蠕虫链模型

✍ dations ◷ 2025-07-29 12:55:35 #高分子化合物,生物物理学,高分子物理学

蠕虫链模型(worm-like chain,WLC)是聚合物物理学中用来阐释半弹性聚合物特性的模型。是Kratky(英语:Otto Kratky)-Porod(英语:Günther Porod)模型的后续版本。

蠕虫链理论模型假设存在一根连续且具弹性的均质棒状物。与自由连接链(英语:Ideal chain)不同的是,他们的弹性仅在独立片段。蠕虫理论特别适用于较坚硬的聚合物,因为此种聚合物的片段拥有一种协同性,大致上会指向同一个方向。依据此理论,在室温下,聚合物的构型会圆滑地弯曲;再绝对零度下( T = 0 {\displaystyle T=0} K),ˋ聚合物则会呈现坚硬的棍状构型。

对于长度 l {\displaystyle l} 的聚合物,将聚合物的路径参数化为 s ( 0 , l ) {\displaystyle s\in (0,l)} 。令 t ^ ( s ) {\displaystyle {\hat {t}}(s)} 为该链再 s {\displaystyle s} 时的单位切线参数,且 r ( s ) {\displaystyle {\vec {r}}(s)} 为该链的位置向量。

得出:

由上可推知此模型的方向相关函数(英语:correlation function)(correlation function)遵守指数衰减:

P {\displaystyle P} 为聚合物的持久长度,即聚合物平均长度的平方:

R 2 = R R = 0 l t ^ ( s ) d s 0 l t ^ ( s ) d s = 0 l d s 0 l t ^ ( s ) t ^ ( s ) d s = 0 l d s 0 l e | s s | / P d s R 2 = 2 P l {\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'\langle R^{2}\rangle =2Pl\left}

蠕虫链理论应用于一些重要的生物性聚合物,包含:

在室温下,聚合物两端的距离会远比原长度 L 0 {\displaystyle L_{0}} 还短。因为热波动会造成聚合物蜷曲,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:

H = H e n t r o p i c + H e x t e r n a l = 1 2 k B T 0 L 0 P ( 2 r ( s ) s 2 ) 2 d s x F {\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 {\displaystyle L_{0}} , the persistence length by P {\displaystyle P} , the extension and external force is represented by extension x F {\displaystyle xF} .

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)):


where k B {\displaystyle k_{B}} is the Boltzmann constant and T {\displaystyle T} is the absolute temperature.

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

H = H e n t r o p i c + H e n t h a l p i c + H e x t e r n a l = 1 2 k B T 0 L 0 P ( r ( s ) s ) 2 d s + 1 2 K 0 L 0 x 2 x F {\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 {\displaystyle L_{0}} , the contour length, P {\displaystyle P} , the persistence length, x {\displaystyle x} the extension and F {\displaystyle 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:

F P k B T = 1 4 ( 1 x L 0 + F K 0 ) 2 1 4 + x L 0 F K 0 {\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:

x = L 0 ( 1 1 2 ( k B T F P ) 1 / 2 + F K 0 ) {\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.

相关

  • 尿常规尿液分析,又称为尿常规,是针对尿液标本所进行的一组医学检验项目,是医学诊断过程中最为常用的方法之一。尿液分析是历史最为悠久的医学检验方法之一,可以反映肾脏和泌尿道等方面
  • 二人转二人转,亦称东北二人转,旧时称为地蹦子、蹦蹦戏、秧歌、小落子、小秧歌、双玩艺、过口、风柳、春歌、半班戏、双条边曲等,1952年定名为二人转。是中国东北地区的走唱类曲艺、地
  • 陶氏杜邦陶氏杜邦(英语:DowDuPont Inc.,NYSE:DWDP)是一家美国化学工业公司。2017年8月31日,由当时陶氏化工和杜邦以换股方式合并,以销售量为全球最大化学企业公司。而在18个月后,该公司按业
  • 埃及第五王朝第 八第 十埃及第五王朝是自前25世纪至前24世纪统治古埃及的一个王朝,历时约150年。埃及第五王朝法老列表:
  • 细川忠兴熊本県熊本市中央区黒髪の泰胜寺迹细川忠兴(1563年11月28日-1646年1月18日)是日本安土桃山时代及江户时代的武将,他是小仓藩的藩祖,细川藤孝的嫡子,曾经是细川辉经的养子,正室为玉
  • 阿尔塞纳·温格阿尔塞纳·温格,OBE,(法语:Arsène Wenger,法语发音:.mw-parser-output .IPA{font-family:"Charis SIL","Doulos SIL","Linux Libertine","Segoe UI","Lucida Sans Unicode","Code
  • 旅游文学旅游文学指内容与旅游相关的创作。当人们去旅游时,以文字形式来记录旅游,就成了“旅游文学”。凡是兼具旅游与文学之成品,如游记、诗等等,都是“旅游文学”。约15.6世纪,地理大发
  • 叶步月叶步月(1907年2月28日-1968年3月23日),台湾作家、医师,本名叶炳辉。台北市大稻埕人。毕业于台北市太平公学校、台湾总督府台北医学专门学校(现国立台湾大学医学院)。以日文创作为主
  • 少年侦探 ~推理之绊~《少年侦探 ~推理之绊~》(日语:スパイラル~推理の絆~)是由城平京原作、水野英多负责作画的日本漫画作品。于史克威尔艾尼克斯漫画杂志《月刊少年GANGAN》平成11年9月号(1999年8月)至
  • 稗田阿礼稗田阿礼(日语:稗田阿礼〔稗田阿禮〕/ひえだの あれ,?-?),是日本奈良时代贵族的杂役,亦是《古事记》的编纂者之一。稗田阿礼的出身至今仍是个谜,同时代编成的《日本书纪》及平安时代编