Interpretation of Mechanochemical Properties of Lipid Bilayer Vesicles

998

Langmuir 1999, 15, 998-1010

Interpretation of Mechanochemical Properties of Lipid Bilayer Vesicles from the Equation of State or Pressure-Area Measurement of the Monolayer at the Air-Water or Oil-Water Interface Si-shen Feng Department of Chemical Engineering, National University of Singapore, and Institute of Materials Research and Engineering, 10 Kent Ridge Crescent, Singapore 119260 Received February 5, 1998. In Final Form: August 3, 1998 There are quite a few types of equation of state and abundant π-a curves of various lipid monolayers at the air-water or the oil-water interface in the literature. However, it has been a problem to interpret mechanochemical properties of bilayer vesicles from the π-a information of the monolayer. In fact, even the bilayer surface pressure has not yet been well characterized although the monolayer surface pressure has already been traditionally defined as the lowering in the surface tension from the clean interfacial tension due to the presence of the monolayer. The monolayer-bilayer correspondence problem, therefore, could not be well defined and completely solved despite its importance in practice to apply the monolayer π-a data to elucidate bilayer vesicle properties. In the present analysis, we thus first define the bilayer leaflet pressure as the intrinsic pressure of the lipid layer-water substrate system. This intrinsic surface pressure should be the same function of the lipid density and the temperature for both the monolayer at an interface and a leaflet of bilayer vesicles. Therefore, the difference between a leaflet of a bilayer and the monolayer at an interface is merely that the latter, but not the former, exhibits a microscopic interfacial tension between the air or the oil and the lipid layer. We show that the value of this intrinsic pressure agrees with that of the traditional monolayer pressure if the macroscopic and microscopic hydrophobic effect assumes the same magnitude. We conclude that the equilibrium pressure between the monolayer and bilayer vesicles is equal in magnitude to the microscopic interfacial tension between the water and the monolayer, which, in first approximation, is equal to the macroscopic oil-water interfacial tension, i.e., ca. 49 mN/m. This conclusion agrees with that briefly derived by pioneers (Gruen and Wolfe, 1982; Nagle, 1976, 1986; Ja¨hnig, 1984). We further develop from mechanics and thermodynamics of membranes a procedure to obtain either analytically from a theoretical or empirical equation of state, or graphically from the π-a curve of the monolayer at an interface, mechanochemical properties of this monolayer and bilayer vesicles. The method is exemplified for the monolayer and bilayer vesicles of dilauroyl phosphatidylethanolamine (DLPE).

Introduction In biological systems, lipids usually organize vesicles by detachment of a portion of a lipid bilayer from a parent plasma or organelle membrane.1 Natural lipid vesicles are involved in many membrane phenomena such as interaction between membrane proteins, interaction between proteins and lipids, external molecule penetration, antibody and ligand binding, membrane channel formation, membrane-membrane interaction, cell fusion, etc. Artificial lipid vesicles are widely applied in cellular and molecular therapy, drug delivery, and artificial blood. Artificial lipid vesicles can be seen as the simplest artificial cells. The simplest artificial blood may be the solution of lipid vesicles with hemoglobins encapsulated inside.2-4 Vesicles are difficult to handle in experiments. The phenomenological measurement of the monolayer at the air-water or oil-water interface, however, can be conveniently carried out under a variety of conditions. It is believed that many important phenomena which occur in bilayer vesicles can, in principle, be elucidated by experiments on the monolayer at an interface. The relationship (1) Alberts, B.; Bray, D.; Lewis, J.; Raff, M.; Roberts, K.; Watson, J. D. Molecular Biology of Cells, 3rd ed.; Garland Publishing: New York, 1994. (2) Djordjevich, L.; Miller, I. F. Exp. Hematol. 1980, 8, 584. (3) Miller, I. F.; Hoag, J. M.; Rooney, M. W. Biomater., Artif. Cells, Immobilization Technol. 1992, 20, 627. (4) Pietrzak, W. S.; Miller, I. F. Biomater., Artif. Cells, Artif. Organs 1989, 17, 563.

between monolayers and bilayer vesicles is therefore essential for interpretation of the bilayer properties from the monolayer experiment. The problem of monolayerbilayer correspondence is first defined, in its narrow and strict meaning, as focusing on whether or not, and if the answer is yes, then under what conditions, a monolayer at the air-water interface can be mechanochemically equivalent to one leaflet of bilayer vesicles in water. Considering the numerous π-a curves of various lipid monolayers at the air-water or the oil-water interface published in the literature, one may want to know what kind of information we can collect from these curves about the monolayer itself and the bilayer vesicles of the same kind of lipids and what practical use these curves may have. The monolayer-bilayer correspondence problem is thus further defined, in its broad and expanded meaning, as how we can interpret the mechanochemical properties of the monolayer and, further, of the bilayer vesicles from the measured π-a curve. In principle, all thermodynamic properties of a monolayer, such as partition function, free energy, surface pressure, surface tension, chemical potential, activity of lipids, etc., are related by principles of membrane mechanics and thermodynamics. The solution of the monolayer-bilayer correspondence problem in its strict definition as stated above would further make it possible to obtain the thermodynamic properties of the bilayer vesicles simply from the π-a curve measurement of the monolayer at the air-water (or oil-water) interface.

10.1021/la980144f CCC: $18.00 © 1999 American Chemical Society Published on Web 01/14/1999

Mechanochemical Properties of Lipid Bilayer Vesicles

Despite its importance, however, the monolayer-bilayer correspondence problem has not been well defined and completely solved, although there have been some important steps in this direction.5-15 Experimentally, the phase transition temperature for monolayers and bilayers is the same only when both of them are under a specific monolayer pressure, which was found to be the equilibrium monolayer pressure, and numerically similar to the interfacial tension between the oil and water.16 This implies that the monolayer at the air-water interface and one leaflet of a bilayer vesicle can have the same thermodynamic properties only under a particular surface pressure.17 Nagle assumed that a bilayer is just two monolayers put together back-to-back with negligible interactions between the two leaflets and that there are only two basic differences between a monolayer and a bilayer. One is that in expansion of a monolayer, an equal area of free water surface is destroyed, which reduces the surface free energy for the monolayer compared to the bilayer by the amount of the area multiplied by the clean air-water surface tension. The second difference is that a hydrocarbon-air interface is created in the monolayer case but not in the bilayer case.11,12 This consideration would have represented an important step forward in search of the possible solution of the monolayer-bilayer correspondence problem, if bilayer pressure had been well characterized and not confused with the bilayer surface tension.12 Ja¨hnig recognized that the difference between a Langmuir monolayer (the monolayer at the air-water interface) and one leaflet of bilayer vesicles is the excess microscopic air-hydrocarbon interfacial tension exhibited by the former. This excess microscopic air-hydrocarbon interfacial tension was approximated by the macroscopic airoil interfacial tension. Gruen and Wolfe examined the correspondence problem between a monolayer at an interface and one leaflet of bilayer vesicles and suggested that a monolayer at the n-alkane-water interface is a closer analogue of one leaflet of bilayer vesicles than the monolayer at the air-water interface. They argued that, due to the roughness of the interface between the air and the hydrocarbon chain surface of the monolayer, the microscopic air-hydrocarbon interfacial tension may be dependent on the surface area of lipid molecules and may not necessarily be equal to the macroscopic air-oil interfacial tension. Thus, in their view, it is only approximately and incidentally the case that the difference in the surface tensions between the Langmuir monolayer and one leaflet of a bilayer vesicle is similar to the bulk air-oil interfacial tension.18 Marsh tried to solve the monolayer-bilayer correspondence problem in its narrow sense.17 He defined the equilibrium bilayer pressure as (5) De Gennes, P. G.; Taupin, C. J. Phys. Chem. 1982, 86, 2294. (6) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: New York, 1992; Chapters 16-17. (7) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. Biochim. Biophys. Acta 1977, 470, 185. (8) Israelachvili, J. N.; Marcelja, S.; Horn, R. G. Q. Rev. Biophys. 1980, 13, 121. (9) Ja¨hnig, F. Biophys. J. 1984, 46, 687. (10) Langevin, D.; Meunier, J. In Micelles, membranes, microemulsions, and monolayers; Gelbart, W. M., Ben-Shaul, A., Roux, D., Eds.; Springer-Verlag: New York, 1994; Chapter 10. (11) Nagle, J. F. Faraday Discuss. Chem. Soc. 1986, 81, 151. (12) Nagle, J. F. J. Membrane Biol. 1976, 27, 233. (13) Schindler, H. Biochim. Biophys. Acta 1979, 555, 316. (14) Schindler, H. FEBS Lett. 1980, 122, 72. (15) von Tscharner, V.; McConnell, H. M. Biophys. J. 1981, 36, 409. (16) Simon, S. A.; Lis, L. J.; Kauffman, J. W.; MacDonald, R. C. Biochim. Biophys. Acta 1975, 375, 317. (17) Marsh, D. Biochim. Biophys. Acta 1996, 1286, 183. (18) Gruen, D. W. R.; Wolfe, J. Biochim. Biophys. Acta 1982, 688, 572.

