Membrane Perturbations Induced by Integral Proteins: Role of

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Membrane Perturbations Induced by Integral Proteins: Role of Conformational Restrictions of the Lipid Chains Sylvio May Institut fu¨ r Molekularbiologie, Friedrich-Schiller-Universita¨ t Jena, Winzerlaer Strasse 10, 07745 Jena, Germany Received March 18, 2002. In Final Form: May 20, 2002 Integral membrane proteins affect lipid bilayers in which they reside. That is, a protein induces an elastic response of the membrane which, for instance, may be caused by the so-called hydrophobic mismatch between the protein and the host bilayer. In addition to that, there are conformational restrictions imposed on those acyl chains that reside in immediate vicinity to the rigid protein surface. We suggest a method how this entropic confinement can approximatively be incorporated into membrane elasticity theory. To this end, we model a (sufficiently large) integral protein by an impenetrable wall, contacting a symmetric, fluidlike lipid bilayer. We represent the flexible lipid chains by fluctuating directors and calculate the entropy loss upon interaction with the wall. The elastic free energy stored in the membrane is calculated from the average director positions. We base elasticity theory on two order parameters, namely, stretching/ compression of the lipid chains and their ability to tilt with respect to the hydrocarbon chain-water interface. We show that conformational restrictions of the lipid chains in the vicinity of the wall modify the microelastic behavior of lipid membranes. In particular, the wall induces an increase in the tilt modulus and a spontaneous tilt. Both quantities depend on the distance to the wall. Our combined model generally predicts nonmonotonic membrane perturbation profiles that are typically characterized by a local membrane thickening 1-2 nm away from the wall. Optimal lipid-protein interaction is achieved for proteins that have a small negative hydrophobic mismatch. Positive spontaneous curvature of the lipids generally lowers the free energy of inserting a rigid inclusion into a lipid membrane.

Introduction Lipid membranes are self-assembled, soft materials that are able to respond to various structural perturbations. Such perturbations arise, for instance, from the insertion of transmembrane proteins (or peptides) into the lipid bilayer. The driving force for protein insertion is provided by the hydrophobic effect1 which strongly couples the hydrophobic span of the protein (and similarly for peptides) to the hydrocarbon core of the bilayer. Partitioning of integral proteins into membranes appears to occur even if the hydrophobic protein thickness does not exactly match that of the host bilayer. The implications arising from this so-called “hydrophobic mismatch”2-3 were intensively studied in the past both experimentally4-7 and theoretically.8 Yet, also membrane-matching proteins perturb membranes. Experimental evidence shows that integral proteins alway affect membrane lipids, at least those in the immediate vicinity of the protein.9 In fact, this is no surprise because the proteins behave as rigid bodies in a fluidlike membrane. Hence, the hydrocarbon chains of the lipids suffer from motional restrictions because they cannot penetrate into the protein interior. Reliable models for lipid-protein interaction and their corresponding energies are of principal interest for (1) Tanford, C. The Hydrophobic Effect, 2 ed.; Wiley-Interscience: New York, 1980. (2) Mouritsen, O. G.; Bloom, M. Biophys. J. 1984, 46, 141-153. (3) Bloom, M.; Evans, E.; Mouritsen, O. G. Q. Rev. Biophys. 1991, 24, 293-397. (4) Webb, R. J.; East, J. M.; Sharma, R. P.; Lee, A. G. Biochemistry 1998, 37, 673-679. (5) Ge, M.; Freed, J. H. Biophys. J. 1999, 76, 264-280. (6) Killian, J. A. Biophys. Biochim. Acta 1998, 1376, 401-416. (7) Dumas, F.; Lebrun, M. C.; Tocanne, J. F. FEBS Lett. 1999, 458, 271-277. (8) Gil, T.; Ipsen, J. H.; Mouritsen, O. G.; Sabra, M. C.; Sperotto, M. M.; Zuckermann, M. J. Biophys. Biochim. Acta 1998, 1376, 245-266. (9) Marsh, D. Mol. Membr. Biol. 1995, 12, 59-64.

understanding membrane partitioning and lateral organization of integral proteins. In the past, two different types of models for lipid-protein interaction were proposed. One is based on some kind of microscopic model, and the other employs membrane elasticity theory. While the former accounts in more or less detail for the modifications of the lipid properties near rigid membrane proteins,8,10-12 the latter describes the membrane as an elastic continuum.13,14 The two approaches do not always lead to similar conclusions as we shall shortly outline in the following. In a recent work, Fattal and Ben-Shaul12 have used a detailed (mean-field level) statistical lipid chain packing theory to calculate the free energy cost of inserting a rigid wall into a lipid bilayer. The thickness of the wall was variable, which allowed the authors to study the consequences of hydrophobic mismatch. Interestingly, the optimal lipid-protein interaction energy was found for a small but notable negative hydrophobic mismatch (where the hydrophobic protein thickness is smaller than that of the membrane). The estimated free energy of the wallinduced bilayer perturbation was ∆Fbl ≈ 0.37 kBTL/Å for a wall of length L (kB is Boltzmann’s constant and T is the absolute temperature). Also, a notable average tilt of the lipid chains in the vicinity of the wall was observed, pointing away from the wall for any choice of the hydrophobic mismatch. The relaxation of the membrane thickness from the wall to the bulk was assumed to be an exponential function. The corresponding decay length was found to be surprisingly small, ξ ≈ 3-6 Å. It is worth mentioning that the results of Fattal and Ben-Shaul have (10) Marcˇelja, S. Biophys. Biochim. Acta 1976, 455, 1-7. (11) Sintes, T.; Baumga¨rtner, A. Biophys. J. 1997, 73, 2251-2259. (12) Fattal, D. R.; Ben-Shaul, A. Biophys. J. 1993, 65, 1795-1809. (13) Goulian, M. Curr. Opin. Colloid Interface Sci. 1996, 1, 358-361. (14) Kim, K. S.; Neu, J.; Oster, G. Biophys. J. 1998, 75, 2274-2291.

