Worm-like chain

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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.

1 Theoretical Considerations
2 Biological Relevance
3 Stretching Worm-like Chain Polymers
4 See also
5 References

[edit] 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. 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.

[edit] Biological Relevance

Several biologically important polymers can be effectively modeled as worm-like chains, including:

double-stranded DNA;
unstructured RNA; and
unstructured polypeptides (proteins).

[edit] 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).

[edit] See also

[edit] References

O. Kratky, G. Porod (1949), "Röntgenuntersuchung gelöster Fadenmoleküle." Rec. Trav. Chim. Pays-Bas. 68: 1106-1123.
J. F. Marko, E. D. Siggia (1995), "Stretching DNA." Macromolecules, 28: p. 8759.
C. Bustamante, J. F. Marko, E. D. Siggia, and S. Smith (1994), "Entropic elasticity of lambda-phage DNA." Science, 265: 1599-1600.

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Categories: Polymer physics | Biophysics