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the sum of the two contributions of the mutual interaction of lipids within the lipid layer and the repulsive interaction between the lipid monolayer and the substrate water. He believed that so-defined surface pressure should be the same for the monolayer at the air-water or oil-water interface and for a leaflet of bilayer vesicles in the substrate water. He thought the hydrophobic interaction between the lipid molecular layer and the water is different for the monolayer at an interface and for a leaflet of bilayer vesicles. He assumed that the monolayer at the air-water or oil water interface and the bilayer vesicles which are in equilibrium state must have the same surface density of lipids under the same surface pressure. However, he did not give the relationship between the bilayer pressure and the traditional monolayer pressure defined in the presence of the clean air-water interface. The monolayerbilayer correspondence problem in its broad sense, i.e., how to obtain the mechanochemical properties of the monolayer and bilayers from the given π-a relationship of the monolayer at the air-water or oil-water interface, was not further pursued in that work.17 To completely solve the monolayer-bilayer correspondence problem, we present a systematic analysis. Our theory discriminates three basic concepts: interfacial tension, surface pressure, and surface tension. Their physical significance is related to intra- and intermolecular interactions. We observe that surface pressure for a monolayer at the air-water or oil-water interface already has its conventional definition as the lowering in its surface tension from the clean air-water or oil-water interfacial tension due to presence of the monolayer at the interface. In contrast, there has been no definition for bilayer surface pressure, which could be comparable with the traditional monolayer pressure. We begin our analysis by defining the surface pressure of one leaflet of a bilayer as the intrinsic pressure of the thermodynamic system composed of the leaflet monolayer and the substrate water. The surface pressure for a monolayer at an interface and for one leaflet of a bilayer at the same lipid density and temperature can be proven to be the same if the microscopic and macroscopic hydrophobic effects can be reconciled as suggested previously by others.19,20 The equilibrium condition between the monolayer at the air-water interface and bilayer vesicles is first used to establish a solution of the monolayer-bilayer correspondence problem in its narrow meaning. We then compare the free energy of the monolayer at the air-water interface with that of one leaflet of a bilayer vesicle; a basic differential equation is obtained to relate the bilayer free energy per lipid to the π-a relationship of the corresponding Langmuir monolayer, leading to a complete solution of the monolayer-bilayer correspondence problem in the broader sense. For the first time, the analytical or empirical equation of state or the measured π-a curve of a monolayer at the air-water or oil-water interface can be used to obtain various thermodynamic properties, such as surface tension, free energy, chemical potential, activity of lipids, etc., of the monolayer as well as of bilayer vesicles. We also present a practical computational example for the monolayer and bilayer vesicles of dilauroyl phosphatidylethanolamine (DLPE) using a theoretical equation of state, in which the thermodynamic properties of a lipid monolayer are referred to the lipid molecular structure and the intra- and intermolecular interaction mechanisms. We find that, for bilayer vesicles, a minimum (19) Sharp, K. A.; Nicholls, A.; Fine, R. F.; Honig, B. Science 1991, 252, 106. (20) Sitkoff, D.; Sharp, K. A.; Honig, B. Biophys. Chem. 1994, 51, 397.

1000 Langmuir, Vol. 15, No. 4, 1999

in the lipid chemical potential occurs at the equilibrium area, whereas a minimum of the surface activity of lipids occurs at a larger molecular area, a consequence of the rapid change in monolayer pressure around the equilibrium value. For a monolayer, however, both the surface chemical potential and the surface activity of lipids decrease from their equilibrium value with increasing molecular area. Finally, the differences in free energy, chemical potential, and activity of lipids between a monolayer and bilayer vesicles are quantitatively analyzed and graphically illustrated. Interfacial Tension, Surface Pressure, and Surface Tension In the monolayer and bilayer literature, as well as in surface science generally, the interfacial tension (equivalently, the bulk interfacial tension, the environmental interfacial tension, or the chemical surface tension) is usually denoted by γ with a subscript to identify the relevant interface. For example, γa/o, γo/w, and γa/w denote, respectively, the interfacial tension between the air and oil, oil and water, and air and water. The interfacial tension between two bulk phases is a macroscopic concept, the magnitude of which depends only on the temperature. In the present analysis, we follow a practice introduced by others and apply the concept of interfacial tension to the microscopic surface between a bulk phase and a lipid monolayer6-9,18-20,21 and denote it by γ˜ . For example, we use γ˜ a/o to denote the microscopic interfacial tension between the air and the hydrocarbon surface of a lipid monolayer. Similarly, γ˜ o/w denotes the microscopic interfacial tension between the polar surface of a lipid monolayer and the water. The interfacial tension between a bulk phase and the monolayer is determined by the chemical properties of the bulk phase and the monolayer in contact. In general, γ˜ a/o and γ˜ o/w may be functions of the surface area of lipid molecules, a, and the temperature, T, i.e., it should be understood that, in general, γ˜ a/o ) γ˜ a/o(a,T) and γ˜ o/w ) γ˜ o/w(a,T).18 Nevertheless, the study of the relationship between monolayers and bilayer vesicles is usually focused on their properties near their equilibrium state, where the monolayer is at such a high-density state that the roughness of the contact area between the bulk phase and the lipid monolayer can be assumed not to significantly affect the magnitude of the microscopic interfacial tension. As a first approximation, then, the microscopic interfacial tension is assumed to be independent of the molecular area of the constituent lipid molecules and is only a function of the temperature, i.e., in the vicinity of the equilibrium state, γ˜ a/o(a,T) z γ˜ a/o(T) and γ˜ o/w(a,T) z γ˜ o/w(T) only. Moreover, it was recently suggested that microscopic and macroscopic surface tensions due to the hydrophobic effect are effectively the same in magnitude if the contact area between the bulk phase and the hydrocarbon layer is considered in molecular detail.19,20 This means that we can reasonably assume that γ˜ a/o(T) z γ˜ a/o(T) and γ˜ o/w(T) z γ˜ o/w(T). This approximation leads to the conclusion which is widely used in the literature, i.e., the difference in surface tension between one leaflet of a bilayer and the monolayer at the airwater interface is merely the interfacial tension between air and oil. However, this conclusion is only approximately true. The microscopic interfacial tension between a bulk phase and a monolayer of molecular thickness is open for further investigation, and a more precise solution of the monolayer-bilayer correspondence problem will then be achieved. The surface pressure for a monolayer at the air-water

Feng

interface, πm, was originally defined by Langmuir22 as the lowering of the surface tension from the clean air-water interfacial tension due to the presence of the monolayer, i.e.,

πm ) γa/w - σm

(1)

where σm denotes the surface tension of the monolayer (Figure 1c).23-27 The magnitude of the monolayer pressure in this definition is recognized as the same as that of the external pressure which is applied to the barrier to prevent the monolayer from spreading over the clean air-water interface in the Langmuir trough. This definition of the monolayer pressure itself suggests a method to measure its value. This definition, however, depends on the clean air-water interface, i.e., the environment of the monolayer. It is not directly related to the interaction between lipids within the monolayer. An alternative definition of this monolayer pressure is based on the reduction of the surface water activity due to the presence of the monolayer at the clean air-water interface, i.e.,

πm ) -nswkT ln asw

(2)

where kT is the Boltzmann constant multiplied by the temperature, nsw is the surface density of water molecules at the interface, asw is the surface water activity at the interface and asw ) 1 for the clean water surface has been assumed.24,26,28 In this definition, the monolayer pressure is also not directly related to the interaction between lipids within the monolayer. The monolayer surface pressure also represents the negative of the partial change of the free energy of the monolayer with respect to the total area of the surface system which includes the interface covered by the lipid layer as well as the clean air-water surface. The free energy of this system, F′m, is equal to the surface free energy in the presence of the monolayer, F′s, minus the surface free energy of the clean air-water interface, F′0, i.e.,

