Electrostatic Model for Mixed Cationic−Zwitterionic Lipid Bilayers

Langmuir 2006, 22, 5129-5136

5129

Electrostatic Model for Mixed Cationic-Zwitterionic Lipid Bilayers Emmanuel C. Mbamala,† Alfred Fahr,‡ and Sylvio May*,† Department of Physics, North Dakota State UniVersity, Fargo, North Dakota 58105-5566, and Department of Pharmaceutical Technology, Friedrich-Schiller-UniVersita¨t Jena, Lessingstrasse 8, D-07743 Jena, Germany ReceiVed January 18, 2006. In Final Form: February 22, 2006 The current interest in mixed cationic-zwitterionic lipid membranes derives from their potential use as transfer vectors in nonviral gene therapy. Mixed cationic-zwitterionic lipid membranes have a number of structural properties that are distinct from the corresponding anionic-zwitterionic lipid membranes. As known from experiment and reproduced by computer simulations, the average cross-sectional area per lipid changes nonmonotonically with the mole fraction of the charged lipid, passing through a minimum at a roughly equimolar mixture. At the same time, the average orientation of the zwitterionic headgroup dipoles changes from more parallel to the membrane plane to more perpendicular. We suggest a simple mean-field model that reveals the physical mechanisms underlying the observed structural properties. To backup the mean-field calculations, we have also performed Monte Carlo simulations. Our model extends Poisson-Boltzmann theory to include (besides the cationic headgroup charges) the individual charges of the zwitterionic lipid headgroups. We model these charges to be arranged as dipoles of fixed length with rotational degrees of freedom. Our model includes, in a phenomenological manner, the changes in steric headgroup interactions upon reorientation of the zwitterionic headgroups. Our numerical results suggest that two different mechanisms contribute to the observed structural properties: one involves the lateral electrostatic pressure and the other the zwitterionic headgroup orientation, the latter modifying steric headgroup interactions. The two mechanisms operate in parallel as they both originate in the electrostatic properties of the involved lipids. We have also applied our model to a mixed anionic-zwitterionic lipid membrane for which neither a significant headgroup reorientation nor a nonmonotonic change in the average lateral cross-sectional area is predicted.

Introduction Cationic liposomes are increasingly being considered as alternative transfection vectors in gene delivery therapy, the introduction of genetic materials (e.g., DNA) into cells for disease treatment.1-3 As opposed to virus-based methods, nonviral vectors offer nonimmunogenity and a tunable capacity to carry various amounts of DNA.4-6 Cationic liposomes (loaded with carboranes) have also been reported to be efficient boron delivery agents in boron neutron capture therapy (BNCT), an anti-cancer treatment based upon neutron absorption by 10B nuclei.7-9 In the procedure for cationic lipid-mediated transfection, the key component, the cationic lipid, serving as the condensing agent of the negatively charged DNA, is usually mixed with an appropriate helper lipid in order to increase the transfection efficiency and reduce toxicity.10-12 Among the commonly used helper lipids are zwitterionic lipids, such as phosphatidyletha* To whom correspondence should be addressed. † North Dakota State University. ‡ Friedrich-Schiller-Universita ¨ t Jena. (1) de Gennes, P.-G. Physica A 1999, 274, 1. (2) Lange, R. Nature 1998, 392, 5. (3) Mo¨nko¨nen, J.; Urtti, A. AdV. Drug DeliVery ReV. 1998, 34, 37. (4) Spector, M. S.; Schnur, J. M. Science 1997, 275, 791. (5) Paukku, T.; Lauraeus, S.; Huhtaniemi, I.; Kinnunen, P. L. J. Chem. Phys. Lipids 1997, 87, 23. (6) Felgner, P. L.; Gadek, T. R.; Holm, M.; Roman, R.; Chan, H. W.; Wenz, M.; Northrop, J. P.; Ringold, G.; Danielsen, M. Proc. Natl. Acad. Sci. U.S.A. 1987, 84, 7413. (7) Larsson, B. In AdVances in neutron capture therapy; Crawford, J., Weinreich, R., Eds.; Elsevier: Amsterdam, 1997. (8) Soloway, A. H.; Tjarks, W.; Barnum, B. A.; Rong, F. G.; Barth, R. F.; Codogni, I. M.; Wilson, J. G. Chem. ReV. 1998, 98, 1515. (9) Barth, R. F. J. Neuro-Onchology 2003, 62, 1. (10) Hui, S. W.; Langner, M.; Zhao, Y. L.; Ross, P.; Hurley, E.; Chan, K. Biophys. J. 1996, 71, 590. (11) May, S.; Harries, D.; Ben-Shaul, A. Biophys. J. 2000, 78, 1681. (12) Ristori, S.; Oberdisse, J.; Grillo, I.; Donati, A.; Spalla, O. Biophys. J. 2005, 88, 535.