10.1021/la025747c CCC: $22.00 © 2002 American Chemical Society Published on Web 06/27/2002

Membrane Perturbations Induced by Proteins

become a frequently used reference for the estimation of the magnitude of lipid-protein interactions.15-19 Membrane elasticity theorysin its most simple versions accounts for the (protein-induced) stretching and for the splay energy of the lipid chains.20-23 Only one single socalled order parameter is employed to fully describe the conformation of the membrane. This type of membrane elasticity theory has frequently been used to calculate the protein-induced spatial relaxation of a perturbed membrane.24-28 Also here, a corresponding decay length can be specified; its value is ξ ≈ 10 Å, clearly larger than in the above-mentioned microscopic model. Despite the usage of only one single-order parameter, the predictions of membrane elasticity theory were found in remarkable agreement with a number of experimental observations. This concerns the estimation of the so-called spring constant29 which describes the response of a lipid membrane to a given hydrophobic mismatch. It also served well to predict the average thinning of a lipid membrane upon insertion of short transmembrane peptides.30 Recently, it was suggested that another energetic contribution should be taken into account when applying membrane elasticity theory to structurally perturbed membranes, namely, the ability of the lipid chains to tilt with respect to the hydrocarbon chain-water interface.31-36 Lipid tilt is possible despite the fact that the membrane is in the fluidlike state. Here, the tilt refers to the average orientation of the flexible chains. The ability to tilt gives rise to a second order parameter which generally lowers both the perturbation free energies and the decay length of the spatial membrane relaxation. The tilt energy is determined by an elastic modulus, the so-called tilt modulus, κt, which was very roughly estimated to be on the order of κt ≈ 0.1 kBT/Å2. With the lipid tilt degree of freedom included, the decay length, ξ, depends on the choice of κt. It becomes similar to the findings of Fattal and Ben-Shaul,12 namely, ξ ) 3-6 Å, only for rather small values of the tilt modulus, where κt < 0.02 kBT/Å2. Yet, for κt ≈ 0.1 kBT/Å2 there is still discrepancy between membrane elasticity theory and the above-mentioned chain-packing theory. (15) White, S. H.; Wimley, W. C. Annu. Rev. Biophys. Biomol. Struct. 1999, 28, 319-365. (16) Ben-Shaul, A.; Ben-Tal, N.; Honig, B. Biophys. J. 1996, 71, 130138. (17) Ben-Tal, N.; Ben-Shaul, A.; Nicholls, A.; Honig, B. Biophys. J. 1996, 70, 1803-1812. (18) Kessel, A.; Cafiso, D. S.; Ben-Tal, N. Biophys. J. 2000, 78, 571583. (19) Huang, J.; Feigenson, G. W. Biophys. J. 1999, 76, 2142-2157. (20) Huang, H. W. Biophys. J. 1986, 50, 1061-1070. (21) Harroun, T. A.; Heller, W. T.; Weiss, T. M.; Yang, L.; Huang, H. W. Biophys. J. 1999, 76, 3176-3185. (22) Nielsen, C.; Goulian, M.; Andersen, O. S. Biophys. J. 1998, 74, 1966-1983. (23) Nielsen, C.; Andersen, O. S. Biophys. J. 2000, 79, 2583-2604. (24) Aranda-Espinoza, H.; Berman, A.; Dan, N.; Pincus, P.; Safran, S. A. Biophys. J. 1996, 71, 648-656. (25) Dan, N.; Pincus, P.; Safran, S. A. Langmuir 1993, 9, 27682771. (26) Dan, N.; Berman, A.; Pincus, P.; Safran, S. A. J. Phys. II 1994, 4, 1713-1725. (27) Dan, N.; Safran, S. A. Biophys. J. 1998, 75, 1410-1414. (28) Dan, N.; Safran, S. A. Isr. J. Chem. 1995, 35, 37-40. (29) Lundbæk, J. A.; Andersen, O. S. Biophys. J 1999, 76, 889-895. (30) Harroun, T. A.; Heller, W. T.; Weiss, T. M.; Yang, L.; Huang, H. W. Biophys. J. 1999, 76, 937-945. (31) MacKintosh, F. C.; Lubensky, T. C. Phys. Rev. Lett. 1991, 67, 1169-1172. (32) Fournier, J. B. Europhys. Lett. 1998, 43, 725-730. (33) Fournier, J. B. Eur. Phys. J. E 1999, 11, 261-272. (34) Hamm, M.; Kozlov, M. M. Eur. Phys. J. B 1998, 6, 519-528. (35) Hamm, M.; Kozlov, M. M. Eur. Phys. J. E 2000, 3, 323-335. (36) May, S. Eur. Biophys. J. 2000, 29, 17-28.

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Different predictions of membrane elasticity and chainpacking theory also arise for the free energy, ∆Fbl, of inserting an inclusion into a membrane. Generally, membrane elasticity yields ∆Fbl much lower28 than microscopic chain-packing calculations (where ∆Fbl ≈ 0.37 kBT L/Å was predicted,12 see above). Clearly, the reason could be the neglect of protein-induced conformational restrictions of the lipid chains. In the present work, we suggest a new approach to calculate nonspecific transmembrane protein-lipid interactions based on membrane elasticity theory. The key point of the present theoretical treatment is the observation that rigid membrane inclusions not only induce elastic membrane perturbations but also conformationally confine the neighboring lipid chains. That is, a fluidlike hydrocarbon chain, which adopts many different conformations in an unperturbed membrane, loses conformational freedom near a rigid, membrane-spanning, protein. We suggest a way how this entropic confinement can approximatively be taken into account in membrane elasticity theory. To this end, we represent a flexible lipid chain by a simple director and calculate the entropy loss upon interaction with a rigid inclusion. We shall employ membrane elasticity theory with the lipid tilt degree of freedom included and supplement itsin a self-consistent waysby the entropic confinement of the lipid chains. We show that our approach gives rise to modifications of the microelastic behavior of lipid membranes. In particular, there appears an increase in the tilt modulus and a spontaneous tilt of the lipid chains close to the wall. To keep our approach as simple as possible, we shall considerslike in the work of Fattal and Ben-Shaul12sa single rigid wall that spans the hydrophobic region of the bilayer membrane. Yet, unlike in ref 12 we shall not assume the membrane profile to follow an exponential function. Instead, we functionally minimize the membrane perturbation. We shall show that our approach leads to predictions that are in good agreement with the detailed molecular model in ref 12. In particular, the insertion energy, ∆Fbl, of proteins into membranes is reproduced. Moreover, as in the work of Fattal and Ben-Shaul12 we always find a tilt of the neighboring lipid chains away from the protein surface. However, there is one important difference. Our approach predicts the membrane height profile to be nonmonotonic, exhibiting a maximum about 1-2 nm away from the rigid wall. This implies that the membrane perturbation extends considerably farther into the membrane than only 3-6 Å as predicted in ref 12. In fact, the modifications of the microelastic properties near the wall increase the spatial extension of the membrane perturbation. (The resulting energetics appears to be well predicted by the most simple version of membrane elasticity theory where the tilt modulus κt is assumed infinitely large.) Our model thus suggests that the consideration of the lipid’s conformational confinement near proteins provides a link between the microscopic theory of chain packing and membrane elasticity theory. An important advantage of the present approach is that more complex geometries than a single, rigid, wall can easily be calculated (which would be extremely time-consuming on the basis of chainpacking theory). Finally, our model gives rise to new predictions: in particular, it states that positive spontaneous curvature of the lipids in a membrane generally lowers ∆Fbl, no matter whether there is hydrophobic mismatch or not.

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u′ ) du/dx and θ′ ) dθ/dx. The result is

∆fel(x) )

Figure 1. Schematic illustration of a perturbed symmetric lipid membrane. Shown is only the outer monolayer. The hydrophobic monolayer thickness at position x is h(x), and the lipid tilt angle with respect to the wall is θ(x).

