CHAPTER 7
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.
All rights reserved
Handbook of Biological Physics
Volume 1, edited by R. Lipowsky and E. Sackmann
359

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
361
2. Planar bilayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
363
2.1. The free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
364
2.2. The probability distribution of chain conformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
367
2.3. Conformational properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
369
2.4. The chain free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
372
3. Elastic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
373
3.1. Stretching elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
374
3.2. Curvature elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
377
4. Lipid-protein interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
388
4.1. The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
389
4.2. The role of hydrophobic mismatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
393
5. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
396
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
396
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
397
360

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 2 l , where l is 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 cylindrical micelles
= c
1
1 /R
1 and c
2
= 1
= 1 /R
> 1 /l , c
2
/R
=
2
. In spherical micelles c
1
= c where R is the radius of the hydrophobic micellar core, with R
6 l . Similarly, for
0; for planar bilayers c
1
= c
2
2
=
=
1 /R
0 and
, 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
1 /c i c
1 and/or c
2 are nonzero, but
= R d where d
6 2 l 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

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 = 2 v/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

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
2 N = 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.

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 ) d z is the volume occupied by segments of a chain in conformation α which are present in the thin layer z , z + d z . 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.
2.1. 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 ) = 2 N f ( 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

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 = 2 N v/A = 2 v/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 > 2 v/a . When d > 2 l the 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
/ 2 N = 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 ' 0.1
kT / A of f s is obtained if
˚ 2 a
, at room temperature [1, 3]. A somewhat better representation is replaced by a
0
= a
−
¯ where a 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
/ 2 N = 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
/ 2 N = γa + C/a = γa 1 − a h a
2
+ const (5) where a h
= ( C/γ ) 1 / 2 is the value of a which minimizes f s
+ f h
. The additive constant is 2 γa h
. The sum, 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

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
≈ 30 A
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
= 2 N f 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
= 2 N f 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 A
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
−
T S t a minimum at some optimal chain packing area, a c
; typically a c decreases, reaching
' ˚ 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
=
X
P ( α ) ε ( α ) + kT
X
P ( α ) ln P ( α )
α α
(6) where P ( α ) is the probability of finding the chain in conformation α , and ε ( 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 ) is the 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
− T s 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].

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 ) =
X
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].
2.2. The probability distribution of chain conformations
P
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 ( ~ ) = 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
, we first define a Cartesian 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 ~ k
− r
0 of the tail segments ( k = 1, . . . n ), relative to the head group position ~
0
. We treat each CH
2 group as one segment of volume ν
' ˚ 3
The terminal, CH
3
, group is usually treated as a segment of volume thus v = ( n
− 1) ν + ν
0
ν
0
[1, 3].
'
2 ν [1, 3],
∼
( 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 ) d z/ν denote the total number of chain segments (whose centers fall) within a narrow shell z , z + d z of the hydrophobic core, parallel to the membrane plane. In other words, Φ ( z ) d z is the volume taken up by chain segments in shell z , z + d z . Henceforth, unless stated otherwise, we shall measure Φ ( z ) d z in units of ν .
(CH
3 groups count as ν
0
/ν
'
2 monomers). Clearly, Φ ( z )
6
A ( z ) where A ( z ) is the area of the above shell. For a planar bilayer A ( z ) = A , i.e.
Φ
( z )
6
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 h
φ
A
( z ) i
+ N
B h
φ
B
( z ) i
= N h
φ
A
( z ) i
+ h
φ
B
( z ) i

368 A. Ben-Shaul where h
φ
A
( z ) i and h
φ
B
( z ) i are the average numbers of segments in z , z + d z , belonging to chains originating at the upper ( A ), and lower ( B ) interfaces, respectively. The averaging is over all possible chain conformations, e.g., h
φ
A
( z ) i
=
X
P
A
( α ) φ
A
( α ; z ).
α
(7)
Here φ
A
( α ; z ) d z is the number of segments of an fall in the region z , z + d z ; see fig. 1.
A -chain in conformation α which
Using the above definitions, the uniform density constraint for a symmetric planar bilayer reads
Φ ( z ) /N = h
φ
A
( z ) i
+ h
φ
B
( z ) i ≡ ¯
( 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 ) = P
B
( ˜ α denoting the mirror image of α (by reflection through the mid-plane). Using the simplified notation z
6 d/ 2,
P
A
( ˜
( α
;
)
− z
=
) and
P ( α
P
A
) and φ
A
( α ; z ) = φ ( α ; z ) we find, for all
− d/ 2
6
¯
( z ) =
X
P ( α ) φ ( α ; z ) + φ ( α ;
− z ) = a (all z ).
α
(8)
Similarly, for the monolayer
X
P ( α ) φ ( α ; z ) 6 a (all z ).
α
(9)
The (functional) minimization of (6), subject to the set of packing constraints (8) yields the desired singlet probability distribution
1
P ( α ) =
Ω exp −
βε ( α ) −
β
Z
π ( z ) φ ( α ; z ) d z (10) with β = 1 /kT . The π ( z ) are the Lagrange parameters conjugate to (6) and Ω is the partition function
Ω =
X exp −
βε ( α ) −
β
α
Z
π ( z ) φ ( α ; z ) d z .
(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
6 z
6 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

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 R π ( z ) →
π
0
( z ) = π ( z ) + constant.
This is because φ ( α ; z ) d z = 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 )
≡
X
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 h
φ ( z ) i
= 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].
2.3. 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 A
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 =
X
P ( α ) 3 cos
2
θ k
( α ) − 1 / 2 (14)
α

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, vector connecting carbons bond orientations, whereas
C e k
S k − 1
= k
= − 2 S k
, measures the average orientation of the and C k + 1
[7–10].
S k
= 0 ( S k
= 0) implies random
1 (or S 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
= 1 for all k .
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 area per head group a = 25 A 2 chains packed in a planar bilayer with
. 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).

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
= 0 and e g
= 500 cal / mole
'
1.7
kT (for a = 25 A a
≈ ˚ 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 (
∼ ˚ ).
Subsequently, the integrals in (10) and (11) are replaced by layer sums. Similarly,
φ ( α ; z )
→
φ i
( α ) and π ( 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 A 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 A 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.

