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Determination of Lipid Spontaneous Curvature
From X-Ray Examinations of Inverted Hexagonal Phases
Michael M. Kozlov
Summary
Structure of a lipid monolayer can be characterized by its spontaneous shape corresponding to a stress-free
state. In the chapter, the rigorous description of monolayer shape in terms of curvature of a Gibbs dividing surface is reviewed, and the notions of the neutral and pivotal surfaces are discussed. Further, the effective and exact
method of finding the position of the pivotal surface and the corresponding monolayer spontaneous curvature of
cylindrically curved monolayers based on the results of the X-ray scattering experiments are presented. Briefly,
some limitations of this method are discussed.
Key Words: Elasticity; inverted hexagonal phase; lipid monolayer; neutral surface; spontaneous curvature;
X-ray scattering.
1. Introduction
Amphiphilic molecules (or amphiphiles) are characterized by dual hydrophobic–hydrophilic
properties. Phospholipids are biological amphiphiles, whose molecules consist of hydrophilic
polar heads, and in most cases, two hydrophobic hydrocarbon tails (1). Phospholipids (referred
in the following as lipids, for simplicity) of different types differ in the chemical structure of
their polar heads as well as in length and degree of saturation of their hydrocarbon chains. In
spite of that, all lipids exhibit a similar behavior in aqueous surrounding. Driven by the
hydrophobic effect (2), lipids self-organize into assemblies referred to as mesophases, which
are characterized by a common feature—the hydrophobic moieties of the lipid molecules are
shielded from water by layers of polar heads. If the amount of water is limited or in special
stabilizing conditions, the lipid mesophases have properties of lyotropic liquid crystal phases
(see refs. 3–7). In excess water, the mesophases can be seen as isotropic solutions of
amphiphilic aggregates.
Building blocks of mesophases are lipid monolayers, which are monomolecular layers
wherein all the lipid molecules are oriented similarly: the polar heads toward the one and the
hydrocarbon chains toward the other side of the layer. As a result, one monolayer side is covered by polar heads, whereas the opposite side is hydrophobic. The monolayer thickness (δ),
determined by the lipid molecular length, equals approx 2 nm. The structures of lipid molecules and interactions between them determine the preferable shapes of monolayers. The
monolayer shape serves for characterization and classification of lipid mesophases. In turn,
lipids can be classified according to shapes of monolayers and the corresponding types of
mesophases they are forming in aqueous solutions (for review, see refs. 8,9). Whereas a large
From: Methods in Molecular Biology, vol. 400: Methods in Membrane Lipids
Edited by: A. M. Dopico © Humana Press Inc., Totowa, NJ
355
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Fig. 1. Inverted hexagonal (HII) phase: schematic illustration and notations. (A) The HII phase consists of infinite prisms packed parallel to each other. The prism cross-sections form a 2D hexagonal
lattice. The first-order Bragg parameter (dhex) is measured by X-ray scattering; (B) cross-section of
one prism of the HII phase. The Luzzati dividing surface is located in the polar head region. It is characterized by radius RW and the area per lipid molecule AW. Two arbitrary dividing surfaces separated
by distance z are denoted by indices 1 and 2.
number of lipid mesophases has been described (for review, see ref. 10), just some characterized by the simplest and most common shapes of their constituting monolayers will be
addressed herein.
The most familiar type of lipid assembly is a planar bilayer consisting of two monolayers,
contacting each other along the hydrophobic planes and whose hydrophilic surfaces cover the
inner and outer bilayer planes. The propensity of lipids to self-assemble into planar bilayers
underlies formation of cell membranes (the universal biological wrappers), which form
boundaries of cells and intracellular organelles. A biological membrane is, basically, a multicomponent lipid bilayer with numerous proteins inserted into the lipid matrix or bound to
the bilayer surface (11). Because of the powerful hydrophobic interactions, the bilayer lipid
matrix provides the cell boundary with mechanical stability. The lyotropic liquid crystal
phase formed by planar lipid bilayers is called the lamellar phase (4). It consists of a stack of
bilayers separated by a few nanometer-thick water layers. Lipids such as lecithins forming the
lamellar phases are often referred to as “lamellar” lipids.
