Stalk Model of Membrane Fusion: Solution of Energy Crisis


Stalk Model of Membrane Fusion: Solution of Energy Crisis

Yonathan Kozlovsky and Michael M. Kozlov
Department of Physiology and Pharmacology, Sackler Faculty of Medicine, Tel-Aviv University, Ramat Aviv, Tel Aviv 69978, Israel

ABSTRACT Membrane fusion proceeds via formation of intermediate nonbilayer structures. The stalk model of fusion
intermediate is commonly recognized to account for the major phenomenology of the fusion process. However, in its current
form, the stalk model poses a challenge. On one hand, it is able to describe qualitatively the modulation of the fusion reaction
by the lipid composition of the membranes. On the other, it predicts very large values of the stalk energy, so that the related
energy barrier for fusion cannot be overcome by membranes within a biologically reasonable span of time. We suggest a new
structure for the fusion stalk, which resolves the energy crisis of the model. Our approach is based on a combined
deformation of the stalk membrane including bending of the membrane surface and tilt of the hydrocarbon chains of lipid
molecules. We demonstrate that the energy of the fusion stalk is a few times smaller than those predicted previously and the
stalks are feasible in real systems. We account quantitatively for the experimental results on dependence of the fusion
reaction on the lipid composition of different membrane monolayers. We analyze the dependence of the stalk energy on the
distance between the fusing membranes and provide the experimentally testable predictions for the structural features of the
stalk intermediates.

INTRODUCTION

The ability to fuse is shared by biological membranes,
consisting of phospholipid bilayers with embedded and
bound membrane proteins, and by artificially formed purely
lipid membranes. Normally, the membranes are mechani-
cally stable. This is guaranteed by powerful hydrophobic
effect (Tanford, 1980), which drives self-assembly of the
lipid molecules into bilayer and maintains the bilayer integ-
rity. Fusion involves drastic although local changes in the
initial membrane structure. Obviously, the membrane con-
figurations emerging at the intermediate stages of fusion
require input of energy and, hence, represent energy barriers
the membranes have to overcome on the way to the new
fused state. These configurations referred to as the fusion
intermediates largely limit the rate of the fusion reaction.

Modeling the lipidic fusion intermediates has a more than
two decades history (Chernomordik et al., 1995a). Al-
though, at the early stages, several different structures have
been suggested, only one of them, called fusion stalk (Gin-
gell and Ginsberg, 1978; Kozlov and Markin, 1983), is
currently recognized to adequately describe the transition
stage of membrane fusion (Chernomordik et al., 1995a;
Jahn and Südhof, 1999).

Fusion stalk is a local lipidic connection between the
proximal (contacting) monolayers of the fusing membranes
(Fig. 1). At the stage of stalk formation, the distal mono-
layers of the membranes are still separated, and the achieved
state is referred to as hemifusion. A physical model of the
fusion process, based on this hypothetical intermediate and

referred to as the stalk model, has been developed in a series
of works (Kozlov and Markin, 1983; Markin et al., 1984;
Chernomordik et al., 1985; Leikin et al., 1987; Kozlov et al.,
1989), further modified by Siegel (1993, 1999) and, re-
cently, by Kuzmin et al. (2001).

A number of important predictions of the stalk hypothesis
have been verified experimentally for fusion of protein-free
lipid bilayers and for some examples of biological fusion.
To start with, the hypothesis suggests that merger of the
proximal monolayers of the membranes precedes merger of
their distal monolayers. Indeed, the existence of a distinct
hemifusion stage has been documented for different exper-
imental systems based on protein-free bilayers (Chanturiya
et al., 1997; Chernomordik et al., 1995a, b; Lee and Lentz,
1997; Pantazatos and MacDonald, 1999) and for fusion of
biological membranes (Chernomordik et al., 1998; Kemble
et al., 1994; Leikina and Chernomordik, 2000; Melikyan et
al., 1995; Song et al., 1991). Although these data substan-
tiated the important positions of the model, more detailed
theoretical analysis revealed major problem with the phys-
ical basis of the approach. Recent estimates (Siegel, 1993,
1999) suggested that the energy of the stalk intermediate is
too high to allow its formation even for lipid membranes
demonstrating very fast fusion, such as bilayers formed
from dioleoylphosphatidylethanolamine (DOPE). Resolv-
ing this “energy crisis” has been of utmost importance for
understanding physical mechanisms of stalk formation and
membrane fusion.

In this work, we suggest a new structure for the stalk
intermediate, which differs in important details from that
considered previously. We demonstrate that the energy of
this modified structure is considerably smaller than that
previously obtained, so that the stalk intermediate can be
formed in the experimental conditions. A remarkable result
of our model is that the stalk energy practically does not
depend on the distance between the fusing membranes. The

Submitted July 16, 2001 and accepted for publication November 16, 2001.

Address reprint requests to Michael M. Kozlov, Tel-Aviv University,
Sackler Faculty of Medicine, Dept. of Physiol. & Pharmacol., Ramat-Aviv,
Tel-Aviv 69978, Israel. Tel.: �972-3-640-7863; Fax: �972-3-640-9113;
E-mail: misha@picard.tau.ac.il.

© 2002 by the Biophysical Society

0006-3495/02/02/882/14 $2.00

882 Biophysical Journal Volume 82 February 2002 882– 895



model accounts for the reported experimental data on mod-
ulation of the fusion reaction by lipid composition of the
membrane monolayers and provides the experimentally
testable predictions for the structural features of the stalk
intermediate.

ENERGY CRISIS OF THE STALK MODEL

Criterion for energy of fusion intermediate

As mentioned above, the fusion intermediates are not sup-
posed to represent the final structures adopted by the mem-
branes. The energy of a fusion intermediate, Fint, may be
larger than the energy of the initial membranes, F0. In the
case where Fint is too large compared to F0, formation of the
fusion intermediate becomes improbable. Estimations based
on the experimental investigations of the electrical break-
down of membranes (Chernomordik and Abidor, 1980;
Weaver and Mintzer, 1981) show (Kuzmin et al. 2001) that
the maximal value of the relative energy, Fbar � Fint � F0,
that the membrane can overcome within a characteristic
time scale of biological fusion is about Fbar � 40kT.

Energies predicted by the previous versions of
the stalk model

Fusion of two initially separated membranes requires a
number of steps, which include a local approach to small
distances; some kind of perturbation of structure of the
apposing monolayers such as, for example, their rupture in
the region of the close contact; merger of the apposing
monolayers; and formation of the lipid stalk between the
membranes (Leikin et al. 1987). Because of complexity of
quantitative evaluation of the early stages of fusion, the
major efforts have been devoted to modeling the lipid stalk,
whose energy is considered as a lower bound to the true
energy barrier of the fusion reaction (Siegel, 1993; Cherno-
mordik et al., 1995a).

Originally, the stalk membrane was assumed to have a
semitoroidal shape (Fig. 1 a) (Kozlov and Markin, 1983).

Depending on conditions, the stalk may have a tendency to
expand, resulting in dimpling of the distal monolayers to-
ward each other and formation of a hemifusion diaphragm
(Kozlov and Markin, 1983; Markin et al. 1984). The inter-
mediate structure of an early stage of stalk expansion was
called the transmonolayer contact (TMC) (Siegel, 1993)
(Fig. 1 b).

