The Bending Rigidity of Phosphatidylcholine Bilayers: Dependences on Experimental Method, Sample Cell Sealing and Temperature G. Niggemann, M. Kummrow, W. Helfrich To cite this version: G. Niggemann, M. Kummrow, W. Helfrich. The Bending Rigidity of Phosphatidylcholine Bilay- ers: Dependences on Experimental Method, Sample Cell Sealing and Temperature. Journal de Physique II, EDP Sciences, 1995, 5 (3), pp.413-425. <10.1051/jp2:1995141>. <jpa-00248169> HAL Id: jpa-00248169 https://hal.archives-ouvertes.fr/jpa-00248169 Submitted on 1 Jan 1995 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destin´ ee au d´ epˆ ot et ` a la diffusion de documents scientifiques de niveau recherche, publi´ es ou non, ´ emanant des ´ etablissements d’enseignement et de recherche fran¸cais ou ´ etrangers, des laboratoires publics ou priv´ es.
Transcript
The Bending Rigidity of Phosphatidylcholine Bilayers:
Dependences on Experimental Method, Sample Cell
Sealing and Temperature
G. Niggemann, M. Kummrow, W. Helfrich
To cite this version:
G. Niggemann, M. Kummrow, W. Helfrich. The Bending Rigidity of Phosphatidylcholine Bilay-ers: Dependences on Experimental Method, Sample Cell Sealing and Temperature. Journal dePhysique II, EDP Sciences, 1995, 5 (3), pp.413-425. <10.1051/jp2:1995141>. <jpa-00248169>
HAL Id: jpa-00248169
https://hal.archives-ouvertes.fr/jpa-00248169
Submitted on 1 Jan 1995
HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, estdestinee au depot et a la diffusion de documentsscientifiques de niveau recherche, publies ou non,emanant des etablissements d’enseignement et derecherche francais ou etrangers, des laboratoirespublics ou prives.
(Received 18 July 1994, accepted in final form 21 November 1994)
Abstract. The bilayer bending rigidities were measured for the same spherical vesicles with
two methods, fluctuation mode analysis and electric deformation. In some of the sample cells
the glue sealing the cell was in direct contact with the sample, in others the sample was shielded
from the glue bya
silicone grease. The values of the rigidity, wlfile being lowest and practicallyequal for the unshielded samples, differed significantly between methods for the slfielded ones.
We also measured the temperature dependence of the bending rigidity. The dependence of the
rigidityon
method seems to suggest the existence of a membrane superstructure.
1. Introduction
The bending rigidities of single fluid bilayers of various phosphatidylcholines and other lipidshave been measured with many methods. In most cases, they were determined from the
mean square amplitudes of bending fluctuation modes. The objects studied were tubular
vesicles [1-4], spherical vesicles [2, 5-8] and sections of very large membranes [4]. In other
experiments, the bending rigidity was obtained from an apparent increase of the membrane
area with lateral tension. This effect is due to the leveling of thermal undulations which at low
enough tensions dominates over real (Hookean) stretching. The tensions were produced either
by sucking a small part of a vesicle into a micropipette [9] or by deforming a vesicle throughelectric fields [10]. Finally, the bending rigidity was derived from strong curvatures produced
by external forces. In those experiments a very thin bilayer tube, called tether, was pulledfrom a vesicle held by a micropipette iii].
The results of these measurements are remarkable in two respects. Firstly, very different
values of the bending rigidity ~ have been found in different sets of experiments. The variation
is the largest for egg yolk phosphatidylcholine (EYPC),a favorite material of the early days,
the reported averages ranging from~ =
2.3 x10~~~ J [1] to 0.24 x
10~~~ J [10]. It is not clear
whether the variation resulted from the change in experimental methods or from differences in
bilayer composition. Natural lipids are variable and even one-component lipids, mostly used
as purchased, need not be completely pure. Additional impurities may enter during sample
preparation, in particular from glues and other materials that seal the sample cell, or may
appear gradually as a consequence of chemical degradation. Angelova et al. [12], studying the
fluctuations of spherical vesicles, found a decrease of the bending rigidity of EYPC bilayersfrom (0.66 + 0.06) to (0.45 + 0.05) x
10~~~ J in the course of two weeks. So far, there have
been no measurements of the bending rigidity in which two different methods were applied to
the same vesicle.
