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Elastic moduli and neutral surface for strongly curved monolayers. Analysis of experimental results
M. Kozlov, M. Winterhalter
To cite this version:
M. Kozlov, M. Winterhalter. Elastic moduli and neutral surface for strongly curved monolayers.
Analysis of experimental results. Journal de Physique II, EDP Sciences, 1991, 1 (9), pp.1085-1100.
�10.1051/jp2:1991206�. �jpa-00247577�
Classification Physics Abstracts
82 65D 87.20C
Elastic moduli and neutral surface for strongly curved
monolayers. Analysis of experimental results
M. M. Ko~ov (I, *) and M. Winterhalter (2)
(') Freie Unlversitat Berlin, FB Physik, WE2, Arnimallee 14, 1000 Berlin 33, Gerraany f) Lehrstuhl fur Biotechnologle, Rontgennng II, 8700 Wurzburg, Gerraany
(Received 20 March 1991, accepted 10 June 1991)
Abstract. In this paper we apply our results concemmg a new view on a mechanical descnption
of strongly curved amphiphllic interfaces to expenmental data available in the literature (Rand
et al, 1990) Our theory predicts that, for cyhndncal interfaces, one has to take into account, three elastic moduh corresponding to area extension, bending and mtxed deformations We
determine the values of the three elastic moduh for the mtemal dividing surface of strongly curved DOPE monolayer m the Hit-Phase Analysing for this purpose the expenmental we recognise lateral compressibility and bending m the same time As a result we recover the moduh that are closer to work done on bilayer than had been determined m precedent papers considenng only bending energy. Using this set of parameters we find the position of the neutral surface and the values of elastic moduh corresponding to the neutral surface We find the quantitative dependence of all the elastic moduh on the pos1tlon of the d1vlding surface and compare them with those previously obtained
In«oduction.
Recently several expenments [1-4] have been performed on interfaces with low surface tension and large curvature. Gruner [I], Rand [2] and coworkers performed a stimulating
investigation of the mechanical properties of the strongly curved lipid monolayers, namely of
the inverted hexagonal phase (Ha of pure lipid DOPE and of mixture between DOPE and
DOPC In one senes of experiments only DOPE and water have been used. In a second senes
a DOPE/DOPC mixture has been used as surfactant; however, to obtain an Hii-Phase, tetradecane has been added If water is in excess, the Hn-Phase has the equihbnum geometncal charactenstic, namely the lattice dimension measured by means of X-ray
diffraction A simple calculation taking into account the known quantity of lipid and water in the system allows to determine the internal radius of lipid cylinder corresponding to the
boundary between water volume and lipid monolayer. The radii of spontaneously formed lipid cylinders are small and show a typical value in the range of 2-3 nm depending on the
(*) Permanent address AN Frulnkln Instituie of Electrochen~istry, USSR Academy of Science, 31Lemnsky Prosp Moscow, l17071, USSR
rum JOURNAL DE PHYSIQUE II bt 9
compositions This means that the radius of curvature is comparable to the monolayer thickness of about approximately 1.5 nm
The expenments by Rand and coworkers [2] have been performed in the following way An
osmotic pressure difference between the mtemal volume of cylinder and the surrounding
water solution was applied. The cylinder was deformed by this pressure difference. The
deformation included the bending the change of mtemal radius of the monolayer- and the
extension-compression the change of area per lipid molecule at the intemal surface of the
monolayer. The values of both deformations were calculated directly from the values measured expenmentally.
The results of these expenments give a rare possibility to determine the elastic moduh of the strongly curved lipid monolayer. However the authors of expenmental investigations [2]
analysed the data only with respect to a change of the curvature of the monolayer as the response to the action of applied pressure difference For this purpose they descnbed the
monolayer in terms of the d1vlding surface situated inside the nonpolar region of the
monolayer and keeping approximately its area constant dunng deformation. In this way any possible effects due to an area extension-compression are not considered. At the same time the expenmental data concemmg area extension-compression of the intemal surface of the
monolayer give important additional information about the mechanical properties of the system.