F′m ) F′s - F′0

(3)

The partial change of eq 3 with respect to the total area, A, gives

( ) ( ) ( ) ∂F′m ∂A

)

N,T

∂F′s ∂A

-

N,T

∂F′o ∂A

N,T

(4)

The first term on the right-hand side is equal to the surface tension of the monolayer, σm. The second term is equal to the clean air-water interfacial tension, γa/w. Comparing eq 4 with eq 1, we have10,26 (21) Tanford, C. J. Phys. Chem. 1972, 76, 3020. (22) Langmuir, I. J. Chem. Phys. 1933, 1, 756. (23) Davies, J. T.; Rideal, E. K. Interfacial Phenomena; Academic Press: New York, 1963; Chapter 1. (24) Defay, R.; Prigogine, I. Surface tension and Adsorption; John Wiley and Sons: New York, 1966. (25) Tanford, C. The Hydrophobic Effects: Formation of Micelles and Biological Membranes; John Wiley & Sons: New York, 1980. (26) Cevc, G.; Marsh, D. Phospholipid bilayers, Physical principles and models; John Wiley and Sons: New York, 1987; Chapter 12. (27) Birdi, K. S. Lipid and biopolymer monolayers at liquid interfaces; Plenum Press: New York, 1989; Chapters 2-3. (28) Evans, E. A.; Skalak, R. Mechanics and thermodynamics of biomembranes; CRC Press: Boca Raton, FL, 1980; Chapter 4.


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simulations of bilayers.29 We shall show that it is the bilayer leaflet pressure defined in this way that may correspond to the above-defined monolayer pressure. To understand the molecular origin of πb, we propose that πb be described from the consideration of free energy partitioning. According to Ja¨hnig,9 the free energy of one leaflet of a bilayer can be partitioned into three components: Fphob, which is due to the hydrophobic interaction between the hydrocarbon layer and the water; Fphil, which comes from the hydrophilic interaction between the polar headgroups of the leaflet and water; and Fint, which results from the interaction between the lipids within the leaflet. Therefore,

Fb ) Fphob + Fphil + Fint

(6)

The hydrophobic interaction tends to decrease the lipidwater contact area and acts as an attractive force between the lipids. The mechanical manifestation of the partial change of Fphob with respect to the total area of the leaflet is the microscopic interfacial tension between the hydrocarbon layer and water, γ˜ o/w. The hydrophilic interaction and the interaction between lipids tend to increase the monolayer-water interfacial area. Thus, both the partial derivatives of Fphil and Fint with respect to the total area contribute to the total repulsive bilayer leaflet surface pressure. From eq 6, we have Figure 1. (a) Schematic drawing showing the traditional monolayer pressure πm, the macroscopic interfacial tension γa/w, the microscopic air-monolayer interfacial tension γ˜ a/o and monolayer-water interfacial tension γ˜ o/w, and the surface tension σm and σb for a lipid monolayer at the air-water interface and for one leaflet of a bilayer vesicle in a Langmuir trough. The surface pressure for a monolayer at an interface has been traditionally defined as the lowering in the monolayer surface tension from the clean air-water interfacial tension due to the presence of the monolayer at the interface. The intrinsic pressure for the thermodynamic system of the lipid monolayer and the substrate water π turns out to be the bilayer leaflet pressure πb. If the bending effects and the interaction between the two leaflets across a bilayer are neglected, the difference between a monolayer at the interface and one leaflet of a bilayer vesicle is merely that the former but not the latter exhibits a microscopic interfacial tension between the air and the hydrocarbon layer. (b) A free body diagram for the barrier of a unit length in a Langmuir trough. The two stresses on the right side, the clean air-water interfacial tension, γa/w, and the external pressure, f, which is applied to prevent the monolayer from spreading over the clean air-water interface and is equal in magnitude to the monolayer surface pressure πm, and three stresses on the left side, the intrinsic pressure π ) πb, the microscopic interfacial tensions γ˜ a/o and γ˜ o/w, are in balance. The force balance consideration leads to eq 15. (c) An alternative of the free body diagram in (b) with the left three forces replaced by their mechanical resultant, the monolayer tension σm. The force balance consideration leads to eq 1, which defines the monolayer surface pressure πm.

( )

πm ) -

∂F′m ∂A

N,T

(5)

Again, this definition depends on the clean air-water interface. In contrast to the monolayer surface pressure, there has been no generally accepted definition of bilayer surface pressure. A well-defined bilayer surface pressure is, however, a prerequisite for defining and solving the monolayer-bilayer correspondence problem. We define the surface pressure of one leaflet of a bilayer, πb, as the intrinsic surface pressure of the thermodynamic system composed of the leaflet monolayer and the substrate water. The same definition was used in molecular dynamics

σb ) γ˜ o/w - πphil - πint

(7)

where σb ) (∂Fb/∂A)N,T is the surface tension within one leaflet of the bilayer and

γ˜ o/w )

(

)

∂Fphob ∂A

N,T

( )

πphil ) -

∂Fphil ∂A

N,T

( )

πint ) -

∂Fint ∂A

N,T

(8)

The two repulsive stresses, πphil and πint in eq 7, are the two components of the bilayer leaflet pressure, πb:

πb ) πphil + πint

(9)

Equation 7 thus becomes

σb ) γ˜ o/w - πb

(10)

This equation states that the surface pressure in one leaflet of a bilayer can also be described as the lowering of its surface tension from the microscopic interfacial tension between the hydrocarbon layer of the opposite leaflet and water. This description of πb is parallel to the traditional definition of the monolayer surface pressure (eq 1). There were similar treatments in characterizing the bilayer surface pressure. Evans and Wough suggested that the bilayer free energy, Fb, be partitioned to two contributions, Fw, which is due to the interaction between the lipid monolayer and the substrate water, and Fp, which is due to the repulsive interaction between the water and the lipids and the kinetic interaction between the lipids within the leaflet, i.e., Fb ) Fw + Fp. Then πb is defined as -(∂Fp/∂A)N,T.26,28,30 The former contribution in this treatment, Fw, is the same as our Fphob, and the latter contribution, Fp, is equivalent to our Fphil + Fint in eq 6. The same treatment, however, was not applied to the monolayer at the air-water or the oil-water interface to (29) Chiu, S. W.; Clark, M.; Balaji, V.; Subramaniam, S.; Scott, H. L.; Jakobsson, E. Biophys. J. 1995, 69, 1230. (30) Evans, E. A.; Wough, R. J. Colloid Interface Sci. 1977, 60, 286.

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Feng

define the intrinsic pressure of the monolayer. Therefore, the monolayer pressure and the bilayer leaflet pressure, which are defined in different ways for different systems, are not comparable, and the monolayer-bilayer correspondence problem could not be further pursued. Marsh characterized the bilayer pressure by the same way as our eq 6 but only for the equilibrium state of the bilayer. However, when the free energy of the monolayer at the air-water or oil-water interface was partitioned in the same way, the contribution due to the hydrophobic interaction between the water and the monolayer was neglected in his work. Therefore, his deduction for the solution of the monolayer-bilayer correspondence problem is different from ours, although his conclusion for the equilibrium pressure is the same as that given by our eq 23.17 The surface tension, σm for a monolayer at the airwater interface and σb for one leaflet of a bilayer, is the total mechanical resultant of the surface pressure and the interfacial (environmental) tension. For a monolayer at an interface, the surface tension can be directly measured as the force acting on a unit length of a plate which penetrates the monolayer at a zero contact angle in a Langmuir trough.23,27 For bilayers, the membrane tension (twice the surface tension for one of its two leaflets) is measured indirectly by other techniques, for example, micropipet aspiration. The bilayer membrane tension also represents the area density of the strain energy stored within a unit area of the deformed membrane. Consequently, its value is referred to the natural, stress-free, undeformed state of the bilayer membrane, which corresponds to the equilibrium state of bilayer vesicles, in which bending and transmembrane osmotic effects are considered negligible. Monolayer-Bilayer Correspondence We now begin the monolayer-bilayer correspondence analysis by finding the relationship between the monolayer surface pressure, πm, and the bilayer leaflet pressure, πb. We are concerned whether the two pressures correspond to each other, i.e., whether the two pressures have the same magnitude under the same area density of lipids and temperature. To do so, it is necessary to apply the free energy partition approach used in defining the bilayer leaflet pressure to the monolayer at the air-water interface to obtain the intrinsic monolayer pressure, π. This intrinsic monolayer pressure is actually an analogue of the bilayer leaflet pressure, πb. We shall explain the meaning of the word intrinsic in this context below. Using an analysis that parallels eq 6, the surface free energy of a monolayer at the air-water interface, Fm, is partitioned into four instead of three contributions. In addition to the three terms on the right-hand side of eq 6, Fphob, Fphil, and Fint, there should be one more contribution, Fa/o, which is due to the presence of the microscopic interface between the air and the hydrocarbon layer, i.e.,

