Fax: + 1-716-829-2344.

posed of lipids and proteins, two possible induction processes have been considered for membrane fusion; fusion induced by proteins of the membranes, and that independent of proteins. As more evidence is accumulated, the former is to predominate in biological membrane fusion. However, in spite of different induction processes of biomembrane fusion, the final stage of all the membrane fusion seems to be in common which involves intermixing of membrane lipids. Since the role of lipid membranes is important for membrane fusion process, in the recent years, a number of workers have studied fusion of lipid membranes [2–4,6,7], particularly, using lipid

0927-7765/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 6 5 ( 9 9 ) 0 0 0 2 2 - 3

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vesicles. Lipid vesicle fusion plays an important role not only in biological cellular processes, but also it serves a practical role in recently developing biotechnology fields: drug delivery, gene transfection into cells, etc. It is generally agreed that vesicle aggregation or adhesion is the essential initial step for two vesicles or membranes to fuse. There are various ways to induce aggregation of lipid vesicles in a suspension solution. In this paper, we would like to limit consideration to vesicle-aggregation induced by ions. Such vesicle aggregation studies have been done by several workers [7 – 9] and the order of ability of ions to induce vesicle aggregation have been found for vesicles of different types of lipids [8–10]. There exist various forces (electrostatic, van der Waals, hydration interactions, etc.) in stability of lipid vesicle suspension. The van der Waals interaction energies of two interacting lipid vesicles in different media have been extensively studied [11]. Uncharged lipid vesicles do not aggregate unless an extreme condition is applied (e.g. very high concentrations of polyvalent ions) because of the hydration repulsive force exerting between two polar membranes. However, for the two adjacent lipid bilayers of multilamellar vesicles, or two parallel lipid membranes which exert a large van der Waals interaction force attributable to a large interacting surface area, there will be a stable adhesion point where the attractive van der Waals forces and hydration repulsive interaction forces make a stable energy minimum point. Such interactions between two deposited or adsorbed neutral lipid bilayers have been studied using an experimental device capable of measuring direct force between interacting flat bilayers as a function of the interlayer distance [12,13]. At the equilibrium bilayer separation, a balance exists between the van der Walls attraction and the steric/hydration repulsion, which gives some information about headgroup properties and the phase state of the hydrocarbon interiors of bilayers. Such measurements have been done down to the interseparation distance of :20 A, ( = 2 nm). Similar experiments mentioned above, using charged lipid membranes have also been done, and it has been confirmed [14,15] that the double layer theory (DLVO) is sufficient to explain the

relation of the interaction energy and intermembrane distances at a relatively large distance. However, the experiments at shorter distances have not been successful. This paper deals with the theory and experiments for aggregation and adhesion of lipid vesicles in both at relatively large distances as well as short distances less than 20 A, .

2. Vesicle interaction and its general theory An obvious initial requirement for membrane fusion to occur is their mutual close approach (membrane adhesion or aggregation). Interaction processes between molecular aggregates or particles are dominated by forces which have long been recognized in colloid chemistry. Colloid stability has been explained by the well-established Derjaguin–Landau–Verwey–Overbeek (DLVO) theory, which accounts for two independent types of long range forces, i.e. electrostatic repulsive forces and van der Waals’ attractive forces for two interacting molecules or particles in suspensions [16,17]. Although the DLVO theory has provided a solid basis for deriving the theoretical framework to describe stability of various colloids, it has recently become apparent that some colloid behaviors cannot always be explained satisfactorily by this theory. In line with interactions between colloidal particles, the interaction forces exerting between charged phospholipid vesicles can be expressed in terms of a repulsion force due to electrostatic interaction, F ele, an attraction force due to van der Waals interaction, F vw and other forces. The sum of these forces determines the free energy and hence we should be able to discuss the ability of vesicles to aggregate in energetical points of view. F tot = F ele + F vw + F other, other

(1)

where F may contain hydration forces, electronic repulsive forces, etc. These forces are rather short range in nature compared with the former two forces, F ele and F vw. For the interacting surfaces at relatively large distances, there may exist a balance between attractive and repulsive forces causing the apposed

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membranes to maintain finite distances. The minimum energy point attained at the balance of electrostatic and van der Waals interaction forces is called the secondary minimum with intermembrane distances ranging from 20 to 100 A, . Aggregation in this minimum is not particularly stable as reflected by the shallow appearance of the energy well. The depth depends upon the presence of salts which reduces the coulombic barrier as the salt concentration increases, and also upon the size of the particles, which attributes mainly to increase in the van der Waals interaction energy with increase in vesicle size. This type of interaction will be discussed with demonstration of the experimental results observed for acidic phospholipid vesicles induced by monovalent cations. On the other hand, strong aggregates are formed at surface separations of 10 A, or less when vesicles aggregate in the so-called primary minimum, which is a condition necessary but not sufficient for membrane fusion to occur. At these small distances, approaching membrane surfaces experience unexpectedly large repulsions, which can be attributed to the hydration of the surface membranes. These kinds of forces were not considered in the original DLVO theory. The forces due to hydration begin to act at intermembrane separations of 20 – 30 A, and increase exponentially as the interseparation distance becomes small. A characteristic decay length of 2–3 A, is seen, which is of the order of the molecular diameter of water. These repulsive forces, which have been called ‘hydration forces’ because their magnitudes are intimately related to the energy required to remove water from the hydrophilic

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surface, dominate over all forces at distances B 25 A, . This was found first in the study of multilamellar lipid membrane systems [18]. The range of the hydration forces is much longer than the hydration shells that surround the lipid headgroup. The hydration repulsive forces depend on the headgroup area and polar nature of lipids, i.e. the larger the headgroup, the larger the hydration repulsion [12–15,19]. Also, later, another type of hydration force was introduced [20]; the force is attractive in nature and due to hydrated water on the hydrophobic surfaces which interact with each other. Therefore, it appears that two kinds of hydration forces exist depending on the nature of two interacting membrane surfaces [21]. In order to discuss the experimental results involving such cases (e.g. polyvalent cation-induced acidic phospholipid vesicle aggregation and fusion), the ordinary DLVO theory must be modified to explain the experimental results, which will be discussed in the later section.

