Lipid membranes with free edges

0.26 MB·English

by  Z. C. Tu

Lipid membranes with free edges
Zhan-chun Tu1,2, and Zhong-can Ou-Yang1,3,
∗ †
1Institute of Theoretical Physics, Academia Sinica,
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P.O.Box 2735 Beijing 100080, China
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2Graduate School, Academia Sinica, Beijing, China
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n 3Center for Advanced Study, Tsinghua University, Beijing 100084, China
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Abstract
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] Lipid membrane with freely exposed edge is regarded as smooth surface with curved boundary.
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o Exteriordifferentialformsareintroducedtodescribethesurfaceandtheboundarycurve. Thetotal
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t free energy is defined as the sum of Helfrich’s free energy and the surface and line tension energy.
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The equilibrium equation and boundary conditions of the membrane are derived by taking the
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n variationofthetotalfreeenergy. Theseequationscanalsobeappliedtothemembranewithseveral
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c freely exposed edges. Analytical and numerical solutions to these equations are obtained underthe
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axisymmetric condition. The numerical results can be used to explain recent experimental results
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0 obtained by Saitoh et al. [Proc. Natl. Acad. Sci. 95, 1026 (1998)].
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5 PACS numbers: 87.16.Dg, 02.40.Hw
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∗Email address: [email protected]
†Email address: [email protected]
1
I. INTRODUCTION
Theoretical study on shapes of closed lipid membranes made great progress two decades
ago. The shape equation of closed membranes was obtained in 1987 [1], with which the
biconcave discoidal shape of the red cell was naturally explained [2], and a ratio of √2 of
the two radii of a torus vesicle membrane was predicted [3] and confirmed by experiment
[4].
During the formation process of the cell, either material will be added to the edge or
the edge will heal itself so as to form closed structure. There are also metastable cup-like
equilibrium shapes of lipid membranes with free edges [5]. Recently, opening-up process
of liposomal membranes by talin [6, 7] has also been observed gives rise to the interest of
studying the equilibrium equation and boundary conditions of lipid membranes with free
exposed edges. Capovilla et al. first study this problem and give the equilibrium equation
and boundary conditions [8]. They also discuss the mechanical meaning of these equations
[8, 9].
The study of these cup-like structures enables us to understand the assembly process of
vesicles. Ju¨licher et al. suggest thatalinetension canbeassociatedwithadomainboundary
between two different phases of an inhomogeneous vesicle and leads to the budding [10]. For
simplicity, however, we will restrict our discussion on open homogenous vesicles.
In this paper, a lipid membrane with freely exposed edge is regarded as a differentiable
surface with a boundary curve. Exterior differential forms are introduced to describe the
surfaceandthecurve. ThetotalfreeenergyisdefinedasthesumofHelfrich’sfreeenergyand
the surface and line tension energy. The equilibrium equation and the boundary conditions
of the membrane are derived from the variation of the total free energy. These equations
can also be applied to the membrane with several freely exposed edges. This is another way
to obtain the results of Capovilla et al. Some solutions to the equations are obtained and
the corresponding shapes are shown. They can be used to explain some known experimental
results [6].
This paper is organized as follows: In Sec.II, we retrospect briefly the surface theory
expressed by exterior differential forms. In Sec.III, we introduce some basic properties of
Hodge star . In Sec.IV, we construct the variational theory of the surface and give some
∗
useful formulas. In Sec.V, we derive the equilibrium equation and boundary conditions of
2
the membrane from the variation of the total free energy. In Sec.VI, we suggest some special
solutionstotheequationsandshow theircorresponding shapes. InSec.VII, weput forwarda
numerical scheme to give some axisymmetric solutions as well as their corresponding shapes
to explain some experimental results. In Sec.VIII, we give a brief conclusion and prospect
the challenging work.
II. SURFACE THEORY EXPRESSED BY EXTERIOR DIFFERENTIAL FORMS
In this section, we retrospect briefly the surface theory expressed by exterior differential
forms. The details can be found in Ref.[11].