Fm ) Fa/o + Fphob + Fphil + Fint

(11)

The partial change of eq 11 with respect to the total surface area, A, at constant temperature, T, and constant total lipid molecule number, N, in the monolayer gives

σm ) γa/o + γo/w - (πphil + πint)

(12)

where the second, third, and fourth terms on the righthand side of this equation have been described in eq 9. The other two terms in eq 12 are

σm )

( ) ∂Fm ∂A

T,N

γ˜ a/o )

( ) ∂Fa/o ∂A

T,N

(13)

Substituting eq 10 into eq 12, we have

σm ) γ˜ a/o + γ˜ o/w - π ) γ˜ a/o + γ˜ o/w - πb

(14)

in which π ) πphil + πint ) πb has been used. By doing so, it has been assumed that the mutual interaction of the lipids within the lipid molecular layer and the hydrophobic and hydrophilic interactions between the lipid molecular layer and the water are all the same for the monolayer at the air-water interface and for a leaflet of bilayers. This may be true around their equilibrium states, where both of them are in their high-density state and the penetration of the water molecules into the hydrocarbon chains is not effectively appreciable.9,18 Indeed, the intrinsic pressure for the monolayer at the air-water interface, π, should be the same as that for one leaflet of a bilayer, πb, at the same temperature and lipid molecular area, because the existence of the air-hydrocarbon interface introduces merely an attractive stress and thus affects only the surface tension but not the surface pressure. Comparing eq 14 with eq 1, we conclude that

πm - πb ) γa/w - (γ˜ a/o + γ˜ o/w)

(15)

Alternatively, eq 15 can also be directly obtained by considering the force balance for a unit length of the barrier in a Langmuir trough. Figure 1b is its free body diagram. It can be seen that the external pressure directed to the left, f, is applied on the right side of the barrier to prevent the monolayer from spreading over the clean air-water interface. Its value is equal to the monolayer pressure πm because this external pressure can be found from Figure 1c to be equal to γa/w - σm. The other force acting on the right side of the barrier is the clean air-water interfacial tension, γa/w. The intrinsic surface pressure π in the monolayer acts on the left side of the barrier and is directed to the right. The other two stresses acting on the left side of the barrier are the two microscopic interfacial tensions γ˜ a/o and γ˜ o/w. The balance of these five surface stresses in the horizontal direction leads to eq 15. As mentioned above, the similarity in magnitude of the microscopic and macroscopic interfacial tensions suggests that19,20

γ˜ a/o z γa/o and γ˜ o/w z γo/w

(16)

Therefore, eq 15 has the following approximate form:

πm - πb z γa/w - (γa/o + γo/w)

(17)

Experimental observation reveals that the interfacial tension between the air and water is roughly equal to the sum of those between the air and oil and between the oil and water, i.e.,

γa/w z γa/o + γo/w

(18)

For example, for bulk n-octane, in which case γa/o z 23 mN/m, γo/w z 49 mN/m, and γa/w z 72 mN/m, eq 18 holds very well. However, this equation is not always valid. The difference between the two sides of the equation was defined by Langmuir as the spreading factor for oil films spreading over the clean air-water interface, which could have a positive, zero, or negative value.22 From eqs 17 and 18, we conclude that


πm(a,T) z πb(a,T)

(19)

i.e., the traditional monolayer pressure and our bilayer leaflet surface pressure are approximately the same under any specified lipid molecular area and temperature. Starting from the next section, therefore, we will not distinguish between πm and πb but will denote both by π. The error due to neglecting the difference between the traditional monolayer pressure and the bilayer leaflet pressure is equal to that due to neglecting the difference between the microscopic and the macroscopic hydrophobic effects. Substituting eq 16 into eq 18 leads to

γa/w z γ˜ a/o + γ˜ o/w

(20)

which gives the interfacial tension between air and clean water as the resultant of the two microscopic interfacial tensions between the air and the monolayer and between the water and the monolayer. Equation 20 will be applied in the following analysis for the solution of the monolayerbilayer correspondence problem in the broad sense, i.e., how we can obtain bilayer properties from the monolayer π-a measurement, although the degree of precision of this approximate equation and its equivalent, eq 19, is open to further investigation. The conclusion that the monolayer pressure and the bilayer leaflet pressure at the same lipid density and temperature should be the same is readily understood if we consider the monolayer-bilayer correspondence problem in a more general way. There are actually three cases in which we have practical interest: the monolayer at the air-water interface, the monolayer at the oil-water interface, and one leaflet of a bilayer. The lipid monolayer and the water substrate are common in all three cases. It is, therefore, appropriate to consider the lipid layer and the water substrate as a thermodynamic system to distinguish the intrinsic properties of this system from its extrinsic properties. The word intrinsic in the present context implies intrinsic to the lipid layer-water system. The difference among these three situations is, therefore, simply a different environmental phase at the nonpolar side of the monolayer, namely, the air phase for the monolayer at the air-water interface, the oil phase for the monolayer at the oil-water interface, and the opposite leaflet for one leaflet of a bilayer. These constitute the possible different environments of the monolayer-water system. The environment dictates the kind of interface that exists between the monolayer and the external world. From this point of view, because interfaces produce interfacial tensions only, the intrinsic pressure of the monolayer-water system should be the same as the bilayer leaflet pressure πb, which for this monolayerwater system happens to be one half of a bilayer. Therefore, we regard πb as an intrinsic property of a bilayer. We observe further that this intrinsic bilayer leaflet pressure should also be the surface pressure for the monolayer at the air-water or the oil-water interface, πm, because the interactions between the hydrocarbon layer and the environment (air or oil) are attractive and can thus affect only the monolayer surface tension, not the monolayer surface pressure. The solution of the monolayer-bilayer correspondence problem addressed in this paper is crucial in obtaining the mechanochemical properties of monolayers and bilayer vesicles from the equation of state or the π-a curve of the monolayer at an interface. There are abundant π-a curves for monolayers of various lipids at the air-water or oilwater interface in the literature. From these π-a curves,

Langmuir, Vol. 15, No. 4, 1999 1003

eq 1 can be used to calculate the monolayer surface tension, σm, which determines the mechanical behavior of the monolayer. The monolayer-bilayer correspondence analysis is pursued by obtaining the relationship between σm and σb. We subtract eq 1 from eq 10 and compare the result with eq 15. It follows that

σm(a,T) ) σb(a,T) + γ˜ a/o

(21)

This equation tells us that a monolayer at the air-water interface is mechanochemically not equivalent to one leaflet of a bilayer, which faces oil instead of air. This situation is depicted in Figure 1a, from which eq 21 can be immediately obtained by comparing the monolayer at the air-water interface with one leaflet of a bilayer vesicle. Equation 21 also states that the difference between the monolayer at the air-water interface and a leaflet of bilayers is merely that the former but not the latter evinces an air-hydrocarbon interfacial tension in its surface tension, which represents the total mechanical effect of all the surface stresses within the molecular layer. This relationship constitutes the basic equation for solving the monolayer-bilayer correspondence problem, i.e., eq 21 can be applied to interpret the thermodynamic properties of the monolayer at an interface and the bilayer vesicles, either analytically from the equation of state or graphically from the pressure-area isotherm of this monolayer. To do so, information of γ˜ a/o is needed. Measurement of microscopic interfacial tensions is challenging for the modern nanotechnology. At the present stage, however, the microscopic interfacial tension γ˜ a/o ) γ˜ a/o(a,T) can be approximated by the macroscopic interfacial tension between the air and oil γa/o ) γa/o(T) in the vicinity of the equilibrium state, where the molecular layers are at their high-density state. We can then pursue an approximate solution of the monolayer-bilayer correspondence problem. The relationship between the monolayer and one leaflet of a bilayer with respect to all other thermodynamic properties can then be deduced from eq 21 by applying thermodynamics and membrane mechanics to the lipid molecular layer. Although eq 21 was obtained in the literature, the monolayer-bilayer correspondence problem, however, was not completely defined and solved.9 We now consider the specific condition under which a lipid monolayer at the air-water interface is in equilibrium with lipid bilayer vesicles in a Langmuir trough. We denote the equilibrium value of a quantity by the same notation but with an asterisk. At the equilibrium state of a ) a*, the free energy of one leaflet of a bilayer must have a minimum; therefore, from eq 10, we have5,9