3. Theory—vesicle interaction at relatively large distances (DLVO theory) In order to describe lipid vesicle interaction at relatively large distances, e.g. 30–100 A, (A, = 0.1 nm), the use of the ordinary DLVO theory seems to be sufficient, since hydration and electronic repulsive interactions are short range forces; less than 20 A, and a few A, , respectively, and these would not contribute significantly to the total interaction energy. Thus, the interaction energy exerting two particles (see Fig. 1) are expressed as:

Fig. 1. Schematic diagram for two lipid vesicles of radius a (b = inner radius) suspended in an aqueous solution at a relatively large separation distance, R.

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G tot =UA +V ele

(2)

where UA is the van der Waals interaction energy and V ele the electric repulsive interaction energy. According to the previous studies [9,22], for two identical spherical lipid vesicles at a separation distance R, the van der Waals interaction energy is expressed by UA =U(a,a) +U(b,b) −2U(a,b),

(3)

where U(a,b)= −(A/6) × {2ab/[(R +2a)2 −(a +b)2/4]

s = (1/e)(oro0kkT) × (1−exp(− y0){[1− (h/3)]exp(y0)+ h/3}1/2 (8) s=−s0{1/[1 + K1n1exp(− y0)+ K2n2exp(− 2y0)]} (9) where K1 and K2 are the binding constants of monovalent and divalent cations to negatively charged sites of the membranes, or is the relative permittivity for the medium and o0 is the permittivity of a vacuum, respectively, and the Debye constant, k, is expressed by

+ 2ab/[R +2a)2 −(a −b)2] k= {[2(n1 + 3n2)e 2]/oro0kT}1/2.

+ ln[(R + 2a)2 −((a + b)2/4)] ×/[(R + 2a)2 −(a −b)2]}

(4)

where a and b are the radii of the outside and inside spheres of the vesicles, respectively, A the Hamaker constant of lipid bilayers which is : 4×10 − 14 dynes [9,23] and R the separation distance of two vesicles as shown in Fig. 1. When the membranes possess negative surface charges, the electrostatic interaction energy is expressed [24] by V ele =(1/k 2)[64pa(n1 +3n2)g 2]exp( − kR),

g ={1/(1 − h/3)}{[(1 −h/3)exp(y0) +h/3]1/2 − 1} (6)

where y0 = ec0/kT is the dimensionless form of the surface potential c0 and h has the following relation with the concentrations of 1:1 and 2:1 electrolytes: h= 3n2/(n1 +3n2).

Therefore, if we give the concentrations of electrolytes, ion binding constants to the membrane charged sites, K1 and K2, and the size of the vesicles, we can calculate the interaction energy for two interacting vesicles as a function of the separation distance, R.

4. Experimental methods and procedures

(5)

where n1 and n2 are the concentrations of 1:1 (uni–uni valent) and 2:1 (bi – uni valent) electrolytes, respectively, k the Debye constant, and g has the following relation with the surface potential c0: ×/{[(1−h/3)exp(y0) + h/3]1/2 +1}

(10)

(7)

The initial surface charge density for the case of non-ion binding and the surface charge density for the case of ion binding of vesicle membranes are denoted by s0 and s, respectively. Then, the surface charge densities should satisfy the following equations:

4.1. Chemicals Bovine brain phosphatidylserine (PS) was purchased from Avanti Polar Lipids, Inc. The lipid showed a single spot on silica gel thin-layer chromatographic plates. Chloride salts were all used and were of reagent grade from Fisher Chemical Co. Hepes [N-(2-hydroxyethyl)piperazine-N%-2-ethanesulfonic acid] was the Ultrol grade from Calbiochem. Co. TbCl3·6H2O (99.9% pure) and dipicolinic acid (pyridine-2,6-dicarboxylic acid= DPA) were obtained from Alfa and Sigma Chemical Companies, respectively. Unless otherwise specified, all vesicle suspensions contained a small amount of EDTA (0.02 mM) to remove divalent and polyvalent cations in vesicle suspensions as possible contaminants. The water used was distilled three times, including the process of alkaline permanganate.

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4.2. Small unilamellar 6esicles

4.4. Turbidity measurements

Small unilamellar vesicles (SUV) were prepared in either (a) 100 mM NaCl, (b) 10 mM TbCl3 and 100 mM sodium citrate or (c) 100 mM dipicolinic acid (DPA) solution, all containing 5 mM Hepes as buffer. The pH of these salt solutions was adjusted to 7.0 with HCl and NaOH. Phospholipids were dispersed in one of the above salt solutions at a concentration of 10 mmol/ml, then vortexed for 10 min, sonicated for 30 min and centrifuged at 100 000× g for 1 h as described in an earlier paper [8]. The supernatant containing small unilamellar vesicles (SUV) was used as the SUV stock suspension. The diameter of such vesicles was determined with a submicron particle sizer (Coulter, N4) and was :300 A, ( = 30 nm) in diameter. Vesicles prepared in (b) and (c) solutions were passed through a Sephadex G-75 column with the elution buffer of 0.1 M NaCl/5 mM Hepes, pH 7.0 (NHB), to remove noncapsulated materials in the vesicle suspension medium. 4.3. Large unilamellar 6esicles

Turbidity of lipid vesicle suspensions as a function of various monovalent cation (Na + , K + , Cs + and TMA + ) and divalent cation (Ca2 + , Mg2 + , Ba2 + ) concentrations was measured at 400 nm for SUV and 500 nm for LUV by use of a Hitachi (100-60) spectrophotometer. The vesicles were suspended at 0.1 mmol lipid/ml in the salt solution, and the ion concentrations were raised step by step by adding small amounts of concentrated salt solutions (3–5 M) and the suspension was stirred well. The absorbance was measured at 2 min after changing the salt concentration and then the salt concentration was increased to the next value. The turbidity is considered as a measure of vesicle aggregation. Turbidity of the large unilamellar vesicle suspension was measured at both 400 and 500 nm in order to account for the effect due to the vesicle size differences, but the relative magnitude of turbidity changes was the same for observations at both wavelengths. For the large vesicles, the vesicle concentration was in the range of 0.03– 0.05 mmol/ml of suspension solutions. The cation concentration corresponding to the maximum increase in the rate of absorbance change was obtained from the curve of the figures and defined as the ‘threshold concentration’ of that cation to induce vesicle aggregation.