Theory We consider a single, rigid inclusion, residing in a symmetric, fluidlike, lipid bilayer. It is convenient to simply represent the inclusion by a wall of (hydrophobic) height 2hP. The length, L, of the wall is sufficiently large to ignore end effects. Mirror symmetry requires the bilayer midplane to remain planar. Let the midplane be located at the plane z ) 0 of a Cartesian coordinate system, and the x-axis point normal to the wall which is located at x ) 0. The monolayer perturbation then depends on the x-coordinate but is invariant in y-direction. We shall characterize the deformation of, say, the outer monolayer in terms of two functional degrees of freedom (order parameters): One is the relative change of the (local) hydrophobic monolayer thickness, u(x) ) h(x)/h0 - 1, where h(x) is the hydrophobic monolayer thickness at position x, and h0 is the equilibrium value of h(x). The secondorder parameter describes the average tilt angle, θ(x), of the lipid chains with respect to the z-axis (see Figure 1). Note that the lipid layer in its liquid-crystalline state is able to access many different conformations of the hydrocarbon chains as evidenced by the quadrupolar splitting of deuterized alkyl chains in 2H NMR experiments.37 Hence, both u(x) and θ(x) result from averaging of the lipid tail positions over many different chain conformations. In the absence of the wall the monolayer is planar and unperturbed, adopting its equilibrium at h(x) ) h0 and θ(x) ) 0. If the wall is present it induces a perturbation of the outer monolayer (and exactly the same perturbation also for the inner monolayer). We write the corresponding excess free energy, ∆F, of the (outer) monolayer as a sum of an elastic contribution, ∆Fel, and a contribution, ∆Fc, resulting from conformational restrictions of the lipid chains in the vicinity of the wall. Both ∆Fel and ∆Fc are measured with respect to a planar, unperturbed monolayer (for which h(x) ) h0 and θ(x) ) 0). In a continuum approach, ∆F is written as an integral over the corresponding area density ∆f ) ∆fel + ∆fc

∫0∞(∆fel + ∆fc) dx

∆F ) ∆Fel + ∆Fc ) L

(1)

Since both monolayers of a symmetric bilayer are entirely equivalent, the excess free energy of a bilayer, ∆Fbl ) 2∆F, is twice the corresponding value of a monolayer. In the following we shall present our approaches to calculate ∆Fel and ∆Fc. Membrane Elasticity Theory. We assume the wallinduced elastic monolayer perturbation to be small. This allows us to expand the area density, ∆fel, of the elastic free energy up to second order in terms of the deformation profile, u ) u(x) and θ ) θ(x), and the first derivatives, (37) Lafleur, M.; Cullis, P. R.; et M. Bloom, Eur. Biophys. J. 1990, 19, 55-62.

κt K 2 κ 2 u + θ′ - κc0θ′ + (θ + h0u′)2 2 2 2

(2)

where K is the compressional modulus of the monolayer, κ is the monolayer bending stiffness, c0 is the spontaneous curvature, and κt is the tilt modulus. Recall that ∆fel(x) is measured with respect to a planar, unperturbed monolayer. Indeed, for u(x) ) 0 and θ(x) ) 0, it is ∆fel(x) ) 0. We proceed with several further comments regarding the expression in eq 2. First of all, there is no term ∼θ′u, implying that the profile u(x) is measured at the so-called neutral surface.38 It is generally believed that this surface is located close to the glycerol backbone of common phospholipids which then is a reasonable measure for the local hydrophobic monolayer thickness. Second, the quantity φ ) θ + h′ (where h′ ) h0u′) is the tilt angle of a lipid director (which represents the average direction into which a given lipid chain points) with respect to the monolayer shape, given by the function h(x). This implies that for the particular choice θ ) -h0u′ the local lipid director points exactly normal to the function h(x). In this case, ∆fel ) Ku2/2 + h02κu′′2/2 + h0κc0u′′ depends no longer on two but only on one single order parameter, namely, the monolayer thickness dilation u. In fact, this approach was the starting point for a number of previous investigations to derive inclusion-induced membrane perturbation energies.20-22,24,25 Below, we shall determine the consequences of neglecting the lipid tilt degree of freedom. Third, in the most general case the three terms ∼θ2, ∼u′2, and ∼θu′ each have its own prefactor as was recently shown by Fournier.33 Writing these terms in the form ∼(θ + h0u′)2 is based on the assumption that the so-called stress profile within the hydrocarbon core of the monolayer acts in layers parallel to h(x). This assumption seems reasonable since it is generally believed that the main contribution to the stress profile is concentrated around the neutral surface.39 We note that eq 2 is in principal agreement with the elastic free energy derived by Hamm and Kozlov.34 We also note that the conservation of the volume of the hydrocarbon core does not affect the quadratic-order expression, eq 2.36 It is common to base the analysis of inclusion-induced membrane perturbations entirely on elasticity theory.13 In all previous approaches it is assumed that the local elastic properties of the membrane are not affected by the presence of the inclusion. That is, the inclusion enters only through the boundary conditions of the Euler equations that minimize the elastic energy ∆Fel. This approach is certainly applicable as long as the characteristic decay length of the monolayer perturbation is sufficiently large compared to all molecular lipid dimensions. In this case, any structural perturbation due to, say, lipid packing or conformational restrictions of the lipid chains, can be absorbed into the boundary conditions of the Euler equations. However, for lipid membranes this is not necessarily the case. In fact, there is experimental evidence that at most a few lipid shells around integral membrane proteins are perturbed.9 In view of that, one may re-think the way to apply membrane elasticity theory. (38) Helfrich, W. Z. Naturforsch. 1973, 28, 693-703. (39) Ben-Shaul, A. Molecular theory of chain packing, elasticity and lipid protein interaction in lipid bilayers. In Structure and Dynamics of Membranes; Lipowsky, R., Sackmann, E., Eds.; Elsevier: Amsterdam, 1995; Vol. 1.


