- Email: [email protected]
Curvature-induced polarization of bilayer lipid membranes
Volume 139, number 3,4
PHYSICS LETTERS A
31 July 1989
CURVATURE-INDUCED POLARIZATION OF BILAYER LIPID MEMBRANES A. DERZHANSKI Liquid Crystal Department, Institute of Solid State Physics, BulgarianAcademy of Sciences, Sofia, Bulgaria Received 3 March 1989; accepted for publication 17 May 1989 Communicated by D. Bloch
The flexoelectric coefficient of a spherically deformed bilayer lipid membrane is considered as the ratio between the deformation-induced transmembrane potential and the curvature. The calculations are carried out on the basis of general electrodynamic considerations without using the analogy with thermotropic liquid crystals.
In 1969 Meyer [1] formulated the idea about the flexoelectricity of liquid crystals as a linear recip rocal correlation between the orientational defor mation and the electric polarization in a liquid crys-
-_______
~
tal medium. He proposed a molecular mechanism of this phenomenon. On the basis of the liquid-crystal concept for biological and model bilayer membranes Petrov [2] suggested in 1975 the existence of a flexoelectric effect in these objects. In a series of subsequent theoretical and experimental investigations of Petrov et al. the existence ofthis phenomenon has been proved. Various molecular mechanisms, leading to its appearance and various sides of its macroscopic manifestations have been studied. The analogy with the flexoelectricity in thermotropic liquid crystals lies at the basis of these investigations. However, the existence of some essential dimensional differences between the bulk thermotropic liquid crystals and the bilayer membranes led to some unconvincing and even debatable results. The aim of this work is to find a correlation between the curvature of a deformed lipid bilayer and the electric potential difference across the membrane resulting from this deformation, using the laws of electrostatics and without analogies with other phenomena. Let us consider an element of a spherically deformed bilayer lipid membrane, which is restricted by a narrow cone with apex in the center of the curvature 0, corresponding to a small space angle 0 (fig. 1). Let R1 and R2 be the radii of two spherical sur170
r
)R
R
L~j
0
)
R2
Fig. 1. A small central element of a spherically-deformed membrane.
faces S1 and S2, with a common center 0 separating the membrane from the surrounding’conductive medium for example an electrolyte solution. Then, in the space outside the membrane, at points at a distance R from the center of the spherical deformation, the field strength E and the density of the space charge a are equal to zero: —
E=0,
a=0
at0
(I)
This condition is used for the definition of the boundary surfaces S1 and S2 [3]. If the membrane contains charged lipids, in the aqueous medium near the hydrophilic heads, a space charge of counterions is formed. In that case condition (I) requires the boundary surface to be situated in the aqueous medium at a distance from the lipid molecules. In this case the membrane thickness R2—R1 will exceed the lipid bilayer thickness by a few Debye lengths. Some uncertainty in defining this thickness is of no great importance since it does not participate considerably in our calculations.
0375-960 1/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 139, number 3,4
PHYSICS LETTERSA
Inside the membrane, when R1
J
EdS=
a(r) dv.
S
(2)
V
Here r is the radial coordinate of a point of the element and S and V denote integration over the surface and the volume of this subelement. Eq. (2) can be represented as R
2ØE(R)~(R)=Ø r2a( R
Ri
r) dr.
(3)
Let R 0 be the radius of a sphere, parallel to the a!ready defined boundary spheres and situated inside the membrane. One may assume that R0= ~(R1 +R2). Now we introduce a reduced2a(r). electricExpressing density q arelated to q, a through R~q(r)=r through we obtain
31 July 1989
Here C is the specific capacity of the membrane. Introducing dZ in (5) we obtain R2R
~U=
J$
q(r) drdZ(R)
Ri Ri
J
R2
=z(R2)
J
R2
q(r) dr—
Ri
q(R)Z(R) dR.
