Thermal shape fluctuations of a quasi spherical lipid vesicle when the mutual displacements of its monolayers are taken into account

Colloids and Surfaces A: Physicochemical and Engineering Aspects 157 (1999) 21 – 33 www.elsevier.nl/locate/colsurfa

Thermal shape fluctuations of a quasi spherical lipid vesicle when the mutual displacements of its monolayers are taken into account I. Bivas a,*, P. Me´le´ard b, I. Mircheva a, P. Bothorel b a

Institute of Solid State Physics, Bulgarian Academy of Sciences, Laboratory of Liquid Crystals, 72 Tzarigradsko Chaussee bl6d., Sofia 1784, Bulgaria b Centre de Recherche Paul Pascal, CNRS, A6. du Dr. Albert Schweitzer, F-33600 Pessac, France Received 30 August 1997; accepted 10 March 1999

Abstract In most of the previous theories dealing with shape thermal fluctuations of quasi spherical vesicles, the membrane was considered as a homogeneous shell characterized by its bending moduli. Our description considers the fluctuations to be followed by a lateral redistribution of the molecules within the bilayer and by an intermonolayer friction. These phenomena change the expressions for both the mean square values and the dynamic behavior of the fluctuation mode amplitudes. The corrections are expressed through the bending moduli of blocked and free exchange of the molecules within the monolayers. Numerical estimations show that such effects can be observed when shape fluctuations of giant quasi spherical vesicles are analyzed. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Bending elasticity; Bilayer; Dynamics; Friction; Monolayer; Shape fluctuations; Stretching elasticity; Vesicle

1. Introduction In the present article, we consider only lipid bilayers in their liquid crystal state. This assumption means that their static shear viscosities are set to zero, the membrane mechanical properties being described by the energies needed for expan French-Bulgarian Laboratory ‘Vesicles and Membranes’ supported by the CNRS (France) and the Bulgarian Academy of Sciences (Bulgaria) * Corresponding author. Tel.: + 2-359-27431, ext: 372; fax: + 2-359-29753-632. E-mail address: [email protected] (I. Bivas)

sion and bend. Let S0 be the area of a flat membrane in its tension free state s = 0, where s is the membrane tension. If S0 is increased by DS, the tension is different from zero: s(DS)= ks

DS , S0

(1)

where ks is the stretching modulus of the membrane [1]. According to Helfrich [1,2], the surface density of the bending energy gc is given by the expression:

0927-7757/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 ( 9 9 ) 0 0 1 1 3 - 2

I. Bi6as et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 157 (1999) 21–33

22

gc =

kc (c1 +c2 −c0)2 +k( cc1c2. 2

(2)

c1 and c2 are the principal curvatures of the bilayer at the point where gc is determined, kc is the bending elasticity and kc the saddle splay elasticity. c0, the spontaneous curvature, is introduced to take into account the membrane asymmetry. The vesicle membrane is a closed surface having the topology of a sphere. Consequently, the integration of the Gaussian curvature c1c2 is a constant (Gauss Bonnet theorem):

7

c1c2ds =4p

(3)

tions through which the spherical harmonics Ym n (u, 8) are defined as: 1 Ym cos(m8)P m when mB 0, n (u, 8)= n (u),

p Ym n (u, 8)= Ym n (u, 8)=

2p 1

p

nmax

Clearly, the saddle splay elasticity kc does not influence the shape fluctuations of a quasi spherical vesicle. The bending modulus kc depends on the mechanisms controlling the molecular displacements in the bilayer [2]. When the exchange of molecules between the two monolayers of the bilayer is free, the bending modulus is k fr c while, in the opposite case (blocked exchange), it is k bl c . We note that the bl inequality k fr c Bk c is always fulfilled, Helfrich’s bl estimation giving k fr c :0.25 ×k c [2]. kc can be determined from the thermal fluctuation analysis of quasi spherical vesicles [3–14]. The theory used herein can be summarized in the following way. Consider a flaccid quasi spherical vesicle characterized by a volume V and its membrane area S. Let XYZ be a laboratory reference frame, the origin O being inside the vesicle (its exact position will be defined later on). Looking at the fluctuating vesicle by a 3-dimensional technique, we define R(u, 8, t) as the modulus of the radius-vector of a point on the surface of the vesicle in the direction (u, 8) (spherical coordinates) at a time t: (4)

u(u, 8, t) being the normalized function describing the shape fluctuations and R0 the radius of a sphere with a volume V. Let P m n (u), n ] m =0, be the normalized Legendre func-

P 0n,

when m=0,

sin(m8)P m n (u),

when m\ 0. (5)

It can be readily demonstrated that the spherical harmonics are orthonormal real functions, so that u(u, 8, t) can be developed in a series of Y m n:

S

R(u, 8, t)= R0[1 +u(u, 8, t)],

1

u(u, 8, t)= %

m % um n (t) · Y n (u, 8).

(6)

n = 0 m  n

In Eq. (6), nmax : 2R0/l, where l is a typical distance between two neighboring molecules. The position of the origin O is then chosen to ensure m um 1 (t)= 0 [8]. As u(u, 8, t) and Y n (u, 8, t) are m real functions, the amplitudes u n (t) are real, i.e. equal to their complex conjugates [u m n (t)]*. One of the main hypotheses used both in the previous theoretical treatments and in this work, is that the vesicle volume V is invariant under shape fluctuations. As a consequence, u 00(t) can be expressed, to a second order of accuracy, by the sum of 2 [u m n (t)] , n] 2. To summarize theses properties: um 1 (t)= 0 m [u m n (t)]*=u n (t)

u 00(t)= −

1

nmax

%

n

%

2 p n = 2 m = − n

2 [u m n (t)] .