The most obvious feature of the stalk and TMC is a
strongly curved shape of their membranes (Fig. 1). There-
fore, the original version of the model (Kozlov and Markin,
1983; Markin et al. 1984; Leikin et al. 1987; Kozlov et al.,
1989) was based on the analysis of the stalk bending energy.
For a membrane composed of a lipid with common prop-
erties such as dioleoylphosphatidylcholine (DOPC), the en-
ergy of stalk was computed to be close to 40kT, whereas
formation of TMC required an additional increase of the
energy. For a membrane consisting of DOPE, a lipid having
an internal tendency to bend and form inverted hexagonal
phase (Gruner et al. 1988; Rand et al., 1990), the stalk
energy was estimated as only a few kT.

The further development of the stalk model by Siegel
(1993, 1999) revealed an additional considerable contribu-
tion to the energy. It is related to the so-called hydrophobic
interstices (Turner and Gruner, 1992), which result from
packing the hydrocarbon chains of lipid molecules around
the hydrophobic regions where the portions of differently
curved lipid monolayers contact each other (Fig. 1). A
tendency of the system to minimize the interstices results in
bending of the distal membrane monolayers, another issue,
which has been neglected in the original stalk model.

The value of the interstice energy can be estimated based
on the experimental data on mesophase behavior of aqueous
solutions of lipids (Siegel, 1993). The resulting overall
energies of the fusion intermediates turned out to be much
larger than originally predicted. For the aqueous distance of
�3 nm between the fusing membranes, which corresponds
to an equilibrium intermembrane spacing in pure lipid sys-
tems (Rand and Parsegian, 1989), the stalk energies for
membranes consisting of DOPC and DOPE were predicted

FIGURE 1 Previous model for stalk intermediates. (a) Fusion stalk. (b) Transmembrane contact (TMC). The structures are axially symmetric.

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Biophysical Journal 82(2) 882– 895



to equal 170kT and 150kT, respectively. Even few times
larger stalk energies have been obtained for the aqueous
distances between the membranes of �10 nm (Siegel, 1993)
corresponding to viral fusion (Skehel and Wiley, 2000).

Recently, another attempt has been undertaken to modify
the stalk model (Kuzmin et al., 2001) based on a more
sophisticated way to estimate the energies of the hydropho-
bic interstices. The authors have chosen to present the
energy with respect to a hypothetical highly stressed dim-
pled state of the membranes. Relating their results to the
initial flat shape of the membranes gives the stalk energy of
more than 220kT, even exceeding the results of the modified
model of Siegel (1999).

Challenge for the present research

Comparing the energies predicted by the modified model
(Siegel, 1993, 1999) with the above criterion for fusion
intermediate leads to the conclusion that the stalk cannot
form in a real system. In contrast, membranes do fuse, and
the major qualitative predictions of the original model (Koz-
lov and Markin, 1983; Markin et al., 1984; Chernomordik et
al., 1985) on modulation of the fusion reaction by the
membrane lipid composition have got a solid experimental
confirmation (Chernomordik et al., 1995a). How to recon-
cile this inconsistency? A straightforward suggestion is that
the structure of the stalk intermediates is different from that
suggested previously. A more optimal structure possessing
a smaller energy, which can be overcome due to thermal
fluctuations or protein refolding has to be found. In the
following, we propose and analyze such a structure.

STRUCTURE OF FUSION STALK

The excessive stalk energy predicted by the earlier analysis
(Siegel, 1993) results from the energy price of the hydro-
phobic interstices, which have been extensively discussed in
the context of packing the hydrocarbon chains of lipid
molecules in the inverted hexagonal (HII) phases (Gruner,
1989; Seddon, 1990; Turner and Gruner, 1992; Kirk et al.,
1984; Tate and Gruner, 1987; Sjölund et al., 1989; Duesing
et al., 1997; Rand and Fuller, 1994). Recently, a model of
the HII phase has been developed proposing that the hydro-
phobic interstices are filled due to tilting the hydrocarbon
chains with respect to the monolayer surface (Hamm and
Kozlov, 1998, 2000). This tilt model succeeded to interpret
quantitatively the phase diagram of the aqueous solutions of
glycolipids (Hamm and Kozlov, 1998).

We suggest the stalk structure based on the tilt model for
the hydrophobic interstices, as illustrated in Fig. 2. To
emphasize the major features of our model, let us compare
Figs. 1 and 2. The hydrophobic void (Fig. 1) is filled (Fig.
2) due to an oblique packing of the hydrocarbon chains with
respect to the membrane surface. This results in a sharp
corner in the monolayer profile. In the cross-section of stalk,

the corner is located between two adjacent molecules (Fig.
2) so that none of them suffers a strong bending. In addition
to tilt of the hydrocarbon chains, the membrane is curved
close to the corner (Fig. 2). The membrane curvature and tilt
of the chains smoothly relax along the membrane surface to
the flat state with vanishing tilt.

The illustration in Fig. 2 implies that several hydrocarbon
chains terminate simultaneously at the center of the stalk
structure. This may lead to chain-packing problems similar
to those revealed by computer analysis of micellar struc-
tures, where only one chain has been shown to occupy the
center of a micelle. However, the stalk structure (Fig. 2) is
similar geometrically to the inverted hexagonal (HII) phases
rather than to micelles. Computer analysis of HII phases
(Gido et al., 1993) confirms a possibility of formation of
sharp corners filled by hydrocarbon chains similar to that
presented in Fig. 2.

Terminologically, the suggested structure (Fig. 2) is the
fusion stalk and the TMC at the same time. Indeed, on one
hand, it corresponds to the very first merger of the proximal
monolayers of the fusing membranes and, on the other, the
distal monolayers are already deformed in a way that they
touch each other, although point-wise.

ELASTIC MODEL FOR STALK INTERMEDIATE

To proceed, we need a physical model describing the elastic
energy of a complex deformation of the membrane mono-
layers, which involves bending of the monolayer surface
and tilt of the hydrocarbon chains of lipid molecules.

Monolayer shape and deformations

We apply the approach developed in detail recently (Hamm
and Kozlov, 1998, 2000), which is, in turn, based on the

H

R

N
nφ

τ
ϕ

FIGURE 2 Tilt model for stalk and TMC. The shape is axially symmet-
ric. Illustration of notations used: N is the unit vector of the normal to the
monolayer surface, n is the director of a hydrocarbon chain, � is the
tangential unit vector, � is the tangential angle, � is the tilt angle; H is the
intermembrane distance; R is the stalk width. The presented relationship
between the monolayer thickness and the dimensions of the hydrocarbon
chains is approximately equal to the real one.

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Biophysical Journal 82(2) 882– 895



ideas of the classical works by Frank (1958) and Helfrich
(1973). We consider a lipid monolayer having, in the initial
state, a flat shape (Fig. 3 a). To characterize the geometry of
the monolayer, we use the dividing surface (Gibbs, 1876/
78), which lies along the interface between the polar groups

and the hydrophobic interior and has been shown for par-
ticular lipids to play a role of the neutral surface (Kozlov
and Winterhalter, 1991a,b; Kozlov et al., 1994; Leikin et al.,
1996). The monolayer shape will be identified with the
shape of its dividing surface. To describe it mathematically,
we define in each point of the dividing surface a unit normal
vector N (Fig. 2).

