Lipid Membranes with Free edges

Z. C. TuInstitute of Theoretical PhysicsChinese Academy of Sciences

Outline

• I. Introduction– The model of cell membranes– Studies for shapes of closed membranes– Previous experimental and theoretical studies for open

membranes

• II. Our new method to deal with the variational problems on a surface– Surface theory expressed by exterior differentials– Stokes theorem and Hodge star– The variation of the surface

Outline

• III. The shape equation and boundary conditions of open membranes– The free energy of an open membrane with an edge– The general equations– The axisymmetrical solutions to the shape equation

and boundary conditions

• IV. Summary

I. Introduction

– The model of cell membranes

– Studies for shapes of closed membranes

– Previous experimental and theoretical studies for open membranes

The model of cell membranes

• Fluid mosaic model (Singer & Nicholson, 1972)carbohydrate

[M. Edidin, Nature Reviews Molecular Cell Biology 4, 414 (2003)]

The model of cell membranes

• Lipid structure

Chemical and schematic structures of the phospholipid

When a quantity of lipid moleculesdisperse in water, they will assemblethemselves into a bilayer in whichthe hydrophilic heads shield thehydrophobic tails from the watersurroundings because of thehydrophobic forces.

Polar head---hydrophilic

Non-polar tails---hydrophobic

Studies for shapes of closed membranes

• Basic concepts– The shape of a cell membrane is determined by its

lipid bilayer– The lipid molecules can be regarded as a polar rod– The thickness of the membrane is much smaller than

its dimension– The bending rigidity of the bilayer is about 20kT– There are some unsymmetrical factors between the

two monolayers– There is a pressure difference between the outer and

inner sides of the membrane

Studies for shapes of closed membranes

• Helfrich free energyThe membrane can be regarded as a smooth surface inmathematical point of view, and liquid crystal phase atthe body temperature in physical point of view.

[W. Helfrich, Z. Naturforsch. C 28, 693 (1973)]

Studies for shapes of closed membranes

• The shape equation of closed membranes

It is called the shape equation orthe generalized Laplace equation

[Z. C. Ou-Yang and W. Helfrich, Phys. Rev. Lett. 59, 2486 (1987)]

Studies for shapes of closed membranes

• Biconcave discoidal shape of normal red cell

• Torus

[http://zh.wikipedia.org/upload/1/13/Redbloodcells.jpg]

[H. Naito, M. Okuda, and Z. C. Ou-Yang,Phys. Rev. E 48, 2304 (1993)]

[Z. C. Ou-Yang, Phys. Rev. A 41, 4517 (1990)]

Exp:[M. Mutz and D. Bensimon, Phys. Rev. A 43, 4525 (1991)]

Experimental and theoretical studies for open membranes

• Opening process of lipid vesicles by Talin

Fig. 1 Fig. 3[A. Saitoh, K. Takiguchi, Y. Tanaka, and H. Hotani,

Proc. Natl. Acad. Sci. 95, 1026 (1998)]

Experimental and theoretical studies for open membranes

• Axisymmetrical case

[J. J. Zhou, PhD thesis]

• General case[R. Capovilla, J. Guven, and J. A. Santiago,

Phys. Rev. E 66, 21607 (2002)]

(Their deduction is too complicated to be understood bycondensed matter physicists)

II. Our new method to deal with the variational problems

– Surface theory expressed by exterior differentials

– Stokes theorem and Hodge star

– The variation of the surface

Surface theory expressed by exterior differentials

• Moving frame method

C

• Structure equations of a surface

Cartanlemma

Surface theory expressed by exterior differentials

[Ref: any differential geometric text bookinvolving in exterior differentials]

Stokes theorem

Hodge star

• Here just list the basic its properties

Variation of the surface

The variation alongwill give the identity.

III. Shape equation & boundary conditions of open membranes

– The free energy of an open membrane with an edge

– The general shape equation and boundary conditions of open membranes

– The axisymmetrical solutions to the shape equation and boundary conditions

Free energy of open membranes

Gauss-Bonnet

Shape equation & boundary conditions

• Shape equation

• Boundary conditions Correspond to theresults ofCapovilla et al.

Be also applied toan open membranewith many edges!

[Z. C. Tu and Z. C. Ou-Yang, Phys. Rev. E 68, 61915 (2003)]

Axisymmetrical solutions

• Only two boundary conditions are independent if the shape equation is valid in axisymmetrical case

• There is no axisymmetrical open membrane with constant mean curvature

Axisymmetrical solutions• Numerical solutions

(1)

(2)

(3)

(4)

Axisymmetrical solutions

[A. Saitoh et al., Proc. Natl. Acad. Sci. 95, 1026 (1998)]

[Z. C. Tu and Z. C. Ou-Yang, Phys. Rev. E 68, 61915 (2003)]

IV. Summary

• Exterior differential forms are introduced to calculate variationalproblems on curved surfaces.

• The shape equation and boundary conditions are obtained in general case.

• Numerical solutions in axisymmetrical case agree with the experimental results.

Acknowledgement

• Prof. Z. C. Ou-Yang (My advisor)– I knew this problem from him.

• Prof. J. Guven (Instituto de Ciencias Nucleares)– He highly praised our method and gave advice on my English.

• Prof. X. Z. Xie (Tsinghua Univ.)– He helped me modify the English.

• Prof. S. S. Chern (Nankai Univ.)– He let me notice the work by Prof. Griffiths (Princeton Univ.)

• SARS (from April to May, last year)– The key idea of our method was born in that horrific period.

Acknowledgement

Nameless heroes

Y. Y Jiang

Nameless heroes

SARS virus

N. S. Zhong