Secondly, it seems to be typical of these studies that the experimental error of the bendingrigidity for an individual vesicle is smaller than the scatter of the results for different vesicles.
In general, the half width of the distribution is more than a third of its mean value. The onlyexceptions from this rule might be the micropipette experiments [9,11]. However, it should
be noted that in the study involving tethers the scatter among vesicles was also larger than
individual errors.
In the following, we report comparative measurements of the bending rigidity of phospha-tidylcholine bilayers. We studied the mean square fluctuation amplitudes and the electric
deformation of the same spherical vesicles made of1-stearoyl-2-oleoyl,sn-phosphatidylcholine(SOPC), 1,2-dioloeoyl-sn-phosphatidylcholine (DOPC) or 1-palmitoyl-2-oleoyl-sn-phosphati-dylcholine (POPC). Some of the sample cells, containing SOPC or DOPC, were sealed with
a glue in direct contact to the water. In the other sample cells, containing DOPC or POPC,the vesicle suspension was shielded from the glue by a barrier of silicone grease. The bendingrigidities were found to be the lowest and insensitive to method when glue and water were in
contact. They were higher and, in the first place, depended markedly on method when glueand water were separated by the silicone grease.
In additional experiments we varied the temperature of the sample. Employing fluctuation
mode analysis of spherical DOPC vesicles, we found the bending rigidity to decrease dramati-
cally with temperature.The discussion of the results centers on the comparative measurements, I.e, the dependence
of the bending rigidity of the same vesicle on experimental method. We argue that the dis-
agreement between methods cannot be explained by existing theories of membrane bendingfluctuations, including the recent hat model [13,14]. Rather, it seems to confirm earlier ideas
of a membrane superstructure. The latter may also be involved in the sensitivity of the rigidityto direct contact between glue and water and to changes of temperature.
2. Materials and Methods
The phospholipids were delivered in chloroform solution (Sigma Chemical, Deisenhofen, Ger-
many) and used without further purification The purity of the lipids indicated by the dis-
tributor was99%. We diluted the lipid solution in chloroform to mg/ml and stored it below
-20 °C. Each lipid was taken every time from the same respective solution in chloroform and
no dependence on storage time of the measured bending rigidities was noted. Giant vesicles
were prepared by standard procedures [15,16]. A small amount of the lipid solution was spread
on the bottom of a beaker or directly on the object slide of the sample cell and left overnightunder vacuum so that the chloroform could evaporate. The lipid started to swell spontaneouslywhen it was covered with water. All the water used in the experiments was ion depleted (Ser-adest) and contained ca. 100 pM NaN3 to prevent bacterial contamination and ensure a small
Debye length.The sample cells for comparative measurements, like those used previously for electric de-
formation only [10], were equipped with two parallel 0.25 mm wide platinum wires serving as
spacers and electrodes. A droplet taken from the vesicle suspension after at least two days of
swelling was transferred from the beaker into a drop of water on the object slide of the sample
N°3 DEPENDENCE OF BENDING RIGIDITY ON METHOD 415
cell. Alternatively, the lipid swelled in the sample cell, forming free vesicles and other mem-
brane structures within hours. The latter procedure often resulted in tight spheres, but visible
fluctuations developed in the course of days. No effect on ~of vesicle preparation was noticed.
A flat ring cut out of a 50 pm thick mylar film was used as spacer when the temperaturedependence was measured. In these experiments, the lipid always swelled in the sample cell.
The gap between object slide and cover slip was sealed with Eukitt (Kindler, Freiburg,Germany) to prevent evaporation of the water. This glue was selected earlier [17] because,in contrast to others, it has little effect on the pH of water, giving pH
=6. However, it
contains xylene besides a polymeric acrylate. Although practically insoluble in water, xylene
may accumulate in the lipid bilayers. In the present experiments contact between glue and
water was avoided in part of the sample cells for comparative measurements by applying a
shield of silicone vacuum grease (Lithelen, Leybold-Heraeus). The mylar ring separated glueand water in the cells built for measuring the temperature dependence.