The average orientation of a hydrocarbon chain is ex-
pressed by a unit vector n referred to as the chain director.
In the initial state, the chains are perpendicular to the
dividing surface, meaning that n coincides with N (Fig. 3 a).
The area of the chain cross section at the dividing surface is
denoted by a. We consider two deformations of the mono-
layer: bending, resulting in deviation of the monolayer
shape from the initial flat one (Fig. 3 b); and tilt of the
hydrocarbon chains, resulting in deviation of the chain
director n from the surface normal N (Fig. 3, c and d). We
will assume the chain area a to remain constant, meaning
that the monolayer does not undergo lateral stretch, because
the corresponding membrane rigidity (Rawicz et al., 2000)
is considerably larger than those of tilt (Hamm and Kozlov,
1998, 2000) and bending (Niggemann et al., 1995).

The volume of a hydrocarbon chain, v, does not change in
the course of membrane deformations, because the hydro-
carbon moiety of a lipid molecule is practically incompress-
ible. A simple geometrical consideration based on Fig. 3 d,
and accounting for constancy of v and a, shows (Hamm and
Kozlov, 2000) that the deformation of tilt results in stretch
of the hydrocarbon chains with respect to their initial length.
At the same time, the monolayer thickness measured per-
pendicularly to the membrane surface remains constant. The
stretch of the chains accompanying their tilt allows for
filling the hydrophobic voids of the stalk intermediate.

Elastic energy

The elastic energy of a monolayer originates primarily from
the hydrocarbon chains (Helfrich, 1973; Ben-Shaul, 1995).
Within the continuous description, the major contributions
to the elastic energy arise, on one hand, from the splay of
the chains (Frank, 1958; Helfrich, 1973) and, on the other,
from uniform tilt of the chains with respect to the monolayer
surface (MacKintosh and Lubensky, 1991).

Energy of splay

Splay of the hydrocarbon chains is illustrated in Fig. 3, b
and c. It can result from two different deformations (Hamm
and Kozlov, 1998, 2000): bending of the monolayer surface,
whereas the hydrocarbon chains remain parallel to the sur-
face normal, n � N, (Fig. 3 b), and variation of the molec-
ular director n along the monolayer surface, whereas the
surface remains flat (Fig. 3 c). This means that the chains
exhibit tilt, n � N, which changes along the monolayer
plane.

FIGURE 3 Deformations of a monolayer. (a) The initial undeformed
state. (b) Bending with vanishing tilt. (c) Splay of the chains resulting from
changing tilt in a flat monolayer. (d) Homogeneous tilt of the chains. (e)
Mutual compensation of bending and tilt resulting in a bent membrane with
vanishing splay of the chains.

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Biophysical Journal 82(2) 882– 895



Obviously, combinations of these two deformations also
result in splay of the chains. Mathematically, splay of the
chains, denoted by J̃, can be presented for the two cases
above and their combinations in the same form (Helfrich,
1973; Hamm and Kozlov, 2000)

J̃ � div n, (1)

where div n denotes the covariant divergence of the director
n along the monolayer surface. In the case, of pure bending,
n � N, the splay (Eq. 1) is given by J̃ � div N, and,
according to the general geometrical relationship, is equal to
the total curvature, J, of the monolayer surface (Thomas,
1961). For the case of varying director, Eq. 1 is analogous
to splay in a liquid crystal (Frank, 1958).

The energy of splay per unit area of the dividing surface
can be expressed as (Hamm and Kozlov, 1998, 2000)

fs �
1
2

� � � �div n � J̃s	
2, (2)

where J̃s is the spontaneous splay and � is the splay modulus
of the monolayer, the two characterizing the monolayer
elastic properties. The spontaneous splay, J̃s, coincides with
the spontaneous curvature, Js, introduced by Helfrich (1973)
for the cases where splay originates solely from the mono-
layer bending. The spontaneous splay is related to the ef-
fective shapes of lipid molecules constituting the monolayer
(Israelachvili et al. 1980; De Kruijff and Cullis, 1980). For
a common lipid such as DOPC, the spontaneous splay is
slightly negative, J̃s � �0.1 nm

�1 (Chen and Rand, 1997);
for DOPE, this value is strongly negative, J̃s � �0.35 nm

�1

(Kozlov et al., 1994; Leikin et al., 1996); whereas, for
lysolipids and, specifically, for lysophosphatidylcholine
(LPC), the spontaneous splay is positive, J̃s � 0.26 nm

�1

(Fuller and Rand, 2001). The spontaneous splay of a mixed
monolayer can be estimated as an average over the mole-
cules constituting the monolayer (Kozlov and Helfrich,
1992).

The splay modulus � coincides with the Helfrich bending
modulus (Helfrich, 1973; Hamm and Kozlov, 2000). Its
value, � � 10kT, measured for monolayers (Niggemann et
al., 1995; Leikin et al., 1996), will be used for calculations
below.

The major assumptions of the model (Eq. 2) are that the
hydrophobic moiety of a lipid monolayer can be described
as an elastic continuum and that the splay is small compared
to the monolayer thickness �, so that �div n� � � 
 1. While
accounting for a combined deformation of bending and tilt,
Eq. 2 represents the Helfrich elastic model (Helfrich, 1973)
for the case of pure bending.

Energy of tilt

Uniform tilt of the hydrocarbon chains is illustrated in Fig.
3 d. Tilt can be quantified by a vector t defined as (Mac-

Kintosh and Lubensky, 1991; Hamm and Kozlov, 1998,
2000)

t �
n

n � N
� N,

which is parallel to the dividing surface and characterizes
the deviation of the director n from the surface normal N
(Fig. 3 d). The corresponding energy per unit area of the
monolayer is given by (MacKintosh and Lubensky, 1991;
Hamm and Kozlov, 2000)

ft �
1
2

� �t � t
2, (3)

where �t is the tilt modulus (Hamm and Kozlov, 2000). The
major contribution to the tilt modulus �t results from the
stretch of the hydrocarbon chains. This follows from com-
parison of the model estimation (Hamm and Kozlov, 2000)
with the value of �t � 0.04 N/m, found from fitting the
experimental results on phase transitions in lipid systems
(Hamm and Kozlov, 1998). The latter value for the tilt
modulus will be used for computations below. The qua-
dratic model (Eq. 3) assumes small absolute values of tilt,
�t� 
 1.

Main equations

We relate the total energy ftot consisting of the energies of
splay (Eq. 2) and uniform tilt (Eq. 3) to that of the initial flat
state with vanishing splay and tilt, J̃ � 0, t � 0. The
resulting energy per unit area of the monolayer is

ftot �
1
2

� � � �div n � J̃s	
2 �

1
2

� �t � t
2 �

1
2

� � � J̃s
2. (4)

To compute the total elastic energy of a monolayer, Fmon,
the energy ftot has to be integrated over the area A of the
dividing surface,

Fmon � �ftot dA. (5)
In Appendix A, we describe the details of calculation of the
stalk energy.

