Langmuir 2000, 16, 7401-7410
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Effects of Chain Structure on Surface Pressure-Area Behavior and Membrane Elasticity of Lipid Monolayers at the Air-Water Interface: Application of an Equation of State§ Si-shen Feng,*,†,‡ Guan-hua Zhu,† and Mei Yin Low† Department of Chemical and Environmental Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 and Institute of Materials Research and Engineering, 3 Research Link, Singapore 117602 Received June 17, 1999. In Final Form: June 15, 2000 A recently developed equation of state (Feng, S.-S.; MacDonald, R. C. Biophys. J. 1995, 69, 460) for lipid monolayers at the air/water or oil/water interface was employed to fit experimental π-A curves of various lipids with different chain number, chain length, and chain unsaturation. In this kind of equation of state the distributional entropy of the lipid headgroups within the membrane plane and the conformational energy of the lipid chains of various unsaturation can be exactly calculated. The three parameters in the equation of state, namely, am/a0, ratio of the minimal molecular surface area to the cross-sectional area of the two chains of a single lipid molecule, γb, the ratio of the effective to the actual projected surface area increment, and π3, the residual part in the total surface pressure, were determined and used to reproduce the experimental data. Such an equation of state can accurately describe the properties of most lipid monolayers in their liquid-expanded state. To overcome the unreliability of experimental data at the collapse pressure, the interfacial elastic modulus of monolayers at their collapse pressure was obtained by extrapolating the reproduced π-A curves to the collapse pressure rather than directly from experimental π-A isotherms. The results indicate that the interfacial elastic modulus increases with the number of double bonds until a peak is reached at the second or the third double bond, corresponding to a peak in the minimal molecular area. Furthermore, the interfacial elastic modulus is lowered if the double bonds are located in the middle of the hydrocarbon chains. For liquid-expanded monolayers of the same degree of unsaturation, the elastic modulus decreases with chain length. A comparison between the interfacial elastic modulus of lipid monolayers at the collapse pressure and that of the same lipid bilayers at their tension free state strongly supports the assertion that a lipid monolayer at its collapse pressure mechanochemically corresponds to a leaflet of large bilayer vesicles of the same lipid in water, which is developed from the monolayer-bilayer correspondence theory (Feng, S.-S. Langmuir 1999, 15, 998).
Introduction A study of the mechanochemical properties of insoluble lipid monolayers at the air/water interface and bilayer vesicles, which serve as model systems for biological membrane, aids in the understanding of the regulation of biological processes and the role of various lipids in biomembranes.1 Moreover, the experimental versatility of lipid monolayers at the air/water interface is useful for constructing self-assembling structures of technological utility.2 The primary characterization of a lipid monolayer usually involves the measurement of surface pressure (π) as a function of lipid molecular area (A), i.e., a π-A curve, from which various mechanochemical properties of the monolayer such as surface tension, chemical potential, and surface activity of lipids can be deduced from thermodynamic and mechanical principles.3,18 It was †
National University of Singapore. Institute of Materials Research and Engineering. § Abbreviations used: PC, phosphatidylcholine; DLPE, dilaureoylphosphatidy-ethanolamine; DPPC, dipalmitoylphosphatidylcholine; POPC, 1-palmitoyl-2-oleoylphosphatidylcholine; PLPC, 1-palmitoyl-2-linoleoylphosphatidylcholine; PLnPC, 1-palmitoyl2-R-linolenoylphosphatidylcholine. ‡
(1) Smaby, J. M.; Brockman, H. L. Langmuir 1992, 8, 563. (2) Huhn, H. Thin Solid Films 1989, 178, 1. (3) Mo¨hwald, H. Annu. Rev. Phys. Chem. 1990, 41, 441. (4) Smaby, J. M.; Brockman, H. L. Langmuir 1991, 7, 1031. (5) Gaines, G. L., Jr. Insoluble monolayers at liquid-gas interfaces; John Wiley and Sons: New York, 1966; pp156-188. (6) Gaines, G. L., Jr. J. Chem. Phys. 1978, 69, 924.
suggested that most acyl lipids found in biological membranes exist in a chain-melted, i.e., liquid-crystalline, state at physiological temperature, which is most closely modeled by a monolayer at its liquid-expanded state.1,2 However, it has been found currently that lyotropic phases formed from the lipid mixtures extracted from biological cells in the physiological temperature are normally in the phase coexistence region between the liquid-crystalline and gel phases also known as LR and Lβ phases. The mixture also often includes some hexagonal phase (H1). Lipid monolayers at the air/water or oil/water interface are ideal model membrane system, from which membrane properties and lipid interactions with the inserts such as proteins or drug molecules can be easily measured under (7) MacDonald, R. C.; Simon, S. A. Proc. Natl. Acad. Sci. U.S.A. 1987, 84, 4089. (8) Menger, F. M.; Richardson, S.; Sherrod, M. J.; Elrington, A. R.; Wood, M. G., Jr.; Zhou, Q. J. Am. Chem. 1988, 110, 6797. (9) Wolf, D. H.; Brockman, H. L. Proc. Natl. Acad. Sci. U.S.A. 1988, 85, 4285. (10) Birdi, K. S. Lipid and biopolymer monolayers at liquid interfaces; Plenum Press: New York, 1989; pp 66, 72, and 178. (11) Feng, S.-S.; Brockman, H. L.; MacDonald, R. C. Langmuir 1994, 10, 3188. (12) Lucassen-Reynders, E. H. In Anionic surfactants: Physical chemistry of surfactant action; Lucassen-Reynders, E. H., Ed.;. Marcel Dekker: New York, 1981; Chapter 1. (13) Feng, S.-S.; MacDonald, R. C. Biophys. J.1995, 69, 460. (14) Ja¨hnig, F. Biophys. J. 1984, 46, 687. (15) Nagle, J. F. Faraday Discuss. Chem. Soc. 1986, 81, 151. (16) Nagle, J. F. J. Membr. Biol. 1976, 27, 233. (17) Marsh, D. Biochim. Biophys. Acta 1996, 1286, 183. (18) Feng, S.-S. Langmuir 1999, 15, 998.
