pubs.acs.org/Langmuir © 2010 American Chemical Society
Electrostatic Bending of Lipid Membranes: How Are Lipid and Electrostatic Properties Interrelated? Sattar Taheri-Araghi and Bae-Yeun Ha* Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada Received May 21, 2010. Revised Manuscript Received July 10, 2010 Electrostatic modification of lipid headgroups and its effect on membrane curvature are not only relevant in a variety of contexts such as cell shape transformation and membrane tubulation but also are intriguingly implicated in membrane functions. For instance, the gating (open vs closed) properties of mechanosensitive channels can be influenced by membrane curvature and ion valence. However, a full theoretical description of membrane electrostatics is still lacking; in the past, membrane bending has often been considered under a few assumptions about how bending modifies lipid arrangements and surface charges. Here, we present a unified theoretical approach to spontaneous membrane curvature, C0, in which lipid properties (e.g., packing shape) and electrostatic effects are self-consistently integrated. For the description of electrostatic interactions, especially between a lipid charge and a divalent counterion, we implement the Poisson-Boltzmann (PB) approach by incorporation of finite ionic sizes, so as to capture both lateral and transverse charge correlations on the membrane surface. Our results show that C0 is sensitive to the way lipid rearrangements and divalent counterions are modeled. Interestingly, it can change its sign in the presence of divalent counterions, thus stabilizing reverse hexagonal (HII) phases. Our results show how electrostatic modification of headgroups influences the preferred structure of lipid aggregates (inverted micelles vs bilayers).
Introduction A lipid bilayer membrane is a self-assembled structure studded with membrane proteins.1-4 Its ability to deform its shape and topology complements its integrity as a “self-sealing” object. This is correlated with the fact that a lipid bilayer is only a particular realization of lipid aggregates. The rich phase behavior of lipid aggregates is not only a direct manifestation of single-lipid properties (e.g., lipid packing) but also a result of external parameters such as salts and temperatures.3-5 Along this line, the electrostatic bending of a possibly asymmetrically charged lipid membrane has been considered for some time.6-8 It not only complements protein-induced bending9 but also has relevance in a variety of different contexts: cell shape transformation,3,10 vesicle budding, and lipid tubulation,6-8 as well as Ca2þ-induced membrane fusion.11-13 An intimately related point is that lipid charges can alter lipid packing stress, which in turn influences membrane functions by modifying the “working” condition for membraneprotein activity (ref 14 and references therein). For instance, the *To whom correspondence should be addressed. E-mail:
[email protected]. (1) Alberts, B. Molecular Biology of the Cell, 5th ed.; Garland Science: New York, 2008. (2) Boal, D. Mechanics of the Cell; Cambridge University Press: New York, 2002. (3) For a pedagogical introduction to lipid membranes and vesicle shapes, see Wortis, M.; Evans, E. Physics in Canada 1997, 53, 281-288. (4) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: New York, 1992. (5) Gruner, S. M.; Cullis, P. R.; Hope, M. J.; Tilcock, C. P. S. Annu. Rev. Biophys. Biophys. Chem. 1985, 14, 211–238. (6) Winterhalter, M.; Helfrich, W. J. Phys. Chem. 1992, 96, 321–330. (7) Mitchell, D. J.; Ninham, B. W. Langmuir 1989, 5, 1121–1123. (8) Chou, T.; Jari, M. V.; Siggia, E. D. Biophys. J. 1997, 72, 2042–2055. (9) Zimmerberg, J.; Kozlov, M. M. Nature Rev. Mol. Cell Biol. 2006, 7, 9–19; also see the supplementary information . (10) Lim H. W., G.; Wortis, M.; Mukhopadhyay, R. Proc. Nat. Acad. Sci. U.S.A. 2002, 99, 16766–16769. (11) Papahadjopoulos, D. Science 1969, 30, 1075–1077. (12) Papahadjopoulos, D.; Portis, A. Ann. N.Y. Acad. Sci. 1978, 308, 50–66. (13) Chanturiya, A.; Scaria, P.; Woodle, M. C. J. Membr. Biol. 2000, 176, 67–75. (14) Bezrukov, S. M.; Rand, R. P.; Vodyanoyc, I.; Parsegian, V. A. Faraday Discuss. 1998, 111, 173–183.
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gating (open vs closed) properties of “mechanosensitive” (MS) channels (as in E. coli) are sensitive to lipid packing or membrane curvature.15 Interestingly, the opening or closure of such channels can be controlled by counterion valence and membrane charges.16 It is worth noting that the electrostatic mechanisms of spontaneous membrane curvature and lipid packing stress share the same physical origin in common, that is, electrostatic modification of lipid headgroups. Indeed, headgroup properties are shown to play a fundamental role in self-assembly of lipid aggregates.17 On the other hand, what remains unclear is the relative roles of electrostatic6-8 and protein-based9,15 mechanisms in “shaping” lipid membranes. In the case of MS channels, for instance, channel shapes are also implicated in their gating properties. Also, cells use a variety of proteins specialized in membrane bending as for membrane vesicle formation.18 Nevertheless, a better understanding of electrostatic bending will be useful for identifying relevant parameters for determining lipid packing and membrane curvature. In fact, the lipid contribution to membrane curvature is influenced by lipid charges, whether proteins are involved or not (see for instance ref 17) and is shown to have nontrivial impact on MS channels.16 Despite much effort, however, the electrostatic bending of a lipid membrane has not been well understood theoretically, owing to the presence of large degrees of freedom such a system presents (e.g., lipid flexibility and the “ionic cloud” forming near a charged surface19). Accordingly, spontaneous membrane curvature has been considered under a few assumptions about how bending modifies lipid arrangements and surface charges.6-8 In this sense, (15) Perozo, E.; Kloda, A.; Cortes, D. M.; Martinac, B. Nat. Struct. Biol. 2002, 9, 696–703. (16) Ermakov, Y. A.; Kamaraju, K.; Sengupta, K.; Sukharev, S. Biophys. J. 2010, 98, 1018–1027. (17) Fuller, N.; Benatti, C. R.; Rand, R. P. Biophys. J. 2003, 85, 1667–1674. (18) Kirchhausen, T. Nat. Rev. Mol. Cell Biol. 2000, 1, 187–198. (19) Gelbart, W. M.; Bruinsma, R. F.; Pincus, P. A.; Parsegian, A. V. Phys. Today 2000, 53, 38–44.
