Chapter 8 – Theoretical Biophysics 345
(8.45)
1
1
2
3
3
2
4
1
2
3 2
2
2
0
2
R
i
j b
r
i
j b
r r dr
H
i
j
=
−
=
−
(
)
−
−
−
∞∫
r
r
π
π
exp
.
=(
)
−
=(
)
−
=(
)
−
−
−
−
−
∫∫
6
6
6
1 2
1
1 2
1 2
1
2
0
0
1 2
π
π
π
b
i
j
b n
di dj i
j
n
n
1 2
1
1 2
2
2
8 3
1
8
3
2
1
8
3
2
0 27
b
n
R
nb
R
R
H
FJC
FJC
−
−
(
)
∴
=
=
≈
π
π
.
Thus
〈
〉>
〈
〉>
>
R
R
R
R
FJC
FJC
G
H
2
2
.
8.3.3 WORMLIKE CHAINS
Biopolymers with relatively long segment lengths are accurately modeled as a wormlike
chain, also known as the continuous version of the Kratky–Porod model. This model can be
derived from the FRC by assuming small θ, with the result that
(8.46)
〈
〉=
−
−
−
R
l R
l
R
l
WLC
p
max
p
max
p
2
2
2
2
1
exp
where
lp is known as the persistence length
Rmax is the maximum end-to-end length, which is equivalent to the contour length from
the FJC of nb
In the limit of a very long biopolymer (i.e., Rmax ≫ lp),
(8.47)
〈
〉≈
−
(
) ≡〈
〉=
∴
≈
R
l R
l
nb
R
nb
l
b
WLC
p
max
p
FJC
2
2
2
2
2
2
2
p
This is the ideal chain limit for a relatively compliant biopolymer. Similarly, in the limit of a
short biopolymer (i.e., Rmax ≪ lp),
(8.48)
〈
〉≈
R
R
WLC
max
2
2
This is known as the rodlike limit corresponding to a very stiff biopolymer.
8.3.4 FORCE DEPENDENCE OF POLYMER EXTENSION
The theory of flexible polymers described previously for biopolymer mechanics results
in several different polymer conformations having the same free energy state (i.e., several
different microstates). At high end-to-end extension close to a biopolymer’s natural contour
length, there are fewer conformations that the molecule can adopt compared to lower values.
Therefore, there is an entropic deficit between high and low extensions, which is manifest as
an entropic force in the direction of smaller end-to-end extension.