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Langmuir 2006, 22, 2916-2919
Comments Comment on 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
There is recurrent interest in the question as to what surface pressure in lipid monolayers best represents the corresponding state in a lipid bilayer vesicle. Such phospholipid vesicles are valuable models for the lipid assembly in biological membranes and, among other applications, serve as a vehicle for drug delivery. Phospholipid monolayers offer an appropriate means to explore the equation of state in these systems, if an equivalence between the two can be established convincingly. My attention was drawn recently to an article by Feng1 on monolayer-bilayer correspondence that was published some time ago in Langmuir. In this article, the author adopts an approach that is based on work by Ja¨hnig2 but which deviates in two important aspects from a similar treatment that I published in an earlier review on lateral pressure in membranes.3 Rather remarkably, the two approaches, although mutually inconsistent, appear to arrive at a similar answer, and one that is in agreement with available experimental data. The purpose of the present comment is to draw attention to a thermodynamic inconsistency in the paper by Feng,1 the consequence of which is that the treatment used in that article would not lead to an equivalence pressure that agrees with experiment, nor even to a viable monolayer-bilayer equivalence. The origins of this discrepancy with experiment are then identified, and the ways in which subsequent results of the article must be modified are presented. Wherever possible without internal contradiction, the notation of Feng’s paper1 is adhered to here, even where the use of corresponding symbols for bilayer and monolayer might be misleading. A notation that preserves real distinctions between bilayers and monolayers is used in my review.3 The problem lies with two incompatible definitions of the variable σm in ref 1, which properly speaking should be the surface tension of the interface in the presence of the monolayer (as opposed to the surface tension of the monolayer). Textbook derivations of Feng’s eq 5 (see, e.g., refs 4 and 5) show clearly that the free energy involved is that of the monolayer alone and therefore is identical to that given by eq 11 of the same article. The inconsistency in ref 1 is then seen immediately by comparing the definition of monolayer surface pressure, πm, given by eq 5 with the definition of σm in eq 13. The result is πm ) -Fm, i.e., that the surface pressure of the monolayer is simply the negative of the surface tension of the interface in the presence of the monolayer; this is obviously wrong. In consequence, the * E-mail:
[email protected]. Tel: +49 551 201 1285. Fax: +49 551 201 1501. (1) Feng, S. Langmuir 1999, 15, 998. (2) Ja¨hnig, F. Biophys. J. 1984, 46, 687. (3) Marsh, D. Biochim. Biophys. Acta 1996, 1286, 183. (4) Defay, R.; Prigogine, I. Surface Tension and Adsorption; Longmans: London, 1966. (5) Aveyard, R.; Haydon, D. A. An Introduction to the Principles of Surface Chemistry; Cambridge University Press: Cambridge, U.K., 1973.
fundamental equation for monolayer-bilayer correspondence of ref 1 (i.e., eq 12) is incorrect.
Definition of Monolayer Surface Pressure at the Air-Water Interface I now develop the treatment given in ref 1, but with compatible definitions of the surface pressure and interfacial tension. Let us start with the textbook definition of eq 3 from ref 1 for the free energy, Fm, of the lipid monolayer
F m ) Fs - F0
(1)
where Fs and F0 are the free energies of the interface in the presence and absence of the monolayer, respectively. It is understood here that the free energies are surface excess quantities. With appropriate definition of the Gibbs dividing surface, Fm reflects only the presence of the insoluble monolayer film and is zero in the absence of the monolayer (see ref 5). Taking the partial derivative of eq 1 with respect to interfacial area, for a closed system at constant temperature, immediately yields the familiar Langmuir definition of monolayer surface pressure4
πm ) γa/w - σm
(2)
where γa/w is the surface tension of the bare air-water interface and σm is the surface tension of the air-water interface in the presence of the monolayer. (This is eq 1 of ref 1.) Following ref 1 further, we now introduce eq 11 from that publication. This represents the partitioning of the monolayer free energy into four components that was suggested originally by Ja¨hnig2
Fm ) Fa/o + Fphob + Fphil + Fint
(3)
where Fphob is the free energy of interaction of the lipid chains with water, Fphil is the hydrophilic free energy of interaction of the lipid headgroups with water, Fint is the free energy of interaction between lipids in the monolayer, and Fa/o is the free energy of interaction of the lipid molecules with the upper phase (i.e., with air). Each term on the right side of eq 3 is linear in the number of lipid molecules, N, in the monolayer (cf. ref 3). Therefore, Fm as defined in eq 3 goes to zero in the absence of the lipid monolayer, as is required by eq 1. Taking the partial derivative of eq 3 with respect to interfacial area, for a closed system at constant temperature, yields the following expression for the monolayer surface pressure
πm ) πphil + πint - γ˜ o/w - γ˜ a/o
(4)
where πphil and πint are components of the repulsive surface pressure that arise from interactions of water with the lipid headgroups and interactions between lipids, respectively; γ˜ o/w is the microscopic hydrophobic free-energy density for lipid chainwater interaction, and γ˜ a/o is the microscopic free-energy density for interaction of the lipids with air. In eq 4, as in eq 2, the
10.1021/la051216n CCC: $33.50 © 2006 American Chemical Society Published on Web 02/16/2006
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Langmuir, Vol. 22, No. 6, 2006 2917
monolayer surface pressure is defined by πm ) -(∂Fm/∂A)N,T, which corresponds to eq 5 in ref 1. (See, for example, refs 4 and 5.)