Published on Web 08/25/2010
DOI: 10.1021/la102052r
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lipid and electrostatic properties are not fully integrated. In fact, the electrostatic interaction between constituent lipids can modify lipid parameters, while the latter can influence the way charged lipids interact with each other. As evidenced later, this interdependence, which has been under-appreciated in the past, is a key feature of lipid assemblies. Furthermore, it has been shown that charge discreteness can play an important role, especially when counterions are multivalent;20-23 this effect is shown to influence appreciably the binding affinity of multivalent counterions for an oppositely charged surface. A seemingly distinct but possibly related observation is that multivalent counterions tend to make surface charges (more) nonuniform. To illustrate this, imagine a duplex formed by a backbone charge and a counterion. When the counterion is multivalent, the duplex carries a net positive charge, possibly surrounded with bare backbone charges of the opposite sign. This effect can induce a negative lateral pressure on a membrane surface13 or even attractions between like-charged molecules (see ref 19 and references therein). This points to the limitation of a conventional meanfield approach, where counterions are assumed to form a laterally uniform ionic cloud near a charged surface,4 implying that the counterion density is a function of the distance from the surface only. Owing to all these complexities, it still remains challenging to describe the electrostatic bending of a lipid membrane consistently, without suppressing its important degrees of freedom. This work is aimed at presenting a unified theoretical approach to the spontaneous bending of lipid membranes (or lipid aggregates) consisting of neutral (zwitterionic) and anionic lipids immersed in a salty solution, possibly containing multivalent counterions. Our approach is distinct from previous attempts6-8 in several respects. First, in our approach the elastic and charge properties of lipids are integrated at the single-lipid level. Accordingly, bending, stretching, and surface charges are taken into account simultaneously and coherently. To this end, we allow lipid parameters to relax at their equilibrium values, which turn out to depend on external parameters within our model, without invoking any further simplification. While our electrostatic analysis is based on the nonlinear Poisson-Boltzmann (PB) equation, it requires a nontrivial generalization so as to capture the aforementioned surface charge discreteness and inhomogeneity. By incorporating charge discreteness into the PB equation, we capture nonuniform charge distributions and thus “lateral” and “transverse” charge correlations.20,21 (See below for the interrelationship between charge discreteness and nonuniformity.) This is particularly important when the solution contains multivalent counterions, which give rise to nonuniform charge distributions on the membrane surface. In fact, it has been shown that Ca2þ can trigger lipid-tail ordering,12 primarily by shrinking lipid headgroups.24 This illustrates limitations of any approach that leaves out charge discreteness or heterogeneity in surface charge distributions. Furthermore, charge discreteness has nontrivial impact on how charged (anionic) lipids interact with their counterions, especially multivalent ones. It tends to enhance lipidcounterion association. Better understanding of the spontaneous (20) Travesset, A.; Vaknin, D. Europhys. Lett. 2006, 74, 181–187. (21) For a recent elaborated effort on ion distributions near a (flat) charged surface, see Travesset, A.; Vangaveti, S. J. Chem. Phys. 2009, 131, 185102-1– 185102-11 (similarly to ref 20, charge discreteness and correlations are much appreciated in this work) . (22) Caleroa, C.; Faraudob, J. J. Chem. Phys. 2010, 132, 024704-1–024704-11 (this work offers numerical evidence of the significance of ionic sizes and valence in determining ion distributions near a charged surface) . (23) Henle, M. L.; Santangelo1, C. D.; Patel, D. M.; Pincus, P. A. Europhys. Lett. 2004, 66, 284–290. (24) Li, Y.; Ha, B.-Y. Europhys. Lett. 2005, 70, 411–417.
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bending of a lipid membrane would necessitate a more consistent treatment of the various effects described above. Our results show how the spontaneous curvature of a membrane, denoted as C0, can be controlled by the elastic and charge properties of lipids and counterion valence.25 While the general picture for C0 emerged from our study in the absence of multivalent counterions is similar to those presented earlier,6-8 it also points to the significance of treating the elastic and charge properties of lipids consistently. Interestingly, its sign can be inverted by the presence of a small concentration of divalent salts. As a result, the membrane tends to bend toward a more highly charged layer, in contrast to what one may expect from meanfieldtype approaches.6-8 This finding is paralleled by the earlier observation that divalent counterions can induce lipid tail ordering12 or a negative lateral pressure on the charged layer.13,24 Also, this is closely related to and thus may offer a quantitative basis for the observed sensitivity of MS channels to counterion valence.16 This electrostatic modulation of membrane curvature can be considered as a particular realization of preferred structures (amphiphilic) lipids form in aqueous solution,4 thus offering a molecular basis for the aforementioned various membrane phenomena, which implicate spontaneous membrane curvature or more generally lipid packing properties. Our results show that electrostatic modification of headgroups is a key determinant of the preferred structure (and phase) of lipid aggregates: inverted micelles versus bilayers (reverse hexagonal vs lamellar phases).
Theoretical Model In this section, we present our molecular model. After introducing a free-energy description of individual lipids forming a monolayer or a bilayer, we develop our electrostatic model for lipid-counterion interactions: their association and its impact on lipid parameters and membrane bending. Single-Lipid Free Energy. The free energy of a lipid aggregate (e.g., a monolayer or bilayer) can be expressed in terms of single lipid parameters. Each lipid (thus its packing shape) is fully characterized by its geometrical parameters: the headgroup area (ah), the area per each lipid at the headgroup-tail interface (ai), and the length of its tail or hydrocarbon chain (lhc), as illustrated in Figure 1. The parameter conjugate to ai is the interfacial tension, γ, arising from the hydrophobicity of hydrocarbon chains or their tendency to avoid contact with water. The resulting free energy per lipid is γai.26 Similarly, the free energy cost for overlapping two headgroups is described as k/ah, where k is a constant characterizing the strength of their repulsion. On the other hand, the free energy of a lipid tail assumes the Hookean form of τl2hc, where τ measures the energy cost for deforming lhc. The total free energy per lipid in an aggregate can be written as27 f ¼
k 2 þ γai þ τlhc ah
ð1Þ
While the first and last term tend to swell the area per lipid, the second term opposes this. Note that different models have (25) One of the main differences between monolayers and bilayers in the context of bending is that any mismatch in the areas of outer and inner layers in the latter can induce spontaneous bending. This effect can cooperate with or compete against the spontaneous curvature arising from local properties of lipids such as lipid packing as detailed later (see subsection Bilayers in the Results). However our description in this section is common for both cases. (26) See ref 3 for a pedagogical estimation of γ. Recall γ is conjugate to area each lipid occupies. This has to be differentiated from the macroscopic counterpart, that is, the one conjugate to the total membrane area, which varies with the number of constituents lipids.28 (27) May, S.; Ben-Shaul, A. Biophys. J. 1999, 76, 751–767.