(7)

If we define Ec(t) as the membrane bending energy, the total energy E(t) of the vesicle can be written as E(t) =Ec(t)+ sS(t), where S(t) is the area of the bilayer. Using the expressions published by Helfrich and by Milner and Safran for the total energy, E HMS(t), and for the time aver2 HMS with age of the square amplitudes, [u m n (t)]  n] 2 (the indices HMS denote that the expressions are deduced by Helfrich and by Milner and Safran [5,7]), we obtain:

I. Bi6as et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 157 (1999) 21–33

kc nmax % % (n − 1)(n + 2) 2 n = 2 m 5 n (c0R0)2 2 n(n+1)+s¯ +2c0R0+ ×[u m n (t)] , 2 kT 2 HMS = [u m n (t)]  ke E HMS(t)=



n

Z(n)= (8)

1 , (9) (n−1)(n+2)[n(n+1)+s¯ +2c0R0+(c0R0)2/2]

×

where k is the Boltzmann constant and T the absolute temperature. s¯ is the reduced membrane tension defined as: s¯ =

s(R0)2 . kc

(10)

In Eq. (9), the membrane tension s is assumed to be time independent. Note also that the bending modulus extracted from the thermal fluctuation analysis corresponds to the free exchange state, k fr c [6,15]. This follows from the fact that the relative slipping of the monolayers is equivalent to the exchange of molecules between them in the limit of small enough fluctuations, compared with the radius of the vesicle. Later on we will confirm this conclusion. The theory describing the dynamics of a fluctuating vesicle was developed by Schneider et al. [3] and by Milner and Safran [7]. They assumed that the vesicle membrane is characterized by a single bending modulus, kc. In the case of a vesicle with a spontaneous curvature c0, their results can be written: MS m% u m n (t)u n% (t+ t) 2 HMS = dnn%dmm%[u m exp( − v MS n (t)]  n t),

(11)

where the upper index MS means the result of Milner and Safran [7]. Using our notations, the is therefore: expression for the frequency v MS n v MS n =

kc h(R0)3

(n−1)(n+2)[n(n+1)+s¯ +2c0R0+(c0R0)2/2] × , Z(n) (12) with

(2n + 1)(2n 2 + 2n+ 1) , n(n+1)

23

(13)

n being the viscosity of the liquid surrounding the membrane (generally aqueous solutions). In several previous works, it was remarked that bending rigidity and out-of-plane membrane fluctuations are influenced by intermonolayer slipping [16–21]. As for the dynamics of the fluctuations, the authors of these works underline that the mutual displacement of the monolayers is much more pronounced for the modes with high enough wave vectors. Seifert and Langer [19] developed a quantitative theory describing the out-of-plane fluctuations of a flat membrane, taking into account the intermonolayer friction as well as the two-dimensional membrane viscosity. Yeung and Evans [21] obtained explicit relations for the fluctuations of the form of a quasi spherical lipid vesicle, influenced by the mutual displacements of the monolayers. comprising its bilayer, for arbitrary values of the fluctuation wave vector. In this work, we present a theory describing the shape fluctuations of a quasi spherical lipid vesicle when the displacements of the two monolayers are allowed. We compare our results with those of the preceding theories of this kind. Numerical estimations are presented, showing that intermonolayer friction may influence shape fluctuations of giant vesicles with a radius of about 10 mm. 2. Bending energy of a fluctuating lipid vesicle when the lateral displacements of its monolayers are taken into account In what follows, we will only consider bilayers whose elastic (bending plus stretching) energy is minimal when they are flat and when the molecular densities of their two monolayers are equal. Let n out and n in be the molecular surface densities of the outer and inner monolayers at some point of the deformed bilayer, dn being the difference: dn= n out − n in.

(14)

When the exchange of molecules between the monolayer (flip-flop) is allowed, the flip-flop coefficient j can be defined through the relation between dn and c1 + c2 [2]:

I. Bi6as et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 157 (1999) 21–33

24

dn=j(c1 +c2),

(15)

where c1 and c2 are the principal curvatures at the same point. Strictly speaking, j depends on the membrane tension s but this dependence will be disregarded in the present work. When Eq. (15) holds, the bending energy density is given by: 1 gc[(c1, c2, dn =j(c1 +c2))] = k fr (c +c2)2 2 c 1 +k( cc1c2,

(16)

where the saddle splay modulus k( c does not depend on the bilayer slipping conditions [22]. In the case dn " j(c1 +c2), gc has to increase. Following Seifert [23], but using a slightly different formalism, we express gc in the following way: gc(c1, c2, dn) z k fr c = (c1+c2)2+ [dn−j(c1+c2)]2+k( cc1c2. 2 2

k bl c (c +c2)2 +k( cc1c2. 2 1

(17)

(18)

fr k bl c −k c . 2 j

(19)

For a given dn, we introduce the quantity c˜0 as the induced spontaneous curvature:



c˜0 = 1−



dn k fr c . k bl j c

(20)

The expression for the bending energy density gc can then be presented as:



7

dn(u, 8, t)ds,

(22)

S(t)

where S(t) is the vesicle area and dN does not depend on the time t. Consequently, the derivative



n

We represent c˜0(u, 8, t) as a series of the spherical harmonics: c˜0(u, 8, t) =



1 ds0 R0 ds nmax

c 00 · Y 00(u, 8)+ %

n

n

m % 6m n (t) · Y n (u, 8) ,

n=1 m=−n

From Eq. (17) and Eq. (18), it follows that: z=

dN=N out − N in =

d S(t) c˜0(u, 8, t)ds /dt is equal to zero.

