Molecular Theory of Chain Packing, Elasticity and Lipid-Protein Interaction in Lipid Bilayers

Transcription

1 CHAPTER 7 Molecular Theory of Chain Packing, Elasticity and Lipid-Protein Interaction in Lipid Bilayers A. BEN-SHAUL The Institute of Advanced Studies, Department of Physical Chemistry and the Fritz Haber Research Center, The Hebrew University, Jerusalem 91904, Israel 1995 Elsevier Science B.V. Handbook of Biological Physics All rights reserved Volume 1, edited by R. Lipowsky and E. Sackmann 359

2 Contents 1. Introduction Planar bilayers The free energy The probability distribution of chain conformations Conformational properties The chain free energy Elastic properties Stretchingelasticity Curvatureelasticity Lipid-proteininteraction The model The role of hydrophobic mismatch Concluding remarks Acknowledgements References

3 1. Introduction The lipid bilayer, which constitutes the basic structural element of biological membranes, is a two-dimensional, self-assembled, aggregate of amphiphilic molecules. The hydrocarbon chains (the tails ) of these molecules comprise the hydrophobic interior of the bilayer, shielded from the surrounding aqueous solution by the lipids hydrophilic heads which are located at the two surfaces of the bilayer. The integrity of the bilayer is due to the hydrophobic interaction, namely the cohesive forces between the hydrocarbon tails, resulting from the tendency to minimize the hydrocarbon-water contact area [1 3]. The planar bilayer is just one of several possible aggregation geometries which satisfy the hydrophobic effect. Other familiar forms include vesicles, small globular (nearly spherical) micelles and elongated (rod-like) micelles. In all these aggregates at least one linear dimension of the hydrophobic core is microscopic, no longer and typically of order 2l, wherelis the length of the amphiphile chain [1, 3 5]. The number of unrestricted dimensions, along which the aggregate can grow, defines its dimensionality. Accordingly, bilayers (in which only the thickness is restricted), cylindrical aggregates and spherical micelles are two-, one- and zerodimensional objects, respectively. Alternatively, the shape of these aggregates can be characterized in terms of the two principal curvatures of their (hydrocarbonwater) interface, c 1 = 1/R 1 and c 2 = 1/R 2. In spherical micelles c 1 = c 2 = 1/R, where R is the radius of the hydrophobic micellar core, with R l. Similarly, for cylindrical micelles c 1 = 1/R 1/l, c 2 = 0; for planar bilayers c 1 = c 2 = 0and in vesicles c 1 = c 2 = 1/R (R l). In the following we shall also be interested in moderately curved bilayers, in which case either c 1 and/or c 2 are nonzero, but 1/c i = R d where d 2l is the bilayer thickness. Another useful characteristic of molecular organization in amphiphilic aggregates is the average area per head group at the hydrocarbon-water interface, a. Simple geometric (surface/volume) packing considerations imply a > kv/l with k = 1, 2 and 3 for the three basic aggregation geometries: planar bilayers, cylinders and spheres, respectively, with v denoting the volume per tail in the hydrophobic core [3 6]. The relative stability of the various possible packing geometries is determined by a delicate balance of forces operating at the interfacial region and within the hydrophobic core of the aggregate. At the interface, head group repulsions, of electrostatic and/or excluded-volume origin, act to increase the average area per molecule, a; a tendency opposed by the hydrocarbon-water surface tension which favors minimal a. Within the hydrophobic core, the attractive interactions between chain segments (monomers) ensure uniform, liquid-like, segment density, comparable to that of bulk liquid hydrocarbons [1, 3 5]. However, since the monomers are connected 361

4 362 A. Ben-Shaul into chains, and since all chains are subject to the boundary condition that their head groups are anchored to the interface, their conformational freedom (entropy) is significantly lower than in a bulk, isotropic, liquid phase. The tight chain packing conditions and the corresponding entropy loss, result in significant inter-chain repulsion (especially in bilayers) whose magnitude depends rather sensitively on the aggregation geometry, i.e. on a, c 1 and c 2. Qualitatively, amphiphiles with large head groups (strong head-head repulsion) and relatively small chains will preferentially pack into high curvature aggregates, such as spherical or cylindrical micelles. On the other hand, the planar bilayer is the optimal geometry for amphiphiles with large tail volume, such as doubly-chained phospholipids [3 6]. The discussion in the following sections will be limited to lipid bilayers in their fluid state, i.e. above the gel to liquid crystal transition temperature. Although in this state the chain segment density is uniform and liquid-like, the (flexible) hydrocarbon tails are highly stretched along the normal to the membrane plane. Thus, the hydrophobic core is anisotropic, resembling in some respects a layer of a smectic liquid crystal. The extent of chain stretching, as reflected by the hydrophobic thickness of the bilayer, d, is inversely proportional to the average cross-sectional area per chain, d = 2v/a. The equilibrium value of a is determined by the balance of forces mentioned above. Similar considerations, involving a balance of the moments, dictate the equilibrium, spontaneous, curvature of the bilayer. From the above qualitative picture it follows that the lipid membrane is an anisotropic medium involving, due to its unique molecular structure, both liquid-like and elastic ( liquid-crystalline ) characteristics. The anisotropy of the hydrophobic medium is revealed in a variety of chain conformational properties such as bond orientational order parameters, segment spatial distributions or the distribution of gauche conformers along the hydrocarbon tail. Some of these single chain characteristics, i.e. properties determined by the singlet probability distribution of chain conformations, can be measured experimentally [3, 7 14]. The conformational properties are closely related to thermodynamic and mechanical properties of the membrane, such as its bending rigidity and stretching elasticity, which reflect the curvature and area dependencies of the bilayer free energy [15 29]. The main purpose of this chapter is to describe and discuss these relationships, starting out from a microscopic, molecular, picture of amphiphile organization in the membrane. Based on a simple statistical-thermodynamic approach we shall first derive (in section 2) an explicit expression for the singlet probability distribution function of chain conformations in the bilayer [30 39]. The only assumption employed in this derivation is that the monomer density within the hydrophobic interior of the membrane is uniform (liquid-like). The geometry of the system, as specified by a, c 1 and c 2, enters through packing constraints on the singlet distribution. Using this distribution one can calculate averages of single chain properties, e.g., bond orientational order parameter profiles and segment spatial distributions, showing generally good agreement with available experimental and computer simulation data. The singlet probability distribution can also be used to calculate, in a mean-field approximation, thermodynamic properties of interest, such as the bilayer free energy, as a function of the area per head group and the interfacial curvature. Appropriate derivatives of the

