Theory of the ripple phase coexistance

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Theory of the ripple phase coexistance

M. Belaya, M. Feigel’Man, V. Levadny

To cite this version:

M. Belaya, M. Feigel’Man, V. Levadny. Theory of the ripple phase coexistance. Journal de Physique

II, EDP Sciences, 1991, 1 (3), pp.375-380. �10.1051/jp2:1991174�. �jpa-00247524�

Classification Physics Abstracts

87.20C

Theory of the ripple phase coexistence

M. L

Belaya ('),

M. v.

Feigel'man

(~) and v G

Levadny

₍₃₎

(')

Instltute of Plant Physiology, Botanicheskaya 35, 127276, Moscow, US S R (2) Landau Instltute for Theoretical Physics, Kosyglna 2, 117940 Moscow, US-S R.

(3) Institute of Cybemetics, U S-S R. Academy of Sciences, Vavilov str. 34, Moscow, U S S R

(Received18 September 1990, accepted13 December 1990)

Abstract. The macroscopic theory of the competition between two different modifications (A- and A/2-phases) _{of Pp~} (npple) phase in lipid bilayers _is presented It _is shown that the _increase of the membrane curvature should lead to the phase transition from the A/2~phase to the A-phase,

the critical value of the curvature lQ ^' is obtained _{as a} function of the geometncal _parameters of the A/2-phase and the free _energy difference AF° _{between both} phases _in the case of planar membranes" _The observed [3] splitting of dtsclinations _in the A/2-phase into well-separated half- integer parts ^connected by linear defects (of the length f~ « A) _is shown to be related wtth the

smallness of AF°. _{The relation bet)}

een R~, f~ ^and geometncal and thermodynamic parameters of the A/2-phase alone _is obtained

The

ripple phase

_Pp~ is known to emst in

hydrated lipid bilayers

_{as an} intermediate

phase

between the low temperature

gel phase

_Lp~ and the

uigh~temperature

fluid

phase

L~

[1,

2].

In the

P~ >-phase,

^the

layers

_are

corrugated

with respect to _{an axis}

parallel

to the

mean

bilayer plane,

_so that _a cross-section of the

bilayer

forms _a sawtooth~hke pattern [2]

(see Fig. I).

In spite of _a

large expenmental (see

_e-g-

[2-6])

and theoretical [3, 7~10] activity, the ongin and properties ^of the

Pp-phase

_are not yet

completely

studied. In

particular,

there _{is no}

microscopic theory,

to _our

knowledge,

which could

explain

the emstence of two different

modifications of the

Pp~phase,

^which differ in the

wavelength

of

npple

modulation with the ratio of

wavelengths

close to 2. The

phase

with

longer wavelength

A

m 230 260

A

_is

usually

denoted _as

A-phase,

whereas the second _{one as}

A/2~phase [2,

3]

The cross~sections of both

P~>~phases

are shown

schematically

in

figure

I. Their symmetnes

were characterized _in reference [3] ^Both

phases

_were observed _on vesicles under rather similar

thermodynamic

conditions

(sometimes

_{even on} the _same

vesicle),

however the A-

phase usually prefers

to emst _on surfaces of moderate curvature

RI

^'

(with

the radius

l~ being

in the range

roughly

500

A

< R

< 5 000

A _[3]),

_{whereas the}

A~phase

emsts _on flat _or

only slightly

curved surfaces. The _ongin of the above~mentioned correlation _seems to be not

explained

_yet.

The second

significant

difference between the two modifications of the

npple phase

refers

to the _nature of their defects. The

principal

defects of the

A-phase

_were shown _to be _a

376 JOURNAL DE PHYSIQUE II N 3

A/~

Fig I Schematic form of cross-sections of the A/2- and A-modifications of Pp~ phase

dischnation of the

strength

_±

1/2 (see

Ref [3] for _a

discussion),

wuich _is consistent with the symmetry ^of the

phase.

The _same kind of symmetry considerations for the

A/2-phase

leads to the conclusion that the minimal disclination's

strength

_{is ±} I, wuich _was indeed

objerved.