σb(a*) )

( ) ∂Fb ∂Ab

Nb,T,a)a*

)

( )

1 ∂Fb Nb ∂a

)

Nb,T,a)a*

γ˜ o/w(a*) - π(a*) ) 0 (22) Experimental measurements have demonstrated that large bilayer vesicles are tension-free under normal conditions.28,31-34 This means that, if bending and transmembrane osmotic effects are negligible, the equilibrium state of bilayer vesicles is their natural, undeformed, stress-free state. The equilibrium value of the bilayer surface pressure is therefore (31) Evans, E. A.; Kwok, R. Biochemistry 1982, 21, 4874. (32) Fettiplace, R.; Andrews, D. M.; Haydon, D. A. J. Membr. Biol. 1971, 5, 277. (33) Hanai, T.; Haydon, D. A.; Taylor, J. Proc. R. Soc. London 1964, A281, 377. (34) Kwok, R.; Evans, E. A. Biophys. J. 1981, 35, 637.

1004 Langmuir, Vol. 15, No. 4, 1999

π* ) π/b ) πb(a*) ) γ˜ /o/w

Feng

where γ˜ /o/w ) γ˜ o/w(a*). For a monolayer of the same kind of lipids at the air-water interface which is in equilibrium with bilayer vesicles in the aqueous substrate, the lipid molecular area must also be a ) a*. This conclusion will be shown in the next section. The equilibrium pressure for the monolayer at the interface is thus π* ) π/m ) γ˜ /o/w. Although the value of the microscopic interfacial tension γ˜ /o/w is not available yet, it can be reasonably approximated by the macroscopic oilwater interfacial tension γo/w, i.e., ca. 49 mN/m, in the vicinity of the equilibrium state. The monolayer-bilayer correspondence problem has been treated by the pioneers in this field.9,11,12,18 Several correspondences have been proposed, which also concluded from quite brief derivation that it is the monolayer with a surface pressure of the value of the oil-water interfacial tension that should be in correspondence with flaccid bilayers. However, as we pointed out before, this conclusion is only approximately valid. The degree of the approximation is equivalent to that of approximation of the microscopic hydrophobic effect by the macroscopic hydrophobic effect. It is observed in experiment that that the equilibrium pressure π* is quite close to the value of the collapse pressure πc of the monolayer under slow compression. The collapse pressure usually varies with the compression rate; here πc is the collapse pressure measured in a pseudostatic compression process. The observation may be explained by the present theory. At equilibrium, the free energy of a leaflet of bilayer vesicles has a minimum. Exchange of lipid molecules, although slow, can occur between bilayer vesicles and the water phase.13 The leaflets of an unconstrained bilayer vesicle are thus in the state of highest lipid density. If more molecules are added to or subtracted from the leaflet of a bilayer vesicle, the bilayer vesicle may change its curvature. However, curvature change is not available for the monolayer at the air-water or oilwater interface because of the constraints of the planar geometry. The monolayer may either reorganize itself (phase transition) or collapse in response to any further compression from its equilibrium state a ) a*. Here, collapse physically means the squeeze-out of constituent lipids by further compression of the monolayer. In either case, a discontinuity of the slopes of the π-a curve would be observed. This reasoning may imply that the equilibrium pressure and the collapse pressure for a monolayer at an interface would most likely be the same. Free Energy, Chemical Potential, and Surface Activity We denote the total surface area and the total number of the lipid molecules by Ab and Nb for a leaflet of bilayer vesicles in aqueous media and by Am and Nm for the monolayer at the air-water interface. If the lipid molecular area is a, the surface density of lipid molecules is then 1/a. We examine the correspondence between a Langmuir monolayer and one leaflet of a vesicle, both having the same lipid molecular area a. For any given value of a,

Ab ) Nba and Am ) Nma

Fb ) Nbφ(a)

(23)

(24)

Consider the free energy for one leaflet of a bilayer. At a constant temperature T, the free energy per lipid molecule is a function of the lipid molecular area a only. Following Ja¨hnig,9 we denote this function by φ(a). The total free energy for one leaflet of a bilayer vesicle is thus

(25)

At a given T and a, the change in the free energy of a bilayer vesicle requires specification of the boundary condition for lipid exchange. Given equilibration of the inner and outer leaflets of a bilayer, then, if bending effects are neglected, a constant pressure boundary condition is appropriate, i.e., the area per lipid molecule remains constant upon alteration of the number of lipids in the bilayer. The chemical potential of lipids in an unperturbed bilayer vesicle is thus defined from eq 25 as

µb(a) )

( ) ∂Fb ∂Nb

) φ(a)

(26)

a

In contrast to one leaflet of a bilayer, the Langmuir monolayer at the air-water interface has one more interface between the air and the hydrocarbon layer. The free energy of this monolayer system should have one term more than that given in eq 25 for one leaflet of a bilayer:

Fm ) Nmφ(a) + γ˜ a/oAm

(27)

As defined in the preceding section, γ˜ a/o represents the microscopic interfacial tension between the air and the hydrocarbon layer. In general, γ˜ a/o should be a function of the lipid density. In the vicinity of equilibrium, however, we assume that γ˜ a/o approximately has a constant value, which can be determined from eq 20 where γ˜ o/w is equal to the collapse pressure πc and thus can be measured from a Langmuir trough experiment. For example, for a dilauroyl phosphatidylethanolamine monolayer, πc ) 49 mN/m at 44.2 °C.35 This gives the value of γ˜ a/o ) 72 - 49 ) 23 mN/m. Using eq 27 and assuming that γ˜ a/o is independent of the lipid surface density in the vicinity of the equilibrium state, an expression for the chemical potential of the Langmuir monolayer is obtained

µm(a) )

( ) ∂Fm ∂Nm

) φ(a) - aφ′(a)

(28)

Am

where ′ denotes d/da. Up to this point, we have followed Ja¨hnig’s procedure.9 With the chemical potential of one leaflet of a bilayer and a monolayer at the air-water interface defined by eqs 26 and 28, respectively, we can show the assertion we made in the preceding section, i.e., the monolayer and the bilayer must have the same lipid density at their equilibrium states. Suppose that the equilibrium value of the lipid molecular area of the monolayer at the air-water interface, a/m, is different from that of the bilayer vesicles in the substrate, a/b. Then the equilibrium value of chemical potential of the bilayer lipids is µb(a/b), which, by definition, is the minimum value of µb(a) in the vicinity of a ) a/b. Consequently, we have

φ′(a/b) ) 0

(29)

and in the vicinity of a ) a/b

φ′′(a) > 0

(30)

The equilibrium value of the chemical potential of the (35) Mo¨hwald, H. Annu. Rev. Phys. Chem. 1990, 41, 441.


lipids of the monolayer at the air-water interface is µm(a/m), which, by definition of equilibrium, should be equal to µb(a/b), i.e.,

µm(a/m) ) µb(a/b)

(31)

Now, let us calculate the chemical potential of the monolayer lipids at a ) a/b; from eqs 28, 29, 26, and finally 31, we can have, step by step,

µm(a/b) ) φ(a/b) - a/bφ′(a/b) ) φ(a/b) ) µb(a/b) ) µm(a/m) (32) Note that eq 32 can be valid if and only if

a/m ) a/b ) a*

(33)

because, from eqs 28 and 30, we have

µ′m(a) ) φ′(a) - φ′(a) - aφ′′(a) ) -aφ′′(a) < 0

(34)

which means that µm(a) monotonically decreases in the vicinity of a ) a/b. The proof for a/b ) a/m ) a* is thus done. This proof, however, seems to be merely a mathematical exercise, which is based on the fundamental assumption that the Helmholtz free energies of the monolayer and bilayers can be simply related by eqs 25 and 27. This assumption almost amounts to assuming that the monolayer and bilayers in equilibrium should have the same area density of lipids. This proof is thus natural and almost trivial. Now let us show that φ(a) can be obtained from the π-a relation function π(a). From eq 25 and the first equation of eq 24, we can obtain the mechanical surface tension in one leaflet of a bilayer vesicle, σb(a), from the free energy function per molecule, φ(a), as

