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Correlations between the form fluctuation modes of flaccid quasispherical lipid vesicles and their role in the
calculation of the curvature elastic modulus of the vesicle membrane. Numerical results
I. Bivas, L. Bivolarski, M. Mitov, A. Derzhanski
To cite this version:
I. Bivas, L. Bivolarski, M. Mitov, A. Derzhanski. Correlations between the form fluctuation modes of flaccid quasispherical lipid vesicles and their role in the calculation of the curvature elastic modulus of the vesicle membrane. Numerical results. Journal de Physique II, EDP Sciences, 1992, 2 (7), pp.1423-1438. �10.1051/jp2:1992210�. �jpa-00247740�
Classification
Physics Abstracts
68.10 82.70 87.20
Correlations between the form fluctuation modes of flaccid
quasispherical lipid vesicles and their role in the calculation
of the curvature elastic modulus of the vesicle membrane.
Numerical results
1. Bivas (I), L. Bivolarski (~), M. D. Mitov (') and A. Derzhanski (1)
(1) Institute of Solid State Physics, Bulgarian Academy of Sciences, Laboratory of Liquid Crystals, 72 Trakia blvd., Sofia1784, Bulgaria
(2) Department of Informatics, Space Research Institute, Bulgarian Academy of Sciences, acad.
G. Bontchev str., block 3, Sofia II13, Bulgaria
(Received J4 May J99J, revised J April J992, accepted J4 April J992)
Rdsum4. Dans cet article nous montrons que [es thdories existantes des fluctuations de la forme d'une vdsicule lipidique quasisphdrique donnent des valeurs trop dlevdes pour l'dnergie moyenne
de dilatation de sa membrane. Une thdorie propre, qui tient compte de la conservation de la surface de la membrane, est proposde. La relation entre [es amplitudes des fluctuations de la fornJe et le module d'dlasticitd de courbure de la membrane est obtenue numdrjquement au
moyen d'une simulation Monte Carlo convenable.
Abstract. It is shown in this paper that the existing theories of the form fluctuations of
quasispherical lipid vesicle give a too high mean stretching energy of its membrane. A more
precise theory taking into account the area conservation of the membrane is proposed. The relation between the amplitudes of the form fluctuations and the curvature elastic modulus of the
membrane is obtained numerically by means of a suitable Monte Carlo simulation.
1. Introduction.
The amplitudes of the therrnal forrn fluctuations of quasispherical (flaccid) lipid vesicles are govemed by the membrane curvature elastic modulus k~ introduced by Helfrich [I]. This is
the reason why they are used for the deterrnination of this modulus [2-12]. There exist
theories perrnitting the calculation of k~ after measuring the fluctuations of the vesicles.
According to them the fluctuations are decomposed in a series of spherical harmonics, and the
amplitudes of the different modes are considered. In the first theories proposed by Engelhard
et al. [2] and by Schneider et al. [3] the fluctuating modes were assumed to be independent,
and the influence of the membrane tension was disregarded. Milner and Safran [5], assuming again the independence of the fluctuations of the different modes, took into account the
presence of this tension. Their theory explains very well the experimental data [6]. In the present work we discuss the validity of the assumption of independent fluctuating modes. We show that one consequence of such an assumption is an unacceptably high mean energy of
stretching of the membrane and propose a theory that avoids this discrepancy through imposing the condition that the surface of the membrane does not fluctuate. As a result of this restriction the different modes are no more independent. A suitable Monte Carlo simulation js proposed permitting us to find numerically the relation between the curvature elastic
modulus and the mean squared values of the amplitudes of the different modes. The results
are compared to those obtained by the existing theory [5]. It is shown that one of tile consequences of the area conservation is the deviation of the distribution of the amplitude lengths from the Gaussian one. A discussion is given concerning the cases when it is worth
using our results.
2. Theory.
We consider a flaccid quasispherical lipid vesicle with a volume V and a surface of the membrane S. Let XYZ be a laboratory frame of reference whose origin 0 is inside the vesicle.
