查看“︁蠕蟲鏈模型”︁的源代码

'''蠕蟲鏈模型'''（worm-like chain，WLC）是[[聚合物物理學]]中用來闡釋半彈性[[聚合物]]特性的模型。是{{le|Otto Kratky|Otto Kratky|Kratky}}-{{le|Günther Porod|Günther Porod|Porod}}模型的後續版本。

== 理論思考 ==
蠕蟲鏈理論模型假設存在一根連續且具彈性的[[各向同性|均質]]棒狀物<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 | access-date=2014-10-07 | archive-date=2019-04-28 | archive-url=https://web.archive.org/web/20190428234717/http://www.kirbyresearch.com/textbook/ | dead-url=no }}</ref>。與{{le|理想鏈理論|Ideal chain|自由连接链}}不同的是，他們的彈性僅在獨立片段。蠕蟲理論特別適用於較堅硬的聚合物，因為此種聚合物的片段擁有一種協同性，大致上會指向同一個方向。依據此理論，在室溫下，聚合物的構型會圓滑地彎曲；再[[絕對零度]]下（<math>T = 0</math> K），ˋ聚合物則會呈現堅硬的棍狀構型。<ref name=Doi/>

對於長度<math>l</math>的聚合物，將聚合物的路徑參數化為<math>s \in(0,l)</math>。令<math>\hat t(s)</math>為該鏈再<math>s</math>時的單位切線參數，且<math>\vec r(s)</math>為該鏈的位置向量。

得出：
:<math>\hat t(s) \equiv \frac {\partial \vec r(s) }{\partial s}</math> ，且頭尾兩端距離為 <math>\vec R = \int_{0}^{l}\hat t(s) ds</math><ref name=Doi/>

由上可推知此模型的方向[[相關函數]]（correlation function）遵守[[指數衰減]]<ref name=Doi/><ref name=Kirby/>：

:<math>\langle\hat t(s) \cdot \hat t(0)\rangle=\langle \cos \; \theta (s)\rangle = e^{-s/P}\,</math>,

<math>P</math>為聚合物的[[持久長度]]，即聚合物平均長度的平方<ref name=Doi/><ref name=Kirby/>：

<math>
\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  
= 2 Pl \left [ 1 - \frac {P}{l} \left ( 1 - e^{-l/P} \right ) \right ]
</math>

* 注意當限制條件<math>l \gg P</math>時，則<math>\langle R^{2} \rangle = 2Pl</math>。此可用於顯示{{le|庫恩長度|Kuhn segment}}（Kuhn length）等於蠕蟲鏈模型持久長度的兩倍<ref name=Rubinstein/>。

== 生物上的應用 ==
蠕蟲鏈理論應用於一些重要的生物性聚合物，包含：

* 雙股[[DNA]]以及[[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>    
* 未結構化[[RNA]]
* 未結構化多肽鏈（[[蛋白質]]）<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>

== 展開蠕蟲链模型 ==
在室溫下，聚合物兩端的距離會遠比原長度<math>L_0</math>還短。因為熱波動會造成聚合物蜷曲，使聚合物任意排列。

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:

<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>.

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>.  
        
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)):    
    
:<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>
     

where <math>k_B</math> is the [[Boltzmann constant]] and <math>T</math> is the absolute temperature.

== 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 <math>K_0</math>, the stretch modulus. For significantly extended polymers, this yields the following Hamiltonian:

<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>,

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.
Several approximations have been put forward, dependent on the applied external force. For the low-force regime (F < about 10&nbsp;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>

<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>.

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>

<math>x = L_0 \left ( 1 - \frac {1} {2} \left ( \frac {k_{B}T}{FP} \right )^{1/2} + \frac {F}{K_0}\right ) </math>.

A typical value for the stretch modulus of double-stranded DNA is around 1000&nbsp;pN and 45&nbsp;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|url=https://archive.org/details/sim_biophysical-journal_1997-03_72_3/page/1335|journal=Biophysical Journal|year=1997|volume=72|pages=1335–1346|bibcode = 1997BpJ....72.1335W |doi = 10.1016/S0006-3495(97)78780-0 }}</ref>

== 參見 ==
* {{le|Ideal chain}}
* [[聚合物]]
* [[高分子物理學]]

== 参考资料 ==

{{Reflist}}

* {{le|Otto Kratky|Otto Kratky|O. Kratky}}, {{le|Günther Porod|Günther Porod|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.&nbsp;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., [https://web.archive.org/web/20071101001439/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.&nbsp;409-413, Vol. 76, No. 1
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[[Category:高分子化合物]]
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[[Category:生物物理学]]
[[Category:高分子物理学]]
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