A more complete description of the mechanics of cylindrical monolayer was made by us in the precedent paper [5] showing that the following important points have to be taken into
account m the analysis of the mentioned expenments First of all the choice of the dividing surface plays an essential role m the mechanical descnption of an interface. In general any arbitrary dividing surface is characterized by three elastic moduli corresponding to area
extension, bending and mixed deformations. The simplest descnption of the monolayer by
means of just two elastic modub (of area extension and of bending) is only possible for the
neutral dividing surface, which by our definition is that of zero modulus of mixed deformation. However, the position of the neutral surface is a priori unknown and in the
general case does not coincide with the position of the dividing surface of constant area.
Secondly, the values of all elastic moduh strongly depend on the position of the dividing
surface. This dependence is especially important for the bending modulus. For strongly curved monolayer the value of bending modulus corresponding to the intemal dividing surface is considerably smaller than that of the outer dividing surface
In the present paper we will revisit the expenmental results concerning the mechanical deformation of lipid cylinders m Hjrphase of DOPE performed by Rand and coworkers [2].
For this purpose we will apply the complete descnption developed by us [5] First, the deformation of the intemal d1vldmg surface of the monolayer will by analysed. From the
expenmental results we will determine three elastic moduh of the mtemal d1vldmg surface corresponding to area extension, bending and mixed deformations. These results allow us to
determine the position of the neutral surface and m a next step to calculate the elastic moduli of the neutral surface The last part of the analysis is the quantitative calculation of the
dependence of the set of elastic modub on the position of the dividing surface. All calculations will be performed without the need of a microscopic model of the structure of the lipid
monolayer
Main equations.
THE SURFACE TENSION AND THE BENDING MOMENT Let US Consider a Cylindncal
monolayer (Fig I). Let us assume a pressure difference between the outer medium and the internal volume of the cylinder, p
= p~~i- pm. For the descnption of the mechanical
z
Lj R~
Fig I. -Geometncal parameters as used in the text
properties of the monolayer we have to choose a dividing surface. A natural choice, is the internal dividing surface, 3~, situated m the region of polar heads. For this dividing surface the Gibbs' excess of water [6] is zero. However, the basic equations will be denved for an arbitrary dividing surface 3 which is parallel to 3~, the distance between both is denoted by z (Fig I).
Let us assume that under the condition of zero pressure difference, P
= 0, the monolayer is
m its spontaneous state. By definition m this state the dividing surface 3 has the spontaneous curvature (
= I/R~, and the spontaneous area per lipid molecule a~. The charactenstic of the spontaneous state is that the surface tension y and the bending moment Ci vanish We denote the elastic moduh of the arbitrary dividing surface 3 by E~~ as the area extension modulus, Ejj as the bending modulus and E~j as the modulus of mixed deformation The applied
pressure difference P causes a monolayer deformation which leads to a corresponding surface tension and bending moment In the case of a linear relation between deformations and force
factors, the surface tension and bending moment are expressed through the bending
I/R I/l~ and the area extension a a~ is the following from
Y = FAA (
+ EAJ(I/R I/J~) (I)
ci
= E~
j
~ ~~
+ E
j Ii /R i/J~) (2)
as
EQUATIONS FOR EQUILIBRIUM. Let the monolayer be m equilibnum with the applied
pressure difference The equihbnum condition in the lateral direction (along the axis of the
cylinder) has the following form [7]
Y.(= -jP (3)
1088 JOURNAL DE PHYSIQUE II bt 9
and the equation for equilibnum in the normal direction, called the modified Laplace equation, has the follov/ing form
y.(-Ci~=-P. (4)
THE DEPENDENCE OF DEFORMATION ON THE PRESSURE DIFFERENCE. inserting (I) and (2)
in (3) and (4) allows us to decouple both kinds of deformations and to obtain two distinct relations for the pressure difference versus bending
~
2(1/R I@)
PR
= ~ ~ (5)
j + ~ (
and versus area extension
2 (
fi
)
~~ ~
EAJ
where D
= E~~ Ejj- Ejj is the determinant of the matnx of the elastic modub.