10.1021/la990781l CCC: $19.00 © 2000 American Chemical Society Published on Web 08/19/2000
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various conditions. However, the usefulness of experimental π-A data for understanding the properties of biological membranes and the regulation of biological processes depends, ultimately, on the degree to which the properties of a monolayer of single lipid species can be defined. This is because the prediction of the behavior of such model systems is usually phenomenological, i.e., it usually relies on the measured properties of monolayers of pure lipids rather than first principles.4 The equation of state for a lipid monolayer at an interface can directly and quantitatively describe the relationship among the intrinsic parameters of the membrane, i.e., the surface pressure, the lipid molecular area, the temperature, and other intensive parameters. It provides essential information concerning the microscopic structure of the surface phase and intra- and intermolecular interactions of surface phase and also contributes useful information on the properties of lipid membranes. A great deal of experimental π-A data of various lipid monolayers have been reported in the literature, from which various equations of state have been proposed.5-11,40-41 Each kind of equation of state has advantage and disadvantage of its own. Any bibliography of this sort is bound to be incomplete because research in this area has produced a steady stream going back more than fifty years. Most equations of state in the literature were usually derived from a two-dimensional kinetic theory of gases or from a phenomenological treatment of a two-dimensional solution, both of which avoid a detail model of molecular structure.5-12,40-41 In general, each model is not perfect in that it is admittedly approximation or based on oversimplified assumptions to limit the number of parameters. Nevertheless, if it can accurately regenerate the experimental isotherm from which it is obtained, this model can be thought to be useful for predicting π from A under a given T and can serve as a basis for deducing other properties of monolayers from principles in thermodynamics and membrane mechanics.18,46 Normally, the (19) Jacobs, R. E.; Hudson, B.; Anderson, A. C. Proc. Natl. Acad. Sci. U.S.A. 1975, 72, 3993. (20) Berde, C. B.; Anderson, H. C.; Hudson, B. S. Biochemistry 1980, 19, 4279. (21) Fowkes, F. M. J. Phys. Chem. 1962, 67, 983. (22) Wolfram, S. The Mathematica, 3rd ed.; Cambridge University Press: New York, 1996. (23) Smaby, J. M.; Momsen, M.; Brockman, H. L.; Brown, R. E. Biophys. J. 1998. (24) Behroozi, F. Langmuir 1996, 12, 2289. (25) Mingotaud, A. F.; Mingotaud, C. M.; Patterson, L. K. Handbook of Monolayers; Academic Press: San Diego, CA, 1993. (26) Evans, R. W.; Williams, M. A.; Tinoco, J. Biochem. J. 1987, 254, 455. (27) Barton, P. G.; Gunstone, F. D. J. Biol. Chem. 1975, 250, 4470. (28) Demel, R. A.; Guerts von Kessel, W. S. M.; Zwaal, R. F. A.; Roelofson, B.; van Deenen, L. L. M. Biochim. Biophys. Acta 1975, 406, 97. (29) Evans, E.; Waugh, R. J. Colloid Interface Sci. 1977, 60, 286. (30) Blume, A. Biochim. Biophys. Acta 1979, 557, 32. (31) Cevc, G.; Marsh, D. Phospholipid bilayers; Wiley-Interscience: New York, 1987. (32) Feng, S.-S.; MacDonald R. C. Biophys. J. 1997, 72, WPMC1. (33) Feng, S.-S. Biophys. J. 1998, 74, A330. (34) Zhelev, D. V.; Needman, D.; Hochmuth, R. M. Biophys. J. 1994, 67, 720. (35) Needman, D.; Evans, E. Biochemistry 1988, 27, 8261. (36) Kwok, R.; Evans, E. Biophys. J. 1981, 35, 637. (37) Evans, E.; Needham, D. J. Phys. Chem. 1987, 91, 4219. (38) Needham, D.; McIntosh, T. J.; Evans, E. Biochemistry 1988, 27, 4668. (39) Needham, D.; Nunn, R. S. Biophys. J. 1990, 48, 997. (40) Fainermam, V. B.; Vollhardt, D. J. Phys. Chem. 1999, 103, 145. (41) Sakamoto, N.; Sakai, K.; Takagi, K. Phys. Rev. 1996, E53, 6164. (42) Albrecht, O.; Matsuda, H.; Eguchi, K.; Nakagiri, T. Thin Solid Film 1999, 338, 252. (43) Schwarz, G.; Taylor, S. E. Supramol. Sci. 1997, 4, 479. (44) Schwarz, G.; Taylor, S. E. Langmuir 1995, 11, 4341.
Feng et al.
fitting parameters obtained have physical significance in the context of the model itself. The equation of state recently developed by Feng and MacDonald13 takes into account the effects of the partition between the headgroups and the water molecules within the interface and the lipid chain conformation. In this kind of equation of state, the phenomenological properties of the monolayer membrane can be referred to its molecular structure. Both of the distributional entropy of the headgroups within the membrane plane and the conformational entropy of the chains can be explicitly calculated for various lipids of different number and position of double bonds of the hydrocarbon chains. This equation of state is based on a statistical mechanical description of monolayers at the air/water interface. It is formulated for the molecular structure and the intra- and intermolecular interactions of lipid monolayers in the liquid-expanded state at the air/water interface. In this model, the total Hamiltonian of the monolayer is assumed to consist of three terms, namely, the two-dimensional mixing entropy of water and lipid molecules at the interface, the chain conformational energy, and a residual part, which includes all other intra- and intermolecular interaction energies. The first two terms were calculated exactly within the limitations of the formulation and gave rise to positive surface pressure. The third term, which is not amenable to calculation, was obtained as the difference between the sum of the two calculated terms and experimental data and was found to represent an approximately area-independent tension.13 It should be noted that this approximation is of acceptable error only for monolayers at their liquid expanded state. This treatment was found to be valid especially for monolayers at a surface pressure, which is close to their collapse pressure. The recent development made by one of the authors shows that it is the monolayer at its collapse pressure that can be considered mechanochemically equivalent to a leaflet of large bilayer vesicles. The equation of state used in the present study can thus have potential applications in the biological and medical research. With appropriate values for the three parameters, which stand for the minimum packing area of lipid, the degree of the overlap of adjacent chains, and the other intra- and intermolecular interactions, the developed equation of state was shown to reproduce the experimental data of DLPE, DPPC, POPC, PLPC, and PLnPC.13 The effects of cis double bonds appear in the theory in two aspects: a bend due to a cis double bond on the chain, which increases the minimal molecular area, and the presence of a double bond in a chain, which decreases the degrees of freedom of the chain. The computations show that the former effect is very significant whereas the later is less significant.13 Apparently, this model has the advantage enabling the effects of chain conformation and the consequences of chain unsaturation to be estimated. However, the limited number of π-A data reproduced by the deduced equation of state raised the questions: (a) Is the equation of state valid to any lipid monolayer? (b) To what range of π-A data it can be applied? To examine the applicability of this equation of state for monolayers of a variety of lipids at the air/water interface, we fit in this paper the proposed equation of state to experimental π-A isotherms of various phaphatidylcholine (PC) molecules with almost all natural double bonds in the hydrocarbon chain. The experimental data (45) Schwarz, G.; Wackerbauer, G.; Taylor, S. E. Colloid Surfaces 1996, A111, 39. (46) Evans, E. A.; Skalak, R. Thermodynamics and Mechanics of Biomembranes; CRC Press: Boca Raton, FL, 1982.
