The worm-like chain (WLC) model in polymer physics is used to describe the behavior of semi-flexible polymers; it is the continuous version of the Kratky-Porod 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, 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
For a polymer of length
It can be shown that the orientation correlation function for a worm-like chain follows an exponential decay:
where
Biological Relevance
Several biologically important polymers can be effectively modeled as worm-like chains, including:
Stretching Worm-like Chain Polymers
At finite temperatures, the distance between the two ends of the polymer (end-to-end distance) will be significantly shorter than the contour length
Here, the contour length is represented by
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 approximates the force-extension behavior with about 15% relative error is (J. F. Marko, E. D. Siggia (1995)):
where
Approximation for the extension-force behavior with about 1% relative error was also reported:
Extensible worm-like chain model
When extending most polymers, their elastic response cannot be neglected. As an example, for the well-studied case of stretching DNA in physiological conditions (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 the material parameter
with
For the higher-force regime, where the polymer is significantly extended, the following approximation is valid:
A typical value for the stretch modulus of double-stranded DNA is around 1000 pN and 45 nm for the persistence length. More accurate interpolation formulas for the force-extension and extension-force behaviors are: