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

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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 $s\; \backslash subseteq\; (0,l)$, allow $\backslash hat\; t(s)$ to be the unit tangent vector to the chain at $s$, and $\backslash vec\; r(s)$ to be the position vector along the chain. Then

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

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

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

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:

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

Biological Relevance

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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

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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

$\backslash frac\; \{FP\}\; \{k\_\{B\}T\}\; =\; \backslash frac\; \{1\}\{4\}\; \backslash left\; (\; 1\; -\; \backslash frac\; \{x\}\; \{L\_0\}\; \backslash right\; )^\{-2\}\; -\; \backslash frac\; \{1\}\{4\}\; +\; \backslash frac\; \{x\}\{L\_0\}$

where $k\_B$ is the Boltzmann constant and $T$ is the absolute temperature (Bustamante, et al, 1994).

See also

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• Polymer
• Polymer physics
• Ideal chain

References

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• 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.

Polymer physics

Biophysics