382 8.6  Rigid-Body and Semirigid-Body Biomechanics

Minimizing the free space around cells minimizes the distance to diffusion for chemical

signals, maximizes the mechanical efficiency in mechanical signal transduction, and results

in a maximally strong tissue. Hence there are logical reasons for cells in many tissues at least

to pack in this way.

8.6.4  MOLECULAR BIOMECHANICS

Molecular biomechanics analysis at one level has already been discussed previously in this

chapter of biopolymer modeling, which can characterize the relation between molecular

force and end-​to-​end extension using a combination of entropic spring approximations with

improvement to correct for excluded volumes and forces of enthalpic origin. In the limit

of very stiff biomolecule or filaments, a rodlike approximation, in essence Hooke’s law, is

valid, but including also torsional effects such as in the bending cantilever approximation (see

Chapter 6). In the case of a stiff, narrow rod of length L and small cross-​sectional area dA, if z

is a small deflection normal to the rod axis (Figure 8.11b), and we assume that strain therefore

varies linearly across the rod, then

(8.128)

∆L

L

z

R

≈

where R is the radius of curvature of the bent rod. We can then use Hooke’s law and inte­

grate the elastic energy density (Y/​2)(ΔL/​L)² over the total volume of the rod, where Y is the

Young’s modulus of the rod material, to calculate the total bending energy Ebend:

(8.129)

E

L

Y

L

L

A

Y

z

R

A

B

R

bend

All rod

All rod

=



^



^

=



^



^

=

∫

∫

2

2

2

2

2

2

∆

d

d

where B is the bending modulus given by YI where I is the moment of inertia. For example,

for a cylindrical rod of cross-​sectional radius r, I =​ πr⁴/​4.

An active area of biophysical modeling also involves the mechanics of phospholipid bilayer

membranes (Figure 8.11c). Fluctuations in the local curvature of a phospholipid bilayer

result in changes to the free energy of the system. The free energy cost ΔG of bilayer area

fluctuations ΔA can be approximated from a Taylor expansion centered around the equilib­

rium state of zero tension in the bilayer equivalent to area A =​ A0. The first nonzero term in

the Taylor expansion is the second-​order term:

(8.130)

∆G

G

A

A A

≈

∂

∂



^



^

=

1

2

2

2

0

A mechanical parameter called the “area compressibility modulus” of the bilayer, κA, is

defined as

(8.131)

κA =

∂

∂



^



^

=

A

G

A

A A

0

2

2

0

Thus, the free energy per unit area, Δg, is given by

(8.132)

∆

∆

g

A

A

≈

(

)

1

2

2

κ