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Figure 1. The packing shape of a lipid in a lipid membrane is characterized by a few geometrical parameters such as ah, ai, lhc, and vhc. There is a simple geometrical relationship between different radii of curvature (see eq 8).
been used28-30 (see ref 30 for comparative studies on a few models). These phenomenological energy terms deserve some discussion. First, the parameters k, γ, and τ reflect the elastic nature of individual lipid molecules, which behave as “molecular springs.” On the other hand, the shape parameters, ah, ai, and lhc are interrelated, since their changes are subject to the constraint that the volume of each tail, vhc, remains invariant,27,28 which is a reasonable assumption for the lipids in the fluid phase, as is the case for a biologically active membrane. Electrostatic Free Energy of a Lipid Membrane. In addition to the free energy in eq 1, one has to include the electrostatic contribution. The lipid membrane we consider here consists of zwitterionic (neutral polar) and anionic lipids such as phosphatidylserine (PS) or phosphatidylglycerol (PG); it is partially charged. The charged lipids interact not only with each other but also with surrounding counterions, especially multivalent counterions. The crudest simplification may amount to smearing out the lipid charges, but this meanfield-type approximation underestimates their attraction with counterions. While their mutual repulsion tends to keep them equidistant from each other, their association with multivalent counterions can modulate their spatial distribution, one counterion may neutralize (bind to) more than one lipid charge. In general, lipid demixing can alter how the membrane interacts with opposite charges, especially when the opposite charges are multivalent. Here we restrict ourselves to the case of monovalent or divalent counterions. In this case, lipid demixing is not expected to be pronounced, since the resulting entropic loss can easily counterbalance the energy gain. In our approach, anionic lipids are considered as forming a hexagonal lattice as depicted in Figure 2. The solution of the PB equation with this arrangement can be used to calculate the electrostatic free energy without suppressing lipid-charge discreteness. Electrostatic effects on lipid parameters can be analyzed by allowing them to relax at new preferred values, by free-energy minimization. For simplicity, we assume that the geometrical parameters (ah, ai, and lhc) take on the same value for both neutral and anionic lipids in the same layer (see below for more general cases). The regularity in the hexagonal geometry allows us to construct and focus on a unit cell, often referred to as a Wigner-
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Figure 2. Charge discreteness and the spatial distribution of anionic lipids on a spherically curved membrane; for simplicity, electrically neutral lipids are not shown. A hexagonal lipid arrangement as well as a Wigner-Seitz cell (the dashed circle) are highlighted. The central lipid, the one at the center of the dashed circle, experiences radially symmetrical interactions on average. It thus suffices to consider the central one explicitly and absorb others into a boundary condition (cf. eq 3).
Seitz (WS) cell.31,32 Below we present our strategy for constructing the WS cell and calculating the electrostatic free energy. The Monovalent Case. In the absence of divalent counterions, each anionic lipid on a hexagonal lattice naturally defines its WS cell; it is placed at the center of the cell (see Figures 2 and 3a), surrounded with electrically neutral ones. Depending on the curvature of the membrane, the WS cell resembles a cylinder, a cone, or an inverted cone.4 The boundary of each WS can be approximated as circular one; on average, each anionic lipid will experience radially symmetrical interactions.31,32 The average electrostatic energy per charged lipid can be obtained by solving the PB equation in the aqueous phase within a WS cell. The PB equation in the presence of a (1:1) salt can be written as8 r2 Ψ ¼ K2 sinhðΨÞ
where Ψ = eΦ/kBT is the reduced electrostatic potential with e as the electronic charge, Φ as the electrostatic potential, kB as the Boltzmann constant, and T as the temperature. The Debye screening length, κ-1, is given by the relation, κ2 = 8πn0ε0εw/ kBT, where ε0 is the permittivity of free space and εw is the dielectric constant of water. The PB equation has to be solved with the following appropriate boundary conditions. • The vanishing normal component of electrostatic fields on the cell boundary to reflect the symmetry of WS cells n 3 rΦðrÞjr ¼ RWS ¼ 0 •
lim ΦðrÞ ¼ 0
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ð4Þ
rf¥
•
ð3Þ
with n the unit vector normal to the WS cell boundary. The vanishing electric potential at infinity
Discontinuity of the electric field across a charged surface with a planar charge density σ DΦðrÞ DΦðrÞ - εl ε0 ¼ σðrÞ ð5Þ εw ε 0 Dr Dr above
(28) Safran, S. Statistical Thermodynamics of Surfaces, Interfaces, and Membranes; Westview Press: Boulder, CO, 2003. (29) See Leibler, S. In Statistical Mechanics of Membranes and Surfaces, 2nd ed.; Nelson, D., Piran, T., Weinberg, S., Eds.; World Scientific: River Edge, NJ, 2004. (30) Petrov, A. G.; Bivas, I. Prog. Surf. Sci. 1984, 16, 369–512. (31) May, S.; Harries, D.; Ben-Shaul, A. Biophys. J. 2000, 79, 1747–1760.
ð2Þ
below
where εl is the dielectric constant of lipids. If R(r) is the local fraction of charged lipids, then σ = -(e/ah) R. (32) Yu., A.; Grosberg, T. T.; Nguyen; Shklovskii, B. I. Rev. Mod. Phys. 2002, 74, 329–345.
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Figure 3. Side view of a Wigner-Seitz cell for the monovalent (a) and divalent case (b). For simplicity, neutral (zwitterionic) lipids are not shown. The radius of curvature Rt (Rh) is measured with reference to the surface of tail ends (headgroups). For the monovalent case (panel a on the left), one anionic lipid defines one WS cell. In the divalent case, however, each WS cell is constructed so as to contain one divalent counterion, which is assumed to form a pair with the central lipid, as illustrated in Figure 3b; each cell can contain more than one anionic lipid. (See below for the meaning of the central ion.) Our electrostatic analysis in this case consists of three steps (see the box on the right). (i) First, we smear out all the backbone charges except the one forming a “pair” with the bound divalent counterion (see (i) in the box on the right). The electrostatic free energy of this (per WS cell) is denoted as F 1 . Since the effect of charge discreteness is the most pronounced for this pair interaction, the simplification introduced in (i) will not alter appreciably the binding affinity of the counterion. This will, however, shift our energy reference with respect to which the bending energy is estimated: if (ii) describes the new reference, (iii) represents the original one. Let F 2 and F 3 be the electrostatic free energy of (ii) and (iii), respectively. Then the difference, ΔF ¼ F 3 - F 2 , is the correction term, which should be subtracted from F 1 . In our approach, the central lipid refers to an anionic lipid forming a pair with a divalent counterion at the center of each cell, for which charge discreteness is preserved (see (i)).