Large unilamellar vesicles (LUV) were prepared by a method modified [25] from the reverse phase evaporation (REV) method [26]. Phospholipids were dissolved in chloroform (2.5 mg lipid/ml) and about 13% (v/v) of water was added to the above lipid– chloroform solution; the mixture was sonicated for 10 min, then was evaporated almost completely to dryness and the remaining solid mixture was rehydrated with one of the buffer solutions mentioned in the SUV preparation; (a), (b) or (c), of 1 ml per 10 mmol phospholipids, and shaken gently to form a completely uniform and milky suspension. This suspension was then passed through a Sepharose CL-2B column (about 30 cm length ×1.6 cm diameter) to fractionate vesicles into different size distributions. Each eluted sample was collected from the column in 2 ml aliquots. The mean size of vesicles in each fraction was determined by a submicron particle sizer (Coulter, N4) and the vesicles of : 200 and 300 nm in diameter were used for the experiments. The average sizes of two successive fractions were distinctly different. The standard error for each sample size was : 20%.

4.5. Vesicle fusion measurements The fusion of small and large unilamellar vesicles induced by cations was followed by a Tb/ DPA assay (the internal content mixing assay). The details are described in the earlier papers [27,28]. Briefly: After the TbCl3 and DPA-encapsulated vesicles of an equal amount (: 0.1 mmol phospholipid each for SUV and 0.05 mmol each for LUV) were suspended in 2 ml of NHB in a quartz cuvette, aliquots of concentrated monovalent or divalent cation solutions (1–3 M) were injected into the vesicle suspensions to change cation concentrations and the solutions were mixed well. The excitation wavelength was 276 nm and the emission fluorescence was measured at 497 nm with a fluorimeter (Perkin–Elmer, LS-

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by release of the contents from the Tb-encapsulated vesicles (the same amounts as in the fusion experiment, except for free from EDTA during the Sephadex G-75 column chromatography) with 0.5% (w/w) sodium cholate. Then, the percentage of maximum fusion, F, was calculated according to the following formula: F= {I− I0)/(I − Itot),

Fig. 2. Turbidity (OD400) for PS-SUV (phosphatidylserinesmall unilamellar vesicle of 30 nm in diameter, ¥ =2a) suspension as a function of concentration of various salts added in 0.1 M NaCl/5 mM Hepes, pH 7.0 (NHB). , Na + ; , K + ; , Cs + ; ×, TMA + .

(11)

where I is the fluorescence intensity measured at 20 s after mixing with added salts, I0 the fluorescence intensity of the suspension before addition of salts, Itot that corresponding to 100% fusion described above. The concentration of a divalent cation at which the fluorescence intensity–concentration curve gave the sharpest increase was defined as the ‘threshold concentration’ of the divalent cation to induce vesicle fusion. Unless otherwise specified, all experiments were performed at 24°C.

5. Results

5.1. Mono6alent cation-induced 6esicle aggregation

Fig. 3. Experimental results similar to those in Fig. 2 except for use of PS LUV (large unilamellar phosphatidylserine vesicles of 300 nm ¥). , Na + ; , K + ; , Cs + ; × , TMA + .

5). The fluorescence was measured at 20 s after the sample was mixed with the added salt solution. The value for 100% fusion was determined as the fluorescence in the presence of 50 mM DPA

5.1.1. Experimental results The turbidity change in the suspension of PSSUV (phosphatidylserine small unilamellar vesicle of 30 nm in diameter) in NHB (0.1 M NaCl/5 mM hepes, pH 7.0) with respect to changes in concentration of various cations (Na + , K + , Cs + and TMA + ) is shown in Fig. 2. The turbidity increase is considered as a result of vesicle aggregation in the suspension solution since monovalent cations used in this experiment do not induce fusion of phosphatidylserine vesicles [28]. The order of effectiveness of monovalent cations to induce the PS vesicle aggregation was Na + \ K + \ Cs + \ TMA + . The threshold concentrations for the monovalent cations to induce vesicle aggregation were 0.7 M for Na + , 1.4 M for K + , 2 M for Cs + and none for TMA + . Experiments similar to the above except for using PS-LUV (phosphatidylserine large unilamellar vesicles of 300 nm in diameter) were done and the results are shown in Fig. 3. The similar results

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have been reported earlier [9]. The threshold concentration for each ion to induce vesicle aggregation was lower than that for small unilamellar vesicles. Another difference between LUV and SUV aggregations was that, for LUV, the turbidity change occurred almost instantly upon addi-

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tion of concentrated salts into the vesicle suspension to change its concentration, but, for SUV, the change in turbidity occurred slowly for some time from 1 min to a few min until it reached to a stable level, depending on the amounts and types of salts added.

Fig. 4. (a – d) Interaction energies (GT) of two interacting PS-SUV (30 nm ¥: a =15 and b = 10 nm) suspended in monovalent salt solutions having different ion binding constants (K1 =0, 0.05, 0.15, 0.7 M − 1) is plotted as a function of separation distance, R, for given salt concentrations (curve 1 (n1 = 0.1 M), curve 2 (n1 =0.2 M), curve 3 (n1 =0.4 M), curve 4 (n1 =0.6 M), curve 5 (n1 = 0.8 M), curve 6 (n1 = 1.0 M) and curve 7 (n1 = 2 M)). The initial surface charge density of PS-SUV, s0, was assumed to be −e/85 A, 2 [37]. (a) K1 =0, (b) 0.05, (c) 0.15, and (d) 0.7 M − 1.