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are taken from the literature.25 The results show that the equation of state can well describe the liquid-expanded monolayer behavior of most of lipid species considered except some of them whose isotherm was obtained at different conditions. The effects of the conformation and the number and position of double bonds of the hydrocarbon chain on the equation of state is in agreement with the statement of Feng and MacDonald.13 In addition, we also have deduced the lateral area elastic modulus at the collapse pressure of PC monolayers, whose liquidexpanded state can be well described by the equation of state. The lateral area elastic modulus for expanded PC monolayers at the collapse pressure is a sensitive function of the number and position of double bonds on the hydrocarbon chain. The comparison between elastic modulus of the monolayer at collapse pressure and that of tension free bilayer vesicles of the same lipid demonstrates that the monolayer at the collapse pressure is indeed mechanochemically equivalent to large unstressed bilayer vesicles in water. This provides an evidence of the monolayer-bilayer correspondence theory.14-18 Theory Equation of State. Following the treatment of Jacobs et al.19 and Berde et al.20 for the thermotropic phase transition of phospholipid bilayer dispersion, Feng and MacDonald13 constructed a statistical mechanical description of monolayers at the air/water interface to investigate chain unsaturation effects. In their model, the total Hamiltonian of the monolayer system was decomposed into three parts: the two-dimensional entropy of mixing between the lipid molecules and the water molecules at the interface, the conformational energy due to changes in chain conformation away from the maximal density state, and a residual term. This term lumps together all other intra- and intermolecular interaction energies. Consider a monolayer system which contains N lipid molecules and each molecule occupies a surface area a and has two hydrcarbon chains. Either or both of the two chains may be saturated, trans-unsaturated, or cisunsaturated. The Hamiltonian function H, the partition function Q, the Helmhotz free energy function F, and the surface pressure function π can then be described as follows.13 2N
H(XN,Y2N) ) Nhm(a) +
hc(Yj,a) + Nh3(a) ∑ j)1
F(N,A,T) ) -kT ln Q ) -kT ln Qm - 2NkT × ln qc - Nh3(a) (3)
( )
Qm ) [(1 + xa/aw) - xam/aw]2N
kT ∂Qm 2kT ∂qc ∂h3 ) + + ) N,T Qm ∂A qc ∂a ∂a πm(a) + πc(a) + π3(a) (4)
In eq 1, Xi (i ) 1, 2, 3...N) are the degrees of freedom of the whole ith molecules, while Yj (j ) 1, 2, 3...2N) are the degrees of freedom of the jth hydrocarbon chain. The hm, hc, and h3 are the Hamiltonians due to the twodimensional mixing entropy for a single lipid molecule, the conformational energy for a single chain, and the residual term of all other interaction energies for a single lipid molecule, respectively. In eqs 2-4, Qm and qc are the partition functions due to the two-dimensional mixing for
(5)
where am is the molecular area at maximal density or the molecular surface area for monolayers at collapse and aw is the surface area occupied by a single water molecule,21 which is taken to be aw ) 9.65 Å2. Accordingly, the partial surface pressure for all lipid molecule densities, πm, which is due to the distribution of the lipid and the water molecules on the surface, can be calculated from
[ (
πm(R)am ) 1/ x1 + R x1 + R - 1 + kT
x )] aw am
(6)
where R ) (a - am)/am is the relative surface area change of the given area a from the maximal density. The minimal molecular surface area, am, can be found from experimental measurement and has a different value for monolayers of lipids with different headgroup size, chain number, chain length, and chain unsaturation. Hence, the number and position of double bonds affect the mixing entropy in this model through the parameter am. For the partial surface pressure due to the chain conformation energy of the two chains on the lipid molecule, i.e., πc, there are two cases considered: saturated chain and unsaturated chain. The partition function for a single saturated chain attributable to conformations at the maximal density is
(1)
Q(N,A,T) ) Qm(N,A,T) × [qc(a,T)]2N × exp[-N h3(a)/kT] (2)
∂F π(N,A,T) ) ∂A
the entire system and to the conformational energy for a single chain, respectively. A ) Na is the total surface area of the system. T is the absolute temperature, and k is the Boltzmann constant. It is clear that each equation includes three terms in its right-hand side, which represent the contribution of the two-dimensional mixing entropy, the chain conformation energy, and the residue part of the molecular interaction energy, respectively. Equation 4 is the equation of state we will employ in the present study. The three components of the total surface pressure at a constant T are denoted by πm(a), πc(a), π3(a), respectively. The partition function for the two-dimensional mixing of lipid and water molecules, Qm, which includes the effects of mixing of the lipid and the water molecules as well as the size difference between the lipid and the water molecules, was formulated as follows
m
qc ) 1 +
∑ i)1
[
( )]
1 + 2 exp -
m-i
× kT π(R)am a0 + γb × 2 exp kT kT am
{[
()
0.14(m - i + 1)
]}
(7)
where ) 500 cal/mol is the expected steric energy for each gauche bond; m is the bond number for which bond rotation generates different conformations; a0 is twice the cross section area per chain in the all-trans state, which can be calculated by a0 ) 2πw2/4 if we take w ) 4.85 Å, the diameter of the projection of an all-trans chain onto the plane of the surface.19,20 For chains with different degrees of unsaturation, the chain conformational partition function qc should differ from eq 7 according to the number and position of trans or cis double bonds. The chain conformational partition function for an unsaturated chain is the following:
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qc ) q1 + q2 + q3
(8)
q1, q2, and q3 should be calculated according to the specific case with the chain structure considered in details. For example, for oleic acid (9-octadecenoic acid),
q1 ) (e-κ/kT + 4e-µ/kT + 4e-λ/kT)
[
7
q2 )
∑ i)1
( )]
2q1 1 + 2 exp -
(9a)
(16-i-3)
× kT π(R)am a0 + γb × 2 exp kT kT am
{[
()
0.14(16 - i + 1) 16
q3 )
∑
i)11
[
( )]
× kT π(R)am a0 + γb × 2 exp kT kT am
{[
(9b)
16-i
2q1 1 + 2 exp -
]}
()
0.14(16 - i + 1)
]}
π3(a) ) πe(a) - {πm(a) + πc(a)} (9c)
for linoleic acid (9,12-octadecenoic acid),
q1 ) (e-κ/kT + 4e-µ/kT + 4e-λ/kT)2
[
7
q2 )
( )]
2q1 1 + 2 exp ∑ kT i)1
{[
2 exp -
kT
×
kT
() am
γb ×
0.14(16 - i + 1) 16
q3 )
[
( )]
{[
2 exp -
+
kT
]}
(10b)
]}
(10c)
16-i
∑ 2q1 1 + 2 exp - kT
i)14
(10a)
(16-i-6)
π(R)am a0
+
×
()
π(R)am a0 kT
am
γb ×
0.14(16 - i + 1)
and for linolenic acid (9,12,15-octadecenoic acid)
q1 ) (e-κ/kT + 4e-µ/kT + 4e-λ/kT)3
7
q2 )
∑ i)1
[
( )]
2q1 1 + 2 exp -
{[
(11a)
(16-i-9)
× kT π(R)am a0 + γb × 2 exp kT kT am
()
0.14(16 - i + 1) q3 ) 0
]}
