Bending elasticity and thermal fluctuations of lipid membranes. Theoretical and experimental requirements

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Submitted on 1 Jan 1989

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Bending elasticity and thermal fluctuations of lipid

membranes. Theoretical and experimental requirements

J.F. Faucon, M. D. Mitov, P. Méléard, I. Bivas, P. Bothorel

To cite this version:

Bending elasticity

and thermal fluctuations

of

_lipid

membranes.

Theoretical and

_{experimental requirements}

J. F. Faucon

_(1),

M. D. Mitov

_(2),

P. Méléard

_(1),

I. Bivas

₍₂₎

and P. Bothorel

₍₁₎

(1)

Centre de Recherche Paul _{Pascal, C.N.R.S.,} Domaine Universitaire, 33405 Talence, France

(2)

Department

of

_Liquid

_Crystals,

Institute of Solid State

_{Physics, Bulgarian Academy}

of

Sciences, Sofia 1784,

Bulgaria

(Reçu

le I S

_février

_1989,

révisé le 25 mai _1989,

_accepté

le 26 mai

₁₉₈₉₎

Résumé. 2014 Les fluctuations

thermiques

de vésicules

_lipidiques

_géantes

ont été étudiées d’un

_point

de vue

_théorique

et

_{expérimental.}

Au niveau

_théorique,

le modèle

_{développé prend}

_{explicitement}

en _compte la conservation du volume de la vésicule et de la surface de la membrane. I1 en résulte que

l’ amplitude

des fluctuations

_thermiques

_dépend

non seulement de l’élasticité de courbure de la _bicouche, mais aussi de la tension de membrane et/ou de la différence de

_pression

hydrostatique

entre l’intérieur et l’extérieur de la vésicule. Au niveau

_{expérimental,}

la détermination du module de

_{courbure kc}

nécessite d’abord

_l’analyse

d’un

_grand

nombre

_(plusieurs

centaines)

de contours afin d’obtenir une bonne

_statistique.

En second lieu, la contribution de l’erreur

expérimentale

sur _{les coordonnées du contour,}

_qui

se _{traduit par} un bruit blanc sur les

amplitudes

de

_Fourier,

doit être _éliminée, et ceci _peut être réalisé

_grâce

à l’utilisation de la fonction d’autocorrélation

angulaire

des fluctuations. Enfin, les

_amplitudes

des

_harmoniques

ayant des _temps de corrélation courts doivent être

_corrigées

de l’effet du _temps

_{d’intégration}

(40 ms)

de la caméra _vidéo,

_qui,

dans le cas _contraire, conduit à une surestimation de

kc.

Toutes ces

_{exigences théoriques}

et

_{expérimentales}

ont été

_prises

en _compte dans

_l’analyse

des fluctuations

_thermiques

de 42 vésicules

_géantes

de

_{phosphatidylcholine}

du

_jaune

d’0153uf. Il _peut être rendu compte du _comportement de cette

_population

de vésicules avec un module de courbure

kc

égal

à 0.4 - 0.5 x

10-19

J, et des tensions de membrane extrêmement _faibles, de moins de 15 x

10-5

mN/m.

Abstract. 2014 Thermal fluctuations of

giant lipid

vesicles have been

_investigated

both

_{theoretically}

and

_{experimentally.}

At the theoretical level, the model

_developed

here takes

_explicitly

into

account the conservation of vesicle volume and membrane area. Under these

_conditions,

the

amplitude

of thermal fluctuations

_depends

_critically

not

_only

on the

bending elasticity

of the

bilayer,

but also on the membrane tension and/or

_hydrostatic

_{pressure difference between the} interior and exterior of the vesicle. At the

_experimental

level, the determination of the

_bending

modulus

_kc

first

_requires

the

_analysis

of a

_large

number

_{(several hundred)}

of vesicle contours to

obtain a

_significant

statistics.

_Secondly,

the contribution of the

_experimental

error on the contour

coordinates, which results in a white noise on the Fourier

_amplitudes,

must be _eliminated, and this can be done

_{by using}

the

_angular

autocorrelation function of the fluctuations.

_Finally,

the

amplitudes

of harmonics

_having

short correlation times must be corrected from the effect of the

integration

time

_{(40 ms)}

of the video camera, which otherwise leads to an overestimation of

kc.

All these theoretical and

_experimental

_requirements

have been considered in the

analysis

of the thermal fluctuations of 42

_giant

vesicles

_composed

_{of egg}

_{phosphatidylcholine.}

The behaviour Classification

Physics

Abstracts 68.10 - 82.70 - 87.20

of this

_population

of vesicles can be accounted for with a

_bending

_{modulus kc}

_equal

to

0.4 - 0.5 x

10-19

J, and

_extremely

low membrane

_{tensions, ranging}

below 15

10-5

mN/m.

1. Introduction.

Biological

membranes are an essential constituent of every

_living

cell. Their mechanical

properties

are

_closely

related to the

_problem

of cell

_stability

and resistance to extemal

influences,

and hence have become the

_subject

of increased attention.

_Considering

the membrane as an

_infinitely

thin

_layer

of

_liquid

_crystal

and

_applying

the ideas of

_liquid

_crystal

physics,

Helfrich

_[1]

has demonstrated that the mechanical state of a membrane element can

be

_completely

characterized

_specifying

its area and

_principal

curvatures. The

_stretching

elastic energy per unit area,

_F,,

is

_{given by}

the

_{expression [1] :}

where v- is the

_bilayer

tension,

_So

is the

_equilibrium

area of the membrane element

(e.g.

the

area at zero membrane

_{tension), S}

is the deformed area of the same

_element,

and

ks

is the

_stretching

elastic constant. The mechanical

_experiments

of Kwok and Evans

_{[2, 3]}

yield

ks

=

(140 ±

16 ) mJ/m2

for egg lecithin

bilayers.

When a small

_piece

of membrane is

_bent,

the

_bending

elastic energy per unit area,

Fc,

is

_{given, according}

to Helfrich

_[1],

_by

the

_{expression :}

where cl and c2 are the

_principal

curvatures of the

membrane,

co is the

spontaneous

curvature

(co :0

0 if the two

_monolayers

have different

_composition

or

_they

face different

_{environments),}

kc

and

_kc

are the elastic constants for

_{cylindrical bending}

and saddle

_{bending, respectively.}

The first

_attempt

to measure the curvature elastic

modulus,

kc,

was made

_by

Servuss et al.

