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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 the continuous version of the [[Otto Kratky|Kratky]]-[[Günther Porod|Porod]] model. | | The writer is called Wilber Pegues. She is really fond of caving but she doesn't have the time lately. My working day job is a travel agent. For a while I've been in Mississippi but now I'm considering other options.<br><br>Feel free to surf to my blog post :: clairvoyant psychic [[http://modenpeople.co.kr/modn/qna/292291 http://modenpeople.co.kr]] |
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| == Theoretical Considerations ==
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| The WLC model envisions an [[isotropic]] rod that is continuously flexible.<ref name=Doi>{{cite book | author=Doi and Edwards | title=The Theory of Polymer Dynamics | year=1999}}</ref><ref name=Rubinstein>{{cite book | author=Rubinstein and Colby | title=Polymer Physics | year=2003}}</ref><ref name=Kirby>{{cite book | author=Kirby, B.J. | title=Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices | url=http://www.kirbyresearch.com/textbook}}</ref> This is in contrast to the [[Ideal chain|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 <math>T = 0</math> K, the polymer adopts a rigid rod conformation.<ref name=Doi/>
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| For a polymer of length <math>l</math>, parametrize the path of the polymer as <math>s \in(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 | |
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| :<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> .<ref name=Doi/>
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| It can be shown that the orientation [[correlation function]] for a worm-like chain follows an [[exponential decay]]:<ref name=Doi/><ref name=Kirby/>
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| :<math>\langle\hat t(s) \cdot \hat t(0)\rangle=\langle \cos \; \theta (s)\rangle = e^{-s/P}\,</math>,
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| 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:<ref name=Doi/><ref name=Kirby/>
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| <math>
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| \begin{align}
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| \langle R^{2} \rangle & = \langle \vec R \cdot \vec R \rangle \\
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| & = \left\langle \int_{0}^{l} \hat t(s) ds \cdot \int_{0}^{l} \hat t(s') ds' \right\rangle \\
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| & = \int_{0}^{l} ds \int_{0}^{l} \langle \hat t(s) \cdot \hat t(s') \rangle ds' \\
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| & = \int_{0}^{l} ds \int_{0}^{l} e^{-\left | s - s' \right | / P} ds' \\
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| \langle R^{2} \rangle & = 2 Pl \left [ 1 - \frac {P}{l} \left ( 1 - e^{-l/P} \right ) \right ]
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| \end{align}
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| </math>
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| * Note that in the limit of <math>l \gg P</math>, then <math>\langle R^{2} \rangle = 2Pl</math>. This can be used to show that a [[Kuhn length|Kuhn segment]] is equal to twice the persistence length of a worm-like chain.<ref name=Rubinstein/>
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| == Biological Relevance ==
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| Several biologically important polymers can be effectively modeled as worm-like chains, including:
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| * double-stranded [[DNA]] and [[RNA]];<ref name=Kirby/><ref name=dekker2005>{{cite journal | author=J. A. Abels and F. Moreno-Herrero and T. van der Heijden and C. Dekker and N. H. Dekker | title=Single-Molecule Measurements of the Persistence Length of Double-Stranded RNA | journal=Biophysical Journal| year=2005| volume=88| pages=2737–2744| doi=10.1529/biophysj.104.052811 }}</ref>
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| * unstructured [[RNA]];
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| * unstructured polypeptides ([[proteins]]).<ref name=lapidus2002>{{cite journal | author=L. J. Lapidus and P. J. Steinbach and W. A. Eaton and A. Szabo and J. Hofrichter | title=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=Journal of Physical Chemistry B| year=2002| volume=106| pages=11628–11640| doi=10.1021/jp020829v}}</ref>
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| == Stretching Worm-like Chain Polymers ==
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| At finite temperatures, the distance between the two ends of the polymer (end-to-end distance) will be siginficantly shorter than the contour length <math>L_0</math>. This is caused by thermal fluctuations, which result in a coiled, random configuration of the polymer, when undisturbed. Upon stretching the polymer, the accesible spectrum of fluctuations reduces, which causes an entropic force against the external elongation.
