Lecture 8. Bio-membranes

Zhanchun Tu (涂展春 )

Department of Physics, BNU

Email: tuzc@bnu.edu.cn

Homepage: www.tuzc.org

Main contents● Introduction

● Mathematical and physical preliminary

● Lipid membrane

● Cell membrane

● Summary and perspectives

§8.1 Introduction

Size and morphology of cells● Size: several to tens of μm● Various shapes

(a) 5 cells of E. coli bacteria

(b) 2 cells of yeast

(c) Human red blood cell

(d) Human white blood cell

(e) Human sperm cell

(f) Human epidermal (skin) cell

(g) Human striated muscle cell (myofibril)

(h) Human nerve cell

Why various shapes?

Animal cell

The shapes of most of animal cells are determined by cytoskeleton

Red blood cell

Human (normal): diameter 8μm, height 2 μm; biconcave discoid (why?)

No inner cellular organelles. Shape is determined by membrane.

Cell membrane

●Timeline: cell-membrane bilayers

[Edidin (2003) Nature Reviews Molecular Cell Biology]

awaiting a new model

lipidmolecule

micelle

bilayer

hexagonal phase

vesicle

Liquid crystal phase.Cannot endure shear strain!

胶囊

● Lipid structures

● Cell membrane is usually in liquid crystal phase

Liquid crystal phase is a necessary condition for cell as an open system

Solid shell ===> cell is dead

Isotropic fluid ==> no difference between inner and outside

of cells in equilibrium

==> cell cannot exist as an basic unit for life

Cancer might be related to the transition from LC to isotropic fluid

取自《从肥皂泡到液晶生物膜》

Some problems we may deal with● How to describe shapes mathematically?● Does there exist a universal equation to govern

the shapes?● Why is human normal RBC a biconcave

discoid?● What is the mechanical function of membrane

skeleton?● To what extend membrane proteins will

influence the shapes of membranes?

§8.2 Mathematical & physical preliminary

Curvature and torsion of a curve

2=1r

Each 3 points determine a curvature circle

2= planeO123∧planeO' 234

s231,2,3,4 close to each other

● Curvature and torsion

● Tangent, normal, and binormal vectort : tangent vectorn : normal vector , point to O

b : binormal vector , b⊥ t ,b⊥nt,n,b right-handed

● Two examples

● Frenet formulas: arc length parameter

Each s corresponds a point in the curve

Different t, n, b at different pointsDifferent κ and τ at different points [ t

nb]=[

0 0− 0 0 − 0][ t

nb]

tn

b

2R

= 1R

, =0

= RR2h /22

=−h /2

R2h /22h

2R

+

x

y

z

nt

b

A curve in a surface● Geodesic curvature

& normal curvatures

● Geodesic torsion

κg: Curvature of C' at P (roughly)

κn: Curvature of C'' at P (roughly)

N: normal vector of surface

g=−N⋅n '

t, n: tangent and normal vectors of C

n': normal vector of C', such that

{t,n',N} right-handed

g=n⋅n '

n= n⋅N

(exactly)

(exactly)

g2n

2=2Obviously,

● Two examples

g=0 ,

g=h /2

R2h/22

n=−R

R2h/22 g=cot

R,

g=0

n=−1R

=R

R2h /22, =− h /2

R2h /22=

1Rsin

, =0

Curvatures of a surface● Principal curvatures

● Mean and gaussian curvatures

norm

al

Rotate 2 normal plane, curvature radii of 2 curves varies.

c1=−1

min{R1}, c2=−

1max {R2}

H=c1c2

2, K=c1c2

● Two examples

RR

c1=−1R

, c2=0

H=c1c2

2=− 1

2 R, K=c1 c2=0

c1=c2=−1R

H=c1c2

2=− 1

R, K=c1 c2=

1R2

Topological invariant of closed surface● Gauss-Bonnet formula

∬K dA=41−g

g=0⇒∬K dA=4 g=1⇒∬K dA=0 g=2⇒∬K dA=−4

Free energy● Min F <=> equilibrium shapes

Configuration space

F

spheretoruscylinder

Finding min F <=> Solving δF=0

?