(8)
Ri
The first term on the right side at (8) is zero due to the electric neutrality of the membrane (J~q(r) dr=0). This expression is mathematically correct, but it lacks physical transparency. We shall transform it to quantities associated with the molecular structureand the supramolecular organization of the lipids building the membrane and which can be determined exwellknown properties of lipid bilayers. Since the inperimentally. For this purpose we consider some tegration is carried out in a narrow region around a sphere with a radius R0, we shall use the variable p defined as R = R0 +p. The new coordinate p expresses the distance to the medium surface S~and its sign on whether point is inside or outside depends that surface. Withouttherestrictions we can represent the function q(p) as a superpositionof an even
R
f
R~ E(R)__R2(R) ~ q(r)dr.
(4)
and an odd function: q=q~+qo, q~(—p)=q~(p),
RI
For the potential jump across the membrane we obtain
q
0 ( —p)
=
—
q0 (p).
(9)
We can assume that the even function qE(p) represents the electric charge distribution in the mem-
R2
J
AU_~ E(R) dR Ri R2
R
R~ 2~(R)~ q(r) drdR.
~R
= Ri
(5)
Ri
Now we introduce a new function Z(R): R~dR dZ(R). R2(R) =
(6)
This function will be referred to as capacitive impedance, since it exhibits the following property: Z(R
2) —Z(RI )
(7)
brane in its undeformed state, while the odd function q0 (p) exhibits the deviation from this symmetric distribution caused by the membrane deformation. The electric neutrality of the membrane as a whole requires a zero integral area of both branches of the function q~(p). For simplicity we assume that the membrane element in its undeformed state has two couples of opposite charges symmetrically situated on the two sides of the neutral surface (fig. 2, top). Obviously, any other distribution of the charges can be represented as a sum or an integral of such couples. We assume as well that on deformation of the membrane this symmetry is broken. This leads to the appearance of 171
Volume 139. number 3,4
PHYSICS LETTERS A
31 July 1989
Now we consider the charge distributions q~(p) and q0(p) expressed through the ô-functions shown in fig. 2. For their superposition q we obtain
~
~LJ~JL
q=Q[~(+p’ )—ô(+p” ) +ö( —p )—~(—p”)] R11
cto ________
-
~ —p’ ) I
R0
—
(10)
-
~•
~
-/‘ ______
j’
,
d
I
d
H
~
Fig. 2. Graphic presentation of a model distribution of the membrane electric density q(p) and its symmetric and asymmetric components qE(p) and q0(p).
two couples of opposite charges, situated symmetrically with respect to the medium surface. On basis of geometric and physical considerations we can assume that the magnitude of these peaks in the odd function q~3(p) is proportional to the membrane curvature expressed by the radius of the medium surface Rç. To the function q~(p)we add one more possible couple of opposite charges, symmetrically situated with respect to the medium surface, which have no analogue in an undeformed membrane. This models the appearance of opposite charges on the two sides of the membrane due to the transfer of lipid molecules from the inner to the outer side of the deformed membrane by means of flip-flop or flow of lipid material through thermally induced pores [4]. In addition we note that the electric charge of the ionized lipid molecules of each monolayer of the membrane is neutralized not only by the adjacent counterion space charge, but partially by the space charge on the other side of the membrane. Therefore the neutrality of the deformed membrane as a whole (the two molecular layers and the adjacent region of space charges) does not exclude the factthat its parts lying on both sides of the neutral surface may be oppositely charged. Any more complex type of q0(p) can be expressed as a sum or an integral of similar elementary models. 172
Here the first term corresponds to q~(p),while the second and the third correspond to q0(p). a and fi are constants. Because of physical considerations they are assumed to be 0
where 1 =—Q[Z(—p’)—Z(—p”)+Z(p’)—Z(p”)] ~, =
—2
Q[Z(
—~
) —z( —~O)
R0
-
+ Z(p —2
) Z(p”)] —
Q*[Z(~)
~, =
z( _d*)
.