5 Molecular theory of chain packing 363 free energy with respect to these variables yield the elastic constants of the system, as demonstrated in section 3. Finally, in section 4, we apply the theory to estimate the contribution of lipid deformation free energy around a rigid hydrophobic solute to lipid-protein interactions in membranes [40 60]. This chapter is not intended to be an exhaustive review of the subject matter; not even the various mean-field theories of molecular organization in bilayers, which in some respects are quite similar and are covered elsewhere. (See, e.g., [61 69]; for reviews, see [32, 67].) Rather, our goal is to describe one consistent approach to the issues mentioned above. Nevertheless, two remarks should be made concerning alternative and complementary approaches. First, it should be mentioned that the most detailed, both structural and dynamical, information on lipid bilayers and other self-assembling aggregates, is provided by large scale computer simulations; mainly molecular dynamics studies. The number and quality of such studies increases steadily, but the number of realistic systems analyzed is still rather limited (see, e.g., [70 76]). Presently, it is hard to anticipate if, and when, these methods will be applied in order to calculate, for instance, elastic properties of membranes, which require systematic simulations subject to varying boundary conditions. It should also be noted that even the most advanced and comprehensive computer simulations to date may encounter nonphysical artifacts [76]. The second remark concerns the calculation of the interactions prevailing in the interfacial, aqueous, region of the membrane. The theoretical approach described in the next sections focuses attention on the chain packing statistics of the hydrocarbon tails within the hydrophobic core. Head group interactions are no less important for the understanding of membrane structure and thermodynamics. However, because of the great variety of lipid polar head groups, the interactions between them are highly specific, depending strongly on their size and charge, as well as on the thermodynamic state of the ambient aqueous solution. (A detailed discussion of electrostatic interactions is given in another chapter in this volume [77]. Additional models and discussions can be found elsewhere, see, e.g., [1, 3, 78 80].) Thus, our treatment of head group interactions in the following discussion will be rather qualitative, and will be based on a simple phenomenological representation of their contribution to the membrane free energy. Since, to a very good approximation, the head and tail contributions to the membrane free energy are separable, this approximation does not detract from the analysis of chain packing statistics inside the hydrophobic region. Clearly, however, a unified theoretical approach which treats simultaneously and on a similar level of accuracy both head group and chain interactions is called for. Several models along this line have recently been suggested [84 86]. 2. Planar bilayers To introduce the basic concepts and quantities that will be encountered in this chapter, let us first consider the simplest system: a planar and symmetric bilayer, composed of 2N = N A + N B lipid molecules, with N A and N B denoting the number of molecules originating from the upper and lower interfaces, respectively, as shown in fig. 1.

6 364 A. Ben-Shaul Fig. 1. Schematic illustration of a planar bilayer (composed of two monolayers A and B) showing a central chain surrounded and compressed by neighboring chains. φ(α; z)dz is the volume occupied by segments of a chain in conformation α which are present in the thin layer z, z + dz. The tight packing conditions induce chain stretching, as compared to an isolated, free chain. The lateral pressure profile, π(z), shown schematically in the figure, represents, for every z, the lateral compressional pressure acting on a given chain by its neighbors. In the symmetric bilayer N A = N B. We adopt here the classical picture of the lipid bilayer, i.e. a thin, liquid-like, hydrophobic film bounded by two flat surfaces the hydrocarbon-water interfaces. The hydrophilic head groups, fluctuating slightly around their equilibrium positions, reside in the aqueous region very close to the interfaces. Due to thermal fluctuations a few chain segments may also, occasionally, protrude into the aqueous region, though at a considerable free energy cost. For the sake of concreteness we may assume that the constituent molecules are single chain amphiphiles of the form P (CH 2 ) n 1 CH 3, with P denoting the polar head group. It should be noted, however, that all the expressions derived below apply to any chain model and can be generalized to more complex systems, such as non-planar bilayers, micelles and mixed (several-component) aggregates The free energy The spatial separation between the hydrocarbon tails and the head groups suggests a corresponding separation of the bilayer free energy into three terms, F = F t + F h + F s (1) representing, respectively, the free energy of the hydrocarbon tails, head group interactions, and an interfacial term accounting for the interactions between the surface of the hydrophobic region and the surrounding solution (including the head groups). Equation (1) involves some minor approximations which have been discussed elsewhere [6]. The Helmholtz free energy F = F (N, A, d, T ) = 2Nf(a,d,T) (2) is a function of the number of molecules per monolayer N, the area of the bilayer A, the thickness of the bilayer hydrophobic core d, and the temperature T. f is the