We regard a membrane with freely exposed edge as a differentiable and orientational
surface with a boundary curve C, as shown in Fig.1. At every point on the surface, we can
choose an orthogonal frame e ,e ,e with e e = δ and e being the normal vector. For
1 2 3 i j ij 3
·
a point in curve C, e is the tangent vector of C.
1
An infinitesimal tangent vector of the surface is defined as
dr = ω e +ω e , (1)
1 1 2 2
where d is an exterior differential operator, and ω ,ω are 1-differential forms. Moreover, we
1 2
define
de = ω e , (2)
i ij j
where ω satisfies ω = ω because of e e = δ .
ij ij ji i j ij
− ·
With dd = 0 and d(ω ω ) = dω ω ω dω , we have
1 2 1 2 1 2
∧ ∧ − ∧
dω = ω ω ; dω = ω ω ; ω ω +ω ω = 0; (3)
1 12 2 2 21 1 1 13 2 23
∧ ∧ ∧ ∧
and
dω = ω ω (i,j = 1,2,3), (4)
ij ik kj
∧
where the symbol “ ” represents the exterior product on which the most excellent expatia-
∧
tion may be the Ref.[12].
Eq.(3) and Cartan lemma imply that:
ω = aω +bω , ω = bω +cω . (5)
13 1 2 23 1 2
3
Therefore, we have
area element: dA = ω ω , (6)
1 2
∧
first fundamental form: I = dr dr = ω2+ω2, (7)
· 1 2
second fundamental form: II = dr de = aω2 +2bω ω +cω2, (8)
− · 3 1 1 2 2
a+c
mean curvature: H = , (9)
2
Gaussian curvature: K = ac b2. (10)
−
III. HODGE STAR
∗
In this part, we briefly introduce basic properties rather than the exact definition of
Hodge star [13] because we just use these properties in the following contents.
∗
If g,h are functions defined on 2D smooth surface M, then the following formulas are
valid:
f = fω ω ; (11)
1 2
∗ ∧
df = f ω +f ω , if df = f ω +f ω ; (12)
2 1 1 2 1 1 2 2
∗ −
d df = 2f, 2 is the Laplace-Beltrami operator. (13)
∗ ∇ ∇
We can easily prove that
(fd dg gd df) = (f dg g df) (14)
ZZ ∗ − ∗ I ∗ − ∗
M ∂M
through Stokes’s theorem and integration by parts.
IV. VARIATIONAL THEORY OF THE SURFACE
the variation of the surface is defined as:
δr = Ω e +Ω e , (15)
2 2 3 3
where the variation along e is unnecessary because it gives only an identity. Furthermore,
1
let
δe = Ω e , Ω = Ω . (16)
i ij j ij ji
−
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Operators d and δ are independent, thus dδ = δd. dδr = δdr implies that:
δω = Ω ω +Ω ω ω Ω , (17)
1 2 21 3 31 2 21
−
δω = dΩ +Ω ω ω Ω , (18)
2 2 3 32 1 12
−
dΩ = Ω ω +Ω ω Ω ω . (19)
3 13 1 23 2 2 23
−
Furthermore, dδe = δde implies that:
i i
δω = dΩ +Ω ω ω Ω . (20)
ij ij ik kj ik kj
−
It is necessary to point out that the properties of the operator δ are exactly similar to
those of the ordinary differential.
