Worm-like chain

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.

Theoretical Considerations

The WLC model envisions an isotropic rod that is continuously flexible.^[1][2][3] 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 $T=0$ K, the polymer adopts a rigid rod conformation.^[1]

For a polymer of length $l$, parametrize the path of the polymer as ${\displaystyle s\in (0,l)}$, allow ${\displaystyle {\hat {t}}(s)}$ to be the unit tangent vector to the chain at $s$, and ${\displaystyle {\vec {r}}(s)}$ to be the position vector along the chain. Then

${\displaystyle {\hat {t}}(s)\equiv {\frac {\partial {\vec {r}}(s)}{\partial s}}}$ and the end-to-end distance ${\displaystyle {\vec {R}}=\int _{0}^{l}{\hat {t}}(s)ds}$ .^[1]

It can be shown that the orientation correlation function for a worm-like chain follows an exponential decay:^[1][3]

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

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:^[1][3]

${\displaystyle \langle R^{2}\rangle =\langle {\vec {R}}\cdot {\vec {R}}\rangle =\left\langle \int _{0}^{l}{\hat {t}}(s)ds\cdot \int _{0}^{l}{\hat {t}}(s')ds'\right\rangle =\int _{0}^{l}ds\int _{0}^{l}\langle {\hat {t}}(s)\cdot {\hat {t}}(s')\rangle ds'=\int _{0}^{l}ds\int _{0}^{l}e^{-\left|s-s'\right|/P}ds'}$

${\displaystyle \langle R^{2}\rangle =2Pl\left[1-{\frac {P}{l}}\left(1-e^{-l/P}\right)\right]}$,

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

Biological Relevance

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

• double-stranded DNA and RNA;^[3][4]
• unstructured RNA;
• unstructured polypeptides (proteins).^[5]

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 $L_{0}$. This is caused by thermal fluctuations, which result in a coiled, random configuration of the polymer, when undisturbed. Upon stretching the polymer, the accessible spectrum of fluctuations reduces, which causes an entropic force against the external elongation. This entropic force can be estimated by considering the entropic Hamiltonian:

${\displaystyle H=H_{\rm {entropic}}+H_{\rm {external}}={\frac {1}{2}}k_{B}T\int _{0}^{L_{0}}P\cdot \left({\frac {\partial ^{2}{\vec {r}}(s)}{\partial s^{2}}}\right)^{2}ds-xF}$.

Here, the contour length is represented by $L_{0}$, the persistence length by $P$, the extension is represented by $x$, and external force is represented by $F$.

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)):

${\displaystyle {\frac {FP}{k_{B}T}}={\frac {1}{4}}\left(1-{\frac {x}{L_{0}}}\right)^{-2}-{\frac {1}{4}}+{\frac {x}{L_{0}}}}$

where $k_{B}$ is the Boltzmann constant and $T$ is the absolute temperature. More accurate approximation for the force-extension behavior with about 1% relative error is:^[6]

${\displaystyle {\frac {FP}{k_{B}T}}={\frac {1}{4}}\left(1-{\frac {x}{L_{0}}}\right)^{-2}-{\frac {1}{4}}+{\frac {x}{L_{0}}}-0.8\left({\frac {x}{L_{0}}}\right)^{2.15}}$

Approximation for the extension-force behavior with about 1% relative error was also reported:^[6]

${\displaystyle {\frac {x}{L_{0}}}={\frac {4}{3}}-{\frac {4}{3{\sqrt {{\frac {FP}{k_{B}T}}+1}}}}-{\frac {10e^{\sqrt[{4}]{900{\frac {k_{B}T}{FP}}}}}{{\sqrt {\frac {FP}{k_{B}T}}}\left(e^{\sqrt[{4}]{900{\frac {k_{B}T}{FP}}}}-1\right)^{2}}}+{\frac {\left({\frac {FP}{k_{B}T}}\right)^{1.62}}{3.55+3.8\left({\frac {FP}{k_{B}T}}\right)^{2.2}}}}$

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 $K_{0}$, the stretch modulus. For significantly extended polymers, this yields the following Hamiltonian:

${\displaystyle H=H_{\rm {entropic}}+H_{\rm {enthalpic}}+H_{\rm {external}}={\frac {1}{2}}k_{B}T\int _{0}^{L_{0}}P\cdot \left({\frac {\partial {\vec {r}}(s)}{\partial s}}\right)^{2}ds+{\frac {1}{2}}{\frac {K_{0}}{L_{0}}}x^{2}-xF}$,

with $L_{0}$, the contour length, $P$, the persistence length, $x$ the extension and $F$ external force. This expression takes into account both the entropic term, which regards changes in the polymer conformation, and the enthalpic term, which describes the elongation of the polymer due to the external force. In the expression above, the enthalpic response is described as a linear Hookian spring. Several approximations have been put forward, dependent on the applied external force. For the low-force regime (F < about 10 pN), the following interpolation formula was derived:^[7]

${\displaystyle {\frac {FP}{k_{B}T}}={\frac {1}{4}}\left(1-{\frac {x}{L_{0}}}+{\frac {F}{K_{0}}}\right)^{-2}-{\frac {1}{4}}+{\frac {x}{L_{0}}}-{\frac {F}{K_{0}}}}$.