(11)
R11 According to the theorem of average values: Z(p” ) Z(p’ ) =Z’ (d) (p’ —p’ ) =Z’ (d)4. —
Here d is the coordinate of a point inside the interval (p’, p”). The following estimations of the quantities in the above formulae can be used: 4 1 A, d~10 A, R~ 1 mm. Using the definition of Z, assuming for simplicity that e ( d) = ( d), replacing Q4 by the electric dipole j.t, and neglecting the terms containing dIR0 of second or higher order we obtain the following expression for I~: —
Volume 139, number 3,4
PHYSICS LETTERS A
“ =—Q[Z’(d)—Z’(—d)]A 2 QRQ
(
1 e(d) (R
2
—
1 (—d) (R
0+d) 2—(R 2 (d) (R0—d) 2 (R 0+d)2 0 +d) 0 d) 4dQR~4p 1 e(d) (R 2(R 2 0+d) 0—d) 4dQ4p 4d~i R 0(d) R0(d)
— —
)
~
0—d)
QR~L1(R —
2
—
(12)
31 July 1989
tween L~Uand the membrane curvature 1 /R0. According to the definition suggested by Bivas [3] the ratio between these two quantities is considered as a flexoelectric coefficient of the membrane. The first term of this expression, proportional to ~i, corresponds to the expressions obtained earlier for the dipole contribution to the flexocoefficient of a bilayer lipid membrane [5]. When a=0, we have the expression for blocked flip-flop. a=l corresponds to the case of entirely free flip-flop. The second term, proportional to Q*, corresponds to the
—
—
charge contribution to the membrane flexoeffect. By
—
The expression for 12 can be obtained in a similar way: ‘2 =
+
R0 Q[Z’ (d) —Z’ (—d) ]4
of nearby situated dipole moments with oppositetwo orientation.
~-~‘
2 / 1 (~~d) (R0 +d)
+ 2ad R0
analogy, the contribution due to quadrupole moments of the lipid molecules may be obtained since the quadrupole moment can be expressed as a sum
quantitative the dipole Finally, weconsiderations offer a short for comment on and (15)charge with contributions. As we have already mentioned, the real distribu-
+ ( —d) (R 0 _d)2)4 2) 2 (R2~+ d R 2ad (R 2 0i(d) QR~4p(R0 +d) 0 —d) 4ad~ R 4 =+ R 0(d) R 0(d)
tion of the couples of asymmetric charges in the membrane structure may in lead the appearance of some similar dipole terms thistoexpression. Each of them has its own value of~and eventually of . Some of them correspond to the intramolecular electric di-
_____
(13)
In accordance with (7), Z(d*)_Z(_d*) is a quantity equal to the reciprocal value of the capacity of the membrane measured between the spherical surfaces with radii R0+d* and R0.~d*.If these surfaces coincide with the boundary of the membrane, C is the membrane capacity measured experimentally. In this case for 13 we obtain 13
=
Q* [Z(d*)—Z( .d*) I
—2 R0
=
—2
R0
Q*
(14)
C~
Adding the obtained expressions for I~,12, 13 and combining the first two of them, we obtain for the potential jump through the deformed membrane:
~u
~
)~
4d~ /3~j*Q* l—a)-—-+2 e(d) c
pole moments. They of aremagnitude due to theofdistribution of charges of an order one electron charge at a distance of about 1 A. These dipoles are situated in the interior of the lipid layers where the value of the dielectric permeability slightly exceeds ~. The electric charges of the ionized lipid molecules and the counterions from the adjacent aqueous layer form dipoles about ten times greater than the former since the distance between the charges is respectively higher. The dielectric constant at the place of their location is also an order of magnitude higher than in the first case. That is why their contribution in the However, overall expression the charge is ofcontribution the same order mayofhave magnitude. a considerable greater value. The electric charges Q* Q are divided by a lipid bilayer 50—100 A thick and with a dielectric permeability e This work was supported by the Bulgarian Mm-
(15)
istry of Culture, Science and Education.
This expression shows the linear correlation be173
Volume 139, number 3,4
PHYSICS LETTERS A
31 July 1989
References
131 1. Bivas, private communication. [41A. Derzhanski,A.G. Petrov and Y.V. Pavloff. J. Phys.
[1]R.B.Meyer,Phys.Rev.Lett.22(1969)918. [21 A.G. Petrov, in: Physical and chemical bases of biological informationtransfer (Plenum, New York, 1975) p. 111.
Lett.42(198l)L1l9. [51A.G. Petrov, Nuovo Cimento D 3 (1984) 174.