V. EQUILIBRIUM EQUATION OF THE MEMBRANE AND BOUNDARY CON-
DITIONS
The total free energy F of a membrane with an edge is defined as the sum of Helfrich’s
free energy[14, 15] F = [kc(2H +c )2 +k¯K]dA and the surface and line tension energy
H 2 0
RR ¯
F = λ dA + γ ds. Here k , k, c , λ and γ are constants. With the arc-length
sl C c 0
RR H
parameter ds = ω , the geodesic curvature k = ω /ds on C and the Gauss-Bonnet formula
1 g 12
KdA = 2π k ds, the total free energy and its variation are given
− C g
RR H
k
F = [ c(2H +c )2 +λ]ω ω +γ ω k¯ ω +2πk¯, (21)
0 1 2 1 12
ZZ 2 ∧ I − I
C C
and
k
δF = k (2H +c )δ(2H)ω ω + [ c(2H +c )2 +λ]δ(ω ω )
c 0 1 2 0 1 2
ZZ ∧ ZZ 2 ∧
¯
+ γ δω k δω , (22)
1 12
I − I
C C
respectively. From Eqs.(17) and (18), we can easily obtain:
δ(ω ω ) = δω ω +ω δω = d(Ω ω ) (2H)Ω ω ω . (23)
1 2 1 2 1 2 2 1 3 1 2
∧ ∧ ∧ − − ∧
Eqs.(5), (17), (18) and (20) lead to:
δ(2H)ω ω = δ(a+c)ω ω
1 2 1 2
∧ ∧
= 2(2H2 K)Ω ω ω +d(Ω ω Ω ω )
3 1 2 13 2 23 1
− ∧ −
+ aΩ dω bdΩ ω +bΩ dω +cdΩ ω . (24)
2 1 2 2 2 2 2 1
− ∧ ∧
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Thus we have:
δF = k (2H +c )[2(2H2 K)Ω ω ω +d(Ω ω Ω ω )
c 0 3 1 2 13 2 23 1
ZZ − ∧ −
+aΩ dω bdΩ ω +bΩ dω +cdΩ ω ]
2 1 2 2 2 2 2 1
− ∧ ∧
k
+ ( c(2H +c )2 +λ)[ d(Ω ω ) (2H)Ω ω ω ]
0 2 1 3 1 2
ZZ 2 − − ∧
¯
+γ [Ω ω +Ω ω ω Ω ] k [dΩ +Ω ω ω Ω ]. (25)
2 21 3 31 2 21 12 13 32 13 32
I − − I −
C C
If Ω = 0, then dΩ = Ω ω + Ω ω , dΩ = Ω ω + Ω ω . On curve C, ω = 0,
2 3 13 1 23 2 3 23 1 13 2 2
∗ −
ω = aω , ω = bω , Ω = Ω . Thus Eq.(25) is reduced to
31 1 32 1 3 C 3C
− − |
δF = [k (2H +c )(2H2 c H 2K) 2λH]Ω ω ω
c 0 0 3 1 2
ZZ − − − ∧
¯
+ k (2H +c )d dΩ γ aω Ω +k (bΩ aΩ )ω (26)
c 0 3 1 3 13 23 1
ZZ ∗ − I I −
C C
In terms of Eqs.(13) and (14), we have:
(2H +c )d dΩ = (2H +c ) dΩ Ω d(2H +c )+ Ω 2(2H +c )ω ω .
0 3 0 3 3 0 3 0 1 2
ZZ ∗ I ∗ −I ∗ ZZ ∇ ∧
C C
Using integration by parts and Stokes’s theorem, we arrive at bΩ ω = bdΩ =
C 13 1 C 3C
H H
Ω db. Thus
− C 3C
H
δF = [k (2H +c )(2H2 c H 2K) 2λH +k 2(2H +c )]Ω ω ω
c 0 0 c 0 3 1 2
ZZ − − − ∇ ∧
¯ ¯
[k (2H +c )+ka]Ω ω Ω [k d(2H +c )+γaω +kdb]. (27)
c 0 23 1 3C c 0 1
−I −I ∗
C C
It follows that
k (2H +c )(2H2 c H 2K) 2λH +k 2(2H) = 0, (28)
c 0 0 c
− − − ∇
¯
[k (2H +c )+ka] = 0, (29)
c 0 C
(cid:12)
[k d(2H)+γaω(cid:12)+k¯db] = 0. (30)
c ∗ 1 C
(cid:12)
(cid:12)
The mechanical meanings of the above three equations are: Eq.(28) is the equilibrium
equation of the membrane; Eq.(29) is the moment equilibrium equation of points on C
around the axis e ; and Eq.(30) is the force equilibrium equation of points on C along the
1
direction of e [8, 9]. It is not surprising that Eq.(29) contains the factor k¯ because it is
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related to the bend energy in Helfrich’s free energy. However, it is difficult to understand
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¯ ¯
why k is also included in Eq.(30). In fact, the term kdb in Eq.(30) represents the shear stress
which also contributes to the bend energy in Helfrich’s free energy.