The rule in composing the conformational partition function for an unsaturated chain is (1) the power index in the expression for q1 is the number of double bonds, nd; (2) the sum for q2 is taken from 1 to k1 - 2, where k1 is the carbon atom number immediately before the first double bond location and the power index in the expression for q2 is (m - i - 3nd), where nd is the number of double bonds in the chain; and (3) the sum for q3 is taken from k2 + 2 to m, where k2 is the carbon atom number immediately after the last double bond location.13 Because the residual part in the Hamiltonian was presumed in the model to include all other intra- and intermolecular interactions between lipids and between lipid and water molecules except for those considered in the first two terms, the corresponding residual part in the total surface pressure, π3, is the difference between the measured surface pressure and the surface pressure due to the two-dimensional mixing and the chain conformational energy. It can be obtained by subtracting the theoretical computation of πm + πc from the experimental measurement of the total surface pressure for a given lipid monolayer of specified chain length and chain unsaturation, πe:
(11b) (11c)
(12)
It has been found from our practice that experimental π-a curves for most lipids can be well fitted by a constant residual component π3(a). This means that the residual component π3(a) can be well approximated by a constant π3: π3(a) ≈ π3. However, it should be noted that the value of this constant π3 could be different for a different temperature T. Determination of Parameters of the Equation of State and Interfacial Elastic Modulus of the Monolayer Membrane. The equation of state involves three parameters, namely am, γb, and π3. The first, am, was obtained directly from the π-a curves at the point of discontinuity of the slopes in the high-pressure region (i.e. at the point of monolayer collapse). On the other hand, γb and π3 were obtained through curve fitting or nonlinear regression. The built-in function “NonlinearRegress” in Mathematica22 finds a least-square fit of data to a model, which is not linear in its parameters. It also returns a list of nonlinear regression diagnostics. It should be pointed out that this equation of state is valid only for lipid monolayers in their liquid-expanded state. Phase transition behavior cannot be investigated by this kind of equation of state since the fitted value of π3 would be different for a monolayer in its different states. Moreover, the constant residual π3 assumption is approximately valid for monolayers at a surface pressure close to their collapse pressure. At lower pressure far from the collapse pressure, this assumption breaks down. π3 would strongly depend on the lipid molecular area a. This is a main limitation of such an equation of state. There have been many models of equation of state in the literature. Each has advantages and disadvantages of its own. For example, two recent models have been developed by Sakamoto et al.41 and Fainerman and Vollhardt,40 respectively. The former is a modification of the van der Waals equation of state, and the latter includes the effects of amphiphile aggregation. Both can predict phase transition behavior of the monolayer. Most of the equations of state are phenomenological, which describe the physical behavior of the monolayer without molecular structure considered in details. In other words, those equations of state are physically orientated, which either tend to do Monte Carlo simulations or density-functional
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Langmuir, Vol. 16, No. 19, 2000 7405
analyses or prefer Landau theories to incorporate electrostatic interactions.50,51 The equation of state applied in the present study can be described as “physicochemical”, in which the phenomenological properties of the monolayer can be calculated from the molecular structure and intraand interactions of the component molecules. In this kind of equation of state, both the headgroup mixing entropy and the chain conformational entropy can be exactly calculated, and the effects of chain unsaturation for various double bond number and position can thus be analytically investigated. The equation of state applied in this research is unique in calculating the double-bond effects. This is an advantage, which makes it over many other models in the literature. This advantage is important especially when applied to biological membranes. Cholesterol component and lipid chain unsaturation are two important factors which strongly affect the cell membrane properties and thus the health of cells. How the membrane properties are affected by the lipid chain unsaturation can be thoroughly investigated by such an equation of state. We now demonstrate the advantages of the present model by comparing fits of this model versus other existing models to the measured π-a data. The equations of state in the literature fall into two major categories depending on whether the substrate water is considered as a component of the surface phase, i.e., on the location of the dividing surface between the monolayer and the bulk phases. The Hill-de Boer equation of state ignores the presence of the substrate water molecules in the surface. It is a two-dimensional analogue of the van der Waals expression for a nonideal gas
(
π+
)
aint a2
(a - am) ) kT
(13)
where aint is the interaction term which is positive for repulsive interaction or negative for attractive interaction. It takes into account both the lipid molecular size and lateral interactions between the molecules.19 Instead, the osmotic type equation of state considers interactions of the water component with the lipid in the monolayer, giving rise to an osmotic description of the monolayer surface pressure, and has the following form4
π)
( ) {[ (
Figure 1. Best fitting of the experimental measurement of surface pressure vs surface molecular area for monolayers of various lipids by the Hill-de Boer equation of state with the fitting values of the parameters listed in Table 1. Symbols represent the experimental values, and the solid curves represent the Hill-de Boer equation of state.19 From left to right: DPPC (C16:0/C16:0 PC) at T ) 42 °C,51 POPC (C16:0/C18:1,n-9 PC), PLPC (C16:0/C18:2,n-6 PC), and PLnPC (C16:0/C18:3,n-3 PC) at T ) 22 °C.52
)]}
ω1 qkT 1 ln 1+ ω1 f a - ω0
(14)
where kT is the Boltzmann constant multiplied by the temperature, a is the lipid molecular area, ω0 is the minimum molecular area of lipid, ω1 ) 9.65 Å2 is the surface area occupied by one water molecule at the interface, and f and q are two fitting parameters which can be referred to the effective surface water activity coefficient and the two-dimensional osmotic coefficient, respectively.11 Figures 1-3 show the measured π-a data for the monolayers of dipalmitoylphosphatidylcholine (DPPC),51 1-palmitoyl-2-oleoylphosphatidylcholine (POPC), 1-palmitoyl-2-linoleoylphosphatidylcholine (PLPC), and 1-palmitoyl-2R-linolenoylphosphatidylcholine (PLnPC),52 (47) Evans, E. A.; Kwok, R. Biochemistry 1982, 21, 4847. (48) Fettiplace, R.; Andrew, D. M.; Haydon, D. A. J. Membr. Biol. 1971, 5, 277. (49) Hanai, T.; Haydon, D. A.; Taylor, J. J. Proc. R. Soc. London 1964, A281, 377. (50) Faure, M. C.; Bassereau, P.; Carignano, M. A.; Szleifer, I.; Gallot, Y.; Andelman, D. Euro. Phys. J. B 1998, 3 (3), 365. (51) Zhao, M. S.; Rice, S. A. J. Phys. Chem. 1999, 111 (5), 2181. (52) Albrecht, O.; Gruler, H.; Sackman, E. J. Phys. 1978, 39, 301. (53) Evans, R. W.; Tinoco, J. Chem. Phys. Lipids 1978, 22, 207.
