348 8.3 Mechanics of Biopolymers
can model these effects in the force dependence as having N-such separate chains attached
in parallel, both with FJC and WLC models. Thus, for N-FLC,
(8.60)
R
R
R
Fb
k T
k T
Fb
i
N
i
i
N
i
i
i
=
=
−
=
=
∑
∑
1
1
max
B
B
coth
,
Similarly, for N-WLC,
(8.61)
F
k T
l
R R
R
R
R
R
B
p i
i
i
i
i
i
N
=
−
(
)
−
+
=
=∑
,
,
,
1
4 1
1
4
1
/
max
max
i
However, in practice, real biopolymers do not consist of segments that are freely jointed as
in the FJC. There is real steric hindrance to certain conformations that are not accounted
for in the FJC model, though to some extent these can be modeled by the additional angular
constraints on the FRC model, which are also embodied in the WLC model. However, all of
these models have the key assumption that different sections of the molecule can transpar
ently pass through each other, which is obviously unphysical. More complex excluded volume
WLC approaches can disallow such molecular transparency.
Also, the assumption that chain segments, or equivalent sections of the chain of lp length
scale, are infinitely stiff is unrealistic. In general, there may also be important enthalpic spring
contributions to molecular elasticity. For example, the behavior of the chain segments can
be more similar to Hookean springs, consistent with an increasing separation of constituent
atoms with greater end-to-end distance, resulting in an opposing force due to either bonding
interactions (i.e., increasing the interatomic separation beyond the equilibrium value, simi
larly for bond angle and dihedral components) and for nonbonding interactions, for example,
the Lennard–Jones potential predicts a smaller repulsive force and greater vdW attractive
force at increasing interatomic separations away from the equilibrium separation, and similar
arguments apply to electrostatic interactions. These enthalpic effects can be added to the
purely entropic components in modified elastically jointed chain models. For example, for a
modified FJC model in which a chain is composed of freely jointed bonds that are connected
by joints with some finite total enthalpic stiffness K,
(8.62)
R
R
Fb
k T
k T
Fb
F
K
B
B
=
−
+
max coth
1
or similarly, for an enthalpic-modified WLC model,
(8.63)
F
k T
l
R R
F K
R
R
F
K
p
=
−
+
(
)
−
+
−
B
1
4 1
1
4
2
/
/
max
max
Deviations exist between experimentally determined values of b and lp parameters and those
predicted theoretically on the basis of what appear to be physically sensible length scales for
known structural information. For example, the length of a single amino acid in a protein, or
the separation of a single nucleotide base pair in a nucleic acid, might on first inspection seem
like sensible candidates for an equivalent segment of a chain. Also, there are inconsistencies
between experimentally estimated values for RG of some biopolymers compared against the
size of cellular structures that contain these molecules (Table 8.1).