7 Molecular theory of chain packing 365 average free energy per molecule, with a = A/N denoting the average area per head group. The assumption of a uniform, liquid-like (or, in brief, compact ), hydrophobic core implies that d = 2Nv/A = 2v/a is not an independent variable; v denoting the specific volume per chain in the liquid state. d is a relevant parameter for swollen bilayers, in which case d>2v/a. Whend>2lthe bilayer is composed of two independent monolayers; l being the length of a fully extended chain. The monolayer limit will be considered in several junctures in the course of the following discussion. In the next section we shall consider non-planar bilayers, in which case F depends also on N A /N B and the membrane curvatures c 1 and c 2. The interfacial contribution f s = f s (a, T ) can be expressed in the simple form f s = F s /2N = γa (3) with γ denoting the effective surface tension of the hydrocarbon-water interface. Due to the presence of the hydrophilic heads and other solutes (e.g., counterions) in the interfacial region γ is not identical to the bare oil-water surface tension. A frequently used estimate based on interpretations of experimental data is γ 50 dyne/cm 0.1kT/Å 2, at room temperature [1, 3]. A somewhat better representation of f s is obtained if a is replaced by a = a ā where ā is the hydrophobic area shielded by the head group. Clearly, however, this would only change (3) by an irrelevant additive constant. As noted in the Introduction, F h is rather complicated and depends on the special characteristics of the hydrophilic heads and the ambient aqueous solution. The theoretical analysis of head group interactions include solutions of the Poisson Boltzmann equation [3, 77, 78, 81 85] when the interactions are predominantly electrostatic, hard-core repulsion models for electrically neutral head groups [84, 86], as well as some phenomenological expressions [1, 3, 4, 86]. One of the simplest and most common representations of F h, based originally on the capacitor model [1, 3, 4] for interfaces composed of ionic or zwitterionic head groups, is f h = F h /2N = C/a (4) where C is a phenomenological constant, generally estimated from experimental data [1, 3, 4]. (Alternatively, (4) may be regarded as the first order term in the expansion of the interaction free energy per molecule in powers of the surface density of head groups ρ 1/a.) The inclusion of head group interactions in the following discussion will be rather limited and qualitative. Whenever they appear we shall use (4). Upon combining (3) and (4) one obtains f s + f h = ( [ ) F s + F h /2N = γa + C/a = γa 1 a ] 2 h + const (5) a where a h = (C/γ) 1/2 is the value of a which minimizes f s + f h. The additive constant is 2γa h.thesum,f s +f h, accounts for the two opposing forces [1, 3 5] operating at the interfacial region: the attractive, surface tension, term which tends to

8 366 A. Ben-Shaul minimize a, and the repulsion between head groups which counteracts this tendency. If these were the only lateral forces acting in the membrane plane, i.e. if F t were independent of a, as assumed in some models of amphiphile self assembly [1, 3 5], then a h would be the equilibrium area per head group. However, the chain packing considerations outlined below show that F t is, in fact, a rather sensitive function of a. In particular, for bilayers in their fluid state, where the typical cross sectional area per chain is a 30A 2, F t provides a major repulsive component to the system free energy. The first term in (1) can be separated into two contributions: F t = F att + F conf. Here F att = 2Nf att accounts for the attractive (Van der Waals) forces responsible for the cohesiveness of the hydrophobic core. Based on the assumption that the core is liquid-like and thus uniformly packed with chain segments, F att can be treated as a constant attractive background term, independent of the shape of the core, and hence independent of a. (Recall that the surface contribution to the free energy is taken care of by (3).) We shall set F att = 0. The second term, F conf = F t = 2Nf t (a,t) is the conformational free energy of the hydrocarbon chains, whose a dependence reflects the effects of inter-chain short range (hard core) repulsive interactions. Due to the high (liquid-like) monomer density within the hydrophobic core, the chains conformational entropy (flexibility) is severely restricted owing to excluded volume interactions between neighboring chains. The conformational entropy loss increases as the chains are more crowded, i.e. as a decreases. For instance, in the limit a a min = v/l ( 20 Å 2 for simple alkyl tails) the chains must fully stretch to their all-trans state, thus minimizing the effects of inter-chain excluded volume repulsions. In this state the internal energy of the chains, E t, is minimal (since there are no gauche bonds), but their conformational entropy (S t 0) is also minimal. As a increases the conformational strain is rapidly relieved and F t = E t TS t decreases, reaching a minimum at some optimal chain packing area, a c ; typically a c 40 Å 2 /chain, as shown in section 3.1. In bilayers, beyond a c, F t increases slowly with a, whereas in monolayers F t stays constant [39]. For the tail free energy we shall use the mean field expression f t = α P (α)ε(α) + kt α P(α)lnP(α) (6) where P (α) is the probability of finding the chain in conformation α, andε(α)is the internal ( trans/gauche ) energy of such a chain. For alkyl chains described by the rotational isomeric state model [87], ε(α) = n g (α)e g,where n g (α)isthe number of gauche bonds along the chain and e g 500 cal/mole is the energy of one gauche bond (relative to that of a trans bond). The two terms in (6) correspond, respectively, to the energetic and entropic contributions to the conformational free energy f t = ε t Ts t. The mean field character of (6) is associated with the fact that its derivation from the exact expression for F t involves a factorization of the many chain distribution function into a product of singlet distributions, P (α 1,...,α N )=P(α 1 )...P(α N ), [6].

