The free energy difference associated with the transfer
of a single cholesterol molecule from the aqueous phase into a lipid bilayer depends on its final location, namely on its insertion depth
and orientation within the bilayer. We calculated desolvation and lipid
bilayer perturbation contributions to the water-to-membrane transfer
free energy, thus allowing us to determine the most favorable location
of cholesterol in the membrane and the extent of fluctuations around
it. The electrostatic and nonpolar contributions to the solvation free
energy were calculated using continuum solvent models. Lipid layer
perturbations, resulting from both conformational restrictions of the
lipid chains in the vicinity of the (rigid) cholesterol backbone and
from cholesterol-induced elastic deformations, were calculated using a
simple director model and elasticity theory, respectively. As expected
from the amphipathic nature of cholesterol and in agreement with the
available experimental data, our results show that at the energetically
favorable state, cholesterol's hydrophobic core is buried within the
hydrocarbon region of the bilayer. At this state, cholesterol spans
approximately one leaflet of the membrane, with its OH group protruding
into the polar (headgroup) region of the bilayer, thus avoiding an
electrostatic desolvation penalty. We found that the transfer of
cholesterol into a membrane is mainly driven by the favorable nonpolar
contributions to the solvation free energy, whereas only a small
opposing contribution is caused by conformational restrictions of the
lipid chains. Our calculations also predict a strong tendency of the
lipid layer to elastically respond to (thermally excited) vertical
fluctuations of cholesterol so as to fully match the hydrophobic height
of the solute. However, orientational fluctuations of cholesterol were
found to be accompanied by both an elastic adjustment of the
surrounding lipids and by a partial exposure of the hydrophobic cholesterol backbone to the polar (headgroup) environment. Our calculations of the molecular order parameter, which reflects the
extent of orientational fluctuations of cholesterol in the membrane,
are in good agreement with available experimental data.
 |
INTRODUCTION |
Cholesterol is a major constituent of the
eukaryotic cell membrane. The concentration of cholesterol largely
varies between membranes of different cells and tissues, and between
the plasma membrane and the internal membranes of the same cell
(Yeagle, 1985
). The effects of cholesterol on lipid bilayers have been studied extensively as a function of concentration, leading to the
understanding that cholesterol mainly affects physical properties of
lipid bilayers (McMullen and McElhaney, 1996). For example, when
present at high concentrations, cholesterol enhances the mechanical
strength of the membrane, reduces its permeability, and suppresses the
main-phase transition of the lipid bilayer. However, in the
low-concentration regime and close to the main-phase transition
temperature, cholesterol acts somewhat oppositely by softening the
bilayer and increasing its permeability (Lemmich et al., 1997; Corvera
et al., 1992).
Besides affecting properties of the host membrane, cholesterol itself
is subjected to restrictions on its motion. In fact, the lipid bilayer
provides a highly anisotropic medium which determines the preferred
location of cholesterol and governs the extent of motional fluctuations
of thermally excited cholesterol. This is reflected, for example, in
the motions of cholesterol along the membrane normal direction:
although the combination of the hydrophobic effect and the
electrostatic desolvation penalty favors the location of the OH group
of cholesterol close to the boundary between the hydrocarbon and the
polar headgroup region, there is still substantial motion perpendicular
to the bilayer normal. This was measured recently by Gliss and
co-workers (1999) who used quasielastic neutron scattering to study the
high-frequency motion of cholesterol in the liquid-ordered phase
(lo-phase) of dipalmitoylphosphatidylcholine (DPPC) membranes
(containing 40 mol % cholesterol). Their study indicates that, at
temperatures higher than 36°C, cholesterol is capable of a
high-amplitude motion parallel to the bilayer normal.
The motional restrictions of the membrane on cholesterol are also
reflected in the magnitude of the molecular order parameter, Smol, of cholesterol, which is a measure of its
orientational fluctuations. An ensemble of rod-like molecules gives
rise to Smol = 0 for unrestricted rotations
of every individual molecule, but yields
Smol = 1 if all molecules are perfectly
aligned in one direction. Cholesterol molecules in lipid bilayers are
aligned roughly along the bilayer normal (e.g., Finegold (1993); also see below) and Smol is a measure of their
fluctuations around the average orientation. Experimentally determined
order parameters of cholesterol are typically found in the range
Smol = 0.70-0.95, depending on the type of
lipid, cholesterol concentration, and temperature. (Taylor et al.,
1981; Dufourc et al., 1984; Murari et al., 1986; Pott et al., 1995;
Kurze et al., 2000; Brzustowicz et al., 1999; Marsan et al., 1999).
