Chap. 5. Biomembranes

林宙晴

Composition of Biomembranes

• Amphiphile• Mesogenes (ex. Liquid crystal) – mesophase

– Form a variety of condensed phases with properties in between those of solids and isotropic fluids

• Single-chain vs Double chain fatty acid– Single-chained molecules assembles into bilayers only at

high concentration (> 50%)

Phospholipids

• Single vs. Double bond

Chain Length of Fatty Acids

• Too short: hard to form bilayers at low concentration

• Too long: too viscous and lateral diffusion within bilayer is restricted

• ~0.1 nm/CH2 & C ~=15-18 bilayer: 4-5 nm

• Mean cross-sectional area of a single chain is 0.2 nm2, surface area occupied is 0.4-0.7 nm2

Favored Phases

• The phase favored by a particular amphiphile partly reflects its molecular shape.

ratio of head group area to cross sectional area of hydrocarbon region

Forming a hole

• Bending a bilayer needs energy

• Hole formation– Forming a hole needs to overcome edge tension

– Effective edge tension is temperature-dependent and vanishes at sufficiently high temperature.

Self-Assembly of Amphiphiles

• CMC: critical micelle concentration

• Competition between hydrophobic region to contact with water and reduction of entropy

• Ebind is defined as the energy required to create the new water/hydrocarbon interface

Ebind = 2ncRhclcc

Sgas = kB{5/2-ln(． [h/{wmkBT}1/2]3}

Fsol ~ Ebind - T Sgas

nclcc

Rhc

: surface tensionSgas: entropy/moleculeFsol: free energy/molecule: molecule density

Aggregation Density (agg)

• Aggregation (agg) occurs at Fsol = 0

agg． [h/{wmkBT}1/2]3 = exp(5/2- Ebind /kB

T)• Estimates from above formula for 10 carbons

– agg(single) 0.3 molar≒

– agg(double) 2≒ ． 10-5 molar

• Experimental values– agg(single) 10≒ -2 - 10-3 molar

– agg(double) 10≒ -3 - 10-5 molar?? CMC == agg

Dependence of CMC on Chain Length• Single chains have uniformly higher CMCs than d

ouble chains• Experimental values (-slope)

– [single double] = [1.15 1.8]

• Theoretical prediction– [single double] = [2 3]

• Selection of values for (surface tension) may produce more compatible values.

• A more rigorous approach (dropping the assumption of two-phase aggregates) produced similar

results.

Effective Cross Sectional Area

• For hydrocarbon part– vhc = 27.4 + 26.9ncx10-3 nm3

– lhc = 0.154 + 0.126nc nm

ahc = vhc/lhc = 0.21 nm2

• For head group– a0 ~ 0.5 nm2

• In the following, packing in several shapes will be discussed.

Shape Factor (vhc/a0lhc or ahc/a0)

• For spherical micelles– 4R2/a0 = (4R3/3)/vhc

R = 3vhc/a0

∵ R l≦ hc vhc/a0lhc 1/3≦• For cylindrical micelles

– 2Rt/a0 = R2t/3/vhc

R = 2vhc/a0

1/3 < vhc/a0lhc 1/2≦

Shape Factor (continued)

• For bilayers– vhc = a0lhc vhc/a0lhc = 1

1/2 < vhc/a0lhc 1≦• For inverted micelles

– vhc/a0lhc > 1

• For amphiphiles in the cell (real situation)– Single chain: ahc/a0 ~ 0.21/0.5 0.4 ≒ micelles

– Double chain: ahc/a0 ~ 0.42/0.5 0.8≒ bilayers

Another advantage of forming bilayers with double chain fatty acids are a low CMC.

Bilayer Compression Resistance

• First model: a homogeneous rigid sheet, such as a thin metallic plate in air

uxx = uyy = S(2/9Kv + 1/6)

= KA(uxx + uyy) =Sdp

KA = dpKv/(4/9 + Kv/3) (uniform rigid plate)

for many materials, Kv ~ 3

KA increases linearly with plate thickness

Kv & : volume compression and shear moduli

KA: area compression modulusuxx + uyy: relative area change

More on KA

ij = ij Kv tru + 2 (uij- ij tru/3)

Under isotropic pressure, ij = -P ij

P = -Kv tru

3D ii = Kv (uxx + uyy + uzz)

2D (plane strain) ii = KA (uxx + uyy)

1D ii = KL (uxx)

• Unrealistic• Both ii uii are defin

ed differently

Bilayer Compression Resistance

• Second model:• E = /a + a

= 2a0 + (/a)(a – a0)2

• E/a0 at a0 ~ [(a – a0)/ a0]2

also = (KA/2)(uxx + uyy)2

KA = 2 (monolayer)

KA = 4 (bilayer)

experimentally, = 0.02-0.05 J/m2

KA = 0.08-0.2 J/m2

E: interface energy/moleculea: mean interface area: surface tension/a: repulsive energy

uxx + uyy = 2 (a – a0)/ a0

Experimentally Measured KA and Kv

• Experimentally,

KA = 0.1-0.2 J/m2

• When carbon number increases,

KA only increases mildly,

KA is independent of dp

• KA vs. cholesterol content

• Kv ~ 2-3x109 J/m3, about the same as water (Kv = 1.9x109 J/m3)

Model 2 is more likely

Bilayer Bending Resistance

• For a given molecular composition, the energy per unit surface area to bend a bilayer increases with the curvature.

• F = (kb/2)(1/R1 + 1/R2)2 + kG/(R1R2)

• E = 4(2kb + kG) (sphere)

• E = kbL/R (cylinder)

F:

F: energy densityE: bending energy

kb: bending rigidity

kG : Gaussian bending rigidity

Experimental Measurements of kb

• Kb is about 10x of kBT

undulate readily

(please refer to p. 28)

• Thus, measurement of bending modulus needs to control undulation.

• KA, app = KA/[1 + KAkBT/(8kb)]

: applied tensionLow T or high : KA Low : 8kb/kBT

kb also rises with cholesterol content

Interpretation of kb

• Many models predict how kb depends on the bilayer thickness dbl.

kb = KAdbl2/

where = 12, 24 or 48.

• If KA is proportional to dbl, then

kb is proportional to dbl3

• Otherwise

kb is proportional to dbl2

From the plot, KA is independent of dbl

There is little experimental support for the rigid-plate prediction

Edge Energy

• : penalty energy for creating a free edge

• No documented results of curvature on • is assumed to be independent of curvature

• Esphere = 4(2kb + kG)

• Edisk = 4Rv

• When Rv begin to > (2kb + kG)/

sphere configuration is favor

(Bending energy)

Estimating • At T> 0, membrane boundary fluctuates,

larger is needed to seal the edge.

• Simulation show that

* = 1.36kBT/b

• Free energy for N plaquettes

open: F 2N≒ b - kBTNln(12.8)

closed: F -k≒ BTNln(1.73)

* = 1.0kBT/b

b: a length scale from the simulation

Membrane Rupture

• At T = 0,

H = E – A

H = 2R – R2

• At the peak, R* = R < R* holes shrink

R > R* holes expand

• When T increases, the energy barrier lowers.

• For planar membranes in two dimensions,

Edge-tensionmin (*) = 1.66kBT/b

R: radius of a hole: two-dimensional tension

Measured Edge Tensions

• For pure lecithin bilayers

= 4x10-11 J/m

• By exp. (shown)

for SOPC (p. 154)

= 0.9x10-11 J/m

for SOPC+30% cholesterol

= 3.0x10-11 J/m

It is estimated that must > 4x10-12 J/m to make the membrane resistant against rupture at ambient temperature.

R