Chapter 8 – Theoretical Biophysics  383

This is useful, since we can model the free energy changes in expanding and contracting a

spherical lipid vesicle (e.g., an approximation to a cell). If we say that the area on the outside

of a bilayer per phospholipid polar head group (see Chapter 2) is A0 +​ ΔA and that on the

inside of the bilayer is A0 − ΔA since the molecular packing density is greater on the inside

than on the outside (see Figure 8.11b), then Δg is equivalent to the free energy change per

phospholipid molecule. The total area S of a spherical vesicle of radius R is 4πR²; thus

(8.133)

∆

∆

∆

∆

A

A

S

S

R

R

R

R

R

=

=

⋅

=

8

4

2

2

π

π

Therefore, the free energy per unit area is given by

(8.134)

∆

∆

g

A^A

R

R

=

(

)

4

0

2

κ

FIGURE 8.11  Modeling mechanics of cells and molecules. (a) The 2D granocentric model of

foam physics—​cells are equivalent to the bubbles in the foam, and the contact angle Φ can be

calculated using trigonometry from knowledge of the touching cell radii. The mean number of

neighbors of a cell is six, as if each cell is an equivalent hexagon (dashed lines). (b) Schematic of

bending of a stiff molecular rod. (c) Schematic of bending of a phospholipid bilayer.