Stable: δ2F>0; unstable:δ2F<0

The meaning of variation

Surface S

Small deformation

Surface S'

F=F [S ' ]−F [S ]

Free energy F[S]

Free energy F[S']

δF=0 => Euler-Lagrange equation(s) describing equilibrium shapes

Euler-Lagrange equation(s) <=> force balance equation(s)

Variational problems on shapes in history● Fluid films

# Soap films ---- minimal surfaces, Plateau (1803)

F=∫ dA

取自《从肥皂泡到液晶生物膜》

Rotation axis

F=0⇒ H=0

Viewed as a surface in mathematics

# Soap bubble ---- sphere, Young (1805), Laplace (1806)

F=∫ dA p∫ dVp

in

pout

p= pout− p in

F=0⇒H= p /2"An embedded surface with constant mean curvature in E3 must be a spherical surface" ---Alexandrov (1950s)

Sphere Cylinder1R=− p

21R=− p

取自《从肥皂泡到液晶生物膜》

● Solid shells

# Possion (1821)

# Schadow (1922)

# Willmore (1982) problem of surfaces

F=∫ H 2dA

F=0⇒∇ 2 H 2 H H 2−K =0

Finding surfaces satisfying the above equation.

Laplace operator

● Lipid bilayer (almost in-plane incompressible)# Spontaneous curvature energy, Helfrich (1973)

Analogy

g=k c

22 Hc0

2−k K

spontaneous curvature

# Shape equation of vesicles, Ou-Yang & Helfrich (1987)

F=∫ g dA∫dA p∫ dV

F=0⇒ p−2H2 kc∇2 Hkc 2 Hc02 H 2−c0 H−2K =0

k c=0⇒ p−2H =0 Young−Laplaceequation

p=0,=0, c0=0⇒∇ 2 H 2 H H 2−K =0 Willmore surfaces

Puzzle from the shape of RBC● Sandwich model (Fung & Tong, 1968)

To obtain the shape of biconcave discoid,

they should assume the thickness of the

membrane is nonuniform in μm scale.

Pinder's experiment (1972): the nonuniform thickness exists only in

molecular (nm) scale. The thickness is uniform in large scale of μm.

● Nonuniform charge model (Lopez, 1968)Nonuniform charge distribution results in the shape of biconcave discoid.

Experiment by Greet & Baker (1970): NO

Solid shell

Solid shell

Fluid film

● Incompressible shell model (Canham, 1970)

Given the area and volume of membrane, the biconcave discoid

minimizes the curvature energy

∫H 2 dA

取自《从肥皂泡到液晶生物膜》

Dumbbell-like Biconcave discoid

Helfrich & Deuling (1975): the

dumbbell-like shape can have the

same curvature energy as the

biconcave disk. But dumbbell-like

shape has never observed in the

experiment.

● Spontaneous curvature model (Helfrich, 1973)

Given the area and volume of membrane, the biconcave discoid

minimizes the spontaneous curvature energy ∫2 Hc02 dA

c00⇒ biconcave discoid is energetically favorable

experiment

Axisymmetric numerical result

Can we give a analytic result from the shape equation?

p−2H2 k c∇2 H k c2 H c02 H 2−c0 H−2 K =0

§8.3 Lipid membranes----Soft-incompressible fluid film

can endure bending but not static shear.

May be asymmetric in inner side and outer one.

H=−1R

, K=1R2

2R

Lipid vesicles● Spherical vesicles

p−2H2 k c∇2 H k c2 H c02 H 2−c0 H−2 K =0

f R≡ p R22kc c02R−2 k c c0=0

R

f(R) p0, c00

R

f(R) p0,c00

R

f(R) p0,c00

Might be related to endocytosis

● Torus [Ou-Yang (1990) PRA]{rcoscos ,rcossin ,sin}

2 H=− r2cosrcos

; K= cosrcos

2 k c /2k c c0

22−12 p2

4 k c c0

22−4 k c c0826 p3

cos

5 k c c0

22−8 k c c01026 p3

2 cos2

2 k c c0

22−4 kc c0422 p3

3 cos3=0

The coefficients of {1, cosj, cos2j, cos3j} should vanish!

