蠕虫链模型

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Otto Kratky-Günther Porod模型的后续版本。

## 理论思考

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。与理想链理论不同的是，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（$T = 0$ K），ˋ聚合物则会呈现坚硬的棍状构型。

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

得出：

$\hat t(s) \equiv \frac {\partial \vec r(s) }{\partial s}$，且头尾两端距离为 $\vec R = \int_{0}^{l}\hat t(s) ds$

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

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

$P$为聚合物的持久长度，即聚合物平均长度的平方：

$$\langle R^{2} \rangle = \langle \vec R \cdot \vec R \rangle <pre> = \left\langle \int_{0}^{l} \hat t(s) ds \cdot \int_{0}^{l} \hat t(s') ds' \right\rangle = \int_{0}^{l} ds \int_{0}^{l} \langle \hat t(s) \cdot \hat t(s') \rangle ds'= \int_{0}^{l} ds \int_{0}^{l} e^{-\left | s - s' \right | / P} ds' \langle R^{2} \rangle </pre> = 2 Pl \left [ 1 - \frac {P}{l} \left ( 1 - e^{-l/P} \right ) \right ]$$

- 注意当限制条件$l \gg P$时，则$\langle R^{2} \rangle = 2Pl$。此可用于显示库恩长度（Kuhn length）等于蠕虫链模型持久长度的两倍。

## 生物上的应用

蠕虫链理论应用于一些重要的生物性聚合物，包含：

- 双股DNA以及RNA
- 未结构化RNA
- 未结构化多肽链（蛋白质）

## 展开蠕虫链模型

在室温下，聚合物两端的距离会远比原长度$L_0$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:

$H = H_{\rm entropic} + H_{\rm external}= \frac {1}{2}k_B T \int_{0}^{L_0} P \cdot \left (\frac {\partial^2 \vec r(s) }{\partial s^2}\right )^{2} ds - xF$.

Here, the contour length is represented by $L_0$, the persistence length by $P$, the extension and external force is represented by extension $xF$.

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)):

$$\frac {FP} {k_{B}T} = \frac {1}{4} \left ( 1 - \frac {x} {L_0} \right )^{-2} - \frac {1}{4} + \frac {x}{L_0}$$

where $k_B$ is the Boltzmann constant and $T$ is the absolute temperature.

## 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:

$H = H_{\rm entropic}+H_{\rm enthalpic}+H_{\rm external} = \frac {1}{2}k_B T \int_{0}^{L_0} P \cdot \left (\frac {\partial \vec r(s) }{\partial s}\right )^{2} ds + \frac {1}{2}\frac {K_0}{L_0} x^{2} - xF$,

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:

$\frac {FP} {k_{B}T} = \frac {1}{4} \left ( 1 - \frac {x} {L_0} + \frac {F}{K_0} \right )^{-2} - \frac {1}{4} + \frac {x}{L_0} - \frac {F}{K_0}$.

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

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

A typical value for the stretch modulus of double-stranded DNA is around 1000 pN and 45 nm for the persistence length.