9 Molecular theory of chain packing 367 It should be stressed, however, that (6) remains an approximation even if P (α) is the exact singlet distribution, namely, if P (α) = α 2,α 3,...,α N P (α, α 2, α 3,...,α N ). In other words, using (6) for f t does not necessarily imply a similar level of approximation for P (α) and related single chain properties (e.g., bond orientational order parameters). Our next aim is to derive an explicit expression for P (α), which later will be used to calculate both single chain and thermodynamic properties. The derivation presented below follows a thermodynamic variational approach, whereby we minimize f t subject to the appropriate (packing) constraints on P (α). An alternative derivation, starting out from the many-chain configurational integral, and demonstrating explicitly the role of inter-chain excluded volume interactions is given elsewhere [6, 30] The probability distribution of chain conformations Apart from the normalization condition α P (α) = 1, the only constraint that will be imposed on P (α) is that it should satisfy the requirement of uniform segment density, i.e. that ρ( r) = ρ = 1/ν is constant throughout the hydrophobic core, with ν denoting the specific volume per chain segment. To formulate this constraint in a form appropriate for the conditional minimization of f t,wefirstdefineacartesian coordinate system whose origin is located at an arbitrary point of the bilayer mid-plane and its z axis is normal to this (xy) plane, see fig. 1. A given chain conformation α is fully specified by the z coordinate of the head group, z 0, and by the coordinates r k r 0 of the tail segments (k = 1,...n), relative to the head group position r 0. We treat each CH 2 group as one segment of volume ν 27 Å 3 [1, 3]. The terminal, CH 3, group is usually treated as a segment of volume ν 2ν [1, 3], thus v = (n 1)ν + ν = (n + 1)v. An alternative characterization of α, appropriate for simple alkyl chains, is α = (z 0, w, b) with w, specified by three Euler angles, denoting the overall chain orientation, and b symbolizing the trans/gauche bond sequence. Let Φ(z)dz/ν denote the total number of chain segments (whose centers fall) within a narrow shell z, z + dz of the hydrophobic core, parallel to the membrane plane. In other words, Φ(z)dz is the volume taken up by chain segments in shell z, z +dz. Henceforth, unless stated otherwise, we shall measure Φ(z)dz in units of ν. (CH 3 groups count as ν /ν 2 monomers). Clearly, Φ(z) A(z) wherea(z)isthe area of the above shell. For a planar bilayer A(z) = A, i.e.φ(z) A. The equality, implying Φ(z)/A = constant, expresses the condition of uniform (z independent) segment density within the core. The inequality applies to monolayers or swollen bilayers [39]. In the compact bilayer Φ(z) = N A φ A (z) + N B φ B (z) = N [ φ A (z) + φ B (z) ]

10 368 A. Ben-Shaul where φ A (z) and φ B (z) are the average numbers of segments in z, z+dz, belonging to chains originating at the upper (A), and lower (B) interfaces, respectively. The averaging is over all possible chain conformations, e.g., φ A (z) = α P A (α)φ A (α; z). (7) Here φ A (α; z)dz is the number of segments of an A-chain in conformation α which fall in the region z, z + dz; seefig.1. Using the above definitions, the uniform density constraint for a symmetric planar bilayer reads Φ(z)/N = φ A (z) + φ B (z) φ(z)=a where a = A/N is the average cross sectional area per chain. In the symmetric system φ A (α; z) = φ B ( α; z)andp A (α)=p B ( α), with the chain conformation α denoting the mirror image of α (by reflection through the mid-plane). Using the simplified notation P A (α) = P (α) andφ A (α;z)=φ(α;z)wefind,forall d/2 z d/2, φ(z) = α P (α) [ φ(α; z) + φ(α; z) ] = a (all z). (8) Similarly, for the monolayer P (α)φ(α; z) a (all z). (9) α The (functional) minimization of (6), subject to the set of packing constraints (8) yields the desired singlet probability distribution P (α) = 1 [ Ω exp βε(α) β ] π(z)φ(α;z)dz (10) with β = 1/kT.Theπ(z) are the Lagrange parameters conjugate to (6) and Ω is the partition function Ω = α [ exp βε(α) β ] π(z)φ(α;z)dz. (11) The function π(z) can be interpreted as the lateral pressure (or stress) acting on the chain by its neighbors. The magnitude of π(z) reflects the extent by which the chain must be compressed (or dilated) in order to fulfill (8). The range of the integrals in (10) and (11) is d/2 z d/2. For the symmetric bilayer π(z) = π( z). The numerical values of the π(z) are determined by solving the set of coupled self consistency equations resulting from the substitution (for all z) of(10) into(8). The

11 Molecular theory of chain packing 369 numerical procedure for evaluating the π(z) is outlined below. As may be noted from (10) and (11) the π(z) are defined to within an arbitrary additive constant, i.e. P (α) is invariant with respect to the transformation π(z) π (z) = π(z) + constant. This is because φ(α; z)dz =v =n+1 (= the chain volume in units of ν, counting the terminal group as two segments) is a constant, independent of α. (Clearly, the distribution of chain segments within the hydrophobic core depends on α, but their sum is a constant.) A physically meaningful choice, consistent with the monolayer case, is obtained by setting π(z = 0) = 0, see below. Minimization of (6) with respect to (9) yields the singlet probability distribution for the monolayer. The resulting functional form of P (α) is again given by (10), but the π(z) are determined by the inequality constraints (9) [39]. These inequalities should be interpreted as follows: If, using the P (α) which minimizes f t, we find, for a given z, that φ(z) P(α)φ(α;z)<a, then (9) is trivially satisfied, making it an irrelevant constraint and, consequently, π(z) 0. On the other hand, for all z where φ(z) = a, i.e. where the constraint is relevant, π(z) 0. In fact, π(z) > 0 since the chain must be laterally compressed at z in order to fulfill the constraint. The special case where all π(z) 0 corresponds to a free (conformationally unperturbed) chain. In this (hypothetical) limit P (α) = P f (α) with P f (α) = 1 Ω f exp [ βε(α) ]. (12) The free energy of a free chain, obtained by substituting (12) into (6) is f t,f = kt ln Ω f. (13) It is not difficult to show that f t,f is a lower bound to the chain free energy, as expected for a free chain [30] Conformational properties Let us digress momentarily from the thermodynamic analysis in order to demonstrate the applicability of the probability distribution, as given by (10), to the calculation of single chain properties. In fig. 2 we show the bond orientational order parameter profile of C 9 (P (CH 2 ) 8 CH 3 ) chains, packed in a planar bilayer, at an average area per chain a = 25 Å 2. The experimental data, obtained via 2 H NMR of selectively deuterated C-H bonds, provide information on the average orientation of the various C-D bonds along the chain [7 9]. More explicitly, the measured quantities are the P 2 order parameters, S k = (3 cos 2 θ k 1)/2 = α P (α) [ 3cos 2 θ k (α) 1 ] /2 (14)