However

integer~strength

dischnations of the

A/2-phase

consist of two

half-integer

discli~

nations

(like

those of the

A~phase)

connected

by

_a defect line of the

length id

_» A

(see

ref. [3]

and

Fig

2)« The

origin

of this

peculiar

defect _structure of

A/2~phase

_was not yet clanfied In tuis paper our main

goal

_is to construct _a

phenomenological theory

of the coexistence of the

A/2~phase

and

A~phase,

which would account for both above~mentioned

peculianties

of these modifications of the

P~,~phases

curvature

dependence

of their relative

stability

and unusual structure of defects. In _{a more} quantitative manner the

problem

_can be formulated _as follows. the free energy difference between A~ and

A/2~phases

_is

obviously

rather small.

AF

=

F~-FA/2«F~,

then, what is the relation

(if any)

between the smallness of the parameter

AF/F~

Ml and two small parameters

A/l~

and A

lid

which charactenze the

macroscopic properties of

P~~~phases

^?

1-j

j

z~

Fig 2 Genenc topological defect of the A/2~phase. A pair of half~integer disclinations

We shall show that these two

apparently

unrelated parameters can be _in fact determined _in

terms of the ratio

AF/F~

and

geometncal

parameters ^{of the}

A/2~phase.

We

begin

_our

analysis

with _a

simple geometncal picture

of

A/2~and A~phases

which _was

proposed

_in

[2,

3],

namely:

the

A~phase

_is

composed by

_{an inversion} of each second

elementary

cell _» of the

A/2-phase,

_as shown _m

figure

3. Then _{we assume}

(and

this _is the

key

point of _our

analysis)

that the

stacking angles

between

straight~lme

segments of the

bflayer

_are

the _{same m} both A~ and

A/2-phases.

This _{means, m}

particular,

that the

angles

denoted _as 2 _y and 2 y' m

figure

3 _{are m} fact

equal

_{: y}

= y'. This important assumption ongmates from _a very

simple

idea _: the

piece~wise

linear nature of the

Pp,-phases (i

e. the emstence of rather

different

length

scales involved _in the

problem)

points to the existence of different free _energy scales

corresponding

to these

length

scales In particular we suppose that the most part of the free energy difference between Pp~ ^and

Lp,-phases

is related with the formation of _{« comers} of the sawtooth~hke structure. The _space scale of these _{« comers » is} much shorter than the

distance between them, therefore the forces which _are

responsible

for their formation _can

hardly

be modified

noticeably

when the

A/2~

to

A~phase

transition takes

place.

8 C

~~ ^a

A b _a

S'

,

C' _.-

i 1

Fig 3. Elementary cell of the A-phase composed from two A/2-phase elementary cells, the angles y and y' _are considered _to be equal (see text)

Now _{one can see}

immediately

that the

A~phase

should be

spontaneously

curved.

simple geometncal

consideration shows that the points A', O', D~ _can be put on the _same circle with the radius

