The WLC model envisions a continuously flexible
isotropic rod. This is in contrast to the
freely-jointed chain model, which is only flexible between discrete freely hinged segments. The model is particularly suited for describing stiffer polymers, with successive segments displaying a sort of cooperativity: nearby segments are roughly aligned. At
room temperature, the polymer adopts a smoothly curved conformation; at T = 0 K, the polymer adopts a rigid rod conformation. For a polymer of maximum length L_0, parametrize the path of the polymer as s \in(0,L_0). Allow \hat t(s) to be the unit tangent vector to the chain at point s, and \vec r(s) to be the position vector along the chain, as shown to the right. Then: :\hat t(s) \equiv \frac {\partial \vec r(s) }{\partial s} and the end-to-end distance \vec R = \int_{0}^{L_0}\hat t(s) ds . The energy associated with the bending of the polymer can be written as: E = \frac {1}{2}k_B T \int_{0}^{L_0} P \cdot \left (\frac {\partial^2 \vec r(s) }{\partial s^2}\right )^{2} ds where P is the polymer's characteristic
persistence length, k_B is the
Boltzmann constant, and T is the absolute temperature. At finite temperatures, the end-to end distance of the polymer will be significantly shorter than the maximum length L_0. This is caused by
thermal fluctuations, which result in a coiled, random configuration of the undisturbed polymer. The polymer's orientation
correlation function can then be solved for, and it follows an
exponential decay with decay constant 1/P: \langle\hat t(s) \cdot \hat t(0)\rangle=\langle \cos \; \theta (s)\rangle = e^{-s/P}\, A useful value is the mean square end-to-end distance of the polymer: \langle R^{2} \rangle = \langle \vec R \cdot \vec R \rangle = \left\langle \int_{0}^{L_0} \hat t(s) ds \cdot \int_{0}^{L_0} \hat t(s') ds' \right\rangle = \int_{0}^{L_0} ds \int_{0}^{L_0} \langle \hat t(s) \cdot \hat t(s') \rangle ds'= \int_{0}^{L_0} ds \int_{0}^{L_0} e^{-\left | s - s' \right | / P} ds' \langle R^{2} \rangle = 2 PL_0 \left [ 1 - \frac {P}{L_0} \left ( 1 - e^{-L_0/P} \right ) \right ] Note that in the limit of L_0 \gg P, then \langle R^{2} \rangle = 2PL_0. This can be used to show that a
Kuhn segment is equal to twice the
persistence length of a worm-like chain. In the limit of L_0 \ll P, then \langle R^{2} \rangle = L_0^2, and the polymer displays rigid rod behavior. The figure to the right shows the crossover from flexible to stiff behavior as the
persistence length increases. == Biological relevance ==