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Chap. 5. Biomembranes

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

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Chap. 5. Biomembranes

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  1. Chap. 5. Biomembranes 林宙晴

  2. 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%)

  3. Phospholipids • Single vs. Double bond

  4. 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

  5. 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

  6. 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.

  7. nclcc Rhc 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= 2pncRhclccg Sgas = kB{5/2-ln(r．[h/{wpmkBT}1/2]3} Fsol ~ Ebind - T Sgas g: surface tension Sgas: entropy/molecule Fsol: free energy/molecule r: molecule density

  8. Aggregation Density (ragg) • Aggregation (ragg) occurs at Fsol = 0  ragg．[h/{wpmkBT}1/2]3 = exp(5/2- Ebind /kBT) • Estimates from above formula for 10 carbons • ragg(single) ≒ 0.3 molar • ragg(double) ≒ 2．10-5 molar • Experimental values • ragg(single) ≒ 10-2 -10-3 molar • ragg(double) ≒ 10-3 -10-5 molar ?? CMC == ragg

  9. Dependence of CMC on Chain Length • Single chains have uniformly higher CMCs than double chains • Experimental values (-slope) • [single double] = [1.15 1.8] • Theoretical prediction • [single double] = [2 3] • Selection of values for g (surface tension) may produce more compatible values. • A more rigorous approach (dropping the assumption of two-phase aggregates) produced similar results.

  10. 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.

  11. Shape Factor (vhc/a0lhc or ahc/a0) • For spherical micelles • 4pR2/a0 = (4pR3/3)/vhc  R = 3vhc/a0 ∵ R ≦ lhc vhc/a0lhc ≦ 1/3 • For cylindrical micelles • 2pRt/a0 = pR2t/3/vhc  R = 2vhc/a0  1/3 < vhc/a0lhc ≦ 1/2

  12. 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.

  13. Bilayer Compression Resistance • First model: a homogeneous rigid sheet, such as a thin metallic plate in air uxx = uyy = S(2/9Kv + 1/6m) t = KA(uxx + uyy) =Sdp  KA = dpKv/(4/9 + Kv/3m) (uniform rigid plate) for many materials, Kv ~ 3m  KA increases linearly with plate thickness Kv & m: volume compression and shear moduli KA: area compression modulus uxx + uyy: relative area change

  14. More on KA sij = dij Kv tru + 2m (uij- dij tru/3) Under isotropic pressure, sij = -P dij  P = -Kv tru 3D sii = Kv (uxx + uyy + uzz) 2D (plane strain) sii = KA (uxx + uyy) 1D sii = KL (uxx) • Unrealistic • Both sii uii are defined differently

  15. Bilayer Compression Resistance • Second model: • E = a/a + ga = 2ga0 + (g/a)(a – a0)2 • DE/a0 at a0 ~ g [(a – a0)/ a0]2 also = (KA/2)(uxx + uyy)2  KA = 2g (monolayer) KA = 4g (bilayer) experimentally, g= 0.02-0.05 J/m2  KA = 0.08-0.2 J/m2 E: interface energy/molecule a: mean interface area g: surface tension a/a: repulsive energy uxx + uyy = 2 (a – a0)/ a0

  16. 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

  17. 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 = 4p(2kb + kG) (sphere) • E = pkbL/R (cylinder) F: energy density E: bending energy kb: bending rigidity kG : Gaussian bending rigidity F:

  18. 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/(8pkbt)] t: applied tension Low T or high t: KA Low t: 8pkbt/kBT kb also rises with cholesterol content

  19. Interpretation of kb • Many models predict how kb depends on the bilayer thickness dbl. kb = KAdbl2/a where a = 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

  20. Edge Energy • l: penalty energy for creating a free edge • No documented results of curvature on l • l is assumed to be independent of curvature • Esphere = 4p(2kb + kG) • Edisk = 4pRvl • When Rv begin to > (2kb + kG)/l sphere configuration is favor (Bending energy)

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

  22. Membrane Rupture • At T = 0, H = E – tA DH = 2pRl – tpR2 • At the peak, R* = t/l R < R*  holes shrink R > R*  holes expand • When T increases, the energy barrier lowers. • For planar membranes in two dimensions, Edge-tensionmin (l*) = 1.66kBT/b R: radius of a hole t: two-dimensional tension

  23. Measured Edge Tensions • For pure lecithin bilayers l = 4x10-11 J/m • By exp. (shown) for SOPC (p. 154) l = 0.9x10-11 J/m for SOPC+30% cholesterol l = 3.0x10-11 J/m l = R/t It is estimated that l must > 4x10-12 J/m to make the membrane resistant against rupture at ambient temperature.

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