The exact position of 0 will be specified later on. We choose Ro to be a radius of a sphere with
the same center 0 and with a volume V equal to that of the vesicle. The fluctuations of the vesicle can be described in the following way [2, 3]. Let the direction specified by the polar angles (R, ~n in a given moment t pierce the surface of the membrane at point A and the
surface of the sphere at point B. We denote R( o, ~n, t)
= OA and Ro(o, ~n = OB. We define
a function u(o, ~n, t) with the property :
R(o, w, t)
= Ro(o, q~)ii + a(o, w, t)i (1)
and use the spherical harmonics Yf(o, q~) as defined in [13] :
Yr(°, 9~)
" (2 7T)~~~~ e~~~~ Pr(°) (2)
where Pf(o) are the normalized associated Legendre functions. The function u(o, q~, t) can be decomposed in a series with respect to the spherical harmonics in the following way :
nm~ n
a(o, ~, t)
= z z a7(t) Y7(o, ~). (3)
n=0 m=-n
In equation (3) n~~~ is of the order of D/A, where D is the diameter of the vesicle and A is a typical distance between two neighboring lipid molecules of the membrane of the
vesicle.
To obtain some relations of the amplitudes uf(t) we use the following properties of the
fluctuating vesicle : a) we choose the origin 0 of the frame of reference in a way assuring that the quantities uf(t) are equal to zero ; b) the quantity u(R, q~, t) is a real one c) during the
fluctuations the volume V of tile vesicle remains constant. As a result :
uf(t)
=
0
iUr(t)i*
# Ui~(t) (4)
U~(t)
=
~f f (Ul(t)(~
2$n=2m=-n
The last of the equations (4) is obtained with a second order of accuracy with respect to the
amplitudes uf(t). It is exactly this equation which takes into account the constancy of the volume during the fluctuations. These properties have been discussed in details elsewhere [5].
To calculate the curvature energy E~ of the membrane of the vesicle we use the following expression for the density g~ of this energy [ii :
~c ~ ~c(Cl + C2 C0)~ + ~cCl C2 (~)
where ci and c~ are the principal curvatures of the membrane at tile point of determination of g~, co is the spontaneous curvature (equal to zero in the case of symmetric membrane
considered by us), k~ and i~ are the curvature elastic modulus, and the saddle splay curvature elastic modulus of the membrane. It is well known that k~ gives a constant curvature energy
contribution for a closed surface, which will not be considered by us.
Let S and So be the actual area of the membrane and the area of the introduced above
sphere w1tll a volume V equal to that of the vesicle. With AS we denote the excess surface of the vesicle, equal to the difference S So. The quantities E~ and AS can be expressed via the
amplitudes uf(t) as follows :
Ec(t)
= kc I I (n i) n(n + i )(n + 2) (l£f(t)(~ (6)
j Am~ n
li$(t)
# (Ro)~ i I (n I )(n + 2) (Uf(t)(~ (7)
~
n=2m=-n
Equations (6) and (7) have been obtained by Helfrich [14] and by MiIner and Safran [5].
In the theories developed up to now a vesicle having a tension
« of its membrane has been
considered and the amplitudes uf(t) of the fluctuating modes have been assumed to be
independent (not correlated). Formally each mode has been considered as an independent
oscillator. Under these assumptions the following mean values am (t) (~ of the squares of the modules of the amplitudes uf(t) have been calculated for n m 2 [5]
m j2
kT
~~ ~
kc (n I )(n + 2) la + n(n + I)1
wh~~~ ~
«(R~)2
k~
3. Development of the theory.
We will show that the assumption about the independence of the fluctuations of the different modes leads to an impossibly high stretching energy. For simplicity we will consider the case
« =
0. From (7) and (8) we obtain :
~ ~~~
(9)
&$ (t
= ~~~ )
~i~~~
l)
When the form of the vesicle fluctuates the area S of its membrane fluctuates too. The
mean value @ of the area is So +@. The
mean square of the fluctuations
iAs(t)12
= is(t) @12 is :
iAs(t)12
t ~~~ )
)~ f ~ + '
~. (io)
To obtain the result (10) we used that the modulus (uf(t)( of the amplitude uf(t) has a Gaussian distribution around zero. This distribution follows from the assumption that each mode can be considered as an independent oscillator.