=r/

=2

p=−2k c c0

2 ,=kc c0 2−

c0

2

=r /

[Mutz-Bensimon (1991) PRA]

10μm

[Evans-Fung (1972) Microvasc. Res.]

[Naito-Okuda-OY (1993) PRE]

● Axisymmetric surface and Biconcave discoid

p−2H2 k c∇2 H k c2 H c02 H 2−c0 H−2 K =0

Axisymmetric

[Hu & Ou-Yang (1993) PRE]

For−ec0B0

describe a biconcave outline[Naito-Okuda-OY (1993) PRE]

Lipid membranes with free edges● Experiment: Opening process of lipid vesicles by Talin

[Saitoh et al. (1998) PNAS] 5μmBar: 2μm

● Previous theories– Derived from axisymmetric variation

● [Jülicher -Lipowsky (1993) PRL]● [Zhou (2002) PhD thesis]

– General case● [Capovilla-Guven-Santiago (2002) PRE]

● [Tu-OuYang (2003) PRE]

Confine the variational problem in a subspace.Unreasonable results exist.

Governing equations of edges are not expressed in the explicit forms of curvature and torsion.

Overcome the above shortages.

● Main results in [Tu-OuYang (2003)PRE]

G=k c

22 Hc0

2−k K

Free energy per area

Total free energy

F=∫G dA∮ ds

F=0⇒shape equationboundary conditions

kc 2 Hc02 H 2−c0 H−2 K −2H2 k c∇2 H=0

Shape equation: force balance in the normal direction

Boundary conditions (curve C satisfies...)

kc 2 Hc0−k kn=0

Force balance equation of points in the edge along normal direction

2 k c∂ H∂b

k n−k g, =0

Moment balance equation of points in the edge around t

G k g=0Force balance equation of points in the edge along b

Axisymmetric analytic solutions

Center of torus Cup-like open membrane

Axisymmetric numerical solutions

Solid squares: experiment data [Saitoh etal. (1998) PNAS]

§8.4 Cell membrane----It is beyond the lipid bilayer.

How can we model it?

shear bendingBilayer NO YESMSK YES NOCM YES YES

Composite membrane model

● Cell membrane = bilayer + membrane skeleton[Sackmann (2002)]

● Basic assumptions

– 1. CM: “smooth” surface

– 2. Polymer in MSK: almost same chain length

– 3. CM: in-plane isotropic, i.e. lipid crystal phase

– 4. Chain length << curvature radius of CM

– 5. Small deformations

– 6. Free energy per area: analytical function

– 7. Invariance of free energy: Strain tensor (+) (-)

● Energy density (energy per area)

Assumption 1-4 => G=G2 H , K , 2 J ,Q

Assumption 5-7 => up to the second order terms

2 J=Tr11 12

12 22 , Q=det11 12

12 22

G=k c

22 Hc0

2−k Kk d /22 J 2−k Q

Contribution from LB

Bending energyContribution from MSK

Compress and shear energy

Shape equation & in-plane strain equation

(Remark: only consider closed cell membrane)

F=∫G dA p∫ dV

F=0

p2 k c [2 Hc02 H 2−c0 H−2 K 2∇ 2 H ]−2H

2H k−k d 2 J − k ℜ:∇ u=0

Especially, if , the above two equations degenerate intok d=k=0

p2 k c [2 Hc02 H 2−c0 H−2 K 2∇ 2 H ]−2H=0

shape equation of lipid vesicles.