Equations (5) and (6) descnbe the deformation of an arbitrary dividing surface under the action of pressure difference One can see that the pressure difference leads to a bending and
area compression of the monolayer simultaneously. Equations (5) and (6) show a much
simpler form if the chosen d1vldmg surface is the neutral surface with zero modulus of mixed deformation, E~j
= 0. In this case the equations of equilibrium have the following simple
form
pR~
= 2 Ej§(I/R~ I/R~~) (7)
~N ~N
pR~
= 2 E(~
~
~ (8)
~s
Expression (7) coincides with the equation used m an earlier analysis of the above mentioned expenments [2], where (7) was applied to the dividing surface keeping the area fixed during
the deformation However, it follows, from a more detailed calculation as presented above, that the neutral surface for which relation (7) is only valid, changes its area under the action of
pressure difference This change of the neutral surface area is given by equation (8)
In order to illustrate the relation of our approach to the earlier analysis [2] we look for the position of the dividing surface which keeps its area approximately constant dunng the first stage of deformation This dividing surface is determined according to (6) by the condition
We would like to emphasize that the dividing surface defined by equation (9) differs from the neutral one with (E~j= 0). Both dividing surfaces coincide only in the case of a flat spontaneous state with R~ - co. The difference is especially important for the interfaces with
strongly curved spontaneous states, as considered below
The properties of elasticity of the internal dividing sudace of the monolayer.
Equations (5) and (6) allow us to analyse the expenmental results obtained by Rand and coworkers [2] for the internal d1vlding surface of the monolayer 3,.
We will analyse below the experimental results dealing with a monolayer of pure DOPE [2]
and presented in figure 2. In figure 2a we demonstrate the data pR~ vs I/R for the intemal
d1vlding surface as obtained m the experiment. In figure 2b the relation between the area per
lipid molecule and the curvature I/R at the mtemal d1vldlng surface is shown From these
curves one can determine with a good accuracy the spontaneous geometncal charactenstics of
monolayer [2] corresponding to the mtemal dividing surface. The spontaneous radius obtained in this way is equal Aj = 2.3nrn, the spontaneous area per molecule is equal
% = 0.53 nm~.
8
_ n
~ D
6 o
,
O
' u
- ~ ~
u
« « °
©v 2
~ a
iJ
o
u,4 0,6 0,8 1.0 I,2
in I lo ~m "'J a)
60
M ~£ii
"~ 50 ~ D
~ ~
m'O
aJ -
40 o
@ o
in I lo ~m -'>
b)
Fig 2 -Data for internal dividing surface of DOPE monolayer as obiained from literature [2].
R-radius, p-pressure difference a) Plot of pR~ vs I/R b) Plot of the area per molecule vs.
I/R
1090 JOURNAL DE PHYSIQUE II bt 9
CHARACTERISTICS OF ELASTICITY WITH RESPECT TO AREA EXTENSION-COMPRESSION AND
TO THE MIXED DEFORMATION. Let uS consider the curve in figure 2a. According to
equation (4) and taking (3) into account one can see that the value PR~ is proportional to the bending moment, C,
=
(PR~)/2. In the framework of traditional theory [8, 9] the bending
moment is proportional to the curvature and the coefficient of proportionality is the bending
modulus. It follows from figure 2a that the dependence PR~
vs. I/R has not a linear character.
Th1s means that the relation between the bending moment and the bending cannot be
descnbed by the simple linear function. One can assume that the deviation of the curve in figure 2a from the straight line is related to increase during the deformation of the effective
bending modulus. But m thJs case the slope of the function PR~
vs. I/R should increase. On the contrary the slope of the experimental curve in figure 2a is decreasing. We can conclude that for the explanation of the data dealing with the mtemal dividing surface of the monolayer
one has to take into account the elastic modulus of mixed deformation E~j and to use
expression (5).