[4].

Analysing

the thermal

_shape

fluctuations of

_long

tubular vesicles

_they

obtained

k, = (2.3

±

_{0.3 )}

x

10- 19

_J,

for egg lecithin membranes. Later on, Sakurai and Kawamura

[5]

evaluated

_kc

= 0.4 _x

10-19

J

by bending myelin figures

in _a

magnetic

field.

Measuring

the

time correlation function of the

_shape

fluctuations of

_long

_cylindrical

tubes

_[6],

as well as

_giant

spherical

vesicles

_[7],

Schneider et al. found

_{kc = (1 - 2)}

x

10-19

J.

_Using

Fourier

_analysis

of

thermally

excited surface undulations of

vesicles,

Engelhardt et

al.

_[8]

obtained

_{kc =}

0.4 x

10- 19

J.

_Recently,

Duwe et al.

[9] reported kc

= 1. 1 _x

10- 19

J.

Introducing

the

angular

correlation function of thermal

_shape

_{fluctuations,}

Bivas et al.

_[15]

measured

_{kc =}

(1.28

±

_{0.25 ) x}

10-19

J.

In contrast to the measurements of

_stretching

elastic constant,

ks,

where a

_single

value has

been

_reported,

there is a

_great

_variety

between the values measured

_by

various authors

2.

_Theory.

To a _very

_{good approximation,}

the vesicle membrane can be considered as a

two-dimensional

_{(geometrical)}

surface.

_{Using spherical polar}

coordinates

_(r,

0,

cp),

with

_origin

0 in the center of the

vesicle,

we can describe a

_slightly

deformed

_spherical

vesicle

_{by writing :}

where R is the radius of a

_{sphere enclosing}

the same volume as that of the

_vesicle,

an

u (03B8 , cp, t)

is a function

_describing

the relative

_{displacements}

_{(the fluctuations)}

of the vesicl wall from the ideal

_{spherical shape,}

0 and ~

being

the

polar angles.

It is assumed in whé

follows that the

_amplitudes

of the fluctuations are _{very small}

_compared

to the vesicl

dimensions,

u (03B8 ,

cp ,

t ) 1

« 1.

As far as we have chosen the

_origin

of the coordinate

_system

in the center of the vesicle w

where

_{Ym1 ( 03B8 , cp )}

are the

_spherical

harmonics defined below. The volume of the

vesicle,

V{u},

is

_{given by}

the

_{expression :}

Here,

cu- 0

denotes the volume of a

_sphere

of radius

_R,

and we have

_kept

_only

the terms of first and second order with

_respect

to u.

Similarly,

the deformed area of the

vesicle,

_{S {u} ,}

is

_{given by}

the

_{expression :}

where uo and

ulp

stand for the

partial

derivatives of u with

_respect

to

_parameters

03B8 and ~,

respectively,

80

is the total vesicle area at zero membrane tension and s is the area in excess over that of a

_sphere

of

_equivalent

volume. Here

_again

we have

_kept

only

the terms of first and second order with

_respect

to u and its derivatives.

The

_{lipid bilayers}

behave like two-dimensional

_liquids,

therefore _{every local variation of} the membrane tension is

_rapidly

relaxed via local flows of the

_lipid

material

_resulting

in

a = _const. all _over the membrane. As _a

The

_bending

elastic _{energy of the}

_{vesicle, { u } ,}

is

_{given by}

the

_{expression :}

where U88 and

_{u~ ~}

stand for the second

partial

derivatives of the function u with

_respect

to

parameters 0

or cp ,

respectively.

When

integrated

over the closed vesicle

surface,

the

Gaussian curvature term

_(the

second term in the

_{equation (3))}

_gives

a constant

_value,

4 irkc,

independent

of the vesicle

_shape

_{[11, 12],}

and it can therefore be omitted. We have

kept again only

the terms of first and second order with

respect

to u and its derivatives.

Usually,

vesicles are

_{poorly permeable}

to water and

_salts,

and to a _very

_{good approximation}

water is an

_{uncompressible}

fluid.

_So,

we shall consider that the vesicle volume does not

change during

the

_{fluctuations,}

and therefore the

_following

condition

_always

holds :

The fluctuations of the vesicle membrane are time

_dependent

thermal

_agitations

around the

equilibrium

vesicle form.

_Thus,

we can consider the vesicle

_shape

at a

_given

_moment,

U ( 8,

cp ,

t ),

as

_composed

of two contributions : a static

_part,

_{Uo ( 8, cp ),}

_{that is the average}

(equilibrium)

vesicle

_shape

and a

_dynamic

_part

or

_{perturbation,}

_&u(0,

cp,

t ),

that

gives

the

deviation from this

_equilibrium

and describes the fluctuations of the vesicle :

The

_angle

brackets in the above

_equation

denote an ensemble or _{time average. It} is natural to

suppose that the

ergodic

hypothesis

holds when there are no

_temporal

drifts or

_changes

in the environmental conditions

_during

the

_experiment.

To find the

_equilibrium

vesicle

_shape,

we have to _{minimize the total energy,}

_keeping

the volume constant. This is a variational

_problem

with a constraint. The usual method to solve it is to

_replace

the

_original

functional and the associated constraint with a new

_functional,

:F {u},

that is a linear combination of them and to treat the last like a functional without

constraint,

so we write :

Here, Ap

is

_Lagrange

_multiplier

associated with the constraint of constant vesicle volume

(physically,

Ap

is the

_hydrostatic

_{pressure difference}

_acting

across the vesicle

membrane).

If

the _{function uo minimizes the functional}

₍₁₃₎

then its first

_variation,

_{B[f {uo,}

_{ÔM} ,}

is zero for all

It is well known that if a function uo satisfies a condition like

_(14),

this function is a solution of

the

_{Euler-Lagrange equation}

associated with this variational

_problem.

The

_explicit

form of the

_{Euler-Lagrange}

_equation

in our case is :

Here, p

and if are the effective dimensionless pressure and

tension,

respectively,

and

co is the

spontaneous

membrane curvature.