nolamine (PE) or phosphatidylcholine (PC).10,13 The preferred equilibrium morphology of a cationic lipid-DNA complex (lipoplex) is dictated by the surface charge density and elastic properties of the constituent lipid layers. Both of these characteristics depend, in turn, on the nature and composition of the cationic-zwitterionic lipid mixture. Modeling the structural properties of mixed cationic-zwitterionic liposomes is the subject of the present study. Specifically, we address the orientational behavior of the zwitterionic lipid headgroups (a frequently ignored property in theoretical studies11,14,15), in the presence of the cationic component. We find that the dipole moment vector (directed from phosphate to the nitrogen, P-N) of the zwitterionic lipid has an average orientation more parallel to the membrane when the membrane is composed of pure zwitterionic lipids. This is well-known from recent experimental work based on an assortment of independent spectroscopic studies, including NMR, X-ray, and neutron scattering.16-21 Although this orientation persists over a wide range of conditions, it can also change to a significantly more perpendicular orientation when the composition of the membrane is dominated by the cationic component. One of our objectives is to investigate the corresponding physical mechanism. We will show that it can be explained on the basis of mean-field electrostatic (or, equivalently, Poisson-Boltzmann) theory if (13) Ryha¨nen, S. J.; Sa¨ily, M. J.; Paukku, T.; Borocci, S.; Maucini, G.; Holopainen, J. M.; Kinnunen, P. K. J. Biophys. J. 2003, 84, 578. (14) May, S.; Ben-Shaul, A. Biophys. J. 1997, 73, 2427-2440. (15) Mbamala, E. C.; Ben-Shaul, A.; May, S. Biophys. J. 2005, 88, 17021714. (16) Seelig, J.; Gally, H. Biochemistry 1976, 15, 5199. (17) Seelig, J.; Seelig, A. Q. ReV. Biophys. 1980, 13, 19. (18) Pearson, R. H.; Pascher, I. Nature 1979, 281, 499. (19) Philips, M. C.; Finer, E. G.; Hauser, H. Biochim. Biophys. Acta 1972, 290, 397. (20) Levine, Y. K.; Bailey, A. I.; Wilkins, M. H. Nature 1968, 220, 577. (21) Bu¨ldt, G.; Seelig, J. Biochemistry 1980, 19, 6170.

10.1021/la060180b CCC: $33.50 © 2006 American Chemical Society Published on Web 04/07/2006

5130 Langmuir, Vol. 22, No. 11, 2006

the dipolar nature of the zwitterionic lipid species is taken into account. Our theoretical model also predicts the headgroup reorientation to be absent if the cationic lipid is replaced with an anionic lipid species. The orientational changes of the zwitterionic lipid headgroups have structural consequences for the mixed membrane. The more parallel headgroup orientation tends to stretch the pure zwitterionic membrane due to strong steric headgroup interactions, leading to a large lateral cross-sectional area per lipid. The lateral area then contracts with addition of cationic lipids and expands again as the mole fraction of the cationic lipid increases further. These results have been obtained recently in a detailed (atomistic) molecular dynamics simulation by Gurtovenko el al.22 for a dimyristoylphosphatidylcholine/trimethylammonium (DMPC/ DMTAP) mixture, showing that the equilibrium average crosssectional area per lipid passes through a pronounced minimum as the composition of the cationic lipid is varied from zero to one. This observation agrees with experimental findings: Zantl et al.23 found for DMPC/DMTAP monolayers a minimum in the area per headgroup at a mole fraction of about φ ) 50% cationic lipids. Also for a system of POPC/SS-1 (SS-1 is a divalent cationic lipid), Sa¨ily et al.24 obtained a minimum in the area per lipid at about φ ) 38%. The minimum in the area per lipid has been interpreted to be a consequence of the change in the PCs headgroup from a more parallel orientation at low cationic lipid composition to a more perpendicular orientation at high cationic lipid composition. Interestingly, cationic-zwitterionic lipid mixtures were recently shown to form defect-free gel-phase supported bilayers,25 a phenomenon which may also relate to the minimum in the area per lipid. In a recent theoretical study, Levadny and Yamazaki26 suggested a model for a mixed cationic-zwitterionic membrane that is based on a lateral polarization of zwitterionic headgroups induced by the cationic lipids. That is, each cationic lipid forms a cluster with neighboring zwitterionic lipids. Due to ion-dipole attraction, these clusters condense and, thus, lower the average area per lipid in the membrane. This condensation also induces the zwitterionic lipids to reorient their headgroups toward the membrane normal direction. The model we develop in the present work suggests that direct electrostatic and steric interactions both contribute in a mutually dependent way to the observed structural membrane properties. The main objective of the present work is to suggest an extended Poisson-Boltzmann model for a mixed cationic-zwitterionic membrane that recovers the experimentally observed structural properties, including the reorientation of the zwitterionic lipid headgroups, the nonmonotonic behavior of the average crosssectional area per lipid, and the absence of these properties for a mixed anionic-zwitterionic membrane. Our extended PoissonBoltzmann model takes the dipolar character of the zwitterionic lipid headgroups into account, as well as the fact that one end of this dipole (the negatively charged phosphate group) is relatively immobile, whereas the other end (the positive charge of the quaternary saturated amine) has considerable conformational freedom. Rather than to model any specific lipid mixture, we keep our model as simple as possible so as to capture the generic properties of a mixed cationic-zwitterionic membrane (22) Gurtovenko, A. A.; Patra, M.; Karttunen, M.; Vattulainen, I. Biophys. J. 2004, 86, 3461. (23) Zantl, R.; Artzner, F.; Rapp, G.; Ra¨dler, J. O. Europhys. Lett. 1999, 45, 90-96. (24) Sa¨ily, V.; Ryha¨nen, S.; Holopainen, J. M.; Borocci, S.; Mancini, G.; Kinnunen, P. K. Biophys. J. 2001, 81, 2135-2143. (25) Zhang, L.; Spurlin, T. A.; Gewirth, A. A.; Granick, S. J. Phys. Chem. B 2006, 110, 33-35. (26) Levadny, V.; Yamazaki, M. Langmuir 2005, 21, 5677-5680.