Figure 2. Best fitting of the experimental measurement of surface pressure vs surface molecular area for monolayers of various lipids by the osmotic type equation of state with the fitting values of the parameters listed in Table 1. Symbols represent the experimental values, and the solid curves represent the osmotic type equation of state.4 From left to right: DPPC (C16:0/C16:0 PC) at T ) 42 °C,51 POPC (C16:0/C18:1,n-9 PC), PLPC (C16:0/C18:2,n-6 PC), and PLnPC (C16:0/C18:3,n-3 PC) at T ) 22 °C.52
which are fitted by the Hill-de Boer equation of state eq 13 (Figure 1) and the osmotic equation of state eq 14 (Figure 2) along with the present model (Figure 3). The function of NonlinearRegress in Package “Statistics” of Mathematica 4.1 (Wolfram Research, Inc., Urbana, IL) was employed for best fitting. The values of the fitting parameters are listed in Table 1. It can be seen from the figures that the fit by the Hill-de Boer equation is very poor. The double-bond effects are very roughly reflected in the magnitude of am and the sign and the magnitude of the parameter aint. The correlation coefficients for the four monolayers are 0.8232, 0.8770, 0.8584, and 0.8426, respectively. Instead, the fitting by the osmotic type of equation of state is quite satisfactory although the fitting by the PLPC and PLnPC monolayers is not good for large molecular areas. The correlation coefficients for the four monolayers are 0.7781, 0.8124, 0.8169, and 0.8389, respectively. However, the double-bond effects are indirectly reflected in the two parameters, q and f, which can be referred to the effective surface water activity coefficient and the two-dimensional osmotic coefficient, respectively. The fit by the present model is the best among these three
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Table 1. Values of Fitting Parameters for the Hill-de Boer Equation of State, Osmotic Equation of State (Smaby-Brockman, 1991), and Double-Bond Equation of State (Feng-MacDonald, 1995) Hill-de Boer
Smaby-Brockman
Feng-MacDonald
aint (×10-28 dyn‚cm3)
q
f
γb
lipid
am (Å2)
π3 (dyn/cm2)
DPPC POPC PLPC PLnPC
45.86 45.00 50.88 60.67
1.5064 -4.3957 -3.9770 -5.1283
3.1475 1.9835 4.7557 6.9133
1.1476 1.0434 1.0987 1.0953
0.6686 0.5145 0.3880 0.8252
14.8122 4.9021 6.2700 9.6117
at high surface pressures by fitting the experimentally obtained π-a isotherms to the proposed equation of state. The Ka values, at the collapse pressure, could then be determined from the fitted isotherms by
Ka* ) (-a*) (dπ/da)a*
(15a)
where a* is the lipid molecular area at the collapse pressure and (dπ/da)a* is the derivative of the surface pressure π with respect to the molecular area a at constant temperature T which is evaluated at a ) a*.24 For the monolayer at the air/water interface, its surface tension σ and surface pressure π are related by the following equation18,31,46 Figure 3. Best fitting of the experimental measurement of surface pressure vs surface molecular area for monolayers of various lipids by the present model with the fitting values of the parameters listed in Table 1. Symbols represent the experimental values, and the solid curves represent the present equation of state.13 From left to right: DPPC (C16:0/C16:0 PC) at T ) 42 °C,51 POPC (C16:0/C18:1,n-9 PC), PLPC (C16:0/C18:2,n-6 PC), and PLnPC (C16:0/C18:3,n-3 PC) at T ) 22 °C.52
types of equations employed in this paper (Figure 3). The correlation coefficients for the four monolayers are 0.8981, 0.8179, 0.8361, and 0.9053, respectively. Moreover, the two-dimensional mixing entropy of the water and the lipid within the monolayers and the conformational energy of the two chains can be directly calculated. The doublebond effects are more directly reflected in the two parameters γb and π3, which represent the degree of the chain overlap and the level of the interaction between the lipid molecules such as van der Waals interactions and electrostatic interactions. An equation of state is a mathematical description of the relationship among various intrinsic state variables of the monolayer at the air/water or oil/water interface. In principle, various mechanochemical properties of the monolayer membrane can be calculated from the equation of state by applying thermodynamics and mechanics. Among various properties of the monolayer, the dilatational elastic modulus or its inverse, area compressibility, is of great importance to determine the behavior of the membrane. There have been limitations associated with experimentally derived values of interfacial elastic modulus Ka for lipid monolayers at high surface pressures (>3035 mN/m).1,23 In essence, Ka values determined directly from an experimental π-a isotherm display a maximum well below the collapse pressure and then diminish at higher surface pressures instead of hyperbolically increasing until the film collapses.23 The maximum of Ka generally occurs at surface pressures greater than 35 mN/m but this depends somewhat upon the lipid or lipid mixture under consideration. Actually, Ka value begins to fall away from the expected hyperbolic-like increase prior to achieving the Ka maximum.23 Such behavior is highly reproducible and has been attributed to intrinsic experimental factors (e.g., trough composition and/or design) but may also reflect a decrease in lipid film stability at high pressures. We thus recommend to obtain the Ka data
πmonolayer ) γa/w - σmonolayer
(15b)
This means that the surface pressure can be defined as the lowering of the surface tension from its value for the clean air/water interface. Substituting eq 15b to eq 15a, we have18,46
Ka* ) a* (dσ/da)a*
(15c)
For bilayers, or large bilayer vesicles with bending effects neglected, how to define it surface pressure has been a problem. However, it has been shown that the monolayer and a leaflet of the bilayer are mechanochemically equivalent as long as the two layers have same molecular density of lipids at the same temperature. Therefore, π ) π(N,a,T) should be the same function for both of them.18 However, since the surface tension represents the total mechanical effects of the surface pressure and the interfacial tension, the surface tension of the bilayer σbilayer would be different from that of the monolayer σmonolayer. This is because the monolayer is at the air/water interface while a leaflet of bilayer is at the interface between the water and the opposite lipid monolayer. With such a monolayer-bilayer correspondence in mind, eqs 15a and 15b can also be used to calculate the elastic modulus of the bilayer from the partial change of its surface pressure or its surface tension. Results Fitting Parameters. All three parameters of the equation of state possess a physical significance and can be determined by fitting the equation of state to the experimental data for a monolayer of specific lipid at the air/water or oil/water interface.13 am/a0 is the ratio of the molecular area for the monolayer at collapse to the crosssectional area of the two chains of a single lipid molecule. The second parameter, γb, is the overlap coefficient, i.e., the ratio of the effective to the actual projected surface area increment due to a gauche bond bend. The third parameter, π3, is the residual part in the total surface pressure, the component due to all intra- and intermolecular interactions excluding the two-dimensional mixing entropy and the chain conformational energy. Table 2 lists the values of am, γb and π3 for the monolayers of various
Application of an Equation of State
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Table 2. Characteristic Parameters of the Equation of Statea lipid
am (Å2)
am/a0
πm,c (mN/m)
γb
π3 (mN/m)
C16:0/C12:0 C16:0/C14:1,n-7 C16:0/C18:1,n-9 C16:0/C18:2,n-6 C16:0/C18:3,n-3 C16:0/C18:3,n-6 C16:0/C22:0 C16:0/C22:1,n-9 C16:0/C22:2,n-6 C16:0/C22:3,n-3 C16:0/C22:4,n-6 C16:0/C22:5,n-6 C16:0/C22:6,n-3 C12:0/C12:0 C14:0/C14:0 C18:0/C18:1,n-9
61.7 65.7 56.4 61.5 71.4 74.3 53.2 60.4 74.3 72.6 69.5 73.0 68.7 58.4 56.1 54.5
1.67 1.78 1.52 1.66 1.93 2.01 1.44 1.63 2.01 1.96 1.88 1.97 1.86 1.58 1.52 1.47
47.5 47.4 50.9 49.2 47.6 47.9 63.6 47.6 46.1 47.3 46.0 45.3 47.5 44.1 36.5 50.3
0.82 0.89 0.53 0.61 0.75 1.01 0.89 0.49 0.65 0.63 0.74 1.40 1.77 1.01 0.53 0.57
-10.9 -13.2 -5.7 -7.5 -9.1 -8.3 -33.2 -8.1 -9.0 -7.7 -10.0 -9.7 -10.8 -16.6 -11.8 -13.9
a As determined by fitting the equation of state to the experimental measurement from ref 16. The πm,c is partial collapse pressure due to the two-dimensional entropy mixing of lipid and water molecules at the air/water interface.