_{Equation (15)}

is linear because of the

_quadratic

approximation

used

_throughout

in this work. If the same

_problem

is treated without

approximations

a forth order nonlinear

partial

differential

equation,

similar to those of

Deuling

and Helfrich

_[11,

_12]

and of Jenkins

_[17],

would be obtained instead. It is

inhomogeneous

because the vesicle radius is

_{given by}

₍₁

+

_uo),

and not

_by

_{uo itself. In the}

spherical

case, uo = _const.,

equation (15)

transforms into the well known

Laplace

law,

p = 2 & / (1 + uo),

thus it can be considered as a natural

_{generalisation}

of

_Laplace

law for the

case of an

_arbitrarily

deformed closed membrane

possessing

curvature and

_stretching

elasticity.

All the functions that are extremals of the functional

(13)

are solutions of the

Euler-Lagrange equation

(15).

In the case _{that uo is} a function that minimizes 37

_{uo}

we must have :

The last term in the above

_equation

_represents

the contributions

_originating

from the fluctuations of the total vesicle area and

_conjugated

to them fluctuations of the membrane tension.

_Generally,

the second variation of a functional like

_(16),

the last term

_excluded,

is a

quadratic

form of the

_{perturbations}

8u and its derivatives

_{(8ue, 8ulp’ 8uee, 8ulplp)}

with coefficients

_depending

on _{the function uo. But it} can be

_always

transformed,

via

_integrations

by

parts,

into a more convenient form :

where C

_{uo}

is a linear differential

_operator

_acting

on the

_{perturbations}

bu. It is convenient to

expand

the

_{perturbations,}

_8u,

in a series of the

_{eigenfunctions}

of the

_operator

_{C {uo}.}

In the

case of our

_quadratic

_{approximation}

the

_dependence

of

_C{Mo}

on _uo

_disappears

and the

eigenfunction equation

obtains the very

simple

form :

The

_{eigenfunctions}

are the well known

_spherical

harmonics,

Y::’(

8,

cp

),

defined as

_{(see [14]) :}

We

_represent

bu in a series of

_{Y:( 6,}

cp

).

But,

when

_writing

such a series one has to take into account that there is a «

_{high frequency}

» cut-off due to the discrete structure of the

membrane,

so we include

_only

the terms _up to _{nmax =}

Ùk

_(N

is the number of

_lipid

molecules

constituting

the

_vesicle).

_{Here nmaX is} so selected that the number of

_independent

_amplitudes

is

_equal

to the number of

_lipid

molecules. The exact _{value of nmax} is not of

_great

_importance

because we shall see further on that the

_physical

_quantities

are

_{only logarithmically dependent}

on it.

_Using

the

_{eigenfunctions}

of

_equation

₍₁₈₎

we can write :

where

_{Un (t )}

are the time

_dependent

coefficients in the

_{expansion. Using}

_{equations (21, 12, 5)}

we obtain :

This

_equation

_clearly

shows that the

_amplitudes

_{Um1 (t )}

are not time

_dependent.

Therefore,

they

do not describe fluctuations.

Moreover,

using

the second

_equation

in

_{(12), (ÔM)}

=

0,

and the

_{orthogonality}

of the

_spherical

harmonics,

one can conclude

_{that ( U§J’(t ) )}

= 0,

and so

do

_{Ui (t )}

= 0. Now we can

_readily

calculate the second variation.

_Substituting

₍₂₁₎

into

_(17),

and thereafter into

_(16),

we obtain :

Expression

(23)

gives

the increase of the vesicle energy due to the thermal fluctuations as a sum of the

_{squared amplitudes}

of the different modes of

_agitation.

The

_ks-term

in it is the

_only

second order contribution

_coming

from the last term in

_{equation (16),}

all the others

_being

of

higher

orders,

and thus

_neglected.

If the coefficients

_{À n (if, p)}

are

_positive,

one can obtain the

mean-squared

value of each

_{amplitude, (1 U:’(t) 12),}

_applying

the

_{equipartition}

theorem to

each mode of

_{agitation :}

If we take kT = 4 _X

1O- 21

J,

ks

= 100

Mj/M2 ,

R

=10- 5

m

_{(10 03BCm ),}

we calculate

( 1 U8(t) 12)

=

10- 10.

This means that the fluctuations of the mean vesicle radius are _very

small.

Now we

_proceed

further and look for a solution of the

Euler-Lagrange

equation (15)

in

where

_Amn

are constants that have to be determined.

_Substituting

the

_expansion

₍₂₅₎

into the differential

_{equation (15),}

one obtains a

_system

of

_{ordinary equations}

_equivalent

to it :

It is

_easily

seen that for

each n #

0,

there are two

_{possibilities :}

_{either 03BBn ( 03C3 p ) = 0}

or

An - 0.

As far

_{as À n}

_depends

on the index n, if for a

given

value n we have

’k n 0,

then for all the others we would

_{have k n (ô:, p) =A 0,}

and therefore

Amn =

0.

_Using

_{equations (25, 22),}

we obtain

_Ai

+

_{Um1 (t )}

= 0 and as far as

Um1 (t )

=

0,

_we

have

_{Ai =}

0 as well.

Thus,

the series

_expansion

₍₂₅₎

is reduced either to a

_single

term

(Ag

only),

and the

_equilibrium

_shape

of the vesicle is

_spherical,

or to a sum of a few terms

(A 0

and the different

_amplitudes

_An

_{of any}

_given

order

_n).

_However,

_starting

from a

_spherical

vesicle and

_increasing

the excess area

_(7),

the n = 2

eigenvalue

À 2 ( if , p)

will first vanish for

given

values of J and

_p.

A

_sphere

to

_ellipsoid

transition should then occur

_leading

to a non-zero _{average value for the n} = 2 mode

(Ar #= 0).

The

_equation

of

_{Euler-Lagrange}

₍₁₅₎

as well as its

_equivalent

form

₍₂₆₎

contains two

parameters, a:

and.

Therefore,

we need two

_{supplementary}

conditions to determine them. These are

_{equations (9)}

and

(11).