The average fraction, denoted as R, is then the number of anionic lipids divided by the number of lipids. The free electrostatic energy of the membrane and the surrounding electrolyte can be written as31 F
elec
ε0 ¼ 2
Z
Z 2
εr ðrÞ½rΦðrÞ dr þ kB T
þ n-
n þ ln
nln - ðn þ þ n - - 2n1 Þ dr n1
nþ n1 ð6Þ
where εr(r) is the dielectric constant at r (e.g., εr = εl in the lipid phase), nþ = nþ(r) (n- = n-(r)) is the concentration of positive (negative) salt ions at the position r, and n1 is the concentration at infinity. This is a general form of electrostatic free energy: the first term accounts for the electrostatic energy, while the second term describes the entropic penalty for redistributing monovalent salt ions. To obtain the free energy of a WS cell, we solve the PB equation in eq 2 subject to the boundary conditions listed above and carry out the first integral over the entire volume of the WS cell and the second integral over the aqueous phase of the WS cell. The resulting free energy or the WS free energy is denoted as F WS . The Divalent Case. The previous WS approach suppresses finite ionic sizes of counterions. It is thus expected to work well for the monovalent case. Monovalent counterions can only form a loose diffusive layer near a charged surface, and their size is not a crucial parameter. However, multivalent counterions interact more strongly with anionic charges,4,20-22,33 and thus charge discreteness plays a more significant role.20-22 Accordingly, we implement our WS approach by incorporation of finite ionic sizes of bound divalent counterions. An important consequence is that the counterion charge overcompensates that of an anionic lipid, thus producing nonuniform charge distributions on the membrane surface. Under the “right” conditions, this leads to the interesting phenomenon of charge reversal or inversion32-34 (also (33) McLaughlin, S.; Mulrine, N.; Gresalfi, T.; Vaio, G.; McLaughlin, A. J. Gen. Physiol. 1981, 77, 445–473. (34) Nguyen, T. T.; Grosberg, A.Yu.; Shklovskii, B. I. Phys. Rev. Lett. 2000, 85, 1568-1–1568-4.
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see ref 21 and references therein). To capture charge discreteness and nonuniformity, we treat bound divalent counterion as charged spheres of some diameter D (D = 3 A˚). (Because of the way we treat divalent ions, ionic sizes and charge nonuniformity are intimately related.) On the other hand, monovalent ions will remain as featureless particles as often assumed in the PB approach. Equation 6 along with the boundary conditions in eqs 3-5 can still be used to calculate the electrostatic free energy. In principle, the PB equation in eq 2 can be modified for the divalent case (see for instance eq 4 in ref 8). This is an essential step, if we want to calculate the spatial distribution of divalent salts within the PB approach. Here, we use a two state model, in which divalent counterions are in one of these two states, “free” and “condensed” (paired with an anionic lipid).21-23 At small bulk concentrations, free ones mainly contribute to the entropy and will be accounted for later (the last term in eq 7).35 The energy of bound ones will be calculated based on the PB equation. The difference between the monovalent and divalent cases is that the WS cell now containsa divalent counterion at its center, as illustrated in Figure 3b. To further proceed with the free energy calculation in the presence of divalent counterions, we “reconstruct” our WS cells so that each cell now contains one divalent counterion paired with a central anionic charge right below, as illustrated in Figure 3b. As a result, each cell can contain more than one anionic lipid (Figure 3b), in contrast to the monovalent case (Figure 3a). The area of each WS cell (or the membrane area per WS cell) and thus the number of anionic lipids per WS cell can be determined by free energy minimization (cf. eq 7). While the central lipid is treated as discrete as before, other lipid charges are assumed to be smeared out on the surface of the membrane (see (i) in the box on the right). (Here, the central lipid simply refers to the discrete lipid charge at the center of each cell). Our motivation here is to simplify each WS (35) What is left out is the electrostatic screening due to free divalent salts, typically in the low millimolar range. To justify this, first note that the surface potential of a highly charged surface is almost independent of bulk salt concentrations (see Chapter 12 of ref 4). On the other hand, in the two-state model, free ions can influence counterion association only through the surface potential (see, for instance, eq 10 in ref 20). It is thus clear that free ions will contribute to counterion condensation mainly through their entropy, more precisely the entropic penalty for bringing them onto the membrane surface.
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cell, while keeping important microscopic details, especially finite ion sizes. To this end, recall that the ion (counterion-central lipid) pair as a whole will not interact strongly with other lipids, since the pair is monovalent and distant from other backbone charges, as is particularly the case for partially charged membranes, which we consider here. This implies that the effect of charge discreteness is the most pronounced for the (attracting) ion pair. Accordingly, the continuum limit taken in (i) will not alter appreciably the attraction of the counterion to the membrane. As a result of this simplification, the WS cell restores cylindrical symmetry, as it should. This will significantly reduce the computational load, similarly to the monovalent case. There is, however, an energy shift, because of this alternation of the backbone charges, which is equivalent to shifting the energy reference. Our WS free energy obtained for the charge distribution in (i) has to be corrected (see Appendix). To understand the physical origin of and compensate for the energy shift, compare the two different backbone charge distributions within each WS cell: a discrete central lipid charge in the uniform background of other lipid charges ((ii) in the box) and a discrete lipid charge distribution ((iii) in the box). The free energy difference between the two is the free energy change caused by the reference shift; see Appendix for a more systematic treatment of this, based on a diagrammatic expansion of the WS free energy. In the discrete case (ii), the free energy of each lipid is calculated following the approach presented in the previous subsection (the Monovalent Case), where a WS cell is defined by one charged phospholipid. The electrostatic free energy of the WS cell in (iii) is the single-lipid WS free energy times the number of anionic lipids in the WS cell, and can thus be obtained. In summary, our free energy calculation consists of a few intermediate steps, as depicted in Figure 3(i)-(iii). First, the free energy, F 1 , is calculated for the charge distribution in (i), which is the same as in Figure 3b except for the backbone charge distribution. The correction term to compensate for the resulting energy shift would be ΔF ¼ F 2 - F 1 . Finally, the free energy of the WS cell in panel b after this correction becomes F WS ¼ F 3 - ΔF ¼ F 3 - F 2 þ F 1 . See Appendix for the details of this procedure.
