346 8.3 Mechanics of Biopolymers
KEY POINT 8.6
There are fewer accessible microstates for high values of end-to-end extension
compared to low values for a stretched biopolymer resulting in an entropic restoring
force toward smaller extension values. The force scale required to straighten an FJC
runs from zero to ~kBT/b.
The force response F of a biopolymer to a change in end-to-end extension r can be
characterized in the various models of elasticity described earlier. If the internal energy of
the biopolymer stays the same for all configurations (in other words that each segment in the
equivalent chain is infinitely rigid), then entropy is the only contribution to the Helmholtz
free energy A:
(8.49)
A R
TS R
k T
p n R
B
( ) = −
( ) =
(
)
ln
,
where p is the radial probability density function discussed in the previous section.
The restoring force experienced by a stretched molecule of end-to-end length R is then
given by
(8.50)
F
A
R
= ∂
∂
with a molecular stiffness k given by
(8.51)
k
F
R
= ∂
∂
Using the result for p for the GC model of Equation 8.44 indicates
(8.52)
A
k TR
R
b
B
= 3
2
2
max
Therefore, the GC restoring force is given by
(8.53)
F
k TR
R
b
GC
B
max
= 3
Thus, the GC molecular stiffness is given by
(8.54)
F
k T
nb
k T
R
GC
B
B
FJC
=
= 〈
〉
3
3
2
2
This implies that a GC exhibits a constant stiffness with extension, that is, it obeys Hooke’s
law, with the stiffness proportional to the thermal energy scale of kBT and to the reciprocal
of b2. This is a key weakness with the GC model; a finite force response even after the bio
polymer is stretched beyond its own contour length is clearly unphysical. The FJC and WLC
models are better for characterizing a biopolymer’s force response with extension. Evaluating
p for the FJC and WLC models is more complicated, but these ultimately indicate that, for
the FJC model,
(8.55)
R
R
F
b
k T
k T
F
b s
≈
−
max
FJC
B
B
FJC
coth