The dynamics of cholesterol in phospholipid bilayers has also been the
focus of recent molecular dynamics (MD) simulations (Tu et al., 1998;
Smondyrev and Berkowitz, 1999; Robinson et al., 1995; Gabdouline et
al., 1996). The results of these simulations showed that the
hydrophobic core of cholesterol is buried in the hydrocarbon region of
the bilayer and that, on average, the molecule is tilted with respect
to the bilayer normal. The simulations also showed that cholesterol
molecules are broadly distributed along the membrane normal, similarly
to the lipids. For example, Tu et al. (1998) found for a DPPC bilayer
containing 12.5 mol% cholesterol (at 50°C), a half-width of ~7 Å for the distribution of the cholesterol's OH group in the membrane
normal direction, which is similar to the corresponding half-width of
the carbonyl oxygens of the lipids. Tu et al. also found the
cholesterol molecules to exhibit an average tilt angle of 14° with
respect to the bilayer normal direction. Even though the short
simulation times do not allow a direct comparison with the NMR-based
measurements of Smol, there is general agreement
between measured and simulated cholesterol orientations in lipid bilayers.
It is the aim of the present work to examine the different components
of the free energy of interactions of cholesterol with lipid bilayers,
and to determine their effects on the preferred orientation and
magnitude of fluctuation of cholesterol in membranes. To this end, we
focus on the limit of small cholesterol concentrations, where all
cholesterol molecules interact independently with the lipid bilayer. By
using phenomenological, approximate treatments for the various free
energy contributions (that are generally on mean-field level) we shall
show, e.g., that cholesterol-induced perturbations of the lipid packing
only marginally contribute to the transfer free energy of cholesterol
from the aqueous phase into the bilayer, but dominate its motional
fluctuations within the bilayer. Our energetic approach to
cholesterol-membrane interactions is nonspecific to cholesterol.
Rather, it is of generic nature and should be applicable in a similar
way to other small membrane inclusions.
| |
FREE ENERGY CONTRIBUTIONS |
We consider the transfer of a single cholesterol molecule from the
aqueous phase into a planar lipid bilayer. The corresponding difference
in the free energy, 
Gtot, is commonly written
as a sum (White and Wimley, 1999; Jähnig, 1983; Ben-Tal et al.,
1996; Engelman and Steitz, 1981; Milik and Skolnick, 1993; Kessel and Ben-Tal, 2001)
|
(1)
|
where Gsolv is the desolvation free
energy, describing the transfer of cholesterol from water into a
hydrocarbon phase. Note that
|
(2)
|
consists of an electrostatic contribution,
Gelec, and a nonpolar term,
Gnp. The second contribution in Eq. 1,
Glip, is the free energy arising from
cholesterol-induced perturbations of the lipid bilayer compared to the
unperturbed state of the bilayer. We decomposed
|
(3)
|
into contributions, Gelast and
Gconf, resulting from elastic lipid bilayer
perturbations and from conformational restrictions of the lipid chains,
respectively. The last term in Eq. 1, Gcf, accounts for conformational changes of cholesterol that are associated with the transfer from the aqueous phase into the membrane. Because cholesterol has a rigid molecular backbone it is reasonable to assume
that its structure is not very sensitive to environmental changes. We
thus assume that Gcf = 0.
The transfer free energy, Gtot, depends on
the final location and orientation of cholesterol within the membrane.
Treating cholesterol as a rigid body with no internal degrees of
freedom, one may describe its relative orientation with respect to the lipid bilayer by three translational and three orientational
coordinates. Owing to the lateral isotropy of the bilayer,
Gtot depends only on one translational
coordinate, namely the penetration depth, h, of the
cholesterol backbone into the bilayer, and another two rotational
coordinates of cholesterol which we may specify as the angle, 
,
between its long axis and the bilayer normal, and the angle, 
, of a
rotation around its long axis. For the present purpose it is sufficient
to treat cholesterol as a cylindrically symmetric, rigid body, allowing
us to neglect the dependence of Gtot on .
This implies Gtot = Gtot(h, ) which is schematically illustrated in Fig. 1.
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FIGURE 1
Schematic illustration of changing cholesterol's
insertion depth (left) and its orientation
(right) in a lipid bilayer. Here, h measures the
insertion depth and is the tilt angle. Cholesterol is depicted
schematically as the shaded figure, the boundaries of the hydrocarbon
region of the bilayer are marked by the two horizontal lines, and the
bilayer midplane is shown as a broken line. The thickness of the
hydrocarbon region of the bilayer is 2b0. It
should be noted that the angle is measured with respect to the
optimal location of cholesterol in the membrane shown in Fig. 2.
|
|
When being transferred into a lipid bilayer, cholesterol may insert
into, say, the upper leaflet of the membrane. Because of its
amphipathic character, cholesterol orients along the bilayer normal,
inserting its hydrophobic backbone into the hydrocarbon region while
maintaining contact between its OH group and the polar headgroup
region. This indicates the existence of a minimum in
Gtot at some position, h = h0, and orientation, = 0. (Of course, an
equivalent minimum will be found for the association of cholesterol
with the opposite monolayer.) Even though the optimal association state
between cholesterol and the bilayer is uniquely defined, one may still
measure h and with respect to an arbitrary reference
within the molecular skeleton of cholesterol. The equilibrium positions, h = h0 and = 0, thus
reflect the specific choice of this reference system.
