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.
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
K, the polymer adopts a rigid rod conformation.
For a polymer of length , parametrize the path of the polymer as , allow to be the unit tangent vector to the chain at , and to be the position vector along the chain. Then
- and the end-to-end distance .
It can be shown that the orientation correlation function for a worm-like chain follows an exponential decay:
where 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 , then . This can be used to show that a Kuhn segment is equal to twice the persistence length of a worm-like chain.
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
of a WLC with contour length
and persistence length
in response to a stretching force
where is the Boltzmann constant and 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 to the above relation:
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).
- 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. PMID 8079175
- M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block (1997), "Stretching DNA with optical tweezers." Biophys. J., 72:1335-1346. PMID 9138579
- C. Bouchiat et al., "Estimating the Persistence Length of a Worm-Like Chain Molecule from Force-Extension Measurements", Biophys J, January 1999, p. 409-413, Vol. 76, No. 1