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Indeed, the present work offers a first step to calculate inclusion-induced modifications of the microelastic behavior. Predictions of Membrane Elasticity Theory. We shall shortly present the results of membrane elasticity theory for an isolated wall. Omitting from ∆F ) ∆Fel + ∆Fc the wall-induced chain conformational confinement, ∆Fc, we must minimize the elastic free energy ∆F ) ∆Fel with respect to the membrane deformation profile, u(x) and θ(x). The Euler equations, h02κtu′′ ) Ku - h0κtθ′ and κθ′′ ) κtθ + h0κtu′, corresponding to eq 2 are solved for appropriate boundary conditions. For an isolated wall the boundary conditions can, most generally, be written as u(0) ) u0, θ(0) ) θ0, and u(∞) ) θ(∞) ) 0. The monolayer deformation profile appears as a damped oscillation

u(x) ) e-ω2x[u0 cos ω1x + g1 sin ω1x] θ(x) ) e-ω2x[θ0 cos ω1x + g2 sin ω1x]

(3)

with

g1 )

g2 ) -

2

ω1(ω1 - 3ω2 )

ω1(ω12 - 3ω22)

and

(

)

K1/2 2h0 1 2h0 (κK)1/2 κt

1/2

(4)

Note that ξ ) 1/ω2 is the decay length of the wall-induced membrane perturbation. Once the deformation profile is known, we can calculate the optimal elastic excess free energy ∆Fel of a monolayer in contact with a rigid wall of length L. To this end, the functions u(x) and θ(x) in eq 3 are inserted into eq 2 and the integration in eq 1 (with ∆fc ) 0) is carried out. The free energy ∆Fel appears as a quadratic function of u0 and θ0 and depends on the elastic monolayer properties

∆Fel ) κL

(

κ K

()

2h0

1/2

+

κ κt

) (( ) 1/2

(( )

2

κ K

K κ

1/2

1/2

+

u02 +

)

κ h0κt

)

(6)

(In fact, this boundary condition was recently interpreted as a lipid-packing constraint by Nielsen and Andersen23 in their analysis of inclusion-induced membrane deformations for circular inclusions in the limit κt f ∞.) The elastic free energy per monolayer for relaxed tilt angle can generally be written in the form

Del* (u0 - u0opt)2 2

∆Fel(u0) ) ∆Felopt +

(7)

where Del* is a spring constant,22 u0opt is the optimal mismatch, and ∆Felopt is the corresponding optimal membrane perturbation free energy. We find

( ()

κc02 ∆Felopt κ )2h0 L 2 K

1/2

(κK)1/2 κ 1/2 κ 2h0 + K κt

( ()

)

κ κt

+

)

1/2

1/2

(8)

and u0opt ) -c0(κ/K)1/2. Clearly, u0 ) u0opt implies that ∆Fel issin addition to θ0salso minimized with respect to u0. Again, according to variational theory this is equivalent to replacing the boundary condition u(0) ) u0 by

ω2(ω22 - 3ω12)θ0 + (h0K/κ)2u0

ω1/2 )

θ′(0) ) c0

Del* ) h0 L

ω2(ω22 - 3ω12)u0 + h02Kθ0/κ 2

condition θ(0) ) θ0 by

θ02 - 2u0θ0 h0

+

c0θ0 (5) Equation 5 is the most general result from membrane elasticity theory, valid for arbitrarily given (but sufficiently small) u0 and θ0. It provides the basis for the analysis of the interaction between a symmetric membrane and an inclusion (of sufficiently large circumference). In particular, it can be used to study the effects of a given hydrophobic mismatch, u0 ) hP/h0 - 1. The question of an appropriate choice for θ0 was discussed in previous investigations. One possibility is θ0 ) 0 at x ) 0 which implies angular matching between the lipid director and the rigid wall. Clearly then, ∆Fel(u0) ∼ u02 is independent of c0 and cannot be lowered beyond Fel(u0)0) ) 0. Another choice for θ0 is that which minimizes ∆Fel for any given u0. From variational calculus40 it follows that minimizing ∆Fel with respect to θ0 corresponds to replacing the boundary

h0u′(0) ) -θ(0)

(9)

implying that the tilt angle, φ ) θ + h′, of the chain director at x ) 0 with respect to the monolayer shape, h(x), vanishes. ∆Felopt in eq 8 is thus the result for minimizing the elastic monolayer energy according to eq 2 with the boundary conditions in eqs 9 and 6. Note that ∆Felopt(c0*0) < 0. Thus, for relaxed tilt angle and optimal mismatch, membrane elasticity theory predicts the monolayer to be able to lower its free energy beyond that of the planar, unperturbed state.28 Entropic Chain Confinement. Lipid chains in fluid membranes are flexible, adopting a large number of different conformations within the hydrophobic core of the membrane. Rigid inclusions are impenetrable for lipid chains; their presence in a fluid membrane thus restricts the number of available conformations of neighboring lipid chains. In this section, we first recapitulate the director model and then present an extended version of it which can be combined self-consistently with membrane elasticity theory. The Director Model. We shall use in our analysis a simple model that was recently suggested to estimate the free energy penalty, ∆Fc, of the conformational chain restrictions.41 The model is based on the representation of a hydrocarbon chain by a fluctuating director, h (of length |h| ) h0), that is anchored at the monolayer interface but is able to adopt all orientations within the hydrocarbon core. That is, the director tip can point to a hemisphere of radius h0 within the hydrophobic core of the lipid layer. Generally, each orientation of the director occurs with a certain probability P(h). One may expect P(h) to be larger (40) Arfken, G. B.; Weber, H. J. Mathematical Methods for Physicists, 5th ed.; Academic Press: New York, 2001. (41) May, S.; Ben-Shaul, A. Phys. Chem. Chem. Phys. 2000, 2, 44944502.

6360


May

for an orientation h parallel to the bilayer normal, compared to a tilted orientation. Yet, to keep our model as simple as possible, we neglect the orientational dependence of P(h). Hence, we assumesfor an unperturbed, planar membraneseach orientation of the director to occur with the same probability. The partition sum of an unperturbed lipid chain is thus q0 ) 2πh02, corresponding to the area of a hemisphere of radius h0. If a rigid wall is present at distance x from the origin of the lipid chain (with 0 e x < h0), then some chain conformations will no longer be accessible. We shall assume here that all those directors that would penetrate into the rigid wall are discarded from the partition sum. The partition sum, q(x), of a chain originating at distance x from the wall is thus given by only that part of the hemisphere (of radius h0) which is outside the wall, namely, q(x) ) πh0(h0 + x). For x g h0 there are no longer any conformational restrictions on the lipid chain and thus q(xgh0) ) q0. The free energy of a director at distance x from the wall is a0fc0(x) ) -kBT ln q(x), where a0 is the cross-sectional area per lipid chain, and fc0(x) is the free energy per unit area. The corresponding excess free energy with respect to an unperturbed director is a0∆fc0(x) ) a0[fc0(x) - fc0(h0)] ) -kBT ln(q(x)/q0) ) -kBT ln[(1 + x/h0)/2]. The overall excess free energy, ∆Fc0, of the lipid chains in the vicinity of a rigid wall is obtained by integrating ∆fc0 over the bilayer midplane. To this end, we assume that the chain directors are distributed uniformly along the x-axis, leading to

∆Fc0 ) L

∫0h

0

dx ∆fc0(x) ) NkBT(1 - ln 2)