ENERGY AND SHAPE OF STALK: RESULTS

Our analysis is aimed at describing the energy of the stalk
intermediates formed in the course of different fusion reac-
tions including, on one hand, fusion of pure lipid vesicles
and, on the other, fusion of cell and viral membranes me-
diated by proteins. In each real system, there are constraints
imposed on the configuration of the fusion site. We char-
acterize these constraints by two parameters (Fig. 2). First,
the distance R from the center of the stalk, to the region
where membranes have to become flat and parallel, is
referred as the width of the stalk base. Second, the distance

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Biophysical Journal 82(2) 882– 895



between the membrane middle planes H is referred to as the
intermembrane distance. In the case of fusion of purely lipid
membranes, the stalk width R is limited by the vesicular
radius, which is usually larger than 100 nm. The intermem-
brane distance H, in this case, cannot become considerably
smaller than its equilibrium value of �6 nm (Rand and
Parsegian, 1989), which is determined by the whole set of
the interbilayer interactions.

In the case of fusion of biological membranes, the con-
straints for both H and R are determined by the protein
architecture of the membrane contact and can vary consid-
erably depending on the type of fusing cells or viruses. For
example, in the case of fusion of influenza virus with target
cell, the intramembrane distance H is determined by the
length of HA molecule, H � 13 nm. The stalk intermediate
is supposed to form within a membrane domain, which does
not contain proteins (Kozlov and Chernomordik, 1998) so
that the dimension of such domain limits the width of the
stalk base R.

To be able to describe all possible types of fusion reac-
tion, we analyze the stalk energy F for a large range of the
intermembrane distances H and the stalk widths R. How-
ever, the calculations illustrating more special features of
the model, such as dependence of the stalk energy on the
spontaneous curvatures of the monolayers, will be performed
for the equilibrium intermembrane distance H � 6.2 nm.

Symmetric membranes

To understand the major features of the fusion stalk, we first
calculated its energy and structure for the case where the
two membrane monolayers have the same spontaneous
splay, J̃s � �0.1 nm

�1, corresponding to the typical lipid
DOPC. The values of the monolayer modulus of splay and
tilt are taken to be � � 4 � 10�20 J and �t � 0.04 N/m
(Hamm and Kozlov, 2000).

Stalk energy

Dependence of the stalk energy F on the stalk width R and
intermembrane distance H has the shape of a canyon, which
descends monotonically with growing R and H. This means
that the stalk has a general tendency to expand its base and
length. If the geometrical constraints limiting R and H are
lifted, what corresponds to fusion of giant vesicles, the stalk
becomes infinitely wide (R 3 �) and long (H 3 �), and its
energy approaches the value of �43kT.

In the case where the expansion of the stalk base is
limited by a specific width R, an optimal intermembrane
distance Hopt exists providing the stalk with a minimal
energy. Dependence of Hopt on the stalk width R is illus-
trated in Fig. 4 by diamonds, and the related variation of the
stalk energy F along the bottom of the canyon is presented
in Fig. 4 by stars. To demonstrate a strong increase of the
elastic energy of the stalk, F, at small R and Hopt, we started

our analysis from Hopt � 5.7 nm, which is slightly below the
equilibrium intermembrane distance H � 6.2 nm (Rand and
Parsegian, 1989). It should be noted that, within the small
range of H between 5.7 nm and 6.2 nm, there is some
positive contribution to the overall stalk energy from the
repulsive interaction between the membranes. As a result,
the whole energy of stalk in this region is slightly larger
than that presented in Fig. 4 (stars) for 6 nm 
 R 
 8 nm.

If the stalk length is limited by a specific intermembrane
distance H, the energy F decreases monotonically with
growing width R, as represented in Fig. 5 for several values
of H. The dependences F(R) can be described by a charac-
teristic stalk width Rc. We define Rc as a width where the
slope of the curve F(R) equals (dF/dR)�Rc � �1 kT/nm. The
energy decreases sharply in a range of small stalk width,
R 
 Rc, whereas, for the value of R exceeding the charac-
teristic width, R 
 Rc, the energy changes slowly and
reaches a plateau. The plateau value of the energy, F*,
depends on the intermembrane distance H, but this depen-

0 10 20 30 40 50 60 70 80
43

44

45

46

47

48

49

50

51

52

R [nm]

F
 [k

T
]

H
op

t [
nm

]

5

6

7

8

9

10

11

12

FIGURE 4 Dependence of the energy F (�) and the optimal intermem-
brane distance Hopt ({) on the width of the stalk base, R.

FIGURE 5 The dependence of the stalk energy F on the width of the
stalk R for different values of the intermembrane distance H, as indicated
in the figure.

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Biophysical Journal 82(2) 882– 895



dence is very weak and F* remains below 47kT for all
relevant values of H. The dependence of the characteristic
width on the intermembrane distance, Rc(H), can be de-
scribed by

Rc � �2.4 � H � 6.2	 nm. (6)

The above definition for the characteristic width Rc is arbi-
trary. However, our numerical estimations show that, due to
a well-pronounced distinction between the regions of a
sharp and slow changes of the energy F(R), the value of Rc
is rather insensitive to the exact slope, (dF/dR)�Rc, chosen
for its determination.

Stalk configuration

The shape of the stalk surface depends on its width R and
the intermembrane distance H. In Fig. 2, a characteristic
shape of the stalk is presented. Change of the stalk shape
with R and H is illustrated in Fig. 6 presenting the line
whose revolution around the z-axis forms the midsurface of
the half stalk. To emphasize the features of the stalk shape,
the scale along the z-axis is chosen a few times smaller than
that along the r-axis. For intermediate values of R and H, the
stalk profile is nonmonotonic, exhibiting a depression be-
fore reaching the flat shape (Fig. 6, a–d). The depression
becomes broader and deeper with increasing R. For an
unconstrained stalk (R 3 �, H 3 �) the surface far from
the stalk center approaches the shape of a catenoid (Fig.
6 e), the axially symmetric minimal surface characterized
by zero total curvature (Nitsche, 1989). This unlimited

catenoid-like shape has the energy of �43kT mentioned
above.

Effects of spontaneous splay

We have computed the influence of the spontaneous splay
of the membrane monolayers on the stalk energy for differ-
ent values of the stalk width R and intermembrane distance
H. Within an experimentally relevant range, the qualitative
character of this influence turned out to be independent of
the specific values of R and H. We present the results (Fig.
7) for the intermembrane distance H � 6.2 nm, correspond-
ing to the equilibrium separation in the lamellar phases of
egg phosphatidylcholine (Rand and Parsegian, 1989), and
the stalk width R � 25 nm, which exceeds the characteristic
width, Rc.