Monolayer-Bilayer Correspondence The effective surface pressure in a monolayer leaflet of a lipid bilayer is defined in ref 1 by
πb ) πphil + πint
(5)
This definition is essentially in accord with that of previous authors.2,3,6 To establish a valid monolayer-bilayer correspondence, it is necessary that the terms on the right side of eq 5 be defined similarly to those on the right side of eq 4, and that all other (“microscopic”) terms appearing in eq 4 correspond to molecular interactions and not to macroscopic interfacial tensions. From eqs 4 and 5, the relation between monolayer and bilayer surface pressures then becomes
πm ) πb - γ˜ o/w - γ˜ a/o
(6)
at the same area per lipid molecule in the monolayer and bilayer. This is very different from the result obtained originally in ref 1, viz., that πm and πb correspond to one another at the same area/molecule, a. In ref 1, it was assumed that the microscopic free-energy densities, γ˜ o/w and γ˜ a/o, in eq 6 are equal to the macroscopic interfacial tensions, γo/w and γa/o, for an oil-water and an air-oil interface, respectively. If the latter were the case, the discrepancy is large and would amount to ca. 72 mN‚m-1. The reason that eq 6 also differs from the conclusions reached in my review3 lies in the interpretation of the contributions to the free energy in eq 3. I reasoned that the contribution of hydrophobic interactions to the free energy of a monolayer at the air-water interface is negligible because the lipid chains will escape energetically unfavorable contacts with water by extending into the upper air phase. Unlike a bilayer, which is immersed in water, there is no energetic or packing requirement that the lipid chains should contact water in a monolayer. Secondly, the Fa/o term in eq 3 is small because it corresponds to the weak van der Waals interactions of air with the lipid chains. Therefore, γ˜ o/w should not enter into eq 6, and the contribution from the freeenergy density of van der Waals interaction with air, γ˜ a/o, is small: it certainly does not correspond to a macroscopic interfacial tension. With this interpretation, πm and πb in eq 6 then become comparable, as required for bilayer-monolayer equivalence. Hence, πm = πb is the correct version of eq 6. Alternatively, for a monolayer at an oil-water interface, the lipid chains are immersed in oil and do not come into contact with water. Therefore, the term γ˜ o/w does not enter into eq 6. The interaction of the lipid chains with the upper phase (i.e., oil) in eq 6 will be larger than that at the air-water interface (γ˜ a/o) but comparable to that for the interaction between monolayers in a lipid bilayer. Thus, the equivalence πm = πb is expected also for a monolayer at the oil-water interface. It will be be seen below (in Figure 1) that results in agreement with experiments on bilayers are obtained from comparison with monolayers both at the airwater interface and at the oil-water interface. Ever since the fundamental work of Evans and Waugh6 on the effective surface pressure in membranes, it is accepted that the tension in a bilayer membrane leaflet is given by
σb ) γ˜ o/w - πb (6) Evans, E. A.; Waugh, R. J. Colloid Interface Sci. 1977, 60, 286.