PCs investigated in this paper. The π-a isotherms were obtained from the literature.25 All of the isotherms chosen were measured at the same temperature of 22 °C, with a few exceptions. Most of the liquid-expanded monolayers of the lipids in Table 2 would collapse at molecular areas between 55 and 75 Å2 and surface pressures between 45 and 50 mN/ m. The effects of chain length and unsaturation are greater on the collapse area, am, than on the collapse pressure, πc, which lies within a narrow range. This narrow range of πc may be applied to monolayers of all PCs being in their LE phase. Therefore, greater insight is gained by studying the effects of chain length and unsaturation on am, which is one of the parameters of the equation of state. The effect found of chain length on am agrees with that from other sources in the literature, whereby an increase in the carbon number of the acyl chains leads to a decrease in am due to the increased van der Waals forces experienced by longer chains. This is seen clearly by comparing C12:0/C12:0 and C14:0/C14:0, whose am values are 58.4 and 56.1 Å2, respectively, or C16:0/C14:1,n-7, C16:0/C18:1,n-9 and C18:0/C18:1,n-9 whose am values are 65.7, 56.4, and 54.5 Å2, respectively. However, am of C16:0/C22:1,n-9 is larger than that of C16:0/C18:1,n-9. This may be due to the slightly higher collapse pressure for C16:0/C18:1,n-9 (about 51 mN/m) relative to that of C16:0/C22:1,n-9 (about 48 mN/m), but is more likely due to the marked difference in chain length between the two acids in the 22-carbon homologue as suggested by Evans et al.26 The effect of an increasing number of double bonds on am differs for 1-palmitoyl PCs with 18 and 22 carbons in the second chain. The value of am increases steadily when the number of cis double bonds increases from one to three for the former. For the latter, am increases from the first (C16:0/C22:1,n-9) to the second cis double bond (C16:0/C22:2,n-6), decreases with the third (C16:0/C22:3,n-3) and fourth double bond (C16:0/C22:4,n-6), increases again for the fifth (C16:0/ C22:5,n-6), and finally decreases for the sixth double bond (C16:0/C22:6,n-3). An explanation for these phenomena could be that the addition of subsequent double bonds can cancel, to some extent, the effects of previous double bonds. However, the am values for the 22-carbon series are still very close in magnitude, with C16:0/C22:2,n-6 having the largest value of 74.3 Å2. In the case of C18:0/C18:1,n-9, the addition of two cis bonds to the first chain and one to the second, increases the value of am from 54.5 to 66.1 Å2.
The position of the cis double bond also affects the minimal molecular area to some degree. A comparison of C16:0/C18:3,n-3 and C16:0/C18:3,n-6 reveals that the latter has a larger value of am. This can probably be explained by the fact that the three double bonds of C16:0/C18:3,n-6 are located more centrally than those of C16:0/C18:3,n-3. As reported by Barton and Gunstone,26 in a series of monoenoic octadecenoyl PCs, the effect of the double bond on the temperature of the gel-to-liquid-crystal phase transition was greatest when it was situated in the middle of the chain. The present case is likely to be similar to their findings. In a similar fashion, the effect of unsaturation on γb appears to follow the same trend as its effect on am. For 1-palmitoyl PCs with 18 carbons in the second acyl chain, an addition of cis double bonds from one to three results not only in the expansion of the monolayer but also an increase in γb values from 0.53 to 0.61 to 0.75/1.01. A plausible reason for this is that, in the same manner as the kinks due to cis double bonds increase the value of am, they prevent the chains from packing so closely that reduces the degree of overlap. For 1-palmitoyl PCs with 22 carbons in the second acyl chain, the trend is again similar to that of am, except that although γb increases for the first two double bonds and decreases slightly for the third, it increases again for the fourth and subsequent double bonds. This could be because the addition of double bonds decreases the degrees of freedom of the chains by fixing the carbon atoms in a rigid position. As such, despite a slight decrease in am, there is less overlapping of chains. The effect of the position of the double bonds on γb also resembles its effect on am, where the γb value of C16:0/ C18:3,n-6 (1.01) is larger than that of C16:0/C18:3,n-3 (0.75). The central location of the double bonds may have bent the two halves of the chain in such a way that the chains are able to pack less closely and the degree of overlap is reduced. The parameter π3 accounts for all other intra- and intermolecular interactions except for those due to the two-dimensional mixing entropy of water and lipid molecules and the chain conformational energy. Therefore, it should encompass van der Waals forces. Hence, we would expect the value of π3 to increase with chain length for the same degree of unsaturation, corresponding to an increase in van der Waals forces. Similarly, we would expect the van der Waals forces and hence the value of π3 to decrease with the greater disorder introduced by the kinks of any cis double bonds, following a trend opposite to that of am. Unfortunately, the values of π3 obtained do not appear to follow the expected trend and are rather erratic, suggesting perhaps that other factors be also involved. Most of π3 values lie between -6 to -15 mN/m. Comparing the effects of chain length by observing the dimensional values of π3 obtained for C12:0/C12:0 and C14:0/C14:0 (-17 and -12 mN/ m) and for C16:0/C14:1,n-7, C16:0/C18:1,n-9, C18:0/C18:1,n-9, and C16:0/C22:1,n-9 (-13, -6, -14, and -8 mN/m respectively) seems to indicate that π3 decreases in magnitude with chain length as with am, C18:0/C18:1,n-9 being the exception. The effects of unsaturation also appear, on the surface, to follow a trend similar to that of am, with some exceptions. For 1-palmitoyl PCs with an 18-carbon chain, the magnitude of π3 increases in the order from one to three cis double bonds. However, the value of π3 for C16:0/C18:3,n-6 (-8 mN/m) is smaller than that of C16:0/C18:3,n-3 (-9 mN/ m). For those with a 22-carbon chain, the magnitude of π3 increases from one to two cis double bonds, decreases slightly for the third double bond, then increases once again for the fourth, fifth, and sixth double bonds. Interfacial Elastic Modulus. The values of the elastic modulus Ka* lie mainly in the range of 130-170 mN/m as
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Feng et al.