The dependence of the stalk energy F on the spontaneous
splay of the monolayers, J̃s, is represented by Fig. 7 (line 1)
and is well described by F � (82 � 261 � J̃s � �) � kT, where
� is the distance between the monolayer neutral surface and
the bilayer midplane, � � 1.2 nm. For the membrane of
DOPC, having J̃s � �0.1 nm

�1 (Chen and Rand, 1997), the
stalk energy is about F � 45kT, which exceeds, by �5kT,
the above criterion of stalk formation due to thermal fluc-
tuations. For DOPE characterized by a spontaneous splay of
J̃s � �0.35 nm

�1 (Kozlov et al., 1994; Leikin et al., 1996)
the stalk energy is strongly negative, F � �30kT, meaning
that, for this lipid, the stalk is energetically favorable and,
hence, does not contribute to the energy barrier of mem-
brane fusion. The spontaneous splay of a mixture of differ-

0 5 10 15 20 25 30 35
6

5

4

3

2

1

0

r[nm]

z[
nm

]

a b c d

e

FIGURE 6 The shape of the stalk profile for constant intermembrane
distance H � 6.2 nm and different values of the stalk width R. To
emphasize the details of the shape, the scale along the z-axis is taken a few
times smaller than that along the r-axis. (a) R � 18 nm; (b) R � 25 nm;
(c) R � 31 nm; (d) R � 37 nm; (e) R and H are unconstrained.

FIGURE 7 Dependence of the stalk energy F on the spontaneous splay
J̃s of the membrane monolayers (H � 6.2 nm, R � 25 nm). (1) Symmetric
membrane: the spontaneous splay is equal in the two monolayers. (2)
Asymmetric membrane: F as a function of the spontaneous splay of the
proximal monolayer, J̃s

p, while the spontaneous splay of the distal mono-
layer is fixed, J̃s

d � �0.1 nm�1. (3) Asymmetric membrane: F as a function
of the spontaneous splay of the distal monolayer, J̃s

d, while the spontaneous
splay of the proximal monolayer is fixed, J̃s

p � �0.1 nm�1.

888 Kozlovsky and Kozlov

Biophysical Journal 82(2) 882– 895



ent phospholipids can be calculated in a good approxima-
tion as a weighted average of the spontaneous splays of the
components (see, for example, Kozlov and Helfrich, 1992).
For example, the energy of a stalk consisting of a DOPE/
DOPC mixture has a value intermediate between the ener-
gies of the pure DOPE and DOPC stalks, as represented in
Fig. 7 by points on the line (1) corresponding to the
weighted spontaneous splays.

Asymmetric membranes

The spontaneous splays of the two membrane monolayers
have different effect on the overall stalk energy F. To
analyze this issue, we performed computations for asym-
metric stalk membrane. Line (2) in Fig. 7 represents the
dependence of F on the spontaneous splay of the proximal
monolayer, J̃s

p, while that of the distal monolayer remains
constant, J̃s

d � �0.1 nm�1. The function F(J̃s
p) practically

coincides with the energy of a symmetric membrane Fig. 7,
(line 1). In contrast, the dependence of the energy F on the
spontaneous splay of the distal monolayer, J̃s

d, turns out to
be very weak. This is illustrated by line (3) in Fig. 7, which
presents F(J̃s

d) at a fixed spontaneous splay of the proximal
monolayer, J̃s

p � �0.1 nm�1. The results, Fig. 7 (lines 2 and
3), show that the major factor influencing the stalk energy is
the spontaneous splay of the proximal monolayer, J̃s

p,
whereas the distal monolayer practically does not contribute
to the effect of the spontaneous splay.

DISCUSSION

We suggest a new structure for the stalk intermediate of
membrane fusion (Fig. 2), which resolves the energy crisis
identified in the earlier models. Our model adequately in-
terprets the experimental results on dependence of fusion on
the lipid composition of different membrane monolayers,
and gives experimentally testable predictions on the config-
uration of the fusion stalk.

Comparison with previous models

Why does the suggested approach succeed in solving the
energy crisis? The major feature of the present model is that
the deformations of the two membrane monolayers are not
limited by bending, as previously assumed (Kozlov and
Markin, 1983; Markin et al., 1984; Siegel 1993, 1999), but
involve also tilt of the hydrocarbon chains (Hamm and
Kozlov, 1998, 2000). The tilt plays a role of an additional
internal degree of freedom of the monolayers and allows
them to relax the overall energy. Specifically, tilt of the
chains results in three effects influencing the monolayer
energy. 1) The hydrophobic voids are filled by the hydro-
carbon material so that no “vacuum” is created in the
system. The stretch of the hydrocarbon chains necessary to

fill the voids is implicated in the chain tilting, and the
related energy is accounted by the tilt elasticity �t. 2) the
overall area of the stalk monolayers is reduced compared to
the initially proposed semitoroidal shape (Fig. 1) due to
creation of a corner in the middle of the structure (Fig. 2).
3) Splay of the hydrocarbon chains resulting from tilt com-
pensates, at least partially, for the splay produced by the
monolayer bending (Fig. 3 e). In the modified stalk model
of Siegel (1993, 1999), the tilt degree of freedom has not
been considered at all and, therefore, the predicted energy
was more than four times larger than that obtained in the
present work.

In the recent model by Kuzmin et al. (2001), the distal
monolayers of the stalk are assumed to undergo a localized
tilt-like deformation, while the hydrocarbon chains of the
proximal monolayer were not allowed to tilt. Because of
these assumptions, the effect of tilt is drastically limited: the
hydrophobic interstices are only partially filled, still leaving
vacuum voids; the area of the strongly bent proximal mono-
layers, which have a semitoroidal shape, is the same as in
the previous models, and, finally, the bending stresses are
not compensated by the tilt-induced splay. If the stalk en-
ergy resulting from the model by Kuzmin et al. (2001) is
related to the initial flat shape of the membrane, it turns out
to exceed the results of Siegel (1999) and to be at least five
times larger than the energy predicted by the present ap-
proach. As a result, the stalk structure suggested in Kuzmin
et al. (2001) is far from being the lowest energy fusion
intermediate.

Effects of different lipids

According to the experimental data accumulated over the
last decades, different phospholipids exhibit distinct prop-
erties with respect to membrane fusion. The lipid DOPC
forms, in aqueous solutions, stable lamellar phases, where
the flat lipid bilayers separated by thin (2- to 3-nm) water
gaps do not demonstrate any tendency to fuse spontaneously
(Rand and Parsegian, 1989). However, addition to these
systems of hydrocarbon solvent induces transformation of
the lamellae into cylindrical membranes of the inverted
hexagonal (HII) phase (Sjölund et al., 1989), the process
including multiple fusion of the initial membranes (Siegel et
al., 1989; Siegel and Epand, 1997; Siegel, 1999). The aque-
ous solutions of DOPE form the HII phase in normal con-
ditions at room temperature without any additions. This
lipid exhibits a temperature-driven spontaneous transition
from lamellar to inverted hexagonal structure at �10°C
(Gawrisch et al., 1992). This implies that, beginning from
this temperature, the DOPE monolayers have a tendency to
fuse spontaneously. Indeed, DOPE and other lipids support-
ing HII phase formation (e.g., unsaturated fatty acids and
diacylglycerol) promote fusion of protein-free bilayers and
protein-mediated fusion (Basanez et al., 1998; Cherno-
mordik et al., 1995a; Chernomordik et al., 1997; Walter et

Stalk Model of Membrane Fusion 889

Biophysical Journal 82(2) 882– 895



al., 1994). In contrast, lysolipids, such as LPC, inhibit
fusion. Addition of LPC to model (Chernomordik et al.
1985; Basanez et al., 1998; Chernomordik et al., 1994) and
biological (Chernomordik et al., 1995a; Chernomordik et
al., 1997; Gaudin, 2000; Melikyan et al., 2000) membranes
considerably inhibited the fusion reaction.