Other type of lipid assembly consists of strongly curved lipid monolayers, whose shape
can be seen within a good approximation as a narrow cylinder with few nanometer radius
(Fig. 1A). The lipid polar heads of such monolayer are oriented toward the internal space of
the cylinder, which is filled with water, whereas the external surface of the cylindrical monolayer is hydrophobic. Within the mesophase, multiple lipid cylinders are oriented in parallel
and contact each other along the hydrophobic surfaces. This leads to a most compact packing
of the monolayers so that the mesophase cross-section perpendicular to the cylinder axes reveals
a hexagonal lattice of cross-sections of the individual cylinders (Fig. 1A). This mesophase is
called the inverted hexagonal (HII) phase, and the corresponding lipids are referred to as the
“hexagonal” ones. A typical example of a “hexagonal” lipid is dioleoyl-phosphatidylethanolamine (see ref. 7).
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The HII phase will be the focus of the present review. At the same time, few other
mesophases should be briefly mentioned in order to illustrate the wide spectrum of possible
shapes of lipid monolayers and the corresponding diversity of the lipid-phase behavior (for a
review see ref. 10 and references therein). Molecules of lysolipids having only one hydrocarbon tail tend to form monolayers, which are strongly curved in the direction opposite to that
of the “hexagonal” lipids. The resulting assemblies are represented by cylindrical or spherical micelles with internal volumes filled by the hydrocarbon chains. In a limited amount of
water the cylindrical micelles pack into hexagonal phases, which in contrast to the inverted
hexagonal phases, are denoted as HI phases. Finally, there are lipids and lipid mixtures forming
bicontinuous cubic phases whose monolayers have shapes of saddles organized into periodic
three-dimensional surfaces.
This short phenomenological consideration aims to illustrate the diversity of shapes
adopted spontaneously by lipid monolayers and the degree of monolayer spontaneous curving, which varies from zero in the case of “lamellar” lipids to high values for the “hexagonal,” “micellar,” and “cubic” lipids. In the following, we will discuss how the spontaneous
shapes of lipid monolayers can be addressed based on X-ray studies of HII phases. First, we
introduce a rigorous description of monolayer shapes in terms of Gibbs dividing surface, and
define the notion of monolayer spontaneous curvature. Further, we introduce the notion of
monolayer elasticity apply it to the cylindrical monolayers of HII phases, and define the neutral
surface. Based on these definitions, we present protocol for measuring the position of neutral
surface and its spontaneous curvature using the X-ray data.
2. Description of Monolayer Shape—Spontaneous Curvature
2.1. Dividing Surface
A few nanometer thickness of a lipid monolayer is, for most membrane structures, few
orders of magnitude smaller than the dimension measured along the monolayer plane.
Therefore, it seems straightforward to describe lipid monolayers as thickness-less surfaces.
Whereas such approach did prove to be productive, there is a delicate issue of definition
of the surface with which the monolayer is identified. This issue becomes especially
important for strongly curved monolayers, such as those forming the inverted hexagonal
or micellar phases, whose radii of curvature are of the same order of magnitude as the
monolayer thickness.
The problem of dealing with layers of small but finite thickness is not specific for lipid
monolayers but rather originated from attempts to treat transition regions between immiscible liquids. Herein the Gibbs method will be used (12), which has been developed for description of the liquid–liquid interfaces, but has a general character and can be applied to any type
of thin layers including lipid monolayers. According to the Gibbs approach, one has to choose
within the lipid monolayer a geometrical surface (called the dividing surface) that lies parallel
to the plane of the polar heads. The monolayer shape is identified with that of the dividing
surface and the monolayer thermodynamic properties are assigned to the dividing surface. To
this end, a reference system is considered where bulk phases are extended with their
unchanged properties up to the dividing surface. The thermodynamic values of the dividing
surface are, by definition, the excess values (e.g., excess entropy [Ss], energy [Us], and masses
of components [ms]) determined as differences between the thermodynamic values of the real
system and those of the reference system.
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The exact position of the dividing surface along the monolayer thickness can be chosen
arbitrarily. However, it is important to realize that both the geometrical and most of the thermodynamic characteristics (except the total free energy) of the system depend on the choice
of the dividing surface, so that once the dividing surface is chosen within the monolayer, the
following description has to be performed consequently for this specific dividing surface (see
ref. 13). Whereas the Gibbs method places no limitations on the choice of the dividing surface,
there are physical reasons for selection of particular dividing surfaces such as Gibbs surface
of tension (12) and neutral surface, which simplify the treatment of monolayer properties and
thermodynamic behavior. The neutral surface is a center of the following discussion.