Our calculations below reveal that the minimum in
Gtot(h, ) is reasonably well
pronounced, which allows an expansion up to quadratic order. Using the
notation G
= Gtot(h0, 0), we write
|
(4)
|
where 
tot is the tilt modulus of cholesterol and
tot is the modulus accounting for vibrations in the
membrane normal direction.
Below we show that changes of Gtot(h,
) near h = h0 and = 0, and
thus also the magnitudes of tot and tot,
are determined by (predominantly elastic) perturbation effects of the
lipid bilayer. We shall see that desolvation effects predict a
different behavior, namely Gsolv(h,
) = G
+ ssolv|h 
h0| + wsolv||, where ssolv and
wsolv are two constants.
Gtot(h, ) thus behaves
according to Eq. 4 as long as |h h0| 
2ssolv/tot and || 2wsolv/tot, for which appropriate
elastic deformations of the lipid membrane suppress changes in the
desolvation contribution to Gtot(h,
) (see Discussion).
In general, Eq. 4 would contain an additional term, accounting for the
mixed derivatives of Gtot. However, we can
measure h0 such that this term vanishes. In
other words, h0 is determined uniquely by the
condition
|
(5)
|
For cylindrically symmetric, rigid bodies of large aspect ratio
(length versus maximal width), Eq. 5 is fulfilled independently of the
specific choice of h0, such that
tot and tot do not depend on
h0. We shall argue below that this is reasonably
the case for cholesterol. We thus can (approximately) characterize the
transfer free energy of a single cholesterol molecule into a lipid
bilayer in terms of three quantities, namely
G, tot, and
tot.
Note that G, tot, and
tot do not only determine the preferred location of
cholesterol and its thermal fluctuations, but they are also related to
the extent of partitioning of a given number of cholesterol molecules
between the membrane and the aqueous phase (Ben-Shaul et al., 1996;
Ben-Tal et al., 1996). In particular, an equilibrium constant
K = Cm/Cs can be defined as the ratio of concentrations of cholesterol in the membrane and in the aqueous solution, respectively. In the dilute limit, the
equilibrium constant is related to the standard free energy difference,
G0, per cholesterol molecule between the
membrane and the aqueous solution via
|
(6)
|
where kB is the Boltzmann constant,
T the temperature, and G0 = G + G
.
Here,
|
(7)
|
is the immobilization free energy, accounting for the restrictions
of the translational and rotational motions of cholesterol within a
lipid bilayer of hydrophobic thickness 2b0
(Ben-Shaul et al., 1996).
In the following two sections we present our models for estimating
Gsolv and Glip (as
defined in Eq. 1) and the corresponding contributions to
G, tot, and
tot (that is, G = G
+ G
+ G
+ G
, etc).
| |
DESOLVATION FREE ENERGY |
Gsolv is the free energy of transfer of
cholesterol from water to a bulk hydrocarbon phase. It accounts for
electrostatic contributions resulting from changes in the solvent
dielectric constant and for van der Waals and solvent structure
effects, which are grouped in the nonpolar term and together define the classical hydrophobic effect. We calculated
Gsolv using the continuum solvent model
(Honig and Nicholls, 1995; Honig et al., 1993; Kessel and Ben-Tal,
2001; Gilson, 1995; Nakamura, 1996; Warshel and Papazyan, 1998; Gilson,
M. 2000. Introduction to continuum electrostatics, with molecular
applications.
http://cbs.umn.edu/biophys/OLTB/channel/Gilson.M.pdf). The
method has been described in detail in earlier studies of the membrane
association of polyalanine -helices (Ben-Tal et al., 1996),
alamethicin (Kessel et al., 2000), and monensin-cation complexes
(Ben-Tal et al., 2000).
In short, the electrostatic contribution,
Gelec, was obtained from finite difference
solutions of the Poisson-Boltzmann equation (the FDPB method)
(Honig et al., 1993), where cholesterol is represented in atomic detail
and the lipid bilayer region is modeled as a slab of dielectric
constant 
lip = 2. The width of the dielectric slab
was chosen as 22.6 Å for consistency with our model of the lipid
chains (see below). However, the results do not depend in essence, on
the slab width, provided that it is larger than the length of
cholesterol's hydrophobic core (data not shown). The nonpolar
contribution to the desolvation free energy,
Gnp = 
A + 
,
is assumed to be proportional to the water-accessible surface area of
cholesterol, A. The values of the surface tension,
0.047 kBT/Å2, and the intercept,
2.9 kBT, were derived
from the measured partitioning of alkanes between water and liquid
alkanes (Sitkoff et al., 1996). The total area of cholesterol
accessible to lipids in a particular configuration was calculated with
a modified Shrake-Rupley algorithm (Shrake and Rupley, 1973).