12 370 A. Ben-Shaul where θ k (α) is the angle between the bisector of the two C-H bonds corresponding to carbon C k and the membrane normal, i.e. the z axis; see fig. 2. Similarly, the skeletal orientational order parameter, Sk = 2S k, measures the average orientation of the vector connecting carbons C k 1 and C k+1 [7 10]. Sk = 0(S k =0) implies random bond orientations, whereas S k = 1(orS k = 1/2) indicates that the (k 1) (k+1) vector lies exactly along the membrane normal (the director ). In particular, for an all-trans chain, oriented normal to the membrane plane, S k = 1forallk. Figure 2 reveals significant alignment of the first few bonds of the chain, followed by a gradual decease in S k towards its terminus, indicating an increasing degree of chain flexibility. The bond order parameter profile calculated using (10) shows very good agreement [30, 31], both with experiment [7] and molecular dynamics simulations [71]. Similar agreement is found with respect to the average chain entropy s t and energy ε t [72], (see also [66]). The calculations based on (10) also show, as could be foreseen, that S k is large in those regions where π(z) is large and that it decreases when π(z) decreases. The orientational order parameter profiles Fig. 2. Bond orientational order parameter profiles for C 9 chains packed in a planar bilayer with area per head group a = 25 Å 2. Triangles experimental results [7]; squares molecular dynamics simulations [71]; circles mean field calculations, based on (10) [30]; diamonds another (quite similar) mean field calculation [66]. The insert shows the angle appearing in the definition of S k, see (14).

13 Molecular theory of chain packing 371 account for various other interesting characteristics of chain packing in membranes, as illustrated for C 12 chains in fig. 3. We see, for example, that the S k s increase as a decreases, reflecting the tighter chain packing conditions; a behavior corroborated by many experiments, see, e.g., [10 13]. It is also seen that the extent of chain stretching is governed, predominantly, by the packing constraints, rather than by the internal chain energy (especially when a is small) [34], as revealed by the similarity between the bond order profiles corresponding to gauche bond energies e g = 0and e g =500 cal/mole 1.7kT (for a = 25 Å). Finally, we note that when a 40 Å 2 the chain behaves as a free chain, i.e. a chain with no neighbors around it. The numerical procedure for calculating P (α) is, roughly, as follows. First the hydrophobic region is divided into L finite layers, of width z = d/l ( 1Å). Subsequently, the integrals in (10) and (11) are replaced by layer sums. Similarly, φ(α; z) φ i (α) andπ(z) z π i with i = 1,...,L. One then generates a large Fig. 3. Bond orientational order parameters of C 12 chains packed at three different areas per head group. The solid circles, triangles and squares correspond to a = 25, 32 and 40 Å 2, respectively. The open diamonds correspond to a free chain (here the finite ordering is due to the presence of the impenetrable hydrocarbon-water interface). The open circles correspond to athermal chains (packed at a = 25 Å 2 ) for which ε(α) 0 for all α. For these chains the energy e g of one gauche bond is e g = 0; in all other cases e g = 500 cal/mole.

14 372 A. Ben-Shaul number of chain conformations α and classifies them into groups according to their segment distributions among the layers, {φ i }. For alkyl chains of length n 20 it is possible to enumerate all possible bond sequences b of the rotational isomeric states. (Self intersecting sequences are discarded.) In most previous calculations each bond sequence has been multiplied by 40 randomly sampled combinations of head group positions (z 0 ) and overall chain orientations (w), yielding a total of 40 3 n 2 conformations α; recall that α = z 0, w, b. The number of conformations belonging to each group {φ i } gives their unperturbed statistical weights (appropriate for a free chain). These weights appear as input data in the set of L coupled self consistency equations obtained by substituting (10) and (11) into (8). The solution of these equations yields the π i and hence P (α) The chain free energy An explicit expression for the chain free energy in terms of the π(z) and φ(z) is obtained by substituting (10) into (6): d/2 f t = kt ln Ω π(z) φ(z) dz d/2 = kt ln Ω 1 2 d/2 d/2 π(z) φ(z)dz (15) = kt ln Ω a d/2 0 π(z)dz where φ(z), which appears in the second equality, is given by φ(z) = φ(z) + φ( z) (16) and represents the total segment density at z, as defined in (8). Note that φ(z) is the average segment density in z due to one chain only, anchored (say) to the upper interface. Thus, the main contribution to the integral in the first equality comes from the upper half of the bilayer (z >0). In passing from the first to the second equality we have used the symmetry property π(z) = π( z). In passing to the third equality we have used the packing constraint φ(z) = a, asin(8). Equation (15) is also applicable to swollen bilayers, where d>2v/a. However, in this case, (8) should be replaced by φ(z) a, implying (as for monolayers) that π(z) 0if φ(z)<aand π(z) > 0if φ(z)=a. In particular, when d>2lthe bilayer splits into two independent monolayers and hence f t is the free energy per chain in a planar monolayer. Indeed, in this limit, chains anchored at the upper interface cannot reach the lower half of the bilayer (z <0), so that φ(z) = φ(z) and the bilayer and monolayer constraints are equivalent. A more detailed treatment of chain