2.2. Spontaneous Cuvature
Once the dividing surface is chosen the monolayer shape can be characterized at each point
by the curvatures of the dividing surface. According to the differential geometry of surfaces
(see refs. 14,15), the local shape is determined by two principal curvatures, c1 and c2, or
equivalently, by the total curvature, J = c1 + c2, and the Gaussian curvature, K = c1 ⋅ c2. The
two latter values represent the two independent invariants of the curvature tensor (14,15) and
are convenient for description of lipid monolayers having properties of two-dimensional (2D)
fluids (16). (Note that in the mathematical literature the notion of the mean curvature H = J/2
rather than the total curvature J is commonly used.) To determine the signs of the curvatures,
the orientation of the normal vector of the dividing surface has to be chosen. It is convenient
and common to define the normal vector to be oriented from the hydrocarbon chains toward
the polar heads. The principal curvatures are defined as positive if the monolayer bulges in
the direction of polar heads and negative in the case of bulging toward the hydrophobic tails.
Herein the monolayer spontaneous curvature (Js) is defined, as the total curvature the
monolayer adopts on spontaneous self-assembly. Note that this definition differs from the
original one, which has been given by Helfrich (16) to quantify the tendency of a flat monolayer to bend spontaneously. The Helfrich spontaneous curvature represents, practically, a
bending stress existing within a monolayer when the latter has a flat shape (indeed, the bending stress equals the Helfrich spontaneous curvature multiplied by the bending modulus).
According to the present definition, the spontaneous curvature is a geometrical characteristic
of a stress-free state of a monolayer.
Whereas the spontaneous curvature is an essentially macroscopic, rather than molecular,
characteristic of a lipid monolayer, the notion of spontaneous curvature of individual lipid molecules is frequently used in the literature. The molecular spontaneous curvature has to be understood as the spontaneous curvature of a monolayer consisting of molecules of a given kind. In
addition to the spontaneous curvature, the stress-free state of a monolayer can be characterized
by a spontaneous area per given macroscopic number of lipid molecules. This area has to be
determined at the dividing surface, and it’s meaning is the area of the monolayer cross-section
by the dividing surface. Commonly, one considers a spontaneous molecular area (As), which is
an effective value equal to the total spontaneous area divided by the number of lipid molecules.
3. Monolayer Elasticity—Neutral Surface
3.1. Elastic Stresses and Energy
To advance toward consideration of the experimental procedure used to determine spontaneous curvatures of individual lipids, a step beyond the spontaneous state of the monolayer has
to made and the monolayer deformations have to be considered. The considerations will be
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limited to cylindrical monolayers of HII phases because this is the system in which the most
productive X-ray measurements have been performed (7,17–25). Geometrical description of a
cylindrical surface is simplified because its Gaussian curvature equals zero, K = 0, and the
absolute value of the mean curvature equals the cylinder radius, |J| = 1/R. There are two types
of deformations (strains) of a cylindrical monolayer: deformation of stretching leading to deviation of the molecular area (A) from its spontaneous value (As), and bending deformation
resulting in difference between the total curvature J and the spontaneous curvature Js.
In the following, for simplicity, positive value 1/R and 1/Rs will be used to characterize the
deformed and spontaneous states of the HII phase monolayers. But it has to be kept in mind
that, according to the earlier convention on curvature sign, the corresponding curvatures are
negative, J = –1/R and Js = –1/Rs. Deformations give rise to elastic stresses and the elastic
energy. A complete consideration of membrane strains, stresses, and relationships among
them is given in ref. 26. Here, instead, a simplified case of cylindrical monolayers is sketched
(27,28). The elastic stresses corresponding to the stretching and bending strains are the lateral
tension (γ) and bending moment (τ), respectively. In the limit of small deformations,
A − As
R − Rs
<< 1 and
<< 1
As
Rs
(1)
the stresses depend linearly on the strains according to:
γ =Γ⋅
and
1 1 
A − As
+ E ⋅ − 
As
 R Rs 
 A − As 
1 1 
τ = κ ⋅ −  + E ⋅
 As 
 R Rs 
(2)
(3)
The coefficients Γ and κ are the monolayer stretching modulus and bending modulus,
respectively. The coefficient E determines coupling between the deformations of stretching
and bending.
The elastic energy of deformation per unit area of the dividing surface is
 A − As 
1 1 
 A − As   1 1 
1
1
F = ⋅Γ ⋅
 + ⋅ κ ⋅ −  + E ⋅
⋅ − 
2
2
 As 
 R Rs 
 As   R Rs 
2
2
(4)
1
→ 0;
Rs
validation of the expression (Eq. 3) for the bending moment and of the curvature-dependent
contributions to the energy (Eq. 4) requires, instead of the condition (Eq. 1), fulfillment of
inequality J ⋅ δ << 1, where δ = 2 nm is the monolayer thickness.