We used the structure of cholesterol as determined by x-ray
crystallography (Shieh et al., 1981). We modified this structure by
replacing the methyl groups on the oxygen and on carbon 23 (Fig.
2) with hydrogens (Insight/Biopolymer),
followed by a short minimization using Insight/Discover (MSI, San
Diego, CA). All the available evidence indicate that cholesterol is
embedded in the hydrocarbon region of the membrane roughly along the
membrane normal with its OH group protruding into the polar headgroup
region of the membrane. Thus, we sampled ~4600 configurations of
cholesterol and the bilayer around this orientation.
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FIGURE 2
Schematic representation of the most favorable
association state between cholesterol and a dielectric slab of
half-thickness b0 = 11.3 Å. The "ball
and stick" model of cholesterol was displayed using InsightII (MSI,
San Diego, CA); carbon atoms are green, hydrogen atoms white, and the
oxygen atom red. The insertion depth of cholesterol is defined as the
distance, h, between the cholesterol oxygen and the bilayer
midplane (dash-dot line). The line connecting the oxygen
atom and carbon atom 23, at an angle of 10° with respect to the
bilayer normal, is shown to demonstrate the somewhat tilted orientation
of cholesterol in its optimal association state. The cholesterol
orientation in this figure defines the orientation = 0, with
respect to which the tilt angle (as defined in Fig. 1) is measured.
Carbon atoms 3 and 23 are marked by arrows.
|
|
The optimal cholesterol-bilayer configuration
The insertion depth and orientation of cholesterol, associated
with the most negative desolvation free energy,
G = G + G = 25 kBT + 0kBT, is
depicted in Fig. 2. In this configuration, the hydrophobic backbone of
the cholesterol molecule is buried in the hydrocarbon core of the
bilayer and the polar OH group penetrates into the headgroup region. We
argue below that lipid perturbation effects are not expected to affect
this association state. Thus, the configuration shown in Fig. 2 defines
the optimal insertion depth h = h0, and
orientation, = 0, with respect to which we expand the free
energy, Gtot(, h) (see Eq. 4).
We note that at = 0 cholesterol exhibits an 10° tilt
angle between the membrane normal and the axis connecting the oxygen
atom and carbon 23.
Insertion of cholesterol into a dielectric slab
Let us vary the insertion depth of cholesterol at fixed
orientation = 0. To this end, we measure h as the
distance between the cholesterol oxygen and the bilayer midplane. The
desolvation free energy, Gsolv(h,
0), for this process and its electrostatic (Gelec) and nonpolar
(Gnp) contributions are shown in Fig.
3. The optimal insertion depth of
cholesterol (shown in Fig. 2) corresponds to the location of the OH
group just above the boundary between the hydrocarbon region of the
bilayer and water (h0 b0 = 11.3 Å) with the hydrophobic backbone fully embedded in the
membrane interior. Pulling cholesterol out of the hydrocarbon region
(by increasing h) leads to an increase in
Gnp, whereas Gelec
remains unaffected. The increase in Gnp is
linear because of the cylinder-like shape of cholesterol. Pushing the
OH group of cholesterol into the hydrocarbon core of the membrane
inflicts an electrostatic energy penalty because the electric dipole of
the OH group interacts unfavorably with the low dielectric medium. Our
calculations reveal Gelec to be a linear
function of h, at least for a sufficiently small deviation
of h from h0 (our calculations yield
|h h0| 
3 Å). The value of
Gnp remains constant in this regime because the water-accessible surface area of cholesterol remains essentially unaffected; the vast majority of the cholesterol molecule is already buried in the bilayer at h = h0.
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FIGURE 3
The desolvation free energy,
Gsolv(h, = 0), of
cholesterol and its two contributions, Gelec
and Gnp, as a function of h, the
distance between the cholesterol OH group and the bilayer midplane. The
two broken vertical lines mark the positions h = b0 and h = b0.
|
|
Combining the linear behaviors for h > h0
and h < h0 it is appropriate to
approximate the desolvation free energy curve of cholesterol by
|
(8)
|
where we extract from Fig. 3 the slopes
ssolv = snp 2 kBT/Å for h > h0
and ssolv = selec 5 kBT/Å for h < h0. The consequences of the linear dependence of
Gsolv on h for the vertical
cholesterol vibrations will be analyzed in the Discussion below. Here
we note that the numerical value for snp can be
very roughly estimated by approximating cholesterol as a cylinder of
radius R = 3.4 Å, corresponding to its cross-sectional
surface of achol 37 Å2
(Lundberg, 1982). The energetic cost of exposing the cylinder surface
to the aqueous environment upon an increase in h is
Gnp = G + 2
R(h h0), which gives rise to
snp = 2R 1 kBT/Å. The value for
snp derived from Fig. 3 is about twice as large
as this estimate because cholesterol is not a cylinder, but has a more
flattened shape that exposes a larger surface area to the aqueous
environment than a cylinder of the same volume.