Promotion and inhibition of fusion by different lipids
correlate with their molecular structure, which can be char-
acterized by the effective molecular shape (Israelachvili et
al., 1980; De Kruijff and Cullis, 1980; Chernomordik et al.,
1985) or, alternatively, the spontaneous curvature, Js (Helf-
rich, 1973). The notion of the spontaneous curvature, Js, has
been recently generalized to the spontaneous splay, J̃s
(Hamm and Kozlov, 1998, 2000). The spontaneous splay of
DOPE measured experimentally (Rand and Fuller, 1994;
Kozlov et al., 1994; Leikin et al., 1996) is strongly negative,
J̃s

DOPE � �1/2.8 nm�1; for DOPC, this value can be esti-
mated (Chen and Rand, 1997) as slightly negative, J̃s

DOPC �
�1/8.7 nm�1; and for LPC, the spontaneous splay is posi-
tive, J̃s

LPC � 1/3.8 nm�1 (Fuller and Rand, 2001). Summa-
rizing the experimental results, the more negative the spon-
taneous splay of a lipid is, the stronger is its propensity to
fuse so that the lipids with the spontaneous splay close to
J̃s � �0.3 nm

�1 exhibit spontaneous fusion.
The stalk model proposed in the present study readily

interprets the dependence of membrane fusion on the lipid
composition. For the DOPC membranes, our model predicts
the energy of the stalk/TMC close to FDOPC � 45kT. This
energy is �5kT larger than Fbar, meaning that fusion should
be an extremely rare event. This agrees with a general
stability of the DOPC lamellar phases (Rand and Parsegian,
1989). In contrast, this energy can be easily decreased to a
value below Fbar to allow for fusion. This can be done by a
suitable change in the system compositions such as addition
of a hydrocarbon solvent (Sjölund et al., 1989), which is
supposed to enter the hydrophobic interstices and partially
relieve the membrane stresses. The energy of a stalk inter-
mediate we predict for the DOPE membranes is negative
FDOPE � �30kT, which explains the strong tendency of
these membranes to fuse. And, finally, a growth of the stalk
energy with increasing spontaneous splay (Fig. 7 (line 1))
explains the effective inhibition of fusion by additions of
LPC to the membrane monolayers (Chernomordik et al.,
1995a).

Effects of different monolayers

Another result of our model is a prediction of a different
influence on stalk formation of the spontaneous splay, J̃s, of
the proximal and distal membrane monolayers. According
to the earlier model (Siegel, 1993), changes in J̃s of the two
monolayers exert a similar effect on the stalk energy F. In
contrast, our calculations show that the dependence of the
energy F on J̃s is related almost solely to the proximal
monolayer (Fig. 7 (lines 2 and 3)). Changes in the sponta-

neous splay of the distal monolayer, practically, do not
influence the energy (Fig. 7 (line 3)). In agreement with our
prediction, the experimental investigations showed that ad-
ditions of LPC to the proximal monolayer considerably
inhibited hemifusion, whereas insertion of the same mole-
cules to the distal monolayer had practically no effect on
membrane hemifusion (Chernomordik et al., 1997; Cherno-
mordik et al., 1998).

Effects of stalk dimensions

Our model predicts that, for every given inter-membrane
spacing H, the stalk tends to expand its base, thus, increas-
ing the width R (Fig. 5). But the force driving this expansion
is strong only as long as the width is smaller than the
characteristic value, R 
 Rc. For larger values of the width,
R 
 Rc, the stalk energy almost does not change and, hence,
the driving force for the expansion of the stalk base is very
small. In practical terms, this means that the stalk tends to
expand up to the characteristic width Rc, which has values
close to the intermembrane distance H (Eq. 6). This result of
the model can be verified experimentally.

How does the stalk energy depend on the intermembrane
distance H? This question is important for comparing fusion
in different systems. Indeed, in the pure lipid systems, the
typical intermembrane spacing is �6 nm (Rand and Parse-
gian, 1989), whereas the distance between fusing cell mem-
branes determined by membrane proteins constitutes up to
12 nm (Skehel and Wiley, 2000). Earlier versions of the
stalk model predicted a drastic increase in the energy with
the intermembrane distance H (Siegel, 1993, 1999). This
would imply that the architecture of the fusion intermediate
in the pure lipidic membrane system should differ from that
formed between cell membranes. In contrast, our model
predicts practically the same energy for the fusion interme-
diates of different intermembrane distances, provided that
all of them can expand laterally reaching the width of Rc.
The energy of a stalk with H � 12 nm turns out to be even
1.5kT smaller than that of stalk with H � 6 nm. Hence, all
the results of our model are valid for different intermem-
brane distances.

If the width of the stalk base is restricted to values smaller
than the characteristic one, R 
 Rc, the stalk energy in-
creases with the intermembrane distance H (Fig. 5). This
can be expected for fusion of influenza virus, whose mem-
brane contains a high density of the fusion protein HA. The
lateral spacing between the HA molecules available for
stalk formation is close to R � 7 nm (Taylor et al., 1987),
whereas the intramembrane distance, as determined by the
length of the HA ectodomains, is H � 13 nm (Bullough et
al., 1994). According to Eq. 6, in this case, the stalk base
does not reach the characteristic width. To avoid a large
energy barrier in such a situation, the stalk formation has to
be preceded by decrease in H due to dimpling of the

890 Kozlovsky and Kozlov

Biophysical Journal 82(2) 882– 895



membranes toward each other, as suggested in Kozlov and
Chernomordik (1998).

Assumptions of the model

The suggested model includes several assumptions, which
have to be discussed. We use a continuum description of a
lipid monolayer, analogous to the Frank theory of liquid
crystals (Frank, 1958) and the Helfrich theory of membrane
elasticity (Helfrich, 1973). Such description requires, in
general, the length scale of deformations to be considerably
larger than the dimension of particles constituting the sys-
tem. The role of such particles in our case is played by the
hydrocarbon chains, and the particle dimension is the diam-
eter of the chain projection on the membrane plane, which
equals �0.6 nm. According to our results (Fig. 2), the
deformation of splay of the stalk monolayers extends over
7– 8 hydrocarbon chains, meaning that its length scale is
only several times larger than the particle dimension. The
deformation of tilt of the stalk intermediate turns out to be
even more concentrated and extends over 2–3 hydrocarbon
chains. Although the macroscopic theories are often applied
in similar cases, the validity of the results is always ques-
tionable and has to be verified by comparison with experi-
mental data. A continuum elastic approach of the present
work has been used previously for analysis of the energy
and the elastic behavior of the inverted hexagonal (HII)
phases (Kozlov and Winterhalter, 1991a,b; Kozlov et al.,
1994; Leikin et al., 1996; Hamm and Kozlov, 1998), where
the ratio between the scale of the monolayer deformations
and the chain dimension is the same as for the stalk inter-
mediate. This approach proved to describe quantitatively
the experimental data on the lamellar– hexagonal phase
transitions and the stress–strain relationships of the HII
phases. This convinces us that, in spite of the general
reservations above, our continuum model adequately de-
scribes the stalk intermediate.

