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 Incoming links: Uniqueness of the packed spherical bilayer radius

Treat the liposome using the usual incompressible-tail packing model of a lipid bilayer: neglect the head-group thickness and let

v

be the volume filling the hydrophobic part of the lipid vesicle. Both membrane leaflets have area

a

per head group. Put

d = v / a

, and let

R

be the unknown outer radius. Counting molecules from the inner and outer surface areas gives

M_{i} = \frac{4 π r _{i}^{2}}{a}, M_{o} = \frac{4 π R ^{2}}{a} .

Equating the total molecular volume to the volume of the spherical shell therefore gives

\frac{4 π}{3} (R^{3} - r_{i}^{3}) = v (M_{i} + M_{o}), R^{3} - r_{i}^{3} = 3 d (R^{2} + r_{i}^{2}) .

To prove uniqueness in the physical domain

R > r_{i} > 0

q (R) = \frac{R ^{3} - r _{i}^{3}}{R ^{2} + r _{i}^{2}} .

q^{'} (R) = \frac{R ^{4} + 3 R ^{2} r _{i}^{2} + 2 R r _{i}^{3}}{( R ^{2} + r _{i}^{2} ) ^{2}} > 0.

Also

q (r_{i}) = 0

and

q (R) \to \infty

R \to \infty

q (R) = 3 d

has exactly one solution with $R > r_{i}$ . This proves the required uniqueness without choosing among ambiguous branches of a cubic formula.

The maximum tail length is a physical admissibility bound, not automatically the actual leaflet thickness. If

r_{m}

is the radius where the two tail regions meet, separate membrane leaflet volume balances give

r_{m}^{3} = r_{i}^{3} + 3 d r_{i}^{2}, R^{3} - r_{m}^{3} = 3 d R^{2} .

The unique candidate must satisfy

r_{m} - r_{i} \leq l, R - r_{m} \leq l .

In particular

R - r_{i} \leq 2 l

is necessary. The premise that a lipid vesicle has assembled supplies this compatibility: arbitrary positive choices of

r_{i}, v, a, l

need not allow any such lipid bilayer. For a very large vesicle the two thicknesses tend to

d

, so

R - r_{i} ≃ 2 v / a

. Setting the thickness equal to

2 l

for every molecule would confuse maximum extension with actual packing. The volume-balance argument and its domain are an example of uniqueness of the packed spherical bilayer radius.