Mbamala et al.

Figure 1. Schematic illustration of a mixed cationic-zwitterionic lipid membrane leaflet. The headgroup of the cationic lipid species carries a single positive charge, whereas the headgroup of the zwitterionic lipid consists of a single dipole of length l with spatially fixed negative-charge base and a mobile positive-charge head. The instantaneous orientation of the dipole is characterized by an angle θ with respect to the positive vertical axis (the z-axis). The positive charge of the dipole is not allowed to penetrate into the hydrocarbon chain region, implying 0 e θ e π/2. Note that w and hc are the dielectric constants of the aqueous and hydrocarbon phases, respectively.

with a minimum of adjustable parameters (in fact, there is none that cannot be reasonably estimated). Furthermore, to backup our mean-field calculations, we present Monte Carlo simulations of a mixed cationic-zwitterionic membrane that support our findings. Finally, we shall also apply our model to a mixed anioniczwitterionic membrane to illustrate the striking differences in the structural behavior of cationic and anionic bilayers. We proceed by describing our extended Poisson-Boltzmann model, followed by the details of the MC simulations. We then present and discuss the results of both theory and simulation.

Theory We employ a mean-field level description in which an extended Poisson-Boltzmann equation will be derived based on functional minimization of the electrostatic free energy. Together with appropriate boundary conditions, our procedure yields equilibrium thermodynamic functions that, upon insertion back into the free energy, predict the equilibrium free energy of the mixed membrane. We consider one single leaflet of a lipid bilayer, containing N lipids of which Nca ) φN are cationic and Nzw ) (1 - φ)N are zwitterionic, where φ is the composition (or mole fraction) of the cationic component. Each cationic lipid headgroup carries a single positive charge whereas each zwitterionic lipid headgroup consists of a single dipole. This dipole has a fixed negativecharge base moiety, and a mobile positive-charge end moiety which can rotate about the base moiety. The two individual charges of the headgroup dipole are separated by a fixed bond length l as schematically illustrated in Figure 1. The average crosssectional area per lipid al is assumed to be the same for both species; it may however vary with composition as discussed below. Because the system is invariant under rotation about the azimuthal angle, only the angle θ between the dipole and the membrane normal (the z axis) needs to be considered; confinement of the dipole’s positive charge implies 0 e θ e π/2. The membrane is in contact with an aqueous symmetric 1:1 salt solution characterized by the following parameters (definitions will follow shortly): the Debye screening length lD (or its inverse κ ) 1/lD) and the Bjerrum length lB, describing the strength of the bare Coulomb interactions in the aqueous medium of dielectric constant

Mixed Cationic-Zwitterionic Lipid Bilayers

Langmuir, Vol. 22, No. 11, 2006 5131

w ≈ 80 (region, z > 0). In contrast, the dielectric constant hc inside the membrane hydrocarbon core (z < 0) is much smaller; hc ≈ 2. The total (mean-field) electrostatic free energy Fel of the mixed membrane can be written as

Fel ) Fc + Ft + Fr

F t ) kB T

∫V dV

[

w 2

∫V dV (∇Φ)2

(1a)

]

n+ nn+ ln + n- ln - (n+ + n- - 2n0) n0 n0 (1b)

Fr ) NzwkBT

∫0π/2 dθ sin θ P(θ) ln P(θ)

(1c)

Fc in eq 1 is the energy stored in the electrostatic field -∇Φ where Φ is the corresponding electrostatic potential; the integration runs over the entire system. In eq 2, Ft accounts for the (ideal) mixing entropy of the mobile salt ions in the aqueous space with n+ and n- being the local concentrations of the positive and negative (salt) ions respectively; n0 is their bulk concentration, kBT is the thermal energy. Finally, Fr in eq 3 describes the orientational entropy of the Nzw dipoles belonging to the zwitterionic membrane lipids, P(θ) is the probability of finding a given dipole at angle θ. The probability distribution is subject to the normalization condition, ∫π/2 0 dθ sin θ P(θ) ) 1. Realizing the fact that the system varies in only one direction (the z axis) and z ) l cos θ, all integrals depend on functions of a single spatial variable, namely z. The total free energy, eq 1, may be written in a more convenient form, fel ) al Fel/AkBT, i.e., in terms of the free energy per lipid

fel )

al 8πlB

∫0∞ dz Ψ′(z)2 + al∫0∞ dz

]

