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Lipid membranes with free edges

a r X i v :c o n d -m a t /0305700v 4 [c o n d -m a t .s o f t ] 7 J a n 2004

Zhan-chun Tu 1,2,?and Zhong-can Ou-Yang 1,3,?

Institute of Theoretical Physics,Academia Sinica,

P.O.Box 2735Beijing 100080,China

Graduate School,Academia Sinica,Beijing,China

Center for Advanced Study,Tsinghua University,Beijing 100084,China

Abstract

Lipid membrane with freely exposed edge is regarded as smooth surface with curved boundary.Exterior di?erential forms are introduced to describe the surface and the boundary curve.The total free energy is de?ned as the sum of Helfrich’s free energy and the surface and line tension energy.The equilibrium equation and boundary conditions of the membrane are derived by taking the variation of the total free energy.These equations can also be applied to the membrane with several freely exposed edges.Analytical and numerical solutions to these equations are obtained under the axisymmetric condition.The numerical results can be used to explain recent experimental results obtained by Saitoh et al .[Proc.Natl.Acad.Sci.95,1026(1998)].

PACS numbers:87.16.Dg,02.40.Hw

I.INTRODUCTION

Theoretical study on shapes of closed lipid membranes made great progress two decades ago.The shape equation of closed membranes was obtained in1987[1],with which the

√biconcave discoidal shape of the red cell was naturally explained[2],and a ratio of

the membrane from the variation of the total free energy.In Sec.VI,we suggest some special solutions to the equations and show their corresponding shapes.In Sec.VII,we put forward a 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.SURF ACE THEORY EXPRESSED BY EXTERIOR DIFFERENTIAL FORMS

In this section,we retrospect brie?y the surface theory expressed by exterior di?erential forms.The details can be found in Ref.[11].

We regard a membrane with freely exposed edge as a di?erentiable 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 e1,e2,e3with e i·e j=δij and e3being the normal vector.For

a point in curve C,e1is the tangent vector of C.

An in?nitesimal tangent vector of the surface is de?ned as

d r=ω1e1+ω2e2,(1) wher

e d is an exterior di?erential operator,andω1,ω2are1-di?erential forms.Moreover,we de?ne

d e i=ωij e j,(2) whereωij satis?esωij=?ωji because of e i·e j=δij.

With dd=0and d(ω1∧ω2)=dω1∧ω2?ω1∧dω2,we have

dω1=ω12∧ω2;dω2=ω21∧ω1;ω1∧ω13+ω2∧ω23=0;(3) and

dωij=ωik∧ωkj(i,j=1,2,3),(4) 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:

ω13=aω1+bω2,ω23=bω1+cω2.(5)

Therefore,we have

area element:dA=ω1∧ω2,(6)

first fundamental form:I=d r·d r=ω21+ω22,(7)

second fundamental form:II=?d r·d e3=aω21+2bω1ω2+cω22,(8)

a+c

mean curvature:H=

Operators d andδare independent,thus dδ=δd.dδr=δd r implies that:

δω1=?2ω21+?3ω31?ω2?21,(17)

δω2=d?2+?3ω32?ω1?12,(18)

d?3=?13ω1+?23ω2??2ω23.(19) Furthermore,dδe i=δd e i implies that:

δωij=d?ij+?ikωkj?ωik?kj.(20) It is necessary to point out that the properties of the operatorδare exactly similar to those of the ordinary di?erential.

V.EQUILIBRIUM EQUATION OF THE MEMBRANE AND BOUNDARY CON-DITIONS

The total free energy F of a membrane with an edge is de?ned as the sum of Helfrich’s free energy[14,15]F H= [k c

(2H+c0)2+λ]ω1∧ω2+γ Cω1?ˉk Cω12+2πˉk,(21)

and

δF=k c (2H+c0)δ(2H)ω1∧ω2+ [k c

Thus we have:

δF=k c (2H+c0)[2(2H2?K)?3ω1∧ω2+d(?13ω2??23ω1) +a?2dω1?bd?2∧ω2+b?2dω2+cd?2∧ω1]

+ (k c

whyˉk is also included in Eq.(30).In fact,the termˉkdb in Eq.(30)represents the shear stress which also contributes to the bend energy in Helfrich’s free energy.

In fact,a=k n and b=τg are the normal curvature and the geodesic torsion of curve C, respectively,and?d(2H)=?e2·?(2H)ω1.Thus Eqs.(29)and(30)become

[k c(2H+c0)+ˉkk n] C=0,(31)

[?k c e2·?(2H)+γk n+ˉk dτg

(2H+c0)2+λ][?d(?2ω1)]+γ C?2ω21?ˉk C K?2ω1.(33)

Otherwise,ω13=aω1+bω2implies that:adω1+db∧ω2+2bdω2?cdω1=?da∧ω1.Thus a?2dω1?bd?2∧ω2+b?2dω2+cd?2∧ω1+d(?13ω2??23ω1)

=?d(a+c)∧?2ω1=?d(2H+c0)∧?2ω1(c0is a constant).(34) Therefore

δF= C[?k c

(2H+c0)2+ˉkK+λ+γk g] C=0,(36)

because ofω21=?k gω1on C.This equation is the force equilibrium equation of points on C along the direction of e2[8,9].

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.

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 in3-dimensional space as r={v cos u,v sin u,z(v)}we obtain

2H=?(sinψ

dv ),K=sinψcosψ

,?(2H)=?r2

(sinψ

)and?2(2H)=

?cosψdv[v cosψd v+cosψdψdv],r2=?r/?v.De?ne t as the direction of curve C and r1=?r/?u.Obviously,t is parallel or antiparallel to r1on curve C.Introduce a notation sn,such that sn=+1if t is parallel to r1,and sn=?1if not.