It is clear from Eq. (17) that the condition dn =0 corresponds to a forbidden molecular exchange between the two monolayers. gc can then be expressed as: gc(c1, c2, dn = 0) =

Let the dependence dn(u, 8, t) be known for a given quasi spherical vesicle. Then the function c˜0(u, 8, t) is defined through Eq. (20). This assumption is justified by the fact that in most of the experiments, related to the form fluctuations of the vesicles, the time of doing the experiment (: 10 min) is much less than the characteristic time of the flip-flop for a lipid bilayer (: 10 h). Assume also that the numbers of lipids contained in the inner and outer monolayers, respectively N in and N out, are fixed. Under these conditions, we can write:

(23)

where ds0 = R0sinudud8 and ds is the vesicle surface element in the solid angle sinudud8. c 00 is time independent because of the fixed molecule number in both monolayers, Eq. (20) and Eq. (22). On the contrary, the amplitudes 6 m n (t) can take arbitrary values without breaking this constraint. The bending energy of the whole vesicle, Ec(t), is written in the form: Ec(t)=

7

gc(c1, c2, dn)ds.

(24)

S(t)

The total energy E(t) of the vesicle membrane remains the same as in the preceding theories:



gc = k bl k bl k bl c c (c1+c2−c˜0)2+ c −1 c˜ 20+k( cc1c2. fr 2 2 k bl c −k c

E(t)=Ec (t)+ s (21)

We conclude that c˜0 is the curvature taken by a membrane deformed as a cylinder with a minimal bending energy for fixed dn.

7

ds.

(25)

S(t)

Remember that s is assumed to be constant. Strictly speaking, this is only an approximation [11], but the corrections are small enough to be ignored further on in this paper.

I. Bi6as et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 157 (1999) 21–33

The total curvature c1 +c2 and the surface element ds can be expressed by using u(u, 8, t), 9u and 92u [7,10], where 9 is the two-dimensional gradient and 92 is the two-dimensional Laplace operator. The obtained expressions are substituted in Eq. (21), Eq. (24) and Eq. (25) using the spherical harmonics decomposition Eq. (6) and Eq. (23). Keeping only the terms up to the secm ond order in u m n , 6 n and their products, we express the energy E(t) in the form (see Appendix A): E(t)



=4pk bl c 2−

1

p

c 00 +

n

1 k bl c (c 00)2 fr 8p k bl c −k c

e=





× 2

+

2 + g[6 m n (t)] }

+ %

g[6 (t)] +4pk( c, m 1

2

2 This equation for [u m n (t)]  is in agreement with the result of Yeung and Evans [21], obtained by using the force method. Comparing Eq. (28) and Eq. (9), we conclude that these equations differ formally by e, this correction appearing when the mutual displacements of the two monolayers are taken into account. Let (c 00)eq be defined by the relation:

(26)

where the coefficients an, bn, and g are defined as follows: an =(n−1)(n +2) s(R0)2 2 0 1 k bl c n(n+ 1)+ + c − (c 00)2 0 fr k bl 8p k bl c c −k c

p +

g=

n

1 k bl c (c 00)2, fr p k bl c −k c

bn=(n−1)(n+2)+

k bl c (c 00), fr k −k

p c 1

bl c

k bl c bl k c − k fr c

(27)

Using Eq. (26), the mean square values 2 [u m n (t)]  can be calculated (see Appendix A): 2 [u m n (t)] =

=

kT 1 2 k bl a −b c n n/g

kT 1 , 2 fr k fr (n−1)(n+2)[n(n+1)+s(R c 0) /k c +e]

where the expression for e is:

fr k bl c −kc . bl kc

(30)

e[(c 00)eq]= 0.

m= −1



(29)

This is the c 00 value minimizing the bracket expression in Eq. (26). We can say that (c 00)eq is the equilibrium value, minimizing the bending energy with respect to c 00. It is evident that:

n

k m % % {a [u m(t)]2 −2bnu m n (t)6 n (t) 2 n=2 m= −n n n

1

n

fr k bl 1 c −kc + c 00 . bl kc 2 p

2

bl nmax c

n

2 fr 1 (k bl k bl 1 c ) c −kc 2 − c 00 fr bl fr bl 2 k c (k c − k c ) kc 2 p

(c 00)eq = 4 p

+4p(R0) s

25

(31)

We conclude that k fr c is truly determined by using the shape fluctuation analysis of a quasi spherical vesicle. Really, after replacing s¯ with s¯ %= s¯ + e, we obtain Eq. (9) with kc = k fr c. 2 The value of [6 m n (t)]  can also be calculated from Eq. (26). The result obtained in Appendix A is: 2 [6 m n (t)] =

kT 1 × , k bl g c

2 [6 m n (t)] =

kT 1 × , 2 k bl g−b c n/an

when n= 1, when n\1.

(32)

As can be seen from Eq. (26) and Eq. (32), 6 m 1 (t) does not depend on the shape fluctuations of the vesicle. So we have to consider only the amplim tudes u m n (t) and 6 n (t) for n] 2.

3. Dynamics of the vesicle shape fluctuations in the presence of intermonolayer displacements (28)

The linearized equations describing the dynamm ics of the amplitudes u m n (t) and 6 n (t) will be deduced in this section. The starting point is Eq.