(n+ + n- - 2n0) +

[

n+ nn+ ln + n- ln n0 n0

1-φ l

∫0l dz P(z) ln P(z),

(2)

along with the normalization condition (1/l )∫l0 dz P(z) ) 1. In writing eq 2, we have made use of the following definitions: A ≡ ∫ dx ∫ dy is the total area of the membrane, Ψ ) eΦ/kBT is the dimensionless electrostatic potential and the Bjerrum length lB ) e2/4πw0kBT (with e, the elementary charge and 0 the permittivity of free space). Note that A/al ) N is the total number of lipids. The free energy is a functional of the functions Ψ(z), n+(z), n-(z), and P(z) and must be minimized to obtain equilibrium properties. This is performed subject to the following boundary conditions:

Ψ′(z)|z)0 )

4πlB (1 - 2φ) al

Ψ′(z)|z)∞ ) 0

F(z) )

(1)

where

Fc )

decoupled from the surface z ) 0 and must thus be included in the net bulk ion density F, implying that

(3a) (3b)

Equation 3a relates the electric field to the net membrane surface charge density, σm ) σca + σzw, where σca () φe/al) and σzw () -(1 - φ)e/al) are the surface charge densities associated with the cationic and zwitterionic lipids, respectively. That is, at z ) 0, the membrane is composed of φN positive charges from the cationic lipids and (1 - φ)N negative charges from the zwitterionic lipids whose positive charge ends have been

{

e e(n+ - n-) + (1 - φ) P(z) 0 < z < l lal e(n+ - n-) lez 0, we chose a random number 0 < R < 1 and if R e e-∆U/kBT, the move is also accepted and otherwise rejected. The system is equilibrated within the first 2000 MC Cycles (MCC) and ensemble average for the zwitterion orientation, 〈θ〉, is calculated for the next 3000 MCC. Here 1 MCC ) N MCS.

Results and Discussions Headgroup Orientation. With the equilibrium probability distribution function, we can directly calculate the average orientation 〈θ〉 of the zwitterions. As discussed in the Introduction, 〈θ〉 is typically found to vary strongly as the composition φ of cationic lipids changes. Recalling z ) l cos θ, we have

〈θ〉 )

∫0lPeq(z) cos-1(z/l ) dz

1 l

∫0lcos-1(z/l )e-Ψ dz ) ∫0le-Ψ dz

Figure 2. Variation of the equilibrium zwitterionic lipid average orientation 〈θ〉 with the cationic lipid composition φ for l ) 5 Å, al ) 65 Å2 and lD ) 10 Å. The full line and the filled circles are PB and MC results respectively as indicated on the graph. The dashed line is the constant value θ0 ) 57.29° for an uncharged membrane. The open circles plot is from the MC simulation based on the simple Yukawa model potential (without image charges).

(9)

For a membrane with no electrostatic interactions (this can be (27) Netz, R. R. Phys. ReV. E 1999, 60, 3174. (28) Netz, R. R. Eur. Phys. J. E 2000, 3, 131. (29) Pink, D. A.; Quinn, B.; Moeller, J.; Merkel, R. Phys. Chem. Chem. Phys. 2000, 2, 4529-4536. (30) Mbamala, E. C.; von Gru¨nberg, H. H. Phys. ReV. E 2003, 67, 031608.

realized by hypothetically switching off the charges or by having the membrane charges completely screened; lD f 0), Ψ ) 0, and eq 9 predicts 〈θ〉 ) θ0 ) 1 (≈57.29°). The same limiting value was obtained in the PB calculations and (approximately) in the MC simulations. Upon switching on the membrane charges (lD ) 10 Å) and increasing the cationic lipid composition from 0 to 1, 〈θ〉 decreases from values well above θ0 to values well below θ0. Figure 2 shows how 〈θ〉 varies with φ for both the PB and MC calculations (indicated on the plot); also shown is the result for the uncharged membrane, θ0 (the straight dashed line). The MC and the PB results are in fairly good agreement, despite the later being based on linearized Yukawa-like potential which generally tends to exaggerate electrostatic effects. The agreement gets better by using the simple Yukawa potential in the MC simulation (open circles). This, however, ignores image-charge contributions and should not be encouraged for this system. In both methods, the variation in 〈θ〉 from low cationic lipid composition, φ ≈ 0, to high composition, φ ≈ 1, is about 40% (PB) to about 45% (MC). This is still considerably lower than about 60% variation found by Gurtovenko et al.22 for their MD simulations under salt-free conditions. The straight dashed line (Figure 2) for the opposite limit of infinite screening (very high salt concentration) suggests an explanation for this difference: the profile should become flatter with increasing salt concentration. Let us consider the two limits of φ. When φ ≈ 0, the membrane is nearly or fully composed of zwitterionic lipids. The attractive electrostatic forces between the mobile positive-charge ends and the fixed negative-charge bases of the zwitterionic lipids lead to a more parallel orientation (〈θ〉 > θ0) as compared to the uncharged membrane where 〈θ〉 ) θ0. In the opposite limit, φ ≈ 1, a zwitterionic lipid is surrounded by a sea of positively charged (cationic) lipids, resulting in an energetically unfavorable repulsive interaction between the mobile positive-charge ends and the charges of the cationic lipids, implying a tendency to keep 〈θ〉 below θ0. Next, we wish to investigate if this behavior has any consequences on the average cross-sectional area per lipid as has been suggested by Gurtovenko et al.22 (31) Metropolis, N.; Rosenbluth, A.; Rosenbluth, M.; Teller, A.; Teller, E. J. Chem. Phys. 1953, 21, 1087.