174
(Paris)
PHYSICS LETTERS A
31 July 1989
CURVATURE-INDUCED POLARIZATION OF BILAYER LIPID MEMBRANES A. DERZHANSKI Liquid Crystal Department, Institute of Solid State Physics, BulgarianAcademy of Sciences, Sofia, Bulgaria Received 3 March 1989; accepted for publication 17 May 1989 Communicated by D. Bloch
The flexoelectric coefficient of a spherically deformed bilayer lipid membrane is considered as the ratio between the deformation-induced transmembrane potential and the curvature. The calculations are carried out on the basis of general electrodynamic considerations without using the analogy with thermotropic liquid crystals.
In 1969 Meyer [1] formulated the idea about the flexoelectricity of liquid crystals as a linear recip rocal correlation between the orientational defor mation and the electric polarization in a liquid crys-
-_______
~
tal medium. He proposed a molecular mechanism of this phenomenon. On the basis of the liquid-crystal concept for biological and model bilayer membranes Petrov [2] suggested in 1975 the existence of a flexoelectric effect in these objects. In a series of subsequent theoretical and experimental investigations of Petrov et al. the existence ofthis phenomenon has been proved. Various molecular mechanisms, leading to its appearance and various sides of its macroscopic manifestations have been studied. The analogy with the flexoelectricity in thermotropic liquid crystals lies at the basis of these investigations. However, the existence of some essential dimensional differences between the bulk thermotropic liquid crystals and the bilayer membranes led to some unconvincing and even debatable results. The aim of this work is to find a correlation between the curvature of a deformed lipid bilayer and the electric potential difference across the membrane resulting from this deformation, using the laws of electrostatics and without analogies with other phenomena. Let us consider an element of a spherically deformed bilayer lipid membrane, which is restricted by a narrow cone with apex in the center of the curvature 0, corresponding to a small space angle 0 (fig. 1). Let R1 and R2 be the radii of two spherical sur170
r
)R
R
L~j
0
)
R2
Fig. 1. A small central element of a spherically-deformed membrane.
faces S1 and S2, with a common center 0 separating the membrane from the surrounding’conductive medium for example an electrolyte solution. Then, in the space outside the membrane, at points at a distance R from the center of the spherical deformation, the field strength E and the density of the space charge a are equal to zero: —
E=0,
a=0
at0
(I)
This condition is used for the definition of the boundary surfaces S1 and S2 [3]. If the membrane contains charged lipids, in the aqueous medium near the hydrophilic heads, a space charge of counterions is formed. In that case condition (I) requires the boundary surface to be situated in the aqueous medium at a distance from the lipid molecules. In this case the membrane thickness R2—R1 will exceed the lipid bilayer thickness by a few Debye lengths. Some uncertainty in defining this thickness is of no great importance since it does not participate considerably in our calculations.
0375-960 1/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 139, number 3,4
PHYSICS LETTERSA
Inside the membrane, when R1
J
EdS=
a(r) dv.
S
(2)
V
Here r is the radial coordinate of a point of the element and S and V denote integration over the surface and the volume of this subelement. Eq. (2) can be represented as R
2ØE(R)~(R)=Ø r2a( R
Ri
r) dr.
(3)
Let R 0 be the radius of a sphere, parallel to the a!ready defined boundary spheres and situated inside the membrane. One may assume that R0= ~(R1 +R2). Now we introduce a reduced2a(r). electricExpressing density q arelated to q, a through R~q(r)=r through we obtain
31 July 1989
Here C is the specific capacity of the membrane. Introducing dZ in (5) we obtain R2R
~U=
J$
q(r) drdZ(R)
Ri Ri
J
R2
=z(R2)
J
R2
q(r) dr—
Ri
q(R)Z(R) dR.
