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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
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