384 8.6  Rigid-Body and Semirigid-Body Biomechanics

Equating ΔR with the membrane thickness 2w therefore indicates that the bending free

energy cost of establishing this spherical bilayer for a membrane patch area A0 is

(8.135)

∆G

A w

A

= 16

0

2

πκ

In other words, there is no direct dependence on the vesicle radius. In practice this can be

a few hundred kBT for a lipid vesicle. However, Equation 8.132 implies that relatively large

changes to the packing density of phospholipid molecules have a small free energy cost (~kBT

or less). This is consistent with phospholipid bilayers being relatively fluid structures.

FEA can also be applied at macromolecular length scales, for example, to model the effects

of mesh network of cytoskeletal filaments under the cell membrane and how these respond

to external mechanical perturbations. This level of mesoscale modeling can also be applied

to the bending motions of heterogeneous semi-​stiff filaments such as those found in cilia,

which enable certain cells such as sperm to swim and which can cause fluid flow around cells

in tissues.

Worked Case Example 8.3: Nucleation of Phase-​Separating Liquid Droplets

a For the formation of biomolecular liquid droplets, if the free energy ΔG is made up pri­

marily of two components of a bulk enthalpic component due to nearest neighbor

attractive interactions of phase-​separating molecules and a surface tension compo­

nent, which scales with the area of droplets to limit their growth, show that you can

model ΔG as –​AN +​ BN^2/​3 where N is the number of phase-​separated molecules inside a

spherical droplet.

What is the energy barrier and the critical number of biomolecules Nc in a droplet in

terms of A and B?

c A protein of ~20 nm effective globular diameter, which was implicated in

neurodegenerative disease, was observed to form liquid–​liquid droplets in live mam­

malian cells. In separate in vitro experiments, the exothermic change in chemical

potential energy upon phase separation was estimated to be 2 × 10^–​3 kBT per molecule,

whereas the surface energy per unit area related to surface tension was equivalent to

4.5 × 10^–​2 kBT per molecule. Predict the radius of droplets at the nucleation activation

barrier assuming tight-​packing of proteins and the ratio of the number of droplets

with this radius \compared to droplets with a 25% larger radius.

d When measurements were performed in a living cell, the total number of these phase-​

separating protein molecules was estimated to be ~7000 molecules per cell. Using

super-​resolution PALM on 10 different cells, from ~5000 droplets detected, a total of

352 had a diameter in the range 140–​160 nm while the number of droplets whose

diameter was in the range 180–​190 nm was 55. Discuss these findings considering

your prediction from part (c).

Answers

a The total free energy change of the bulk enthalpic interaction will be propor­

tional to the number of molecules present N, assuming just nearest neighbor

interactions, and will be negative for an attractive (i.e. exothermic) interaction, so

will be –​AN where A is a positive constant. The free energy change associated with

surface tension is proportional to droplet area. Assuming droplet density remains

the same, its volume is proportional to N, so its radius is proportional to N^1/​3, hence

its surface area is proportional to N^2/​3. Since surface tension will oppose droplet

growth the associated surface tension free energy change is positive, hence +​BN^2/​3

where be B is a positive constant. Therefore, the net free energy change is

-​AN +​ BN^2/​3.

KEY BIOLOGICAL

APPLICATIONS:

RIGID–​SEMIRIGID BODY

MODELING TOOLS

Cell pattern formation in devel­

opmental biology; Mechanical

signal transduction analysis.