Changing the orientation of cholesterol in the dielectric slab
Upon tilting cholesterol, the desolvation free energy,
Gsolv, adjusts, in general, both its
electrostatic and nonpolar contributions. However, as long as the OH
group of cholesterol remains outside the dielectric slab,
Gelec remains essentially unaffected. The value of Gsolv Gnp is then dominated by a tilt-induced exposure of
some hydrophobic residues of cholesterol to the polar environment (that
is, into the region of high dielectric constant w,
corresponding to the headgroup region or water; see Fig. 4). Our
calculations indeed showed that Gsolv is
minimal if cholesterol tilts around the OH group, avoiding penetration
of the polar group into the hydrophobic core of the bilayer. We also
found that Gsolv(h0, ) is not very sensitive with respect to the exact choice of
h0. Shifting h0 from the
OH group to carbon 3 did not result in a notable change in
Gsolv (see Fig. 4). This is consistent with the fact that cholesterol has a rather large aspect ratio (length versus width).
Approximating cholesterol as a cylinder of radius R = 3.4 Å, we can estimate
Gnp(h0, ). At
= 0 the cylinder mantle is fully inserted into the hydrocarbon
region of the bilayer. If the cylinder is tilted (with tilt angle ,
see Fig. 4) an area 2R2|tan | of its
mantle protrudes out of the dielectric slab, which leads to a free
energy penalty of Gnp = G + 2R2|tan |. Fig.
4 compares the prediction from the simple
cylinder representation of cholesterol with the full atomic-level
calculations of Gnp as described above. Free
energy decomposition (data not shown) indicates that, indeed, the
electrostatic contribution to the desolvation free energy nearly
vanishes for all .
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FIGURE 4
The desolvation free energy of cholesterol,
Gsolv(h0, ), as a
function of the tilt angle, . Cholesterol was tilted around the
oxygen atom of the molecule ( ) and around carbon 3 of its backbone
( ). The solid line marks the approximative result,
Gnp = G + 2R2|tan |, which was obtained using a
representation of cholesterol as a cylinder of radius R = 3.4 Å. The dielectric constants inside the bilayer and in the
polar region are denoted by lip and w,
respectively.
|
|
Limitations of the model
A detailed discussion of the limitations of the model used for
calculating Gsolv is given in Ben-Tal et al.
(1996). In the following we remark on the two limitations that we
consider the most important for the cholesterol-membrane system. The
description of the lipid bilayer as a slab of low dielectric constant
obscures all atomic details of the cholesterol-bilayer interactions,
i.e., electrostatic, nonpolar, and steric interactions, as well as the ability of cholesterol and lipids to interact via hydrogen bond formation. Although this is a standard representation of the
hydrocarbon region of lipid bilayers (Ben-Tal et al., 1996, 1997, 2000;
Kessel et al., 2000; Bernèche et al., 1998; Biggin et al., 1997;
Efremov et al., 1997), our work does take into account additional lipid bilayer perturbation effects (at least on a phenomenological level; see
next section). As we shall see, these effects are predicted to govern
the magnitudes of tot and tot.
Another approximation of our model results from the complete neglect of
the (polar) headgroup region of the bilayer and the step-like decay of
the dielectric constant from w = 80 in the aqueous
phase to lip = 2 in the hydrophobic bilayer
interior. The corresponding sharp change in hydrophobicity may
generally lead to an overestimation of Gsolv
which, however, does not affect our principal conclusions. Within our
treatment it is most appropriate to regard the headgroup region as
being part of the aqueous phase because the dielectric constant there
was estimated to range between 25 and 40 (Ashcroft et al., 1981). We
note that, in principle, one could incorporate an interfacial region of
varying dielectric constant into the Poisson-Boltzmann equation
(Blackburn and Kilpatrick, 1996). However, even if the dielectric
profile in this region was known, calculation of Gsolv
would still require knowledge on the local values of the surface
tension of cholesterol with the corresponding parts of interfacial
(headgroup) region. This information is currently not available and,
hence, cannot be incorporated into the model.
| |
PERTURBATION OF THE LIPID BILAYER |
There are two obvious nonspecific mechanisms by which a rigid
hydrophobic solute (like cholesterol) may perturb a lipid membrane. Both mechanisms are intimately related to the packing of the lipid chains in the vicinity of a rigid inclusion. First, the solute may
induce an elastic perturbation of the lipid bilayer. This elastic
perturbation is a consequence of the solute's shape and size, which
the lipid bilayer tends to adapt because of the strong hydrophobic
coupling between the solute and the membrane. An experimentally (Dumas
et al., 1999; Killian, 1998) and theoretically (Mouritsen and Bloom,
1984; Dan et al., 1993; Aranda-Espinoza et al., 1996; Nielsen et al.,
1998; Fattal and Ben-Shaul, 1993) well-studied example is the so-called
hydrophobic mismatch, where the hydrophobic height of a transmembrane
protein or peptide differs from that of the host membrane. Yet, the
deviation of a solute's shape from that of a cylinder (Fournier, 1998;
May and Ben-Shaul, 1999) or the tilt of a cylindrical inclusion are
also expected to induce an elastic membrane deformation. The latter
case, which serves us as a model for changing the orientation of
cholesterol, will be investigated in the first part of this section.
