Bending Membranes Flashcards
Packing Parameter
Definition
-with knowledge of the optimal are per head group, ao, we can roughly determine the structures that will form
Ns = v/ao*lc
Ns = packing parameter ao = area of head group lc = alkyl chain length v = volume of alkyl chain v = lc(ao+bo)/2 bo = base area
Packing Parameter
Ns ~ 1
- each molecule is a cylinder
- combining them forms a bilayer sheet
Packing Parameter
1/2 < Ns < 1
- each molecule is a cone, tail slightly narrower than head
- combining them forms a liposome/vesicle
Packing Parameter
Ns ≤ 1/3
- each molecule is a cone, head wider than tail
- combining them forms a spherical micelle
Aggregation Number
Definition
-number of surfactant molecules in the aggregate
Packing Parameter
Ns > 1
- each molecule is an inverted cone, tail wider than head
- combining them forms inverse micelles
Bending of Membranes
- the lowest energy state of a lamellar phase bilayer is perfectly flat, planar
- membranes are typically not flat rigid sheets but often display curvature when they form vesicles and organelles
- when membranes are asymmetric they may curve spontaneously
Spontaneous Curvature
-the shape factor of a lipid molecule leads directly to the spontaneous or preferred curvature of a membrane
Ns < 1/3 => flat
Ns ~ 1 => positive curvature
Ns > 1 => negative curvature
Induced Curvature
- inserting lipids (or proteins) with different Ns in to a membrane to force curvature
- BAR proteins placed around the membrane can physically curve it to match the shape of the membrane
Characterising Curvature
-bending is characterised by its radius of curvature, R, or its curvature, c=1/R
Gaussian Curvature
-since in a bilayer there are typically two principle curvatures, c1 and c2, that may be different Gaussian curvature, a measure of total curvature:
κ = c1c2
Mean Curvature
-in a bilayer with two principle curvatures:
H = c1+c2 / 2
Curvature and Free Energy
-to curve the membrane one has to provide free energy
-the free energy density of a membrane, g, is given by:
g = 1/2Kb(c1+c2-c0)² + Kgc1c2
-where:
Kb = bending modulus
Kg = Gaussian curvature modulus
c0 = spontanteous curvature
How bendy are biological molecules?
- the stiffness of biological membranes has been engineered by nature to prevent corrugation by thermal fluctuations, they are relatively stiff on a small length scale
- but at the same time the energy for gross overall changes in shape (i.e. cell crawling) are in the few hundred kbT range, requiring only a few dozen ATP molecules
- SO membranes are tough but still flexible enough
Bending Modulus
Outline
-consider bending a 1D monolayer of thickness d to radius of curvature R
-this requires stretching ahead by Δa:
Δa = ahead*d/R
-assume energy cost for this:
ΔF = Co + C1Δa + C2(Δa)^2 + …
Bending Modulus
Steps
ΔF = Co + C1Δa + C2(Δa)^2 + …
-drop Co (no Δa)
-considering a second monolayer as well (a bilayer) at any particular bend, the other monolayer will be moving in the other direction i.e. compressing rather than expanding
-hence C1Δa will drop out balanced by C1(-Δa) on the opposing monolayer
-for small bends, higher terms (>C2) will be negligibly small
-rename C2=k/2 where k is a bending modulus, 1/2 to account for one molecule depth representing half of the bilayer, half of total contribution to modulus:
elastic energy per phospholipid molecule = 1/2 k (Δa)² = 1/2 k (ahead*d/R)²
Bending Modulus
Free Energy Cost to Bend a Bilayer Membrane
-total number of head groups per unit area is 2/ahead
-so total energy:
2 * 2/ahead * 1/2 k (aheadd/R)^2
= 2kaheadd²/R²
-reduce the top of the equation to bend stiffness, κ=2kahead*d²
-we can say that the free energy cost per unit area to bend a bilayer membrane into a cylinder of radius R is of the form κ/2R²
Bending Modulus
Higher Dimensions
- to bend into a spherical shape, ahead is stretched in 2D not just 1D so Δa is twice as great so 4x the energy is required, 2κ/R² per unit area
- the surface area of a sphere is given by 4πR², so bending energy of a vesicle is 8πκ
- no R value so the total bending energy is INDEPENDENT of radius