26

I. Bi6as et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 157 (1999) 21–33

(26) and the related ones, Eq. (A.6) and Eq. (A.8) from Appendix A. According to Schneider et al. [3] and Milner and Safran [7], the modes u m n (t) are not correlated (see Eq. (11)). We note that this assumption is independent from the assumption that a not fluctuating tension s exists, permitting us to express the total energy of the membrane in the form of Eq. (25). When a vesicle bilayer fluctuates, viscous forces corresponding to different motions can be defined, including the motions of the inner and outer volumes and the lateral and normal displacements of the two monolayers. To obtain explicit, although approximate, solutions, we introduce three simplifying assumptions (not disturbing the main properties of the system): m 1. the inequalities u m n (t)  1 and 6 n (t)  1 are fulfilled; 2. the bilayer stretching modulus ks, Eq. (1), is high enough to neglect any surface density fluctuation within the lipid bilayer; 3. the main viscous forces in our model originate from two phenomena: (i) viscous friction, resulting from the membrane normal motions, between the volume elements of the liquid medium inside and outside the membrane, as they are discussed by Schneider et al. [3] and by Milner and Safran [7], and (ii) intermonolayer friction. The third assumption implies that we are neglecting the viscous forces of the liquid inside and outside the vesicle as a result of the tangential displacements of the monolayers and the two-dimensional viscosity within the monolayers (it has to influence very high wave vectors only [19]). Giving u(u, 8, t) with a known c˜0(u, 8, t), we will demonstrate that we obtain the same restoring force as in the case of a forbidden intermonolayer displacement (i.e. when the intermonolayer friction is infinitely high). As a result of the induced restoring force, each membrane element ds will be translated after a time dt, keeping its area and its induced curvature c˜0. So let us consider now a reference state with zero fluctuation amplitude, u(u, 8)= 0 and a given c˜0(u, 8). This state is characterized by an ensemble of amplitudes (6 m n )0. Suppose a fluctuation u(u, 8) appears, keeping the local area and induced

curvature of the reference state and excluding any vesicle rotation (one necessary condition for this is that the initial reference state is restored when u returns to 0). Then this fluctuation has an ensemble of amplitudes u m n . Evidently, because of the lateral displacements of the membrane elements, a new ensemble of amplitudes 6 m n (u) will appear m with the property: 6 m n (u)= (6 n )0[1+ O(u)]. Using Eq. (26) valid to a second order of accuracy with m respect to u m n , 6 n , and their products, we deduce m that the energy of the reference state (u m n , 6 n ) will be the same as the energy of a vesicle with a m fluctuation (u m n , (6 n )0). Consequently, the depending on the magnitude of this energy restoring elastic force for a vesicle with forbidden intermonolayer displacements is the same as in the case when the ensemble of amplitudes 6 m n (u, 8) is fixed. Assuming also the mutual independence of the fluctuation modes as in Eq. (11), we can limit our next considerations to the analysis of only m one fluctuation u m n (t), 6 n (t) with fixed n, m: n]2, − n5 m5 n. To obtain the dynamic equation for u m n (t), we use the results of Milner and Safran [7]. Calling (E HMS)m n (t) the contribution to the vesicle total energy E HMS(t) in Eq. (8) depending only on the amplitude u m n (t),we can write: (E HMS)m n (t)=

kc HMS m 2 a [u n (t)] , 2 n

(33)

where



= (n− 1)(n + 2) a HMS n s(R0)2 (c R )2 × n(n+1)+ +2c0R0+ 0 0 . kc 2

n

(34)

Relating Eq. (12) to Eq. (34), we conclude that can also be written: the frequency v MS n v MS n =

kc a HMS n × , h(R0)3 Z(n)

(35)

with Z(n) from Eq. (13). Now compare Eq. (A.8) (obtained from Eq. (26), Appendix A) with Eq. (33). If 6 m n is time independent, both equations have the same form. In this case, a new frequency vm n can be defined:

I. Bi6as et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 157 (1999) 21–33

vm n =

k bl an c , × 3 h(R0) Z(n)

(36)

where Z(n) and an are taken from Eq. (13) and Eq. (27), respectively. This frequency is related to the time auto correlation function of u m n (t)− bn6 m n /an in the following way:

#

um n (t)−

bn m 6n an



um n (t + t) −

bn m 6n an

$

 exp(−v m n t).

(37)

The relation Eq. (37) is a solution of the first order time derivative equation:



n

bn m d m m 6 n =y m [u (t)]+ v m n u n (t) − n (t). dt n an

(38)

The function y m n (t) has the properties of a white noise, its amplitude being determined later on. Eq. (38) is the first of the dynamic equations m relating the amplitudes u m n and 6 n . Strictly speaking, Eq. (37) and Eq. (38) are valid only when m 6m n =0. But if this condition is not fulfilled, v n in Eq. (37) and Eq. (38) must be replaced by v m [1+ n O(6 m n )]. Because of the first assumption given at the beginning of this section, a dynamic equation of the type of Eq. (38) should be linear with m respect to the amplitudes u m n and 6 n , and the m corrections to v n can be omitted. Consider now the symmetrical case of a restoring force acting on the amplitude 6 m n when a given m fluctuation (u m (t), 6 (t)) appears. This force is the n n same as if the amplitudes u m are frozen (do not n depend on time). In this case, Eq. (A.6) from Appendix A shows that 6 m n will satisfy the following equation: (bn)2 m d m m (39) [6 n (t)]+ Vm u n =z m n 6 n (t) − n (t), g dt where the function z m n (t) has properties similar to m those of y m (t). The functions ym n n (t) and z n (t) are m not correlated. The frequency Vn depends on the amplitude u m n , the dependence being of the kind m Vm [1+O(u n n )]. As with Eq. (38), we are looking for the linearized form of Eq. (39), so this dependence can be neglected. Consequently, it is enough to determine the frequency Vm n in the case out in b b of u m =0. Let V (u, 8, t) and V (u, 8, t) be the n velocities of the outer and inner monolayers of the



n

27

(spherical) vesicle at the point (u, 8) at the moment t. We omit the indices n and m in the above velocities, keeping in mind that we are considering fluctuations of the mode (n, m). Then we can write: d d [dn(u, 8, t)]= [n out(u, 8, t)−n in(u, 8, t)] dt dt 1 1 = 0 div [Vb out(u, 8, t)−Vb in(u, 8, t)], (40) s0 R0 where div is the two dimensional divergence, defined on the unit sphere (with radius r= 1), and s 00 is the mean area per molecule in the monolayers. We introduce a length d0 to relate the flip-flop coefficient j from Eq. (15) to s 00: d0 j= 0. s0

(41)

m If a fluctuation 6 m n with u n = 0 appears, it will change in time, tending to its equilibrium value 6m n = 0. According to the third assumption at the beginning of this section, this relaxation is governed by the restoring force as a result of the energy stored in the fluctuation and by the viscous force as a result of the friction between the monolayers. This friction can be characterized by the friction coefficient bs. If two flat monolayers slip one over the other with a relative velocity Vb , then bs is the proportionality coefficient between the friction force per unit area, fb , and this relative velocity:

fb = − bsVb .