We make an unusual assumption that the profile of the
stalk membrane exhibits a sharp corner. To a first glance,
this corresponds to an infinite membrane curvature (J 3 �),
which is concentrated within a vanishing area (a 3 0).
However, consideration of the structure of the corner in
terms of deformations of tilt and splay of the hydrocarbon
chains shows that the chains do not exhibit infinite splay at
the corner (Fig. 2). At the same time, the finite splay,
analogous to finite curvature, and finite tilt of the hydrocar-
bon chains in the region of the corner are taken into account
by integrating the energy (Eq. 4) up to the corner. Yet, an
additional energy related to the sharp corner may originate
from a repulsive interaction between the polar groups of the
lipid molecules, resulting, for example, from hydration re-
pulsion (Rand and Parsegian, 1989; Leikin et al., 1993) or
the electrostatic interaction. Because this energy is a part of
the energy of membrane deformations, to calculate its value,
one needs a theory accounting for contribution of the hydra-

tion and electrostatic forces to the elastic characteristics of
membranes.

We are not aware of any systematic treatment of the
influence of hydration forces on the elastic moduli of mem-
branes. However, a general idea about these effects can be
based on the existing models for the hydrations forces
(Appendix C). Based on the qualitative consideration of
Appendix C, we assume that the hydration of the polar
heads provides a negligible contribution to the stalk elastic
energy.

A model for contributions of the electrostatic interaction
to the membrane elasticity has been developed only for
small curvatures of the membrane surface (Winterhalter and
Helfrich, 1988), and, hence, it cannot describe the energy of
the corner. We do not consider these effects, assuming the
corner to be situated between adjacent uncharged polar
heads.

The elastic model (Eq. 4) assumes the deformations of
splay and tilt to be smaller than one, �div n � �� 
 1, �t� 
 1,
where � is the monolayer thickness. In reality, in our cal-
culations, the deformations of tilt and dimensionless splay
approach but never exceed the value of one in a very narrow
region close to the corner of the stalk structure. To assure
fulfillment of this condition, we limited the ranges of vari-
ations of parameters in our numerical procedure.

Another assumption of our calculation is that, on one
hand, the neutral surfaces of the two membrane monolayers
are parallel to the membrane midplane, and, on the other,
the molecular areas on the neutral surfaces are constant
along the whole membrane. Because of the relationship
among the molecular volume, molecular area, and the splay
of the hydrocarbon chains (Hamm and Kozlov, 2000), this
assumption means that the molecular volume is not strictly
conserved in the regions with splay. However this effect
turns out to be of a higher order with respect to the dimen-
sionless splay. Contributions of this order are not taken into
account by our Hamiltonian (Eqs. 2 and 4). Nevertheless, to
estimate the error related to these assumptions, we have
numerically computed the deviations of the monolayer sur-
faces from the supposed shape, as required by conservation
of the molecular volume. We have calculated the related
change of the monolayer curvatures and the resulting en-
ergy. The determined maximal error of our results consti-
tutes 5%.

We did not account in our model for the energy of the
saddle-splay deformation (Frank, 1958; Helfrich, 1973) of
the stalk monolayer, which is determined by a combination
of the deformations of monolayer bending and tilt of the
hydrocarbon chains (Hamm and Kozlov, 2000). The saddle
splay is analogous in the case of simple bending to the
Gaussian curvature of the monolayer surface. The value of
the elastic modulus, �� , determining the energy of the saddle-
splay deformation of a monolayer is largely unknown (Ben-
Shaul, 1995). We are not aware of any experimental inves-
tigations that have directly attempted to obtain �� of a

Stalk Model of Membrane Fusion 891

Biophysical Journal 82(2) 882– 895



monolayer. The numerical calculations have been per-
formed for contribution of the hydrocarbon chains to �� and
have shown that its value constitutes less than 10% of the
bending modulus � (Szleifer et al., 1990). Based on these
results, our estimation of a possible contribution of the
saddle-splay deformation to the stalk energy did not exceed
few percents of the energy of splay.

CONCLUSIONS

To conclude, the present work resolves the energy crisis of
the stalk model of membrane fusion. We suggest a specific
structure of the stalk intermediate where the major roles are
played by deformations of tilt and splay of the hydrocarbon
chains. We demonstrate that the stalk intermediates have
energies much smaller than those predicted previously and
can be formed in real systems due to thermal fluctuations of
the membranes. The obtained dependence of the stalk en-
ergy on the lipid compositions of the membrane monolayers
agrees with the existing experimental data. We predict that
the stalk energy, practically, does not depend on the in-
tramembrane distance, provided that the stalk base is suffi-
ciently free to expand in the lateral direction.

APPENDIX A

Detailed form of equations and
boundary conditions

Lipid monolayers forming the stalk have axially symmetric shapes (Fig. 2).
To present mathematically the dividing surface of such a monolayer, we
use the radial coordinate r and the tangential angle � between the radial
direction r and the unit vector � tangential to the surface as illustrated in
Fig. 2. The dependence �(r) determines the shape of the dividing surface.

To express the tilt of the hydrocarbon chains, t, we determine the tilt
angle �(r) between the chain director n and the normal to the surface N
(Fig. 2). The absolute value of the tilt vector is

�t� � tan�t. (A1)

Splay (Eq. 1) of the hydrocarbon chains can be expressed as (Appendix B)

div n � cos� �
1

r

d�rsin�	

dr
� cos� �

1

r

d�rsin�	

dr
.

(A2)

According to Eq. A2, for the case of pure bending (Fig. 3 b) with vanishing
tilt angle, � � 0, the splay adopts a familiar form of the total curvature of
an axially symmetric surface,

div n �
1

r

d�rsin�	

dr
,

(see for example Helfrich, 1990). In the case of a flat surface, � � 0 (Fig.
3 c), splay is generated by the change of the tilt angle,

div n �
1

r

d�rsin�	

dr
.

And, finally, the tangential angle and the tilt angle can cancel each other,
� � ��. As a result, the bent surface is characterized by a vanishing splay

of hydrophobic chains, div n � 0. This configuration is illustrated in Fig.
3 e.

The elastic energy per unit area of the dividing surface of a monolayer,
accounting for Eqs. 4, A1, and A2, is

ftot �
1

2
� � � �cos� � 1r d�rsin�	dr � cos� � 1r d�rsin�	dr

� Js�2 � 12 � �t � �tan�	2 � 12 � �Js2. (A3)
Aim of analysis

The shape of a lipid bilayer is represented by its middle plane and
described by the dependence of the tangential angle of the middle plane �B
(Fig. 2) on the radial coordinate, �B(r). The dividing surfaces of the two
lipid monolayers are assumed to be parallel to the middle plane. Hence, the
tangential angles of the proximal, �p(r), and distal, �d(r), monolayers are
directly related to �B(r) of the middle plane (Appendix B). The tilt angles
in the proximal, �p(r), and distal, �d(r), monolayers are independent
functions. The energy of a bilayer is the sum of the energies (Eq. 5) of the
two monolayers. The aim of our analysis is to compute the stalk energy, F,
and to determine the shape of the stalk membrane, �B(r), and distribution
of tilt in its monolayers, �p(r) and �d(r).