Figure 4. Values of the lateral area elastic modulus for PC monolayers at the collapse pressure.
shown in Figure 4. The comparison of C12:0/C12:0 and C14: 0/C14:0, and of C16:0/C14:1,n-7, C16:0/C18:1,n-9, C18:0/C18:1,n-9, and
C16:0/C22:1,n-9 yields some interesting results. For C12:0/C12:0 and C14:0/C14:0, the effect of chain length on am is dominant, leading to a smaller value of Ka* for the latter (126 mN/m) relative to the former (160 mN/m). Hence, C14:0/C14:0 monolayers and bilayers, in equilibrium with each other, are more fluid than those of C12:0/C12:0. For C16:0/C14:1,n-7, C16:0/C18:1,n-9, C18:0/C18:1,n-9, and C16:0/C22:1,n-9, the respective values of Ka* are 169, 152, 173, and 155 mN/m. Again, the trend seems to follow that of am, with the exception of C18:0/C18:1, showing the effects of am to be dominant. The effects of (dπ/da)a* are significant only in the case of unsaturation. For C16:0/C18:1,n-9, C16:0/C18:2,n-6, and C16:0/C18:3,n-3, the effects of am are still dominant with an increase in Ka* from 152 to 156 to 164 mN/m. Interestingly enough, however, despite having a larger value of am (i.e., 74.3 vs 71.4 Å2), the value of Ka* is much smaller for C16:0/C18:3,n-6 (138 mN/m) than for C16:0/C18:3,n-3 (164 mN/ m), illustrating the effects of (dπ/da)a*. In fact, its value is even lower than that for C16:0/C18:1. This is again evident for the series of 1-palmitoyl PCs with a second chain of 22 carbons. The values of Ka* for the fourth, fifth, and sixth double bonds fall below that of the first double bond, despite having larger am values. In the case of C18:0/C18:1,n-9, the addition of one and two double bonds to the unsaturated and saturated chains of C18:0/C18:1,n-9 respectively gives rise to an increase in am but this again does not override the effect of the gentler slope of C18:2/C18:2. This leads to C18:2/C18:2 having a lower Ka* value (142 mN/m) as compared to C18:0/C18:1 (173 mN/ m). Applicability of Equation of State. As pointed out by the two original authors,13 the equation of state employed in the present study is only applicable to liquid-
expanded monolayers at the air/water or oil/water interface. Therefore, it fails to fit those π-A curves of monolayers displaying slope discontinuities due to phase transition and those in other phases, for example, the liquid-condensed phase, most of which are involved in π-A curves of monolayers with two saturated chains of more than 14 carbons at room temperature. These includes PCs of C16:0/C14:0, C16:0/C16:0, C16:0/C18:0, C18:0/C18:0, and C16:0/C22:0. However, the equation of state fitted the π-A curves of C16:0/C16:0 obtained at 42 °C because lipid is at liquid-crystalline phase at that temperature.13 In Figures 5-7 the experimental data of surface pressure versus surface area for monolayers of various lipids are fitted by the theoretical prediction of the equation of state with the values of the parameters listed in Table 2. It was found that the equation of state provides a good fit to most of the π-a isotherms except for C12:0/C18:1,n-9, C14:0/C18:1,n-9, C18:1,n-9/C18:1,n-9, C16:0/C16:1,n-7, and C16:0/C10:0. A reason π-a curves of C12:0/C18:1,n-9 and C14:0/C18:1,n-9 monolayers cannot be well fitted by the equation of state is that the subphase used is not water at pH 5.1, like most of the other isotherms, but a buffer at pH 7.4. It has been found that increasing the pH value of the suspending solution leads to lipid headgroup deprotonation, increasing lipid polarity and lowering chain melting transition temperatures. As well as the reproducibility of the experimental π-a relationship, the equation of state can be applied to interpret thermodynamic and mechanical properties of bilayer vesicles of the same lipid molecule. The monolayer-bilayer correspondence problem, i.e., “under what conditions can a monolayer at the air/water interface or oil-water interface and one leaflet of bilayer vesicles in water be considered mechanochemically equivalent”, has remained a subject of controversy for the past decades. Some of investigators have concluded that the monolayer
Application of an Equation of State
Figure 5. The experimental data of surface pressure versus surface area for monolayers of short-chain PC molecules at the air/water interface are fitted by the equation of state. Dots represent the experimental data, and solid curves represent the corresponding to theoretical predictions of the equation of state with the fitting parameter values: 1, C16:0/C10:0; 2, C16:0/C12:0; 3, C16:0/C14:0; 4, C16:0/C14:1,n-7; 5, C12:0/C18:1,n-9; 6, C14:0/C18:1,n-9; 7, C12:0/C12:0; 8, C14:0/C14:0.
Langmuir, Vol. 16, No. 19, 2000 7409
Figure 7. The experimental data of surface pressure versus surface area for monolayers of C16:0/C22:x PC molecules at the air/water interface are fitted by the equation of state with the values of parameters. Dots represent the experimental data, and solid curves represent the corresponding to theoretical predictions of the equation of state with the fitting parameter values: 1, C16:0/C22:1; 2, C16:0/C22:2; 3, C16:0/C22:3; 4, C16:0/C22:4; 5, C16:0/C22:5; 6, C16:0/C22:6.
Discussion
Figure 6. The experimental data of surface pressure versus surface area for monolayers of C16:0/C18:x and C18:x/C18:x PC molecules at the air/water interface are fitted by the equation of state. Dots represent the experimental data, and solid curves represent the corresponding to theoretical predictions of the equation of state with the fitting parameter values: 1, C16:0/ C18:1,n-9; 2, C16:0/C18:2,n-6; 3, C16:0/C18:3,n-3; 4, C16:0/C18:3,n-6; 5, C18:1/C18:1; 6, C18:2/C18:2.