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
6
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
P ( α ).
π i and hence
2.4. The chain free energy
An explicit expression for the chain free energy in terms of the π ( z ) and h φ ( z ) i is obtained by substituting (10) into (6): f t
=
=
−
− kT kT ln ln
Ω
Ω
−
−
Z d/ 2
π ( z ) h
φ ( z ) i d z
1
− d/ 2
Z d/ 2
π ( z )
¯
( z ) d z
2 − d/ 2
=
− kT ln Ω
− a
Z
0 d/ 2
π ( z ) d z
(15) where
¯
( z ), which appears in the second equality, is given by
¯
( z ) = φ ( z ) + φ (
− z ) (16) and represents the total segment density at z , as defined in (8). Note that h
φ ( z ) i 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
¯
( z ) = a , as in (8).
Equation (15) is also applicable to swollen bilayers, where d > 2 v/a . However, in this case, (8) should be replaced by
¯
( z )
6 a , implying (as for monolayers) that
π ( z ) ≡ 0 if
¯
( z ) < a and π ( z ) > 0 if
¯
( z ) = a . In particular, when d > 2 l the 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
¯
( z ) = h
φ ( z ) i and the bilayer and monolayer constraints are equivalent. A more detailed treatment of chain

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 respect to π ( z ) yields, with
¯
( 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
δf t
π ( z ) =
−
δ
¯
( z )
= kT
δ ln Z
δ
¯
( 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
¯
( z ) = a for all z .
( a , T ) from being only a function of a to f t
= f t
( { ¯
( z ) } , T ), which is a function of the ‘specific areas’
{
φ ( z ) } . Of course, for the planar bilayer
= f t
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 = N a ) dependence of the free energy, i.e.
the stretching elasticity of planar bilayers and then consider curvature deformations.

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
/a 2 , where a 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
6 a c where a c is the chain area at which f t reaches its minimum. The value a c
'
40 A 2 /chain is larger than the typical equilibrium areas per chain in fluid bilayers, a
0
≈
30 A
2
/chain. In fact, a c
'
2 a min
= 2 v/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 h
φ ( z ) i in a densely packed monolayer [39]. More explicitly, these calculations show that h
φ ( z ) i is characterized by a flat region where h
φ ( z ) i
= 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 h
φ ( z ) i , 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 h
φ
A
( z ) i
+ h
φ
B
( z ) i 6 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
| 6
∆z ) we find π ( z ) = 0.
Then, using (11), and noting that h
φ ( z ) i
= a wherever π ( z ) > 0, it is easily shown that
Z
π t
=
−
∂ f t
∂ a
= π ( z ) d z ( a < a c
).
(22)
Thus, π t
> 0 for a < 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

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 = 0.12
kT / A 2 and a h
= 20 A 2 tail. The dashed line represents f s
.
f
+ t f is calculated using (15), n denoting the length of the hydrophobic h
. The dotted line (shown only for n = 16) corresponds to a planar monolayer. (b) The free energy per molecule
Note that the equilibrium area per molecule, to a
0
'
34 A 2 f a
0 for
= varies only slowly with n f t
=
+ f h
+ f s as a function of a , for n = 12, 14 and 16.
n
16. (After [39, 60].)
, from a
0
' ˚ 2 for n = 12

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 = 2 v/a , yields
π t
= δf t
/δa = ∂ f t
/ ∂ a
−
∂ f t
/ ∂ d (2 v/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 = 0.12
kT / A 2 , is based on a common estimate [3, 4]. The value of chosen so as to obtain a
0 a h
( a h
= 20 A 2 ) used to calculate f h
= γa
2 h
/a was
' 32 A 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 = 0.08
kT / A
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 =
−
π t
+ 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
.

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
∂
2 f
∂ a
2 a = a
0
( a
− a
0
)
2
=
1
2
κ
A a
0
δa a
0
2
(23) with the second equality serving as the definition of the area compressibility modulus κ
A
; κ
A
= a
0
( ∂
2 f / ∂ a 2 ) a
0
[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
, is given by
κ
A
=
− a
0
2
∂
∂ a
Z d/ 2
− d/ 2
π ( z ) d z + 2 γ a a h
0
2
.
(24)
Both terms on the right hand side are positive. Numerical estimates based on calculations of the kind shown in fig. 4 yield κ
A
≈
0.5
kT /
˚ 2 for a
0
≈
30 A 2 , with the main contribution coming from κ
A,t and increases with n .
. The chain contribution to κ
A decreases with a
0
3.2. 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.
3.2.1. 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/ 2 N = χ
A f
A
+ χ
B f
B
.
(25)

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
/ 2 N and χ
B
= 1 −
χ
A
= N
B
‘mole fractions’ of A -type and B -type molecules, and 2 N = N
A
+ N
B
.
f
/ 2 N are the
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 ) and P
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 ≡
N a ( z ) (26) with z = d/ 2 and
− d/ 2 denoting the positions of the upper ( A ) and lower ( 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
= 2 N
−
N
A between the two leaflets of the bilayer. The areas per head group at the two interfaces
) are a
A
= A ( z = d/ 2) /N
A
= N/N
A a ( z = d/ 2) and a
B
= A ( z =
− d/ 2) /N
B
= N/N
B a ( z =
− d/ 2).
Thus a
A is given by a
A
=
1
2 χ
A a (0) 1 +
1
2
( c
1
+ c
2
) d +
1
4 c
1 c
2 d
2
(27) with a
B given by a similar expression (with d replaced by − d and χ for the symmetric planar bilayer a
A
= a
B
= a (0).
A by χ
B
). Only
The packing constraints on P
A
( α ) and P
B
( α ) in the curved bilayer are obtained by a straightforward generalization of (8), namely
χ
A
X
P
A
( α ) φ
A
( α ; z ) + χ
B
α
X
P
B
( α ) φ
B
( α ; z ) =
α
1
2 a ( z ).
(28)
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
= independent of head group density and interfacial curvature, we find that
γ
B
= γ is f s
≡
F s
/ 2 N = γ χ
A a
A
+ χ
B a
B

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 s
0
=
1
8
γa (0) c
1 c
2 d
2 ≡
¯ s c
1 c
2
(29) where f s
0 = γa (0) is the surface free energy of a planar bilayer with the same total mid-plane area A (0) = N a (0).
¯ 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
− f
0 h 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 ˜ ( d ) from the interface, so that f
A,h a
A
= C/ ˜
A is given by (27) but with d/ 2 replaced by ˜ 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 of the tail free energy, 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) = N a (0), is given by f t
=
−
χkT ln Ω
A
− (1 −
χ ) kT ln Ω
B
−
1
Z
2
π ( z )
¯
( z ) d z (30) where now Ω
A
, Ω
B z is now, cf. (28), and π ( z ) depend on a (0), c
1
, c
2 and χ . The segment density at
¯
( z ) = χ h φ
A
( z ) i + (1 − χ ) h φ
B
( z ) i =
1
2 a ( z ) (all z ).
(31)
The second equality corresponds to a compact bilayer.
Ω
A and Ω
B are given by (15) with φ
A
( α ; z ) and φ
B
( α ; z ) replacing φ ( α ; z ). The numerical algorithm for calculating the π ( z ), and hence P
A
( α ), P
B
( α ) and f 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.

