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.
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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 <math>T = 0</math> K, the polymer adopts a rigid rod conformation.
For a polymer of length <math>l</math>, parametrize the path of the polymer as <math>s \subseteq (0,l)</math>, allow <math>\hat t(s)</math> to be the unit tangent vector to the chain at <math>s</math>, and <math>\vec r(s)</math> to be the position vector along the chain. Then
<math>\hat t(s) \equiv \frac {\partial \vec r(s) }{\partial s}</math> and the end-to-end distance <math>\vec R = \int_{0}^{l}\hat t(s) ds</math> .
It can be shown that the orientation correlation function for a worm-like chain follows an exponential decay:
<math>\langle \cos \; \theta (s)\rangle = e^{-s/P}\,</math>,
where <math>P</math> is by definition the polymer's characteristic persistence length. A useful value is the mean square end-to-end distance of the polymer:
<math> \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} </math>
- Note that in the limit of <math>l >\! > P</math>, then <math>\langle R^{2} \rangle = 2Pl</math>. 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 are known to behave as worm-like chains, including:
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 <math>x</math> of a WLC with contour length <math>L_0</math> and persistence length <math>P</math> in response to a stretching force <math>F</math> is
<math>\frac {FP} {k_{B}T} = \frac {1}{4} \left ( 1 - \frac {x} {L_0} \right )^{-2} - \frac {1}{4} + \frac {x}{L_0}</math>
where <math>k_B</math> is the Boltzmann constant and <math>T</math> is the absolute temperature (Bustamante, et al, 1994).
See also
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.