In fact, a = k and b = τ are the normal curvature and the geodesic torsion of curve C,
n g
respectively, and d(2H) = e (2H)ω . Thus Eqs.(29) and (30) become
2 1
∗ − ·∇
¯
[k (2H +c )+kk ] = 0, (31)
c 0 n C
(cid:12) dτ
[ k e (2H)+γk(cid:12) +k¯ g] = 0, (32)
c 2 n
− ·∇ ds (cid:12)
(cid:12)C
(cid:12)
(cid:12)
respectively.
If Ω = 0, then dΩ = Ω ω +Ω ω Ω ω = (Ω bΩ )ω +(Ω cΩ )ω = 0. It
3 3 13 1 23 2 2 23 13 2 1 23 2 2
− − −
leads to Ω = bΩ and Ω = cΩ .
13 2 23 2
δF = k (2H +c )[aΩ dω bdΩ ω +bΩ dω +cdΩ ω +d(Ω ω Ω ω )]
c 0 2 1 2 2 2 2 2 1 13 2 23 1
ZZ − ∧ ∧ −
k
+ [ c(2H +c )2 +λ][ d(Ω ω )]+γ Ω ω k¯ KΩ ω . (33)
0 2 1 2 21 2 1
ZZ 2 − I − I
C C
Otherwise, ω = aω +bω implies that: adω +db ω +2bdω cdω = da ω . Thus
13 1 2 1 2 2 1 1
∧ − − ∧
aΩ dω bdΩ ω +bΩ dω +cdΩ ω +d(Ω ω Ω ω )
2 1 2 2 2 2 2 1 13 2 23 1
− ∧ ∧ −
= d(a+c) Ω ω = d(2H +c ) Ω ω (c is a constant). (34)
2 1 0 2 1 0
− ∧ − ∧
Therefore
k
δF = [ c(2H +c )2Ω ω k¯KΩ ω λΩ ω +γΩ ω ]. (35)
0 2 1 2 1 2 1 2 21
I − 2 − −
C
It follows that:
k
[ c(2H +c )2 +k¯K +λ+γk ] = 0, (36)
0 g
2 (cid:12)
(cid:12)C
(cid:12)
because of ω21 = kgω1 on C. This equation is the force (cid:12)equilibrium equation of points on
−
C along the direction of e [8, 9].
2
Eqs.(28), (31), (32) and (36) are the equilibrium equation and boundary conditions of
the membrane. They correspond to Eqs. (17), (60), (59) and (58) in Ref.[8], respectively.
In fact, these equations can be applied to the membrane with several edges also, because in
above discussion the edge is a general edge. But it is necessary to notice the right direction
of the edges. We call these equations the basic equations.
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VI. SPECIAL SOLUTIONS TO BASIC EQUATIONS AND THEIR CORRE-
SPONDING SHAPES
In this section, we will give some special solutions to the basic equations together with
their corresponding shapes. For convenience, we consider the axial symmetric surface with
axial symmetric edges. Zhou has considered the similar problem in his PhD thesis [16].