(42)

As shown in Appendix B, the frequency Vm n can then be expressed through the already introduced quantities: Vm n =

fr k bl c −kc n(n+ 1). bs(d0R0)2

(43)

This equation is one more complete form of the similar result, obtained by Yeung and Evans [21]. The system Eq. (38) and Eq. (39), together with the relations Eq. (28), Eq. (32), Eq. (36), and Eq. (43), lead to the auto correlation function f m n (t): m m fm n (t)= u n (t)u n (t+t).

(44)

I. Bi6as et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 157 (1999) 21–33

28

As it is shown in Appendix C, f m n (t) can be expressed as: fm n (t) 2 [u m n (t)]  0m = m [C m ˜m n exp(−V n t)+exp (−v n t)], (45) C n +1

where Cm n =

2 m 2 m m 0m ˜m v ˜m (V0 m n n ) −(Vn ) +V nv n −Vn v n × m2 . 2 m m m m m V0 n (Vn ) +Vn v n −V0 n v ˜ n −(v ˜m n)

(46) 9 iV0 m ˜m n and 9iv n (i is the imaginary unity), m m V0 n ] v ˜ n ] 0, are the four imaginary roots of the biquadratic equation;





m 2 m m v 4 + (Vm n +v n ) −2Vn v n 1 −



2 m 2 +(Vm 1− n ) (v n )

(bn)2 ang

n

(bn)2 ang

n

v2

2

=0.

(47)

(the proof that Eq. (47) has four purely imaginary roots follows from the condition of an energy minimum for E from Eq. (26) with respect to u m n and 6 m n ). Eq. (45) and Eq. (46) are the main result of our investigation concerning the fluctuating mode dynamics of a quasi spherical flaccid vesicle. They show that the auto correlation function f m n (t) contains two exponentially decaying terms. Such a form of the auto correlation function is obtained by Yeung and Evans [21]. Our results for the preexponentials and for the frequencies V0 m n and v ˜m of the auto correlation function are not identin cal with the results of these authors [21]. The differences are essential in the region of wave lengths (more precisely the values of the index n m m of the mode u m n ) for which Vn 5v n . The numerical values of the differences are considered in the following section. The reason for the differences between our results and these of Yeung and Evans [21] originates from the different mathematical treatment of the problem. These authors have made the assumption that (the paragraph after Eq. (24) in [21]) ‘…the shape-perturbing impulses imparted to the vesicle have durations that are much shorter than the characteristic relaxation times of a 9 …’. In our calculations we

have not imposed such unnecessary constraint on the spectrum of the random forces. In the following section, we will estimate C m n MS and the difference between v ˜m calcun and v n lated according Eq. (12) with kc = k fr c as they represent a measure of the effect of the intermonolaver displacement in the auto correlation function f m n (t).

4. Numerical estimations and discussion Let us estimate first the values of c 00, Eq. (23) and e, Eq. (29). This can be carried out starting from the relation that follows from Eq. (26) and the equipartition theorem (long enough time intervals are considered, so that the time dependence of c00 can no longer be neglected):

#

1

2 p

c 00 − 2

fr k bl c −kc bl kc

$ 2

=

fr kT(k bl c −kc ) . bl 2 4p(k c )

(48)

bl Following Helfrich [2], we assume now k fr c = k c /4. − 12 Taking k fr  10 erg and T 300 K, we obtain c  c 00  5.4 and  e  0.2. If the vesicle has a molecular density difference of its two monolayers close to its equilibrium value, then c 00 and e can be important for the dynamics of the fluctuation modes with n= 2. For vesicles far from equilibrium, this influence can spread over higher values of n, too. As a consequence, c 00 must be one of the adjustable parameters when the experimental data of the dynamics of the form fluctuations of quasi spherical vesicles are analyzed [8,10]. To estimate the influence of the intermonolayer displacement on the vesicle shape fluctuations, we m will start with the simpler case Vm n /v n  1. This inequality is fulfilled for long enough wave lengths of the fluctuations. Then from Eq. (47), we de2 m m duce V0 m and v ˜m n : Vn n : v n [1− (bn) /(ang)]. 0 0 eq m Moreover, if c 0 = (c 0) , Eq. (30), v ˜ n can be MS expressed through n, k fr , h, and R c 0 while v n , Eq. (12), is written using n, kc, h, and R0. The quantity C m n is then much less than unity and the auto correlation function f m n (t) is practically the same as Milner and Safran’s expression, Eq. (11). Another limiting case, appearing when n takes m high enough values, is Vm n /v n  1. Evidently, this is the description of the fluctuations with short

I. Bi6as et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 157 (1999) 21–33

wave lengths. In this case we obtain that V0 m n : 2 m m fr bl vm ˜m n, v n : Vn [1 −(bn) /(ang)] and C n :k c /(k c − fr k c ) (according to Yeung and Evans [21], in this case C m n should be equal to zero). Consequently, for short wave lengths the correlation function consists of two exponentially decaying terms with very different decrements, but with comparable preexponential factors. This result can be important for the explanation of the dynamics of the out-of-plane fluctuations of lamellar phases [20]. Numerical values of C m n are obtained using the − 12 following numerical values: k fr erg, k bl c =10 c = fr 4k c , h =1 cp, R0 =10 mm (a typical value of the radius of giant quasi spherical lipid vesicles whose fluctuations have been experimentally studied); for d0 we take the value of the bilayer thickness, i.e., d0 = 50 A, ; for bs, we use the experimental values of Merkel et al. [25] determined on the base of the model developed by Evans and Sackmann [24]; bs is between 2.7× 106 dyn s/cm3 and 107 dyn s/cm3 when the friction is measured for chains of different lengths in a disordered state. A typical accuracy of the experimental determination of the mean square amplitudes of the modes for n between 2 and 20 is 10% (for details see [8,10]). So the mutual displacement of the monolayers is not important if its influence is smaller than this accuracy, i.e. when C m n 50.1. For the lowest measured value of bs, our predicted influence of the intermonolayer friction in this interval of values of n is practically negligible. If bs =107 dyn s/cm3, then Cm n 50.1 for n ]16, and for all n ] 2 there are important differences between the frequency v ˜m n calculated on the base of the results in this paper, and that calculated with the expression of Milner and Safran with kc =k fr c (see Table 1). For the numerical values of the elastic and