Outline of analysis

The energy F, as obtained by integrating Eq. A3 over the areas of the
membrane monolayers, depends on the functions �B(r), �p(r), and �d(r).
To find the membrane shape, �B(r), and the distribution of tilt of the
hydrocarbon chains, �p(r) and �d(r), we have to minimize the energy
F[�B(r), �p(r), �d(r)] with respect to these functions.

In the course of minimization, several conditions have to be satisfied:

• To ensure that the hydrophobic interstices are filled by the hydrocarbon
chains, we set the tilt angles in the center of the stalk, r � 0, as �d(r �
0) � �	/4, and, �p(r � 0) � 	/4 (Fig. 2). This also accounts for finite
values of splay in the corner.

• We require that, at the distance r � R from the center of the stalk, the
membranes become flat and parallel with the distance between the
membranes’ middle planes equal to H.

• Far from the stalk center, we require that the tilt of the hydrocarbon
chains vanishes, t(r 3 �) � 0.

We calculate the energy and the structure of the stalk intermediate as a
function of the following parameters dependent on the membrane proper-
ties, which can be modified in the experiments: the spontaneous curvatures
Js of the membrane monolayers, and the geometrical constraints imposed
on the membranes, namely, the distance H between the membranes and the
stalk width R. We perform the calculation numerically by the method of
finite elements, which is equivalent to solving the Euler–Lagrange equa-
tions. The natural length scale of our calculation is determined by the
thickness of the hydrophobic part of lipid monolayer, �. We take egg
phosphatidylcholine as a typical lipid and use for calculations the values of
its geometrical parameters, and, specifically, � � 1.2 nm (Rand and
Parsegian, 1989).

APPENDIX B

Geometrical relationships

Expression for splay

In this section, we express the splay of the hydrocarbon chains, div n,
through the tangential angle of the membrane profile, �, and the tilt angle,

892 Kozlovsky and Kozlov

Biophysical Journal 82(2) 882– 895



� (Fig. 2). We consider two lateral unit vectors in the meridional, e�, and
equatorial, e
, directions. The vector e� coincides with the tangential vector
� defined in Appendix A (Fig. 2).

The lateral covariant divergence of the chain director representing the
splay can be calculated according to

div n �
dn
ds�

e� �
dn
ds


e
, (B1)

where s� and s
 are the arc lengths measured along the meridional and
equatorial directions, respectively. The chain director can be presented as

n � cos � � N � sin � � e�, (B2)

so that its differential is

dn � cos � � dN � sin � � de� � sin � � N � d�

� cos � � e� � d�. (B3)

The differential of the normal and the tangential meridional unit vectors are

dN � e� � d� � sin � � e
 � d
, (B4)

de� � �N � d� � cos � � e
 � d
. (B5)

The differentials of the arc length in the meridional and equatorial direc-
tions are give by

ds� �
dr

cos �
(B6)

and

ds
 � r � d
. (B7)

Inserting Eqs. B3–B7 into Eq. B1, we obtain for the splay

div n � cos � � cos � �
d�� � �	

dr
�

sin�� � �	

r
.

(B8)

The expression (B8) can be presented in a slightly different form (Eq. A2).

Relationship between the tangential angles of the
monolayers and the bilayer midplane

The tangential angles of the proximal, �p(r), and distal, �d(r), monolayers
are related to that of the bilayer middle plane �B(r) by

�p�r � � � sin ��r		 � �B�r	,

�d�r � � � sin ��r		 � ��B�r	. (B9)

APPENDIX C

Qualitative consideration of effects of
hydration forces

According to the established view (Marčelja and Radić, 1976; Gruen and
Marčelja, 1983; Leikin et al. 1993), the hydration layer of a membrane is
a layer of polarized water, which tends to decay exponentially in the
direction perpendicular to the membrane surface. This layer has been
described by a liquid crystal-like model (Marčelja and Radić, 1976) with
water polarization playing a role of the order parameter. This approach

successfully treated the hydration repulsion between apposing membranes
(Gruen and Marčelja, 1983). Extending, in the spirit of Frank (1958), this
liquid– crystalline model to description of deformations of splay and tilt of
the hydration layer, one obtains a Hamiltonian analogous to Eq. 4, where
the elastic coefficients account for the interactions between the polarized
water molecules.

This model predicts that, in the vicinity of the corner, the water
polarization can avoid infinite splay by tilting with respect to the mem-
brane surface, analogously to the deformation of the hydrocarbon tails we
considered above. This allows avoiding a seemingly infinite energy of
hydration layer in the corner, in full analogy with the energy of the
hydrocarbon chains.

Within this model, the hydration layer undergoes deformations similar
to those of the layer of the hydrocarbon chains, and its energy can be
computed analogously to that of the lipid monolayer. At the current stage,
we have no information available on the elastic moduli of the hydration
layer, but we can draw some general conclusions. First, for deformation of
simple bending of a lipid membrane, the layer of the hydrocarbon chains
and the hydration layer undergo similar splay. Therefore, the measured
bending modulus includes contributions from the chains and from the polar
head. According to the experimental results (Leikin et al., 1996; Fuller and
Rand, 2001), the values of the bending modulus practically do not depend
on the type of polar heads. Hence, the hydration of the heads contributes
very little to the membrane splay elasticity. Second, the elastic moduli of
a layer scale as powers of its thickness (Helfrich, 1990; Hamm and Kozlov,
2000). The thickness of a hydration layer is about ten times smaller than
that of a lipid monolayer (Rand and Parsegian, 1989; Leikin et al., 1993).
Hence, the elastic moduli of hydration layer are expected to give a less than
10% contribution to the total moduli of a lipid monolayer.

We are grateful to Leonid Chernomordik for stimulating discussions,
critical reading of the manuscript, and constructive suggestions. We thank
David Andelman, Samuel Safran, and Michael Schick for helpful discus-
sions. M. K. is grateful to the Israel Science Foundation funded by the
Israel Academy of Science and Humanities and the Human Frontier Sci-
ence Program Organization for the financial support.

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	Stalk Model of Membrane Fusion: Solution of Energy Crisis
	Introduction
	Energy crisis of the stalk model
	Criterion for energy of fusion intermediate
	Energies predicted by the previous versions of the stalk model
	Challenge for the present research


	Structure of fusion stalk
	Elastic model for stalk intermediate
	Monolayer shape and deformations
	Elastic energy
	Energy of splay
	Energy of tilt

	Main equations

	Energy and shape of stalk: results
	Symmetric membranes
	Stalk energy
	Stalk configuration
	Effects of spontaneous splay

	Asymmetric membranes

	Discussion
	Comparison with previous models
	Effects of different lipids
	Effects of different monolayers
	Effects of stalk dimensions
	Assumptions of the model

	Conclusions
	Acknowledgement
	Appendix ADetailed form of equations andboundary conditions
	Aim of analysis
	Outline of analysis
	Appendix BGeometrical relationships
	Expression for splay


	Relationship between the tangential angles of the monolayers and the bilayer midplane
	Appendix CQualitative consideration of effects ofhydration forces
	References