Stability of charged membranes

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Stability of charged membranes

D. Bensimon, F. David, S. Leibler, A. Pumir

To cite this version:

689

LE

JOURNAL

DE

_PHYSIQUE

Short Communication

Stability

of

_charged

membranes

D.

_{Bensimon(1 ),F. David(2),S. Leibler(2)}

and A.

_Pumir(1)

(1)

Laboratoire de

_Physique

_Statique,

ENS,

24 rue

Lhomond,

Paris

75005,

France

(2)

Service de

_{Physique Théorique,}

CEN

_Saclay,

91191 Gif/Yvette

Cedex,

France

(Reçu

le

_18janvier

_{1990, accepté} le

_{9 février}

₁₉₉₀₎

Résumé. 2014 La contribution

_{électrostatique}

au module d’élasticité est calculée _pour une bicouche

phospholipide chargée

en milieu

_ionique.

Cette

_{contribution,}

_identique

_pour une membrane conduc-trice comme _isolante, est

_toujours

stabilisante. On montre _que cette stabilité des membranes libres

est une

_conséquence

de l’absence de tension

_{superficielles.}

Abstract. 2014 The electrostatic contribution to the

bending

elastic modulus of

_{charged phospholipid}

bilayers

in an ionic solution is

_computed.

It is found to be the same for

_conducting

and

_{non-conducting}

membranes and is

_always

_stabilizing.

This

_stability

for free membranes is shown to be a

_simple

conse-quence of the

vanishing

of the

_physical

surface tension.

1

_Phys.

France 51

_{( 1990)}

689-695 15 AVRIL _1990,

Classification

Physics

Abstracts 87.10-87.20E

1. Introduction.

1fie

_{physical properties}

and

_stability

of

_phospholipid

_bilayers

and the

_{objects they}

form

_(cells

and

_vesicles)

is an

_{important problem}

in

_biology

and

_{pharmacology.}

Various

_aspects

of these

properties

have been studied over the _years and some are

_by

now well

established,

in

_particular

the Van der Waals and electrostatic intermembrane forces which form the basis of the DLVO

theory[l].

An artificial menbrane consists of a

_bilayer

of

_phospholipid

molecules with a

_polar

head

(which

may be

_charged)

in contact with the

_aqueous

medium and two

_hydrophobic

tails which form the bulk of it. In absence of external stresses

_(e.g.

osmotic

_pressures,

film

_tension,

_etc.)

the

area _per head of

_phospholipid

molecules in a fluid membrane is

_optimized,

i.e. the free _energy is minimized

_[2,3].

That results from a balance between

_hydrophilic

and

_entropic

interactions which

690

tend to increase the area

_per

head and the

_hydrophobic

interactions between the

_lipidic

chains and the water which tend to reduce it.If the

_{compressibility}

of the membrane is low the area

_per

head will be fixed at this

_optimal

value and does not fluctuate much. In the

_following

we

_suppose

that the

_{compressibility}

is

_strictly

zero; the total area of the membrane is thus fixed and the free

energy X

is then

_essentially

of elastic

_origin

_{[4 - 6] :}

Where

_Rl,R2

are the

_principal

radii of _{curvature, Ro} a

_{possible spontaneous}

curvature and x, /~

the normal and Gaussian

_bending

elastic moduli.

In the

_following,

we will

_investigate

the

_stability

of a membrane with surface

_charge

_density

_0-0 in an ionic solution characterized

_by

a

_{Debye screening length given by}

x -1. We

shall be interested

mostly

in the

_{long wavelength}

limit

_(X

» k

=

27r/A),

where the electrostatic contribution _{to the}

free

_energy

amounts to a modification of some

_{local parameters,}

such as the elastic moduli. In that limit the

_only

relevant

_lengthscale

is

_x-land

sinuez and /~ have units of _{energy (typically:} x m

10- 19J

[7 - 12],

_by

_{straightforward}

dimensional

_analysis

_[13]

the electrostatic contribution to

the elastic modulus x is ôx =

CO"Õ/X3ê

with C a constant of

_O(1)

and e the dielectric constant of the solution. There are two reasons one

_{may want}

to _go

_beyond

this

_simple

estimation.