Mixed Cationic-Zwitterionic Lipid Bilayers

Langmuir, Vol. 22, No. 11, 2006 5133

Membrane Structure and Area per Lipid. The average crosssectional area per lipid, al, is one of the most fundamental properties that can be measured for a model membrane. It plays a major role for the structural behavior of a lipid bilayer and determines the average lipid shape.32,33 Under various conditions, including temperatures ranging from 25 °C to 50 °C, al has been measured for lipid membranes and generally ranges from 40 Å2 to more than 70 Å2.34-38 Our aim is to set up a typical situation (close to physiological conditions) and then determine the equilibrium average area per lipid 〈al〉 as a function of the cationic lipid composition φ in a binary mixture with zwitterionic lipid, using the present meanfield electrostatic model. As a first step, we must characterize how the energy of an uncharged membrane changes as a function of al. To this end, we use the familiar “opposing forces” model of Israelachvili.33 Within this model, the nonelectrostatic contribution to the free energy per lipid in a membrane (either mixed or one-component) can be written as

B fne ) γal + al

(10)

The first term accounts for the unfavorable polar-apolar interface; γ ≈ 0.12 kBT Å-2 is the surface tension. This term tends to decrease al. The second contribution adds an opposing repulsive energy between the headgroups; it can arise through hydration forces, steric headgroup repulsion, and chain interactions.33 The interaction strength is characterized by B. At equilibrium, ∂fne/ ∂al ) 0, and a0 ) al (min) ) (B/γ)1/2, where a0 is the optimal or equilibrium area per lipid for the uncharged membrane where the average orientation is θ0 ) 57.29°; see Figure 2. Substituting for B in eq 12, we obtain the equilibrium nonelectrostatic free energy per molecule

( )

fne ) γal 1 +

a02 al2

(11)

For a charged membrane both the nonelectrostatic and electrostatic free energies are added together, f ) fne + fel. To obtain the equilibrium area per lipid for a charged membrane, we minimize the net free energy per molecule f with respect to al

∂f ∂fne ∂fel ) + )0 ∂al ∂al ∂al

(12)

and solve for 〈al〉 ≡ al(min). Note that fel is the electrostatic free energy per lipid given in eq 7. Orientation Dependent Steric Interaction. So far (see eq 10), we have assumed that the area per lipid is the same for both the cationic and the zwitterionic lipids and that any change in the average cross-sectional area of the membrane as a function of φ arises entirely due to direct electrostatic interactions. However, it is plausible that there is a second mechanism which (32) Boal, D. Mechanics of the cell; Cambridge University Press: New York, 2002. (33) Israelachvili, J. N. Intermolecular and surface forces; Academic Press: London, 1992. (34) Lewis, R. N.; Tristram-Nagle, S.; Nagle, J. F.; McElhaney, R. N. Biochim. Biophys. Acta 2001, 1510, 70-82. (35) Nagle, J. F.; Tristram-Nagel, S. Biochim. Biophys. Acta 2000, 1469, 159195. (36) Petrache, H. I.; Dodd, S. W.; Brown, M. F. Biophys. J. 2000, 79, 31723192. (37) Costigan, S. C.; Booth, P. J.; Templer, R. H. Biochim. Biophys. Acta 2000, 1468, 41-54. (38) Nagle, J.; Zhang, R.; Tristram-Nagle, S.; Sun, W.; Petrache, H. I.; Suter, R. M. Biophys. J. 1996, 70, 1419-1431.