For the higher-force regime, where the polymer is significantly extended, the following approximation is valid:^[8]

${\displaystyle x=L_{0}\left(1-{\frac {1}{2}}\left({\frac {k_{B}T}{FP}}\right)^{1/2}+{\frac {F}{K_{0}}}\right)}$.

A typical value for the stretch modulus of double-stranded DNA is around 1000 pN and 45 nm for the persistence length.^[9] More accurate interpolation formulas for the force-extension and extension-force behaviors are:^[6]

${\displaystyle {\frac {FP}{k_{B}T}}={\frac {1}{4}}\left(1-{\frac {x}{L_{0}}}+{\frac {F}{K_{0}}}\right)^{-2}-{\frac {1}{4}}+{\frac {x}{L_{0}}}-{\frac {F}{K_{0}}}-0.8\left({\frac {x}{L_{0}}}-{\frac {F}{K_{0}}}\right)^{2.15}}$

${\displaystyle {\frac {x}{L_{0}}}={\frac {4}{3}}-{\frac {4}{3{\sqrt {{\frac {FP}{k_{B}T}}+1}}}}-{\frac {10e^{\sqrt[{4}]{900{\frac {k_{B}T}{FP}}}}}{{\sqrt {\frac {FP}{k_{B}T}}}\left(e^{\sqrt[{4}]{900{\frac {k_{B}T}{FP}}}}-1\right)^{2}}}+{\frac {\left({\frac {FP}{k_{B}T}}\right)^{1.62}}{3.55+3.8\left({\frac {FP}{k_{B}T}}\right)^{2.2}}}+{\frac {F}{K_{0}}}}$

See also

• Ideal chain
• Polymer
• Polymer physics

References

1. Doi and Edwards (1988). The Theory of Polymer Dynamics.
2. Rubinstein and Colby (2003). Polymer Physics.
3. Kirby, B.J. Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices.
4. J. A. Abels and F. Moreno-Herrero and T. van der Heijden and C. Dekker and N. H. Dekker (2005). "Single-Molecule Measurements of the Persistence Length of Double-Stranded RNA". Biophysical Journal. 88: 2737–2744. doi:10.1529/biophysj.104.052811.
5. L. J. Lapidus and P. J. Steinbach and W. A. Eaton and A. Szabo and J. Hofrichter (2002). "Single-Molecule Effects of Chain Stiffness on the Dynamics of Loop Formation in Polypeptides. Appendix: Testing a 1-Dimensional Diffusion Model for Peptide Dynamics". Journal of Physical Chemistry B. 106: 11628–11640. doi:10.1021/jp020829v.
6. Petrosyan, R. (2016). "Improved approximations for some polymer extension models". Rehol Acta. doi:10.1007/s00397-016-0977-9.
7. Marko, J.F.; Eric D. Siggia (1995). "Stretching DNA". Macromolecules. 28: 8759–8770. Bibcode:1995MaMol..28.8759M. doi:10.1021/ma00130a008.
8. Odijk, Theo (1995). "Stiff Chains and Filaments under Tension". Macromolecules. 28: 7016–7018. Bibcode:1995MaMol..28.7016O. doi:10.1021/ma00124a044.
9. Wang, Michelle D.; Hong Yin; Robert Landick; Jeff Gelles; Steven M. Block (1997). "Stretching DNA with Optical Tweezers". Biophysical Journal. 72: 1335–1346. Bibcode:1997BpJ....72.1335W. doi:10.1016/S0006-3495(97)78780-0. PMC 1184516. PMID 9138579.

• Kratky, O.; Porod, G. (1949). "Röntgenuntersuchung gelöster Fadenmoleküle". Rec. Trav. Chim. Pays-Bas. 68: 1106–1123.
• Marko, J. F.; Siggia, E. D. (1995). "Stretching DNA". Macromolecules. 28: 8759–8770. Bibcode:1995MaMol..28.8759M. doi:10.1021/ma00130a008.
• Bustamante, C.; Marko, J. F.; Siggia, E. D.; Smith, S. (1994). "Entropic elasticity of lambda-phage DNA". Science. 265: 1599–1600. Bibcode:1994Sci...265.1599B. doi:10.1126/science.8079175. PMID 8079175.
• Wang, M. D.; Yin, H.; Landick, R.; Gelles, J.; Block, S. M. (1997). "Stretching DNA with optical tweezers". Biophys. J. 72: 1335–1346. Bibcode:1997BpJ....72.1335W. doi:10.1016/S0006-3495(97)78780-0. PMC 1184516. 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