(8)
Ri
The first term on the right side at (8) is zero due to the electric neutrality of the membrane (J~q(r) dr=0). This expression is mathematically correct, but it lacks physical transparency. We shall transform it to quantities associated with the molecular structureand the supramolecular organization of the lipids building the membrane and which can be determined exwellknown properties of lipid bilayers. Since the inperimentally. For this purpose we consider some tegration is carried out in a narrow region around a sphere with a radius R0, we shall use the variable p defined as R = R0 +p. The new coordinate p expresses the distance to the medium surface S~and its sign on whether point is inside or outside depends that surface. Withouttherestrictions we can represent the function q(p) as a superpositionof an even
R
f
R~ E(R)__R2(R) ~ q(r)dr.
(4)
and an odd function: q=q~+qo, q~(—p)=q~(p),
RI
For the potential jump across the membrane we obtain
q
0 ( —p)
=
—
q0 (p).
(9)
We can assume that the even function qE(p) represents the electric charge distribution in the mem-
R2
J
AU_~ E(R) dR Ri R2
R
R~ 2~(R)~ q(r) drdR.
~R
= Ri
(5)
Ri
Now we introduce a new function Z(R): R~dR dZ(R). R2(R) =
(6)
This function will be referred to as capacitive impedance, since it exhibits the following property: Z(R
2) —Z(RI )
(7)
brane in its undeformed state, while the odd function q0 (p) exhibits the deviation from this symmetric distribution caused by the membrane deformation. The electric neutrality of the membrane as a whole requires a zero integral area of both branches of the function q~(p). For simplicity we assume that the membrane element in its undeformed state has two couples of opposite charges symmetrically situated on the two sides of the neutral surface (fig. 2, top). Obviously, any other distribution of the charges can be represented as a sum or an integral of such couples. We assume as well that on deformation of the membrane this symmetry is broken. This leads to the appearance of 171
Volume 139. number 3,4
PHYSICS LETTERS A
31 July 1989
Now we consider the charge distributions q~(p) and q0(p) expressed through the ô-functions shown in fig. 2. For their superposition q we obtain
~
~LJ~JL
q=Q[~(+p’ )—ô(+p” ) +ö( —p )—~(—p”)] R11
cto ________
-
~ —p’ ) I
R0
—
(10)
-
~•
~
-/‘ ______
j’
,
d
I
d
H
~
Fig. 2. Graphic presentation of a model distribution of the membrane electric density q(p) and its symmetric and asymmetric components qE(p) and q0(p).
two couples of opposite charges, situated symmetrically with respect to the medium surface. On basis of geometric and physical considerations we can assume that the magnitude of these peaks in the odd function q~3(p) is proportional to the membrane curvature expressed by the radius of the medium surface Rç. To the function q~(p)we add one more possible couple of opposite charges, symmetrically situated with respect to the medium surface, which have no analogue in an undeformed membrane. This models the appearance of opposite charges on the two sides of the membrane due to the transfer of lipid molecules from the inner to the outer side of the deformed membrane by means of flip-flop or flow of lipid material through thermally induced pores [4]. In addition we note that the electric charge of the ionized lipid molecules of each monolayer of the membrane is neutralized not only by the adjacent counterion space charge, but partially by the space charge on the other side of the membrane. Therefore the neutrality of the deformed membrane as a whole (the two molecular layers and the adjacent region of space charges) does not exclude the factthat its parts lying on both sides of the neutral surface may be oppositely charged. Any more complex type of q0(p) can be expressed as a sum or an integral of similar elementary models. 172
Here the first term corresponds to q~(p),while the second and the third correspond to q0(p). a and fi are constants. Because of physical considerations they are assumed to be 0
where 1 =—Q[Z(—p’)—Z(—p”)+Z(p’)—Z(p”)] ~, =
—2
Q[Z(
—~
) —z( —~O)
R0
-
+ Z(p —2
) Z(p”)] —
Q*[Z(~)
~, =
z( _d*)
.