Figure 3. Variation of the scaled equilibrium cross-sectional area per lipid 〈al /a0〉 as function of the composition φ of cationic lipids in the membrane leaflet. The different curves correspond to different screening lengths lD that vary between 0 and 20.0 Å. All cases refer to λ ) 0 (see eq 14) for which the change in steric interactions upon reorientation of the dipolar headgroups is neglected.

influences the average cross-sectional area: the change in orientation of the zwitterionic headgroups with φ can be expected to modify the steric interactions between the headgroups, thus affecting the average cross-sectional area. Clearly, this mechanism is nonelectrostatic in nature but is mediated through electrostatic interactions (as those affect the average headgroup orientation 〈θ〉). In other words, the repulsion parameter B will depend on 〈θ〉. We seek to formulate a minimal model for this dependence. First, we recall our assumption that when electrostatic interactions are (hypothetically) switched off, 〈θ〉 ) θ0, and a0c ) a0z ) a0, where a0c and a0z are the optimal area per lipid for the cationic and zwitterionic lipids, respectively. A more general model (where a0c and a0z are not necessarily equal to each other) would weight the contributions according to the mole fractions of cationic and zwitterionic lipids; implying aj0 ) φa0c + (1 φ)a0z. When electrostatics is switched on, the average dipole orientation changes, 〈θ〉 deviates from θ0, so that up to first order in (〈θ〉 - θ0) we may write a0z/a0c ) 1 + λ(〈θ〉 - θ0). The prefactor, λ has the value cot θ0 (see Appendix), but can, more generally, be interpreted as a phenomenological parameter that determines how strong the electrostatically mediated orientational change in headgroup orientation affects the lateral cross-sectional area of the zwitterionic component. Since a0c is not affected by orientation, a0c ) a0 and a0z ) a0[1 + λ(〈θ〉 - θ0)]. Thus

aj0 ) a0[1 + (1 - φ)λ(〈θ〉 - θ0)]

(13)

Replacing a0 in eq 11 with aj0, we obtain

fne ) γal +

γa02 [1 + (1 - φ)λ(〈θ〉 - θ0)]2 al

(14)

which is our final expression for fne. Solving eq 12, with fne as in eq 14 instead of eq 11 gives us the average equilibrium area per lipid, accounting for both direct electrostatic interactions and for electrostatically mediated changes in steric headgroup interactions. In the numerical solution of eq 12, we set a0 ) 65 Å2 and vary φ between 0 and 1. All other parameters remain as assigned previously. Figure 3 shows plots of the scaled equilibrium area per lipid 〈al /a0〉 versus the cationic lipid composition, φ. Note

5134 Langmuir, Vol. 22, No. 11, 2006

Figure 4. Same as in Figure 3 but for fixed, lD ) 10.0 Å, and λ is switched from 0.0 to 0.25 and cot θ0 ) cot(1) ≈ 0.64. The straight horizontal line marks the equilibrium area per lipid for an uncharged membrane where a0c ) a0z ) a0 ) 65.0 Å2.

that Figure 3 is calculated for λ ) 0 (for which eq 14 reduces to eq 11). That is, the changes in steric headgroup-headgroup interaction upon reorientation of the dipolar headgroups are neglected. Even with this restriction we see that 〈al〉 passes through a minimum as φ is increased between 0.0 and 1.0. That is, the membrane leaflet first contracts as the number of cationic lipids is increased and then starts to expand as φ is increased further. The value of φ corresponding to the minimum 〈al〉 shifts from φ ≈ 0.2 for lD ) 10 Å to φ ≈ 0.5 for lD ) 1 Å. Also, the curves exhibit saturation as lD increases, showing little difference between 10 and 20 Å. As lD decreases, the membrane-charge screening becomes more effective and the tendency for membrane expansion or contraction wanes. For lD ) 0, the membrane charges are completely screened, 〈al /a0〉 is constant at 1.0, corresponding to a constant cross-sectional area a0 for the uncharged membrane (recall that a0 being independent of composition is a convenient assumption as stated above, the model can easily be generalized). Let us now investigate the influence of the steric headgroup interactions on the average cross-sectional area. To this end, we show in Figure 4 〈al〉 as function of φ for different choices of λ but fixed screening length, lD ) 10 Å. Recall that λ tunes the orientation-dependent steric headgroup interaction strength (see eq 14). The choice λ ) cot(θ0) ) cot(1) ) 0.64 is perhaps the most reasonable one as it corresponds to the steric interaction model presented in the appendix. Figure 4 clearly shows that the orientation-dependent changes in steric headgroup interactions lead to a more pronounced minimum in the 〈al /a0〉 versus φ curves. The most significant influence of a nonvanishing λ is the increased expansion of the membrane leaflet when it is dominated by the dipolar PC component (small φ). Figure 5 summarizes how the two components of the total free energy, f ) fel + fne, contribute to the behavior of the curves shown in Figures 3 and 4. When the membrane has no charges (or equivalently, completely screened charges), the average orientation, 〈θ〉 ) θ0, and the area per lipid 〈al /a0〉 is constant with φ (curve 1). For curve 2, electrostatic interactions are switched on (e.g. lD ) 10 Å) but the steric influence of the zwitterionic lipid’s orientational headgroup changes when the cross-sectional area is neglected, i.e., λ ) 0, even though 〈θ〉 * θ0. Curve 3 is obtained by assuming PC reorientation (〈θ〉 * θ0 and λ * 0) without accounting for the direct electrostatic influence (that is we set ∂fel/∂al ) 0 in eq 12). This behavior is hypothetical because the changes in headgroup orientation are mediated

Mbamala et al.