J~~=

^A

A(ab-c~)

~~~~

( (fl

_"

)/2)

^~

2(b

_{a c}

(I)

where A

/2

_is the

wavelength

of

A/2-phase

and other parameters are defined _in

figure

3 the second

(approximate) equality

_in equation

(I)

is valid if (as we shall

assume)

the asymmetry is

weak

(fl a)

Ml- Let _us emphasize that _in the limit of _zero asymmetry

(a

_-

fl)

the spontaneous curvature llj~^' goes ^to zero, as it

obviously

should. at _{a -} fl _any ^difference between

A/2-

and

A-phases disappears. ~Note

also that small

(macroscopic)

curvature

lli

^' should not be confused with much

higher

local curvature that could be defined e-g- using points B', C' and O' _in

figure 3)

Thus _we conclude that the radius of the spontaneous

378 JOURNAL DE PHYSIQUE II N 3

curvature for the

A-phase

_{is gJven} by

llj

The

curvature-dependent

free

energies

for both

Pp>-phases

can be wntten as follows _:

FA/2(R)

₌

Fi12

+

~°

~

(2)

F~(R)=F(+~ ((-~)~- _{~~=F(+ ~~-}

^~

₍₃₎

o

2Ro

R Ro

where K _is the

bending ngldity

of the

bilayer (considered

to be the _{same in} both

Pp,-phases)

^{In the} case of flat

bilayer (R

- oJ

)

the

A/2-phase

_was the

only

_one observed, _so

we conclude with equations

(2), (3)

that

F(

~ F(~~. ^{On the other} hand, the

A-phase

should have _a lower free energy when the curvature of the

bilayer

_is

equal

to the spontaneous

curvature

llj,

_i-e-

F(

_F

(~~ <

K/R(. ₍₄₎

Thus _{one can} conclude that at _some intermediate curvature

Ep

^' _<

Ri

^' the first-order

phase

transition between

A/2~phase

and

A~phase

should take

place;

the transition curvature

1Q

^' _is ^determined

by

the relation

F~~2(R~)

=

F~(R~),

which leads to

~

^{K K 2}

c(b

_a

)

(l~i _{l~i12) R0} (l~i l~i12) ~(ab

C~) ~~~

Thus _we have found the relation between the free _energy difference

AF°

=

F(~~ F(

_and

the « transition curvature radius A~. ^However it seems to be difficult _to check equation

(4) expenmentally,

due _to the absence of

directly

obtained data for the value of

AF°.

Nevertheless, _{as we} shall show, this value _can be estimated from the consideration of the defect structure of the

A/2-phase.

The appearance of the one~dimensional

penodicity,

^associated with the

npple phases,

means the spontaneous

breaking

of both translational

(m

the direction of

npple)

and

onentational symmetnes existing in the

Lp,~phase.

The

topologiial

_defects, associated with these types ^of spontaneous symmetry

breaking,

_are dislocations and disclinations

respectively.

The

phenomenological

free _energy associated with the

inhomogeneous npple

state can be written as follows

az(w )

=

d2xjA _{( (vw}

₎₂

_{(2 «IA}

_{)2)2 + B}

(v2w )2j (6)

where A and B _are

phenomenological

_{constants, p} is the

phase

of scalar order parameter

~ = ~o

f(p), f(p

₊ 2

ar)

=

f(p),

which descnbes the

npple~phase

modulation.

Unper-

turbed

npple phase

_is descnbed

by

_p

(x)

distribution

p(x)

=

2 arnxA, where _{n is} the unit vector

determining

the direction of the modulation. For the

elastically perturbed npple

state

n ₌ Vp ^The form of the free energy

(6)

is _a genenc one for the systems ^with one~dimensional modulation with _an

arbitrary

direction n (determined

by

the spontaneous symmetry

breaking) [11].

It _is easy ^to show

~but

_we shall not dwell _upon this point, see e-g-

[12])

that the free energy of _an isolated dislocation calculated with the free _energy

(6)

is finite

(contrary

to the

case of usual

crystal

where it would be

loganthmically divergent).

Tuerefore the

equilibrium density

of dislocations _is finite and _so the constraint rot _n

= 0 _can be relaxed

Thus _{we can} rewrite the free _energy of

long~wavelength

modulations

(6) (with

dislocations taken into

account)

in terms of _an

arbitrary 2~component

vector n :

fF(n

=

J/2 d~x(V~

n

)~. ₍₇₎

Now _we should remind the difference between structures of defects _in the A/2~ ^{and A~}

modifications of the

npple-phase

[3].

A-phase

is symmetnc with respect to the inversion

n - n, therefore the minimal

strength

of

A~phase

disclinations _{is ±}

1/2,

which

corresponds

to the solutions for _n of the form _n

= (cos

o/2,

_{± sin}

0/2)

with 0 being _a

polar

coordinate

angle

defined around the dlsclination. The disclinations with the _same

sign

of the _n rotation

repel

each other

(as

well _as s1mllar

topological

defects _: dislocations _in

crystals,

vortices _in

superfluid liquid, etc.)

_so that the free _energy of two

+1/2

disclinations

depends

_on the distance f _{between them}