Note that in case of vanishing spontaneous curvature: J s = 0 or, equivalently,
3.2. Elastic Moduli for Different Dividing Surfaces
All elastic and structural characteristics of a monolayer (κ, Γ, E, As, and Rs) depend on the
position of the dividing surface. Consider an arbitrary dividing surface whose characteristics
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will be indicated by subscript 1, and another dividing surface, indicated by superscript 2,
which is shifted by z with respect to the first toward the external surface of the cylindrical
monolayer (Fig. 1B). The relationships between the radii of spontaneous curvatures and
spontaneous molecular areas corresponding to the two dividing surfaces are given by
Rs2 = Rs1 + z
(5)

z 
As2 = As1 ⋅ 1 +

 Rs2 
(6)
A thorough analysis of relationships between the elastic moduli related to different dividing surfaces is given in refs. 24,27,28. Here, only simplified equations are presented. For the
case of vanishing spontaneous curvature (Js = 0), the set of elastic moduli (κ1, Γ1, and E1) is
related to the corresponding set (κ2, Γ2, and E2), by
Γ 2 = Γ1
(7)
κ 2 = κ1 + z 2 ⋅ Γ1 − 2 ⋅ z ⋅ E1
(8)
E2 = E1 − z ⋅ Γ1
(9)
3.3. Neutral Surface
Analysis of the elastic moduli dependences on the position of the dividing surface shows that
the bending and stretching moduli adopt only positive values. Along with the spontaneous geometrical characteristics As and Rs, they describe macroscopically the monolayer structure and
elasticity, and represent material properties of membrane monolayers, which within the approximation (Eq. 1), do not depend on deformations. At the same time, the coupling modulus E can
adopt positive or negative values depending on the position of the dividing surface. For a special dividing surface, called the neutral surface, this coefficient vanishes: E = 0 (26–28), and the
whole description of the membrane elastic properties (2–4) simplifies. Specifically, for the neutral surface the lateral tension γN does not depend on the bending deformation J–Js, and the
bending moment τN is independent of the deformation of stretching A–As.
It is instructive to present (Eqs. 2 and 3) in the form that determines the membrane
deformations 1 − 1 and A − As , which result from stresses γ and τ applied to the membrane
R Rs
by external sources:
 γ E⋅τ
A − As
1
=
 −

(10)
As

E2   Γ Γ ⋅ κ 

1 −
 κ⋅Γ
 τ E⋅γ
1 1
1
− =
 −

2
R Rs 
E  Γ Γ ⋅ κ

1 −
 κ⋅Γ
(11)
Following from Eqs. 10 and 11, a special property of the neutral surface (E = 0) is that its
molecular area (AN) does not change and remains equal to the spontaneous value (AsN) if only
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a bending moment τ is applied to the monolayer. Deviation of AN from AsN requires application
of tension γ. Analogously, the curvature of the neutral surface (J N =1/RN ) is independent of γ and
changes only on application of τ.
The neutral surface is the most convenient to treat the elastic behavior of lipid monolayers.
On the other hand, the position of the neutral surface within a monolayer depends on the
monolayer structure and is usually unknown. The Eq. 9 provides a key for finding the neutral
surface. Assume that there is a way to measure the whole set of the elastic moduli (κ1, Γ1, E1)
for a certain dividing surface with a known position. The distance zN between this dividing
E
surface and the neutral surface is given, according to (Eq. 9) by zN = 1 . Hence, measurement
Γ1
of the elastic moduli for one of the dividing surfaces gives enough information for determination of the location of the neutral surface, and calculation of the corresponding elastic moduli
according to the relationships (Eqs. 7–9) or the more general ones presented in refs. 26–28. In
the following, a specific protocol for finding the dividing surface and the related elastic and
structural characteristics of a monolayer based on X-ray studies of the inverted hexagonal (HII)
phases is presented.
4. Structural Studies of HII Phases
4.1. Directly Measurable Characteristics
An HII phase can be seen as a 2D hexagonal lattice formed by axes of infinitely long and parallel regular prisms (Fig. 1A). The water core of each prism is lined with the lipid polar groups,
and the rest of the prism volume is filled with the hydrocarbon chains. The essence of the experimental approach that is being described in this review is variation of the water content of HII
phase at fixed and known lipid content, and measurement of the related changes in the structural parameters of the HII prisms. The measurements are performed by X-ray diffraction.