_Using

the series

_expansions

(21, 25),

these

_{supplementary}

conditions satisfied

_by

the functions

_{uo ( 0, ç )}

and

_{8u ( (J,}

cp,

t )

are transformed into relations

between the

_amplitudes

_An

and

_{Un (t ) :}

where s is the excess area defined in

(7).

The above two

_equations

must hold for each moment

of

_time,

and

_therefore,

the membrane

_{tension, J (t ),}

is a function of time determined

_by

equation

(28).

These fluctuations of the membrane tension

_give

rise to the last term in the second variation

₍₁₆₎

as

_{already pointed.}

We see that as far as

_{equation (27)}

holds,

Uoo(t)

is of second order instead of first

_(as

_expected

a

priori),

and

therefore,

it may be

always

dropped

when

_squared.

Because

_Um(t)

= 0 _as

already

mentioned,

the summation _on n in

equations

(27, 28)

can start from n = 2.

Furthermore,

since the

equilibrium

shape

of the

vesicle is time

_independent

we can take a time _{average of both}

_{equations :}

We have to stress now that it is the mean value of the membrane

_tension,

if

_{= if (t) ,}

that

30)

one can

_finally

obtain the relations that determine the

_equilibrium

_shape

of the vesicle. In

the case of a

_single

term we have :

Equation

(28)

contains the excess area, s, as a

_parameter

and the average

_shape

of the vesicle

depends

critically

on its value. The first two

_equations

in

_{(31) impose}

an

_implicit

_dependence

of the mean membrane

tension, u (s),

and the

_hydrostatic

_pressure,

p(s),

on the excess _area,

s, and

therefore k,, (u (s ), p (s ) ) is

determined via

(20)

as well. The third of them

gives

the

value of

_Aô.

_Here,

_{An -}

0

_{(for n _ 1)}

and the _{average vesicle}

_shape

is

_spherical.

At a

_given

critical value of the excess area, scr,

(determined

from the

condition : k 2 (a; (Sc,), p (Sc,»

=

0)

the

_amplitudes

_A2

are not zero _{anymore, and}

thereafter,

for any value

_satisfying

the relation s > _scr we have :

Here

_again

the first two

_equations

of

₍₃₂₎

_impose

an

_{implicit dependence}

of the membrane

tension, 03C3 (s ),

and the

_hydrostatic

_pressure,

_{p (s ),}

on the excess _area, s, and the last two of them determine the vesicle

_shape

via the

_amplitudes

_Aô

and

_Ai.

In the case s > _{scr the vesicle}

m=+2

fluctuates around an

_{elliptical equilibrium shape}

m = £ + 2

I Am2

#

_{0 .}

We

_point

out that it is

Îm = -

2

/

m=+2

03A3

1 Ail2

that is determined

_by

the last of the above

_equations,

the

_sign

and the relative

m=-2

proportion

of different m-modes

_being

undetermined. This is a _consequence of the

_quadratic

approximation

used.

_Higher

order terms must be included to discriminate between

_prolate

and oblate

_ellipsoids

_{(see [10]}

for more detailed

discussion)

and to fix the relative

_proportion

of different m-modes

_(to

be

_published).

As far as the vesicle

tension, 03C3(s ),

and

hydrostatic

_pressure,

p (s ),

depend

on the excess

area, s, it follows from

(24)

that the mean

squared

value of the

fluctuations, (1

( Umn (t )

12) ,

will

depend

on it as well :

For a

_spherical

_vesicle,

as

_long

as the fluctuations are not too

_big

_(e.g.

when the

_quadratic

therefore,

it can be omitted. Then

_equation

₍₃₃₎

is transformed into the

_expression

₍₃₄₎

previously

obtained

_by

Milner and Safran

_{[10] :}

The

_neglected

term in

₍₃₃₎

becomes of

_importance

in the case of an

_ellipsoidal

vesicle

_(e.g.

relatively

bigger

excess

_area).

In this case J + 6 =

0,

according

_to Milner and Safran

[10],

and

therefore the membrane tension does not _{vary when the} excess _{area, s,} is

_changed.

The

amplitudes

of the

_elliptical

modes

_(n

=

2,

m

1

-- 2) adjust

themselves _to fit the available excess area. In our model it is

_Àz(u,p)

that

_vanishes,

the membrane

_{tension, u(s),}

and the

hydrostatic

pressure,

p (s )

are functions of the excess _{area, s.} We

_{have U = - 6 (p = - 12),}

both

_becoming

more and more

_negative

when the excess area is increased. This result is in

agreement

with the conclusions made

_{by Deuling}

and Helfrich

_{[11, 12]}

as well as those of

Jenkins

_[17],

both obtained on the basis of numerical solution of the exact nonlinear

equivalent

of our

_{Euler-Lagrange}

_equation

_(15).

The static

_amplitudes

of the second

m=+2

harmonies, 03A3

+ 2

Am 2,

adjust

themselves,

according

to

_(32),

to fit the excess area as well.

m=-2

We need to find a

_practical

method for

_determining

the mean membrane

_tension,

ô.

_According

to

_{equation (34)}

the

_{product :}

is

_independent

of n as seen from the

_right-hand

sides of the above

_equations.

This result can

be

_effectively

used to determine a:

_{by comparing}

the

_products

in the left-hand side of

_equation

(35)

for different

_harmonics,

n. The value thus obtained is the exact membrane tension we are

looking

for. The

_bending

elastic

_modulus,

_kc,

can be calculated thereafter from

_equation

_(35).

The

_amplitudes

of the fluctuations of

_giant

vesicles are

_extremely

sensitive to _{the presence of} a

very small membrane

tension,

as

_already

mentioned

_by

Schneider et al.

_[7].

For a vesicle of radius R

=10- 5

m

(10 03BCm)

and

_kc

=

10- 19

J _a tension of u = 2 _x

10 - 5

mN/m leads _{to a}

two-fold decrease of the

_amplitudes

of second

_harmonics,

_{U2 (t ).}

For a small vesicle of radius

R

=10- 8

m

₍₁₀₀

Â)

the tension

_producing

the same effect is a = 20

mN/m,

_a value _one order

of

_{magnitude higher}

than the membrane

_rupture

_tension,

2 - 3

_mN/m,

as

_{reported by}

Kwok and Evans

_[2].