First,

there

are now some rather accurate mesurements

_{[7 - 12]}

of the

_{bending rigidity,}

K, for

phospholipid

membranes and it therefore becomes

_possible

to make

_precise

measurements of the electrostatic contribution to x for

_charged

_{phospholipid bilayers.}

The surface

_charge

_density

can be controled

by

changing

the

pH

(which

determines the

_degree

of dissociation of the

_{polar groups)}

and the

Debye screening length by changing

the ionic concentration. The second reason is to determine the

_sign

of béc. If it is

_positive

the membrane is

_rigidified,

but if it is

_negative

the membrane _may be destabilized at

_{long wavelengths by}

the electrostatic interactions.

In the

_following

we shall consider two cases: membranes for which the

_{in-plane conductivity}

of

_charges

is much smaller than the

_conductivity

of

_charges

_through

the solution

_{(non-conducting}

or

_{insulating membranes),}

or membranes for which the

_in-plane

_conductivity

is much

_larger

than the bulk ionic

_conductivity

of the solution

_{(conducting membranes).}

For

_insulating

membranes the local

_charge

is constant

_during

a

_deformation,

whereas for

_conducting

membranes it is the surface

_{potential (and possibly}

the total

_charge)

which is constant

_during

the deformation. For

insulating

membranes one

_expects

a

_bending

deformation to increase the free

_energy

of the mem-brane and thus the

_sign

of 6/c is

_expected

to be

_positive:

the membrane is

_{rigidified by}

the

elec-trostatic interactions. For a

_{conducting mem6rane,}

however one _may

_expect

the existence of an electrostatic

_instability

similar to the electrostatic

_instability

of a

_charged

surface

_[14].

This is

how-ever not the case, as the full calculation

_(to

be described

_below)

shows. In fact

_charged

_insulating

and

_conducting

membranes have

_{identical positive 6K !}

t

2. Electrostatic contribution to the free _energy of a membrane.

Consider a

_charged

infinite membrane of thickness d in contact with an ionic solution

charac-terized

_{by a Debye screening length:}

_x-1.

For

_conducting

membranes and for

_insulating

ones with

equal

charges

on both

_sides,

we

_may

without loss of

_generality

consider the

_limiting

case: d = 0.

We

_suppose

that the electric fields

_4>:f:on

both sides of the membrane

₍₊

above and -

_below)

satisfy

the Poisson-Boltzmann

_equation,

or its linearized version: the

_{Debye-Hückel equation}

_[1]

For a monovalent solution of ionic concentration no the

screening length

is

given

by:

x2

=

87rnoe2/kBT,W,

where eW is the relative dielectric constant of water

_{(eW m 80).We}

shall

con-sider two

_types

of

_boundary

conditions. If the membrane is

_insulating

and

_{incompressible}

with a

constant surface

_charge

_density

_0"0, the fields on the membrane have to

_satisfy

Gauss’law

where n is the normal to the interface. If the membrane is

_conducting

its surface is an

equipoten-tial :

Let

_{((.c, y)be}

a small deformation of a flat membrane. Our

_purpose

is too calculate the net

_change

in the

_free

_{energy 6:F} of the membrane as a result of this deformation. The

_procedure

is

straight-forward :

1)

Solve

_{perturbatively équations (2,}

₃₎

for the field

0±

up

to 0

_{(c3) .}

2) Compute

the electrostatic

_energy[16]:

And

_identify

the

_change

6C in the electrostatic

_energy

due to the deformation.