Membrane Free Energy Monolayers. In this section, we construct the total free energy of a lipid bilayer membrane (per lipid), as a function of a few independent parameters. Our preferred membrane parameters are the membrane curvature (C) and the headgroup area (ah) from which other parameters are derivable. In the divalent case, an important electrostatic parameter is the planar density of divalent counterions, which sets the area of each WS cell, AWS. Notice that C0 is specified only with respect to a reference surface. A convenient choice is the so-called “neutral surface”, where bending and stretching are decoupled27,29,30 (this should not be confused with an electrically neutral surface). For a monolayer, the location of the neutral surface can vary appreciably as the elastic properties of constituent lipids are altered, as evidenced later (see Figure 5). The monolayer free energy per lipid (possibly in the presence of divalent counterions) can be written as FðC, ah , AWS Þ ¼
k 2 þ γai þ τlhc þF ah
WS
ah aion þ kB T ln AWS n2 AWS vion Langmuir 2010, 26(18), 14737–14746
ah AWS ð7Þ
where aion and vion are the cross sectional area and volume of divalent ions, respectively, and n2 is the bulk density of divalent ions. (Recall ai is the average area per lipid water interface, lhc is the hydrocarbon length; k, γ, and τ are corresponding conjugate parameters characterizing the elastic properties of phospholipid molecules.) The second last term accounts for the electrostatic free energy per lipid molecule. The last term in eq 7 represents the entropic penalty for bringing divalent ions from the solution and confining them to the membrane surface. For n2 = 0, the last term should be dropped; the meaning of AWS is then different, as discussed earlier. Other parameters such as ai and lhc are not independently changeable but are derivable from C, ah, and vhc, the volume of each lipid tail (assumed to be a constant). To see this, note that ah and ai subtend the same solid angle with respect to the common origin but represent different radii of curvature (they are “parallel surfaces” of each other28,30), that is, Rh = 1/C þ lhc þ rh and Ri = 1/C þ lhc, respectively. With the convention that the curvature C of the monolayer in Figure 1 is negative, the area ai can be written as ai ¼ ah
1=C þ lhc 1=C þ lhc þ rh
2 ð8Þ
To relate lhc to other geometrical parameters, consider the volume of a spherical shell specified by its outer and inner radii, Rh and Ri, respectively. The volume vh is then the shell volume divided by 4πRh/ah. This consideration leads to ai
vhc ¼ 2 1 þ lhc 3 C
"
1 þ lhc C
3
3 # 1 C
ð9Þ
This equation can be solved for lhc in terms of vhc, C, and ah. The relations in eqs 8 and 9 enable us to express the free energy of a lipid monolayer in terms of C, ah, and AWS, as assumed in eq 7. Bilayers. In principle, our free energy analysis can be extended to the case of a bilayer membrane. Imagine coupling two monolayers into a bilayer and bending it. Some subtlety arises from the fact that the bilayer coupling represents a global constraint. How this constraint is felt by individual lipids is model dependent.2,8,30,36 A few molecular models for lipid arrangements in a bilayer have been known. For instance, “connected” and “unconnected” bilayers are often used and compared with each other.2,3 The connected bilayer model assumes that the two layers are not allowed to slide against each other; they are “glued” together. In the unconnected bilayer model, each layer is permitted to slide past the other. Not surprisingly, there is no unique way of analyzing lipid arrangements caused by bending, and thus the computation of bending moduli replies on a specific model.2,30 On the other hand, the (local) spontaneous curvature of a lipid bilayer membrane reflects any asymmetry in molecular “shapes” of lipids between the two layers and is considered as a local quantity (unless the two layers are physically coupled by any mechanism).9 It suffices to use the unconnected model in the computation of C0.25 In contrast to the case of monolayers, the neutral surface of a symmetric bilayer membrane always coincides with its midplane. A charge imbalance between the two layers, for instance, can alter this picture. Nevertheless, one can argue that this effect is minor; the electrostatic effect can be considered as renormalizing γ, (36) Miao, L.; Seifert, U.; Wortis, M.; D€obereiner, H.-G. Phys. Rev. E 1994, 49, 5389–5407.
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which has a minor effect on the neutral surface, as evidenced later (see Figure 5). With this simplification, we measure C of a bilayer membrane with respect to the midplane, whether the membrane is symmetrically charged or not,37 while we explicitly construct the neutral surface of each layer. In our approach, the radius of curvature is always measured from the end of the hydrocarbon tails of lipid molecules (see Figure 1). For a monolayer, a more convenient choice is its neutral surface, which is significantly different from the surface formed by the tail ends. However, our monolayer analysis is only an intermediate step in our approach to a bilayer. For a bilayer, however, it proves useful to use the tail-end surface as a reference, which approximately represents the neutral surface as discussed earlier. Even in the unconnected case, the two layers should remain attached to each other, and their curvatures with reference to their interface are the same in magnitude but are opposite in sign. This is the only constraint imposed on each layer. This implies that the total free energy of the bilayer per lipid can be written as out in in in Fbl ¼ F out ðC, aout h , AWS Þ þ F ð - C, ah , AWS Þ
ð10Þ
where the superscripts “out” and “in” refer to the inner and outer layer, respectively. This free energy is to be minimized with respect out in out to five parameters: C, ain h , ah , AWS, and AWS, as detailed in the next section.
Results Monolayers. Optimal Headgroup Area. We have first calculated the equilibrium or optimal headgroup area a0 of a monolayer by free energy minimization for a planar surface (a0 = ah = ai) in the presence or absence of divalent counterions and have plotted our results in Figure 4. This effort illustrates how lipid and electrostatic parameters influence each other. We have chosen R = 0.3 and used various choices of lipid parameters as depicted in different colors (see the legend): γ = 0.06,0.12kBT/ A˚2 2,3,26,38 and τ = 0,0.004kBT/A˚2.27 Throughout this paper, T = 300 K. In all cases here and below, we have chosen k such that a0 = 64.9 A˚2 for the corresponding electrically neutral surface (R = 0, not to be confused with the “neutral surface”), as marked by the dotted line. This explains why a0 values tend to the dotted line as n1 increases, that is, as the electrostatic interaction becomes more screened. In the absence of divalent counterions (the top four curves with unfilled symbols), lipid charges enlarge the headgroup more effectively at lower salt concentrations, as expected. In the presence of as small a concentration as 5 mM of divalent counterions (the bottom four curves with filled symbols), however, the headgroup shrinks compared to the corresponding uncharged case (the dotted line). Intriguingly, the general trend observed for the monovalent case is reversed. This is not unexpected, since in this case nonuniform charge distributions on the membrane surface can induce a negative lateral pressure, which tends to shrink the area occupied by each lipid. An important consequence of this is that the presence of a small concentration of multivalent counterions can reverse the sign of C0 of a lipid membrane, as evidenced later. Importantly, the headgroup shrinkage in this case is well correlated with the (37) Note that since the thickness is not necessarily the same for the two layers if bent, the interface between the layers is not always identical to the midplane. But the difference is minimal, especially for small C. (38) For a recent measurement of area stretch moduli (KA), see Rawicz, W.; Olbrich, K. C.; McIntosh, T.; Needham, D.; Evans, E. Biophys. J. 2000, 79, 328– 339. There is a simple relationship between KA and γ: KA = 4γ, from which γ can be extracted.