The samples were observed with a phase contrast microsope, the numerical aperture of the
objective being 0.75. The object stage could be brought to the desired temperature with an
accuracy of 0.1 °C. No measurements were done above 45 °C because of strong convection
currents at elevated temperatures.The images of the fluctuating vesicle were taken with a CCD camera (Pullnix, TM 700,
England) and could be recorded on a VHS video tape recorder or a video printer. An imageprocessing system (Engel und Stiefvater, Karlsruhe, Germany) was connected to a parallelpersonal computer (Proteus, Karlsruhe, Germany) with a 68030 Host Processor (Motorola)and seven parallel processors IT 800). This system digitized the pictures from the CCD camera
into 512 x 512 pixels with 8 bit resolution for the gray levels and stored them in the memory
of a computer. One pixel corresponded to about 127 nm. The video integration time of an
image was 40 ms. The time interval between subsequent pictures was made long enough to
ensure negligible correlation for the slowest fluctuation mode Depending on vesicle radius, it
was usually several seconds. The number of pictures for each measurement was in general 500
and never less than 300. We developed a special algorithm to determine the contour. Spuriousnoise of the contour was reduced by a method similar to that of Faucon et al. [7]. In this
procedure the experimental contour was approximated by its first four or five Fourier modes.
Only those signal points were taken into account which were closer than a certain distance to
this artificial contour.
The approximately 500 images of a vesicle were used to obtain the autocorrelation function of
its contour which was then decomposed into Legendre polynamials. We calculated the bendingrigidity ~(l) for each of these compound modes from the coefficient El of the I-th polynomialby use of [7]
~~~~ 4~iIi 1)(1/~i
+l(1 + 1)]
~~~
The parameter n=
~R~/~ introduces alateral tension ~
of the membrane, R being the vesicle
radius. The values of lateral tension and pixel noise were chosen such as to obtain a plateau of
~(l) extending over the first 10 to 20 modes. Another adjustment was made, when necessary,
for the video integration time. In all this, we closely followed Faucon et al. [7]. The method
of fluctuation mode analysis was checked by analyzing numerically simulated fluctuations of a
spherical vesicle.
The electric field deforming the spherical vesicles had a frequency of 2 kHz and ranged
up to 100 V/cm. In order to evaluate the deformations we did not use photographs as in
the previous experiments [10]. Instead, we recorded five contours of the vesicle for each field
strength and computed from them the major axes of the contour which was practically an
416 JOURNAL DE PHYSIQUE II N°3
ellipse. The lateral tension acting on the undulations was obtained as before [10] from the
equatorial Maxwell stress
~~rr~eq ~jfwf0~0'
~~
acting on the membrane from the outside of the nearly spherical vesicle, and the normal force
equation
(Cl + C2)eq~h (Trr)eq" (Cl + C2)pole~h
Here ew =80 is the dielectric constant of water, Eo is the r,m.s. strength of the homogeneous
electric field far away from the vesicle, ci and c2 are the principal curvatures of the membrane
taken either at the equator (eq) or the pole, and ~h is the homogeneous lateral tension to be
computed. The apparent relative increase in area, 1hA IA, caused by the leveling of undulations
obeys [18]
where the ositive parameter~o is obtained by extrapolation and
can belarger
than the ctual
lateral ension at zero field strength. the bending rigidity is given by theslope
AA IAplotted
uers~s the of ~h. inor orrections of this amounting to
less than 5% arisefrom
a real tretching of the bilayer and aweakening
of the
(Trr)~q with thetheir description [10].
The initial lateral tension, either ~ or ao, had to be low in both kindsWe used only vesicles where
hese tensions were smaller than io~s mN/m. In addition, only
free vesicles withdiameters between
8 and 80 pm wereselected.
For very vesicle,
analysis wasdone
first andollowed
immediately by electricdeformation.