(10)

where N ) Lh0/a0 is the number of chains in the monolayer being in contact with a wall of length L. Using h0 ) 12.5 Å and a0 ) 32.5 Å2, we find for the excess free energy of a bilayer 2∆Fc0/L ) 2 kBT(1 - ln 2)h0/a0 ) 0.24 kBT/Å. This value is already on the same order as the lipid-protein interaction energy, derived by Fattal and Ben-Shaul12 in their molecular chain packing calculations where ∆Fbl/L ) 0.37 kBT/Å was found for -(CH2)13-CH3 chains. We note that the director model can be used to calculate the average tilt angle, θ0c, with respect to the membrane normal. For a director, located at distance x e h0 away from the wall, the result is (for a derivation see eq 18)

sin θ0c(x) ) -

(

)

1 x 12 h0

(11)

Thus, at x ) 0 the magnitude of the average tilt angle, θ0c ) -30°, whereas it vanishes for x ) h0. To summarize, the director model provides an approximate way to describe conformational restrictions of the lipid chains imposed by rigid membrane inclusions. However, the results in eqs 10 and 11 completely neglect the mutual interactions between different chain directors. These interactions can be taken into account by membrane elasticity theory. One should thus seek to (self-consistently) combine both models. To this end, we need to extend the director model which is the subject of the following part. Extension of the Director Model by Addition of an External Field. With elasticity theory included, the tilt angle profile, θ(x), will deviate from that given in eq 11. Any deviation of θ(x) from θ0c(x) further increases the free energy, ∆fc above ∆fc0 as given in eq 10. For sufficiently small differences of θ(x) - θ0c(x), we may expand the free energy density, ∆fc, up to quadratic order. The result can

Figure 2. The conformational space of a director at distance x from the wall (thick line). Without an external field, all positions 0 e xj e x + h0 of the director tip have the same probability, leading to an average tilt angle of θ0c according to eq 11. If the average tilt along the x-axis is constrained to be θ (broken arrow), then an appropriate external field, H (see eq 18), must be supplied.

be written in the form

∆fc(x) ) ∆fc0(x) +

κtc(x) [θ(x) - θ0c(x)]2 2

(12)

where κtc(x) is a contribution to the tilt modulus resulting from conformational chain restrictions. In the following we derive an expression for κtc(x). To this end, we apply an external field H(x), acting along the x-direction such that the average tilt angle, θ(x), of the director is a conserved quantity. This is illustrated in Figure 2, showing a director originating at x whose tip is at xj. Generally, the external field, H(x), can be used to force the average director to adopt a given angle θ(x) (that is, to coincide, say, with the broken arrow in Figure 2). In this case, the probability distribution, P(xj), to find the director tip at position xj is no longer uniform. Let P(xj) be normalized according to

πh0

∫0x+h

0

dxjP(xj) ) 1

(13)

Then, the average position of the director tip is

〈xj〉 ) πh0

∫0x+h

0

dxjP(xj)xj

(14)

With that, the free energy density, fc(x), of a director originating at x is given by

fc(x) πh0 ) kBT a0

∫0x+h

0

dxjP(xj)[lnP(xj) - H(h0 sin θ + xj - x)] (15)

From eq 15 it follows that the external field is conjugate to the constraint 〈xj〉 ) x - h0 sin θ. In equilibrium, the probability distribution is given by P(xj) ) exp(Hxj)/q where the partition sum is

q ) πh0

∫0x+h

0

dxjeHxj )

πh0 H(x+h0) [e - 1] H

(16)

After insertion of P(xj) into eq 15, we obtain a0fc/kBT ) -ln q - H(h0 sin θ - x). Hence, the excess free energy, ∆fc(x) ) fc(x) - fc(x)h0), with respect to an unperturbed director (with corresponding partition sum q0 ) 2πh02) is

()

a0∆fc q ) -ln - H(h0 sin θ - x) kBT q0

(17)

The correct choice of the external field is given by the solution of the equation ∂fc/∂H ) 0. For sufficiently small H(x) one finds

H)

12 sin θ + (1 - x/h0)/2 h0 (1 + x/h )2 0

(18)


Langmuir, Vol. 18, No. 16, 2002 6361

From this relation the optimal tilt, θ ) θ0c for vanishing field follows immediately and is given in eq 11. The expression for H(x) in eq 18 may be inserted into eq 17. Expanding the resulting expression in terms of θ - θ0c up to quadratic order gives rise to eq 12 with

κtc(x) 12 ) for 0 e x e h0 kBT a0(1 + x/h0)2

(19)

For x g h0 it is κtc(x)h0) ) κt,0c ) 3 kBT/a0. Note that κt,0c is the contribution of the director model to the tilt modulus of a planar, unperturbed (and inclusion-free) lipid bilayer. It results from the modifications of the probability distribution, P(xj), upon imposing a tilt angle φ. Note that for vanishing field, eqs 17 and 16 recover the expression a0∆fc(H)0) ) a0∆fc0 ) -kBT ln[(1 + x/h0)/2]. Euler Equations and Boundary Conditions. According to our approach, the local free energy, ∆f ) ∆fel + ∆fc, of the membrane close to a rigid wall is determined by an elastic term (∆fel) and conformational restrictions of the lipid chains (∆fc). On the basis of the analysis above, we see that both contributions favor different monolayer conformations. This is most clearly seen for a membranematching wall (u0 ) θ0 ) 0) and vanishing spontaneous curvature, c0, of the lipid layer: While the elastic contribution is optimal for a planar, unperturbed monolayer, restrictions of the lipid chain conformations would induce a large tilt of the lipid directors, as given in eq 11. If both energetic contributions are taken into account, the optimal monolayer perturbation will be a compromise between the two competing effects. The Euler equations for ∆F ) ∆Fel + ∆Fc, following from eq 2 and eq 12 are

h02κtu′′ ) Ku - h0κtθ′

(20)

κθ′′ ) κt(x)θ - κtc(x)θ0c(x) + h0κtu′ The effects of the conformational restrictions of the lipid chains enter only the second one of eqs 20. In fact, there are two notable modifications of the Euler equations. First, there appears to be an (x-dependent) tilt modulus κt(x) ) κt + κtc(x). It is useful to rewrite this effective tilt modulus in the form κt(x) ) κt0 + ∆κtc(x) with κt0 ) κt + κt,0c and ∆κt(x) ) κtc(x) - κt,0c. Then, the first contribution (κt0) is the tilt modulus of a monolayer in the absence of the rigid wall. It consists of a part, κt ) κtst, which is related to the stretching of the lipid chains upon a tilt deformation (for an estimate see Appendix), and a contribution, κt,0c, from the conformational chain restrictions. The second part, ∆κtc(x), is the wall-induced modification of the tilt modulus. It is given by

∆κtc(x) )

[

kBT 12 -3 a0 (1 + x/h )2 0

]