29

rheological constants, used in the calculation of m the data in Table 1, the relation Vm n = v n is fulfilled when n= 12, corresponding to a wave length of 2.5 mm. For this value of n the ratio V0 m ˜m 12/v 12 is equal to 14.4. According to Yeung and Evans [21], this ratio should be equal to 1 in this case. In the existing experimental methods analyzing the optically observed fluctuations of giant vesicles [8,10], corrections to the measured values using Eq. (11) are introduced to take into account the finite integration time (40 min) of the video camera. If the friction coefficient bs is of the order of 107 dyn s/cm3, these corrections have to integrate the existence of the two exponential functions in the auto correlation function f m n (t), using the results from Eq. (47), Eq. (45), Eq. (46). This offers, in principle, the possibility to evaluate the 2 bl fr 2 factor bs(d0k bl c ) /(k c − k c ) , another method being a direct measurement of the functions f m n (t) [9,10]. In the light of the presented here calculations, the earlier results for the dynamics of the form fluctuations of quasispherical lipid vesicles [9,10] should be corrected. Evidently, this is necessary if the corrections are more important than the statistical errors of the measurements. The magnitude of the corrections could be evaluated only if fr the quantity bs and the ratio k bl c /k c are measured by independent experiments. The predicted effects will be more pronounced when the friction coefficient between the two monolayers, bs, increases. Different possibilities exist to obtain such an effect. If the chains of the two monolayers form interdigitated structures, the friction between them may increase [25]. These

Table 1 MS Numerical calculations of the coefficients C m ˜m for the pair values of n in the interval between n from Eq. (46) and the ratios v n /v n 2 and 20a n

2

4

6

8

Cm n MS v ˜m n /v n

0.00 0.90

0.02 0.78

0.03 0.70

0.04 0.63

10 0.06 0.58

12 0.07 0.53

14 0.09 0.49

16 0.10 0.45

18 0.11 0.42

20 0.12 0.40

The value of the coefficient of friction between the monolayers is bs =107 dyn s/cm3. The quantity v ˜m n is calculated from Eq. MS fr (47), and v n from Eq. (12) with kc = k c . The values of s¯ %=s¯ +e (see the text after Eq. (31)) and c 00 are zero and (c 00)eq, respectively. The values of the other quantities used in the calculation are given at the beginning of this section. a

I. Bi6as et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 157 (1999) 21–33

30

structures would appear when the two aliphatic chains of the lipid forming the bilayer have different lengths. Additives like cholesterol, that stiffen the hydrophobic part of the monolayer and make it more structured, should increase the friction too. Experiments with such molecules should lead to a verification of the above theory and can give one more illustration of the relation between the elastic and rheological properties of the studied objects. Finally, we should like to note that the used in the present article approach can be generalized for membranes, built up of different (not identical) molecules. For this aim, flip-flop coefficients, similar to the introduced by Eq. (15), have to be defined for each kind of molecules in the membrane.

Appendix A. Energy of the vesicle bilayer as a m function of the amplitudes u m n and 7 n The expressions for the curvature c1 + c2 and the surface element ds in the solid angle sin udud8 as a function of the fluctuation u(u, 8, t) are [7,10]: 1 (2−2u−92uq + 2u 2+2u92u), R0

c1+c2=



(A.1)

n

1 ds=(R0)2 sinu 1+2u+u 2+ (9u)2 , 2

(A.2)

where 9 is the two-dimensional gradient and 92 is the two dimensional Laplace operator, both defined on the unit sphere. When the fluctuation u is small enough, then:



n

2

ds0 =(R0) sinudud8.

(A.4)

After the integration of Eq. (A.2) over the whole surface, the area S(t) is obtained: p

S(t)=(R0)

u=0

× sinudud8.

2p

8=0

1

=4pk bl c 2−

p

n

1 k bl c (c 00)2 bl 8p k c −k fr c

c 00+

+ 4p(R0)2s

!

n

nmax n (b )2 k bl c 2 % % an − n [u m n (t)] 2 n = 2 m = -n g 2 bn m +g 6m u n (t) n (t)− g

+



1

n"

2 g[6 m 1 (t)] + 4pk( c.

+ %

n

1 1+2u+u + (9u)2 2 2

The equation (Eq. (A.6)) for the energy E(t) leads 2 to the expression for [u m n (t)] , Eq. (28). This mean value is obtained because the energy has a quadratic form. In deriving Eq. (28), we also use the relation:



k bl c an −

(bn)2 g

n



n

s(R0)2 +e . k fr c

(A.7)

where e is given by Eq. (29). 2 To calculate [6 m n (t)] , Eq. (32), we present E(t) from Eq. (26) in the following way: E(t)



1

=4pk bl c 2− (A.5)

(A.6)

m= −1

(A.3)

with

& & 



E(t)

= k fr c (n−1)(n+2) n(n+1)+

ds0 1 = 1−2u+3u 2− (9u)2 , ds 2

2

Replacing Eq. (A.1) in Eq. (21), Eq. (24) and Eq. (25), Eq. (A.5) in Eq. (25), expressing u(u, 8, t) through the spherical harmonics and the amplitudes u m n (t) from Eq. (6), writing the induced curvatures c˜ 00 through the spherical harmonics and the amplitudes c 00 and 6 m n (t) from Eq. (23), and keeping the terms up to the second order with m respect to the amplitudes u m n , 6 n and their products, we obtain Eq. (26). This equation can be written in the form:

+ 4p(R0)2s

p

n

1 k bl c (c 00)2 bl 8p k c −k fr c

c 00+

I. Bi6as et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 157 (1999) 21–33

+

! n "

nmax n k bl bn m c % % an u m 6 (t) n (t) − 2 n = 2 m = -n an n



+ g− 1

+ %

n

2

d dis [E (t)]= dt

(bn)2 2 [6 m n (t)] an

2 g[6 m 1 (t)] +4pk( c.