Due to the

_{relation p -- 2 ôF,}

the same two-fold decrease of

_U2(t)

can

alternatively

be

_{produced by}

osmotic effects of as low as 2 x

10- 9

M concentration difference for a

_giant

vesicle

(10 03BCm),

while 2 M are necessary for a small

₍₁₀₀

Â)

one.

_Therefore,

the small vesicles are

always

_fluctuating

even under osmotic stresses that lead to membrane

rupture !

3. Some

_{expérimental}

_quantities.

Now we have to find a relation between the

_{amplitudes, (1}

Un (t ) 2> ,

and an

_{experimentally}

measurable

_quantity.

It is believed that what is seen under a

_phase

contrast

_microscope

is the cross-section of the vesicle membrane with the focal

_plane

of the

_{objective. Considering}

the

The mean radius of the

_equatorial

_{cross-section,}

_{p (t ),}

at a

_given

moment, t, is :

Using

the series

_{(25, 21),}

after some

_algebra

we obtain :

where :

Equation

(38)

shows that the

_equatorial

cross-section radius fluctuates. This effect is

_mainly

due to the vesicle deformations

_changing

its

_shape

from oblate to

_prolate

and vice-versa. The time

_averaged

_radius,

_p =

( p (t)>,

is

easily

calculated because the last _term in

(38)

vanishes

when

_averaged

over the time.

_Using

_equation

_{(29) (for}

a

_spherical

vesicle

_An

= 0 when

n> 1),

we

_finally

have :

We see that the mean cross-section radius is

_equal

to the mean vesicle

radius,

both

_being

always

smaller than the radius of a

_sphere

of

_equivalent

volume,

an effect

_purely

due to the existence of thermal fluctuations.

Following

Bivas et al.

_[15]

we calculate the normalized

_angular

autocorrelation function of the vesicle

_{radius, 03BE ( y, t ),}

at a

_given

moment of time :

Using

the series

_expansions

_{(25, 21),}

after some

_algebra

we obtain the time

_averaged

autocorrelation function :

According

to

_{equations (24, 34),}

the

_{amplitudes, (1}

_{Umn (t )}

_{12) ,}

are not

_dependent

on the index m.

_{Taking advantage}

of this fact and

_using

the addition theorem for the

_spherical

harmonics

[14],

the sum on m can be

_{explicitly performed}

and the above

_expression

is transformed into

the form :

equations

(34).

The term

_{(2 n + 1 )}

reflects the fact that for each n there are

_{(2 n + 1 )}

statistically independent

m-modes. The last term in

₍₄₂₎

_compensates

for the addition of all

missing m

= 0 modes to

equation

(41)

which _are otherwise _{necessary for the} addition

theorem.

Up

to now we were

supposing

that we knew the

_position

of the vesicle center

_0,

whose coordinates are

_given

_{by equation (5).}

But

_observing

_{(and measuring)}

_only

the

_equatorial

cross-section it is not

_possible

to calculate them because the

_complete

_{u ( 03B8, ~ , t )}

_{is necessary.} The best we can do is to calculate the

coordinates,

(xÓ(t), yÓ(t)),

of the center

0’ of the observed

_equatorial

contour.

Let

_{(p, cp )}

and

_{(p’, ç ’ )}

be the

_polar

coordinates measured with

_respect

to

_point

0 and 0’

_{respectively. By}

definition we have :

Let us denote the distance O’O

_by

_Rw,

the

_angle

it makes with the X axis

_by

t/J,

and the

_angle

it is seen from a

_point

on the contour

_by

_~,

_{(Fig. 1).}

The

_primed

_quantities

can be

_expressed

as functions of

_nonprimed

ones and

_developed

in series with

_respect

to

w « 1

and 0

«

1,

keeping

only

the terms _up to the second order. The

_quantities

w =

w (t) and 03C8 = 03C8 (t)

are both unknowns that have to be determined from

_{equations (43).}

We calculate the

radius,

p’ (t ),

of the

_equatorial

cross-section defined

_{by equation}

_(37),

but

now we use

_{(p’ ( q; " t ), q;’)}

instead of

_{(p ( cp, t ), cp ) :}

Fig.

1. - Cross-section of

a vesicle in the

_plane

_{(X, Y).}

0 is the center of the vesicle,

The first term in the above

_equation

coincides with

_expression

₍₃₉₎

and the last is a

consequence of the fact that the center of the cross-section is

_fluctuating

around the vesicle

center. The

_displacement

itself is a first order

effect,

given by

U; l(t)

amplitudes,

but its influence on the cross-section radius is of second order.

Using

the same

_approach

we calculate the

_experimental

autocorrelation

function,

03BE ( y , t),

given by

the relation

_(40),

but now we use

(p’ (Cf", t), ~)

instead of

(p (cp, t), cp).

Keeping only

the terms _up to the second order we calculate the

_experimental

autocorrelation

function,

1 ’ ( y ) =

03BE ( y, t) :

We see that the

_{experimentally}

calculated autocorrelation function has the same form as the

theoretical one

_except

the extra

_{P 1 (cos y )-term. Equation}

₍₄₅₎

shows that the

_angular

autocorrelation

function,

03BE’ ( y ),

is a series of

_{Legendre polynomials,}

Pn(cos y ),

with

theoretical coefficients

_{Bn(03C3, kc) :}

Comparing

with

_equation

₍₂₄₎

we conclude that

En (if, kc)

= 2 n + 1

Un (t ) I 2 , >

Thus we

4 03C0

have obtained the needed direct

_{correspondence}

between the

_amplitudes

of the vesicle

fluctuations, Un (t ) I 2> ,

and the

_{experimentally}

measured vesicle

radius,

p’(~,t),

via the autocorrelation

function,

03BE’(y).

To find the value of the

_bending

elastic

modulus,

kc,

we calculate the

_experimental

autocorrelation

_function,

_{03BE’(y, t),}

for each contour

_and,

_decomposing

it into series of

Legendre polynomials,

we

_get

the

_experimental

_{coefficients,}

Bn (t ).

We evaluate the mean

values,

_{Bn = (Bn(t),}

and estimate the

_{corresponding dispersions}

_{(standard deviations),}

Dn . Using

and

_kc

as

_fitting

_parameters

we minimize the

_function,

M(03C3, kc) :

where N is the number of

_{amplitudes (harmonics)}

used for the

_fitting.