We checked in some
cases that a subsequent repetition of luctuation mode analysis gave
3. Results
Comparative measurements of the bending rigidity, subjecting the same spherical vesicle to
fluctuation mode analysis and electric deformation, were made on samples without and with
silicone shield. Figure i shows as an example the results obtained with an unshielded sample.The values ~(l) computed from the mean square fluctuation amplitudes of the Legendre modes
are displayed in Figure ia. They are corrected as described above for lateral tension and pixelnoise. In this example the radius (8 pm) and, thus, the fluctuation amplitudes were rather
small. The modes beyond the twelfth could not be adequately resolved and were therefore not
considered in calculating the corrections. The data of the electric deformation experiment are
plotted in Figure 16. The slope of the straight line as obtained by a least squares fit gives the
bending rigidity when inserted in equation (2).Table I lists the bending rigidities found in comparative measurements with unshielded sam-
ples of SOPC or DOPC. Inspection shows that for each of the five vesicles the two values of the
bending rigidity are in good agreement. One of the SOPC vesicles was apparently bilamellar,and we divided its bending rigidity by 2 when taking the averages. For both materials, the
results were confirmed by several vesicles whose bending rigidity could be measured with one
N°3 DEPENDENCE OF BENDING RIGIDITY ON METHOD 417
~
0.4
I~
0.3
~'Clflf'~b0~
~ii
IJ~
0.0
1a)
418 JOURNAL DE PHYSIQUE II N°3
Table I. Bilayer bending rigidities ~ of SOPC and DOPC vesicles meas~red with fl~ct~ationmode analysis and electric deformation. T is temperat~re, R is vesicle radi~s,
~and ~o repre-
sent lateral tensions obtained by adj~stment and extrapolation, respectwely (see text). Sampleswitfi direct contact of gl~e to water (no silicone shield). The asterisk marks the exceptional
of the two methods only. Previous determinations of the bending rigidities of SOPC, DOPC
and POPC bilayers by electric deformation, done in exactly the same type of sample cells,yielded 0.26, 0.18 and 0.23 in units of10~~9 J from 6, 8 and 10 vesicles, respectively, the errors
being +20 to +30% (17]. The old data for SOPC and DOPC agree within the errors with the
new results, while those for POPC have no counterpart in the present study.Table II lists the results of comparative measurements with silicone shielded samples of
DOPC and POPC. In contrast to unshielded samples, the bending rigidities now depend sig-nificantly and systematically on method. They are larger with electric deformation than with
fluctuation mode analysis, the ratio of the two values being on average 2.5 for DOPC and 1.5
for POPC.
The bending rigidity of DOPC was measured not only by both methods but also in shielded
and unshielded samples. Moreover, the fluctuation mode analysis was done with and without
silicone barrier also for POPC if we include the previous results (see above). For both lipids,mode analysis yielded larger values (by factors of I-s for DOPC and 1.7 for POPC) with
barrier than without it. However, the emphasis of the present work is on the comparativemeasurements, I.e. the results obtained by the two experimental methods from the same
vesicle.
In the case of DOPC we measured the bilayer bending rigidity also as a function of temper-ature, preparing the vesicles in sample cells with a mylar spacer and using fluctuation mode
analysis only. The data for a particular vesicle are plotted in Figure 2. Evidently, the bendingrigidity drops strongly with temperature, decreasing by factors of five between 13 and 34 °C.
This very strong dependence cannot be a pretransitional effect as the range of temperatures
was far above the main transition of DOPC at -21 °C (21]. An Arrhenius plot of the same
data, shown in Figure 3, yields an apparent activation energy Eo"
0.8 x10~~~ J for the
N°3 DEPENDENCE OF BENDING RIGIDITY ON METHOD 419
Table II. Bilayer bending rigidities of POPC and DOPC vesicles meas~red with fl~ct~ationmode analysis and electric deformation. See Table I for symbols. Samples with a barrier ofsilicone uac~~m grease between gl~e and water. The asterisk marks a
bilamellar vesicle whose
rigidity ual~es were dimded by 2 before calc~latmg the mean ual~es.
Table III. Temperat~re dependence of the bilayer bending rigidity ~ of flue DOPC vesicles.
R is the vesicle radi~s and Eo is the activation energy of the flexibility (inverse r~gidity). The
last two coi~mns show the uanation of~ m
the range of "room temperat~re". The asterisk
marks a bilameiiar vesicle.
