Worm-like chain

The model of the worm-like chain (also WLC model from English worm-like chain , more rarely Porod-Kratky chain ) is a model for the physical description of rigid polymers . It is more complex than the free-moving chain model and is suitable for rigid polymers such as double-stranded DNA , double- and single-stranded RNA, and proteins .

properties

The WLC model approximately describes a continuous, isotropic rod that is flexible at defined points . In contrast, the staff in the Freely Jointed Chain model is only flexible in some places. In the WLC model, the segments point in roughly the same direction. At room temperature the polymer adopts a curved conformation , at absolute zero the rod adopts a rigid conformation.

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

${\displaystyle {\hat {t}}(s)\equiv {\frac {\partial {\vec {r}}(s)}{\partial s}}}$and the endpoint distance . ${\displaystyle {\vec {R}}=\int _{0}^{l}{\hat {t}}(s)ds}$

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

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

with as characteristic persistence length . A typical value is the square of the mean endpoint distance of the polymer: ${\displaystyle P}$

${\displaystyle {\begin{aligned}\langle R^{2}\rangle &=\langle {\vec {R}}\cdot {\vec {R}}\rangle \\&=\left\langle \int _{0}^{l}{\hat {t}}(s)ds\cdot \int _{0}^{l}{\hat {t}}(s')ds'\right\rangle \\&=\int _{0}^{l}ds\int _{0}^{l}\langle {\hat {t}}(s)\cdot {\hat {t}}(s')\rangle ds'\\&=\int _{0}^{l}ds\int _{0}^{l}e^{-\left|s-s'\right|/P}ds'\\\langle R^{2}\rangle &=2Pl\left[1-{\frac {P}{l}}\left(1-e^{-l/P}\right)\right]\end{aligned}}}$

In the borderline case is . This can be used to show that the Kuhn length is twice the persistence length of a WLC polymer. ${\displaystyle l\gg P}$ ${\displaystyle \langle R^{2}\rangle =2Pl}$

Stretching of polymers

At a given temperature, the distance between the two ends of the polymer will be significantly shorter than the contour length . This is caused by thermal fluctuations that result in a coiled, random conformation at rest. By stretching the polymer, the available spectrum of fluctuations is reduced, which creates an entropic force against the external elongation. The entropic force can be estimated using the following Hamilton equation : ${\displaystyle L_{0}}$

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

with the contour length , the persistence length , the elongation and the force in . ${\displaystyle L_{0}}$ ${\displaystyle P}$ ${\displaystyle xF}$

By atomic force microscopes and optical tweezers , the force-dependent elongation of the polymer can be determined. The following interpolation was developed by JF Marko, ED Siggia (1995) to approximate the force-extension behavior:

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

with as Boltzmann constant and as absolute temperature . ${\displaystyle k_{\mathrm {B} }}$ ${\displaystyle T}$

Extensible worm-like chain model

With increasing stretching, polymers become elastic. For example, when stretching DNA at a neutral pH value , an ionic strength of 100 mM and room temperature, the agreement with the model must be considered enthalpically using the tensile modulus : ${\displaystyle K_{0}}$

${\displaystyle H=H_{\text{entropisch}}+H_{\text{enthalpisch}}+H_{\text{extern}}={\frac {1}{2}}k_{\mathrm {B} }T\int _{0}^{L_{0}}P\cdot \left({\frac {\partial {\vec {r}}(s)}{\partial s}}\right)^{2}ds+{\frac {1}{2}}{\frac {K_{0}}{L_{0}}}x^{2}-xF}$

with the contour length , the persistence length , the extent and the external force . This contains the entropic term, which considers the change in conformation, and the enthalpic term, which considers the expansion under the action of an external force. The enthalpic term is described here as a linear Hooke's spring . ${\displaystyle L_{0}}$ ${\displaystyle P}$ ${\displaystyle x}$ ${\displaystyle F}$

Different approximations are based on the force applied. The following interpolation was formulated for the areas of small forces below ten piconewtons:

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

In the case of greater forces, at which the polymers are clearly stretched, the following approximation applies:

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

A typical value for the tensile stiffness (under tension) of double-stranded DNA is about 1000 pN, and about 45 nm for the persistence length.

literature

• O. Kratky , G. Porod : X-ray investigation of dissolved thread molecules . In: Rec. Trav. Chim. Pays-Bas. , 1949, 68, pp. 1106-1123.
• C. Bouchiat et al .: Estimating the Persistence Length of a Worm-Like Chain Molecule from Force-Extension Measurements . In: Biophys J , 1999, 76 (1), pp. 409-413; doi: 10.1016 / S0006-3495 (99) 77207-3 .
• JF Marko, ED Siggia: Stretching DNA . In: Macromolecules , 28, 1995, p. 8759.
• C. Bustamante, JF Marko, ED Siggia, S. Smith: Entropic elasticity of lambda-phage DNA . In: Science , 265, 1994, pp. 1599-1600, PMID 8079175 .
• MD Wang, H. Yin, R. Landick, J. Gelles, SM Block: Stretching DNA with optical tweezers . In: Biophys. J. , 72, 1997, pp. 1335-1346, PMID 9138579 .

Individual evidence

1. ↑ P. Kratochvíl, UW Suter: Definitions of terms relating to individual macromolecules, their assemblies, and dilute polymer solutions (Recommendations 1988) . In: Pure and Applied Chemistry . tape 61 , no. 2 , 1989, pp. 211-241 , doi : 10.1351 / pac198961020211 ( iupac.org [PDF; 1,3 MB ; accessed on February 18, 2014]).
2. ↑ ^{a b c d} B.J. Kirby: Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices .
3. ↑ JA Abels, F. Moreno-Herrero, T. van der Heijden, C. Dekker, NH Dekker: Single-Molecule Measurements of the Persistence Length of Double-Stranded RNA . In: Biophysical Journal . 88, 2005, pp. 2737-2744. doi : 10.1529 / biophysj.104.052811 .
4. ↑ LJ Lapidus, PJ Steinbach, WA Eaton, A. Szabo, J. Hofrichter: Single-Molecule Effects of Chain Stiffness on the Dynamics of Loop Formation in Polypeptides. Appendix: Testing a 1-Dimensional Diffusion Model for Peptide Dynamics . In: Journal of Physical Chemistry B . 106, 2002, pp. 11628-11640. doi : 10.1021 / jp020829v .
5. ↑ ^{a b c d e} Doi, Edwards: The Theory of Polymer Dynamics 1999.
6. ↑ ^{a b} Rubinstein, Colby: Polymer Physics 2003.
7. ^ JF Marko, Eric D. Siggia: Stretching DNA . In: Macromolecules . 28, 1995, pp. 8759-8770. bibcode : 1995MaMol..28.8759M . doi : 10.1021 / ma00130a008 .
8. ^ Theo Odijk: Stiff Chains and Filaments under Tension . In: Macromolecules . 28, 1995, pp. 7016-7018. bibcode : 1995MaMol..28.7016O . doi : 10.1021 / ma00124a044 .
9. ↑ Michelle D. Wang, Hong Yin, Robert Landick, Jeff Gelles and Steven M. Block: Stretching DNA with Optical Tweezers . In: Biophysical Journal . 72, 1997, pp. 1335-1346. bibcode : 1997BpJ .... 72.1335W . doi : 10.1016 / S0006-3495 (97) 78780-0 .