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5.1.2. Comparison of the experimental results with calculated results Fig. 4 shows the interaction energies (GT) calculated from Eq. (2) with Eqs. (3) – (10) for two PS-SUV (small unilamellar phosphatidylserine vesicles of its diameter of 30 nm) suspended in various salt solutions having different binding constants for cations to the charged groups of the membranes (i.e. K1 =0, 0.05, 0.15, 0.7 M − 1). These values correspond to those (K1 =0 M − 1 for TMA + , 0.05 for Cs + , 0.15 for K + and 0.7 for Na + ) reported previously [29,30]. The energies are plotted as a function of the separation distance R for various salt concentrations (in n1 (M)) of monovalent salts. For the case where the monovalent cation does not bind to the negatively charged sites of the membrane (i.e. K1 =0) the depth of the secondary minimum energy is much less than kT even at the salt concentration of 0.8 M. Therefore, there will be no adhesion of vesicles at that concentration. Even, 1M salt concentration produces only a shallow minimum of :0.2 kT at around 20 A, . Therefore, no vesicle aggregation should be expected to occur in such a situation (Fig. 4a). This case is observed for the PS-SUV suspended in solutions at various concentrations of tetramethylammonium chloride (see Fig. 2, TMA + ). Therefore, the experimental data support the calculated results well. In the cases where counter-monovalent ions bind to the charged sites of membranes, the net surface charge of the membrane will be reduced and the well of the secondary minimum energy points will be deepened which depend on the strength of ion binding. The calculated results for such cases are also shown in Fig. 4(b – d) (K1 = 0.05, 0.15, 0.7 M − 1). As the ion binding constant as well as the salt concentration become greater, the magnitude of energy at the secondary minimum becomes greater. When the magnitude becomes comparable to or more than kT, two interacting vesicles will be relatively stable at the distance of the secondary minimum point. In such a case, therefore, it is possible for vesicle aggregation to occur at the secondary minimum point, which is at 20 – 30 A, distance, since the hydration repulsive force begins to act at about 20 A, or less

distance. The experimental results (Fig. 2) seem to be parallel to those of the calculated results (Fig. 4). In the case of larger vesicle sizes, the calculations similar to the above can be done. The calculated interaction energies of two interacting phosphatidylserine vesicles having 300 nm in diameter which are suspended in various concentrations of monovalent salt solutions are shown in Fig. 5(a–e) with different binding constants of counter cations to the fixed charge of the membrane (e.g. K1 = 0, 0.05, 0.15, 0.7 M − 1). In this case also, the calculated results (Fig. 5) compare well to those of the experimental results (see Fig. 3). In order to compare the experimental results and calculated results, the plot of a quantity, exp[−(Gmin/kT)−0.3], which is the quantity expressing the degree of aggregation of two vesicles (a distribution ratio of two interacting vesicles between the secondary minimum energy state and dispersed state), is shown in Fig. 6(a,b) as a function of salt concentrations for various salt suspensions. In the case where this quantity is equal to 1 or greater, significant amounts of vesicle aggregation would occur where the values of 0.3 in the exponential was rather arbitrarily taken judging from the experimental data. The graphs (Fig. 6a,b) have similar tendency to those of the experimental results (Figs. 2 and 3). Monovalent cations easily and spontaneously induced aggregation of large charged lipid vesicles (LUV) at higher concentrations than 0.2 M, while it requires more than 0.7 M for aggregation of small charged vesicles (SUV). The order of its effectiveness for cations to induce vesicle aggregation is Na + \ K + \ Cs + \ TMA + for both phosphatidylserine LUV and SUV. Aggregation studies using other charged lipid vesicles (e.g. phosphatidic acid) have been performed in our laboratories, and similar analyses for the PA vesicle aggregation in terms of interaction energies have been done but the results are not reported here. The agreement between the calculation and experimental results is good and similar to that for the PS vesicle case presented above. It appears that aggregation phenomena of negatively charged lipid vesicles induced by monovalent salts

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Fig. 5. (a – d) Calculated results similar to those in Fig. 4, except for use of PS LUV (300 nm ¥: a = 150 nm and b =147 nm). The initial surface charge density of PS-LUV was assumed to be − e/68 A, 2 [28].

seem to be satisfactorily explained by use of the DLVO theory. However, the DLVO theory is not necessarily applicable to all lipid vesicle aggregation phenomena. Such an example can be seen with phosphatidylcholine (PC) vesicle systems. Since phosphatidylcholine is electrically neutral (zwitter ionic), according to the DLVO theory, there should be a stable vesicle aggregation occurring

due to the van der Waals attractive forces. However, the experiments showed that there is no aggregation among the PC (suv) vesicles even at very high concentrations of any ionic salts. The reason may be due to that since phosphatidylcholine has a large and strong polar group, the large hydration repulsive interaction force may be exerted onto interacting vesicles at a relatively longer distance (25 A, ), and therefore, the at-

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tractive van der Waals force does not create a deep enough secondary minimum well compared to kT. Therefore, there is no vesicle aggregation for these cases. However, it should be noted that the case of two interacting parallel PC membranes is considered, there is a stable intermembrane separation possible, because of balance of a large van der Walls force attributed to the large interacting area and hydration repulsion interaction at a relatively distant point which has been mentioned in the Introduction. Another example is the case where counter ions strongly bind to the negatively charged sites of the membranes as seen in divalent and polyvalent ions, the DLVO theory analysis is not sufficient to explain the experimen-

Fig. 6. Quantity, exp[ −((G min T /kT)− 0.3)], is plotted as a function of salt concentration of various monovalent cations for (a) PS-SUV (30 nm ¥) and (b) PS-LUV (300 nm ¥) suspensions. , Na + ; , K + ; , Cs + ; × , TMA + .

tal results involving such ions. Such examples will be given in the following section with phosphatidylserine vesicle aggregation induced by divalent cations. Then, the detailed discussion regarding the hydration interaction energy will be given in the later section.