If expressing the surface in 3-dimensional space as r = vcosu,vsinu,z(v) we obtain
{ }
2H = (sinψ+cosψdψ), K = sinψcosψdψ, (2H) = r2 d (sinψ+cosψdψ) and 2(2H) =
− v dv v dv ∇ −sec2ψdv v dv ∇
cosψ d [vcosψ d (sinψ + cosψdψ)], where ψ = arctan[dz(v)], r = ∂r/∂v. Define t as the
− v dv dv v dv dv 2
direction of curve C and r = ∂r/∂u. Obviously, t is parallel or antiparallel to r on curve
1 1
C. Introduce a notation sn, such that sn = +1 if t is parallel to r , and sn = 1 if not.
1
−
Thus e = sn r2 and e (2H) = sncosψ ∂ (sinψ +cosψdψ) on curve C. For curve C,
2 secψ 2 ·∇ − ∂v v dv
k = sinψ, τ = 0, and k = sncosψ. Thus we can reduce Eqs.(28), (31), (32) and (36)
n − v g g − v
to:
sinψ dψ 1 sinψ dψ 1 sinψ dψ 2sinψcosψdψ
k ( + cosψ c )[ ( +cosψ )2 + c ( +cosψ ) ]
c 0 0
v dv − 2 v dv 2 v dv − v dv
sinψ dψ cosψ d d sinψ dψ
λ( +cosψ )+k [vcosψ ( +cosψ )] = 0, (37)
c
− v dv v dv dv v dv
sinψ dψ sinψ
¯
k ( +cosψ c )+k = 0, (38)
c 0
(cid:20) v dv − v (cid:21)
C
d sinψ sinψ
¯
snkcosψ ( )+γ = 0, (39)
(cid:20)− dv v v (cid:21)
C
k¯2 sinψ sinψcosψdψ cosψ
( )2 +k¯ +λ snγ = 0. (40)
(cid:20)2k v v dv − v (cid:21)
c C
In fact, in above four equations only three of them are independent. We usually keep
Eqs.(37), (38) and (40) for the axial symmetric surface. For the general case, we conjecture
that there are also three independent equations among Eqs. (28), (31), (32) and (36).
Eq.(37) isthesame astheequilibrium equationofaxisymmetrical closed membranes [17, 18].
In Ref [18], a large number of numerical solutions to Eq.(37) as well as their classifications
are discussed.
Next, Let us consider some analytical solutions and their corresponding shapes. We
merely try to show that these shapes exist, but not to compare with experiments. Therefore,
we only consider analytical solutions for some specific values of parameters.
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A. The constant mean curvature surface
The constant mean curvature surfaces satisfy Eq.(28) for proper values of k , c , K, and
c 0
¯
λ. But Eqs.(31), (32) and (36) imply 2H +c = 0, k = 0, and kK +γk = 0 on curve C if
0 n g
¯
k , k, and γ are nonzero.
c
For axial symmetric surfaces, k = 0 requires sinψ = 0. Therefore K = 0 which requires
n
k = 0. Only straight line can satisfy these conditions. It contradicts to the fact that C is
g
a closed curve. Therefore, there is no axial symmetric open membrane with constant mean
curvature.
B. The central part of a torus
When λ = 0, c = 0, the condition sinψ = av +√2 satisfies Eq.(37). It corresponds to
0
a torus [15]. Eqs.(38) and (40) determine the position of the edge v = √2(kc+k¯), where
e −a(2kc+k¯)
a = γ(kc+k¯)√2(2kc2+4kck¯+k¯2). If we let k = k¯ and kc = 2√14 (unit: length dimension),
− (2kc+k¯)kck¯ c γ 3
it leads to 1/a = 1 and v = 2√2 (unit: length dimension). Thus the shape is the
− e 3
central part of a torus as shown in Fig.2. This shape is topologically equivalent to a ring as
shown in Fig.3.