380 A. Ben-Shaul
3.2.2. 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) = N a (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 b c
2
+
+
1
4
2
+
− c
2
−
+
1
2
λ χ
−
1
2
2
+ ω χ
−
1
2 c
+
.
(32)
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 the bilayer is treated as two independent monolayers: For c and λ can be derived if
1
= c
2
= 0, we have
χ = χ
A
= a (0) / 2 a
A
, χ
B
= 1 −
χ = a (0) / 2 a
B
, cf. (27). Thus,
δχ = ( χ
−
1 / 2) =
− a (0) / 2 a
2
A
δa
A
=
−
(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
'
4 a
0
κ
A
.
(33)

Molecular theory of chain packing 381
Equation (32) allows for the fact that the surface densities of head groups, as measured by χ , may be coupled to the change in the bilayer curvature. A change in
χ in the course of a curvature fluctuation may be due to lateral diffusion of molecules within each monolayer (into and out of the section of bilayer under discussion).
Another, much less likely, mechanism on the time scale of membrane fluctuations, is a ‘flip-flop’ exchange between the monolayers [15–19]. There are cases, e.g., when the hydrophilic head groups are chemically polymerized, that the exchange is ‘blocked’ and χ is a constant, independent of c
1
, c
2
. The opposite limit of ‘free exchange’ corresponds to the case where χ adjusts freely to the momentary curvatures so as to minimize δf . Thus, generally, one can treat χ as a function of c
1 and c
2
, which to first order in c
1
, c
2 is a function of c
+
= c
1
+ c
2 since
χ
−
1
2
=
∂ χ
∂ c
1 0,0 c
1
+
∂ χ
∂ c
2 0,0 c
2
=
∂ χ
∂ c
+ 0,0 c
+
= ηdc
+
(34) with the derivative evaluated at the planar geometry, and with the second equality serving as the definition of η . (In passing to the second equality we made use of the fact that for a laterally isotropic bilayer ( ∂ χ/ ∂ c
1
) = ( ∂ χ/ ∂ c
2
) = ( ∂ χ/ ∂ c
+
).) η is a dimensionless concentration-composition coupling parameter which depends on the mode of deformation.
Using (34) we rewrite (32) in the form
1 a (0)
δf =
1
2
κc
2
+
+
1
4
2
+
− c
2
−
(35) with the rescaled bending constant
κ = κ b
+ ωdη +
1
2
λd
2
η
2
.
(36)
The special case of ‘blocked exchange’ corresponds to η = 0, cf. (34), and hence
κ = κ b
. The minimum value of κ as a function of is obtained when η =
−
ω/λd , in which case
η , corresponding to ‘free exchange’,
κ = κ b
− ω
2
/ 2 λ .
(37)
A third special case of interest corresponds to a bending deformation during which the areas per head group, at both interfaces, remain constant a
A this case, δf
'
δf t
= a
B
= a (0). In is due, almost entirely, to (‘splay-like’) chain conformational distortion. Using (27) we see that, to first order, a
A
= a (0) implies χ
− 1 / 2 = dc
+
/ 4, corresponding to η = 1 / 4 in (34).
A more general form of the deformation free energy, allowing for the case that the planar bilayer is not the equilibrium geometry (but still serving as a reference state)

382 A. Ben-Shaul is obtained by adding to (35) a linear term in c
+
. Writing this term as
−
κc
0 c
+
, we obtain
1 a (0)
δf =
1
2
κ ( c
+
− c
0
)
2
+
1
4
=
1
2
κ c
1
+ c
2
− c
0
2
2
+
− c
2
−
κc
1 c
2
−
1
2
−
1
2
κc
2
0
κc
2
0
(38) with the second equality expressing the familiar Helfrich form [15–19]. The last term ensures that δf = 0 for the planar bilayer. Note that the χ dependence of δf is absorbed into κ . The constants κ , ¯ and c
0 are the familiar splay modulus, saddle splay modulus and the spontaneous curvature [15–21]. As is well known, and easy to show, the equilibrium curvatures are given by c eq
= c eq
1
= c eq
2
= c eq
+
/ 2 = κc
0
/ (2 κ κ ), c eq
−
= 0.
Thus, c eq = c
0
= 0 is the equilibrium condition for the planar bilayer. Thermodynamic stability requires that κ >
−
κ/ 2 > 0.
From (38) it follows immediately that
1
κ = a (0)
(
∂
2 f
∂ c
2
+
+
∂
2 f
∂ c
2
−
)
, (39)
∂
2 f
¯ =
−
2 a (0) ∂ c
, (40)
κc
0
=
−
1 a (0) ∂
∂ c f
+
=
1 a (0) c eq
+
∂
2 f
∂ c
2
+
(41) with all derivatives evaluated at the planar geometry, c
+
= c − = 0 ( c
1
= c
2
= 0).
Notice that f = f ( c
+
, c
−
) is treated here as a function of c
+ and c
− only; the χ dependence has been absorbed into f through (34).
3.2.3. Molecular theory
Application of the general thermodynamic relations (39)–(41) to the tail free energy f t
, as given by (30), yields explicit expressions for the chain contribution to the curvature elastic constants. After some algebra, involving the use of (11) for Ω
A and Ω
B and of the packing constraint (31), one finds [37]
Z
κc
0
= π ( z ) z d z , (42)
¯ =
−
Z
π ( z ) z
2 d z (43)

Molecular theory of chain packing 383 with the integration limits extending over the bilayer thickness. More precisely, the tails contribution to the elastic constants (which is the contribution explicitly considered here) comes from
− d/ 2
6 z
6 d/ 2. By extending the integrals to the aqueous regions one obtains also the interfacial (head group repulsion and surface tension) contributions to the κc
0 and ¯ (see below). Similarly so for the integrals appearing in the expressions for κ as outlined next.
The expression obtained for κ (using (39) and (30)) involves three terms[37]:
κ =
−
Z
1
+
16
∂ π
∂ c
+
η 2 d 2 a (0)
χ z d z
−
Z
∂ π
∂ χ
1
2
ηd
Z
∂ π
∂ χ
φ
A
( z ) − φ
B
( z z d
) z d z
(44) with all derivatives evaluated at the planar geometry, c
+
, c
−
= 0, and it should be noted that the derivative appearing in the first integrand is evaluated for a bilayer with constant ‘composition’ χ . The three terms in (44) correspond, respectively, to the three terms in (36). The first and third terms, representing κ b and λd
2
η
2
/ 2 are positive, while the second (‘coupling’) term is negative. This follows from the fact that upon increasing c
+
(at constant χ ) the area per molecule increases for z > 0 and decreases for z < 0, implying that ( ∂ π/ ∂ c
+
)
χ is negative at z > 0 and positive at z < 0. Similarly, upon increasing χ = χ
A at constant curvature ( c
+
= c
−
= 0) the area per molecule decreases ( π ( z ) increases) in the upper monolayer ( z > 0), and increases (lower π ( z )) in the lower monolayer. Thus the second integral in (44) is obviously positive. The third is positive because h φ
A
( z ) i − h φ
B
( z ) i is positive at z > 0 and negative at z < 0. It should be noted that ¯ , unlike κ , is independent of
η , i.e. independent of the mode of deformation [37].
Analogous expressions to (42) and (43), with σ ( z ) =
−
π ( z ), representing the stress profile in the bilayer, have originally been derived by Helfrich based on thermodynamic-mechanical considerations [17]. (Equation (44) was derived in [37] and a similar expression in [97].) In Helfrich’s expressions the integrations extend from −∞ to +
∞ and thus include the contributions to the elastic constants arising from the interactions prevailing in the interfacial region, f h
+ f s in our notation.
The integration limits in (42)–(44) can be extended similarly, provided we interpret π ( z ) = π t
( z ) + π s
( z ) + π h
( z ) as a sum of tail, surface tension and head group terms [19].
π t
( z ), defined between − d/ 2 and + d/ 2, is the chain lateral pressure appearing in (10) and (11). For f s
, using again the simple form (3) we should set
π ( z ) =
−
γa
A
δ ( z
− d/ 2)
−
γa
B
δ ( z + d/ 2), with a
A and a
B given by (27). Similarly, the simple model (4) for f h implies
π h
( z ) = C/ ˜
2
A
δ ( z
− d/ 2 − ˜
) + C/ ˜
2
B
δ ( z + d/ 2 +
˜
),