The second mechanism derives from the flexibility of the lipid chains
in the fluid state. The presence of a rigid solute reduces the
conformational freedom of the neighboring lipid chains. In other words,
because the lipid chains cannot penetrate into the rigid solute, the
number of accessible chain conformations and orientations is smaller in
the vicinity of the solute than far away from it. The corresponding
free energy penalty (loss of entropy) will be estimated in the second
part of this section.
Although the present work treats elastic membrane perturbations and
chain conformational confinements separately, it should be kept in mind
that both mechanisms are not strictly independent of each other.
Rather, one may suspect that rigid solutes already induce an elastic
membrane perturbation through their effects on the conformational
freedom of the neighboring lipid chains. This indirect mechanism is
neglected here, but can roughly be estimated to be of secondary
importance to the overall lipid perturbation effects (May, 2000).
Elastic lipid layer perturbation
We estimate the elastic response of a lipid layer, induced by
either a tilt angle, , of cholesterol with respect to the bilayer midplane, or by a displacement, h h0,
along the bilayer normal direction. The response of the lipid bilayer
is reflected by the magnitudes of elast and
elast. Both quantities will be calculated here on the
basis of a number of approximations. This allows us to apply a simple
continuum theory of elasticity that was recently used for studying
protein-induced membrane deformations (May, 2000).
Membrane elasticity theory
Let us consider first how tilting the cholesterol backbone
affects the membrane. We shall represent cholesterol as a rigid cylinder of radius R and height b0
(with b0 
R), residing in the upper leaflet
of a lipid bilayer. The tilt angle between the long axis of the
cylinder and the bilayer normal direction is . Qualitatively, the
perturbation of the lipid layer involves different deformation modes
along the tilt direction of the cylinder and perpendicular to it. Along
the tilt direction, the dominant deformation mode is a splay
of the lipid chains. Perpendicular to it, the lipid chains exhibit a
twist (Frank, 1958). Note that splay and twist refer to the
directors of the lipid chains that result from an average
over a sufficiently large number of different chain conformations. The
perturbation of the lipid bilayer does, in general, involve tilt of the
lipid molecules. The fact that this possibility exists even in fluid
bilayers is well-recognized (Helfrich, 1973; Helfrich and Prost, 1988;
MacKintosh and Lubensky, 1991; Fournier, 1998, 1999) and has recently
been shown to be equivalent to lipid layer deformations induced by
curvature (Hamm and Kozlov, 1998, 2000). Fig.
5 illustrates the splay and twist of the
lipid directors caused by the cylinder tilt. Each of the two
deformations decays over a characteristic length, denoted by
1 and 2 for the splay and twist
deformations, respectively. The magnitudes of 1 and
2 depend on the properties of the lipid bilayer. It is
generally accepted that, despite their fluid-like character, lipid
bilayers exhibit a small but notable rigidity against a splay
deformation (Helfrich, 1973). Much less is known about the rigidity
against a twist deformation. Most likely, the response of a lipid
bilayer to a twist deformation is less pronounced compared to a splay
deformation (M. Kozlov, personal communication). As opposed to ordinary
liquid crystals, lipid bilayers consist of very flexible chains whose
packing properties (rather than van der Waals interactions) determine
the energy of the bilayer perturbation. We argue that although along
the cylinder tilt direction the chain packing must adapt to the tilt
angle, , imposed by the cylinder, virtually no such chain
conformational adjustment is necessary normal to the tilt direction,
where the lipids experience a twist deformation. It is therefore
reasonable, as a first approximation, to assume that there is no
appreciable twist rigidity. Adopting this approximation, we note that
only the lipids in the cylinder tilt direction suffer from a
tilt-induced perturbation. All other lipids remain in the same state as
for = 0, implying that the characteristic length
2 vanishes. Our approximation 2 = 0 allows us to reduce the problem to that of a tilted wall
residing in a lipid layer. The solution of this one-dimensional problem
gives us
for an appropriately chosen length of the wallthe
deformation of the lipid layer in the direction of the cylinder tilt.
We shall only briefly outline the basic notion of the theory; further
details of the underlying model have been presented recently (May and Ben-Shaul, 1999; May, 2000) and are related to the previous treatments of Hamm and Kozlov (1998, 2000) and Fournier (1998, 1999).