Let us compare the theoretical dependence of the bending moment PR~
vs. the curvature
I/R given by expression (5) with the expenmental curve presented in figure 2a. This
companson is related to the determination of two free parameters, namely the quotient of two
elastic moduh and the determinant of the matrix of elastic moduli, E~~/D, E~j/D To
overcome this difficulty we will make the assumption concerning the relative value of elastic moduli, E~~ »E~j/~ which will be checked later. Expression (5) has the following
approximate form convenient for the companson with the expenmental results :
pR~
=
~ ~ +
~ ~ l +
~ ~~~ ~ ~~~~ ~
FAA Rs FAA FAA fir R EAA2 R (10)
In the nght hand side of the approximate expression (10) the first term is related only to
E~~ ID and does not depend on E~j/D. Due to this fact one can determine independently the values of E~~/D and E~j/D through the comparison of the theoretical curve (10) with the expenmental results (Fig. 2a). Th1s fit is presented in figure 3. The spontaneous radius was
assumed to be R~ = 2.3 nm The values of quotients of elastic moduh and determinant D are equal to
E~~/D = 5 8 x 10'~ [J-'], E~j/D
= 3.4 x 10'° [N~~]. (l1)
The relation E~j/(E~~ R~) = 0.2, which justifies our approximation
CHARACTERISTIC OF ELASTICITY WITH RESPECT TO BENDING. A Similar comparison for
(6) allows us to determine the quotient of bending modulus and the determinant,
EjjD. In this case EjjD is a singular free parameter since the parameter E~j/D was determined before.
However, for the determination of EjjD we also need the relation between the change m area per lipid molecule a and the curvature I/R presented in figure 2b. We fitted the
expenmental dependence (Fig. 2b) by the expression a = 0 73 5.23/R + 15.581/R~. In
figure 4 we show the theoretical dependence given by expression (6) with use of extrapolation
curve for the function a( I/R). The best fit for the elastic parameter under consideration is the
following
EjjD = 45 [m/Pfl (12)
THE COMPARISON OF THE RELATIVE VALUES OF ELASTIC MODULI. Tile companson of the
theoretical functions (10) and (6) with the experimental data (Fig 2a, b) gave the quotients of
~
~°
~
~
©k
2
o u.4
0,6
in
iu92 JOURNAL DE PHYSIQUE II N 9
The relation of the elastic modulus E~j to the elastic modulus of bending Ejj is equal to
E~jR~
=
EJJ 1.7
It follows from this expression that for the intemal dividing surface of the monolayer of DOPE the elastic modulus of mixed deformation E~j can play a more important role than the elastic modulus of bending Ejj.
Neutral surface.
THE POSITION oF THE NEUTRAL SURFACE. As already mentioned, the neutral surface is
defined by that of zero modulus of mixed deformation E~j
= 0. Th1s condition allows us to
use the simplified relations between the extemal force factors and the monolayer deformation (7), (8) including only the modulus of area extension E~~ and the bending modulus
Ej~ The position of the neutral surface is a priori unknown. In the same time the values of parameters characterising the elasticity of the intemal dividing surface, found before, give the
possibility to determine the distance z] between the intemal dividing surface and the neutral surface in the spontaneous state. For this purpose one has to use the equation for
z] previously derived [5] which allows us to determine z~ in terms of quotients of the elastic moduli and the spontaneous curvature of the intemal dividing surface :
iA IiiiJ
i
IIlllill~l
Ill
llllii2 ~~~~
The numencal solution of this equation for the values of parameters obtained before gives the following value
zf
= 6 15 h (15)
which differs significantly from that used in the earlier analysis [2]. The presented
determination of zf
is free from the use of any model of membrane structure [8, 9].
THE ELASTIC MODULI OF THE NEUTRAL SURFACE. Once the pos1tlon of the neutral surface
(15) is known, the bending modulus and the area extension modulus for this surface may easily been found by reanalysing the data
However, in the analysis of the expenmental results we have to take into account the
conservation during the deformation of the volume between the intemal dividing surface and
the neutral surface The expression relating the distance between these surfaces in the deformed state z~ to the value m spontaneous state z] has the following form
z'~(R)
= lR~ + 2 I Rzf l + ( R (16)
a
where R and a are the radius and the area per molecule on the internal d1vlding surface in the
deformed state. The radius of the neutral surface is equal to R~= R+z~(R) The
dependence of PR~~
vs. I/R~
is presented m figure 5 The expenmental curve might be
reasonably fitted by a straight line The corresponding theoretical function is given by
expression (7) and is also the straight line with a slope proportional to the bending modulus.
~ 2
°~
'
m C
«
ck
0 0.3
i'j
