Worm-like chain

The worm-like chain (WLC) model in polymer physics is used to describe the behavior of semi-flexible polymers; it is sometimes referred to as the Kratky-Porod worm-like chain model.

Theoretical Considerations
The WLC model envisions an isotropic rod that is continuously flexible; this is in contrast to the freely-jointed chain model that is flexible only between discrete segments. The worm-like 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 $$s \subseteq (0,l)$$, allow $$\hat t(s)$$ to be the unit tangent vector to the chain at $$s$$, and $$\vec r(s)$$ to be the position vector along the chain. Then


 * $$\hat t(s) \equiv \frac {\partial \vec r(s) }{\partial s}$$ and the end-to-end distance $$\vec R = \int_{0}^{l}\hat t(s) ds$$.

It can be shown that the orientation correlation function for a worm-like chain follows an exponential decay:


 * $$\langle\hat t(s) \cdot \hat t(0)\rangle=\langle \cos \; \theta (s)\rangle = e^{-s/P}\,$$,

where $$P$$ is by definition the polymer's characteristic persistence length. A useful value is the mean square end-to-end distance of the polymer:

$$ \begin{matrix} \langle R^{2} \rangle & = & \langle \vec R \cdot \vec R \rangle \\ \\ \ & = & \langle \int_{0}^{l} \hat t(s) ds \cdot \int_{0}^{l} \hat t(s') ds' \rangle \\ \\ \ & = & \int_{0}^{l} ds \int_{0}^{l} \langle \hat t(s) \cdot \hat t(s') \rangle ds' \\ \\ \ & = & \int_{0}^{l} ds \int_{0}^{l} e^{-\left | s - s' \right | / P} ds' \\ \\ \ \langle R^{2} \rangle & = & 2 Pl \left [ 1 - \frac {P}{l} \left ( 1 - e^{-l/P} \right ) \right ] \end{matrix} $$


 * Note that in the limit of $$l >\! > P$$, then $$\langle R^{2} \rangle = 2Pl$$. This can be used to show that a Kuhn segment is equal to twice the persistence length of a worm-like chain.

Biological Relevance
Several biologically important polymers can be effectively modeled as worm-like chains, including:
 * double-stranded DNA;
 * unstructured RNA; and
 * unstructured polypeptides (proteins).

Stretching Worm-like Chain Polymers
Laboratory tools such as atomic force microscopy (AFM) and optical tweezers have been used to characterize the force-dependent stretching behavior of the polymers listed above. An interpolation formula that describes the extension $$x$$ of a WLC with contour length $$L_0$$ and persistence length $$P$$ in response to a stretching force $$F$$ is
 * $$\frac {FP} {k_{B}T} = \frac {1}{4} \left ( 1 - \frac {x} {L_0} \right )^{-2} - \frac {1}{4} + \frac {x}{L_0}$$

where $$k_B$$ is the Boltzmann constant and $$T$$ is the absolute temperature (Bustamante, et al., 1994; Marko et al., 1995).

In the particular case of stretching DNA in physiological buffer (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 by adding a stretch modulus $$K_0$$ to the above relation:


 * $$\frac {FP} {k_{B}T} = \frac {1}{4} \left ( 1 - \frac {x} {L_0} + \frac {F}{K_0} \right )^{-2} - \frac {1}{4} + \frac {x}{L_0} - \frac {F}{K_0}$$

where a typical value for the stretch modulus of double-stranded DNA is around 1000 pN and 45 nm for the persistence length (Wang, et al., 1997).