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| This entropic force can be estimated by considering the entropic Hamiltonian:
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| <math> 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</math>.
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| Here, the [[contour length]] is represented by <math>L_0</math>, the persistence length by <math>P</math>, the extension and external force is represented by extension <math>xF</math>.
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| 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 is (J. F. Marko, E. D. Siggia (1995)):
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| :<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>
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| where <math>k_B</math> is the [[Boltzmann constant]] and <math>T</math> is the absolute temperature.
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| == Extensible Worm-like Chain model ==
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| 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 paramter <math>K_0</math>, the stretch modulus. For significantly extended polymers, this yields the following Hamiltonian:
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| <math> 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 </math>,
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| with <math>L_0</math>, the contour length, <math>P</math>, the persistence length, <math>x</math> the extension and <math>F</math> 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.
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| Several approximations have been put forward, dependend on the applied external force. For the low force regime (F < about 10 pN), the following interpolation formula was derived:<ref>{{cite journal|last=Marko|first=J.F.|coauthors=Eric D. Siggia|title=Stretching DNA|journal=Macromolecules|year=1995|volume=28|pages=8759–8770|bibcode = 1995MaMol..28.8759M |doi = 10.1021/ma00130a008 }}</ref>
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| <math>\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}</math>.
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| For the higher force regime, where the polymer is significantly extended, the following approximation is valid:<ref>{{cite journal|last=Odijk|first=Theo|title=Stiff Chains and Filaments under Tension|journal=Macromolecules|year=1995|volume=28|pages=7016–7018|bibcode = 1995MaMol..28.7016O |doi = 10.1021/ma00124a044 }}</ref>
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| <math>x = L_0 \left ( 1 - \frac {1} {2} \left ( \frac {k_{B}T}{FP} \right )^{1/2} + \frac {F}{K_0}\right ) </math>.
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| A typical value for the stretch modulus of double-stranded DNA is around 1000 pN and 45 nm for the persistence length.<ref>{{cite journal|last=Wang|first=Michelle D.|coauthors=Hong Yin, Robert Landick, Jeff Gelles and Steven M. Block|title=Stretching DNA with Optical Tweezers|journal=Biophysical Journal|year=1997|volume=72|pages=1335–1346|bibcode = 1997BpJ....72.1335W |doi = 10.1016/S0006-3495(97)78780-0 }}</ref>
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| == See also ==
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| * [[Ideal chain]]
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| * [[Polymer]]
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| * [[Polymer physics]]
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| == References ==
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| {{Reflist}}
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| * [[Otto Kratky|O. Kratky]], [[Günther Porod|G. Porod]] (1949), "Röntgenuntersuchung gelöster Fadenmoleküle." ''Rec. Trav. Chim. Pays-Bas.'' '''68''': 1106-1123.
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| * J. F. Marko, E. D. Siggia (1995), "Stretching DNA." ''Macromolecules'', '''28''': p. 8759.
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| * C. Bustamante, J. F. Marko, E. D. Siggia, and S. Smith (1994), "Entropic elasticity of lambda-phage DNA." ''Science'', '''265''': 1599-1600. PMID 8079175
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| * 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
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| * C. Bouchiat et al., [http://www.biophysj.org/cgi/content/abstract/76/1/409 "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
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| <!-- this is probably more about physics than chemistry, since no reactivity is involved -->
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| [[Category:Polymer physics]]
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| [[Category:Polymers]]
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| <!-- I see no reason this should be excluded from Biophysics category -->
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| [[Category:Biophysics]]
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| <!-- [[Category:Biology]] too general -->
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| <!-- [[Category:Physics]] too general -->
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The writer is called Wilber Pegues. She is really fond of caving but she doesn't have the time lately. My working day job is a travel agent. For a while I've been in Mississippi but now I'm considering other options.
Feel free to surf to my blog post :: clairvoyant psychic [http://modenpeople.co.kr]