Figure 5. Summary of contributions from the various components of the free energy f ) fel + fne. Curve (1) shows 〈al〉 ) a0. Curve (2) redisplays the curve for lD ) 10 Å in Figure 3. Curve (3) is derived for lD ) 10 Å and λ ) 0.64 but neglecting the derivative ∂fel/∂al ) 0 in eq 12. Finally, curve (4) redisplays the curve for λ ) 0.64 in Figure 4.

through electrostatics; the curve still shows that this isolated effect alone would not account for the deep minimum in 〈al〉. Finally, curve 4 shows that this deep minimum is obtained if both contributions, fel and fne, are present at the same time. Lipid Orientation and Area per Lipid. Our study confirms the previous suggestion22 that the orientation of the zwitterionic lipid headgroups correlate with the pronounced nonmonotonic behavior of the 〈al〉 versus φ curves (Figures 3-5): For a membrane composed purely of zwitterionic lipids, electrostatic and even more steric interactions keep the lipid headgroups in a more parallel orientation, resulting in a large cross-sectional area per lipid. At low cationic lipid composition (φ j 0.5 in Figure 5), the cationic lipids induce two different effects. First, they reduce the lateral electrostatic pressure by neutralizing some of the negatively charged ends of the zwitterionic dipoles (this effect is expressed in the term ∂fel/∂al in eq 12). Second, due to the repulsion with the positively charged ends of the dipoles, the zwitterionic headgroups reorient toward the membrane normal direction (lowering 〈θ〉). This leads to weaker steric interaction of the zwitterionic lipids (this effect is expressed in the term ∂fne/∂al in eq 12). Both effects lower 〈al〉. As the composition φ increases further (0.5 j φ in Figure 5), the electrostatic repulsion among the cationic lipids becomes more dominant and the membrane expands again. This expansion is mostly a direct electrostatic effect; it is only weakly counteracted by the reorientation of the remaining zwitterionic headgroups. See Figure 6 for a schematic illustration. To demonstrate that the above picture is indeed the mechanism for membrane compression and expansion and also that it is unique to cationic membranes, we compare, in Figure 7, the already discussed cationic-zwitterionic with an equivalent anionic-zwitterionic membrane system. To realize such a system in our PB model, for example, we need only to modify the boundary condition at the membrane surface (eq 3a), where now the membrane “surface charge density”, σm ) σan + σzw ) -e/ al; the membrane is fully charged at z ) 0. Figure 7a shows the variation of 〈θ〉 with φ for the cationic-zwitterionic membrane system as in Figure 2 (the lower set of curves) and for the anioniczwitterionic membrane (the upper set). Here φ represents the cationic lipid composition and also the anionic lipid composition. The plots show that, in contrast to the cationic-zwitterionic

Mixed Cationic-Zwitterionic Lipid Bilayers

Figure 6. Schematic illustration of membrane expansion and contraction as played out by lipid headgroup electrostatic and steric interactions.

Langmuir, Vol. 22, No. 11, 2006 5135

composition” at z ) 0. In Figure 7b, we compare the (scaled) average cross-sectional area 〈al /a0〉 for a mixed cationiczwitterionic (broken lines) with a mixed anionic-zwitterionic membrane (solid lines). Two sets of curves are shown, one (λ ) 0) ignoring and the other one (λ ) cot(θ0) ) cot(1) ) 0.64) accounting for steric headgroup interactions. Note that for φ ) 0 (only zwitterionic lipids) the average cross-sectional area depends considerably on λ: large steric headgroup interactions increase 〈al /a0〉. Also, as expected, for φ ) 1, the average crosssectional area is the same for a pure cationic or anionic membrane. However, intermediate for 0 < φ < 1, the two membranes (mixed cationic-zwitterionic and mixed anionic-zwitterionic) behave very differently, a behavior which is even more emphasized by the presence of steric headgroup interactions (λ ) 0.64). Although for the mixed anionic-zwitterionic membrane 〈al /a0〉 exhibits a monotonic change, the mixed cationic-zwitterionic membrane contracts for medium compositions. The latter is a consequence of the interplay between reduced lateral electrostatic pressure and a relief in steric headgroup repulsion due to headgroup reorientation.