3)

Keeping

the surface

_charge

or the membrane

_potential

constant

_(depending

on the case

considered) compute

the electrostatic contribution to the

_free

_{energy Xe}

_{by integrating}

the

ther-modynamic

relation[4,17]:

2.1 NON-CGNDUCTING MEMBRANES. -

_{Let((x, y)}

be a small

_{(0(e) )}

déformation of a flat

mem-brane

_{(Eq. (x,}

y, z =

((.c, y))

and (

(kx, ky)

its Fourier transform:

We look for a

_perturbative

solution of

_equation(2):

Inserting equations (6, 7)

into the

_boundary

condition

_{equation (3a)}

and

_solving

the

_resulting

692

Where 1Îl

(k"" ky)

is the Fourier transform of

_{e(x, y)}

=

X2(2/2 - (V ( ) 2

/2 -

(V2(.

We _may now

compute

the electrostatic _energy

_{C, équation (4):}

In the case of a surface in contact with an

_electrolytic

solution

_{(x # 0), part}

of the electrostatic

energy is "used" to lower the

_entropy

of the solution

_by

_{partially segregating}

the ions within the

Debye

screening

layer.

The relevant

_{thermodynamic quantity}

is the

_free

_energy

_Fe,

_{equation (5),}

which is obtained

_by

_{integrating equation (9):}

Where we have used:

2.2 CONDUCTING MEMBRANES - The

_{case of conducting}

membranes held at constant

_potential

4>0

(or

which total

_charge

_Q

is

_constant)

is similar to the case of

_insulating

membranes treated

previously. Inserting equations (6, 7)

into the relevant

_boundary

condition,

_{equation (3b), yields:}

Where,(fi

(kxky)

is the Fourier transform of

_{0 (x, y)}

_{= _X(2 /2}

+

_{( f}

d2kq(k eik. p.

For a membrane held at a constant

_{potential4>o (by patch clamp}

_techniques,

for

_example),

the electrostatic

_energy

is: --- - ...-

-The second term

_represents

the work done in order to

_keep

the

_potential

of the membrane

con-stant

_[16],

i.e. in order to

_bring

a

_charge

_Q

= f ud 2p

from a "reservoir" at

_potential

_4>0.

_Using

equation (11)

in

_{equation (12)}

one obtains:

Let us note that in the

_limitez

0 and 0"0 =

-WXOO/2ir

=

_const.,

we recover the result for the

In the case of

_{electrolytic solutions x #}

_0,

the free _energy

0e

becomes:

3.

_Vanishing

surface tension and

_stability

of the membrane.

The

_{k2 terms present}

in

_equation

₍₁₅₎

has a

_{negative sign.}

This _may

_naively

_suggest

that the

conducting charged

membrane is unstable with

_respect

to

_{long-wavelength}

modes

_{(in opposite}

to

the

_insulating

case, in which

the k2

term,

_Eq.(10),

is

_positive).

On the other hand the

k2

term

is

_usually

associated with the

_surface

tension

_which,

as we have mentioned in the

introduction,

is

strictly

zero for the free membrane. It is therefore _necessary to reconsider the

_significance

of the

vanishing

of the surface tension and its

_{çonsequences}

in our

_problem.

The size of a

_fluctuating

membrane can be in

_general

described

_by

two distinct variables: the total area A and the

_"projected"

area

_Ap,

i.e. the area of the

_projection

of the membrane on

the

_plane

_{(x, y).}

Whereas for the

_{incompressible}

_membrane,

whose number of molecules does

not _vary, the total area A is constant, this is not the case for

_Ap.

Indeed,

for the

_free

membrane there is no constraint on the value of

_Ap

and it fluctuâtes around its mean value

_Ap

> . As a

consequence

the intensive

_{thermodynamical}

variable

_coupled

to

_{Ap -}

the

_physical

surface tension

r - vanishes

_identically

[18]:

- 1

Here 0 is the total free energy of the

membrane,

which is the sum of the three contributions:

_(i)

the elastic _energy

_{(1), (ii)}

the electrostatic term

₍₁₀₎

or

₍₁₅₎

and

_(iii)

the chemical

_potential

term

for the total area A:

ro is the chemical

potential

of the

amphiphilic

molecules. We have thus

where _

for

_conducting

membranes for

_insulating

membranes

Since we consider in this

_paper

the case of

_free,

_{unconstrained,}

_membranes,

the coefficient

694

equation (18) identically

vanishes! Therefore there is no

_instability

and the lowest term which contributes to the free

_energy

of the

_charged

membrane is the last term of

_{equation (18).}

It is the same both for

_conducting

and

_insulating

membranes and is

_always

_positive.