As far as

Bn

values are an arithmetic mean of a _very

_large

number of

experimental

Bn (t )

data

(usually

400 or

more),

their distribution is

Gaussian, therefore,

the

quantity

M ( 03C3 , kc)

_obeys

a

X 2

distribution with

_{(N - 2 ) degrees}

of freedom. This fact can be used to

_verify

the

_quality

of the fit

_{by comparing}

the calculated

_{M(03C3, kc)}

value with the value of

_{X 2-distribution}

with

There is one more

_point

that deserves a comment. The

_image

formed

_by

the

_microscope

is

projected

onto the

_target

of a video camera and is accumulated on it for the time between two

successive

_{scans, ts}

= 40 _ms. This

integration

leads to

_smearing

of the

_{fastly moving}

_parts

of the _contour, and

_degradation

of the resolution. If one _{suppose that the}

_{experimentally}

measurable

_radius,

_{p’ ( ~’, t )}

is the mean value over the scan time :

the autocorrelation

_function,

_{03BE’ ( y ),}

_averaged

over the

_period

of the

_observation,

to,

would be

_{given by}

the

_{expression :}

This is in fact a

_{triple integral}

with

_respect

to the time in which we

_change

the order of

integration.

Using

the series

_expansions

_{(25, 21)}

and

_making

first the

_integration

over the

period

of

observation,

to,

one obtains an

_expression

similar to

_(41),

but now we have :

The

_integral

in the last

_equation

is the

_correcting

factor due to the camera

_integration.

If there was no such effect

_(ts

=

0 )

this factor would be 1 and _we would obtain the

previous

result

(41).

The

_quantity

_Tn

is the correlation time of the

_respective

modes calculated

_by

Milner and Safran

_[10].

_Comparing

the

_expressions

for the correlation time

_{(see [10])}

with that for the

mean

_squared

_{amplitudes (34)}

one can see that

_they

are related :

where q is the

viscosity

of the medium

surrounding

the vesicle membrane. We see

_that,

as a

result of the camera

_integration,

the value of the

_{experimentally}

measured

_quantities,

Bn,

is modified

_by

a factor

_depending

on the ratio between the camera scan

_time,

ts,

and the correlation time of the

_respective

modes :

Using

an iterative

_procedure,

the best values of the correction factor and

Bn

were calculated from the

_{experimental amplitudes}

_Bn,

and the corrected

values,

Bn

were used in

_equation

₍₄₇₎

to calculate the elastic

modulus,

kc.

4.

_Sample

_preparation.

Egg-yolk

phosphatidylcholine (EPC)

was

_{prepared according}

to the method of

_{Singleton et}

al.

[18],

and dissolved in

_{CH30H/CHCl3 (1:9 v/v)}

at a concentration of

_{0.5 mglml.}

A small

amount of this solution was

_sprayed

on a

_microslide,

then the solvent was removed under

except

a

_watertight

_material,

about 0.1 mm

_thick,

_{made up of the silicon}

_product

CAF4

(Rhône-Poulenc, France)

was used as a _{spacer. Both the microslide and the} cover

_slip

were

previously

treated with

_{trimethylchlorosilan (Sigma, U.S.A.)}

for

_making

them

_hydrophobic.

The cell was filled with deionized water

_(Millipore

_MQ,

_U.S.A.),

and then sealed to avoid any

evaporation.

Giant vesicles were formed

_{spontaneously}

in the

cell,

and

_{generally they}

were studied one

_day

to one week after the

_sample

_preparation.

5. Vesicle observation and

_{image processing.}

Giant vesicles were observed

_using

an inverted

_phase

contrast

_microscope

IM35

_(Carl

_Zeiss,

F.R.G.) (objective

PH 100 x,

NA = 1.25).

The thermal fluctuations were monitored

vi a a

_{contrast-enhancing}

video camera C2400-07

(Hamamatsu

Photonics,

Japan)

and

recorded on a U-matic video

_tape

recorder

(Sony, Japan)

for at least

_eight

minutes. Video

images,

taken at

_regular

time intervals of 1 s, were

_digitized

with a Pericolor 2001

_image

processing

system

(Numelec, France).

Digitized images

were constituted of 512 x 512 8-bit

pixels,

and,

with the

_{magnification}

used, 1

pixel roughly corresponded

to 0.1 03BCm. An

out-of-focus

_image

was

_usually

subtracted from that of the vesicle to eliminate mottle or

background

heterogeneities. Digitized images

were then transferred to a VAX-8600

_computer

_(Digital,

U.S.A.)

for the determination of the contour coordinates. This was

_performed

as in

_[8]

_by

searching

for the minimum

_{intensity along}

the vesicle radius in

_given

directions. In the conditions

_used,

the contours

_appeared

like a dark line of about 5

_pixels

width at

_half-height.

In order to _{increase the accuracy} on the contour

_{determination,}

the vesicle radius in a

_given

direction was calcùlated as the

_{intensity-weighted}

_{average of the radii of these five}

_points.

The

number of contour

_points

so determined was

_{roughly equal}

to the number of

_pixels

constituting

the contour, i.e. between 500 and 1000

_depending

on the vesicle size. An

_example

of what can be obtained is shown in

_figure

2,

where calculated contours have been

superimposed

to the

_{digitized images}

of vesicles.

Fig.

2. -

The second

_step

of the treatment was to eliminate the contours

_{corresponding}

to

_noisy

records,

which can

_originate

from a bad

_{microscope focusing,}

for

_example.

_{For this purpose,} the

_experimental

contours were

roughly represented by

a Fourier series limited to the first five terms. All the

_experimental

_{points being}

farther than a chosen critical distance

(=

2

_pixels)

from that

_{representation}

were deleted. If the number of deleted

_points

for a

_given

contour exceeded some value

_{(= 5 %),}

the whole contour itself was eliminated.

_Usually,

less than 5 % of the total number of

_{digitized images}

was lost after such

_{manipulation.}

The rest

were used for the determination of the

_bending

elastic constant.

