Chapter 7. Molecular theory of chain packing, elasticity and lipid-protein interaction in lipid bilayers

A. Ben-Shaul
The Institute of Advanced Studies, Department of Physical Chemistry
and the Fritz Haber Research Center, The Hebrew University, Jerusalem 91904, Israel

1. Introduction

The lipid bilayer, which constitutes the basic structural element of biological membranes, is a two-dimensional, self-assembled, aggregate of amphiphilic molecules. The hydrocarbon chains (the 'tails') of these molecules comprise the hydrophobic interior of the bilayer, shielded from the surrounding aqueous solution by the lipids hydrophilic 'heads- which are located at the two surfaces of the bilayer. The integrity of the bilayer is due to the 'hydrophobic interaction', namely the cohesive forces between the hydrocarbon tails, resulting from the tendency to minimize the hydrocarbon-water contact area [1-3]. The planar bilayer is just one of several possible aggregation geometries which satisfy the hydrophobic effect. Other familiar forms include vesicles, small globular (nearly spherical) micelles and elongated (rod-like) micelles. In all these aggregates at least one linear dimension of the hydrophobic core is microscopic, no longer and typically of order 2l,wherelis the length of the amphiphile chain [1, 3-5].

The number of unrestricted dimensions, along which the aggregate can grow, defines its dimensionality. Accordingly, bilayers (in which only the thickness is restricted), cylindrical aggregates and spherical micelles are two-, one- and zero-dimensional objects, respectively. Alternatively, the shape of these aggregates can be characterized in terms of the two principal curvatures of their (hydrocarbonwater) interface, c₁ = 1/R₁ and c₂ = 1/R₂. In spherical micelles c₁ = c₂ = 1/R, where R is the radius of the hydrophobic micellar core, with R =< l. Similarly, for cylindrical micelles c₁ = 1/R > 1/l, c₂ = 0; for planar bilayers c₁ = c₂ = 0 and in vesicles c₁ = c₂ = 1/R (R >> l). In the following we shall also be interested in moderately curved bilayers, in which case either c₁ and/or c₂ are nonzero, but 1/c_i  =  R >> d where d =< 2l is the bilayer thickness. Another useful characteristic of molecular organization in amphiphilic aggregates is the average area per head group at the hydrocarbon-water interface, a. Simple geometric (surface/volume) packing considerations imply a > kv/l with k = 1, 2 and 3 for the three 'basic' aggregation geometries: planar bilayers, cylinders and spheres, respectively, with v denoting the volume per tail in the hydrophobic core [3-6].

The relative stability of the various possible packing geometries is determined by a delicate balance of forces operating at the interfacial region and within the hydrophobic core of the aggregate. At the interface, head group repulsions, of electrostatic and/or excluded-volume origin, act to increase the average area per molecule, a; a tendency opposed by the hydrocarbon-water surface tension which favors minimal  a. Within the hydrophobic core, the attractive interactions between chain segments (monomers) ensure uniform, liquid-like, segment density, comparable to that of bulk liquid hydrocarbons [1, 3-5]. However, since the monomers are connected 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 interchain repulsion (especially in bilayers) whose magnitude depends rather sensitively on the aggregation geometry, i.e. on a, c₁ and c₂. 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 ydrophobic
thickness of the bilayer, d, is inversely proportional to the average cross-sectional area per chain, d = 2v/a. The equilibrium value of a is determined by the balance of forces mentioned above. Similar considerations, involving a balance of the moments, dictate the equilibrium, 'spontaneous', curvature of the bilayer.

From the above qualitative picture it follows that the lipid membrane is an anisotropic medium involving, due to its unique molecular structure, both liquid-like and elastic ('liquid-crystalline') characteristics. The anisotropy of the hydrophobic medium is revealed in a variety of chain conformational properties such as bond orientational order parameters, segment spatial distributions or the distribution of gauche conformers along the hydrocarbon tail. Some of these 'single chain' characteristics, i.e. properties determined by the singlet probability distribution of chain conformations, can be measured experimentally [3, 7-14]. The conformational properties are closely related to thermodynamic and mechanical properties of the membrane, such as its bending rigidity and stretching elasticity, which reflect the curvature and area dependencies of the bilayer free energy [15-29]. The main purpose of this chapter is to describe and discuss these relationships, starting out from a microscopic, molecular, picture of amphiphile organization in the membrane.

Based on a simple statistical-thermodynamic approach we shall first derive (in section 2) an explicit expression for the singlet probability distribution function of chain conformations in the bilayer [30-39]. The only assumption employed in this derivation is that the monomer density within the hydrophobic interior of the membrane is uniform (liquid-like). The geometry of the system, as specified by a, c₁ and c₂, 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 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].

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