384 A. Ben-Shaul with ˜ denoting the distance of the head group layer from the hydrocarbon-water a
A is given by (27) with d replaced by d/ 2 + δ
˜
.
Although we have so far been explicitly concerned with single-component and symmetric ( χ = 1 / 2) bilayers, it should be noted that (42)–(44) apply just as well to mixed and/or non-symmetric systems. The composition and concentration dependencies of κ , ¯ and c
0 enter through the π ( z ) profile. The lateral pressure π ( z ), in turn, is dictated by packing conditions of the form (31) which can easily be extended to mixed bilayers. The only assumptions here are that the compositions (but not necessarily the head group areas) and the (random) lateral distributions in each monolayer are not allowed to vary in the course of a curvature deformation. Including these variables as additional degrees of freedom (thus also allowing for lateral segregation in each leaflet), would result in additional terms in (44).
From (42) we see immediately that for a symmetric bilayer, where π ( z ) = π (
− z ), the equilibrium curvatures are c eq
+
= c eq
−
= c
0
= 0, as expected. Using (43) one can evaluate the saddle-splay constant, ¯ , using the lateral pressure profile π ( z ) of the planar bilayer. Numerical calculations of that it is negative [37]. Its magnitude, |
¯ with chain length, n , and decreases very steeply ( ∼ a
− b per chain, a , increases (lower π ( z t
¯ t
, the chain contributions to ¯ , reveal
| , increases moderately (roughly linearly)
)). Typical values of |
¯ t
| with b
∼ 10) as the area
, e.g., when a
' ˚ 2 range from ∼ 3 kT for n = 8 to ∼ 20 kT for n = 16. In a mixed bilayer of,
, say, C
8 and C
16 chains, ¯ t varies roughly linearly with composition. The surface tension contribution to ¯ can be estimated using (29). From (29) and (38) we get
κ
κ s
γ s
=
'
γd
0.1
2 than
−
¯ t
/ 8 = γν
∼ n 2
− g , with g ranging between 0 and 1/3.) For n
'
16, a
'
30 A kT / A
2
2 ( n/a ) one finds
. Ignoring ¯ h
2 / 4. (At equilibrium, as noted in section 3.1, a
∼ n d
'
, one finds
˚ and thus
¯ = ¯ t
κ s
¯ s
∼
∼ −
10
10 kT kT g , hence
˚ 2 and
, comparable but smaller
. As noted above,
−
κ is expected to decrease steeply as a increases. Under certain conditions ¯ may become positive, violating the bilayer stability condition and favoring spontaneous saddle-like structures [20, 21, 92–94]. It should be noted, however, that all model calculations of
¯ reported so far, including the above, involve considerable uncertainties, reflecting the high sensitivity of the results to the details of the molecular model used (especially for head group interactions [19, 84]). The estimates of the splay bending constant,
κ , or more precisely its tail component κ t seem more reliable.
The bending modulus, κ , can be calculated using (46). The curvature and concentration derivatives of π ( z ) appearing in this equation can be evaluated by solving
(numerically) a set of integral equations containing these derivatives, which are obtained by differentiation of the packing constraint (1.31). Additional details are given in [37].
Figures 5 and 6, both taken from the work of Szleifer et al. [37], show two sets of calculated (chain part, κ t of) κ for three types of bending deformations. Figure 5 demonstrates, for a pure bilayer composed of C
16 chains, how κ varies with the average area per chain, a (0) (= a in the planar geometry) for the three modes of bending deformations mentioned in connection with (36): a) ‘Blocked exchange’, corresponding to η = 0 in (36) and (44), in which case χ = 1 / 2 is constant and
κ = κ b is maximal, since no concentration relaxation (e.g., via lateral diffusion)

Molecular theory of chain packing 385
Fig. 5. The chain part of the bending modulus as a function of the average area per chain (in the planar bilayer). a), b) and c) correspond to: blocked exchange, constant area and free exchange deformations, respectively [37] (see text).
Fig. 6. The chain part of the bending modulus as a function of chain length (dashed lines, upper scale), and as a function of the short chain fraction X s in a mixed bilayer of C
8 and C
16 chains (full lines, lower scale) [37]. The three cases considered are the same as in fig. 5 (see text).
accompanies the bending deformation; b) ‘Constant area’ deformation. In this case
χ changes in the course of the deformation, ensuring that a
A
= a
B
= a (0) stays constant at all curvatures. As noted earlier this deformation mode corresponds to
η = 1 / 4. Physically this special case is characteristic of a bilayer in which the equilibrium area per head group is fully governed by the balance between head group repulsion and surface tension (i.e. by π h and π s in (21), implying a
0
= a h
), with the chains adjusting to the area prescribed by the interfacial interactions. c) ‘Free exchange’, in which case χ adjusts freely at each curvature, so as to minimize κ
(more precisely, κ t
). Here κ = κ b
−
ω
2
/ 2 λ , cf. (37). In all cases κ increases steeply as a decreases (below a
≈ 30 A 2 ), reflecting the strong increase in the magnitude of the