15 Molecular theory of chain packing 373 packing in (polymeric) monolayers, accounting explicitly for the role of the solvent, has recently been presented [88]. Interpreting π(z) as a lateral pressure profile, Ω = Ω({π(z)}, T ) should be regarded as an isothermal-isobaric partition function, cf. (11), and g t = kt ln Ω (17) as the Gibbs free energy per chain. Using (11), the (functional) derivative of g t with respect to π(z) yields, φ(z) = δg t δπ(z) = kt δ ln Ω δπ(z). (18) It then follows from the second line of (15) that f t and g t are related by a Legendre transformation, as is usually the case for these free energies. (For the free chain f t = g t.) To complete this analogy we note, using (15) and (11), that π(z) = δf t δ ln Z = kt δ φ(z) δ φ(z) (19) with the canonical partition function Z defined by the usual relation f t = kt ln Z. (20) Note that in the last two equations we have generalized f t = f t (a, T ) from being only a function of a to f t = f t ({ φ(z)}, T ), which is a function of the specific areas {φ(z)}. Of course, for the planar bilayer φ(z) = a for all z. All the relations obtained so far can be generalized to more complex systems including mixed bilayers and curved aggregates such as micelles or bent bilayers [35 37]. In these systems f t depends also on the molecular composition of the aggregate and its principal curvatures. These dependencies modify the packing constraints, but the functional form of P (α) is still given by the simple expression (10). The composition and/or curvature dependencies enter, implicitly, through the Lagrange parameters (i.e. the π(z)) conjugate to the packing constraints. 3. Elastic properties In this section we consider the free energy changes associated with elastic deformations, such as stretching and bending, of the planar bilayer. The discussion will be based on the molecular approach outlined in the previous section. Our main goal is to relate the microscopic characteristics of molecular packing in bilayers to the elastic constants appearing in the phenomenological (continuum) theories of these systems. We shall first discuss the area (A = Na) dependence of the free energy, i.e. the stretching elasticity of planar bilayers and then consider curvature deformations.

16 374 A. Ben-Shaul 3.1. Stretching elasticity The equilibrium area per molecule in a (tensionless) planar bilayer, a 0, is determined by the minimum condition f a = f t a + f h a + f s a = π t π h π s = 0 (21) with the second equality representing the tail, head and surface contributions to the lateral pressure. Using the phenomenological expressions (3) and (4), we find π s = γ and π h = C/a 2 = γa 2 h /a2,wherea h is the quantity defined in (5). The lateral pressure arising from chain conformational distortion (chain-chain repulsion) π t, is given by the area derivative of (15). It should be noted, however, that for compact (incompressible) bilayers, the integration limits in (15) depend on a, since d/2 = v/a. Explicit numerical calculations of f t, of the kind shown in fig. 4, reveal that for a wide range of relevant a values, f t is the same for monolayers and bilayers. More precisely, this similarity persists for all a a c where a c is the chain area at which f t reaches its minimum. The value a c 40 Å 2 /chain is larger than the typical equilibrium areas per chain in fluid bilayers, a 0 30 Å 2 /chain. In fact, a c 2a min = 2v/l corresponds to a bilayer in which a chain originating at one interface can reach the opposite interface, a situation of interest only for strongly interdigitating bilayers. (The slow increase of f t at a>a c, in bilayers, reflects the (minor) loss of conformational entropy due to collisions with the opposite interface.) Thus, to a very good approximation, in most cases of interest, the bilayer can be regarded as two (partly interdigitating but otherwise independent) monolayers, which have been brought into contact with each other. In other words, for a<a c, the packing constraints in a bilayer and a monolayer are essentially equivalent. This conclusion is supported by the results shown in figs 3 and 4 as well as by detailed calculations of the density profile φ(z) in a densely packed monolayer [39]. More explicitly, these calculations show that φ(z) is characterized by a flat region where φ(z) = a is constant, followed by an approximately linearly decreasing tail. The range of the plateau region, where most chain segments are found, is z v/a. The tail of φ(z), of range l z, contains the dangling chain ends. The free energy penalty associated with the coupling of two monolayers into a bilayer is negligible, since at the overlap regime the packing constraint φ A (z) + φ B (z) a is easily satisfied. From the above analysis it follows that for a<a c, π(z)>0 in the range z < z <d/2, whereas around the bilayer s midplane (i.e. for z z)wefindπ(z)=0. Then, using (11), and noting that φ(z) = a wherever π(z) > 0, it is easily shown that π t = f t a = π(z)dz (a<a c ). (22) Thus, π t > 0fora<a c, consistent with the numerical results shown for f t in fig. 4. (Notice that π t and π(z) bear different units.) The (slow) increase of f t with a for

17 Molecular theory of chain packing 375 Fig. 4. (a) The tail, head and surface contributions (f t, f h and f s) to the free energy of a molecule in a planar bilayer as a function of the average cross sectional area per molecule, in units of kt. (The origins of the energy scales are arbitrary). The free energies f s = γa and f h = γa 2 h /a are calculated for γ = 0.12kT/Å 2 and a h = 20 Å 2. f t is calculated using (15), n denoting the length of the hydrophobic tail. The dashed line represents f s +f h. The dotted line (shown only for n = 16) corresponds to a planar monolayer. (b) The free energy per molecule f = f t + f h + f s as a function of a, forn=12, 14 and 16. Note that the equilibrium area per molecule, a 0 varies only slowly with n, from a 0 32 Å 2 for n = 12 to a 0 34 Å 2 for n = 16. (After [39, 60].)