5.2. Poly6alent cation-induced 6esicle aggregation 5.2.1. Experimental results The turbidity and the extent of vesicle fusion in the PS-SUV suspension measured as a function of concentrations of various divalent cations in NHB (0.1 M NaCl/5 mM Hepes, pH 7.0) are shown in Fig. 7(a). As seen from the figure, the addition of mM amounts of divalent cations to vesicle suspensions of NHB induced both vesicle aggregation and fusion and the order of effectiveness to induce vesicle aggregation and fusion was the same for both experiments: Ba2 + \ Ca2 + \ Mg2 + . The threshold concentration to induce aggregation was always slightly lower than that for fusion. Similar results have been reported in refs. [25,27,28]. Experiments similar to the above, except for using PS-LUV (PS large unilamellar vesicles of 200 nm in diameter), were also done. The results are shown in Fig. 7(b). In this case also, both aggregation and fusion were observed for the addition of mM amounts of divalent cation to the vesicle suspension of NHB, except for no vesicle fusion observed with Mg2 + up to 20 mM. 5.2.2. Comparison between experimental and calculated results The interaction energy of two interacting PS vesicles suspended in NHB as a function of divalent cation concentrations using the ordinary DLVO theory is shown in Figs. 8 and 9 for both PS-SUV and PS-LUV cases, respectively, with different binding constants (K2 = 10 and 40 M − 1) for divalent cations, where K2 = 10 M − 1 corresponds to the case for Mg2 + , 30 for Ca2 + and 40 for Ba2 + [28,30–32]. The calculation indicates that there should be no aggregation of both SUV and LUV with addition of divalent cations up to 20 mM in 0.1 M NaCl solution (NHB) as seen from Figs. 8 and 9, while the experiments resulted

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Fig. 7. Aggregation and fusion of (a) PS-SUV (30 nm ¥) and (b) PS-LUV (200 nm ¥) in NHB with respect to various divalent cation concentrations. Open symbols refer to the turbidity increase in vesicle suspensions, and closed symbols refer to vesicle fusion. , Ba2 + ; , Ca2 + ; , Mg2 + .

in aggregation of both vesicles, SUV and LUV, at the same ionic conditions as for the calculation. Therefore, these results indicate that the use of the ordinary DLVO theory is not sufficient to describe vesicle aggregation behaviors for these (divalent cation) cases. In addition to aggregation, the addition of divalent cations in mM range induced fusion of PS vesicles of both sizes (SUV and LUV). A theory to explain such aggregation

phenomena will be described in the following section.

6. Theory—vesicle interaction at a close distance (modified DLVO theory) When two membranes interact at a close distance (e.g. 10–20 A, or 1–2 nm) (see Fig. 10a), the

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hydration energy contribution to the total energy becomes important. Therefore, the total interaction energy, G tot, may be expressed as a sum of the van der Waals attractive interaction energy UA, the electrostatic repulsive interaction energy, V ele and the hydration interaction energy, V hd: G tot(R)=UA +V ele +V hd

(12)

If the separation distance is small enough (e.g. B 20 A, ) compared to the size of the sphere (e.g. the radius of the vesicle is usually \ 200 A, and the thickness of the vesicle membrane is about 50 A, ), the interaction energy of the interacting membranes may be replaced approximately with that of two interacting hydrocarbon plates of infinite

Fig. 8. Calculated interaction energies (GT) per kT are plotted as a function of separation distance, R, for PS-SUV (30 nm ¥) suspended in NHB containing different divalent cations (K2 =10 M − 1 (a) and 40 M − 1 (b)) at various concentrations (curve 1 (n2 = 0.1 mM), curve 2 (n2 = 0.5 mM), curve 3 (n2 = 1 mM), curve 4 (n2 =2 mM), curve 5 (n2 =5 mM), curve 6 (n2 =10 mM) and curve 7 (n2 = 20 mM)).

Fig. 9. Similar calculation results as shown in Fig. 8, but using PS-LUV (200 nm ¥).

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AP = pAH + (1− p)Aw where p is the degree of hydrophobicity of the surface layer (05 p5 1); p= 0 corresponds to the hydrophobicity of the water-like phase and p=1 that of the hydrocarbon phase. The physical meaning of p is given in the later section. Thus, the van der Waals energy can be calculated as a function of the separation distance R for a set of given parameters: Hamaker constants for various regions, hydrophobic index, p, and the thickness, h, of the polar layer.

6.2. Electrostatic energy Electrostatic energy of the same system as above having surface charges is expressed [17,24] by V ele(R)= (1/k)(64kTn1 + 3n2g 2)exp(− kR) Fig. 10. (a) Schematic diagram for two lipid vesicles of radius a interacting each other in an aqueous solution at separation distance R. (b) Schematic diagram of two closely adhered membranes separated by an aqueous phase at a distance, R.

thickness having the surface layer, h, (see Fig. 10b).

6.1. 6an der Waals interaction energy The van der Waals energy for two interacting membranes (as shown in Fig. 10b) per unit membrane area is then expressed by UA(R)=



1 −1 ( AP − AW)2 2 12p R

+

2( AP − AW)( AH − AP) (R + h)2

+

( AH − AP)2 (R + 2h)2

n

(13)

where A is the Hamaker constant, R the interlayer separation distance, h the thickness of the surface layer and Aw is the Hamaker constant of water phase which equals 3.0× 10 − 13 erg and AH that of hydrocarbon which is 6.5× 10 − 13 erg [20,33]. The Hamaker constant of the polar phase, AP, is assumed to be expressed as the mixture of the water and hydrocarbon phases.

(14)

where c0, k and g are the same quantities as defined above. The Debye constant, k, is known for a given experimental solution. The surface potential can be obtained for a given salt concentration and a given binding constant of the ions. Eq. (14) is valid when kR \ 1. Therefore, in our case, the electrostatic interaction energy evaluated at shorter distances than 10 A, is not accurate. However, it will be shown later that, for low surface potential, the electrostatic interaction energy is smaller than the other two terms (hydration and van der Waals energies), and, therefore, makes only a small contribution to the total energy. The calculated values of the van der Waals interaction energies for two interacting parallel membranes having a surface layer thickness of h= 3 A, for various surface hydrophobic indexes (p= 0.0, 0.2, 0.4, 0.5, 0.6, 0.7) are shown in Fig. 11, as a function of the interseparation distance, R. It can be seen that the van der Waals attractive energy has some dependency of the surface hydrophobic index at a short separation distance but not so much at a large separation distance. It also does not depend so much on the thickness of the surface polar layer (the results are not shown). The electrostatic interaction energies are calculated with different surface potentials of the mem-