(21)

which is a positive quantity, depends on the distance x to the wall, and vanishes for x g h0. Hence, the presence of the wall enhances the tilt modulus of a lipid layer. The second modification of the Euler equations (see eqs 20) is the appearance of a spontaneous tilt angle, θ0c(x) (given in eq 11). Also the spontaneous tilt angle depends on the distance x to the wall. It is a negative quantity which expresses the energetic preference of the lipid chains to tilt away from the wall. It is important to note that in our approach the presence of the rigid wall does directly enter into the Euler equations

because the wall alters the microelastic properties of neighboring lipids. This is the main difference to previous approaches where the wall acts only through the boundary conditions of the Euler equations.20-25,28,33,36,42 The need to incorporate the wall directly into the Euler equations follows from the typical magnitude of the decay length, ξ, which is generally found on the same order as the molecular length of the lipids. Structural perturbations due to packing constraints and, in particular, due to chain conformational restrictions are expected to act on the same molecular length scale and should thus appear in the Euler equations. Since we consider an isolated wall, the boundary conditions for large x are u(∞) ) θ(∞) ) 0. At the wall, the lipid director optimizes its average direction; hence θ′(0) ) c0 as presented in eq 6. The last boundary condition is u(0) ) u0 or h0u′(0) ) - θ(0) (see eq 9), depending on whether the wall has a fixed hydrophobic height or whether we are interested in the optimal hydrophobic mismatch, respectively. Results and Discussion We present numerical results for the wall-induced monolayer perturbation profile and free energy and investigate the influence of hydrophobic mismatch and spontaneous curvature. The perturbation free energy, ∆F ) ∆Fel + ∆Fc, accounts for both the elastic response of the lipid layer (∆Fel) and for conformational restrictions (∆Fc) due to the presence of the rigid wall. In all calculations we shall use h0 ) 12.5 Å for the hydrophobic equilibrium thickness of the monolayer as well as K ) 0.2 kBT/Å2 and κ ) 10 kBT for the elastic monolayer properties. These choices are in general agreement with experimentally obtained values.43-47 For the effective tilt modulus in absence of an inclusion, κt0, there is no experimental determination available so far. We thus have to rely on theoretical estimates. Previous molecular models34-36,42 relate the tilt modulus entirely to the stretching of the lipid chains upon imposing a tilt angle. This results in a prediction for κtst, which is roughly κtst ) 0.1 kBT/Å2. For one particularly simple molecular model, we shall shortly recall in the Appendix the derivation of κtst. Besides the chain stretching, lipid tilt will also confine the available configurational space of the lipid chains. We have seen that the director model provides a simple framework to estimate this contribution (see eq 19 for x ) h0): it results in κt,0c ) 3 kBT/a0, which for a0 ) 65 Å2 yields κt,0c ) 0.1 kBT/Å2. Hence, the sum κt0 ) κtst + κt,0c results in the estimate κt0 ) (0.1 + 0.1)kBT/Å2 ) 0.2 kBT/Å2, which we shall use in the present work. No Hydrophobic Mismatch. Figure 3 shows the monolayer perturbation profile derived for a vanishing hydrophobic mismatch, u0 ) 0, and a vanishing spontaneous curvature, c0 ) 0. Since the boundary condition θ′(0) ) c0 ) 0 is fulfilled (see eq 6), there is no splay deformation of the lipid directors directly at the inclusion boundary. However, there is a pronounced tilt, θ0 ) -0.286, of the director at x ) 0, quite close to the spontaneous tilt angle θ0c ) -0.3. This finding is in principle agreement with recent molecular chain-packing calculations12,41 where the (42) May, S.; Ben-Shaul, A. Biophys. J. 1999, 76, 751-767. (43) Niggemann, G.; Kummrow, M.; Helfrich, W. J. Phys. II 1995, 5, 413-425. (44) Evans, E.; Rawicz, W. Phys. Rev. Lett. 1990, 64, 2094-2097. (45) Evans, E.; Needham, D. J. Phys. Chem. 1987, 91, 4219-4228. (46) Meleard, P.; Gerbeaud, C.; Pott, T.; Fernandez-Puente, L.; Bivas, I.; Mitov, M. D.; Dufourcq, J.; Bothorel, P. Biophys. J. 1997, 72, 26162629. (47) Lipowsky, R.; Sackmann, E., Eds.; Structure and Dynamics of Membranes; Elsevier: Amsterdam, 1995.

6362


May

Figure 3. The monolayer perturbation profile for K ) 0.2 kBT/ Å2, κ ) 10 kBT, c0 ) 0, κt0 ) 0.2 kBT/Å2, h0 ) 12.5 Å, and u0 ) 0. The corresponding monolayer perturbation energy is ∆F/L ) 0.291 kBT/Å, and the resulting lipid tilt angle at the wall is θ0 ) -0.286. Some lipid directors are shown as solid lines at arbitrary positions. The broken line marks h0.

orientational dependence of P(h) was calculated rather than assumed to be uniform. This fact provides an additional motivation for assuming P(h) to be constant as we do in the present work. It is important to realize that the tilt θ0 < 0 at x ) 0 is not in contradiction with uniform packing of lipid chain segments in the vicinity on the wall. The lipid tilt away from the wall does not imply the creation of a void region. This is because the lipid director only represents the average location of the lipid chain. It does not specify the packing properties of the lipids. This fact is most clearly seen in the chain-packing calculations of Fattal and BenShaul12 where a lipid tilt, similar to the present results, was found for a lipid bilayer of uniform chain segment density. Another prediction of Figure 3 is a local thickening of the monolayer in the vicinity of the wall. Recall that membrane elasticity theory alone would predict an unperturbed, planar monolayer (with ∆F ) 0) for the system shown in Figure 3. Thus, the monolayer thickening has its origin in the chain conformational restrictions and reflects the coupling between ∆Fel and ∆Fc. For the monolayer in Figure 3 we find the wall-induced perturbation free energy (per length L of the wall) ∆F/L ) 0.29 kBT/Å. The corresponding bilayer value is thus ∆Fbl ) 2∆F ) 0.58 kBTL/Å, somewhat larger but on the same order as the estimate of 0.37 kBT/Å by Fattal and BenShaul12 using their molecular chain-packing theory (we cannot attempt a direct comparison because the elastic properties of the bilayer membrane were not calculated in ref 12). Let us comment on the decay length, ξ, of the monolayer perturbation for the system in Figure 3. Assume first that the wall would not impose any conformational restrictions on the lipid chains (this would be the case if the monolayer perturbation originates, say, from a membrane adsorbed particle). Then ∆κt(x) ≡ 0 (implying constant κt(x) ) κt0), and we can use eq 4 to calculate ξ ) 1/ω2. For the material parameters in Figure 3, we find ξ ) 11.7 Å. This is only slightly smaller than the decay length in the limit κt f ∞ for which eq 4 predicts ξ ) 13.3 Å. Hence, for our choice of κtst ) 0.1 kBT/Å2 (implying κt0 ) 0.2 kBT/Å2) the membrane behaves not so much different compared to κt f ∞. Only if κt is much smaller will ξ then decrease significantly. (For example, κt ) 0.01 kBT/Å2 results in ξ ) 5.1 Å.) While these estimates of ξ are based entirely on elasticity theory, we note that with the conformational chain restrictions included (where ∆κtc(x) no longer vanishes), there is actually not a simple exponential decay of the monolayer perturbation profile. Near the wall, the tilt modulus is effectively increased, and a spontaneous tilt is present, forcing the monolayer perturbation to penetrate farther into the membrane than is predicted by an exponential decay with the decay length ξ. The onset of the exponential decay is only at x ) h0 where the lipid chains no longer suffer from conformational restrictions.