R EoK
[pm] (10~~~j 18 °C 25 °C
14 5 1 00 0.29 0.18
17.5 0 82 0.19 0.ll
12 8 1.18 0.62 0.30 *
16.0 0.81 0 29 0,17
14.9 1.10 0 27 0.15
420 JOURNAL DE PHYSIQUE II N°3
~n0.C0
~c~I" 0 30
o.20
o. i o
12 16 20 2C 28 32 36
T/°c
Fig. 2. Bilayer bending rigidityas a
function of temperature for a DOPC vesicle (mylar spacer,
fluctuation mode analysis).
$~.
,
" C5 0
~.
-
3.2~
N°3 DEPENDENCE OF BENDING RIGIDITY ON METHOD 421
producing a discontinuity in the temperature dependence of the bending rigidity. As a rule,two or more buds were not separate but arranged in a string of spheres connected to each other
and the big sphere by optically irresolvable constrictions. The buds appeared to be stable while
the pictures were taken at a given temperature.
4. Discussion
Small fractions of dissolved impurities are unlikely to have a large effect on the intrinsic
bending rigidity of afluid membrane. However, theories predict that impurity molecules even
at rather small concentrations greatly decrease the effective bending rigidity if they possess a
local spontaneous curvature very different from that of the host. The spontaneous curvature
of a symmetric bilayer is zero by definition. (It remains zero on average in the presence of
impurities producing local spontaneous bilayer curvature if their concentrations are equal on
both sides of the bilayer.) The effective bending rigidity of mixed monolayers and bilayers has
been treated in terms of continuum models [22, 23] and the so-called hat model [13,14].The hat model has been worked out in some detail for membranes with a flat
mean surface
such as symmetric bilayers. In this model each molecule in a monolayer, or pair of oppositemolecules in a bilayer, is taken to be a spherical cap. The curvature of this cap fluctuates about
its spontaneous value. The spherical cap and a smoothly attached axisymmetric brim of zero
mean curvature together make up a hat. Like the nonlocal bending modes of the continuum
model, the hats can be superimposed as long as the fluctuating membrane makes small enoughangles with the flat surface about which it fluctuates.
The decrease of the effective bending rigidity is calculated from the adjustment of surfactant
concentrations to membrane curvature in the continuum model. It is derived from the en-
hancement of local bending fluctuations by the diffusion of surfactant molecules with different
spontaneous cap curvatures in the hat model. The two models yield essentially the same results
for monolayers. The hat model has the advantage of providing a microscopic description of
mixed monolayers, including their athermal roughness. Also, it permits a treatment of local
bending irustration and athermal roughness in mixed bilayers [14].Based on those theories, one expects the purest bilayer of a given lipid to have the highest
bending rigidity. For comparison, we note that other authors found~ =
0.9 x10~~~ J from
apparent stretching [9] and ~ =l.2 x10~~~ J from the pulling of tethers ill] for SOPC as
well as ~ =lA x
10~~~ J from fluctuation mode analysis for both POPC and SOPC [8]. A
high concentration of impurity molecules producing rather pointed hats in the bilayer would be
needed to reduce the effective bending rigidity to the much smaller values which we measured
in unshielded samples of SOPC and (previously) POPC. This can be seen from the estimates
given in the Appendix.However, impurities cannot explain why, in shielded samples, we found distinctly different
values of the bending rigidity with the same vesicle when we changed the experimental method.
One may wonder whether the discrepancies are removed if the hat model is employed instead
of the more familiar continuum model. We will show in the Appendix that this is not so.
The natural alternative to hats are saddles. A bilayer superstructure consisting of cooperativelocal saddles has already been postulated a few years ago [13]. Disordered arrays of saddles seem
suited to explain an extraordinary membrane roughness which was inferred from experimentalstudies of the mutual adhesion induced by lateral tension as observed with PC and other lipidmembranes [24-26]. Hats were ruled out since they not only store area but also reduce the
bending rigidity. The capability of the bilayer to form saddles, which should automatically be
cooperative, was attributed to higher order bending elasticity. Some pictures of grainy egg yolkPC membranes obtained in transmission electron cryomicroscopy seem to show a disordered
422 JOURNAL DE PHYSIQUE II N°3
superstructure [27] and prompted, when seen for the first time, the proposal of cooperative
saddles as its elements.