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FIGURE 5
A tilted cylinder in a lipid layer causes a deformation
with the two characteristic perturbation lengths, 1 and
2. The filled circles and corresponding solid lines
represent lipid headgroups and chain directors. The latter result from
an average over many chain conformations.
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The lipid layer is characterized by two functional
degrees of freedom. One is the (average) lipid tilt angle,
(x), with respect to the normal direction of the planar
bilayer midplane, and the other one is the local effective (average)
chain length, b(x). Because we consider a one-dimensional
model, both quantities depend only on the distance, x, to
the wall. This is schematically illustrated in Fig.
6. Any two functions, b(x) and
(x), define the structure of the lipid layer. For
example, the hydrophobic thickness of the lipid layer at position
= x + b(x) sin (x) is given
by h() = b(x) cos (x).
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FIGURE 6
A tilted wall in a lipid layer. A lipid at position
x is characterized by a tilt angle, /2 , with
respect to the x-axis and a local effective chain length,
b(x). The equilibrium hydrophobic thickness is
b0. The tilt angle of the wall is . The tilt
angle of the lipid director with respect to the hydrocarbon-water
interface is  .
|
|
Consider the elastic excess free energy per molecule,
gelast, in terms of the tilt angle and
the relative dilation of the effective chain length s = b/b0 1, where b0 is the
equilibrium hydrophobic monolayer thickness. For small perturbations
one can expand gelast(, s)
around the equilibrium, 
0 and s 0, up
to first order in , s, and their first derivatives, '
and s'
|
(9)
|
where a0 is the equilibrium cross-sectional
area per lipid in an unperturbed planar layer. Requiring
incompressibility of the molecular chain volume, 
, gives rise to the
relation = a0b0. In Eq. 9,
K, 
, 
0, 
, and
kt are constants that characterize the elastic
properties of the lipid layer. Specifically, K is the
stretching modulus of a lipid layer. The coefficients ,
0, and describe a splay (') deformation
of the lipids. They can be related to the commonly used (Helfrich,
1973) bending modulus, k, spontaneous curvature,
c0, and the position of the so-called neutral surface, where bending and stretching deformations
decouple (Hamm and Kozlov, 2000). Note finally that the lipids may be
tilted with respect to the hydrocarbon-water interface. The
tilt angle is = + b0s' (see
also Fig. 6). The coefficient kt is the tilt modulus of the lipids with respect to changes in . The appearance of
a single term ~( + b0s')2
(instead of three independent terms ~2,
~s'2, and ~s') results from the
additional assumption that the lateral stress profile in the lipid
layer acts only within surfaces that are parallel to the
hydrocarbon-water interface.
The overall elastic excess free energy is
|
(10)
|
where the integration runs over all N = 
dn lipids that are perturbed by the presence of the
wall. In equilibrium, the two functions b(x) and
(x) will adjust such that
Gelast adopts a minimum. When the inclusion
is untilted ( = 0) the lipid layer does not experience a
deformation (s(x) 0 and (x) 0), implying G = 0. For
0 the tilt angles, (x), must adopt
nonvanishing values because hydrophobic coupling between the wall and
the lipid layer requires (0) = (see Fig. 6). Note at this
point that we assume the thickness of the wall to be small, which is
motivated by the fact that the width of cholesterol is small compared
to its length. Even though there is no wall-induced chain
stretching/compression (that is, s(x = 0) = 0),
the function s(x) will adopt nonvanishing values for
x 0 because of the coupling of chain dilation and
tilt. Far away from the inclusion the lipid layer is unperturbed (that
is, s(
) = () = s() = () = 0). The determination of the optimal lipid layer configuration,
as expressed through s(x) and (x), corresponds
to solving an appropriate set of Euler-Lagrange equations with boundary
conditions at x = 0 and x 
± as given
above (for an explicit formulation of the Euler-Lagrange equations, see
May (2000)). Because the present description of the lipid layer
perturbation is based on a quadratic expansion of
Gelast it will also be valid only up to
quadratic order in the tilt angle, , of the wall. Yet, this yields
exactly the elastic contribution to the tilt modulus as appearing in
|
(11)
|
Minimizing the lipid layer perturbation energy with respect
to s(x) and (x) thus allows us to calculate
elast. The final result can conveniently be expressed in
terms of the quantities
|
(12)
|
and is given by
|
(13)
|
where L is the length of the wall (whichas argued
aboveneed not be large compared to the size of the lipids). Let us
shortly discuss the expression for elast. It
monotonously increases with kt, reflecting the
rigidification of the lipid layer upon confinement of the lipid tilt
degree of freedom. In the limit of a large lipid tilt modulus, namely
for kt , we find g3 , implying that elast converges to some finite
value. In fact, if we further set = 0 = 0, we obtain
elast = 4k/1 with
= 4b
/K.