(7)
Figure 1. Surface tension of a monolayer (σm) and of one leaflet of a bilayer (σb) as a function of area/lipid molecule, a, deduced from the π-a isotherms of dilauroyl phosphatidylethanolamine (DLPE) at the air-water interface at 44.2 °C (solid lines, ref 7) and of dioleoyl phosphatidylcholine (DOPC) at the oil-water interface at 3.1 °C (dashed lines, ref 12).
where γ˜ o/w ) (∂Fphob/∂A)N,T is the microscopic hydrophobic freeenergy density. (This is eq 10 of ref 1.) The equilibrium condition for a tension-free bilayer in its natural state is therefore πb(a*) ) γ˜ o/w, where a* is the equilibrium area per lipid molecule in a bilayer. The latter expression corresponds to the monolayerbilayer equivalence pressure, which is in the region of 30-35 mN‚m-1,3 but this is not the result that would be predicted from eq 6 as given above. It also does not correspond to the collapse pressure of the monolayer, as was suggested in ref 1. Figure 1 (solid lines) compares the tension in a lipid bilayer leaflet predicted by eq 7 with the surface tension of an air-water interface in the presence of a lipid monolayer at the same area per lipid molecule, a, for dilauroyl phosphatidylethanolamine (DLPE) at 44.2 °C. As in ref 1, these values are obtained from the experimental π-a isotherm of DLPE that is given in ref 7. The surface tension, σm, of the monolayer-covered interface that is given in Figure 1 reproduces the original experimental data by using eq 2 with γa/w) 72 mN‚m-1 and is identical to that presented in ref 1. The tension, σb, in the bilayer leaflet that is obtained by using γ˜ o/w ≈ 32 mN‚m-1 for the hydrophobic freeenergy density, or equivalence pressure, however, is considerably smaller than that calculated in ref 1. In particular, the equilibrium area/lipid molecule at which the bilayer tension is zero, a* ) 0.52 nm2 (Figure 1), is similar to that measured for DLPE bilayers at 35 °C (0.51 nm2) by X-ray diffraction8,9 and to that for the ether analogue didodecyl phosphatidylethanolamine at 45 °C (0.55 nm2), measured also by X-ray diffraction.10 Thus, at the equivalence pressure of γ˜ o/w ≈ 32 mN‚m-1, the area/lipid in the DLPE monolayer corresponds to that in tension-free bilayers, as required for valid monolayer-bilayer equivalence. In contrast, the value of a* ) 0.46 nm2 that is obtained at σb ) 0 for DLPE in ref 1 is not in accord with X-ray diffraction results. This disagreement with experiment arises from an inappropriate choice for the value of γ˜ o/w. It should be noted that the range of areas per lipid molecule that is given in Figure 1 corresponds to that over which the (7) Mo¨hwald, H. Annu. ReV. Phys. Chem. 1990, 41, 441. (8) McIntosh, T. J.; Simon, S. A. Biochemistry 1986, 25, 4948. (9) Nagle, J. F.; Tristram-Nagle, S. Biochim. Biophys. Acta 2000, 1469, 159. (10) Seddon, J. M.; Cevc, G.; Kaye, R. D.; Marsh, D. Biochemistry 1984, 23, 2634.
2918 Langmuir, Vol. 22, No. 6, 2006
Comments
monolayer π-a isotherm was measured. For lipid bilayer vesicles, however, such high values of area/molecule cannot be achieved practically before the applied tension required causes vesicle rupture. Typically, area extensions of only up to ca. 5% can be sustained before rupture of giant vesicles.11 For comparison with DLPE, the dashed lines in Figure 1 give corresponding data deduced from the π-a isotherm of dioleoyl phosphatidylcholine (DOPC) at the oil-water interface at 3.1 °C.12 The latter isotherm was used for illustrative purposes in my review.3 The surface tension in the presence of the DOPC monolayer is much less than that for DLPE, in large part because of the different interfaces: oil-water as opposed to air-water. The area/lipid at which the bilayer tension is predicted to be zero, a*) 0.71 nm2 (Figure 1), is greater than that for DLPE because of the unsaturated chains and greater headgroup hydration of DOPC (see, e.g., ref 13). Again, the value obtained for a* is in good agreement with X-ray diffraction: measurements on DOPC bilayers yield a* ) 0.70 nm2 at 2 °C14 and a* ) 0.72 nm2 at 30 °C.9