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Figure 4. Relaxed headgroup area lipids in a monolayer as a function of n1 (the concentration of monovalent positive salt ions in bulk), in the presence (filled squares) or absence (unfilled squares) of divalent counterions. Here R = 0.3, lipid parameters γ and τ are adjusted together with k such that a0 = 64.9 A˚2 for R = 0 (uncharged case). In the presence of 5 mM divalent counterions, the headgroup area shrinks compared to the corresponding uncharged case.
observed MS channel closing induced by trivalent counterions (Gd3þ);16 the main difference is that Gd3þ is expected to shrink lipid headgroups more effectively than Mg2þ or Ca2þ. In both cases (filled and unfilled symbols), the electrostatic effect is less significant if γ is larger (thus the monolayer is stiffer). This is already hinted in our finding in Figure 4 that the τ-dependent tail elasticity is less important for larger γ. Our results in Figure 4 illustrate how the elastic and charge properties of lipids are interrelated. Neutral Surface. Our analysis in this section so far is limited to a flat surface. The free energy of a monolayer (or a bilayer) if bent is most conveniently expressed with respect to its neutral surface, which will not suffer from stretching upon bending. Imagine bending a uniform elastic sheet, which has constant material properties and thickness. By symmetry, the geometric midplane coincides with its neutral surface. However, this picture does not necessarily apply to a lipid aggregate, except when it is a symmetrical bilayer. In our approach, it is straightforward to find the neutral surface. For a flat layer, ah = ai = a0. Upon bending, ah will no longer remain the same as a0 but its equilibrium value can be obtained by free energy minimization; similarly, the equilibrium lhc can be estimated. For spherical bending we consider here, the neutral surface can be located from a purely geometrical consideration. Imagine translating the equilibrium headgroup area ah in the normal direction; the neutral surface is where the cross sectional area of lipid is the same as a0. Let δN be the location of the neutral surface for a monolayer, defined as the distance from the end of lipid tails (see Figure 1). To examine the dependence of δN on the elastic and charge properties of lipids, we have plotted δN (as well as lhc) in Figure 5 in the presence or absence of divalent counterions. We have chosen a few combinations of γ and τ (see the legend). Let us compare the two cases, γ = 0.06kBT/A˚2 and γ = 0.12kBT/A˚2 (with the same τ = 0.004kBT/A˚2). The neutral surface for the “stiffer” case (larger γ) is closer to the headgroup-tail interface, as expected from the following picture. As γf¥, the neutral surface is expected to coincide with the interface at which the interfacial tension operates. As τf0, the neutral surfaces for the stiffer and softer cases tend to collapse onto each other. This is not surprising, since Langmuir 2010, 26(18), 14737–14746
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Figure 5. Location of neutral surface (δN) and thickness (lhc) of a lipid monolayer, as a function of the curvature C. We have chosen n1 = 50 mM and R = 0.3. Our results here show how δN implicates lipid elastic parameters γ and τ. Intriguingly, δN is almost independent of salt ions. Despite the seeming gradual change in δN with C, the location of the neutral surface remains invariant if “corrected” for thickness change (see lhc curves).
the location of the neutral surface is solely determined by the competition between the headgroup repulsion and the surface tension; for our particular choice of the monolayer free energy in eq 7, one can show that for τ = 0 the neutral surface coincides with the headgroup region,30 independently of γ, as also shown in our results for τ = 0. The gradual change of δN is to reflect the thickness change. If the hydrocarbon chain deforms uniformly, the “relative” position (or the position along the contour of the chain) of the neutral surface is invariant. Importantly, the position of the neutral surface is almost insensitive to salts. This implies that it is mainly determined by nonelectrostatic contributions. At first glance, this finding may sound puzzling but can be understood as follows. The electrostatic contribution to the free energy in eq 7 can be considered as renormalizing γ. As indicated above, for τ = 0, δN is independent of γ30 (electrostatic effects as well), in good agreement with our results in Figure 5. For τ > 0, δN changes as γ changes, but the change is only moderate (∼10%), even when γ doubles. We expect the change to be more pronounced for larger τ. It is also conceivable that the dependence of δN on electrostatic effects may be model dependent. (Another commonly used model is the “harmonic-spring” model for a lipid aggregate.29,30) We believe that experimentally more accessible quantities such as C0 are not quite model dependent, as is particularly the case for a bilayer, where the nonelectrostatic contributions to C0 of the two layers balance out. This together with our results in Figure 5 allow us to choose the midplane of a bilayer as its neutral surface, even if the bilayer is asymmetrically charged, as long as the nonelectrostatic properties of the two layers are the same (see the next subsection for related comments on asymmetrical bilayers). This does not mean that the C0 of a bilayer is not sensitive to charge asymmetry as shown below. Spontaneous Curvature of a Monolayer and the Formation of HII Phases. The preferred structure of lipid membranes is controlled by packing shapes of the constituent lipids4 and thus by the ionization status of headgroups.17 Indeed, a recent experiment on PS-containing membranes shows that at low pH (R = 0) the membrane prefers to form reverse hexagonal (HII) phases (thus C0 < 0), while the sign of C0 is inverted at neutral or high pH. Recall that we only consider spherical bending so as to utilize the Langmuir 2010, 26(18), 14737–14746
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symmetry assumed in Figure 2. However, this will not limit the applicability of our results. What our approach predicts is the preferred structure or morphology of lipid aggregates, which will eventually dictate phases they form. The only structural requirement for the formation of HII phases is the inverted cone shape,5 which translates into C0 < 0. To offer a theoretical basis for the observation with PScontaining membranes, we have calculated the spontaneous curvature of a monolayer for a few choices of R and plotted our results in Figure 6, as a function of n1. The lipid parameters have been chosen so as to mimic PS, negative C0 when R = 0: γ = 0.12kBT/A˚2 and τ = 0.004kBT/A˚2. Our results (open symbols) are illuminating, since they imply that at low pH (R = 0) PScontaining membranes tend to form HII phases (see the illustration), while at higher pH lamellar phases (or positively curved structure) are stabilized by headgroup repulsions. Our results also offer an alternative mechanism of HII-phase formation at neutral pH, that is, HII phases stabilized by charge correlations due to divalent counterions, as illustrated in Figure 6. Charge correlations reduce the optimal area of charged headgroups, as evidenced in Figure 4. This theoretical prediction is paralleled with the longstanding observation that divalent counterions induce HII phases of lipids, which would otherwise form lamellar phases.5 Bilayers: Spontaneous Curvature of Asymmetrically Charged Bilayers. In contrast to the case of a lipid monolayer, the preferred curvature of a lipid bilayer, that is, the value of C at which the membrane free energy is minimized, is determined by asymmetries between the two layers. As a result, a perfectly symmetrical bilayer has a vanishing preferred curvature. There are two kinds of asymmetry (see refs 9, 10, and 36 and references therein). First, any asymmetry in packing shapes between the constituent layers results in a nonzero spontaneous curvature. This reflects local properties of the bilayer. Second, any mismatch in relaxed areas of the two layers can induce membrane bending. Here the relaxed areas refer to the neutral surfaces, and are invariant upon bending if the two layers are “unconnected”.39 The resulting preferred curvature has a global or nonlocal character. For a bilayer, we consider here (one consisting of two identical layers except for charge properties), the nonlocal out in out preferred curvature can be expressed as Cnl 0 = (a0 - a0 )/[(a0 þ in 9 a0 )δN]. Note here that δN is the location of the neutral surface of each layer for C = 0 and is essentially the same for both layers. In our approach, we mainly focus on the computation of C0, the (local) spontaneous curvature; Cnl 0 can be readily read off from our results in Figure 4. Furthermore, in a more general case, Cnl 0 is also influenced by the number of lipids in each layer.9,10,36 In this sense, Cnl 0 is a less intrinsic quantity than C0. Also, as it turns out, the interrelationship between lipid properties and bending is much less obvious for C0 (cf. Figure 8), and we focus on calculating C0. The simultaneous presence of both local and global effects makes it challenging to determine C0 and Cnl 0 separately. To focus on C0, we allow the bilayer to relax at its preferred area in difference per lipid, that is, Δa = aout 0 - a0 . This is equivalent to minimizing the free energy of each layer independently of the other layer with respect to the curvature of the bilayer. To be specific, we have considered a bilayer, in which the inner layer is neutral, while the outer layer contains 30% (R = 0.3) charged lipids, as for the monolayer case considered in the previous (39) This does not mean that two neutral surfaces are needed for describing bilayer bending. A common neutral surface can be constructed as discussed here and in ref 36. In our case, however, the midplane essentially coincides with the bilayer neutral surface, as discussed in subsection Neutral Surface in section Results.