386 A. Ben-Shaul
π ( z ), as noted already with respect to stretching deformations, see e.g., fig. 4. (An approximate scaling argument, explaining qualitatively the a and n dependence of κ is given below). For typical values of the area per chain in phospholipid bilayers, a
∼ 30 − 35 A 2 ( ∼ 60 − ˚ 2 per head group of doubly chained lipids), the results in fig. 5 show that for C
16 chains κ t varies between ∼ 70 kT in case (a) to ∼ 3 kT in case
(c). These estimates should be regarded as lower bounds to κ , since the calculations do not include κ h
– the head group contribution to κ . (The surface term, at least according to (29), is negligible.) Estimates of κ h
, based on electrostatic or excluded volume interaction models, are typically on the order of few kT , or less [81–85]. For bilayers composed of (or containing) short chains (see below), or at relatively large head group areas, this contribution to κ can be most significant, especially in the case of free exchange. Typical experimental values of κ for phospholipid bilayers are
∼
10–50 kT , [15–19, 22–27]. Considerably smaller bending constants, κ
∼
1 kT , were measured for bilayers containing short chain amphiphiles [28, 29], or ‘bola’ lipids [25–27]. A possible explanation of these observations is provided by chain packing considerations, as outlined next.
Figure 6 displays the variation of κ = κ t with the amphiphile chain length
( n = 8–16), for a fixed value of the average area per chain in the planar bilayer, a = 31.6 A 2 . Also shown in this figure is the dependence of κ on the mole fraction of short chains in a binary bilayer of randomly mixed C
16 latter calculation the composition ( C
8
/C
16 and C
8 chains. In the ratio) is the same in both monolayers, and is not allowed to change as a function of curvature. As in fig. 5, the three modes of deformation considered are: a) Blocked exchange; b) Constant area, and c) Free exchange. In all cases corresponding to the single-component bilayers, κ rapidly decreases with n , approximately according to κ
∼ n α ( α
'
3), reflecting mainly the increase with n in the range (
− d/ 2
6 z
6 d/ 2) over which π ( z ) > 0. The addition of small amounts of short chains ( C
8
) to a bilayer composed of longer chains ( C
16
) leads to a more dramatic lowering of κ , in qualitative agreement with experiment
[25–29]. This behavior reflects the substantial decrease of π ( z ), or more precisely of the range over which π ( z ) is large, attendant upon the addition of short chains to the membrane, as illustrated schematically in fig. 7. The addition of short chains of, say, n s segments relieves much of the lateral stress on the last ∼ n l
− n s segments of the long chains, i.e. those which need not compete for the available volume with the short chains.
The conclusions derived from fig. 5 regarding the n and χ dependence of κ should be subjected to the assumption that a (0) = a is the same in all cases considered.
However, this is not necessarily the equilibrium area per chain a
0
(in the planar bilayer) for all cases. Although we have concluded that a
0 varies only weakly with n , this dependence may be amplified in κ due to the strong dependence of κ on a , cf. fig. 5. Suppose κ
∼ n α /a β and a
0
∼ n g then, at a
0
, κ
∼ n γ with γ = α
−
βg . A simple model, outlined below, suggests α
'
3 and β
'
5 (the numerical calculations suggest β
' 7 [37]). We have concluded earlier that g varies between 0 and 1/3, hence γ may be as low as 4/3 (or even 2/3 if one uses β = 7) if g = 1 / 3, i.e. if a
0 is determined only by the balance between chain repulsion and surface tension while head group repulsion is negligible. In this case the lowering of κ by the addition of

Molecular theory of chain packing 387
Fig. 7. Schematic illustration of the lateral pressure profile in a single component bilayer (left), and a mixed bilayer of short and long chains. Note the decrease in the bilayer thickness, and the decrease in the tail contribution to π ( z ).
short chains is also expected to be less dramatic. Indeed, recent calculations of κ for a monolayer of diblock copolymers (where only chain repulsion is relevant) suggest that for a mixture of symmetric diblock chains κ varies approximately linearly with composition [98].
We close this section with an approximate scaling argument which, based on a simple ‘compressional model’ [36, 99], can explain qualitatively why and how κ increases with n and decreases with a . Consider a cylindrical deformation ( c
1
= c , c
2
= 0) of a symmetric planar bilayer, with a (0) = a
0 denoting the average area per head group in the planar geometry. From (39), for this deformation, κ =
(1 /a
0
)( ∂
2 f / ∂ c
2
). Suppose for concreteness that the bending takes place under the condition of blocked exchange. Upon bending the bilayer, the average area per head group in the ‘outer’ (convex, A ) monolayer changes from a
0 to a
A
= a
0
(1 + cd/ 2) = a
0
(1 + cl
0
).
Similarly, a
B
= a
0
(1
− cd/ 2) = a
0
(1
− cl
0
) with d/ 2 = l
As in section 3.1, let l c
∼ n chain. Then,
0 f
=
∼ v/a
( l
0
0
/l
∼ n/a c
) 2
0 denoting the average chain length in the planar bilayer.
1 / 2 denote the average length of the ‘free’(unstretched) is the free energy per chain in the planar geometry ( f = f t
).
In the curved geometry the average chain lengths are given by l
A
= l
0 a
0
/a
A
∼ l
0
(1
− cl
0
) and l
B
= l
0 a
0
/a
B
∼ l
0
(1 + cl
0
).

388
Thus, the average free energy change per molecule upon bending is
A. Ben-Shaul
δf = δf
A
+ δf
B
/ 2 ∼ l
A
/l c
2
+ l
B
/l c
2 − 2 l
0
/l c
2
= l
0
/l c
2 l
2
0 c
2 ∼ n
3
/a
4
0 c
2
.
Hence (the tail part of) the bending constant is
κ = (1 /a
0
) ∂
2 f / ∂ c
2 ∼ n
3
/a
5
0
.
The arguments given above are obviously rather crude since a bending (splay) deformation involves a change in the average shape of the chain (from a ‘cylinder’ to a ‘truncated cone’) and not only in its average cross sectional area. Notwithstanding this proviso, the approximate model can be used to derive a simple relationship between κ and the area compressibility modulus κ
A defined in (23), (hence the term
‘compressional model’). For the cylindrical deformation,
δa
A
= a
A
− a
0
= a
0 cl
0 and
δa
B
= a
B
− a
0
=
− a
0 cl
0
.
Now, using (23) for each of the two monolayers, and noting 2 κ
A
κ
A
(bilayer) we find
(monolayer) =
δf = δf
A
+ δf
B
= κ
A a
0 l
2
0 c
2
/ 2 = κa
0 c
2
/ 2 with the last equality expressing the bending energy for cylindrical deformation. We thus find κ
A
∼
κ/l
2
0 and hence κ
A
∼ n/a
3
0 since l
0
= v/a
0
∼ n/a
0
.
4. Lipid-protein interaction
The presence of a rigid hydrophobic solute, such as an integral protein or a cholesterol molecule in the membrane, introduces additional boundary conditions on its lipid environment – beyond the usual packing constraints prevailing in the ‘unperturbed’ (solute-free) membrane. The rigid solute, say the hydrophobic part of a trans-membrane protein, restricts the conformational freedom (entropy) of the lipid chains surrounding it. Furthermore, if the hydrophobic thickness of the protein, d
P is different from that of the lipid bilayer, d
0
L
, then neighboring lipid chains should stretch out (if d
P
> d
0
L
) or compress (if d
P
< d
0
L
), in order to minimize the contact area between hydrophobic groups and the surrounding aqueous solution. These
, and other factors, such as the detailed shape of the protein, differences between the
‘hydrophobicity’ of the lipid chains and the protein amino acid residues, and the