6. Numerical simulations.

The

_only

_way to check the

_validity

of the theoretical model is to use the function

₍₄₇₎

with as

many different

harmonics,

Bn,

as

_possible.

It is obvious

_however,

that due to the

_experimental

limitations the

_{number, N,}

of the harmonics that can

_actually

be observed and

_investigated

is

limited. At least three

_important

factors have to be considered :

_(i)

the limited number of

analysed images

(contours), (ii)

the

_precision

on the contour

_coordinates,

and

_(iii)

the time resolution of video

_{(40 ms)}

_compared

to the correlation times of the fluctuations modes. An easy way to estimate the effect of these different factors is to

_perform

_computer

simulations.

In a first

_step,

simulated contours were created without

_taking

into account the third

_factor,

i.e. video time. The theoretical values of the

_mean-squared

Fourier

_{amplitudes corresponding}

to a

_given

set of values for

_{kc, if,}

and the mean vesicle radius were calculated as in

[8].

The

squared

Fourier

_amplitudes

for a

_given

contour were obtained

_by

_multiplying

these theoretical values

_by

random numbers

_obeying

a _X

2-distribution.

These

_amplitudes

were then

randomly decomposed

into the sine and cosine

_components

of the Fourier

_series,

which allowed the

_angular

_dependence

of the radius of the simulated contour to be

_computed.

A Gaussian noise could then be added to the vesicle

radius,

with a chosen standard

deviation,

in order to account for the

_experimental

errors on the contour coordinates. These simulated

contours were further

_analysed

in the same _way as the

_experimental

ones.

An

_example

of such a simulation is shown in

_figure

3. In this case, 40 contours, which

roughly

corresponds

to the number used in the

_previous

works

_{[8, 15],}

have been

_generated

Fig.

3. -

Simulated data : values

_{of kc}

versus the order n of fluctuations. 40 contours were

_generated,

using

k,

=

10- 19

J, Ir = 0, and _{a mean} vesicle radius

equal

_to 100

pixels.

No noise _was added _to the

without any added noise. It is clear that the obtained values of

kc

present

large

variations

versus n, the order of the harmonics.

However,

when the number of

generated

contours is

equal

to

_400,

_kc

remains rather constant _up to n =

_20,

as seen in

_figure

4,

even _{in the presence}

of a Gaussian noise of 1 % on the vesicle radius. Such a noise

_corresponds

to an error of 1

_pixel

on the radius for a 20 _>m diameter

vesicle,

i. e. to the

_expected

error in normal

experimental

conditions.

Fig. 4.

- Simulated data : autocorrelation functions

_{(left Fig.) and}

dependence of kc versus the

order

n

_{(right Fig.).}

Four hundred contours were

_generated,

_{in the presence}

_{(full line)}

or absence

_{(dotted line)}

of a Gaussian noise

_equal

to 1 % on the vesicle radius. Entered values for

_{simulations: kc}

=

10-19

J, 03C3= _0, and mean vesicle radius = 100

_pixels.

Another

_{interesting point}

is worth

_mentioning.

It can be seen that the autocorrelation

functions obtained either with or without noise are almost

identical,

_except

in one

_point,

y = 0.

So,

the noise contributes to the autocorrelation function

_mainly

in the form of a 8-function at y -

0,

as

_{already pointed}

out in

_[15],

and it has

_practically

no effect on the values of

_kc

calculated from the

_{Legendre amplitudes, Bn,}

of the autocorrelation function.

When the

_analysis

of these simulated data is

_{performed by}

direct Fourier

_{decomposition}

of the contours, as in

_[8],

_{the presence of noise} leads to an

_important

decrease of

kc

for

_{wavevectors q}

8

_{(Fig. 5).}

The noise contribution can be

_considerably

reduced in this

case as

_well,

but an

_appropriate

white noise have to be subtracted from the Fourier

amplitudes,

as shown in

_figure

5.

Finally,

we should mention that the accurate value of

_kc

can be recovered from simulated

contours

_generated

with initial values of J

_ranging

from 0 to

_200,

even in the presence of a

1 % Gaussian noise on the vesicle radius

_(data

not

_shown).

All the

_preceding

simulations have been

_{performed assuming}

that the time resolution of video is much

_higher

than the

_frequency

of the

_{fluctuations,}

which is

_obviously

not the case

when

_high

order harmonics are considered.

_So,

an

_attempt

has been done to

_quantify

this effect

_{by using}

the correction factor

₍₅₂₎

derived above. As a first

_{approximation,}

the

Fig.

5. -

Analysis

of simulated data

by

Fourier

_{decomposition}

of the contours :

_(---)

without added noise on the vesicle _{radius ;}

(-)

in the presence of a Gaussian noise of 1

% ; (- - - -)

in

the presence of noise, after subtraction of an

_appropriate

white noise from the Fourier

amplitudes.

Entered values for the simulations were the same as in

figure

3.

Tn

of the

_respective

modes. An

_example

of what can be obtained is shown in

_figure

_6,

the entered values

_being

in this case :

_kc

= 0.5 x

10-19

J, J

= 50,

vesicle radius = 100

pixels,

Gaussian noise = 1

pixel.

When the fit is done _as

usual,

using only

the

X 2 --

1

criterion,

12

modes can be fitted

(Fig.

6A,

solid

line),

and this leads to

_{kc =}

_{(0.71 ± 0.05 )}

x

10-19

J and

J = 35 ±

4,

i.e. _to values

clearly

different from the entered _ones.

Moreover,

_a

significant

increase of

_kc

is observed when n 13. On the

_contrary,

if the fit is limited to the first few

modes,

whose correlation times are

_larger

than the video

time,

the obtained values are

_nearly

equal

to the entered

ones :

for

_example,

when n =

2, ... , 7,

(corrélation

times - 0.1

s),

_one

obtains kc = (0.49 ± 0.09)

x

10-19

J and if = 57 ± 15

(Fig. 6A,

dotted

line).

Finally,

when

the correction factor is introduced in the

_analysis

of simulated data

_(Fig.

6A,

dashed

_line),

kc

remains constant _up to n =

20,

and the recovered value

kc = (0.43 ± 0.03 )

_x

10-19

J,

is

again

very close to the entered one.