[Tu-OuYang, J. Phys. A (2004), J. Comput. Theor. Nanosci. (2008)]

Spherical cell membrane and its stability

11=22= ,12=0Homogenous in-plane strain

pR222 k d−k Rk c c0c0 R−2=0

Stable <=> δ2F >0 <=> p pl≡2 k 2 kd−k

[k d l l1−k ]R

2 k c

R3 [l l1−c0 R ] ,l1

Critical osmotic pressure pc=min {pl}

pc=2 k c 6−c0 R

R3 when k=0

pc=4 k /k d 2 k d−k k c

R2 whenk k d 2 k d−k R2

k c6 k d−k 21

[return to the result of lipid vesicle,OuYang- Helfrich (1987) PRL]

Typical parameters for cell membranes

k≃k d≈4.8N /m [Lenormand et al. (2001) Biophys. J]

k c≈10−19 J [Duwe et al. (1990) J. Phys. Fr.]

R≈5m

⇒ pc=4 k /k d 2k d−k k c

R2 ≈0.1 Pasatisfyk kd 2k d−k R2

k c 6k d−k 21

If no MSK, i.e., lipid vesicle, k=0⇒ pc=2 kc 6−c0 R

R3 ≈0.008 Pa

Reveals a mechanical function of MSK: highly enhances stability of CM

§8.6 Summary and perspectives

Summary● Mathematical description of shape of

membranes.● Physical meaning of variation; History of

variational problem in shapes.● Helfrich spontaneous curvature model.● Lipid vesicles and lipid membrane with free

edges.● Composite membrane model for cell

membrane.

Perspectives● Lipid domains

Cell membrane consists of many different

kinds of lipid molecules which usually

form micro-domains as shown in left Fig

at physiological temperature. Each

domain contains one or several kinds of

lipid molecules.

[Edidin (2003) Nature Reviews Mol. Cell Biol.]

Lipid rafts

– Special domains

– depleted in unsaturated

phospholipids

– enriched in cholesterol,

sphingolipids and lipid-

anchored proteins

[Simons & Ikonen (2000) Science]

Cholesterol/unsaturated phospholipid/sphingolipid bilayer

lo: liquid-ordered phase

lc: liquid-disordered phase

[Brown & London (2000) J. Biol. Chem.]

Morphology of model membrane with raft and non-raft domains

Red: liquid-disordered phase

“non-raft” domain

Blue: liquid-ordered phase

“raft” domain

[Baumgart et al. (2003) Nature]

We need to develop a new theory to explain various shapes of vesicles.

● Interaction: membrane proteins and lipid bilayers

Ion-channel function and membrane properties

Ion-channel open probability as a function of pipette pressure (<=> surface tension) for mechanosensitive channels in lipids with different tail lengths

[Phillips (2009) Nature]

Membrane doping and membrane protein function

Insertion of various molecules can alter the protein-membrane interaction: (1) Asymmetrical insertion of lysolipids produces a torque on the protein. (2) Introduction of toxins can alter the boundary conditions between the protein and the surrounding lipids. (3) Small rigid molecules can stiffen the membrane. [Phillips (2009) Nature]

Structure and energy at the protein–lipid interface [Phillips (2009) Nature]

Challenge: extending Helfrich's model to include the protein-lipid interactions

Further reading● 欧阳钟灿 , 刘寄星 , 从肥皂泡到液晶生物膜 ( 湖南教育出版

社 1992).

● 谢毓章，刘寄星，欧阳钟灿 , 生物膜泡曲面弹性理论 ( 上海科

学技术出版社 2003)

● Z. C. Tu & Z. C. Ou-Yang, Elastic theory of low-dimensional continua and its applications in bio- and nano-structures, J. Comput. Theor. Nanosci. 5 (2008) 422-448

● T. Baumgart, S. T. Hess & W. W. Webb, Imaging coexisting fluid domains in biomembrane models coupling curvature and line tension, Nature 425 (2003) 821-824

● R. Phillips et al., Emerging roles for lipids in shaping membrane-protein function, Nature 459 (2009) 379-385