Conclusions

Figure 7. Comparison of the cationic-zwitterionic membrane with an equivalent anionic-zwitterionic one. (a) average zwitterionic headgroup orientation 〈θ〉 versus membrane composition, φ. (b) Scaled average cross-sectional area, 〈al /a0〉 versus φ. Here φ is used to represent the cationic as well as the anionic composition. For all curves, lD ) 10 Å and a0 ) 65 Å2.

membrane, 〈θ〉 remains almost constant for the anioniczwitterionic membrane. It increases instead, but only slightly, by about 8% when φ (anionic) is increased between 0 and 1. This is hardly surprising because the replacement of the zwitterions with the anions (φ f 1) does not change the membrane “ionic

In this study, we have employed the Poisson-Boltzmann formalism and Monte Carlo simulations to model and investigate the structural behavior of membranes composed of cationic lipids (of, e.g., TAP headgroup) and zwitterionic lipids (e.g., PC headgroup at neutral pH). In our model, the cationic lipid headgroup is represented by a single positive charge and the zwitterionic lipid is modeled by a dipole that has one charge spatially fixed and the other mobile. Both the immobile dipole charge and the charge of the cationic lipid reside within the same plane at the membrane’s polar-apolar interface (z ) 0). The mobile dipole charge is allowed to rotate about the fixed end but must remain within the polar region (z > 0). We summarize our main results: (i) At low composition of the cationic lipids, the average orientation of the zwitterionic (PC) headgroup dipoles is more parallel to the membrane interface, whereas at high composition, it tends to be more perpendicular (see Figure 2). (ii) There are two mechanisms that determine the change in average cross-sectional area of the membrane leaflet with composition. One is the change in lateral electrostatic pressure and the other is the change in steric headgroup repulsion due to dipole reorientation. The first mechanism is directly electrostatic, whereas the second is only mediated by electrostatics but realized through steric interactions. The combination of both mechanisms gives rise to a pronounced minimum in the average cross-sectional area per lipid as function of membrane composition. (iii) To show that this structural behavior is unique to cationic lipid membranes, we also investigated the behavior of an equivalent system of anionic-zwitterionic membrane. We argue that, for this system, the PC headgroup orientation remains essentially unaffected and the average cross-sectional area per lipid changes monotonically. (iv) Our electrostatic mean-field calculations are backed up by MC simulations that make the same qualitative predictions. The minor quantitative differences between the two may arise from both correlation effects or from the small variations in the interaction potentials (recall that in the MC simulations each lipid charge has a nonvanishing steric size whereas PB theory is based on pointlike charges). As our structural predictions principally agree with the much more detailed results of a recent MD simulation,22 it is worth emphasizing that our model is based on mean-field electrostatics

5136 Langmuir, Vol. 22, No. 11, 2006

Mbamala et al.

and a small number of structural assumptions. Among these assumptions are the fixed dipole length l, the abrupt change in the dielectric constant at z ) 0, the sterically unrestricted accessibility of the salt ions into the headgroup region, and the firm attachment of both cationic lipid charges and of the negative dipole charges at a single plane (at z ) 0). These assumptions were made to facilitate the extraction of the physical mechanisms that underly previous experimental results.23,24 Indeed, the agreement of our predictions with established experimental observations suggests that the model is robust. Acknowledgment. S.M. and E.C.M. acknowledge financial support from ND EPSCoR through NSF Grant No. EPS-0132289. S.M. and A.F. thank Joachim and Anna Seelig for illuminating discussions.

Appendix In obtaining eq 14 for the orientation dependent nonelectrostatic free energy of the mixed membrane system, we stated that the ratio of the optimal area per lipid corresponding to the zwitterionic molecules, a0z, to that corresponding to the cationic molecules, a0c, can be written as

a0z/a0c ) 1 + λ(〈θ〉 - θ0)

(15)

This appendix shows that λ ) cot θ0 where θ0 is the average orientation of the zwitterionic headgroups in absence of electrostatic interactions. We recall our assumption that, when there is no electrostatic interaction on the membrane (〈θ〉 ) θ0), a0z ) a0c ) a0 (see eq 11). In this case, the average crosssectional area per lipid is fixed by the orientation of the zwitterionic PC head. When electrostatic interactions are switched on (i.e., fel * 0), the dipolar headgroup can be found in an average orientation 〈θ(φ)〉 with 0 e 〈θ〉 e π/2, where φ is the composition

Figure 8. Estimating the optimal area per lipid.

of the cationic lipid component. This reorientation can change a0z but not a0c. If it does, we must replace a0 in eq 11) with the average aj0 ) φa0c + (1 - φ)a0z and since a0c ) a0

aj0/a0 ) φ + (1 - φ)a0z/a0

(16)

We can estimate a0 and a0z by estimating the projected area of the PC head on the membrane plane. If we assume such an area to be rectangular as shown in Figure 8, with sides x ) l sin〈θ〉 and y, then a0 ∼ yl sin θ0 and a0z ∼yl sin〈θ〉, so that

a0z/a0 ) sin〈θ〉/sin θ0

(17)

Expanding eq 17 around θ0 to linear order in 〈θ〉 gives

a0z/a0 ≈ 1 + cot θ0 (〈θ〉 - θ0)

(18)

and comparison with eq 15 reveals λ ) cot θ0. The uncertainties in this estimate can be amended by adjusting the value of λ. For example, λ ) 2 cot θ0 if x ) l sin〈θ〉 would be the radius of a circular area projection of the dipolar headgroup. On the other hand, if there would be overlap of neighboring projected rectangular areas, then, λ < cot θ0. LA060180B