Thus e2=sn r2

?v (sinψ

)on curve C.For curve C,

k n=?sinψ

.Thus we can reduce Eqs.(28),(31),(32)and(36) to:

k c(sinψ

dv?c0)[

+cosψ

dψ

c0(

sinψ

2sinψcosψ

]?λ(sinψdv)+k c cosψdv[v cosψd v+cosψdψ

+cosψ

dψ

v C=0,(38) ?snˉk cosψd v)+γsinψ

2k c

(

sinψ

dψ

v C=0.(40)

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)is the same as the equilibrium equation of axisymmetrical closed membranes[17,18]. In Ref[18],a large number of numerical solutions to Eq.(37)as well as their classi?cations 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 speci?c values of parameters.

A.The constant mean curvature surface

The constant mean curvature surfaces satisfy Eq.(28)for proper values of k c ,c 0,K ,and

λ.But Eqs.(31),(32)and (36)imply 2H +c 0=0,k n =0,and ˉkK

+γk g =0on curve C if k c ,ˉk

,and γare nonzero.For axial symmetric surfaces,k n =0requires sin ψ=0.Therefore K =0which requires k g =0.Only straight line can satisfy these conditions.It contradicts to the fact that C is 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=0,the condition sin ψ=av +

√

2(k c +ˉk

)

2(2k 2c +4k c ˉk +ˉk 2)

=2√

(unit:length dimension),

it leads to 1/a =?1and v e =2√3

(unit:length dimension).Thus the shape is the

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 sin ψ=Ψ,according Hu’s method [19]Eq.(37)reduce to:

(Ψ2

?1)d 3Ψ

dv 2d Ψ2(

d Ψ

v d 2Ψ2v

(

d Ψ

2c 0Ψ

k c

3Ψ2?2

+(c 20

k c

1v +

Ψ3

dv 3+1dv 2?

2k c c 0

if ˉk

=?2k c .If let β=1,v 0=1/c 0=1(unit:length dimension)

and γ?k c c 0,we obtain v/v 0≈1and its corresponding shape likes a cup as shown in Fig.4.

This shape is topologically equivalent to a disk as shown in Fig.3.

VII.AXISYMMETRICAL NUMERICAL SOLUTIONS

It is extremely di?cult to?nd analytical solutions to Eq.(37).We attempt to?nd the numerical solutions in this section.But there is a di?culty 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)cos u,v(s)sin u,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)+[(c20+2λ/k c)ψ′?(ψ′)3?2ψ′′′]v3

+[(c20+2λ/k c)sinψ?4c0sinψψ′+3sinψ(ψ′)2?4cosψψ′′]v2=0,(43)

k c(c0?sinψv

C =0,(44)

ˉkc0sinψ2k

c )

sin2ψ

v C=0.(45)

We can numerically solve Eqs.(42)and(43)with initial conditions v(0)=0,ψ(0)=0,ψ′(0)=αandψ′′(0)=0and then?nd 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 di?erential equation[8,9,21],but we still use the third order di?erential equation(43)in our numerical scheme.

In Fig.5,we depicts the outline of the cup-like membrane with a wide ori?ce.The solid line corresponds to the numerical result with parametersα=c0=0.8μm?1,λ/k c=0.08μm?2,γ/k c=0.20μm?1andˉk/k c=0.38.The squares come from Fig.1d of Ref.[6].

In Fig.6,we depicts the outline of the cup-like membrane with a narrow ori?ce.The solid line corresponds to the numerical result with parametersα=c0=0.86μm?1,λ/k c= 0.26μm?2,γ/k c=0.36μm?1andˉk/k c=?0.033.The squares come from Fig.3k of Ref.[6].

Obviously,the numerical results agree quite well with the experimental results of Ref.[6]. VIII.CONCLUSION

In above discussion,we introduce exterior di?erential forms to describe a lipid membrane with freely exposed edge.The total free energy is de?ned 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

also be applied to the membrane with several freely exposed edges.A numerical scheme to give some axisymmetric solutions and their corresponding shapes do agree with some experimental results.

The method that combines exterior di?erential forms with the variation of surface is of important mathematical signi?cance.It is easy to be generalized to deal with and to simplify the di?cult variational problems on high-dimensional manifolds.

Although we give some axisymmetric numerical solutions that agree with experimental results obtained by Saitoh et al,up to now,we still cannot?nd any unsymmetrical solution.

A large number of unsymmetrical shapes are found in experiments,which will be a challenge to the theoretical study.

IX.ACKNOWLEDGMENTS

We are grateful to the instructive advice of Mr.J.J.Zhou,H.J.Zhou,R.Capovilla and J.Guven.We thank Prof.Y.Z.Xie and Dr.L.Q.Ge for their critical reading our manuscript.

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FIG.1:The surface with an edge C.At every point of the surface,we can construct an orthogonal frame e1,e2,e3,where e3is the normal vector of the surface.For a point on curve C,e1is the .

tangent vector of C

FIG.3:A ring(left)and a disk(right).

FIG.4:A cup.

v (m)

FIG.5:The outline of the cup-like membrane with a wide ori?ce.The solid line is the numerical result with parametersα=c0=0.8μm?1,λ/k c=0.08μm?2,γ/k c=0.20μm?1andˉk/k c=0.38. The squares come from Fig.1d of Ref.[6].z-axis is the revolving axis and v is the revolving radius.

v (m)

FIG.6:The outline of the cup-like membrane with a narrow ori?ce.The solid line is the numerical result with parametersα=c0=0.86μm?1,λ/k c=0.26μm?2,γ/k c=0.36μm?1andˉk/k c=?0.033.The squares come from Fig.3k of Ref.[6].z-axis is the revolving axis and v is the revolving radius.