In order to exp1aln a dependence of the bending rigidity on method, we may speculate that
a superstructure with short-range order gives rise to the formation of relatively rigid "floats"
in the fluid bilayer. They could weaken undulations of wavelengths smaller than their size. For
a tentative assessment of a suitable size of the floats we make use of the formula
(~=
~/~ (3)
which relates the deflection length ( to bending rigidity and lateral tension of the membrane
[24]. On one hand, the floats should not be much smaller than the deflection length ( so that a
range of fluctuation modes with wavelengths near ( are significantly reduced by their presence.
The lateral tensions occurring in the electric deformation of vesicles are on the order of10~~
mN/m. Inserting this value and ~ =10~~~ J into (3), ~ve compute (
=3 ~m. On the other
hand, the floats should be smaller than the wavelengths used in the fluctuation mode analysis,
so that this method gives the same values of the bending rigidity for all modes accessible to it,
I-e- for all wavelengths larger than ca. 4 ~m. We niay conclude from these estimates that the
two conditions to be satisfied by the size of the floats are approximately compatible.
If the envisaged superstructure of the bilayers does exist, the bending rigidity could be very
sensitive to temperature changes and impurities, for instance through the size and shape of
the floats. Rigid floats should increase the stiffness over that of the fully fluid membrane if
they are flat. However, if they are bent they may in addition act as hats so that the effective
bending rigidity can, in principle, be raised or lowered by their presence.
Since practically equal rigidities were measured with fluctuation Diode analysis and electric
deformation in unshielded samples, any floats there would have to be much smaller than the
deflection length (. The absence of large floats appears consistent with the fact that the bendingrigidity was minimal in unshielded samples. In order to explain the dramatic increase of the
bending rigidity of DOPC bilayers with temperature, one could try to employ the hat model
of mixed membranes without invokinga superstructure. However~ hats created by impurity
molecules are not likely to change in concentration or cap curvature to the extent required for
a fivefold increase of the flexibility over a temperature interval of only 20 K.
5. Conclusion
The comparative measurements in samples with a silicone shield between glue and water have
shown that electric deformation yields larger values of the bilayer bending rigidity than fluc-
tuation mode analysis when the two methods are applied to the same DOPC orSOPC vesicle.
The average ratios between the two rigidity values are 2.5 for DOPC and 1.5 for SOPC. In
contrast, no dependence on method was noted m DOPC or POPC samples with direct contact
of glue and water, a situation in which the bending rigidities are minimal. We have discussed m
detail why a dependence of the bending rigidity on experimental method cannot be understood
m terms of existing theories, which points to the existence of a superstructure of the bilayers.The very strong effect of temperature on the bilayer bending rigidity, as found by fluctuation
mode analysis for DOPC vesicles, suggests an activation energy of 1 x10~~~ J. The associated
decrease of the rigidity by a factor of two between 18 to 25°C, I e. in the range of room
temperature, might explain part of the experimental scatter observed in measurements of the
bending rigidity.T'he emphasis of the present study is on the comparative measurements and the evidence
for a membrane superstructure which they seem to provide. The other observations were
N°3 DEPENDENCE OF BENDING RIGIDITY ON METHOD 423
included because they too are difficult to interpret in terms of theories of mixed membranes.
We have introduced the concept of floats in an attempt to explain the dependence of the
bending rigidity on experimental method. The cooperative saddles underlying our model ofthe superstructure, when they are fully ordered, should form a pattern similar to an egg carton
instead of uncorrelated floats [13,28]
Appendix A
Let us first briefly review the hat model [13,14]. Instead of some set of nonlocal modes, the hat
model employs local bending modes to describe the bending fluctuations of fluid membranes.
Each molecule in a monolayer, or pair of opposite molecules in a bilayer, is thought to be a
spherical cap surrounded by a smoothly attached brim of zero total curvature, I.e. ci +c2=
0.
Local bending fluctuations are produced by the fluctuations of the cap curvatures and by the
diffusion of molecules with different spontaneous cap curvatures.