40

S. Ohki, H. Ohshima / Colloids and Surfaces B: Biointerfaces 14 (1999) 27–45

brane as a function of the interseparation distance (c0 = − 85 mV for the PS membrane in 0.1 M NaCl, pH 7.0 and c0 = −45 mV for the PS membrane in 0.1 M NaCl/l mM Ca2 + , pH 7.0, the latter of which corresponds to the condition of the threshold of small vesicle fusion) as a function of the interseparation distance which are shown in dashed curves in Fig. 11. As is shown in Fig. 11, the magnitudes of electrical repulsive interaction energy are quite small for the surface potential smaller than the order of magnitude of kT/e.

where P0 is a constant having the dimension of pressure and P is the force exerting on the interacting bilayer surfaces per unit area at a separation distance R. The energy of bringing two interacting surfaces from the infinite separation distance to a distance R will be:

6.3. Hydration interaction energy

V hd R (R)= −

6.3.1. Repulsi6e hydration interaction energy The hydration repulsive interaction force between two interacting lipid bilayers having strongly hydrophilic surfaces has been studied extensively [18,19,34]. It is found that the hydra-

The energy to bring two membranes to molecular contact (which is ca. equal to the energy to remove the interlayer water) per unit area would be

tion repulsive interaction force decreases exponentially with respect to the separation distance with a short decay constant l (l= 1.7–2.5 A, ): P= P0e − R/l

V hd R (0)= −

(15)

&

R



&

P dx = P0l exp(−R/l).

(16)

0



P dR= P0l.

(17)

According to published work [25], P0 is estimated to be 7× l09 dynes/cm2 for phospholipid membranes (e.g. PC) in 0.1 M NaCl solution. If the value of l is taken to be 2.0 A, , the energy necessary to remove the interlayer water or to adhere two membrane closely at R= 0, would be P0l= 7× 109 dynes/cm2 × 2×10 − 8 cm=140 dyne/cm (see also the end of Section 7).

6.3.2. Attracti6e hydration interaction energy However, if the membrane surfaces are hydrophobic, the hydration interaction force exerting between two membrane surfaces would be attractive due to the rearrangement of hydrated water on the membrane surface. In this case, the force decays exponentially but with a longer decay constant l%= 10 A, [20]. Such a force is expressed by P%= P %0l% exp(− R/l%)

Fig. 11. van der Waal interaction energies (solid lines) calculated for various surface hydrophobic indexes (p =0, 0.2, 0.4, 0.5, 0.6, 0.7) and the electrostatic interaction energies (dashed lines) as a function of the separation distance, R. The surface potentials (c0 = − 85 and − 45 mV) correspond to those of a phosphatidylserine membrane in 0.1 M NaCl, pH 7.0 and 0.1 M NaCl/1 mM CaCl2, pH 7.0, respectively.

(18)

where P %0 is the negative value. When the hydrophobic index p= 1, the value of P %0 is about −60 dynes/cm, which will be discussed later. Similarly, the energy to bring such surfaces to a distance R would be V hd A (R)= P %0l% exp(− R/l%) =VA(0)exp(− R/l%) (19)

S. Ohki, H. Ohshima / Colloids and Surfaces B: Biointerfaces 14 (1999) 27–45

We assume that any membrane surface consists of a mixture of hydrophobic and hydrophilic surfaces in fractional ratio. Then, the total hydration interaction energy will be expressed as the sum of the hydrophilic and hydrophobic interaction energies. hd V hd(R)= (1− p)V hd R (R) + pV A (R)

= (1−p)P0l exp( − R/l) + pP %0l% exp( − R/l%).

(20)

According to the previous work [35], the energy to remove water from the interlayer space is expressed in terms of interfacial tensions: E hd = 2(gm − gm/w)

(21)

where gm and gm/w are the surface tensions of the membrane and the interfacial tension of the membrane against the water phase, respectively. This energy should be equal to V hd(0): E hd = V hd(0).

(22)

The quantities of gm and gm/w are not easily obtainable, although the latter, gm/w, can be measured to a limited degree. Thus, for simplicity, we assume that the surface of the membrane consists of the mixtures of water-like phase and hydrocarbon-like phase at a certain fractional ratio, which would vary depending on the nature of the membrane surface. gm = pgH +(1 −p)gw,

(23)

where p is the fractional value (0 5p 5 1) which we call the hydrophobic index, representing the degree of hydrophobicity of the membrane surface, and gm and gw are the surface tensions of hydrocarbon and water-like phases, respectively, which are known quantities. The hydrophobic index of lipid membranes can be defined as follows: when the interfacial tension of the membrane against the aqueous phase is zero dyne/cm, the hydrophobic index, p, of the membrane surface is zero, and when it is 50 dynes/cm which is the interfacial tension of the hydrocarbon/water interface, the hydrophobic index is 1.0. The hydrophobic index of lipid membranes can be deduced from the interfacial tension measurements of lipid monolayers formed either at the air/water or the oil/water interface. Under a

41

given salt environment and at an appropriate area per molecule of the monolayer which corresponds to that of the lipid bilayer, the interfacial tension of the lipid monolayer has been measured as a function of fusogenic ion concentration [3,28]. From these values, the hydrophobic index of the lipid bilayer can be deduced. Some estimation for such quantities is given in the later section. Thus, the dehydration energy E hd is rewritten in terms of gm and gw with Eq. (21)Eq. (23). E hd = 2(pgH/a + (1− p)gw/a − pgH/w) = 2p(gH/a − gH/w)+ 2(1− p)gw/a

(24)

where 2p(gH/a − gH/w) and 2(1−p)gw/a are the hydrophobic and hydrophilic terms, respectively, and gH/a  20 dynes/cm, gw/a  72 dynes/cm, and gH/w  50 dynes/cm [36]. Incidentally, when p =0, E hd = 144 dynes/cm2 is calculated from Eq. (24), which is approximately equal to V hd(0)= 140 dynes/cm2 obtained from Eq. (17). This agreement supports that our estimate of hydration energy of the membrane in terms of interfacial tensions described above is reasonable. In the case of p=1.0, the E hd A (0) would be − 60 dynes/cm. Therefore, V hd(0)= − 60 dynes/cm from Eq. (22) in such a case. With Eqs. (16), (19), (20), (22) and (24), the hydration interaction energy V hd(R) can be expressed by the interlayer separation distance R and the known surface tension values for a given hydrophobic index, p, of the membrane surface: V hd(R)= (1− p)V hd R (0)exp(− R/l) +pV hd A (0)exp(− R/l%) = 2(1− p)gw/a exp(− R/l) + 2p(gH/a − gH/w)exp(− R/l%)

