Chapter 8 – Theoretical Biophysics 347
Similarly, for the WLC model,
(8.56)
F
k T
l
R R
R
R
WLC ≈
−
(
)
−
+
B
p
max
max
1
4 1
1
4
2
/
The latter formulation is most popularly used to model force-extension data generated
from experimental single-molecule force spectroscopy techniques (see Chapter 6), and this
approximation deviates from the exact solution by <10% for forces <100 pN. Both models
predict a linear Hookean regime at low forces. Examples of both FJC and WLC fit to the same
experimental data obtained from optical tweezers stretching of a single molecule of titin are
shown in Figure 8.5b.
By integrating the restoring force with respect to end-to-end extension, it is trivial to esti
mate the work done, ΔW in stretching a biopolymer from zero up to end-to-end extension R.
For example, using the FJC model, at low forces (F < kBT/b and R < Rmax/3),
(8.57)
∆W
k T
nb R
k T
R
R
=
=
〈〉
3
2
3
2
2
2
2
B
B
In other words, the free energy cost per spatial dimension degree of freedom reaches kBT/2
when the end-to-end extension reaches its mean value. This is comparable to the equivalent
energy E to stretch an elastic cylindrical rod of radius r modeled by Hooke’s law:
(8.58)
∆
∆
∆
∆
R
R
Y
F
F
E
f
R
F
R
R
=
=
∴
=
(
) =
(
)
∫
σ
0
0
0
2
2
d
where
Y is the Young’s modulus
σ is the stress F/πr2
F0 is πr2Y
However, at high forces and extensions, the work done becomes
(8.59)
∆W
C
k T
b
R
R
B
≈
−
=
−
ln
max
1
where C is a constant. As the biopolymer chain extension approaches its contour length Rmax,
the free energy cost in theory diverges.
8.3.5 REAL BIOPOLYMERS
As discussed earlier, the GC chain model does not account well for the force response of
real molecules to changes in molecular extension, which are better modeled by FJC, FRC,
and WLC models. In practice, however, real biopolymers may have heterogeneity in b and lp
values, that is, there can be multiple different structural components within a molecule. We