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/CH 2 & C ~=15-18  bilayer: 4-5 nm Mean cross-sectional area of a single chain is 0.2 nm 2, surface area occupied is nm 2

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 E bind is defined as the energy required to create the new water/hydrocarbon interface E bind = 2  n c R hc l cc  S gas = k B {5/2-ln(  ． [h/{w  mk B T} 1/2 ] 3 } F sol ~ E bind - T S gas n c l cc R hc  : surface tension S gas : entropy/molecule F sol : free energy/molecule  : molecule density

Aggregation Density (  agg ) Aggregation (  agg ) occurs at F sol = 0   agg ． [h/{w  mk B T} 1/2 ] 3 = exp(5/2- E bind /k B T) Estimates from above formula for 10 carbons –  agg (single) ≒ 0.3 molar –  agg (double) ≒ 2 ． molar Experimental values –  agg (single) ≒ molar –  agg (double) ≒ molar ?? CMC ==  agg

Dependence of CMC on Chain Length Single chains have uniformly higher CMCs than double chains Experimental values (-slope) –[single double] = [ ] 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 –v hc = n c x10 -3 nm 3 –l hc = n c nm  a hc = v hc /l hc = 0.21 nm 2 For head group –a 0 ~ 0.5 nm 2 In the following, packing in several shapes will be discussed.

Shape Factor (v hc /a 0 l hc or a hc /a 0 ) For spherical micelles –4  R 2 /a 0 = (4  R 3 /3)/v hc  R = 3v hc /a 0 ∵ R ≦ l hc  v hc /a 0 l hc ≦ 1/3 For cylindrical micelles –2  Rt/a 0 =  R 2 t/3/v hc  R = 2v hc /a 0  1/3 < v hc /a 0 l hc ≦ 1/2

Shape Factor (continued) For bilayers –v hc = a 0 l hc  v hc /a 0 l hc = 1  1/2 < v hc /a 0 l hc ≦ 1 For inverted micelles –v hc /a 0 l hc > 1 For amphiphiles in the cell (real situation) –Single chain: a hc /a 0 ~ 0.21/0.5 ≒ 0.4  micelles –Double chain: a hc /a 0 ~ 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 u xx = u yy = S(2/9K v + 1/6  )  = K A (u xx + u yy ) =Sd p  K A = d p K v /(4/9 + K v /3  ) (uniform rigid plate) for many materials, K v ~ 3   K A increases linearly with plate thickness K v &  : volume compression and shear moduli K A : area compression modulus u xx + u yy : relative area change

More on K A  ij =  ij K v tru + 2  (u ij -  ij tru/3) Under isotropic pressure,  ij = -P  ij  P = -K v tru 3D  ii = K v (u xx + u yy + u zz ) 2D(plane strain)  ii = K A (u xx + u yy ) 1D  ii = K L (u xx ) Unrealistic Both  ii u ii are defined differently

Bilayer Compression Resistance Second model: E =  /a +  a = 2  a 0 + (  /a)(a – a 0 ) 2  E/a 0 at a 0 ~  [(a – a 0 )/ a 0 ] 2 also = (K A /2)(u xx + u yy ) 2  K A = 2  (monolayer) K A = 4  (bilayer) experimentally,  = J/m 2  K A = J/m 2 E: interface energy/molecule a: mean interface area  : surface tension  /a: repulsive energy u xx + u yy = 2 (a – a 0 )/ a 0

Experimentally Measured K A and K v Experimentally, K A = J/m 2 When carbon number increases, K A only increases mildly,  K A is independent of d p K A vs. cholesterol content K v ~ 2-3x10 9 J/m 3, about the same as water (K v = 1.9x10 9 J/m 3 ) 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 = (k b /2)(1/R 1 + 1/R 2 ) 2 + k G /(R 1 R 2 ) E = 4  (2k b + k G ) (sphere) E =  k b L/R (cylinder) F:F: F: energy density E: bending energy k b : bending rigidity k G : Gaussian bending rigidity

Experimental Measurements of k b K b is about 10x of k B T  undulate readily (please refer to p. 28) Thus, measurement of bending modulus needs to control undulation. K A, app = K A /[1 + K A k B T/(8  k b  )]  : applied tension Low T or high  : K A Low  : 8  k b  /k B T k b also rises with cholesterol content

Interpretation of k b Many models predict how k b depends on the bilayer thickness d bl. k b = K A d bl 2 /  where  = 12, 24 or 48. If K A is proportional to d bl, then k b is proportional to d bl 3 Otherwise k b is proportional to d bl 2 From the plot, K A is independent of d bl 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 E sphere = 4  (2k b + k G ) E disk = 4  R v When R v begin to > (2k b + k G )/ sphere configuration is favor (Bending energy)

Estimating At T> 0, membrane boundary fluctuates, larger is needed to seal the edge. Simulation show that * = 1.36k B T/b Free energy for N plaquettes open: F ≒ 2N b - k B TNln(12.8) closed: F ≒ -k B TNln(1.73)  * = 1.0k B T/b b: a length scale from the simulation

Membrane Rupture At T = 0, H = E –  A  H = 2  R –  R 2 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-tension min ( *) = 1.66k B T/b R: radius of a hole  : two-dimensional tension

Measured Edge Tensions For pure lecithin bilayers = 4x J/m By exp. (shown) for SOPC (p. 154) = 0.9x J/m for SOPC+30% cholesterol = 3.0x J/m It is estimated that must > 4x J/m to make the membrane resistant against rupture at ambient temperature.  R 