With E~ we denote the energy of stretching of the membrane. It can be expressed via the
area S(t) and the area S( of the unstretched membrane :
E~(t)
= k~
~~~~~ ~~~~
2 S( (ll)
where k~ is the stretching elastic modulus of the membrane. For the case «=0,
S( is equal to @. The mean value @ of the stretching elastic energy is :
@
= k~
~~~~~~~
= k~ ~~~~~~~~
(12)
2 So 2 So
From (10) and (12) :
E~(t)
= k~
~~°~ ~~ ~ f ~ ~ ~
16 k~ (13)
~~
n~(n + 1)~
We can estimate the magnitude of E~(t). For this aim, the following values of the quantities taking part in (13) are used : Ro
= lo ~Lm ; k~ = 140 dyn/cm k~ = 10~ ~~ erg
kT
= 4 x10~~~ erg if the diameter D of the vesicle is equal to 2Ro = 20 ~Lm and the intermolecular distance A is of the order of lo I then
n~~~ =
D/A
=
2 x 10~. The numerical result for @ is :
E~(t) = 10~ kT. (14)
Obviously, one degree of freedom (the area of the vesicle) cannot contain such an enormous energy of its fluctuations. Consequently the assumption that the various modes have the
behavior of uncorrelated oscillators cannot be correct. The estimate (14) is valid for a = 0 but this is not an essential restriction. For each if having an acceptable value the
quantity @ will remain much greater than kT (@
= 10~ kT for a
= loo, which is much greater than the experimentally observed values of this quantity [6]).
To avoid the very high value of the mean energy of stretching we propose the following
model for description of the fluctuations. We divide all the modes into two groups. The first of them includes all the modes with high enough index n, whose independent (not correlated)
fluctuations require a mean energy of stretching not greater than kT for all the modes of this group. We assume that in this group the modes fluctuate independently. The second group includes all the other modes. On the modes of the second group we impose the requirement
that their fluctuations should be correlated in such a way that the area of the membrane remains constant. In this way we assure that the mean energy of stretching does not exceed kT.
We determine no with the property all the modes af(t) with n in the interval
no ~n ~n~~~, if they are independent, to require a mean energy of stretching equal to kT, no plays the role of « cut off » for the variable n. The number no defined in this way must
satisfy the condition :
k~
~~°~ ~~ ~ f ( ~ ~
~ =
kT. (15)
16 kc =~~
n (n + 1)
To obtain no we consider n as the continuous variable, replace ~j with ldn, and put
n~~ = m. The result is :
Ii k~ kT (R~)~ 1/2
no =
~
(l6)
' 6 " (k~
If we use the numerical values, preceding equation(16) we obtain the estimation
no = 300. Strictly speaking this value depends on if too but this dependence is very weak.
Later on we shall disregard it.
The condition imposed on the amplitudes af(t) with nw no can be presented in the
following form :
~~~ ij f
(n I )(n + 2)(U~(t)(~
= h$o (17)
n=2 m=-n
where ASO is a parameter that depends on the experimental conditions of formation of the vesicle. ASO in our model corresponds to that part of the excess area of the vesicle, which is
due to the fluctuations of the amplitudes of the spherical harmonics with nwno.
ASO does not depend on time.
The relation (17) determines a surface in the space of the modules of the amplitudes
(uf(t)(. In the model used by us the phases of the amplitudes do not take part in the expressions for the energy and for the excess surface of the vesicle. This is the reason why only
the modules of the amplitudes are considered. The partition function Z of the system,
expressed via the modules (uf(t)( is :
no n ~~s
8 ~j ~j (n-I)(n+2)(uf(~-
~ ~~ n=2m=-n
(Ro)
~"~ox d(~2 (d(~2 d~~no~ ~
~~ ~
~
2 ~j ~j (n 1)~ (n + 2)~ am
(~
n=2m=-n
jk~~f jj (n-I)n(n+I)(n+2)(af(~
xexp ~"~~"~~
~~ (18)
where Zo does not depend on (af(t)(. The integration over (af(t)( in (18) is carried out from 0 to m. Starting from this partition function we can calculate the mean values
(af(t)(~ of the quantities (uf(t) for different values of ASO. As we could not obtain an
analytical form of the integral (20) we calculated these values using a proper Monte Carlo simulation described later on.