Molecular theory of chain packing 389 head group interactions prevailing in the interfacial region, combine to determine the nature and the extent of the lipid-protein interaction free energy.
The importance of lipid-protein interactions in controlling the biological activity of certain membranal proteins and in modifying the physico-chemical properties of biological membranes [41, 47, 100–102] has motivated the development of many theoretical models of these phenomena [40–60]. Some of these models are based on
Landau-type expansions of the interaction free energy, with the ‘hydrophobic mis-
match’, d
P
− d 0
L
, or related quantities serving as the thermodynamic order parameters in the free energy expansion [40, 41, 43–46]. Other authors have formulated continuum models, of the kind used in the elastic theory of (smectic) liquid crystals, representing the influence of the protein [54–56] by additional boundary conditions.
There are also some computer simulation studies [57–59]. Only a few theoretical studies have addressed the issues of lipid-protein and lipid mediated protein-protein interactions from a molecular, statistical thermodynamic, approach. The latter include the seminal, mean-field, approach developed by Marˇcelja (for the case d
P
= d 0
L
[42], Pink’s ‘ten-state model’ [48] and the ‘mattress model’ of Mouritsen, Bloom
) and coworkers [41, 50, 51] which has been extensively applied to a variety of issues, notably to investigate the role of the hydrophobic mismatch in membrane phase transitions. Several comprehensive reviews of the theoretical approaches to lipid-protein interaction are available [40, 41, 100], and there is no reason to repeat their analysis here. Thus, in this section we focus attention on one very recent and rather simple model [60] constituting a natural extension and application of the concepts developed in the previous sections.
4.1. The model
As in most previous models of lipid-protein interaction the protein is treated in this model as a smooth and rigid solute embedded in the bilayer hydrophobic core, as illustrated schematically in fig. 8. For concreteness, the hydrophobic part of the protein may be envisioned as a cylinder of height d a
0
P and radius R a
1 / 2
0
, with denoting the average area per chain in the unperturbed bilayer. Thus, to the lipid chains surrounding it, the protein presents an essentially flat and impenetrable wall. The protein and the lipid chains are assumed to have similar hydrophobicities, so that lipid-lipid and lipid-protein attractions are the same. Interactions between the lipids polar heads and hydrophilic groups of the protein are not included in the model. Consequently, the lipid-protein interaction free energy is due entirely to the boundary conditions on lipid conformational freedom imposed by the protein wall, and (when d
P
= d
0 ), to the elastic deformations of lipid chains associated with the
L adjustment of the bilayer thickness to that of the protein.
Let d
L
( x ) denote the bilayer hydrophobic thickness at distances x from the protein. The condition of hydrophobic matching at the lipid-protein interface requires d d
L
L
( x = 0) = d
P
, see fig. 8. The decay of d
L
( x
→ ∞ ) = d
0
L
( x ) to the unperturbed bilayer thickness,
, is assumed to be exponential d
L
( x ) = d
0
L
+ d
P
− d
0
L exp ( − x/ξ ) (45)

390 A. Ben-Shaul
Fig. 8. Schematic illustration of the lipid-protein interaction model. Negative hydrophobic mismatch
(left) results in bilayer compression and hence an increase in the average area per lipid head group near the protein. Positive mismatch results in chain stretching in the vicinity of the protein [60].
with ξ denoting the ‘coherence length’ of the perturbation. (The range of the perturbation is ∼ 3 ξ .) The model treats ξ as a variational parameter, determined by minimization of the lipid-protein interaction free energy ∆F . The exponential form (45) is predicted by the phenomenological (Landau-type) models of lipid-protein interaction. It should be mentioned, however, that other functional forms for d
L
( x ) have been inferred by other approaches [53–56]. The model described here is not capable of predicting the analytic form of d
L convenient reasonable parametrization.
( x ), so that (45) should be regarded as a
Unlike the case of uniform bilayers, the presence of the protein implies that lipid molecules anchored at different distances from the protein are characterized by different conformational properties and, when d
P
− d
0
L densities. Accordingly, the free energy per molecule,
= 0, by different head group f ( x ) = f t
( x ) + f h
( x ) + f s
( x ) (46) is now a function of x . The three terms on the right hand side of (46) describe, as in (1), the tail, head and surface contributions to f .

Molecular theory of chain packing 391
Consider a ‘slab’ of the bilayer of width L y
(parallel to the plane of the protein wall) and length L along the x direction; both L y and L x are taken to be large on a molecular scale, say L y a 1 / 2 and L x
ξ . (As we shall see below, ξ is typically of the order of several molecular diameters, i.e.
ξ is several times larger than a 1 / 2 ). Let d N
A
= σ
A
( x ) L y d x denote the number of chains anchored to the upper, A , interface, within the distance interval x , x + d x . d N
B
= σ
B
( x ) L y d x is the number of chains originating, within the same interval, from the lower interface.
Using d V ( x ) = d
L find σ
B bilayer,
( x ) + σ
A
( x )
( x
=
) L d y
L
( d x x
) to denote the volume of the membrane ‘slice’
/v , with v
L y d x , we denoting the volume per chain. In a symmetric
σ
A
( x ) = σ
B
( x ) = σ ( x ) = d
L
( x ) / 2 v .
For x ξ , σ ( x ) →
σ
0
= d
0
L
/ 2 v = 1 /a
0
. Note however that, whenever d
0
= d
P the
L interface is curved and, for small x , σ ( x ) = 1 /a ( x ), where a ( x ) is the local head group area, see below.
The free energy of the lipids in the above slab, per unit length of the protein wall
(i.e. the free energy divided by L y
) is given by
Z
F = σ
A
( x ) f
A
( x ) + σ
B
( x ) f
B
( x ) d x = 2
Z
σ ( x ) f ( x ) d x (47) with the second equality holding for symmetric bilayers. Equation (47) can be generalized to the case of mixed lipid bilayers, by adding the contributions of the different molecular species, and by adding a ( x dependent) mixing entropy term.
This is an interesting case, especially because of the possibility of protein induced lipid demixing, i.e. enhanced concentration of one (or more) species in the vicinity of the protein. However, for the sake of simplicity the following discussion will be limited to symmetric, single component, membranes.
The lipid-protein interaction free energy is defined by
Z
∆F = F
−
F
0
= 2 σ ( x ) f ( x ) − f
0 d x (48) with f f
0
0
= F
0
/ 2 N denoting the free energy per molecule in the protein free bilayer;
= f ( x
→ ∞
).
By a straightforward generalization of the phenomenological models for f h and f s from section 2.1, we obtain f s
( x ) + f h
( x ) = γa ( x ) + C/a ( x ) = γa ( x ) 1
− a h a ( x )
2
+ const (49) with C = γa 2 h is given by
. The average local area per head group, at distance x from the protein,
2 v a ( x ) = d
L
( x )
"
1 + d
0
L
(
2 x ) 2
#
1 / 2
(50)