as.

E~(I)

=

jJln(ilA) (8)

with the

npple wave-length

A

being

the short~scale cut-off for the effective free energy

(7).

Turning

to the

A/2~phase

we note that _now the symmetry ^with respect to n _~ n is absent, _so that the

m1nimaLstrength

disclinations _are those with

strengths

_± I. However, the observed [3]

objects

are not

point-like

± I dischnations, but

couples

of ₊

1/2

_or

1/2

disclinations with

a line defect between them. Tuis is not too surpnsing, because the free energy difference between two

Pp ^<-phase

modifications _is rather low and it _can be relevant _on

sufficiently large

scales

only. Being

_{more precise, we} can now estimate the free energy of two

+1/2

disclinations _in the

A/2-phase

as

EA/~(f)

₌

^~

^{J In}

((IA

_{) + at} At _AF

(9)

where the second term in equation

(9)

_is just the free energy of _a linear defect of the

A/2-

structure which should emst between ₊

1/2

disclinations due _{to n}

- n asymmetry ^{of this}

structure. Here AF _is the free _energy

density

difference

F~-F~~2 (that

_is

equal

to

AF°

_if

we consider the _case of

planar bilayer)

and _at f

(aj

_{is some} number of the order of

I)

_is

the _estimate for the disturbed _area. The minimization of

E~j~(f)

^with respect to f leads

immediately

to the

following

result for the

equilibrium

_spacing

id

between

halGinteger

disclinations _:

id

=

$

^~

(10)

4 ai A

(F~

_F~~~)

To estimate the value of the _« stiffness J

(with

the

dlmdnsionality

of

energy)

_{one can} note that J _is

roughly

the _core energy of _a

minimal-strength

dischnation, _on the other hand the

core of such _a disclination _can be descnbed _{as a}

region (of

_{an area m} A where _any

ripple

structure is broken. Therefore _{we can} estimate the value of J _as

a~(F~(

A^~

(a~

is some

numencal coefficient of the order of

I)

where

F~

is a free energy

density

associated with the formation of _a

npple

structure. For the value of

F~

_{one can} take difference _in free energies between

Pp>-phase (irrespectively

of its

modifications)

and the continuation of the

Lp>-phase

at the _same temperature i e. we estimate

FR

as

F~ = a~

~~ AH I I

Tcj

where AH _is the

enthalpy

_jump at the

Lp,

_-

Pp,

transition, a~ m I,

J~,

is the temperature ^of this transition and T

= T~, ⁺ AT _is actual temperature ^of

Pp~phase. ^Finally

we can rewrite equation

(9)

_as

f~

=

(~

^~~

(~

^~~

~' cj

(FA

_I~A/2)

380 JOURNAL DE PHYSIQUE II N 3

With equations

(5)

and

(I I)

we can exclude _an unknown value of

(F( _F(j~)

_{and find the}

relation between measurable parameters

only.

In the

simplest

_case when _we consider the defect structure _on

planar bilayers,

the result _is

R

lid

a j

~'

_~

a

C(b

_a

) (i

_K

~ Ro AT AH A ^' ~~ _~²

~

_{~ ~~} 2

(12)

where

aj=4ajla~a~.

Thus _we have obtained the relation between the parameters

l~

^and

id,

which charactenze _a competition between

A/2

and

A-phases,

_in terms of

geometncal

and

thermodynamic

parameters ^of the

A/2-phase only,

_in order _to calculate the value of the numencal parameter _aj some more detailed

theory

should be

developed.

To

conclude,

_we have

developed

_a

phenomenological theory

of

phase

coexistence between the

A/2~

and the A~modifications of the

ripple (Pp>) phase

^of

lipid bilayers.

We have shown that the

A-phase

_{possesses a} spontaneous curvature whose value

lli~

_can be

expressed

_in

terms of the

geometrical

_parameters of the

A/2~phase.

The cntical value of _curvature

A~~^'

(which

makes the

A~phase

_more stable than the

A/2-phase) depends

_on lli ^' and the free energy difference

AF°

=

F(

F

(~~ for the

planar

membranes

(see Eq. (5)).

It _was also shown that the smallness of

AF°

_{leads to}

an unusual nature of disclinations _in the

A/2~phase,

consisting ^of two

half-integer

dischnations

separated by

_a

large

distance

id

_{» A~,~} The

increase of AF leads _to the decrease both of A~ ^and

f~,

the

A~/fd

can be

expressed (see

Eq (12))

_in terms of

llj

and

thermodynamic

_parameters of

A/2-phase (it

is important for

expenmental implications

that all these parameters can be obtained

by

measurements _on the

A/2-phase

alone, which exists _in multilamellar

form)

These results _{are in}

qualitative

agreement with the

expenmental

observations [3] that the addition of _a small concentration of

some

impurities

leads both to the

disappearance

of the

A-phase

and to the decrease of the

id

value,

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