Lipid membranes with free edges
Zhanchun Tu ( 涂展春 )Department of Physics, Beijing Normal University
Email: tuzc@bnu.edu.cn
Homepage: www.tuzc.org

Outline●●Introduction
Theretical analysis to an open lipid membranewith free edge(s)●Numerical solutions●SummaryIntroduction●RBC

Human (normal): diameter 8μm, height 2 μm; biconcave discoid (why?)
No inner cellular organelles. Shapes are determined by membranes.

●Cell membrane

Fluid mosaic model[Singer & Nicolson (1972) Science]

Shape determined mainly by lipid bilayer.
Lipid bilayer in liquid crystal phase●Spontaneous curvature model [Helfrich (1973)]
kc2g = 2 H c 0 − k K
2Mean curvature

Gaussian curvature

Analogyspontaneous curvature●Shape equation of vesicles [Ou-Yang & Helfrich (1987)]
F =∫ g dA ∫ dA p ∫ dV

 F =022

 p−2 H 2 k c ∇ H k c  2 H c 0  2 H −c 0 H −2 K =0
Describes the equilibrium shapes. called Shape equation●

Only 3 Analytical solutions (lipid vesicles)# Spherical surface
22Shape equation=>  p R 2 k c c 0  R−2 k c c0 =02R

=r /# Torus [Ou-Yang (1990) PRA]

10μm

Shape equation=>rD= 2 ⇒ =  21≈2.4
d[Mutz-Bensimon (1991) PRA]

# RBC[Naito,Okuda, Ou-Yang (1993) PRE]
Shape equation[Naito,Okuda,Ou-Yang][Evans&Fung (1972) Microvasc Res]

●Experiment: lipid vesicles opened by Talin
Experimental facts(1) Talin opens the closed lipid vesicles(2) Talin adheres to the free edge(s)(3) The size of hole is enlarged withincreasing the concentration of talin(4) The process is partially reversible
if decrease the concentration of talin●Motivation of our work
(1) Can we derive the equation(s)to describe the equilibrium
configurations of lipid membranes
with free edges?

[Saitoh et al. (1998) PNAS](2) Can we find analytical solutions?
(3) Numerical solutions to explain experimental results

5μmTheretical analysis to an open
lipid membrane with free edge(s)●Model# Free energy

F =∫ G dA∮ dskc
G= 2 H c 02 k K 2Smooth surface with boundary curve C
Orthogonal moving frame : Line tension, related to the concentration of talin in the experiment F =0 ⇒shape equationboundary conditions

●Governing equations
Shape equation: force balance in the normal direction
22k c 2 H c 02 H −c 0 H −2 K −2  H 2 k c ∇ H =0

Boundary conditions (curve C satisfies...)k c 2 H c0  k k n =0
Moment balance equation of points in the edge along normal directiond g∂H−2 k c k n  k=0∂ e2dsForce balance equation of points in the edge around e1kc
2k K  k g=0
 2 H c 0  2Force balance equation of points in the edge along e2
Note: above equations are also valid for an open membrane with more than one edges.
[Capovilla, Guven, Santiago (2002) PRE; Tu & Ou-Yang (2003) PRE]

●Analytical solutions# Trivial case: planar disk

 R=0R# Axisymmetric nontrivial cases
Outline profile
Shape Eq=>

BCs=>


k = k , =, =
kckckc

=±1 : orientation of boundary curve

Note: shape Eq and BCs are highly nonlinear!

Procedure of finding analytical solutions:(1) Find a surface satisfying the shape equation(2) Find find a curve C on the surface satisfying the BCs(3) The domain M enclosed by C on the surface is the solution
MKey point:
For a given surface satisfying the shape equation, we may notalways find a curve C on that surface satisfying the BCs.

Compatibility condition:
The shape equation is integrable, from which we arrive at the reduced shape Eq.

The points at the free edge should satisfy not only the reduced shape Eqbut also three Bcs. Thus we derive
Compatibility condition:

0=0[Tu (2010) JCP]CDoes there exist an axisymmetric open membrane being apart of a torus or a biconcave discoid?

Generation curves: sin =   2 , ≠0

0=−≠0sin =c 0  ln /B  , c 0≠00=−2 c0≠0Compatibility condition is violated! Thus we have
Theorem 1. There is no axisymmetric open membrane being a part of torus
generated by a circle expressed by sin =  2 .
Theorem 2. There is no axisymmetric open membrane being a part of a biconcave
discodal surface generated by a planar curve expressed by sin =c 0  ln /B .

Does there exist an axisymmetric open membrane being apart of a constant mean curvature surface?
H=const. satisfies the Compatibility condition. Howerver, compatibility conditionis necessary but not sufficient condition for existence of proper solutions.
In axisymmetric case, we can easily find that the shape equation and BCs
cannot simultaneously be satisfied if H=const.
Theorem 3. There is no axisymmetric open membrane being a part of a constantmean curvature surface (including also spherical cap and short cylinder)

Direct consequence:R

It is almost hopeless to find analyticalsolutions to the shape equation and BCs
for lipid membranes with free edge(s).

(impossible!)

We have to invoke the numerical method!

Axisymmetric Numerical solutions# Reduced SEq and BCs with the compatibility condition[Tu (2010) JCP]

SEq:

2 BCs:# Result from shoot method
Numerical results: solid, dash, dot lines
Experimental data: squares, circles, triangles[Saitoh et al. (1998) PNAS]

Common parameters:−1k =−0.122, c0=0.4  mBoth results agree well with each other,which implies that line tension negatively
correlate with the concentration of talin[See also: Umeda et al.(2005)PRE based on the area difference model]

Summary●●●●

Investigate lipid membranes with free edges
based on Helfrich spontaneous curvature model
Derive the shape equation and BCs
Elucidate three theorems of nonexistence, which
implies hopeless to find analytical solutions
Numerical results are in good agreement withthe experimental results.

Acknowledgments●Organizer●Z. C. Ou-Yang (Chinese Academy of Sciences)●X. H. Zhou (Fourth Military Medical University, China)●National Natural Science Foundation of China●Foundation of National Excellent Doctoral Dissertation of China

Thank you for your attention!