蠕蟲鏈模型

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Otto Kratky-Günther 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均質棒狀物。與理想鏈理論不同的是，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（$T = 0$ K），ˋ聚合物則會呈現堅硬的棍狀構型。

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

得出：

$\hat t(s) \equiv \frac {\partial \vec r(s) }{\partial s}$，且頭尾兩端距離為 $\vec R = \int_{0}^{l}\hat t(s) ds$

由上可推知此模型的方向相關函數（correlation function）遵守指數衰減：

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

$P$為聚合物的持久長度，即聚合物平均長度的平方：

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

- 注意當限制條件$l \gg P$時，則$\langle R^{2} \rangle = 2Pl$。此可用於顯示庫恩長度（Kuhn length）等於蠕蟲鏈模型持久長度的兩倍。

## 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
- 未結構化RNA
- 未結構化多肽鏈（蛋白質）

## 展開蠕蟲鏈模型

在室溫下，聚合物兩端的距離會遠比原長度$L_0$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:

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

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.