392 with d
0
L
( x ) = d[ d
L
( x )] / d x . If d
L
( x ) is given by (45) then d
0
L
( x ) = d
L
( x ) − d
0
L
/ d
P
− d
0
L
.
The local chain free energy is given by f t
( x ) =
X
P ( α ; x ) ε ( α ) + kT
α
X
P ( α ; x ) ln P ( α ; x ) = ε t
( x ) −
T s t
( x )
α
A. Ben-Shaul
(51) with P ( α ; x ) denoting the local singlet distribution of chain conformations. Again, the ‘actual’ P ( α ; x ) can be determined through minimization of
Z
F t
= 2 σ ( x ) f t
( x ) d x subject to the relevant packing constraints on
{
P ( α ; x )
}
. In formulating these constraints it should be noted that, unlike the case of an unperturbed bilayer, here the system, and hence the singlet distribution, is not invariant to translations in the xy plane but, rather, only to translation along y .
Let ρ (
~
) denote the chain segment density at an arbitrary point
~
= X , Y , Z within the hydrophobic interior of the bilayer. As usual, we assume that the segment density is uniform and liquid-like throughout the hydrophobic core: ρ (
~
) = ρ = 1 /ν , where ν is the specific volume per segment in a bulk liquid hydrocarbon. Again,
ρ (
~
) = ρ can be expressed in the form of a packing constraint on
{
P ( α ; x )
}
. It should be noted that the density at
~ involves contributions from all chains, on both interfaces, which are within reach of this point. It is not difficult to show that the appropriate form of the packing constraint for a symmetric bilayer (of width L y
= 1 along the y axis) is
Z
σ ( x )
X
P ( α ; x ) φ
A
( α ,
~
; x ) + φ
B
( α ,
~
; x ) d x = ρ (all
~
).
(52)
α
In this equation
φ
A
( α , S ; x ) d S/ν
~
= X , Z denotes an arbitrary point in the xz plane of the membrane.
is the number of segments belonging to a chain in conformation
α , anchored to the A interface at distance x from the protein, whose X , Z coordinates fall within the small area element d X d Z around X , Z (regardless of their Y coordinates).
The minimization of F t subject to (52) yields
P ( α ; x ) =
1
Ω ( x ) exp
−
βε ( α )
−
β
Z
λ (
~
) φ ( α ,
~
; x ) d (53) with
{
λ (
~
)
} denoting the set of Lagrange parameters conjugate to (52). Their values are determined, as usual, by substitution of (53) into the packing constraints (52) and solving, numerically, the resulting self-consistency equations. The numerical

Molecular theory of chain packing 393 effort required here is considerably larger than for the unperturbed bilayer. First, because the number of equations that need to be solved is bigger: the XZ plane is now divided into many ‘boxes’ ∆X∆Z , rather than into several ‘layers’ of thickness
∆Z . Second, because the segment density in box ∆X∆Z collects contributions from
(non-equivalent) chains originating at several different points x . Nevertheless, these calculations can easily be performed using ordinary work stations.
Using (47) and (51)–(53), one finds
F t
=
−
2 kT
Z
σ ( x ) ln Ω ( x ) d x
−
ρ
Z
λ (
~
) d
~
.
(54)
This, as well as all previous equations in this section reduce to those of the unperturbed bilayer when P ( α ; x ) = P ( α ) and σ ( x ) = σ are independent of x .
4.2. The role of hydrophobic mismatch
As in section 2, the expressions derived in section 4.1 can be used to calculate both single chain (conformational) properties and thermodynamic functions of interest.
Some of the qualitative conclusions can easily be explained by reference to fig. 8. For instance, in the immediate neighborhood of the protein the calculated orientational order parameters reveal enhanced orientational order when d
P
> d
0
L and a lower degree of orientational order in the case of negative hydrophobic mismatch, d
P
< d
0
L
.
This, obviously, is a direct reflection of the enhanced chain stretching in the former case as opposed to chain compression in the latter. These results are consistent with experiment [11]. It has been suggested, based on qualitative theoretical consideration and various experimental observations, that the chains in the vicinity of the protein should show a finite tilt angle of their ‘director’ (the average end-to-end vector) [10].
The results obtained from calculations based on (53) confirm this prediction revealing also that the tilt angle is small when d
P
A finite tilt is also observed when protein wall [60].
d
P
> d
= d
0
L
0
L and relatively large when d
P
< d
0
L
, resulting from chain repulsion by the
.
Figure 9, taken from the work of D. Fattal [60], shows the results obtained for the lipid-protein interaction free energy as a function of the hydrophobic mismatch d
P
− d per chain in these calculations is of 2 a
0
0
L
, for a membrane composed of (saturated) C
14 chains. The asymptotic area
∼
˚ 0 a
0
= 32 A if the bilayer is regarded as composed of double-chain lipids. The hydrophobic thickness of the bilayer corresponding to this value of
L
The two curves shown in fig. 9b correspond to two different choices of the interfacial interaction parameters: γ = 0.12
kT / A 2 , a h
=
(no head group repulsion); both choices yield a
20 A
0
2 and
∼
32 A
γ = a
0
0.08
is d kT /
0
A 2
=
,
24.5 A a h
= 0
˚ 2 for the equilibrium head group area in the unperturbed bilayer.
The results show that ∆F is minimal around d
P
− d
0
L
' 0, yet ∆F > 0 even if the hydrophobic thickness of the lipid and the protein match exactly. In the latter case
( d
0
= d
P
) the head group and surface tension components of ∆F are zero and, hence,
L
∆F is due entirely to chain distortion. The major contribution to ∆F in this case is

394 A. Ben-Shaul
Fig. 9. a) The tail ( ∆F t
, squares) and interfacial ( ∆F s
+ ∆F h
, triangles) contributions to the total lipid deformation free energy (solid circles), as a function of the hydrophobic mismatch [60]. The results correspond to the lipid-protein interaction free energy per unit length of the protein perimeter. The lipids are modeled as C
14 a h chains with head group and surface interaction parameters:
= 20 A 2 . The unperturbed bilayer thickness is d interaction free energy (per 1 A
0
L
= 24.5 A a
0
' 32 A
γ = 0.12
kT / A 2 and
˚ of protein perimeter) corresponding to the above case (solid circles), and to the case γ = 0.08
kT / A 2 , a h
= 0 (open circles). In both cases d
0
L
= 24.5 A