The bending fluctuations of a cap obey the equipartition theorem in the form
~~ ~~~ ~°~~~bend~° ~~ ~~'~~
Here J= ci + c2 is the total curvature, Jo is the spontaneous curvature of the spherical cap,
vanishing for the symmetric bilayer, and Ao is the area of the molecular cross section. We
omit the Gaussian curvature, assuming its elastic modulus to be the same for all surfactant
molecules. The effect of a nonuniform modulus of Gaussian curvature on hat formation has
been briefly discussed elsewhere [14].Solving (A.1) for the flexibility 1/~ and adding the fluctuations of the local curvature due
to diffusion, we obtain the effective flexibility
efl'~ ~~
°~~~bend~ ~~~ ~°~~~difl' j~) ~~'~~
According to this equation, the flexibility consists of an intrinsic part which is related to the
bending fluctuations of the caps and adiffusive part which can enter only in the case of mixed
membranes. The spontaneous curvature Jo is the average of the molecular spontaneous curva-
tures. The intrinsic part is an average over the molecular flexibilities and readily generalized
to the case of different molecular areas. The diffusive part is easy to handle only for random
mixing of molecules of equal area Ao. It is a basic assumption of the hat model that the super-
position principle applies to the normal displacements associated with all the hats. The hat
model and continuum theory [22, 23] predict the same bending rigidities for mixed membranes.
Next, we want to calculate the extra area1hAh stored by a single hat in an otherwise flat
membrane. Introducing the radius ro and the boundary angle po "(J/2)ro of a spherical cap
of total curvature J, we may write, for p( < 1,
1 R ~2l~Ah
"~@( 27Tr dr (A.3)2
~r
Herer is the distance from the center of the hat and arR2 the area of the membrane imagined
to be a circle of radius R. Integration of (A.3) yields
/lAh"
w(arr(In(R/ro) (AA)
For a mixed membrane with Jo=
0, equation (A.2) may be rewritten in the form
424 JOURNAL DE PHYSIQUE II N°3
=
~~~~~~ (A.5)~efl kT
where lpi) combines the contributions of cap bending and cap diffusion to the mean square
boundary angle. Inserting (A.5) and Ao=
Jrr( in (All and squaring the argument in the
logarithm lead to
/hAh kT~~~~~ ~
~° ~~~~~~~°~ (A.6)
In the presence of lateral tension, R has to be replaced by the deflection length ( (or,more
correctly, by ( times an unknown numerical factor near unity). Subtracting 1hAh without
tension from 1hAh with tension, one immediately recovers from (A.6) and (3) the logarithmicdependence of the apparent area dilation on lateral tension as given by (2).
Equation (A.6) holds only if the lateral tension does not deform the spherical cap of the hat,
I.e. for ( » ro. This condition was easily satisfied in the experiments because the deflection
length ( is of the order of 3 ~m for ~ =10~~~ J and ~ =
10~5 mN/m. The diffusive contribution
to the flexibility, I.e. the flexibility of mixed membranes, has so far been considered for uniform
curvatures. It seems obvious that the results are valid for a periodic deformation only if its
wavelength is much larger than the average spacing of the molecules of each kind. The typicalvalues of the wavelength estimated above are about 4 pm. This is indeed much larger than
the 40 nm spacing of impurities as computed, e.g., from an impurity concentration of I% and
a lipid molecular cross section of 0.7 nm2.
Let us estimate the r-m-s, boundary angle in a pure one-component bilayer, putting ~ =
x10~~~ J and kT
=4 x
10~~~ J (room temperature). We obtain from (A.1)
Note that ro cancels out in the calculation. If impurity molecules are to double this flexibilityby inducing a local spontaneous curvature in the bilayer, they must have a boundary angle
po "10° at a relative concentration of10%,
as may be derived from (A.2) and (A.5). Note
that the square of the required curvature is inversely proportional to the relative impurityconcentration if the latter is much smaller than unity.
Acknowledgments
This work ~vas supported in part by the Deutsche Forschungsgemeinschaft through Sfb 312.
We thank E. Sackmann for sending us a copy of W. Hickl's Diplomarbeit.
References
iii Servuss R.M., Harbich W. and Helfrich W., Biochim Biophys. Acta 463 (1976) 900
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