σb(a) )

( ) ∂Fb ∂Ab

)

Nb

( )

1 ∂Fb Nb ∂a

) φ′(a)

(35)

Nb

From eq 29, it follows that σb(a*) ) 0. Chemical potential considerations thus lead to the conclusion that bilayer vesicles must be stress-free at equilibrium if bending effects and transmembrane osmotic stress are negligible. Moreover, eq 35, combined with eq 10, gives an expression relating the monolayer pressure π to the function φ(a) as

π(a) ) γ˜ o/w - φ′(a)

(36)

This result can also be obtained by considering the relationship between the surface tension, σm, and the free energy, Fm, for the Langmuir monolayer. σm is the partial change of Fm with respect to Am under the condition that Nm is kept constant, so from eq 27

σm(a) )

( ) ∂Fm ∂Am

)

Nm

( )

1 ∂Fm Nm ∂a

) γ˜ a/o + φ′(a)

(37)

Nm

which, when combined with eq 1 and then eq 20, yields

π(a) ) (γa/w - γ˜ a/o) - φ′(a) ) γ˜ o/w - φ′(a)

(38)

Each of eqs 36 and 38 results in a differential equation for φ(a):

φ′(a) ) γ˜ o/w - π(a)

(39)

which, for any specific value of the lipid molecular area

Langmuir, Vol. 15, No. 4, 1999 1005

a, relates the lipid molecular free energy function for a bilayer vesicle to the surface pressure of a Langmuir monolayer. This surface pressure as a function of the lipid molecular area, π(a), is usually known and specified either analytically as a theoretical or empirical equation of state, π ) π(a,T), or graphically as an experimental π-a curve at a specified temperature T. Hence, the deduction of eq 39 is the key step in the solution of the monolayer-bilayer correspondence problem. An immediate application of this equation is that it can be used to obtain the unique value of the surface pressure at which the monolayer at the air-water interface and bilayer vesicles are in equilibrium. Equation 39 is valid for any a. From eq 29, φ′(a*) ) 0. Therefore, we find once again that the equilibrium monolayer pressure is equal in magnitude to the microscopic interfacial tension between the water and the hydrocarbon layer, i.e., π* ) π(a*) ) γ˜ o/w. This is the same conclusion as that given in eq 23, which was obtained from the requirement that bilayer vesicles at equilibrium have zero surface tension. Equation 39 can thus be rewritten as

φ′(a) ) π(a*) - π(a)

(40)

Another application of eq 39 involves the analysis of a wide variety of pressure-area isotherms available in the literature; these data can be used to obtain the relative chemical potential of the lipids in bilayer vesicles. Integrating eq 40 from the equilibrium area a* to any specific value of the lipid molecular area a, we obtain

µb(a) - µb(a*) ) φ(a) - φ(a*) )

∫a*a [π(a*) - π(a)] da

(41)

To find the expression for the chemical potential of a Langmuir monolayer, we invoke eq 28 first at any specified a and then at the equilibrium value a*, subtract the two equations at these two specified values a and a*, and combine the resultant equation with eqs 40 and 41. We then obtain the following equation which relates the chemical potential of lipids in the Langmuir monolayer as a function of a in the vicinity of the equilibrium state:

µm(a) - µm(a*) )

∫a*a [π(a*) - π(a)] da - [π(a*) - π(a)]a

(42)

From eqs 41 and 42, the difference in the chemical potential of lipids between the Langmuir monolayer and the corresponding bilayer vesicles is thus

µb(a) - µm(a) ) [π(a*) - π(a)]a

(43)

Figure 2 shows how eq 41 can be applied to a π-a curve of a Langmuir monolayer to calculate the molecular free energy of the bilayer vesicles, φ(a) - φ(a*), which is also the chemical potential of the lipids in bilayer vesicles, µb(a) - µb(a*), at any specific value of the lipid molecular area a. In Figure 2, the length of the horizontal segment A1A2 represents (a - a*) and the length of the vertical line A2C2 represents π* ) π(a*), i.e., the equilibrium pressure. The integral on the right-hand side of eqs 41 and 42 represents the area of the triangle-like plane figure C1B2C2 which has the π-a curve as its lower boundary. The area of this plane figure can also be obtained by subtracting the area of trapezoid-like plane figure C1CBB2 with the π-a curve as its right boundary from the area of rectangle CC2B2B. The negative value of the area of this trapezoidlike plane figure C1CBB2 is represented by the integral of the same π-a curve with respect to the variable π from

1006 Langmuir, Vol. 15, No. 4, 1999

Feng

Figure 2. Chemical potential of lipids within a monolayer and within bilayer vesicles is obtained from the experimental π-a curve of the monolayer at the air-water interface. According to eq 41, we can graphically treat the chemical potential of lipids in bilayer vesicles, µb(a) - µb(a*), as the positive value of the graphical area of the figure C1B2C2 with the π-a curve as one of its three sides, C1B2. Correspondingly, according to eq 42, the chemical potential of lipids within the monolayer, µm(a) - µm(a*), can be treated as the negative value of the graphical area of the figure C1CBB2 with its right side C1B2 being a segment of the π-a curve. The difference in the chemical potential of lipids between the monolayer at the air-water interface and bilayer vesicles is thus graphically represented by the area of the rectangle C2CBB2 at any given molecular area a. See text for details.

π* to π at the specific value of a:

µb(a) - µb(a*) ) φ(a) - φ(a*) ) (π* - π)a -

∫ππ* a(π) dπ

From the analysis given above, we can see that if an experimental π-a curve for a Langmuir monolayer is available, eqs 41 and 42, or alternatively eqs 44 and 45, can be applied to obtain the chemical potential of lipids at any given molecular area a for the monolayer and the bilayer by measuring the area of the figures C1B2C2 and C1CBB2. If an equation of state for a monolayer at the air-water interface is known, eqs 41 and 42, or alternatively, eqs 44 and 45, can then be applied to obtain the analytical expressions for the chemical potential of lipids in the monolayer and bilayer vesicles. Many properties of lipid membranes such as water permeability, protein penetration, channel opening and closing, interactions between integral proteins and lipids, lipid exchange, membrane fusion, etc., are affected by the surface activity of the constituent lipid molecules. Although a specific, two-dimensional osmotic pressure-type equation of state for lipid monolayers at the air-water interface has been developed,36,37 from which the surface activity of water molecules can be calculated, obtaining the surface activity of the lipid molecules has been problematic. However, the present monolayer-bilayer correspondence analysis alters that situation. This is because it allows calculation of the surface activity of lipid molecules as the function of lipid molecular area for both bilayers, Rb ) Rb(a), and monolayers, Rm ) Rm(a), from the expression of the lipid chemical potential. The chemical potential of lipids at any molecular area a can be related to their surface activity by

µm(a) ) µm(a*) + kT ln Rm(a) + π(a)(a - a*) for a Langmuir monolayer and by

µb(a) ) µb(a*) + kT ln Rb(a) + π(a)(a - a*) (47) for a bilayer vesicle, respectively. In these two equations, µm(a*) and µb(a*) are respectively the chemical potential of lipids for a Langmuir monolayer and bilayer vesicles at equilibrium. The surface activity of lipids at any specified lipid molecular area a, relative to their equilibrium values, is thus obtained as

(44) Rm(a)

This equation can also be obtained from eq 41 by performing integration on the right-hand side by parts. These two equations are thus equivalent. Figure 2 also shows how eq 42 can be applied to obtain, from the π-a curve of a monolayer at the air-water interface, the chemical potential of lipid, µm(a) - µm(a*). The length of segment CB represents π(a*) - π(a), and the length of the side AA2 represents the value of the molecular area a. The area of rectangle CBB2C2 thus represents µb(a) - µm(a), the difference between the chemical potential of lipid in a bilayer vesicle and that of the corresponding Langmuir monolayer at a specific lipid molecular area a. Consequently, the relative chemical potential of lipids in a Langmuir monolayer can be obtained as the negative value of the graphical area of the trapezoid-like figure C1CBB2, the right boundary of which, C1B2, is the π-a curve. We thus have

(46)

Rm(a*)

) exp

[

]

µm(a) - µm(a*) π(a)(a - a*) kT kT

(48)

for a monolayer and

Rb(a) Rb(a*)

) exp

[

]

µb(a) - µb(a*) π(a)(a - a*) kT kT

(49)

(45)

for a bilayer. In sum, the π-a relationship of a Langmuir monolayer at the air-water interface, either as an equation of state or as an experimental π-a curve, implicitly contains all of the thermodynamic information for this monolayer and, according to our monolayer-bilayer correspondence analysis, for the corresponding bilayer as well. The monolayerbilayer correspondence analysis is thus the theoretical basis in applying the π-a data for a Langmuir monolayer to obtain other thermodynamic properties of this monolayer and the properties of its corresponding bilayer.