(A.8)

m= −1

7

S0

31

bs(grad2 B+rot2 Ab )ds.

(B.3)

If we use now the minimum energy dissipation principle stating (from a thermodynamic point of view) that the relaxation of the mode 6 m n (t) is a quasi-static process, we conclude that rot Ab =0. The scalar function B(u, 8, t) can then be presented as: m B(u, 8, t)= B m n (t) · Y n (u, 8),

Appendix B. Calculation of the frequency Vm n as presented by Eq. (43) Consider a sphere with a radius R0, a surface S0 and a homogeneous molecular distribution in the two monolayers, i.e. for all n ] 2, − n 5 m5 n, 6m ˜ 0 that only one 6 m n =0. Then modify c n is different from 0. Waiting some time, this amplitude will vanish too. The corresponding relaxation frequency Vm n can be determined if the fluctuation amplitude is big enough and the restoring force as a result of the energy stored in this fluctuation is significantly higher than the random forces as a result of thermal agitation. We present the vector field dVb =Vb out −Vb in (see Eq. (40) and the text above it) in the form: 1 1 9B + rot Ab , (B.1) R0 R0 where the velocities are related to the n, m mode, 9 is the two-dimensional gradient, rot is the planar component of the rotation, both operators defined on the unit sphere, B and Ab are scalar and vector fields, both defined on the sphere S0, Ab being perpendicular to the surface of the sphere. In the case of a spherical vesicle, the velocities Vb out and Vb in are evidently tangential to the surd dis face of the sphere. Let [E (t)] be the energy dt dissipated per unit time by the viscous forces. This quantity can be expressed through dVb : dVb = −

d dis 1 [E (t)]= dt (R0)2

7

S0

bs (dVb )2ds,

(B.2)

where bs is defined in Eq. (42). From Eq. (B.1) and Eq. (B.2)), we can write:

(B.4)

the spherical harmonic Y m n (u, 8, t) having the property: 1 (R0)2

7

2 [9Y m n (u, 8, t)] ds=n(n+ 1).

(B.5)

S0

From Eq. (B.5) Eq. (B.4), Eq. (B.3), Eq. (B.1), Eq. (41) and Eq. (40), the following result is deduced:



d dis k bl [E (t)]=bs d0R0 bl c fr dt k c −k c



2

2 [d6 m n (t)/dt] n(n+1)

(B.6)

But d[E dis(t)]/dt is equal to the opposite of the time derivative of the energy Eq. (26). When a single amplitude 6 m n (t) is nonzero, this time derivative is: d d d m m [E(t)]=− [E dis(t)]=k bl [6 (t)]. c g6 n (t) dt dt dt n (B.7) From d m Eq. (B.7) and Eq. (B.6), we obtain: [6 (t)] dt n 2 k bl d[6 m n (t)]/dt m d0R0 bl c fr + k bl c g6 n (t) = 0. kc −kc n(n+ 1) (B.8)

!

n

"

The expression (Eq. (43)) for Vm n is then a consequence of (Eq. (B.8).

Appendix C. Deduction of Eq. (45) and Eq. (46) m m We express the quantities u m n (t), 6 n (t), y n (t), m and z n (t) as Fourier integrals with respect to the time t:

32

um n (t)= 6m n (t)= ym n (t)= zm n (t)=

& & & &

I. Bi6as et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 157 (1999) 21–33

−

−

m m 4 Dm n =(Vn +v n )

um n (v) exp(ivt)dv, 6m n (v) exp(ivt)dv, ym n (v) exp(ivt)dv,



−

zm n (v) exp(ivt)dv,

(C.1)

where i is the imaginary unity. Replacing Eq. (C.1) in Eq. (38) and Eq. (39), we obtain the system of equations for the amplitudes u m n (v) and 6m n (v):

 

n n

bn m m m ivu m 6 n (v) =y m n (v)+v n u n (v)− n (v), an bn m u (v) =z m n (v). g n

m m iv6 m n (v)+Vn 6 n (v)−

m The moduli y m n (v) and z n (v) do not depend on m v because the functions y m n (t) and z n (t) have the properties of a white noise. We denote these m moduli by y m n and z n , respectively. The square 2 m modulus u n (v) calculated from (Eq. (C.2)) is:

  n

2 m2 m 2 [v 2+(Vm n ) ] y n +(v n )

=



v + (V +v ) −2V v

m n

(b )2 +(V ) (v ) 1− n ang

2

4

m 2 n

m n

m 2 n

m 2 n

m n



n

bn 2 ( z m n ) an (b )2 1− n v2 ang 2

,

(C.3)

the random functions y(t) and z(t) being uncorrelated. The auto correlation function f m n (t) from 2 (v) in the Eq. (44) is expressed through u m n following way [26]: fm n (t)=2p

&



−

2 u m n (v) exp(ivt)dv.

