一維和二維半柔性高分子之力學性質

Mechanical property of one and two dimensional semiflexible biopolymers

吳浩澐(Hao-Yun Wu)

指導教授 ： 周子聰

淡江大學/理學院/物理學系碩士班/碩士(2013年)

https://doi.org/10.6846/TKU.2013.00804

摘要

協同連接鏈(Cooperative Jointed Chain - CJC)模型是描述半柔性高分子性質的基本模型。我們用精確計數(exact enumeration)法研究端點約束條件對有限長度的CJC之力學性質的影響。我們固定一維鏈的頭、尾兩端方向，計算力與相對伸長的關係，並與無限長鏈的已知理論結果比較，以研究端點約束條件對有限長度的CJC之力學性質的影響。一維情況下，固定(向左或向右)或不固定(隨機)鏈兩端方向產生共九種可能的組合，扣去簡併會有六種結果。我們的結果顯示所有組合在鏈很長時都會趨近於無限長鏈極限的結果。但短鏈會有三種類型的結果，分別是力不大時伸長量很接近、較小、較大於無限長鏈極限的結果；其中兩端棒不固定時最接近無限長鏈極限的結果、兩端棒向左時在力不大時伸長量最小、兩端棒向右時在力不大時伸長量最大。我們將所有的結果都做到鏈長N=200，而結果顯示N>100時兩端點的約束條件(固定端點棒的方向)對結果影響已經可以忽略。
　　我們用精確計數法研究了兩個看起來不同的用來描述半柔性高分子性質的二維模型，其中一個是CJC模型，另一個是具有固有曲率(intrinsic curvature)c的模型。CJC模型只有一個參數，即彎曲剛度(bending rigidity)k1。模型二則有兩個參數，即彎曲剛度k2與固有曲率。我們發現當模型二的c為負時，完全等價於CJC模型；c為正但比較小時也等價於CJC模型。在這裡，等價指的是對於任意給定的k2與c，我們總可以找到一個k1，使得兩個模型x方向的相對伸長量、端點座標的y分量的平方之平均值相同、及平均能量做為力的函數僅差一個常數。因此在這些情況下模型二可以完全由模型一取代。我們也發現在等價的狀態下，固定模型二的k2時，CJC模型的k1與模型二的c具有線性關係。

關鍵字

半柔性高分子 ； 力學性質 ； 協同連接鏈模型 ； 約束條件 ； 固有曲率

並列摘要

Cooperative jointed chain (CJC) model is a basic model to describe the property of semiflexible polymer. We use the exact enumeration method to study the effect of constraints on the mechanical properties of a CJC with finite length. We fix the direction of both ends of a one-dimensional chain, calculate the relationship between force and relative extension, and compare them with the existed results of an infinite long chain. In 1D case, fixing (directed to left or right) or freeing (in randomly) both ends of the chain result in nine possible situations in which three are degenerate. Our results show that with increasing length of chain, in all cases the results get closer and closer to the result of infinite long chain. But there are three types of results for short chain, i.e., the relative extensions become very close, smaller or larger than that of an infinite long chain. When force is small, the extension with free ends is closest to the result of a infinite long chain. Meanwhile, fixing both ends to left results in smallest extension, and fixing both ends to right leads to largest extension. We have done all results from N=3 to N=200 with N being the length of chain. Our results show that when N>100, effects of constraint conditions (fixed both ends of the bar) affected can be disregard.
We use the exact enumeration method to study two two-dimensional models of semiflexible polymer. One of models is the CJC model, and the other is a model with intrinsic curvature c. CJC model has only one parameter which is bending rigidity k1, but the second model has two parameters which are bending rigidity k2 and intrinsic curvature. We found that when c is negative, the second model is equivalent to the CJC model. When c is positive but small, the second model is also equivalent to the CJC model. Here the “equivalence” means that for any k2 and c, we can always find a k1 such that the relative extensions and the both models are the same, where yN is the component of position vector of the right end, and the mean energies of two models are differed by a constant only. Therefore, in these cases, the second model can be replaced completely by CJC. We also find that in equivalent state and with a given k2, k1 and c have linear relation.

並列關鍵字

semiflexible biopolymers ； mechanical property ； cooperative jointed chain(CJC) ； endpoint constraints condition ； inherent curvature

參考文獻

[33] 戴偉帆, 一個二維半柔性生物高分子模型的基本物理性質, 淡江大學 (2011).

[1] Philip Nelson, Biological Physics: Energy, Information, Life, (W. H. Freeman and Co, New York and Basingstoke, 2004).

[5] N. C. Seeman, Nanotechnology 17, 437 (1999).

[6] Y. F. Maa and S. H. Chen, Macromolecules 20, 138 (1987).

[12] G. Binnig, C. F. Quate, and Ch. Gerber, Phys. Rev. Lett. 56, 930 (1986).