at surface pressures in the 30-35 mN/m range produces conditions similar to those found in each half of the bilayer,17,27-31 while others believed that the monolayer at collapse pressure corresponds to the bilayer state at the equilibrium hydration.9,18,32,33 In any case, this issue can be resolved if we develop a proper approach to compare monolayer and bilayer data. Although only two of the lipids studied here have been investigated by micropipet aspiration techniques, they allow us to compare the properties of monolayers and bilayers at similar conditions. The monolayer Ka* values of C18:0/C18:1,n-9 PC and C14:0/C14:0 at the collapse pressure are 173 and 126 mN/m, respectively, whereas the corresponding unstressed bilayer values are 180 ( 2 and 144.9 ( 10.5 mN/m,34,35 respectively. It is clear that the elastic modulus data of monolayers at the collapse pressure are quite close to those of the tension free bilayers. This agreement in the area modulus for PC monolayers and bilayer vesicles, therefore, strongly supports the monolayer-bilayer correspondence theory, which asserts that the monolayer at its collapse pressure mechanochemically corresponds to a leaflet of large bilayer vesicles, which are tension free in equilibrium.14,17,18
The present study applies a recently developed equation of state, which can exactly calculate the mixing entropy of the lipid headgroups with water molecules at the interface and the conformational entropy of the hydrocarbon chains with different degree of chain unsaturation. The objective is to know the structure-function relation of the monolayer and to calculate various properties, especially area elastic modulus of the monolayer of various lipids of almost all natural double bonds. The method is to fit the proposed equation of state to the experimental data, which are obtained from the literature.25 Although the π-a curves were found to be quite repeatable for most lipids with different chain unsaturation, which include dilaureoylphosphatidylethanolamine (DLPE), dipalmitoylphosphatidylcholine (DPPC), 1-palmitoyl-2-oleoylphosphatidylcholine (POPC), 1-palmitoyl-2-linoleoylphosphatidylcholine (PLPC), and 1-palmitoyl-2R-linolenoylphosphatidylcholine (PLnPC), it was quite common to have discrepancy of the experimental π-a measurement. The differences were usually ascribed to structural fault of the trough, e.g., leakage between the barrier and the trough wall, and to contamination of the monolayer. Another problem is the possible effect of the lipid solubility in the substrate water. Recent reports have demonstrated that although sparing, the solubility of lipids in the bulk water phase could make significant differences to the measured π-a isotherms.43-45 This effect, however, can be excluded by a sophisticated analysis, which can obtain the lipid concentration in the bulk water.45 Different amount of lipids can be spread over the air/water interface under a constant surface pressure. The plot of molecular area of lipids vs number of the lipid molecules turned out to be a straight line. Its slope represents the molecular area of the lipids under such a given pressure while the intersection represents the lipid solubility in the substrate water.45 For each straight line, a pressure-area datum pair is obtained. If the value of the constant pressure is changed, many straight lines can be made, which actually provides another way to draw the π-a curve of the lipid. In the present study, since the π-a curves used were widely applicable and they were found to be quite repeatable, the problem of the lipid solubility was not raised. From the earlier work involving a few selected PC molecular species, it is obvious that the mechanochemical
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properties of lipid assemblies, such as the lateral compressibility or its inverse function, the area dilation modulus, express the structure features of lipids in very sensitive ways.36,37 The micropipet aspiration technique is a particularly effective method to investigate the elasticity of lipid bilayer vesicles. It provides insightful information, regarding not only the bending and shear forces that bilayers can withstand but also the changes in the in-plane elastic packing interactions that occur in lipid bilayers as acyl chain composition is varied or as cholesterol component is involved.38,39 However, this method requires the lipids to form large, stable bilayer vesicles. Its utility is therefore limited with respect to lipid type and mixing compositions. The Langmuir film balance technique for lipid monolayers is an alternative means for investigating the inplane elastic packing interactions of lipids. An important advantage of using the monolayer approach is that subtle alterations in intermolecular behavior can be detected under various conditions to avoid mesophasic structural changes that often occur in bilayer model membrane when the lipid phase state is changed. However, there are still some limitations in the monolayer approach. For instance, it was found that accurate experimental measurement of elastic modulus is possible only for monolayer at the surface pressure up to approximately 35 mN/m. The π-a measurement will be disrupted at high surface pressure due to technical difficulties in the trough construction. Therefore, the elastic modulus values at high surface pressures cannot be obtained directly from experimental data. An extrapolation of the measured π-a isotherm is thus needed. In our case, the collapse pressure obtained from the experimental measurement is definitely beyond that pressure range. The elastic modulus, therefore, can only be obtained by fitting the π-a isotherms to a monolayer equation of state. The common perception is that a greater number of cis double bonds would lead to a larger minimal molecular area, am. However, the results showed that this is not entirely true. Although an increase in the degree of unsaturation does initially increase the minimal molecular area, this effect tends to reach a peak at around two to three double bonds. Subsequent increases in the degree of unsaturation did not affect the minimal area much, and in fact even may decrease it slightly due to new kinks in the chain, which may cancel, in certain degree, the effects of the previous kinks. Our finding agrees with the study conducted by Evans et al.26 Of greater interest and significance, however, is the effect of the degree of unsaturation on the lateral area elastic modulus, Ka*. Our results indicate that an increase in the degree of unsaturation from one up to three cis double bonds tends to increase the value of Ka* and to make the film less rather than more compressible. This is true for the 1-palmitoyl PC series with 18 carbon atoms, the values of Ka* being 152, 156, and 164 mN/m for C16: 0/C18:1,n-9, C16:0/C18:2,n-6, and C16:0/C18:3,n-3 respectively. Likewise, in the case of the 1-palmitoyl PC series with 22 carbon atoms, the values of Ka* are 155, 167, 165, 138, 146, and 144 mN/m for C16:0/C22:1,n-9, C16:0/C22:2,n-6, C16:0/ C22:3,n-3, C16:0/C22:4,n-6, C16:0/C22:5,n-6, and C16:0/C22:6,n-3, respectively. An increase in the number of double bonds therefore decreases the compressibility of the monolayer membrane relative to that of C16:0/C22:1 until the fourth and subsequent double bonds are added. The results above
Feng et al.
make it evident that the effects of chain unsaturation on am are less dominant past its peak at two to three double bonds, where the effects of a gentler π-a isotherm begin to exert themselves. Another important aspect is the effect of the position of cis double bonds on the value of Ka*. The value of Ka* is much smaller for C16:0/C18:3,n-6 (138 mN/m) than for C16:0/C18:3,n-3 (164 mN/m), despite having a larger value of am (i.e., 74.3 vs 71.4 Å2). In fact, it dips below that of C16:0/C18:1,n-9, C16:0/C18:2,n-6, and C16:0/C18:3,n-3. This emphasizes that double bonds introduced toward the center of fatty acid chains create more disorder and disrupt intraand intermolecular interactions to a greater extent than those in other positions. This gives rise to a much gentler π-a isotherm and to a much more compressible film. The effect of (dπ/da)a* is hence able to significantly override that of am, resulting in a more compressible film with a lower elastic modulus. Conclusion We applied a recently developed equation of state for lipid monolayers at the air/water interface, which explicitly calculates the mixing entropy of the water and lipid molecules at the interface and the conformational energy of the chains with or without chain unsaturation, to investigate effects of the number and position of double bonds of the hydrocarbon chain on the pressure-area behavior and membrane compressibility of the monolayer. We found that the equation of state can reproduce most experimental pressure-area isotherms (π-a curves) of lipid monolayers at the air/water interface with appropriate values of the three parameters, which were obtained by fitting the experimental data to the equation of state. The interfacial elastic modulus of monolayers at their collapsed pressure increases with the number of double bonds until a peak is reached at the second or the third double bond and decreases with the chain length for monolayers with the same degree of unsaturation. Furthermore, the interfacial elastic modulus is lowered if the double bonds are located in the middle of the hydrocarbon chain. We compared the calculated data of the area modulus of lipid monolayers to the experimental value of large bilayer vesicles. The results obtained show that, as predicted by the monolayer-bilayer correspondence theory,14,17,18 the monolayer at its collapse pressure can be considered to be mechanochemically corresponding to a leaflet of large bilayer vesicles. Bilayer vesicle properties can thus be elucidated by the monolayer measurement, which can be easily carried out under various conditions. The monolayer may thus be applied to understand the behavior of bilayer vesicles, which are widely used either as a model biomembrane in experiment or in practical applications such as drug delivery, gene therapy, and artificial blood. Acknowledgment. The authors are greatly indebted to the unknown reviewers for their constructive comments and suggestions as well as for bringing to us quite a few critical references. This work is supported by NUS Research Grant RP 970637 and IMRE Equipment Fund. G.-H. Zhu thanks NUS for providing a research scholarship. LA990781L