The results obtained

_by

direct Fourier

_analysis

of these simulated data are

_reported

in

figure

6B. The solid line

_represents

the values of

_kc

versus the wavevector _{q of}

_{fluctuations,}

without

_taking

into account the effect of if and the white noise contribution. As

_expected,

the entered

_positive

value of if leads to

_large

values of

_{k, at}

low wavevectors. A

_plateau

is observed

_{for q}

= 10 with

kc

= 0.8 _x

10 -19

J,

then

k,

decreases

for q .

19. It is

interesting

to

note that the decrease

_{of kc at}

_{high q}

is much less

_pronounced

than in the case of

_figure

5,

Fig.

6. -

Simulated data : effect of the

_integration

time of the video camera.

_(A)

Values of

kc

derived from the autocorrelation function :

_full

line :

_using

the

_{XR ,1}

criterion ; dotted line :

considering only

modes with n = 5; dashed line : after

_introducing

the correction factor

_{(Eq. (52)).}

(B)

Values

_{of kc}

derived from the Fourier

_analysis

of the contours :

_full

line :

_{rough analysis ;}

dotted line : after white noise subtraction and

_{using J}

= 35.

contributions of the Gaussian noise and the effect of video

_integration

time act in

_opposite

way on the final result.

_However,

in this case, the noise contribution

prevails

over the effect of

correlation

_times,

so that a decrease of

_{kc is}

still observed.

The dashed line in

_figure

6B

_represents

the values of

_kc

obtained after white noise subtraction and after

_{introducing J}

= 35. A constant value

of kc

0.75 _x

10 -19

J,

_once

again

very different from the entered one, is then obtained

for q

12. 7.

_Experimental

results and discussion.

7.1 EXAMPLES OF GIANT EPC VESICLES. - An

example

of

_experimental

data is shown in

figure

7,

for a vesicle of 19.4 03BCm in diameter : 384 contours have been

_analysed,

either

_using

the autocorrelation function and its

_{decomposition}

into

_{Legendre amplitudes Bn}

_(Fig.

_7A,

_B),

or

_by

direct Fourier

_analysis

of the contours

_{(Fig. 7C).}

As can be seen in

_figure

7B,

different behaviours are observed

_depending

on the criteria

used to fit the

_{Legendre amplitudes, Bn,}

of the autocorrelation function. When the

_only

criterium used is

_{X R ,1,}

_up to 16

_amplitudes

can be fitted

_{(Fig. 7B,}

solid

_line),

and

kc = (0.69 ± 0.03 )

x

10-19

J with J = 26 _± 2 is obtained. It is

noteworthy

that

kc

increases

for orders n --

14,

as

_expected

if the video time is

_larger

than the correlation times of these modes. When the

_analysis

is limited to the first few

_amplitudes,

_having

correlation times

larger

than 0.16 s

_(i.e.

at least fourfold

_larger

than the video

_time),

the dotted line in

figure

7B,

with kc =

(0.44 ± 0.1 )

x

10-19

J and u = 47 ± 15 is obtained.

Finally,

when the

correction factor is used as described in the

_preceding

section,

a

_good

_fip

_up to

n = 21 _can be

obtained,

with

kc = (0.40 ± 0.02)

x

10-19

J and J = 55 ± 5.

The Fourier

_analysis

of the contours of the same vesicle is

_presented

in

_figure

7C. When

excess area of the vesicle. Then a

_plateau

is reached

_{around q =}

10,

and

_finally,

an increase is

observed

_{for q >}

15. In this _case,

_therefore,

the noise contribution to the measured Fourier

amplitudes

is smaller than the effect due to the video

_integration

time. The

_comparison

of the

experimental

data with the simulations for a vesicle of

_nearly

the same

_{characteristics,}

in the presence of 1

pixel

Gaussian noise

_{(Fig. 6B),}

seems to indicate that the

_experimental

noise on

the vesicle radius is

_{significantly}

less than the indicated value.

Using

=

26,

the dotted line in

figure

7C is

obtained,

and

k,

= 0.7 _x

10-19

J remains

practically

constant

_{for q ‘}

13. The behaviour is

_quite

similar to that obtained from the

Legendre amplitudes

of the autocorrelation function

_(Fig.

7B,

solid

_line).

Fig.

7. -

Example

of

_experimental

data obtained with a vesicle of 19.4 ktm in diameter, and

_using

384 contours :

_(A)

autocorrelation

_{function e (y)}

of the fluctuations ;

(B)

dependence

of

_kc

versus the

order n, deduced from e

( y ) : full

line :

_using

the

_{X2 R --}

1 _{criterion ;} dotted line :

_{considering only}

modes with correlation times

_larger

than 0.16 s ; dashed line : after

_introducing

the correction factor

_(Eq.

7.2 EFFECT OF VIDEO INTEGRATION TIME : SUM OF DIGITIZED IMAGES. - It is clear from the above

_example

that the video

_integration

time constitutes a serious limitation in the

_study

of

fluctuating

vesicles. Another mean to illustrate this effect is to increase at wish the characteristic time of

_video,

which can be

roughly

done

_{by adding}

successive

_digitized

_images

before

_performing

the

_analysis.

Of course, it would be much better to act in the

_opposite

_way, i. e. to decrease the

_integration

time.

_{Unfortunately,}

this is not so _easy.

The result from the

_analysis

of contours obtained after such additions of

_{digitized images}

is

presented

in

_figure

8. A vesicle of radius 14 itm was used in this case. The

_analysis

has been

performed

either as

_usual,

on

_{single images}

(solid line),

or after

_summing

two

_{(dotted line)}

or

four

_{(dashed line)}

successive

_images.

Figure

8A shows the

_dependence

of

_kc

on n when the

only

criterion taken into account is

x 2

1. In this case, the

kc

value obtained increases from 1.0 x

10-19

J to 1.4 x

10-19

J and to

2.0 x

10- 19

J for a

single,

sum of two and sum of four

_images,

_{respectively.}

Fig.

8. - Values

of kc

versus the order n of fluctuations obtained from the usual

_analysis

(full line),

or

after

_summing

2

_{(dotted line)}

or 4

_{(dashed line) digitized}

_{images :}

_(A)

_using

the

_{x2 =}

1 _{criterion ;}