There are two methods used to change the water content (see refs. 7,24,29–31 and references therein for detailed descriptions). According to the first, referred to as the gravimetric
method, dry lipid samples are hydrated in water vapor atmosphere to different degrees quantified by the weight fraction of water within the samples. The volume fraction of water in the
sample (φw) is then calculated using specific molecular volumes of water, lipids, and lipid
polar groups. In the second approach, called the osmotic pressure method, a lipid sample is
placed in an excess amount of aqueous polyethylene glycol solution of known osmotic pressure (Π). The amount of water within HII phase is then controlled by the value of Π.
The structural dimension of the HII phase that is measured in X-ray diffraction experiments,
is the first order Bragg-spacing dhex illustrated in Fig. 1A. Usually, an HII phase is characterized
by at least three X-ray spacings bearing ratios to dhex, of 1, 1 3 , 1 4 , 1 7 , and so on.
These measurements yield dhex with an accuracy of ±0.01 nm. The direct outcome of the
gravimetric studies is the dependence of dhex on the volume fraction of water in the HII-phase.
The osmotic pressure studies give dhex as a function of Π. The two experimental functions,
dhex (φw) and dhex (Π), contain sufficient information for determining the structural-elastic
parameters κ, Γ, E, as, and Rs of the HII phase monolayers.
4.2. Luzzati Plane
The cross-section of each prism of HII phase is hexagonal at the hydrocarbon chain boundary. The cross-section of the prism-water core is represented by a hexagon with smoothened
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vertexes (7,32–34). Analysis shows that, as far as determination of spontaneous curvature,
bending and stretching moduli is concerned, the water core can be approximated by a circular
cylinder. According to the method introduced by Luzzati (35), a prism can be subdivided into
lipid and water compartments. The boundary between the two compartments is represented by
the Gibbs dividing surface of zero excess of water, which means that it encloses a volume equal
to the total volume of water within the prism. This dividing surface (referred to as the Luzzati
surface) lies within the region of polar heads, and by convention, has a cylindrical shape.
The characteristics of the cylindrical Luzzati plane are its radius (RW) and the area (AW)
per lipid molecule (Fig. 1B). Both of them can be determined from the measured values of
dhex and φW using the relationships (see ref. 24)
RW = dhex ⋅
AW =
2 ⋅ φw
π 3
2 ⋅ φw ⋅ Vl
(1 − φw ) ⋅ RW
(12)
(13)
where Vl is the volume of a lipid molecule. It is assumed that the value of dhex determines
unambiguously the values of RW and aW independently of whether the gravimetric or osmotic
pressure method is applied. This allows one to determine, based on combination of the gravimetric and osmotic data, the characteristics of the Luzzati plane as functions of the osmotic
pressure, RW (∏) and AW (∏) (29,30). The determination is performed by analyzing the
monolayer mechanical equilibrium on pressure ∏, and using the relationships between the
elastic and structural characteristics of different dividing surfaces (see above). Description of
this procedure in full detail and its application to specific experimental data are described in
refs. 27,28. Although such treatment is feasible in principle, the attempts to perform it have
demonstrated that the accuracy of experimental determination of variations in AW is not sufficiently high to guarantee reliable estimation of the whole set of parameters characterizing
the neutral surface. Below, determination of the elastic parameters of a so-called pivotal plane
is described, which only approximately describes the neutral surface but allows for a more
exact characterization based on the HII phase dehydration experiments.
4.3. Pivotal Plane as Approximation for the Neutral Surface—Determination
of Spontaneous Geometrical Characteristics
The pivotal dividing surface is defined as the dividing surface for which the molecular area
A does not change in the course of deformation (ref. 24 and references therein). As aforementioned, when the monolayer deformation is solely generated by application of τ, the neutral surface keeps its molecular area, and therefore, represents the pivotal plane. In the dehydration
experiments, both a bending moment and a compressing lateral stress are applied to the cylindrical monolayers. Hence, the neutral surface must decrease its area and deviate from the
pivotal plane. The pivotal plane is a dividing surface for which the area changes related to bending and compression mutually compensate. However, a thorough analysis shows that deviation
of the pivotal plane from the neutral surface is relatively small (24), and differences between the
structural characteristics related to the neutral and the pivotal surfaces are of the order of paramκ
eter γ =
. The values of this parameter are: γ ≈ 0.03 − 0.1 , and hence, the pivotal plane
Γ ⋅ Rs2
describes the neutral surface with an accuracy higher than 10%.