DOI: 10.1021/la102052r
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Figure 6. Spontaneous curvature of a charged lipid monolayer as a function of n1 for a few choices of R: R = 0, 0.1, 0.3. The lipid parameters have been chosen to mimic PS (phosphatidylserine): γ = 0.12kBT/A˚2 and τ = 0.004kBT/A˚2. When the charge on the lipid is turned off by lowering pH, the spontaneous curvature C0 is negative. When the fraction of charged lipids increases, the sign of C0 can be inverted, as is particularly the case for R > 0.2. In the presence of 5 mM divalent counterions in solution, however, C0 < 0 for the entire range of n1. Divalent counterions invert the sign of C0 more effectively for larger R.
subsection. Our results can readily be extended to the case in which the inner layer is charged, simply by changing the sign of C0. Except for the charge properties, the two layers are assumed to be identical. In principle, our approach can be generalized to the case where nonelectrostatic parameters are different for the two layers, by introducing different lipid parameters for the two layers. Alternatively, the C0 of a bilayer can readily be obtained, once C0 of each layer is known.36 Strictly speaking, this strategy works well if the two layers are electrically decoupled or weakly coupled, that is, the electrostatic properties of one layer is unperturbed or weakly perturbed by the presence of the other. For the parameters we use in this paper, εl/εwκd , 1 (d = 2lhc), indicating weak electric coupling. This is most obvious within the linearized PB approach to a flat membrane (see, for instance, ref 8, especially eq 13, where the solution of the linearized PB equation is expanded in powers of this ratio). Figure 7 displays C0 as a function of monovalent salt concentration, n1, in the presence or absence of divalent counterions. In the absence of divalent counterions (unfilled symbols), the electrostatic repulsion between charged lipids induces a positive curvature. In other words, the membrane tends to bend toward the electrically neural, inner layer. This is paralleled by our finding that the repulsion enlarges the headgroup area (see Figure 4). However, it is worth noting that our C0 results reflect both inplane and out-of plane deformations of the membrane, while only in-plane deformations are taken into account in our a0 calculations. Curvature can be induced not only through in-plane lipid deformations (that is, a0 changes in the outer layer) but also through the modification of the ionic cloud of the outer layer. Upon bending toward the inner layer, the ionic cloud expands, similarly to the case of in-plane stretching. It is this entropic gain that induces a positive C0.8 On the other hand, the presence of 5 mM of divalent ions inverts the sign of spontaneous curvature for the entire range of n1 shown in the figure, as already hinted in Figure 4; the nonuniform charge distribution on the outer layer in this case means that the layer can lower the electrostatic free energy by curving inward (via both in-plane and out-of plane deformations), inducing a negative spontaneous curvature. As n1 increases, however, the electrostatic effect diminishes as indicated in the figure. 14744 DOI: 10.1021/la102052r
Figure 7. Spontaneous curvature of an asymmetrically charged lipid bilayer (R = 0 for the inner layer and R = 0.3 for the outer layer) as a function of n1. The presence of 5 mM divalent counterions in solution inverts the sign of C0. In other words, the bilayer tends to bend toward the charged layer.
In both monovalent and divalent cases, the electrostatic effect is more pronounced for smaller γ or τ, since the membrane is more easily deformable in that case; it is worth noting that τ is also implicated in C0. So far we have determined C0 and δN simultaneously and systematically without using any ad hoc assumption about surface charges, which may obscure the physical picture of electrostatic bending. Our approach has enabled us to determine such parameters as δN and a0 consistently with electrostatic interactions. In the past, however, simplification has often been invoked, which amounts to using prechosen δN for a membrane without the benefit of derivation.8 To test this, we have used prechosen values of the position of the neutral surface δN and plotted the resulting C0 in Figure 6. Here θ describes the relative position of the neutral surface, For instance, θ = 1 means that the neutral surface coincides with the headgroup region, while θ = 0 corresponds to the connected case; for θ = 0.5, the neutral surface lies halfway Langmuir 2010, 26(18), 14737–14746
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Figure 8. Spontaneous curvature of an asymmetrically charged lipid bilayer (R = 0 for the inner layer and R = 0.3 for the outer layer) as a function of θ, the relative position of the neutral surface prechosen for each monolayer. These results show that C0 is sensitive to θ, demonstrating the significance of determining the location of the neutral surface consistently, by free energy minimization. (a-c). Illustrations of a few hypothetical models of bilayer bending, corresponding to a few choices of θ. While (a) represents the connected model, (c) can be realized if tails are flexible (or headgroups are “bulky”); these two limiting features are combined in (b).
between the head-tail interface and the midplane of the bilayer. As θ changes between 0 and 1, C0 changes appreciably and nontrivially. (In our full analysis, θ ≈ 0.7-0.8 for τ = 0.004kBT/ A˚2 and θ ≈ 0.8-0.9 for τ = 0.) Our results in Figure 8 clearly suggest that θ has to be determined according to the energetics of each layer. Interestingly, the peak of C0 appears to occur around our estimated θ value. This is not unexpected, since each layer tends to bend with respect to its neutral surface.