(25)

Fig. 12 shows the hydration energies for interacting two membrane surfaces having various surface hydrophobic indexes as a function of the interlayer distance. In the case of acidic lipid membranes in a given monovalent salt solution with various divalent cations, there is apparently one to one correspondence between the surface potential and surface hydrophobicity of the membrane at different divalent cation concentrations [3,28]. The magnitude of hydration energy for two interacting membranes is much larger compared with two

42

S. Ohki, H. Ohshima / Colloids and Surfaces B: Biointerfaces 14 (1999) 27–45

Fig. 12. Hydration interaction energy for two interacting parallel membranes with various hydrophobic indexes (p =0, 0.2, 0.3, 0.4, 0.5, 0.6, and 0.7) as a function of separation distance R. The values of l, l% and h were 2, 10 and 3 A, , respectively.

is estimated to be p= 0.14 in 0.1 M NaCl, considering the effect of curvature of the vesicle membrane. In this case, also, the electric interaction energy and the hydration repulsive energy is quite large and the minimum interaction energy at the distance of 25 A, is too small (0.5 dynes/cm) for two membranes to have a close approach. However, when the hydrophobic index p is 0.28 which corresponds to the interfacial tension of the membrane/ water interface being about 14 dynes/cm, the interaction energy of two membranes has a minimum energy 6 dynes/cm at about 8 A, separation distance. With this energy, the two vesicles are able to have a close contact but they are still separated by a relatively high energy barrier so that the molecules of two interacting membranes would not intermix. A lipid bilayer having interfacial tension of 14 dynes/cm would correspond to that of a small unilamellar phosphatidylserine vesicle in 0.1 M NaCl solution containing 0.7 mM Ba2 + , 1 mM Ca2 + or 8 mM Mg2 + , pH 7.0, which is the fusion threshold condition for the small unilamellar phos-

other interaction energies (van der Waals and electrostatic energies) at a short separation distance (B 10 A, ) and depends greatly upon the degree of surface hydrophobicity of the membrane. It is seen from Fig. 12 that the hydration interaction energy varies from positive values to negative values as the surface hydrophobic index increases from zero. The calculated total interaction energy, GT = UA(R) +V ele(R) + V hd(R), for the two interacting membranes are shown in Fig. 13 as a function of the separation distance R in the cases of various degrees of the surface hydrophobicity (p =0, 0.2, 0.3, 0.4, 0.5 and 0.6). It is seen from Fig. 13 that, in the case of p= 0, the hydration repulsive energy is quite large (see Fig. 12), and the two vesicles will not make a close contact. This corresponds to that of phosphatidylserine large vesicles suspended in TMA + solution. For small phospholipid vesicles suspended in 0.1 M NaCl, the surface hydrophobicity

Fig. 13. The total interaction energies per unit membrane area of interacting parallel membranes for various hydrophobic indexes (p =0, 0.2, 0.3, 0.4, 0.5 and 0.6) as a function of the separation distance R.

S. Ohki, H. Ohshima / Colloids and Surfaces B: Biointerfaces 14 (1999) 27–45

phatidylserine vesicles. Therefore, even at the fusion threshold condition, the two parallel membranes are not energetically ready to mix. However, at the hydrophobic index \0.5, there will be a large change in the total interaction energy; under such conditions, the energy barrier per unit membrane area is reduced greatly and can even become negative, and the two membranes at a close molecular distance could mix.

7. Discussion According to the theoretical analysis described above, when the membrane surface is sufficiently polar and hydrophilic, e.g. the interfacial tension of the membrane/water interface (gm/w) is nearly zero, a large hydration repulsive force would be exerted onto the two interacting membranes at a short separation distance which prevents close approach of the two membranes. The hydration repulsive interaction energy sharply increases at around the separation distance of 10 A, (see Fig. 13, p= 0) as the distance reduces. In addition to hydration repulsive force, when the two interacting membranes have surface charges of the same sign, the electrostatic repulsive interaction force is exerted onto two membranes which is a long range interaction. With this and another long range interaction attractive force, van der Waals forces, vesicles or particles could find a minimum energy point, the secondary minimum point, at a relatively large separation distance (e.g. in the range 20–100 A, ) under a certain condition (see Figs. 4 and 5). Therefore, if vesicles or particles aggregate at the secondary minimum at an interseparation distance \20 A, , we do not need to include the hydration interaction term in discussion for stability of these vesicles where the DLVO theory is sufficient. This situation occurs for two acidic lipid membranes in monovalent salt solutions (e.g. phosphatidylserine vesicles suspended in any monovalent salt solutions). As shown above, the phosphatidylserine vesicles do not aggregate in 0.1 M monovalent salt solutions (Figs. 2 – 5) However, if an ion has the ability to bind (or adsorpb) to the membrane charged sites, it will reduce the effective surface charge of the negatively charged membranes. If the

43

effective surface charge is reduced, the electrostatic repulsive interaction force will be reduced. Therefore, when enough reduction of the electrostatic interaction energy is reduced, which depends on the strength of ion binding (or adsorption) and, particularly, its high ion concentration, it makes the secondary minimum deep, and if the depth is comparable or deeper than the magnitude of kT, it would result in vesicle aggregation (Figs. 4 and 5). In general, ions, which have stronger binding (or adhesion) affinity to the surface charge sites of colloid particles, and also its higher concentration of ions, can induce particle aggregation easier, which is known as the Schulze–Hardy rule in colloid science. This has been clearly shown in the section of monovalent cation-induced aggregation of phosphatidylserine vesicles in this paper. We have observed the same results not only for the phosphatidylserine vesicle but also for the phosphatidic acid vesicle suspensions. Monovalent cations can induce aggregation of charged lipid vesicles at higher concentrations, and with the same cation, vesicles having larger diameters can aggregate more easily, rapidly and strongly than smaller diameter vesicles do, because of larger contribution of van der Waals attractive forces attributed to a larger size of vesicles (see Fig. 5). However, even when such membranes were suspended in monovalent salt solution of even at high concentrations (e.g. 1.0 M NaCl for the phosphatidylserine membranes) and a massive vesicle aggregation occurred, we did not observe fusion of the vesicles as observed for divalent cation cases. Because, even in the case where lipid vesicles aggregate at high concentration (e.g. 1 M) of monovalent salt solution as discussed above, vesicle membrane surfaces are still enough hydrophilic (no increase in interfacial tension from that in 0.1 M NaCl solution) according to the studies of surface tension measurements of lipid monolayers [3,28]. Therefore, large repulsive hydration interaction forces must exert onto the two interacting membranes, the magnitude of which is much greater than the attractive forces exerted from the long range van der Waals interaction forces (see Fig. 12, p=0 and Fig. 11). Even if a minimum energy well is formed at the relatively large distance, the hydration repulsive force prevents for two mem-