The

k4term

_gives

the effective

_rigidity

of the

_charged

membrane Ketf. It can thus be written

as

where

where ew m 7 x

10-10 F/m

is the dielectric constant of water.

Thus both

_insulating

and

_conducting

membranes are

_{rigidified by}

electrostatic interactions. Our results

_{equation (20)}

are in

_agreement,

in the

_{long wavelength}

_limit,

with a similar and

inde-pendent

perturbative analysis

of reference

_[19]

for

_conducting

_membranes,

but

_{they disagree}

with calculations of 6", for the

_{nonconducting}

case of reference

_[20].

We believe that the

_discrepancy

is related to the fact that in reference

_[20]

the membrane has a finite thickness and a much smaller attenuation

_length

inside the

membrane,

so that the electric

_potentials

on both sides of the

mem-brane are

_decoupled.

It is not clear to us whether such an

_assumption

is valid for

_purely

dielectric médium. We have checked that the

_procedure

of reference

_[20],

based on the

_comparison

of the

electrostatic

_energies

_per

unit area for a

_{charged sphere,}

a

_{charged cylinder}

and an infinite

_plane,

leads,

both for

_insulating

and

_conducting

membranes,

to the same results for the electrostatic

con-tribution to the

_bending

modulus

_{of elasticity K}

than our

_perturbative

analysis. (1 )

It allows also to

assess the electrostatic contribution to the Gaussian modulus of

_{elasticity R}

which is

_opposite

to

6 "’.

4. discussion

Our

result,

_{equation (20),}

is

_only

valid within the

_{Debye-Hückel}

_{approximation,}

i.e. for

values of the membrane

_potential

_4>0

25mV. Since the values of the

_{Debye screening length}

x-1

vary

between

10-6m

in

pure

water

no

=

10-7M)

and

10-9m

in

_{physiological}

conditions

(no m

0.1

_M),

_{for Oo - 10 mv,bn}

_{may vary between}

0.2 x

10-19

J in

_pure

water

_(where

it should be

_measurable)

and 2 x

10-13

J in

_{physiological}

conditions,

where it is thus not

_expected

to be

relevant. We also remark that the "bare" electrostatic contribution to the surface tension

_(the

co-efficient of the first term in

_{equation (15):}

_6"(

=

^X02)

is

always

very small

7

x

10 -8

_-by

7 x

10-SN/m

7b measure ô x one could extend the work of

_Safinya

et al

_[11]

to

_charged

phos-pholipids

(e.g.

DMPG).

In such a

_system

_{by controlling}

the concentration of surfactants and

_ions,

one should be able to achieve a situation where 6", > K, i.e. where the electrostatic contribution to the

_bending

_rigidity

is dominant.

(1)

Note added in

_{proof :}

This corrects a statement made in a

_preprint

version of this letter. These

re-sults were

_{independently}

found

_by

several authors. See: P

_Higgs

and J.-E

_{Joanny (private communication);}

B.

_Duplantier,

R.E.

Goldstein,

A.T Pesci and V Romero-Rochin

_(in

_{preparation),}

B.

_{Duplantier (Saclay}

Acknowledgements.

We would like to

_acknowledge

R.E. Goldstein and FLB. Williams for useful discussions. The work of D.B. was

_supported

in

_part

_by

the DRET under contract

no.89/1327.D.B.

_acknowledges

B.

_Duplantier

for useful discussions.

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