Conclusions In conclusion, we have presented a unified approach to the electrostatic modification of lipid headgroups and its impact on the spontaneous curvature of a lipid membrane. Our effort is distinct from the existing approaches in two major respects. First, in our approach, the elastic and charge parameters are combined in a more coherent manner. This is accomplished by free-energy minimization with respect to lipid parameters for given C (in a salt-dependent manner). Accordingly, the lipid parameters are allowed to relax at their equilibrium values for given C. As a result, lipid molecules can rearrange themselves upon bending so as to be optimally packed, in the sense that the monolayer they comprise is ensured to be deformed with respect to its neutral surface. While this has not been well appreciated in the past, our results illustrate its significance (see Figure 8). Consequently, our approach does not rely on any assumption about how bending influences surface charges. Second, our approach captures lateral and transverse charge correlations on the surface of a charged layer. This is accomplished by incorporating finite ionic sizes into the Poisson-Boltzmann approach, especially for describing the association of a divalent counterion with an anionic lipid (see Langmuir 2010, 26(18), 14737–14746
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Figure 3b). An interesting consequence is that the binding of divalent (or more generally multivalent) counterions brings about nonuniform surface charge distributions on their binding layer. Indeed, the key to understanding the conformational and phase properties of lipid systems in a salty solution is a consistent treatment of their elastic and electrostatic properties. It should be, however, emphasized that the way lipid charges interact with each other can easily be influenced by the surrounding salt ions, especially those forming an ionic cloud around the lipid charges. The reason is that the ionic cloud will be rearranged in response to any change in the lipid charge distribution (see for instance ref 8 as well as ref 19 for a similar issue). Interestingly, for the monovalent case the main contribution to spontaneous membrane curvature is the entropy “stored” in the ionic cloud, as also discussed in ref 8. The ionic cloud tends to expand, but it can expand only if the associated lipid membrane also expands or bends away from the ionic cloud (as indicated in Figures 4, 6, and 7 for the entire range of salt concentrations shown). In our approach, this effect is reflected in increased AWS. This entropic picture can easily break down in the presence of multivalent ions. In this case, the entropic effect should compete with that of charge correlations arising from nonuniform surface charge distributions, which tends to shrink lipid headgroups (see Figure 4). The presence of a few millimolar concentration of divalent counterions suffices to invert the sign of spontaneous membrane curvature (see Figures 6 and 7). This is also responsible for the formation of HII phases of PS (phosphatidylserine)containing lipids, which would otherwise favor lamellar phases or positively curved structures, as shown in Figure 6. A general picture that has emerged from our approach is paralleled by the experimental observation that the electrostatic modification of lipid headgroups is one of the key determinants of lipid packing, which in turn influences membrane functions16 or the structure and phase of lipid aggregates.17 On the other hand, the relative role of electrostatic6-8 and protein-based bending9 is unclear. Nevertheless, our results reported here can offer a quantitative basis for various experiments with pure lipid membranes (e.g., Ca2þ-induced membrane fusion11-13) or those with biological membranes where ion valence is a key parameter (e.g., ref 16). In a recent experimental study,16 for instance, multivalent counterions such as Gd3þ are described as effectively adjusting the lateral packing of PS, especially by shrinking a PS-containing membrane; they can thus block mechanosensitive channels imbedded in the membrane. This observation is aligned well with the general picture presented here (e.g., Figure 4), except that Gd3þ is expected to be more potent than divalent ions. In principle, our approach can be extended to the analysis of other membrane parameters such as (both mean and Gaussian) bending moduli as well as to the study of Ca2þ-induced lipid ordering and lipid phase transitions. A related (but more involved) problem is membrane perturbations by cationic antimicrobial peptides (CAPs).40-43 CAPs are known to selectively disrupt bacterial (cytoplasmic) membranes, initially by asymmetrical incorporation into the outer layer, carrying a large fraction of anionic lipids (PG). Interestingly, they can significantly soften their binding membranes,44 likely through the combined effects: membrane thinning and charge-correlations. Both effects can soften the membrane. We leave this membrane-softening mechanism for future consideration. (40) (41) (42) (43) (44)
Zasloff, M. Nature 2002, 415, 389–395. Brogden, K. A. Nat. Rev. Microbiol. 2005, 3, 238–250. Taheri-Araghi, S.; Ha, B.-Y. Phys. Rev. Lett. 2007, 98, 168101-1–168101-4. Taheri-Araghi, S.; Ha, B.-Y. Soft Matter 2010, 6, 1933–1940. Bouvrais, H.; et al. Biophys. Chem. 2008, 137, 7–12.
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Acknowledgment. B.Y.H. acknowledges the financial support of NSERC (Canada).
Appendix In this Appendix, we present detailed intermediate steps that lead to the free energy correction discussed at the end of section Theoretical Model. To this end, we carry out a diagrammatic expansion of the electrostatic energy of a Wigner-Seitz cell for the divalent case. This effort will offer a transparent physical picture for the origin of an energy shift caused by our simplification that amounts to replacing Figure 3b by Figure 3i. Our WS cell approach requires a rotational symmetry about the axis joining the central backbone charge and the origin with respect to which the radius of curvature is measured. To satisfy this requirement, we smear out the backbone charges except the “central” one interacting with a divalent counterion. (See Figure 2 for the illustration of the central ion as well as the caption for Figure 3.) Since the effect of charge discreteness is the most pronounced for the central charge-counterion pair, this simplification will not alter appreciably the binding affinity of the counterion to lipid charges. On the other hand, the energy reference and the electrostatic bending energy will be modified. The electrostatic free energy of each Wigner-Seitz cell has to be corrected. To understand the aforementioned free-energy shift more systematically, consider the diagrams in Figure 9. Here, the double sum is to be carried out over all charge pairs: counterion-lipid as well as lipid-lipid pairs. (In principle, the electrostatic free energy can be expressed as a sum over the positions of all ions. For simplicity, mobile ions are not shown in the diagram.) On the other hand, the single sum refers to the interaction of the counterion with lipid charges. In the first step, we decompose all the pair interactions into two diagrams: a
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Figure 9. Diagrammatic expansion of the electrostatic free energy of a Wigner-Seitz cell for the divalent case (see Figure 3b): the continuum limit and the resulting origin of a free-energy shift. The top left one illustrates one WSC defined by a divalent counterion and a few lipid charges it interacts with. For simplicity, neutral lipids are not shown and bending is suppressed.
single-sum diagram and a double-sum diagram as indicated by the first equality. The single-sum diagram can be approximated by the one right below it, where all the charges except the pair shown are smeared out. Note that the electrostatic term in eq 6 (the first term) includes both single-sum and double-sum interactions. To recast our diagrammatic expansion into a more illuminating form, we add a “canceling pair”, as indicated in the figure, and absorb the upper one into the single-sum diagram, as described by the bottom diagram. Finally, the total interaction energy is the sum of three diagrams labeled as (i), (ii), and (iii), which correspond to the illustration (i), (ii), and (iii) in Figure 3, respectively.
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