Figure 4. The monolayer perturbation profile for optimal hydrophobic mismatch, corresponding to the boundary condition h0u′ ) -θ(0). The different diagrams are derived for c0 ) 0 (top), leading to ∆F/L ) 0.254 kBT/Å, c0 ) 0.02 (bottom, left) where ∆F/L ) 0.161 kBT/Å, and c0 ) -0.02 (bottom, right) with ∆F/L ) 0.316 kBT/Å. Some lipid directors are shown as solid lines at arbitrary positions. The broken lines mark h0 ) 12.5 Å. It is furthermore K ) 0.2 kBT/Å2, κ ) 10 kBT, and κt0 ) 0.2 kBT/Å2.

Figure 5. The excess free energy, ∆F, of a perturbed monolayer and its contributions ∆Fc and ∆Fel (all in units of LkBT/Å) as a function of u0 for h0 ) 12.5 Å, K ) 0.2 kBT/Å2, κ ) 10 kBT, κt0 ) 0.2 kBT/Å2, and c0 ) 0. The boundary condition for the tilt angle is θ′(0) ) c0.

Optimal Hydrophobic Mismatch. Figure 4 shows the monolayer perturbation for optimal hydrophobic thickness of the wall and different values of the spontaneous curvature c0 ) 0. Recall that the boundary conditions for this case are given by eqs 6 and 9. The diagrams in Figure 4 clearly show that the membrane generally prefers a negative hydrophobic mismatch. Again, this prediction of the present theory is in agreement with the conclusions of the molecular chain-packing theory by Fattal and Ben-Shaul.12 The preferred magnitude of the negative mismatch, |hP - h0|, depends somewhat on c0 but does generally not exceed 4 Å. What differs from the predictions of Ben-Shaul and Fattal12 is the spatial extension of the bilayer perturbation. While the molecular chain-packing calculations were based on an exponential decay with a resulting decay length of ξ ) 3-6 Å, we find a notable membrane thickening even 2 nm away from the wall (see also Figure 3). This perturbation, of course, is nonmonotonic and would not exist if an exponential membrane shape was imposed. Hence, it would be interesting to see the predictions of the molecular chain-packing calculations without the restriction of the membrane shape to follow an exponential function. Spring Constant. The excess free energy, ∆F/L ) 0.254 kBT/Å in Figure 4 for c0 ) 0 is of course smaller than the corresponding value ∆F/L ) 0.291 kBT/Å in Figure 3 because the mismatch u(x)0) ) u0 is optimal in the former case while it is not in the latter case. Figure 5 shows how ∆F and its two contributions, ∆Fel and ∆Fc, vary as a function of the hydrophobic mismatch u0. Since the


Langmuir, Vol. 18, No. 16, 2002 6363

Figure 6. Spring constants (all in units of LkBT/Å) as a function of the tilt modulus κt0 for h0 ) 12.5 Å, K ) 0.2 kBT/Å2, and κ ) 10.0 kBT. Shown is the numerically derived value of D/L, its contribution, Del/L and Dc/L, and Del*/L according to eq 8 (c). The broken lines indicate the values for κt0 f ∞. The spring constants are derived for c0 ) 0 but turn out to be virtually independent of the spontaneous curvature c0.

changes of ∆F as a function of u0 are quadratic, it is convenient to define a spring constant, D, in analogy to eq 7

∆F(u0) ) ∆F

opt

D + (u0 - u0opt)2 2

(22)

The spring constant, D ) Del + Dc, consists of an elastic contribution (Del), and one that results from conformational restrictions of the lipid chains (Dc). Figure 5 suggests that Dc is much smaller than Del. This is indeed the case as shown in Figure 6 where the spring constants and its contributions are displayed as a function of the tilt modulus κt0 (Figure 5 corresponds to κt0 ) 0.2 kBT/Å2 in Figure 6). Also shown in Figure 6 is the spring constant, Del*/L according to eq 8, where no conformational restrictions of the lipids are present. The values for Del/L and Del*/L are not very different from each other even though neither the membrane perturbation profile nor the optimal boundary value for θ0 correspond to that predicted by membrane elasticity theory with ∆Fc ) 0. Thus, we conclude that, despite its large influence on the optimal perturbation profile and free energy, the contribution ∆Fc only slightly affects the membrane energetics. We also note that the value of the spring constant, D, is virtually independent of the spontaneous curvature (calculations not shown), in agreement with the prediction form pure membrane elasticity theory where Del* does not depend on c0 at all. Finally, Figure 6 suggests that the values of the spring constants strongly depend on the tilt modulus κt0. However, if indeedsas we have assumed heresthe magnitude of the tilt modulus is κt0 ) 0.2 kBT/Å2, then the difference to the usually adopted limit κt0 f ∞ is not very significant. In fact, we find D(κt0 ) 0.2 kBT/Å2) ≈ Del*(κt0 f ∞), which suggests that the simple expression (see eq 8)

()

h0 Del*(κt0 f ∞) ) L 2

1/2

κ1/4K3/4

(23)

well describes the energetics of varying the hydrophobic mismatch of sufficiently large membrane inclusions. Variations of the Spontaneous Curvature. Recall that Figure 4 predicts a negative hydrophobic mismatch to be energetically favorable for the insertion of rigid inclusions into bilayer membranes. Figure 4 also indicates a positive splay deformation of the lipids close to the wall. Such a deformation suggests that lipids with positive spontaneous curvature, c0, should interact more favorable with integral proteins than those with negative c0. Using our present approach, we can calculate the perturbation free energy, ∆F (and its contributions ∆Fel and ∆Fc) as function of c0.

Figure 7. Lipid-protein interaction free energy (per monolayer), ∆F, and its contributions, ∆Fel and ∆Fc (all in units of kBT per 1 Å length of the protein wall), as a function of the spontaneous curvature, c0, for K ) 0.2 kBT/Å2, κ ) 10.0 kBT, κt0 ) 0.2 kBT/Å2, and h0 ) 12.5 Å. All calculations are performed for θ′(0) ) c0. The solid lines correspond to optimal hydrophobic mismatch (h0u′ ) -θ(0)); the broken lines are derived for positive hydrophobic mismatch of u0 ) 0.1. Note, the lipid-protein interaction free energy for a bilayer membrane is given by ∆Fbl ) 2∆F.

