Lecture 8. Bio-membranes

Zhanchun Tu ( 涂展春 )

Department of Physics, BNU

Email: [email protected] 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

No inner cellular organelles. Shape is determined by membrane.

Human (normal): diameter 8μm, height 2 μm; biconcave discoid (why?) Cell membrane ●Timeline: cell-membrane bilayers

awaiting a new model

[Edidin (2003) Nature Reviews Molecular Cell Biology] ● Lipid structures

micelle 胶囊

bilayer vesicle lipid molecule Liquid crystal phase. Cannot endure shear strain!

hexagonal phase ● 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

取自《从肥皂泡 到液晶生物膜》

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

● Curvature and torsion Each 3 points determine a curvature circle 1 2= r

 planeO123∧ planeO' 234 2= 1,2,3,4 close to each other s23

● Tangent, normal, and binormal vector t : tangent vector n: normal vector , point to O b : binormal vector , b⊥t ,b⊥ n t,n,b right-handed ● Two examples z b + t R b n = 2 2 R h/2 2R n t h h/2 =− R2h/22 1 = , =0 2R R y

● Frenet formula x s: arc length parameter Each s corresponds a point in the curve t˙ 0  0 t Different t, n, b at different points n˙ = − 0  n Different κ and τ at different points [b˙ ][0 − 0][b] A curve in a surface ● Geodesic curvature & normal curvatures

κ : Curvature of C' at P (roughly) g

g=n⋅n ' (exactly) κ : Curvature of C'' at P (roughly) n

n= n⋅N (exactly)

2 2 2 Obviously, gn= t, n: tangent and normal vectors of C N: normal vector of surface ● Geodesic torsion n': normal vector of C', such that {t,n',N} right-handed g=−N˙ ⋅n ' ● Two examples

R h/2 1 = , =− = , =0 R2h/22 R2h/22 Rsin 

R cot  1 g =0 , n=− 2 2  = ,  =− R h/2 g R n R h/2 g= g =0 R2 h/22 Curvatures of a surface

● Principal curvatures

l

a

Rotate 2 normal plane, curvature radii of 2 m

r

curves varies. o

n

1 1 c1=− ,c2=− min{R1 } max {R2 }

● Mean and gaussian curvatures

c c H = 1 2 ,K=c c 2 1 2 ● Two examples

R R

1 1 c1=c2=− c1=− ,c2=0 R R c c 1 c c 1 1 H = 1 2 =− ,K=c c =0 H = 1 2 =− ,K=c c = 2 2 R 1 2 2 R 1 2 R2 Topological invariant of closed surface

● Gauss-Bonnet formula ∬ KdA=41− g

g=0⇒∬ KdA=4 g=1⇒∬ KdA=0 g=2⇒∬ KdA=−4 Free energy

● Min F <=> equilibrium shapes

F

Configuration space cylinder torus ? sphere Finding min F <=> Solving δF=0

Stable: δ2F>0; unstable:δ2F<0 The meaning of variation

Surface S Free energy F[S]

Small deformation

Surface S' Free energy F[S']

 F=F [S ' ]−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 Viewed as a surface in mathematics # Soap films ---- minimal surfaces, Plateau (1803)

F =∫ dA

 F=0⇒ H =0

Rotation axis

取自《从肥皂泡到液晶生物膜》 # Soap bubble ---- sphere, Young (1805), Laplace (1806) F =∫ dA p∫ dV p  F=0⇒ H = p/2 in p "An embedded surface with constant mean out curvature in E3 must be a spherical surface" ---Alexandrov (1950s)  p= pout− pin

1 − p 1 − p Sphere = Cylinder = R 2 R 

取自《从肥皂泡 到液晶生物膜》 ● Solid shells

# Possion (1821) F =∫ H 2dA

# Schadow (1922)  F=0⇒∇2 H 2 H H 2 −K =0

Laplace operator

# Willmore (1982) problem of surfaces Finding surfaces satisfying the above equation. ● Lipid bilayer (almost in-plane incompressible) # Spontaneous curvature energy, Helfrich (1973) k g= c 2 H c 2−kK 2 0

spontaneous curvature Analogy

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

F =∫ gdA∫ dA p∫ dV

2 2 F=0⇒ p−2H2 kc ∇ Hkc 2 Hc02H −c0 H−2K=0

k c=0⇒ p−2H =0 Young−Laplaceequation

2 2  p=0,=0,c0=0⇒∇ H 2 H H −K =0 Willmore surfaces Puzzle from the shape of RBC

● Sandwich model (Fung & Tong, 1968)

To obtain the shape of biconcave discoid, Solid shell they should assume the thickness of the Fluid film membrane is nonuniform in μm scale. Solid shell

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 ● Incompressible shell model (Canham, 1970)

Given the area and volume of membrane, the biconcave discoid minimizes the curvature energy ∫ H 2 dA