Equation 45 can also be obtained from eq 42 by performing the integration on the right-hand side by parts; hence, eqs 42 and 45 are mathematically equivalent.

(36) Wolfe, D. H.; Brockman, H. L. Proc. Natl. Acad. Sci. U.S.A. 1988, 85, 4285. (37) Feng, S. S.; Brockman, H.; MacDonald, R. C. Langmuir 1994, 10, 3188.

µm(a) - µm(a*) ) -

∫ππ* a(π) dπ


Figure 3. Experimental π-a data of the DLPE monolayer (C12:0/C12:0) at the air-water interface35 can be well fitted by the suggested equation of state, eq 50, with the parameters a* ) 49 Å2, γb ) 0.685, and γres ) -15 mN/m, where a* is the minimum lipid molecular area to which the lipid monolayer can be compressed, γb represents the overlap coefficient for the chain projection area onto the membrane plane, and γres is the residual pressure, which incorporates all other intra- and intermolecular interaction effects than the distributional entropy of lipids on the membrane plane and the configurational entropy of lipid chains.

Langmuir, Vol. 15, No. 4, 1999 1007

Figure 4. Surface tension in units of mN/m vs the lipid molecular area a in units of Å2 for the monolayer of DLPE at the air-water interface and for one leaflet of bilayer vesicles in water phase at T ) 44.2 °C. At equilibrium, the bilayer vesicles have zero surface tension. The surface tension for a Langmuir monolayer is always greater than that for one leaflet of bilayer vesicles. As shown in eq 21, the difference in the magnitudes is equal to the microscopic interfacial tension between the air and the DLPE monolayer γ˜ a/o, which can be approximated as the value of the collapse pressure of the monolayer in compression at the air-water interface and was measured to be 49 mN/m.35

Computational Example This section illustrates the application of the present monolayer-bilayer correspondence analysis to calculate the thermodynamic properties of the monolayer and bilayer vesicles from a π-a relationship using dilauroyl phosphatidylethanolamine (DLPE) as an example. A DLPE molecule has two saturated chains, each containing 12 carbon atoms. The experimental π-a curve is taken from Mo¨hwald.35 We illustrate the analytical approach by fitting this curve with an equation of state, which was developed by Feng and MacDonald on the basis of molecular structure and intra- and intermolecular interactions of the lipid monolayer.38 This equation of state has the following form

π(a) )

kT ∂qt 2kT ∂qc + + γres qt ∂a qc ∂a

(50)

in which the total monolayer pressure is decomposed into three components. The first term on the right-hand side of eq 50 describes the tendency of lipid molecules in twodimensional mixtures with surface water molecules to maximize their translational (or distributional) entropy. The partition function for this component is qt. The second term on the right-hand side of eq 50 represents the tendency of the lipid molecules to maximize their chain configurational entropy. The corresponding partition function is qc. The third term on the right-hand side of eq 50 incorporates together residual interactions not included in the first two terms.38 There are three unknown parameters in eq 50: the minimum molecular area of lipid (38) Feng, S. S.; MacDonald, R. C. Biophys. J. 1995, 69, 460.

to which the monolayer can be compressed, a*; the overlap coefficient of the chain projections on the monolayer plane, γb; and the residual pressure, γres, which can be determined by fitting eq 50 to the experimental π-a curve. In the case of the DLPE monolayer, a* ) 49 Å2, γb ) 0.685, and γres ) -15 mN/m provide a satisfactory fit (Figure 3). Given the analytical form of the equation of state, eq 50, for the DLPE monolayer at the air-water interface at T ) 44.2 °C, we can immediately obtain from eqs 1 and 10 the surface tensions for the Langmuir monolayer and for one leaflet of DLPE bilayers, as shown in Figure 4. At equilibrium, the bilayer vesicle has zero surface tension. It is also seen that the surface tension for the Langmuir monolayer is greater than that for one leaflet of the bilayer, the difference being the interfacial tension between the air and the DLPE hydrocarbon layer, γ˜ a/o. By substituting the specified equation of state for the lipid monolayer, eq 50, into eq 41, the free energy per lipid molecule in the bilayer vesicle can be shown to be

φ(a) - φ(a*) ) -kT ln

qt q/t

- 2kT ln

qc

+ q/c (π* - γres)(a - a*) (51)

where q/t ) qt(a*) ) 1, q/c ) qc(a*), and π* ) π(a*). Note that the free energy given by eq 51 is not absolute, but relative to the equilibrium value. Using eqs 25 and 27, we can compare the free energy per lipid for the Langmuir monolayer with that for the bilayer vesicle. In Figure 5, the free energy per lipid relative to the equilibrium value for the monolayer and

1008 Langmuir, Vol. 15, No. 4, 1999

Feng

Figure 5. Free energy per lipid, which is related to its equilibrium value of lipids in bilayer vesicles and in units of kT, plotted as a function of the lipid molecular area a in units of Å2 for the monolayer of DLPE molecules at the air-water interface and for bilayer vesicles in the substrate water at T ) 44.2 °C. The free energy per lipid in a Langmuir monolayer is always greater than that per lipid in bilayer vesicles. According to eqs 25 and 27, the difference is equal to γ˜ a/oa, which increases from 2.4kT, or 1.5 kcal/mol, at the equilibrium pressure to 5.1kT, or 3.2 kcal/mol, at near-zero monolayer pressure if γ˜ a/o ) 23 mN/m is assumed to be constant in the lipid molecular area a.

for the bilayer vesicle is plotted respectively as the function of the lipid molecular area a. The free energy per lipid for a Langmuir monolayer is always greater than that for the bilayer vesicle. The difference, according to eqs 25 and 27, is equal to γ˜ a/oa. This difference is due to air-hydrocarbon surface energy. If γ˜ a/o z γa/o ) 23 mN/m is assumed, then this difference is computed to increase from 2.4kT, or 1.5 kcal/mol, at the equilibrium pressure for a ) a* ) 49 Å2 to 5.1kT, or 3.2 kcal/mol, at near-zero monolayer pressure for a ) 96.6 Å2. Although this calculation provides a rough estimate of the free energy difference, it should be recognized that γ˜ a/o may not remain constant over this large area variation. From eq 51, a dimensionless expression for the chemical potential of lipids within the bilayer vesicle can be obtained:

µb(a) - µb(a*) ) kT qt qc a*(π* - γres) a -ln / - 2 ln / + - 1 (52) kT a* qt qc

(

)

Similarly, by substituting eq 51 into eq 42, a dimensionless expression for the chemical potential of lipids in the

Figure 6. Chemical potential of lipids, related to its equilibrium value µm(a*) ) µb(a*) in units of kT, plotted as a function of the lipid molecular area a in units of Å2 for the monolayer of DLPE molecules at the air-water interface and for bilayer vesicles in bulk water. The two curves intersect at a ) a* z 49 Å2, where the monolayer and bilayer are in equilibrium and thus have the same chemical potential. The chemical potential of lipids of bilayer vesicles always has a positive value and has a minimum of zero at a ) a*. The chemical potential of lipids for a monolayer, however, is monotonic and always has a negative value. The difference in the chemical potential of lipids between monolayer and bilayer vesicles of DLPE molecules decreases from 10.1kT (6.4 kcal/mol) to 0 upon compression from nearzero pressure to the equilibrium state.