Chemical Potential and Activity Expressions for the monolayer and bilayer thermodynamic functions, in terms of monolayer π-a isotherms, that are obtained in ref 1 are not entirely dependent on the inconsistencies in definition of surface pressure. Given the correct equations, the numerical values are, nevertheless, still affected by the values assumed for γ˜ o/w and γ˜ a/o in ref 1 and consequently by the values derived for the equilibrium area/lipid, a*, in a tension-free bilayer, as discussed already above. As derived by Ja¨hnig2 and used in refs 1 and 3, the chemical potential of a lipid in a bilayer, µb ) (∂Fb/∂N)a,T, is equal also to the free energy per lipid, φb ) Fb/N, where N is the number of lipid molecules. This is because the area/molecule, and not the total area, remains constant on introducing an additional lipid molecule (i.e., the bilayer tension remains constant). Using the definition of bilayer tension, σb ) (∂Fb/∂A)N,T, eq 7 can be rewritten as
( ) ∂φb ∂a
N
) π(a*) - π(a)
(8)
where the substitution πb(a*) ) γ˜ o/w has also been made. This is eq 40 of ref 1, and subscripts for π are omitted to indicate monolayer-bilayer equivalence. Integration of eq 8 then yields
µb(a) - µb(a*) ) φb(a) - φb(a*) ) π(a*) (a - a*) -
∫a*a π(a) da
(9)
for the lipid chemical potential in the bilayer, which is eq 41 of ref 1. The free energy per lipid in a monolayer, on the other hand, is simply given by
φm(a) - φm(a*) ) -
∫a*a π(a) da
(10)
This standard thermodynamic result follows immediately from the definition of monolayer surface pressure, πm ) -(∂Fm/∂A)N,T, but it is not that used in ref 1. (11) Needham, D.; Nunn, R. S. Biophys. J. 1990, 58, 997. (12) Yue, B. Y.; Jackson, C. M.; Taylor, J. A. G.; Mingins, J.; Pethica, B. A. J. Chem. Soc., Faraday Trans. 1 1976, 72, 2685. (13) Cevc, G.; Marsh, D. Phospholipid Bilayers: Physical Principles and Models; Wiley-Interscience: New York, 1987. (14) Gruner, S. M.; Tate, M. W.; Kirk, G. L.; So, P. T. C.; Turner, D. C.; Keane, D. T.; Tilcock, C. P. S.; Cullis, P. R. Biochemistry 1988, 27, 2853.
Figure 2. Chemical potential (left-hand ordinate) of lipids in a monolayer (µm) and in a bilayer (µb) deduced from the π-a isotherms of DLPE at the air-water interface at 44.2 °C (solid lines, ref 7) and of DOPC at the oil-water interface at 3.1 °C (dashed lines, ref 12). Values at area/lipid molecule a are given relative to those at area/lipid molecule a*, which corresponds to the area/lipid in a tension-free bilayer. The free energy per lipid, φb, for a bilayer (right-hand ordinate) is identical to µb, whereas for a monolayer φm differs considerably from µm.
Unlike the situation for the bilayer, the chemical potential in the monolayer, µm ) (∂Fm/∂N)A,T, is not equal to the mean free energy per lipid. Allowance must be made for the change in area/molecule at constant total area of the monolayer2
µm(a) ) φm(a) - a
( )
∂φm ) φm(a) + π(a)a ∂a
(11)
where it is important to retain the subscript for φ (compare eq 9 with eq 10). Referred to the area/lipid at the equivalence pressure, the chemical potential in the monolayer is therefore given by
µm(a) - µm(a*) ) φm(a) - φm(a*) + π(a)a - π(a*)a* (12) Substitution from eq 10 then yields
µm(a) - µm(a*) ) π(a)a - π(a*)a* -
∫a*a π(a) da
(13)
As a result of a fortuitous canceling error, eq 42 of ref 1 agrees with this result, although the expression for the free energy, φm(a) - φm(a*), in ref 1 does not. Figure 2 gives the free energy per lipid and the lipid chemical potential for lipid bilayers and monolayers at the same area/ lipid. These results are obtained from the π-a isotherms of DLPE (solid lines) and DOPC (dashed lines) monolayers by using eqs 9, 10, and 13. The results for the chemical potential of DLPE in the bilayer (µb) and in the monolayer (µm) are similar to those obtained in ref 1, except that the point at which both are equal is specified by the more realistic value for the bilayer equilibrium area/lipid of a* ) 0.52 nm2 that was deduced from Figure 1. As noted already, the agreement for the monolayer results from compensating errors. This is seen very clearly from the values of the monolayer free energy/lipid, φm(a) - φm(a*), which are totally different from those in ref 1. Whereas the free energy per lipid and the chemical potential decrease continuously with increasing area per lipid in monolayers, both values in bilayers have a minimum at the equilibrium area/lipid. This is