This is shown in Figure 7 for two different cases of the hydrophobic mismatch: the solid lines correspond to optimized (that is, negative) hydrophobic mismatch (with the boundary condition h0u′ ) -θ(0) of the Euler equations, see eq 9), and the broken lines are derived for a positive hydrophobic mismatch (with u0 ) 0.1). Figure 7 suggests that, indeed, a positive spontaneous curvature appears to considerably lower ∆F. Interestingly, ∆Fel becomes negative for sufficiently large c0 > 0 but not for c0 < 0, in contrast with the predictions of eq 8. This is a result of the wall-induced tilt of the lipid directors away from the rigid surface of the wall which strongly favors the packing of lipids with positive c0. Figure 7 shows qualitatively similar results both for optimal (that is, negative) hydrophobic mismatch (solid lines) and positive mismatch (broken lines). Hence, even if the hydrophobic mismatch is positive, the insertion of rigid membrane inclusions is more favorable for positive spontaneous curvature (c0 > 0) of the lipids. On the other hand, lipids with negative spontaneous curvature (c0 < 0) generally render lipid-protein interaction energies more unfavorable. Available experimental results indeed support this notion. For example, Lewis and Cafiso48 have found the membrane partitioning of alamethicin to increase with c0. Scarlata and Gruner49 suggest that phosphatidylethanolamine (which has c0 < 0) destabilizes the contact region between a lipid membrane and gramicidin A. On the other hand, various different lyso-phospholipids (c0 > 0) were consistently found to increase the dimerization constant of membrane-bound gramicidin A.50 Finally, also modifying the electrostatic stress of charged lipid membranes affects the spontaneous curvature.51 In particular, screening of the mutual repulsions between charged lipid headgroups (say, by addition of salt) decreases c0 and is thus expected to destabilize the gramicidin A dimer. Here again, experimental evidence of gramicidin A in phosphatidylserine/n-decane bilayers supports this prediction.52 Note again that our conclusion about the dependence of ∆F on the spontaneous curvature, c0, differs qualitatively from previous findings that are based on membrane elasticity theory without including conformational chain (48) Lewis, J. R.; Cafiso, D. S. Biochemistry 1999, 38, 5932-5938. (49) Scarlata, S.; Gruner, S. M. Biophys. Chem. 1997, 67, 269-279. (50) Lundbæk, J. A.; Andersen, O. S. J. Gen. Physiol. 1994, 104, 645-673. (51) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: San Diego, CA, 1992. (52) Lundbæk, J. A.; Maer, A. M.; Andersen, O. S. Biochemistry 1997, 36, 5695-5701.

6364


restrictions. While with ∆Fc ) 0, we find ∆F(c0) ) ∆Fel(c0) to exhibit a minimum for c0 ) 0, the present approach provides a simple physical mechanism that leads to a decrease of ∆F with c0. That is, the negative spontaneous tilt, θ0c(x), of a lipid layer in the presence of a rigid membrane inclusion energetically favors a positive spontaneous curvature, c0, of the lipids. Concluding Remarks We have provided a theoretical attempt to include into membrane elasticity theory conformational restrictions of the lipid chains in the vicinity of rigid proteins. This gives rise to modifications in the microelastic membrane properties, namely, an increase in the tilt modulus of the lipids and the appearance of a spontaneous tilt angle. The predictions of the model compare well with more detailed molecular chain-packing calculations. In particular, the lipid chains are, on average, tilted away from the rigid membrane inclusion, and optimal interaction between lipid bilayers and membrane inclusions takes place for negative hydrophobic mismatch. The major contribution to the perturbation free energy, ∆Fbl, of membranes upon the insertion of rigid membrane proteins arises from the restrictions of the chain conformational freedom. Positive spontaneous curvature of the lipids appears to generally lower ∆Fbl. Above that, our calculations reveal a nonmonotonic membrane deformation profile, that decays a few nanometers away from the membrane inclusion. The latter finding is in agreement with previously applied versions of membrane elasticity theory that are based only on one order parameter. It should be noted that membrane elasticity theory employs a number of approximations. Perhaps the most serious one is the neglect of the discrete lipid structure. On the other hand, when compared to experimental results, membrane elasticity theory often provides a reasonable theoretical comparison. This motivates further development of this theory. The present approach to include chain conformational restrictions is an attempt in this direction. It can easilyswithout much computational effortsbe generalized to calculate membranemediated protein-protein interactions for arbitrarily shaped proteins. Acknowledgment. I would like to thank A. Ben-Shaul for many helpful discussions. This work is supported by TMWFK. Appendix We recall a simple molecular model that can be used to estimate the magnitute of the tilt modulus κt. This model was recently used to relate the tilt modulus to molecular

May

lipid properties.42 The principal idea is that a pure tilt deformation of a membrane induces the hydrocarbon chains to increase their effective length. Hence, according to our notion we actually attempt to calculate κtst from a given molecular model. A similar increase in the lipid chain length can be observed for a common stretching deformation of a lipid membrane (with the corresponding stretching modulus K). It should thus be possible to directly relate K and κtst. To this end, consider a planar monolayer that is part of a symmetric membrane. The hydrophobic monolayer thickness is b. Incompressibility of the lipid chains implies ab ) ν, where a is the cross-sectional area per lipid and ν is the (conserved) lipid chain volume. We shall employ a simple molecular model that contains expressions for the free energies of the headgroups, chainwater interfacial region, and hydrocarbon chain region. Within this model, the free energy per lipid is given by

f ) γa +

B + τb2 a

(24)

where γ is the interfacial tension acting at the chainwater interface, B is a headgroup repulsion parameter, and τ is a chain repulsion parameter. The model in eq 24 extends the opposing forces model51 by the free energy of lipid chain compression/extension. It can easily be shown42 that the molecular model in eq 24 gives rise to the relation

κtst ) K - 2γ

(25)

Since γ > 0, we find K > κtst. This general property can be understood as follows: A pure stretching deformation of a lipid bilayer modifies both the effective length of the lipid chains and the interfacial hydrocarbon chain-water region. On the other hand, a pure tilt deformation of a lipid bilayer only affects the lipid chain length but not the hydrocarbon chain-water region. Hence, for the simple mechanical model in eq 24, the energy of a pure stretching deformation should be larger than that of a pure tilt deformation. The macroscopic value for the interfacial tension of alkane-water interfaces is given by γ ) 0.1 kBT/Å2. However, on a microscopic scale γ appears to be smaller.53 A rough estimate is γ ) 0.05 kBT/Å2. Together with K ) 0.2 kBT/Å2, which we have used throughout the present work, we obtain κtst ≈ 0.1 kBT/Å2. While the value κtst ≈ 0.1 kBT/Å2 was also estimated by Hamm and Kozlov,35 the present calculation suggests how sensitive the estimate of κtst is with regard to the molecular model and the chosen material parameters, K and γ. LA025747C (53) Sitkoff, D.; Ben-Tal, N.; Honig, B. J. Phys. Chem. 1996, 100, 2744-2752.

Membrane Perturbations Induced by Integral Proteins: Role of

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