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S. Ohki, H. Ohshima / Colloids and Surfaces B: Biointerfaces 14 (1999) 27–45

branes to come close distance. In the case of small lipid vesicles, they are in an expended molecular state than that of the flat bilayer membrane. The area per molecule of such an expanded state is estimated to be 85 A, per molecule [37]. According to the earlier study, the interfacial tension of small lipid vesicle (e.g. SUV) is larger than a large lipid vesicle (e.g. 300 nm in diameter). The initial interfacial tension of the outer monolayer of such vesicles is about 7 dynes/cm from the measurement of interfacial tension of lipid monolayer formed at the oil/water interface [3,38]. The interfacial tension of 7 dynes/cm is equivalent to hydrophobic index of p= 0.14. However, according to the theory described above, there will be still a large repulsive hydration force exerting between two such membranes (SUV). Also, although adhesion between the two membranes may occur at high monovalent salt concentrations, the separation distance is relatively large which is about 25 A, (see Fig. 5), and no molecular intermixing does occur and, thus, no fusion. No vesicle fusion has been observed at this condition experimentally [9,25]. As we have mentioned above, monovalent cations do not alter the negatively charged membrane surface nature (to more hydrophobic surface) greatly, except for H + , but they induce only vesicle aggregation at a relatively long distance and the energy of the secondary minimum is small. However, since divalent cations and polyvalent cations, e.g. trivalent cations can bind or adsorbed onto the membrane surfaces, and cause not only a large reduction in membrane fixed charges, but also more importantly alter the nature of the membrane surface to more hydrophobic. At a certain concentration of polyvalent cations, they induce a close adhesion of the two membranes as seen above. When a small amount of polyvalent cations is present in negatively charged vesicle suspension solution, polyvalent cations bind strongly to the negatively charged sites of the membranes, and also the binding by these ions causes the increase in surface hydrophobicity or increase in interfacial tension of the membrane. For example, at 1 mM Ca2 + in 0.1 M NaCl, the interfacial tension of phosphatidylserine membrane increases about 7 dynes/cm from that in 0.1 M NaCl solution [28,35]. As mentioned above, small vesicle membranes are

in the expanded state. Therefore, in 0.1 M NaCl/1 mM Ca2 + /pH 7.0, the total interfacial tension of the outer layer of small phosphatidylserine vesicles is about 14 dynes/cm. The interfacial tension of 14 dynes/cm (7 dynes/cm due to the expanded membrane and 7 dynes/cm due to divalent cation binding to the membrane) corresponds to the surface hydrophobic index of p= 0.28 according to our definition described above. At p=0.28, the interseparation distance at the interaction energy minimum point becomes small (i.e. :8 A, ) and the adhesion energy (minimum interaction energy) is about 6 dynes/cm and two vesicle closely adhered. If we assume the contact area of two membranes to be 20 × 20 A, 2, the energy due to the interaction is : − 6 kT, which would make two membranes to adhere to each other. At this point, however, there is still a relatively large repulsive energy barrier between the two interacting membranes (see Fig. 13), and therefore, direct molecular mixing between two interacting membranes would not occur through the closely adhered flat bilayer regions. However, due to a strong adhesion, the region at the rim area of two interacting membranes, membrane deformation may occur. If the membranes at the rim area become to have enough high surface tension or high surface hydrophobicity due to the deformation of membranes, these membrane molecules at these regions can exchange each other and becomes the initiation site of membrane fusion. The threshold value of surface hydrophobicity to cause membrane molecular exchange seems to be in the range of p= 0.45–0.5, judging from the experimental data and the calculated results. When the size of vesicles becomes larger, the interfacial tension due to the curved membranes is reduced, and in order to maintain the same attractive energy, the hydrophobicity of the membrane surface induced by ions have to be increased, e.g. the threshold concentration of divalent cation to induce vesicle aggregation has to be increased. This situation has been observed experimentally. As the size of vesicle increases, the tension at the boundary region or the surface hydrophobicity of the boundary region of the adhered membranes decreases. The experimental data also indicate the same tendency. The details account including the

S. Ohki, H. Ohshima / Colloids and Surfaces B: Biointerfaces 14 (1999) 27–45

above consideration for the fusion mechanism of lipid vesicles is in progress. It should be noted that some workers who measured the interaction force between interacting lipid vesicles at various separation distances by use of the atomic force microscope method [39,40], or the interaction energy between cell – cell by use of a contact angle measurement [41], obtained the decay length for hydrophilic repulsive hydration forces to be about 6 – 10 A, , which is a similar magnitude as that of the hydrophobic attractive forces. If we use in our calculations these values (6–10 A, ) instead of 2 A, for the decay length of hydration repulsive forces, the point of the energy minimum will be at larger distances and the depth of energy minimum will be slightly smaller than those shown in Figs. 12 and 13. However, the essential argument regarding vesicle aggregation described above will not be altered; Phenomenon of acidic phospholipid vesicle aggregation caused by monovalent cations except for H + is explained well in terms of the DLVO theory, and that caused by divalent cations is not explained by the DLVO theory but is explained by the modified DLVO theory which includes hydration interaction forces. As far as the decay length of hydrophilic repulsive forces is concerned, the shorter one seems to fit well for the analysis of membrane fusion.

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