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. Dumbbell-like Biconcave discoid 取自《从肥皂泡到液晶生物膜》 ● Spontaneous curvature model (Helfrich, 1973)

Given the area and volume of membrane, the biconcave discoid minimizes the spontaneous curvature energy 2 ∫2 H c0 dA

c00⇒ biconcave discoid is energetically favorable

experiment

Axisymmetric numerical result

Can we give a analytic result from the shape equation? 2 2  p−2H2kc ∇ H kc2H c02 H −c0 H−2K =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. Lipid vesicles

● Spherical vesicles

2 2  p−2H2kc ∇ H kc2H c02 H −c0 H−2K =0

2 2 f R≡ pR 2kc c0R−2k c c0=0 2R 1 1 f(R) f(R) H =− ,K= 2 R R  p0,c00  p0,c00 f(R)  p0,c00 R R

R

Might be related to endocytosis ● Torus [Ou-Yang (1990) PRA] =r/

{rcoscos,rcossin,sin}

r2cos cos 2H=− ;K= rcos rcos

2k 2 k c2 2 1 2 p 2 c /  c  0  −    r 2 2 2 3 = / 4k c  −4 k c 8 6 p  c 0 c 0 cos  2 2 2 3 5 k c c0  −8k c c0 10 6 p 2  cos  10μm 2 2 2 2 3 2 k c c0  −4 kc c0 4 2 p 3  cos =0 3

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

=2 2k c c c 0 2 0 [Mutz-Bensimon (1991) PRA]  p=− ,=kc c0 − 2  2  ● Axisymmetric surface and Biconcave discoid

2 2  p−2H2kc ∇ H kc2H c02 H −c0 H−2K =0 Axisymmetric

[Hu & Ou-Yang (1993) PRE]

For−ec0 B0

[Naito-Okuda-OY (1993) PRE]

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

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] 5μm Bar: 2μm ● Previous theories – Derived from axisymmetric variation

● [Jülicher -Lipowsky (1993) PRL]

● [Zhou (2002) PhD thesis] Confine the variational problem in a subspace. Unreasonable results exist.

– General case

● [Capovilla-Guven-Santiago (2002) PRE] Governing equations of edges are not expressed in the explicit forms of curvature and torsion.

● [Tu-OuYang (2003) PRE] Overcome the above shortages. ● Main results in [Tu-OuYang (2003)PRE]

Free energy per area

k G= c 2 H c 2−kK  2 0

Total free energy F =∫GdA∮ ds  F =0⇒shape equationboundary conditions Shape equation: force balance in the normal direction

2 2 kc 2 H c02 H −c0 H −2 K −2 H 2k c ∇ H =0

Boundary conditions (curve C satisfies...)

kc 2 H c0−kkn=0 Force balance equation of points in the edge along normal direction

∂ H , 2 k k −k  =0 c ∂ b n g Moment balance equation of points in the edge around t

Gk g=0 Force 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? Composite membrane model [Sackmann (2002)]

● Cell membrane = bilayer + membrane skeleton

shear bending Bilayer NO YES MSK YES NO CM YES YES ● 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=G2 H , K , 2 J ,Q

    2 J =Tr 11 12 ,Q=det 11 12 12 22 12 22

Assumption 5-7 => up to the second order terms k G= c 2 H c 2−kK k /22 J 2−kQ  2 0 d

Contribution from LB Contribution from MSK Bending energy Compress and shear energy Shape equation & in-plane strain equation

F =∫GdA p∫ dV

 F =0 (Remark: only consider closed cell membrane)

2 2  p2k c [2 H c02 H −c0 H −2 K  2∇ H ]−2 H

2H k −k d 2 J −k ℜ: ∇ u=0

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

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

2 2  p2k c [2 H c0 2 H −c0 H −2 K  2∇ H ]−2 H =0 shape equation of lipid vesicles. Spherical cell membrane and its stability

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

2   pR 22k d −k Rk c c0c0 R−2=0

2k 2 k k 2k 2  d −  c Stable <=> δ F >0 <=>  p pl ≡  3 [l l1−c0 R] ,l1 [k d l l1−k ] R R

Critical osmotic pressure pc=min {pl }

2k c 6−c0 R [return to the result of lipid vesicle, pc= when k =0 R3 OuYang- Helfrich (1987) PRL]

    2 4k /k d 2k d−kk c kkd 2k d −k R pc= 2 when 2 1 R k c6 k d−k  Typical parameters for cell membranes

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

−19 k c≈10 J [Duwe et al. (1990) J. Phys. Fr.] R≈5 m

  2   kkd 2k d−k R 4k/k d 2k d −k k c satisfy 2 1⇒ pc= 2 ≈0.1 Pa k c 6k d −k  R

2 k 6−c R If no MSK, i.e., lipid vesicle, c 0 k=0⇒ pc= ≈0.008 Pa R3

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