2 Topology in Biology: From DNA Mechanics to Enzymology 7Intrinsically straight wormlike chains, those whose minimum-energy con-
formations are that of a perfectly straight rod, have tractable closed-form
expressions for the mean-square end-to-end distance,〈h^2 〉,
〈
h^2〉
=〈h·h〉=2P[
L−P
(
1 −e−L/P)]
, (2.2)
wherehis the end-to-end vector of the polymer chain,Lis its contour length,
andPis a parameter calledthe persistence length.P is a quantity with di-
mensions of length that characterizes the bending rigidity of the wormlike
chain model and measures the tendency of an intrinsically straight chain to
propagate in the direction specified by the tangent to the chain at one end.
It is useful to consider the asymptotic behavior of the expression for〈h^2 〉in
the limit of both small and largeL, namely
lim
L→ 0〈
h^2〉
L^2
= 1 and lim
L→∞〈
h^2〉
L^2
=
1
L
. (2.3)
Thus, in the limit of small wormlike chains, the polymer’s end-to-end vector
tracks the contour whereas the displacement of the ends of large chains grows
with anL^1 /^2 dependence, identical to that of a nonself-avoiding random walk.
Pis a kind of correlation length for the chain, which can be appreciated from
the formula for the average component of the end-to-end vector in the initial
chain direction,〈hz〉
〈hz〉=〈h·ˆz〉=1
X
(
1 −e−X)
, (2.4)
whereX=L/P. For a wormlike chain withL=P (equal to about 50 nm
or 150 base pairs for DNA under physiological conditions), the component of
the end-to-end vector in the chain’s initial direction is only 63% of the overall
contour length. The fact that chains of this length do not behave as rigid rods
is shown in Fig. 2.3, which shows the extent of conformational space occupied
by several Boltzmann-sampled chain conformations propagating initially in
the same (z−) direction.
The concept of persistence length as a measure of intrinsic bending flex-
ibility makes sense only for wormlike chains whose minimum-elastic-energy
conformation is that of a straight rod. Wormlike chains that have distorted
elastic minima deviate from a rodlike structure even in the absence of thermal
fluctuations and thus have a structural or static component to the persistence
length. Thermal fluctuations, the magnitude of which depend on the local
bending modulus of the chain and the absolute temperature, are superposed
on this minimum-energy shape. The value of the bending energy constant in
(2.1),αi, can be related to the persistence length of an intrinsically straight
molecule with the same flexural elasticity, the so-called dynamic persistence
length,Pd
αi=α=2kBTPd/, (2.5)