18 376 A. Ben-Shaul a a c, implying π t < 0, is due to the non-trivial d dependence of f t in this range. A qualitative explanation of this behavior is obtained by noting that δf t = ( f t / a ) δa + ( f t / d ) δd which, using d = 2v/a, yields π t = δf t /δa = ( f t / a ) ( f t / d ) (2v/a 2 ). Both f t / a and f t / d are negative since increasing either a or d allows for more conformational freedom. When a is small (a <a c ), f t / a is large and f t / d is small, implying π t > 0. On the other hand, when a is large f t / a is small (since, laterally, the chains are not severely restricted) whereas f t / d is relatively large, resulting in π t < 0. Returning to fig. 4, we note that a 0 is determined by the balance of chain repulsion (π t > 0) and head repulsion (π h > 0) on the one hand, and the surface tension (π s = γ <0) on the other hand. The calculation of f t shown in the figure is based on (15) and involves no adjustable parameters. The value of the surface tension used for calculating f s (see (3)), γ = 0.12kT/Å 2, is based on a common estimate [3, 4]. The value of a h (a h = 20 Å 2 ) used to calculate f h = γa 2 h /a was chosen so as to obtain a 0 32 Å 2 (for n = 12), which is a common value of the area per chain in lipid bilayers. The results reveal that, at a = a 0, the repulsive force balancing the surface tension is mainly due to chain repulsion. In fact, using a smaller value for γ, say,γ=0.08kt/å 2 would imply π h 0, so that π t = π s suffices to balance the surface tension. Similar conclusions about the important role of chain repulsion have been obtained from analysis of thickness fluctuations of membranes [89]. Nevertheless, more reliable estimates of both π s and π h are required before drawing more quantitative conclusions on the relative importance of the various lateral forces acting in the membrane. Figure 4 shows also that a 0 depends only weakly on the chain length l (l n). This also means that d v/a n/a is nearly proportional to n, as corroborated by experiment [3, 90]. An easy explanation of this fact is obtained if one completely ignores chain repulsion, because if this were the case, then a 0 = a h, independent of l. We have just concluded, however, that π t plays an important, possibly decisive, role in determining a 0. It is therefore interesting to estimate the n dependence of a 0 in the opposite limit, when π h = 0. This can be done based on approximate scaling arguments, as follows. When π h = 0, a 0 is determined by ( f t + f s ) / a = πt + γ = 0. Let l c denote the average (end-to-end) length of the chain, when packed at the optimal area a c. Treating this, least strained, chain as an ideal chain (in the meltlike [91] interior of the bilayer) we have l c n 1/2. Upon packing at smaller area, a = v/l < a c (l>l c ), the chain is stretched, implying f t ( l/l c ) 2 (v 2 /n)/a 2 n/a 2.

19 Molecular theory of chain packing 377 Thus, π t n/a 3 0, and hence a 0 (n/γ) 1/3 increases only slowly with n, in qualitative agreement with the results shown in fig. 4. Adding head group repulsion would lead to an even weaker dependence of a 0 on n. The free energy cost of area fluctuations around the equilibrium (tensionless) state, to second order in δa = a a 0, are given by δf = f f 0 = 1 ( 2 ) f (a a 0 ) 2 = 1 ( ) 2 δa 2 a 2 a=a 0 2 κ A a 0 (23) a 0 with the second equality serving as the definition of the area compressibility modulus κ A ; κ A = a 0 ( 2 f/ a 2 ) a0 [15 19]. Writing f = f t + f h + f s and using the previous expressions for the three free energy components, we find that the surface contribution is zero and κ A = κ A,t + κ A,h,isgivenby κ A = a 0 2 a d/2 d/2 ( ) ah 2 π(z)dz+2γ. (24) a 0 Both terms on the right hand side are positive. Numerical estimates based on calculations of the kind shown in fig. 4 yield κ A 0.5kT/Å 2 for a 0 30 Å 2, with the main contribution coming from κ A,t. The chain contribution to κ A decreases with a 0 and increases with n Curvature elasticity The molecular theory of chain packing statistics developed in section 2 for a single component, symmetric and planar bilayer can easily be extended to more complex aggregates, including pure and mixed micelles of different shapes as well as asymmetric (N A N B ) and/or non-planar bilayers. The generalizations required in order to treat these systems involve straightforward modifications of the expressions for the free energy of the system and the geometric packing constraints. In all cases, the singlet probability distributions of chain conformations which appear in F t, preserve the simple functional form (10) for all components [6, 30 39]. The dependencies of the P (α) s on the area, curvature and composition of the aggregate, enter through the lateral pressure profile. In this section we shall be explicitly interested in the free energy changes associated with curvature deformations of single- and two-component bilayers, focusing mainly on the tail contribution to the deformation free energy Free energy and packing constrains Consider a piece of a uniformly curved single-component bilayer, of total (midplane) area A = A(0) and principal curvatures c 1 = 1/R 1 and c 2 = 1/R 2, measured at the mid-plane. Allowing for different numbers of molecules in its two leaflets, the bilayer free energy is given by F/2N = χ A f A + χ B f B. (25)

20 378 A. Ben-Shaul A similar representation applies also to the components of F,i.e.F t, F h and F s. f A and f B denote the free energy per molecule of amphiphiles anchored to the upper and lower interface, respectively. χ A = N A /2N and χ B = 1 χ A = N B /2N are the mole fractions of A-type and B-type molecules, and 2N = N A +N B. f A and f B are no longer equal because of the different surface density of head groups and different curvatures (opposite in sign) characterizing the two interfaces. For the same reason, the conformational probability distribution P A (α) andp B (α) appearing in f A,t and f B,t are different. The variation in the area of the bilayer as a function of the distance z from the mid-plane is given by A(z) = A(0) [ 1 + (c 1 + c 2 )z + c 1 c 2 z 2] Na(z) (26) with z = d/2 and d/2 denoting the positions of the upper (A) andlower(b) interfaces, respectively. Small curvature deformations correspond to c 1 d and c 2 d 1. The sign convention in (26) is such that c i > 0 indicates that the bilayer (as shown in fig. 1) is convex upwards. Note that a(z) is a purely geometric characteristic of the membrane, independent of the distribution of molecules (N A, N B = 2N N A ) between the two leaflets of the bilayer. The areas per head group at the two interfaces are and a A = A(z = d/2)/n A = ( N/N A ) a(z = d/2) a B = A(z = d/2)/n B = ( N/N B ) a(z = d/2). Thus a A is given by a A = 1 a(0) [ χ (c 1 + c 2 )d + 14 ] c 1c 2 d 2 A (27) with a B given by a similar expression (with d replaced by d and χ A by χ B ). Only for the symmetric planar bilayer a A = a B = a(0). The packing constraints on P A (α) andp B (α) in the curved bilayer are obtained by a straightforward generalization of (8), namely χ A P A (α)φ A (α; z) + χ B P B (α)φ B (α; z) = 1 a(z). (28) 2 α α Equations (25) (28) provide the basis for extending the expressions derived in section 2 for f and its components, f t, f h and f s, to the more general case of a curved and/or asymmetric bilayer. Thus, for example, assuming that γ A = γ B = γ is independent of head group density and interfacial curvature, we find that f s F s /2N = γ ( χ A a A + χ B a B )