(11)
R11 According to the theorem of average values: Z(p” ) Z(p’ ) =Z’ (d) (p’ —p’ ) =Z’ (d)4. —
Here d is the coordinate of a point inside the interval (p’, p”). The following estimations of the quantities in the above formulae can be used: 4 1 A, d~10 A, R~ 1 mm. Using the definition of Z, assuming for simplicity that e ( d) = ( d), replacing Q4 by the electric dipole j.t, and neglecting the terms containing dIR0 of second or higher order we obtain the following expression for I~: —
Volume 139, number 3,4
PHYSICS LETTERS A
“ =—Q[Z’(d)—Z’(—d)]A 2 QRQ
(
1 e(d) (R
2
—
1 (—d) (R
0+d) 2—(R 2 (d) (R0—d) 2 (R 0+d)2 0 +d) 0 d) 4dQR~4p 1 e(d) (R 2(R 2 0+d) 0—d) 4dQ4p 4d~i R 0(d) R0(d)
— —
)
~
0—d)
QR~L1(R —
2
—
(12)
31 July 1989
tween L~Uand the membrane curvature 1 /R0. According to the definition suggested by Bivas [3] the ratio between these two quantities is considered as a flexoelectric coefficient of the membrane. The first term of this expression, proportional to ~i, corresponds to the expressions obtained earlier for the dipole contribution to the flexocoefficient of a bilayer lipid membrane [5]. When a=0, we have the expression for blocked flip-flop. a=l corresponds to the case of entirely free flip-flop. The second term, proportional to Q*, corresponds to the
—
—
charge contribution to the membrane flexoeffect. By
—
The expression for 12 can be obtained in a similar way: ‘2 =
+
R0 Q[Z’ (d) —Z’ (—d) ]4
of nearby situated dipole moments with oppositetwo orientation.
~-~‘
2 / 1 (~~d) (R0 +d)
+ 2ad R0
analogy, the contribution due to quadrupole moments of the lipid molecules may be obtained since the quadrupole moment can be expressed as a sum
quantitative the dipole Finally, weconsiderations offer a short for comment on and (15)charge with contributions. As we have already mentioned, the real distribu-
+ ( —d) (R 0 _d)2)4 2) 2 (R2~+ d R 2ad (R 2 0i(d) QR~4p(R0 +d) 0 —d) 4ad~ R 4 =+ R 0(d) R 0(d)
tion of the couples of asymmetric charges in the membrane structure may in lead the appearance of some similar dipole terms thistoexpression. Each of them has its own value of~and eventually of . Some of them correspond to the intramolecular electric di-
_____
(13)
In accordance with (7), Z(d*)_Z(_d*) is a quantity equal to the reciprocal value of the capacity of the membrane measured between the spherical surfaces with radii R0+d* and R0.~d*.If these surfaces coincide with the boundary of the membrane, C is the membrane capacity measured experimentally. In this case for 13 we obtain 13
=
Q* [Z(d*)—Z( .d*) I
—2 R0
=
—2
R0
Q*
(14)
C~
Adding the obtained expressions for I~,12, 13 and combining the first two of them, we obtain for the potential jump through the deformed membrane:
~u
~
)~
4d~ /3~j*Q* l—a)-—-+2 e(d) c
pole moments. They of aremagnitude due to theofdistribution of charges of an order one electron charge at a distance of about 1 A. These dipoles are situated in the interior of the lipid layers where the value of the dielectric permeability slightly exceeds ~. The electric charges of the ionized lipid molecules and the counterions from the adjacent aqueous layer form dipoles about ten times greater than the former since the distance between the charges is respectively higher. The dielectric constant at the place of their location is also an order of magnitude higher than in the first case. That is why their contribution in the However, overall expression the charge is ofcontribution the same order mayofhave magnitude. a considerable greater value. The electric charges Q* Q are divided by a lipid bilayer 50—100 A thick and with a dielectric permeability e This work was supported by the Bulgarian Mm-
(15)
istry of Culture, Science and Education.
This expression shows the linear correlation be173
Volume 139, number 3,4
PHYSICS LETTERS A
31 July 1989
References
131 1. Bivas, private communication. [41A. Derzhanski,A.G. Petrov and Y.V. Pavloff. J. Phys.
[1]R.B.Meyer,Phys.Rev.Lett.22(1969)918. [21 A.G. Petrov, in: Physical and chemical bases of biological informationtransfer (Plenum, New York, 1975) p. 111.
Lett.42(198l)L1l9. [51A.G. Petrov, Nuovo Cimento D 3 (1984) 174.
174
(Paris)