Lipid Membranes with Free edges Z. C. Tu

Transcription

1 Lipid Membranes with Free edges Z. C. Tu Institute of Theoretical Physics Chinese Academy of Sciences

2 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

3 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

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

5 The model of cell membranes Fluid mosaic model (Singer & Nicholson, 1972) carbohydrate [M. Edidin, Nature Reviews Molecular Cell Biology 4, 414 (2003)]

6 The model of cell membranes Lipid structure Polar head---hydrophilic Non-polar tails---hydrophobic When a quantity of lipid molecules disperse in water, they will assemble themselves into a bilayer in which the hydrophilic heads shield the hydrophobic tails from the water surroundings because of the hydrophobic forces. Chemical and schematic structures of the phospholipid

7 Studies for shapes of closed Basic concepts membranes 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

8 Studies for shapes of closed membranes Helfrich free energy The membrane can be regarded as a smooth surface in mathematical point of view, and liquid crystal phase at the body temperature in physical point of view. [W. Helfrich, Z. Naturforsch. C 28, 693 (1973)]

9 Studies for shapes of closed membranes The shape equation of closed membranes It is called the shape equation or the generalized Laplace equation [Z. C. Ou-Yang and W. Helfrich, Phys. Rev. Lett. 59, 2486 (1987)]

10 Studies for shapes of closed membranes Biconcave discoidal shape of normal red cell [ [H. Naito, M. Okuda, and Z. C. Ou-Yang, Phys. Rev. E 48, 2304 (1993)] Torus [Z. C. Ou-Yang, Phys. Rev. A 41, 4517 (1990)] Exp:[M. Mutz and D. Bensimon, Phys. Rev. A 43, 4525 (1991)]

11 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)]

12 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, (2002)] (Their deduction is too complicated to be understood by condensed matter physicists)

13 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

14 Surface theory expressed by exterior differentials Moving frame method C Structure equations of a surface Cartan lemma

15 Surface theory expressed by exterior differentials [Ref: any differential geometric text book involving in exterior differentials]

16 Stokes theorem

17 Hodge star Here just list the basic its properties

18 Variation of the surface The variation along will give the identity.

19 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

20 Free energy of open membranes Gauss-Bonnet

21 Shape equation & boundary Shape equation conditions Boundary conditions Correspond to the results of Capovilla et al. Be also applied to an open membrane with many edges! [Z. C. Tu and Z. C. Ou-Yang, Phys. Rev. E 68, (2003)]

22 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

23 Axisymmetrical solutions Numerical solutions (1) (2) (3) (4)

24 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, (2003)]

25 IV. Summary Exterior differential forms are introduced to calculate variational problems on curved surfaces. The shape equation and boundary conditions are obtained in general case. Numerical solutions in axisymmetrical case agree with the experimental results.

26 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.

27 Acknowledgement Nameless heroes Nameless heroes SARS virus N. S. Zhong Y. Y Jiang