Note that in this case 1 is the decay length of the
perturbation profile as indicated in Fig. 5.
Molecular lipid model
Equation 13 provides an expression for the elastic tilt modulus,
elast, in terms of the phenomenological parameters,
K, , 0, , and
kt, appearing in Eq. 9. To specify these
parameters we use a simple molecular lipid model that has been used in
this (May, 2000) or in modified (May and Ben-Shaul, 1995, 1999)
versions to predict various elastic properties of lipid layers. The
molecular model expresses the free energy per lipid,
gelast, in terms of the effective chain length,
b, and its cross-sectional areas, ai
and ah, measured at the hydrocarbon-water
interface and at the headgroup region, respectively,
|
(14)
|
The first term is the interfacial energy; 
= 0.12 kBT/Å2 is the surface tension
exerted at the hydrocarbon-water interface. (We note that corresponds to create a planar oil-water interface and is more than
twice as large than , which is derived from alkane
partitioning. The difference reflects the curvature dependence of the
nonpolar contribution to the desolvation free energy; for a discussion
see Southall and Dill (2000).) The second term in Eq. 14 accounts for
the (usually) repulsive headgroup interactions; B > 0
is the headgroup repulsion parameter. The model for the headgroup
energy is based on the assumption that the headgroups interact only
within a given surface located at fixed distance lh above (and parallel to) the hydrocarbon-water
interface. The first two terms in Eq. 14 compose the well-known
opposing forces model (Israelachvili, 1992). The last term
in Eq. 14 extends the opposing forces model by taking into account the
conformational freedom of the lipid chains. The corresponding
conformational free energy depends (for essentially planar membranes)
only on the effective chain length b. The parameter 
characterizes the rigidity against changes of the optimal effective
chain length lc. We note that
ai and (similarly) ah are
coupled to b and owing to the incompressibility of the
lipid chain volume .
It can be shown how the molecular interaction parameters in Eq. 14
relate to the phenomenological material parameters in Eq. 9. To this
end, it is convenient to define the reduced (dimensionless) quantities
|
(15)
|
The molecular area a of a planar lipid layer
(with (x) 0) is characterized by a = ah = ai = /b. A
simple calculation shows that the relation 
= (1 
)/(2(1 
c)) ensures
that b0 = /a0 is the
hydrophobic thickness of a planar lipid layer in equilibrium. With
that, the relations between the molecular constants and the
phenomenological parameters are (May, 2000)
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|
(16)
|
In our model, the physical origin of the rigidity with respect to
lipid tilt (as expressed through kt) is the fact
that during a pure tilt deformation (with s 0 and
= const) the lipid chains become stretched (whereas
ai and ah remain
constant); hence kt = 0 for = 0. We note that the expression for kt in Eq. 16 is likely to provide only a lower bound of the tilt modulus because all
contributions to the tilt modulus beyond that of pure chain stretching
are not accounted for. This may concern, for example, a headgroup
contribution to kt or the confinement of the
chain conformational freedom upon a tilt deformation. In our numerical estimates, presented next, we shall therefore consider the two limits
kt = (1 ) and
kt .
Numerical estimates
The relations in Eqs. 16 open the possibility to calculate
elast in terms of molecular interaction parameters. A
reasonable choice for these parameters is
|
(17)
|
The values for and lc are calculated
from statistical mean-field chain packing calculations of
C-14 chains (May, 2000). Together with the values for B and
lh they give rise to a vanishing spontaneous
curvature (c0 = 0) of the lipid monolayer,
a corresponding bending rigidity of k = 7.5 kBT, a hydrophobic half-thickness of
b0 = 11.3 Å for an unperturbed bilayer,
and a monolayer stretching modulus of K = 0.55 kBT/Å2 (May, 2000). All these
values are in agreement with typical experimental observations. (A more
quantitative comparison with experiment is not attempted because our
lipid model in Eq. 14 is not specific to a particular lipid.)
With our numerical choices in Eq. 17 we obtain from Eq. 12, 13, and 16 elast = 0.9 L kBT/Å
rad2. Due to the uncertainties regarding the magnitude of
the tilt modulus we also investigate the limit
kt (that is, we do not use the
expression for kt in Eqs. 16 but instead we use
kt ). We then obtain
elast(kt 
TOP
ABSTRACT
INTRODUCTION
![&kgr;<A><AC>c</AC><AC>˜</AC></A><SUB>0</SUB>=<UP>−</UP>&ggr; <FR><NU>b<SUB>0</SUB></NU><DE>2</DE></FR> [1−<A><AC>B</AC><AC>&cjs1171;</AC></A>(1+2<A><AC>l</AC><AC>&cjs1171;</AC></A><SUB><UP>h</UP></SUB>)]](/content/vol81/issue2/fulltext/643/img034.gif)