Molecular theory of chain packing 395 associated with the loss of conformational freedom (
−
T ∆S t
> 0) experienced by the chains in the immediate vicinity of the protein wall. This follows from the fact that many of the conformations α available to the chains in a protein free bilayer become forbidden once they are anchored near the protein; namely, all those conformations which ‘penetrate’ into the protein region. The lipid-protein interaction free energy for a system characterized by d
0
L
The conclusions are similar.
= d
P has originally been studied by Marˇcelja [42].
| d
P
A qualitative explanation for the increase of ∆F with the hydrophobic mismatch,
− d 0
L
|
, is provided by the schematic illustration in fig. 8. When d
P
> d 0
L the chains in the vicinity of the protein are highly stretched, losing more of their conformational entropy, in addition to the loss implied by the presence of the wall. Thus, d 0
L
) > ∆F t
( d
P
∆F t
( d
P
>
= d 0
L
). As the chains are stretched their average cross sectional area,
1 /σ ( x ), decreases. However, since the interface is curved, the decrease in a ( x ) and, consequently, the change in ∆F h
+ ∆F s is marginal, see fig. 9a. Hence, for d
P
> d
0
L
, ∆F
'
∆F t
' −
T ∆S t
[60]. On the other hand, when d
P
< d
0
L
, the bilayer is compressed (in the vicinity of the protein), implying an increase in the average cross sectional area per chain (1 /σ ( x ) > 1 /σ
0
= a
0
) and an even larger increase in the interfacial area per head group, a ( x ) > 1 /σ ( x ) > a
0
. Thus, the chains recover some of their lost conformational disorder: ∆F t
( d
P
< d
0
L
) < ∆F t
( d
P
= d
0
L
). However, this gain in the tails free energy is generally over-compensated by the concomitant increase in the surface free energy ∆F s
. ( ∆F h decreases, but to a considerably lesser extent. Note, though, that strong head group repulsion may shift the minimum of
∆F to a slightly negative for a h
= 20 A
2 and a h
= d
P
− d
0
L value, as shown in fig. 9 by comparing the curves
0.) These trends are confirmed by the results shown in fig. 4 for the planar bilayer. Namely, when a increases beyond the equilibrium value a
0
, the tail free energy decreases rather slowly, whereas the surface tension contribution increases linearly with a . Thus, for d 0 > d
P
, the excess lipid-protein interaction free
L energy is due, mainly, to the increase in ∆F s
, see fig. 9a.
For all the data points shown in fig. 9 the value of ξ has been optimized by minimizing ∆F . It turns out that in all cases, the range of the perturbation (
∼
3 ξ ) sult suggests that a microscopic, molecular, approach to the problem may be more appropriate than a phenomenological continuum theory.
In view of the relatively small number of lipid molecules affected by the presence of the protein, detailed molecular dynamics or Monte Carlo simulations of lipid-protein systems seem feasible. Considering the highly specific nature of lipidprotein interactions, simulation methods seem also to constitute the most appropriate approach to this problem. Only very few studies of this kind have so far been published [57, 59].
The model outlined in this section can be extended to bilayers containing a mixture of lipids, and to other shapes of hydrophobic proteins or other solutes. One interesting system which can be studied using the above methods is a lipid-cholesterol bilayer.
On the other hand, as far as lipid-protein systems are concerned, it should be kept in mind that only few proteins can be regarded as simple, rigid, hydrophobic solutes.
Furthermore, the model described above, like most previous models of lipid-protein

396 A. Ben-Shaul interaction, has relied on various simplifying assumptions restricting its applicability to other, more general and possibly more interesting, systems. In view of this fact, the model should be regarded as a small step towards understanding the intricacy of biological membranes.
5. Concluding remarks
Biological membranes, the subject matter of this volume, are extremely complex physico-chemical systems, not to mention their biological aspects. Even the ‘model’ amphiphilic bilayers, which have been considered here and which serve to mimic real membranes are also very complex many-body systems. In addition to their biological relevance, these systems are interesting and challenging theoretically, due to the special coupling between their microscopic and macroscopic behaviors and due to their self-organizing characteristics. Most of the relevant information on either biological or model membranes is, naturally, experimental. Nevertheless, the various theoretical models, ranging from highly qualitative phenomenological pictures to detailed molecular dynamics simulations also contribute to the understanding of their intricate nature. In this chapter we have outlined an intermediate approach to certain issues pertaining to lipid bilayers. Namely, a mean-field theory which takes into consideration some of the basic molecular interactions governing molecular organization in lipid bilayers but treating approximately the cooperative, thermodynamic, properties.
As we have seen, the mean field approach is capable of predicting quite well certain single-chain properties (e.g., bond orientational order parameter profiles), as well as some thermodynamic-mechanical trends, e.g., the role of short chain amphiphiles in reducing the bending rigidity of bilayers.
Undoubtedly, in the near future computer simulations will become increasingly more detailed, reliable and efficient, and will yield relevant information on membrane structure and dynamics. However, computer simulations can not come up with the ultimate answers to all the interesting issues. For instance, it is hard to imagine, at least presently, a series of comprehensive molecular dynamics simulations of a mixed bilayer at different curvatures, performed in order to derive the bending modulus of the membrane. Also, in computer simulations one studies one set of parameters at a time. On the other hand, analytical theories, such as mean field models, often provide the explicit parameter dependence. Furthermore, mean field theories may also be useful in suggesting which parameters and conditions should be studied via simulation. Thus, there is still room and need for elaborating upon the existing approximate approaches. A major improvement towards this direction would be to treat, simultaneously, and on equal grounds the interactions governing chain organization within the hydrophobic region and the electrostatic and/or excluded volume forces prevailing at the membrane interface. As noted earlier, some work along this line has already been published and more work is in progress.
Acknowledgements
The molecular theory of chain packing statistics in amphiphilic aggregates reviewed in this chapter has been developed jointly with several collaborators: Igal Szleifer

Molecular theory of chain packing 397 and W.M. Gelbart are full partners in most of the published (and unpublished) work.
Diego Kramer, Didier Roux and Sam Safran took part in the study of elastic constants.
The analysis of lipid-solute interaction is a part of the Ph.D. Thesis of Debbie Fattal, who also performed some work on other issues mentioned in the text. I would also like to thank Erich Sackmann and Zhen-Gang Wang for many helpful and illuminating discussions, and Reinhard Lipowsky for his critical comments. The
Yeshaya Horowitz Association, the Israel Science Foundation and the US–Israel
Binational Science Foundation are acknowledged for financial support. The Fritz
Haber Research Center, of which A.B.-S. is a member, is supported by the Minerva
Gesellschaft f¨ur die Forschung, mbH, Munich, Germany.
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