21 Molecular theory of chain packing 379 is given by f s = γa(0)[1 + c 1 c 2 d 2 /8], independent of χ A. Hence δf s = f s f 0 s = 1 8 γa(0)c 1c 2 d 2 κ s c 1 c 2 (29) where fs 0 = γa(0) is the surface free energy of a planar bilayer with the same total mid-plane area A(0) = Na(0). κ s is the surface contribution to the Gaussian (or saddle splay ) bending modulus defined below. The head group contribution to the deformation free energy, δf h = f h fh 0 can be estimated similarly, using for example the approximate representation (4). The resulting explicit expression is somewhat more involved and, unlike (29), depends on χ A in the curved and planar geometries. In applying (24) allowance can be made for the fact that the plane of head group repulsion is located at a finite distance δ( d) from the interface, so that fa,h = C/ã A where ã A is given by (27) but with d/2 replaced by d/2 = d/2 + δ. Detailed analysis of δf = δf h +δf s based on this approximate model, combined with (29) for δf s,have been presented elsewhere [19] for different modes of bending deformations, and will not be repeated here. More quantitative treatments of the electrostatic contributions to the bending free energy appear in another chapter in this volume [77]. Thus, in the following discussion we shall be mainly concerned with the curvature dependence ofthetailfreeenergy,f t. The average free energy per chain in a curved bilayer characterized by (i) concentrations χ A = χ, χ B = 1 χ, (ii) curvatures c 1 and c 2, and (iii) mid-plane area A(0) = Na(0), is given by f t = χkt ln Ω A (1 χ)kt ln Ω B 1 2 π(z) φ(z)dz (30) where now Ω A, Ω B and π(z) depend on a(0), c 1, c 2 and χ. The segment density at z is now, cf. (28), φ(z) = χ φ A (z) + (1 χ) φ B (z) = 1 a(z) (all z). (31) 2 The second equality corresponds to a compact bilayer. Ω A and Ω B are given by (15) with φ A (α; z) andφ B (α;z) replacing φ(α; z). The numerical algorithm for calculating the π(z), and hence P A (α), P B (α) andf t, as outlined in section 2.3 for the planar symmetric bilayer, can be repeated for an arbitrary set of parameters, a(0), c 1, c 2 and χ, which enter the calculation as boundary conditions. By a systematic variation of these parameters one can obtain the curvature, area and composition dependence of f t. The numerical effort involved is reasonable since all chain conformations are generated only once, at the outset of the calculation, and enumerated and classified into groups of φ(α; z) s as mentioned in section 2.3 [36, 38]. Yet, a simpler and more elegant procedure is available for small curvature deformations, i.e. when c i d 1 [37]. In this case the deformation free energy can be expressed in terms of the π(z) characterizing the undeformed bilayer.

22 380 A. Ben-Shaul Bending moduli Let f 0 = f (c 1 = 0, c 2 = 0, χ = 1/2, a = a(0)) denote the free energy per molecule in the planar symmetric bilayer. We assume that this is the equilibrium state of the membrane, an assumption which may be removed later on. Consider now a small bending deformation in which the mid-plane area A(0) = Na(0) remains constant. This defines the mid-plane as the surface of in-extension (or the neutral surface). Let δf = f ( c 1, c 2,1/2, a(0) ) f(0,0,1/2, a(0) ) denote the free energy change associated with the deformation. Transforming, for convenience, from c 1, c 2 to the sum and difference curvatures c + = c 1 + c 2, c = c 1 c 2, we find that, to second order in c +, c and χ, the free energy change (per unit area) can be expressed in the form 1 a(0) δf = 1 2 κ bc ( 4 κ( c 2 + ) 1 c2 + 2 λ χ 1 ) 2 + ω ( χ 1 ) c +. (32) 2 2 The expansion coefficients, representing elastic moduli of the membrane are determined by the appropriate second derivatives of δf, see below. The curvatures appearing in (32) are measured at the mid-plane. Since δf is symmetric with respect to c 1, c 2 c 2, c 1 (c c ) the expansion cannot contain a linear (or any odd) term in c. The choice of the mid-plane as the neutral surface, allows to express the free energy change associated with an arbitrary curvature-area deformation as a sum of a stretching term of the form (23), and a bending term of the form (32), without a mixed term (a a(0))(c 1 +c 2 ). The choice of the neutral surface enters, implicitly, into the definition of the elastic constants. The coupling between stretching and bending elasticities of membranes is a rather intricate issue involving, apart from the choice of the neutral surface (or surfaces), a careful specification of the conditions under which the deformation takes place [15 19, 92 96]. The constant λ appearing in the third term of (32) is closely related to the area compressibility modulus κ A defined in (22). This follows from the fact that a change in χ at constant area and curvature, corresponds to changing the head group areas in the two monolayers. A simple relationship between κ A and λ can be derived if the bilayer is treated as two independent monolayers: For c 1 = c 2 = 0, we have χ = χ A = a(0)/2a A, χ B = 1 χ = a(0)/2a B, cf. (27). Thus, δχ = (χ 1/2) = [ a(0)/2a 2 A] δaa = (1/2)(δa/a 0 ) since at equilibrium a A = a(0) = a 0. Thus, comparing the expressions for δf obtained using (23) and (32), we find λ 4a 0 κ A. (33)