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Verification of the existence of the pivotal plane and determination of its position with
respect to the Luzzati plane can be done based on the pure geometrical relationships. The
molecular A and the radius of curvature R at any cylindrical dividing surface, which is separated from the Luzzati plane by a volume V per lipid molecule, are given by
A
A2 = AW2 + 2 ⋅ V ⋅ W
(14)
RW
R = RW ⋅ 1 +
1 − φW V
⋅
φW Vl
(15)
The relationship (Eq. 14) can be rewritten in the form
AW2 = A2 + 2 ⋅ V ⋅
AW
RW
(16)
AW
can serve to make a “diagnosis,” i.e., to find out whether the pivotal plane
RW
exists. Indeed, this plot is linear if A does not change with dehydration, which means that the
corresponding dividing surface is a pivotal plane. The intercept of this linear plot gives the
square value of the molecular area at the pivotal surface (Ap), whereas the slope determines
the molecular volume (Vp) between the Luzzati and the pivotal planes.
Finding the values of RW and AW for the Luzzati plane at full hydration (according to Eqs. 12
and 13), determining the position of the pivotal plane by means of the diagnostic plot (16), and
finally, calculating Rp and Ap for the pivotal plane using expressions (Eqs. 15 and 16), is the way
1
to determine the corresponding spontaneous curvature J sp = − , and molecular area Asp = Ap.
Rp
4.4. Determination of the Bending Modulus
2
A plot AW vs
Once the pivotal plane is found, osmotic pressure measurements allow for determination
of the monolayer bending modulus κ. Based on the expression (Eq. 2), taking into account
constancy of the area of the pivotal plane, and considering the mechanical equilibrium of a
monolayer under pressure, the relationship between the osmotic pressure Π and the radius of
the pivotal plane Rp is found (30,31):
1
1 

∏ ⋅ Rp2 = 2 ⋅ κ p ⋅  −

R
R
 p
ps 
(17)
From the intercept and the slope of the plot Π⋅ Rp vs
1
one obtains the spontaneous
Rp
curvature −
1
and the κp related to the pivotal plane.
Rps
4.5. Discussions and Conclusions
The described protocol for analysis of the X-ray diffraction data has been used during the last
decade in studies of HII phases formed by mixtures of dioleoyl-phosphatidyl-ethanolamine with
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numerous lipids (5,7,17–22,24,25,28–31,34). The major goal was to determine the spontaneous
curvatures of physiologically relevant lipids. This could be done because of the fact that the
spontaneous curvatures of different lipid species are additive. It has been shown experimentally
(17,20–22,24,25) and predicted theoretically (36) that, in many cases, the total spontaneous curvature of a mixed monolayer is a weighted average of the spontaneous curvatures of the individual lipid components. A table collecting the values of the lipid spontaneous curvature
measured to date can be found in ref. 8.
Spontaneous curvatures of different lipids vary in a broad range depending on the lipid
molecular structure. At the same time, positions of the pivotal planes, and hence, of the neutral
surfaces of lipid monolayers proved to be fairly similar for different lipid species and their mixtures. The pivotal surfaces have been always found close to the plane of glycerol backbones of
lipid molecules (see refs. 17,20,21,24). This result can be explained by considering the internal
monolayer factors that determine the position of the neutral and pivotal planes. The two planes
have to be located in the most rigid region within the monolayer. The narrower this region is,
the closer the two surfaces are to each other. Most probably, the local rigidities are distributed
nonhomogeneously through the monolayer thickness, and the region of the glycerol backbones
is the most rigid one, independently of the monolayer composition. This would determine the
locations of the neutral and pivotal surfaces next to each other and to the glycerol backbone.
In conclusion, it has to be mentioned that whereas the described protocol proves to be productive in determining the radii of spontaneous curvatures of lipids, the results obtained for
the bending modulus are, in many cases, not accurate enough to find the contributions of individual lipids. Moreover, attempts to determine the values of the stretching modulus give only
order of magnitude estimations rather than exact values. Reaching a better quality of the elastic
moduli determination requires improved accuracy in the X-ray diffraction measurements.
Acknowledgments
The work of MMK is supported by the Israel Science Foundation (ISF), The Binational
USA-Israel Science Foundation (BSF), and Marie Curie Network “Flippases.”
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