Comments
Langmuir, Vol. 22, No. 6, 2006 2919
because, in lipid bilayers, the energetically unfavorable intermolecular repulsions dominate for a < a*, on the one hand, and the energetically unfavorable exposure of lipid chains to water dominates for a > a*, on the other hand. Yet again, this is an illustration of Tanford’s principle of opposing forces.15 In contrast, as already discussed, the hydrophobic effect is absent in lipid monolayers. To obtain an expression for the lipid activity, R, it is helpful to go back to the definition of chemical potential in a monolayer (see, e.g., ref 5, and cf. eq 11)
µm )
( ) ( ) ( )( ) ∂Fm ∂N
)
A
∂Fm ∂N
-
π
∂Fm ∂A
N
∂A ) ∂N π o µm + kT ln Rm + πa (14)
where µ°m is the standard monolayer chemical potential and the partial molecular area is assumed to be equal to the area/molecule, a. Expressing the chemical potential once more relative to that at area/lipid a* therefore gives
µm(a) - µm(a*) ) kT ln
Rm(a) Rm(a*)
+ π(a)a - π(a*)a* (15)
which differs significantly in the last term on the right from eq 46 that is given in ref 1. (Note, however, that eq 46 in ref 1 would differ yet more radically from eq 15 if the contentious first definition in eq 13 of the same article had been adopted.) Combining eqs 13 and 15 finally leads to
Rm(a) Rm(a*)
(
) exp -
)
∫a*a π(a) da kT
(16)
which expresses the lipid activity in terms of the π-a isotherm. As might be anticipated, this differs from the corresponding result (eq 48) that was obtained in ref 1. Unlike the monolayer, the bilayer leaflet does not experience a net lateral pressure but only an applied lateral tension. Hence, the analogue of eq 14 for a bilayer must be expressed in terms of the bilayer leaflet tension, σb. (This difference from monolayers, discussed also in ref 3, was not recognized in ref 1.) The chemical potential in the bilayer is hence given by
µb )
( ) ( ) ( )( ) ∂Fb ∂Fb ) ∂N a ∂N
-
σb
∂Fb ∂A
N
∂A ∂N
)
σb
µob + kT ln Rb - σba (17) The bilayer analogue of eq 15 then becomes
Rb(a) - σb(a)a ) Rb(a*) Rb(a) kT ln + (π(a) - π(a*))a (18) Rb(a*)
µb(a) - µb(a*) ) kT ln
where we have used σb ) π(a*) - π(a) ) 0 for a bilayer in the (15) Tanford, C. The Hydrophobic Effect; Wiley: New York, 1980.
Figure 3. Chemical activity of lipids in a monolayer (Rm) and in a bilayer (Rb) deduced from the π-a isotherms of DLPE at the air-water interface at 44.2 °C (solid lines, ref 7) and of DOPC at the oil-water interface at 3.1 °C (dashed lines, ref 12). Values at area/lipid molecule a are given relative to those at area/lipid molecule a*, which corresponds to the area/lipid in a tension-free bilayer.
natural state with a ) a*. Combining eqs 9 and 18 leads to
Rb(a) Rb(a*)
) exp
(
π(a*)(2a - a*) - π(a)a kT
)
∫a*a π(a) da
(19)
for the lipid activity in a bilayer leaflet. Understandably, this is not the expression (eq 49) that was obtained in ref 1. Figure 3 gives the lipid activity in bilayers and monolayers, at the same area/molecule, that is obtained from the π-a isotherms of DLPE (solid lines) and DOPC (dashed lines) monolayers by using eqs 16 and 19. The results for the chemical potential of DLPE in the bilayer (Rb) and in the monolayer (Rm) differ considerably from those obtained in ref 1. The difference is not simply that the point at which bilayer and monolayer activities are equal is specified by the bilayer equilibrium area/lipid of a* ) 0.52 nm2, as opposed to a* ) 0.46 nm2. Rather, it corresponds to certain fundamental differences in the definition of chemical potential, as already discussed. Lipid activity in the monolayer decreases continuously with increasing area/molecule. This is at least partly a dilution effect. In contrast, lipid activity in the bilayer increases sharply and monotonically with increasing area/ molecule. Unlike the prediction of ref 1, and also unlike the chemical potential, the activity in the bilayer is not biphasic and does not exhibit a minimum. This represents the dominating effect of the extensive bilayer tension, which potentiates the incorporation of lipid molecules. Derek Marsh*
Max-Planck-Institut fu¨r Biophysikalische Chemie, Abt. Spektroskopie, Am Fassberg 11, 37077 Go¨ttingen, Germany ReceiVed May 6, 2005 In Final Form: NoVember 16, 2005 LA051216N
