The WLC model envisions an isotropic rod that is continuously flexible. This is in contrast to the freelyjointed chain model that is flexible only between discrete segments. The wormlike chain model is particularly suited for describing stiffer polymers, with successive segments displaying a sort of cooperativity: all pointing in roughly the same direction. At room temperature, the polymer adopts a conformational ensemble that is smoothly curved; at
T
=
0
K, the polymer adopts a rigid rod conformation.
For a polymer of length
l
, parametrize the path of the polymer as
s
∈
(
0
,
l
)
, allow
t
^
(
s
)
to be the unit tangent vector to the chain at
s
, and
r
→
(
s
)
to be the position vector along the chain. Then
t
^
(
s
)
≡
∂
r
→
(
s
)
∂
s
and the endtoend distance
R
→
=
∫
0
l
t
^
(
s
)
d
s
.
It can be shown that the orientation correlation function for a wormlike chain follows an exponential decay:
⟨
t
^
(
s
)
⋅
t
^
(
0
)
⟩
=
⟨
cos
θ
(
s
)
⟩
=
e
−
s
/
P
,
where
P
is by definition the polymer's characteristic persistence length. A useful value is the mean square endtoend distance of the polymer:
⟨
R
2
⟩
=
⟨
R
→
⋅
R
→
⟩
=
⟨
∫
0
l
t
^
(
s
)
d
s
⋅
∫
0
l
t
^
(
s
′
)
d
s
′
⟩
=
∫
0
l
d
s
∫
0
l
⟨
t
^
(
s
)
⋅
t
^
(
s
′
)
⟩
d
s
′
=
∫
0
l
d
s
∫
0
l
e
−

s
−
s
′

/
P
d
s
′
⟨
R
2
⟩
=
2
P
l
[
1
−
P
l
(
1
−
e
−
l
/
P
)
]
,
Note that in the limit of
l
≫
P
, then
⟨
R
2
⟩
=
2
P
l
. This can be used to show that a Kuhn segment is equal to twice the persistence length of a wormlike chain.
Several biologically important polymers can be effectively modeled as wormlike chains, including:
doublestranded DNA and RNA;
unstructured RNA;
unstructured polypeptides (proteins).
At finite temperatures, the distance between the two ends of the polymer (endtoend distance) will be significantly shorter than the contour length
L
0
. This is caused by thermal fluctuations, which result in a coiled, random configuration of the polymer, when undisturbed. Upon stretching the polymer, the accessible spectrum of fluctuations reduces, which causes an entropic force against the external elongation. This entropic force can be estimated by considering the entropic Hamiltonian:
H
=
H
e
n
t
r
o
p
i
c
+
H
e
x
t
e
r
n
a
l
=
1
2
k
B
T
∫
0
L
0
P
⋅
(
∂
2
r
→
(
s
)
∂
s
2
)
2
d
s
−
x
F
.
Here, the contour length is represented by
L
0
, the persistence length by
P
, the extension is represented by
x
, and external force is represented by
F
.
Laboratory tools such as atomic force microscopy (AFM) and optical tweezers have been used to characterize the forcedependent stretching behavior of the polymers listed above. An interpolation formula that approximates the forceextension behavior with about 15% relative error is (J. F. Marko, E. D. Siggia (1995)):
F
P
k
B
T
=
1
4
(
1
−
x
L
0
)
−
2
−
1
4
+
x
L
0
where
k
B
is the Boltzmann constant and
T
is the absolute temperature. More accurate approximation for the forceextension behavior with about 1% relative error is:
F
P
k
B
T
=
1
4
(
1
−
x
L
0
)
−
2
−
1
4
+
x
L
0
−
0.8
(
x
L
0
)
2.15
Approximation for the extensionforce behavior with about 1% relative error was also reported:
x
L
0
=
4
3
−
4
3
F
P
k
B
T
+
1
−
10
e
900
k
B
T
F
P
4
F
P
k
B
T
(
e
900
k
B
T
F
P
4
−
1
)
2
+
(
F
P
k
B
T
)
1.62
3.55
+
3.8
(
F
P
k
B
T
)
2.2
When extending most polymers, their elastic response cannot be neglected. As an example, for the wellstudied case of stretching DNA in physiological conditions (near neutral pH, ionic strength approximately 100 mM) at room temperature, the compliance of the DNA along the contour must be accounted for. This enthalpic compliance is accounted for the material parameter
K
0
, the stretch modulus. For significantly extended polymers, this yields the following Hamiltonian:
H
=
H
e
n
t
r
o
p
i
c
+
H
e
n
t
h
a
l
p
i
c
+
H
e
x
t
e
r
n
a
l
=
1
2
k
B
T
∫
0
L
0
P
⋅
(
∂
r
→
(
s
)
∂
s
)
2
d
s
+
1
2
K
0
L
0
x
2
−
x
F
,
with
L
0
, the contour length,
P
, the persistence length,
x
the extension and
F
external force. This expression takes into account both the entropic term, which regards changes in the polymer conformation, and the enthalpic term, which describes the elongation of the polymer due to the external force. In the expression above, the enthalpic response is described as a linear Hookian spring. Several approximations have been put forward, dependent on the applied external force. For the lowforce regime (F < about 10 pN), the following interpolation formula was derived:
F
P
k
B
T
=
1
4
(
1
−
x
L
0
+
F
K
0
)
−
2
−
1
4
+
x
L
0
−
F
K
0
.
For the higherforce regime, where the polymer is significantly extended, the following approximation is valid:
x
=
L
0
(
1
−
1
2
(
k
B
T
F
P
)
1
/
2
+
F
K
0
)
.
A typical value for the stretch modulus of doublestranded DNA is around 1000 pN and 45 nm for the persistence length. More accurate interpolation formulas for the forceextension and extensionforce behaviors are:
F
P
k
B
T
=
1
4
(
1
−
x
L
0
+
F
K
0
)
−
2
−
1
4
+
x
L
0
−
F
K
0
−
0.8
(
x
L
0
−
F
K
0
)
2.15
x
L
0
=
4
3
−
4
3
F
P
k
B
T
+
1
−
10
e
900
k
B
T
F
P
4
F
P
k
B
T
(
e
900
k
B
T
F
P
4
−
1
)
2
+
(
F
P
k
B
T
)
1.62
3.55
+
3.8
(
F
P
k
B
T
)
2.2
+
F
K
0