C. A cup
If we let sinψ = Ψ, according Hu’s method [19] Eq.(37) reduce to:
d3Ψ d2ΨdΨ 1 dΨ 2(Ψ2 1)d2Ψ 3Ψ dΨ
(Ψ2 1) +Ψ ( )3 + − + ( )2
− dv3 dv2 dv − 2 dv v dv2 2v dv
c2 2c Ψ λ 3Ψ2 2 dΨ c2 λ 1 Ψ Ψ3
+( 0 + 0 + − ) +( 0 + ) + = 0. (41)
2 v k − 2v2 dv 2 k − v2 v 2v3
c c
Now, we will consider the case that Ψ = 0 for v = 0. As v 0, Eq.(41) approaches to
→
d3Ψ + 1d2Ψ 1 dΨ + Ψ = 0. Its solution is Ψ = α /v+α v+α vlnv where α = 0, α and
dv3 v dv2 − v2 dv v2 1 2 3 1 2
α are three constants. If λ = 0 and c > 0, we find that Ψ = sinψ = β(v/v0)+c vln(v/v )
3 0 0 0
satisfies Eq.(37). The shapes of closed membranes corresponding to this solution are fully
discussed by Liuet al.[20]. Eqs.(38) and(40) determine the positionof theedge that satisfies
tanψ(v) = γ if k¯ = 2k . If let β = 1, v = 1/c = 1 (unit: length dimension)
−2kcc0 − c 0 0
and γ k c , we obtain v/v 1 and its corresponding shape likes a cup as shown in Fig.4.
c 0 0
≫ ≈
This shape is topologically equivalent to a disk as shown in Fig.3.
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VII. AXISYMMETRICAL NUMERICAL SOLUTIONS
It is extremely difficult to find analytical solutions to Eq.(37). We attempt to find the
numerical solutions in this section. But there is a difficulty that sinψ(v) is multi-valued.
To overcome this obstacle, we use the arc-length as the parameter and express the surface
as r = v(s)cosu,v(s)sinu,z(s) . The geometrical constraint and Eqs.(28), (31) and (36)
{ }
now become:
v (s) = cosψ(s), z (s) = sinψ(s), (42)
′ ′
(2 3sin2ψ)ψ v sinψ(1+cos2ψ)+[(c2 +2λ/k )ψ (ψ )3 2ψ ]v3
− ′ − 0 c ′ − ′ − ′′′
+[(c2 +2λ/k )sinψ 4c sinψψ +3sinψ(ψ )2 4cosψψ ]v2 = 0, (43)
0 c − 0 ′ ′ − ′′
sinψ sinψ
¯
k (c ψ ) k = 0, (44)
c 0 ′
(cid:20) − v − − v (cid:21)
C
sinψ k¯ sin2ψ cosψ
¯ ¯
kc k(1+ ) +λ snγ = 0. (45)
(cid:20) 0 v − 2k v2 − v (cid:21)
c C
We can numerically solve Eqs.(42) and (43) with initial conditions v(0) = 0, ψ(0) = 0,
ψ (0) = α and ψ (0) = 0 and then find the edge position through Eqs.(44) and (45). The
′ ′′
shape corresponding to the solution is topologically equivalent to a disk as shown in Fig.3.
In fact, Eq.(43) can be reduce to a second order differential equation [8, 9, 21], but we still
use the third order differential equation (43) in our numerical scheme.
InFig.5,wedepictstheoutlineofthecup-likemembranewithawideorifice. Thesolidline
corresponds to the numerical result with parameters α = c = 0.8µm 1, λ/k = 0.08µm 2,
0 − c −
γ/k = 0.20µm 1 and k¯/k = 0.38. The squares come from Fig.1d of Ref.[6].
c − c
In Fig.6, we depicts the outline of the cup-like membrane with a narrow orifice. The
solid line corresponds to the numerical result with parameters α = c = 0.86µm 1, λ/k =
0 − c
0.26µm 2, γ/k = 0.36µm 1 and k¯/k = 0.033. The squares come from Fig.3k of Ref.[6].
− c − c
−
Obviously, the numerical results agree quite well with the experimental results of Ref.[6].
VIII. CONCLUSION
In above discussion, we introduce exterior differential forms to describe a lipid membrane
with freely exposed edge. The total free energy is defined as the Helfrich’s free energy plus
the surface and line tension energy. The equilibrium equation and boundary conditions of
the membrane are derived from the variation of the total free energy. These equations can
10