As shown before, no
= 300. This great value should create some technical difficulties in the Monte Carlo simulation. On the other hand in tile performed experiments up to now, the maximum value of n for which the amplitudes UC(t) ~ have been measured does not exceed 20. We show that the replacement of no with 20 changes negligibly the values of
am (t) ~ for n w 20. At the restriction imposed by us for constancy of the area of the vesicle membrane, the amplitude (af(t) of a given mode with n w 20 is exactly determined if the
amplitudes of all the other modes are given. If only the amplitudes of the modes with
n w 20 are given, then the amplitude of the mode under consideration will fluctuate around
some mean value with a magnitude, determined by the fluctuations of the vesicle's are due to the fluctuations of the modes with n between 21 and no. If this magnitude is much less than the fluctuation of the mode under consideration, then the error in the deterrnination of the mean square of this mode is negligible. We estimate quantitatively the error in the mean squared amplitudes am (t) ~ due to the substitution of no with some other value n(, n( w no. Evidently, ASO can be presented in the forrn :
Aso
= Asj(t) + Asj'(t) (19)
where AS( is the excess area due to the fluctuations of the amplitudes of the spherical
harrnonics with 2wnwn(, while ASI'- the same quantity for n(~nw no. For fixed
ASO, AS( and ASS will fluctuate around some mean values @ and @. One estimate of these
fluctuations can be obtained if the fluctuation modes with n in the interval n( ~ n w no are
assumed to be independent and if is assumed to be zero. We denote 85((t) = AS((t) @
and 8Sl'(t)
= ASI'(t) @. From (19) it follows that [85((t)]~
= [8Sl'(t)]~. Under the just
mentioned assumption [8Sl'(t)]~ can be calculated by an expression, similar to lo), namely :
~~~°~~~~~
~
[~~ 1 ~
~
Ii,
n
Iii )~ ~~°~
The fluctuations of Sl'(t) are of the order of [8Sl'(t)]~)~~~ The value of @ is :
&sj(t) = (21)
~~~ ) ~~/~ )
Let Uf(AS() be the quantity (af(t)(~ calculated when AS( is fixed, and Vf(ASO) be the
quantity (af(t)(~ calculated when ASO is fixed. The following relation between these
quantities exists :
Vr(hS0)
~
Ur(@) + (@)~ ~ ~~ ~~)~ ~~~~#~~~
+ (22)
~~~'l' ~~') ' ~
Using that (AS(-@)~= [8Sl'(t)]~
we can estimate the error of the correct value
Vf(ASO) when AS( is given instead of ASO. For n(
= 20 (the value used by us in the Monte Carlo simulation) and no = 300 the numerical value of the expression in tile parentlleses in
(22) is estimated to be 2 x lo ~ [this value does not exceed 5 x 10~~ for all the values of a in the interval (- 5,25)]. After the presentation of the results of the simulations, we will show that the second derivative on the r.h,s, of (22), multiplied by (@)~, is smaller than 1,
2
that will prove that the quantities Vf are precise enough when n(
= 20 is used instead of no.
To facilitate the simulations we change the variables that take part in (20) in the following
way. For each n of the interval 2 w n w n( and for each m of the interval n w m w n we
choose functions ff(A[,A(,.,,A(~), each function ff having 2n arguments A~~,
I
= 1, 2,
,
2 n, with the properties :
n
~j ff(A /))~
=
l for each A/, 2 w n w n(, I
= 1, 2,
,
2 n (23)
m -n
Then we introduce the variables w~.
ff(A (, A(,
...,
A(~)
~~~~ ~/(n i )(n
+ 2) ~'~~'
~ ~ ~ ~ ~~ ~ ~'~ ~ ~' ~~~~
The determinant of transformation D
=
~~~~ ~~
can be presented in the
a(w~,
/, ~.
,
A(~)
form :
D
#
ij ((Wn)~~ Y'n(Al'Al'
'
Aln)) (25)
where the functions P~(A [, A(,
,
A(~) do not depend on w~. In the statistical sum (18) we
change the variables according to (24) and no with n( and obtain :
l~' ~ 2ASo
Z
= Z ( dw~dw~ dw~, 8 ~j (w~) x exp x
~
n 2 (R0)~
ni ni nb
k~ ~j n (n + I) (w~)~ + kT In £ (n I) (n + 2) (w~)~ 2 kT ~j (n In (w~))
~
n 2
~
n 2 n 2
~ kT
(26) where Z( does not depend on w~. Changing once again the variables :
k~ 1/2
v~ = w~ (27)
kT from (26) and (27) we obtain :
~ ~ ,, ~~ ~~ ~~ ~
~°
~~ ~~
k~ 2 ASO l~
~ ~ ~ ~~
~2
~ kT (R0)~
x exp
(
n (n + I) (v~)~ I
In (
(n I) (n + 2 (v~)~ +
( (n In (v~))1(28)
n=2
~
n=2 n=2
where Zl' does not depend on v~. Forrnally the expression that is the argument of the
exponent can be considered as a generalized Hamiltonian of the system and the Boltzmann