(C.4)

The denominator in the right-hand side of Eq. (C.3) can be written as: [v + (V0 ) ][v +(v ˜ ) ], 2

m 2 n

2

(C.5)







n







n

2 (V0 m n) = 1 (bn)2 m 2 m m (Vm + D m n + v n ) − 2Vn v n 1− n 2 ang

2 (v ˜m n) = 1 (bn)2 m 2 m m (Vm − D m n + v n ) − 2Vn v n 1− n . 2 ang (C.6)

Using Eq. (C.4), the relation:

(C.2)

2 u m n (v)

n

(bn)2 . ang

The positive sign of D m n is a consequence of the condition for the vesicle stability [1 −(bn)2/ (ang)]]0, (Eq. (26)). The explicit expressions for 2 2 (V0 m ˜m n ) and (v n ) are:



−



m 2 m m − 4(Vm n +v n ) Vn v n 1−

m 2 n

where V0 m ˜m n and v n are already defined, Eq. (47). m Let D n be the discriminant of the biquadratic equation (Eq. (47)):

&

exp(− ivt) p dv = exp(− at), 2 2 a v +a −

(C.7)

and the expressions for V0 m ˜m n and v n , we calculate m the auto correlation function f n (t): fm n (t) =

!

2(p)2 2 m 2 0 (Vn ) − (v ˜m n)

2 2 2 m 2 m2 m 2 0m y m n [(V n ) −(Vn ) ]− z n (v n ) (bn/an) exp(−V0 m n t). m V0 n

+

2 2 2 m 2 m2 m 2 ˜m z m n (v n ) (bn/an) + y n [(Vn ) −(v n) ] m v ˜n exp(−v ˜m t) n

"

(C.8)

2 m Evidently, f m n (0)=  u n (t) . From this relation and (Eq. (C.8)), the following equation can be obtained: 2  u m n (t)  2(p)2 = m m m V0 n v ˜ n (V0 n + v ˜m n)

!

 "

m 2 m2 m 2 [V0 m ˜m nv n + (Vn ) ] y n + (v n )

bn an

2

2 z m . n

(C.9) Using the same approach, we can deduced for the amplitude 6 m n (t):

I. Bi6as et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 157 (1999) 21–33 2  6 m n (t) 

=

2(p)2 0m V0 m ˜m ˜m nv n (V n +v n)

!   2 (Vm n)

bn g

2

"

2 m 2 m2 0m y m ˜m . n +[V nv n +(v n ) ] z n

(C.10) The system of Eq. (C.9) and Eq. (C.10) leads to 2 m2 the calculation of the squares y m n and z n for m m the unknown amplitudes y n and z n . Doing this and replacing the results in Eq. (C.8), we obtain Eq. (45) and Eq. (46). References [1] W. Helfrich, Z. Naturforsh. 28c (1973) 693. [2] W. Helfrich, Z. Naturforsh. 29c (1974) 510. [3] M.B. Schneider, J.T. Jenkins, W.W. Webb, J. Phys. Fr. 45 (1984) 1457. [4] H. Engelhardt, H.P. Duwe, E. Sackmann, J. Phys. Fr. Lett. 46 (1985) L395. [5] W. Helfrich, J. Phys. Fr. 47 (1986) 321. [6] I. Bivas, P. Hanusse, P. Bothorel, J. Lalanne, O. AguerreChariol, J. Phys. Fr. 48 (1987) 855. [7] S.T. Milner, S.A. Safran, Phys. Rev. A 36 (1987) 4371. [8] J.-F. Faucon, M.D. Mitov, P. Meleard, I. Bivas, P. Bothorel, J. Phys. Fr. 50 (1989) 2389. [9] P. Meleard, M.D. Mitov, J.F. Faucon, P. Bothorel, Europhys. Lett. 11 (1990) 355.

.

33

[10] M.D. Mitov, J.F. Faucon, P. Meleard, P. Bothorel, in: G.W. Gokel (Ed.), Advances in Supramolecular Chemistry, JAI Press, Greenwich, 1992, Vol. 2, p. 93. [11] I. Bivas, L. Bivolarski, M.D. Mitov, A. Derzhanski, J. Phys. II Fr. 2 (1992) 1423. [12] M.D. Mitov, P. Meleard, M. Winterhalter, M.I. Angelova, P. Bothorel, Phys. Rev. E 48 (1993) 628. [13] L. Fernandez-Puente, I. Bivas, M.D. Mitov, P. Meleard, Europhys. Lett. 28 (1994) 181. [14] G. Niggemann, M. Kummrow, W. Helfrich, J. Phys. II Fr. 5 (1995) 413. [15] T.M. Fischer, Biophys. J. 63 (1992) 1328. [16] E.A. Evans, R. Skalak, CRC Crit. Rev. Bioeng. 3 (1979) 181. [17] M. Bloom, E. Evans, in: L. Peliti (Ed.), Biologically Inspired Physics, Plenum Press, New York, 1991, p. 137. [18] R.E. Waugh, J. Song, S. Svetina, B. Zeks, Biophys. J. 61 (1992) 974. [19] U. Seifert, S.A. Langer, Europhys. Lett. 23 (1993) 71. [20] W. Pfeiffer, S. Konig, J.F. Legrand, T. Bayerl, D. Richter, E. Sackmann, Europhys. Lett. 23 (1993) 457. [21] A. Yeung, E. Evans, J. Phys. II Fr. 5 (1995) 1501. [22] W. Helfrich, in: R. Balian (Ed.), Lcs Houches Session XXX V Physics of Defects, North-Holland Publishing Company, Holland, 1981, p. 716. [23] U. Seifert, Z. Physik B 97 (1992) 299. [24] E. Evans, E. Sackmann, J. Fluid Mech. 194 (1988) 553. [25] R. Merkel, E. Sackmann, E. Evans, J. Phys. Fr. 50 (1989) 1535. [26] L.D. Landau, E.M. Lifshitz, Course of Theoretical Physics, vol. 5, Statistical Physics, Pergamon Press, London-Paris, 1959.