Langmuir monolayer is obtained as

( )[

]

µm(a) - µm(a*) µb(a) - µb(a*) a*π* a π(a) ) 1kT kT kT a* π* (53) Figure 6 plots the dimensionless chemical potential of lipids for the bilayer vesicle, [µb(a) - µb(a*)]/kT, and for the corresponding Langmuir monolayer, [µm(a) - µm(a*)]/ kT of DLPE molecules as functions of the lipid molecular area, a. We see from this figure that (1) the two chemical potential curves intersect at a ) a* ) 49 Å2, where the lipids in the bilayer vesicle and in the corresponding monolayer have the same chemical potential; (2) the chemical potential of lipids for the bilayer vesicle, µb(a) µb(a*), has a minimum value of 0 at a ) a* and increases in either direction from a*, while the chemical potential of lipids for the corresponding Langmuir monolayer, µm(a) - µm(a*), monotonically decreases from the equilibrium zero value to negative values as a increases from a*; (3) upon compression from near-zero pressure to the collapse pressure, the chemical potential of lipids, relative to the equilibrium value, decreases from 3.6kT to 0 for the bilayer vesicle and increases from -6.5kT to 0 for the Langmuir monolayer. The difference between the chemical potential of lipids in the bilayer vesicle and that of lipids in the Langmuir monolayer increases from 0 at equilibrium monolayer pressure to 10.1kT, i.e., 6.4 kcal/mol, at nearzero monolayer pressure. Using values of the collapse pressure πc ) π* ) γ˜ o/w z γo/w) 49 mN/m and the molecular area at near zero pressure a|πf0 z 91 Å2 for the DLPE monolayer, the value of 10.1kT (or 6.4 kcal/mol) for the


Langmuir, Vol. 15, No. 4, 1999 1009

monolayer is always less than unity and decreases as a increases. (4) As the chemical potential of lipids in the bilayer vesicle has a minimum at a ) a*, the surface activity of DLPE in the bilayer vesicle also has a minimum; however, this minimum does not occur at the equilibrium molecular area a ) a*. This is because the surface pressure decreases more rapidly than the chemical potential increases as a increases from a*. At about 52 Å2 for DLPE bilayers, these two effects together produce a minimum value of the relative surface activity of about 0.7. The above calculation for the thermodynamic properties of a DLPE monolayer and bilayer illustrates the application of the present monolayer-bilayer correspondence analysis to a specific π-a relationship. It is not limited to a specific type of equation of state; different types of equations of state may have particular advantages for understanding the structure-function relationship of monolayers and bilayers. Among other applications, the type of eq 50 could be used to probe the effects of chain unsaturation on the behavior of biomembranes.38 Conclusion

Figure 7. Surface activity of DLPE molecules for bilayer vesicles, Rb(a)/Rb(a*), and that for the corresponding Langmuir monolayer at the air-water interface, Rm(a)/Rm(a*), which are related to their equilibrium value, plotted as function of the lipid molecular area, a, in units of Å2. At the equilibrium area a ) a*, the lipid in bilayer vesicles and in the monolayer has the same surface activity, which is assumed as unity. For the DLPE monolayer, the surface activity of lipid molecules is always less than unity and decreases to 0 as a increases, because both the chemical potential of lipids and the monolayer pressure decrease as a increases. For bilayer vesicles, although the chemical potential has a minimum at a ) a*, the minimum of the lipid surface activity does not occur at the equilibrium position a ) a*. As a increases from a*, the effect of the increase in the chemical potential of lipids on the surface activity is overcome by that of the sharp decrease in surface pressure on the surface activity. At a certain value, i.e., about a ) 52 Å2, the two opposite effects are balanced and the surface activity thus has a minimum of about 0.7 there. The surface activity of the DLPE molecules then increases as a further increases.

chemical potential difference is easily verified using the graphical method described in the preceding section. Figure 7 shows the plotting of the surface activity of DLPE molecules for the bilayer vesicle, Rb(a)/Rb(a*), and for the Langmuir monolayer, Rm(a)/Rm(a*), both relative to their equilibrium value, as a function of the lipid molecular area a. Activity of lipid molecules is affected by two factors. One is the chemical potential of the lipids; the larger the chemical potential, the larger the surface activity. Another is the surface pressure; the larger the surface pressure, the smaller the surface activity. Adopting the convention that the activity of lipid molecules at equilibrium is defined as unity, we have the following four observations from Figure 7. (1) At equilibrium, the DLPE molecules have the same surface activity for the lipids in the Langmuir monolayer and in the bilayer vesicle. (2) Away from the equilibrium molecular area a*, the surface activity of DLPE molecules in the bilayer vesicle is always greater than that of the lipids in the Langmuir monolayer at the same lipid molecular area. (3) The activity of DLPE molecules in the Langmuir

Solution of the monolayer-bilayer correspondence problem is the key to obtaining mechanochemical properties of lipid bilayer vesicles from experimental measurement of the monolayer at the air-water or the oil-water interface. Traditionally, the surface pressure of the monolayer at the air-water interface was defined as the lowering of the surface tension from the clean air-water interfacial tension due to the presence of the monolayer at this interface. This definition suggests a method to measure the surface pressure as the external pressure applied on the barrier to prevent the monolayer from spreading over the clean air-water interface. This definition and measurement method depend on a clean airwater interface and are thus not applicable to bilayer vesicles. Also, this traditional definition does not reveal the mechanism of the surface pressure as an intrinsic property of the monolayer. It is thus inappropriate to use this definition to solve the monolayer-bilayer correspondence problem, although it has been widely accepted for the past decades. There are actually three situations of practical interest: the monolayer at the air-water interface, the monolayer at the oil-water interface, and one leaflet of bilayer vesicles. They have a common monolayer and a substrate water system. The only difference among them is the environment. Because surface pressure is an intrinsic property of the monolayer, it should not be affected by the environment. That is, surface pressure should have the same value for monolayers in all three situations as long as the surface density of lipids and the temperature are the same. We therefore define the surface pressure as the total sum of the repulsive surface forces per unit of interaction line within the membrane plane for the monolayer-water substrate system. The difference for monolayers in different environments is merely an interfacial tension between the environment and the hydrocarbon layer. By applying thermodynamics and mechanics of membranes to the monolayers in different environments, the monolayerbilayer correspondence problem is then successfully solved. We thus conclude that, if the interaction between the two leaflets of a bilayer is negligible and the vesicles are large enough with negligible bending effects, the mechanochemical difference between the monolayer at the airwater interface and a leaflet of bilayer vesicles with the same surface density of lipids and at the same temperature is merely that the former but not the later exhibits an extra microscopic interfacial tension between the air and

1010 Langmuir, Vol. 15, No. 4, 1999

the lipid chain layer. Accordingly, the difference between the monolayer at the oil-water interface and one leaflet of bilayer vesicles is merely that the monolayer at the oil-water interface evinces an extra interfacial tension between the oil and the lipid monolayer. If the penetration of the oil into the lipid chain layer can be neglected, however, the monolayer at the oil-water interface would be mechanochemically equivalent to, and thus could be an ideal analogue of, one leaflet of bilayer vesicles. The exact solution of the monolayer-bilayer correspondence problem can be achieved provided the microscopic interfacial tensions become available either from delicate experimental measurement or from molecular mechanics calculation. Nevertheless, an approximate solution has been made in the present study under the assumption that the microscopic and the macroscopic interfacial tensions can be reconciled. The present theory makes it possible to interpret such mechanochemical properties as surface tensions, free energies, chemical potentials, and surface activities for a monolayer and bilayer vesicles analytically from any equation of state, or graphically from the measured π-a curve of the monolayer at the

Feng

air-water or the oil-water interface. There have been abundant equations of state and π-a curves published in the literature. The present monolayer-bilayer correspondence theory asserts that all mechanochemical properties of the monolayer at an interface as well as those of bilayer vesicles are actually included in the equation of state or the π-a curve of the monolayer at the interface. Interpretation of the mechanochemical properties of bilayer vesicles from the measurement of the monolayer at an interface is a task which is theoretically important in the study of biomembranes and practically useful for applying lipid vesicles to medical and pharmarceutical practice. Acknowledgment. The author thanks the reviewers for their constructive suggestions. He is also indebted to Prof. Robert MacDonald at Northwestern University for useful discussion. This work was supported by NUS Grant RP970637 and IMRE Equipment Fund, Singapore. The early stage of this work was supported by NIH Grant 1 PO 1 HL45168. LA980144F

Interpretation of Mechanochemical Properties of Lipid Bilayer Vesicles

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