臺灣博碩士論文加值系統

我們發展出可快速計算晶格交互作用性自迴避行走模型(ISAW) 準確配分函數的演算法，並將此演算法應用到數個子題上。首先，我們在二維正方和三維立方晶格上精確計算(exact enumeration) ISAW 的所有可能結構數，並將結構數與能量的分布轉換成配分函數，由配分函數在複數平面上的根，推論其相變行為。接著，我們分別檢驗所有在二維正方和三維立方晶格上的結構，計算其兩端點距離(end-to-end distance) 隨溫度變化的關係。最後，我們引進了雙鏈帶電荷HP 模型，希望藉由對於最低能態的分析，了解蛋白質沉澱與摺疊的特性。

Ideas and methods of statistical physics have been shown to be useful for understanding many physical, chemical, biological and industrial systems. The interacting self avoiding walks (ISAWs) on a lattice is the simplest model of homopolymers, which can serve as the framework of lattice proteins. We develop an efficient algorithm to compute the exact partition functions of ISAWs and use this algorithm to explore three issues. First, we propose a method based on partition function zeros which considers both the loci of partition function zeros and the thermodynamic functions associated with them. This method is applied to the ISAWs with up to 28 monomers on the simple cubic lattice. A clear scenario for the collapse transition and the freezing transitions can be obtained by this approach. Second, we compute the average end-to-end distance as a function of temperature and find it''s not a monotonically increasing function with some magic numbers of monomers on the simple cubic lattice. Third, we investigate the ground states of a charged HP protein model and find that protein aggregation in this model might not be related to protein misfolding.

口試委員會審定書 i
致謝 ii
中文摘要 iii
Abstract iv
Contents v
List of Figures viii
List of Tables x
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Exact partition functions for ISAWs on lattice . . . . . . . . . . . . . . . 2
2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.1 The lattice setting and data structures . . . . . . . . . . . . . . . . . . 8
2.1.2 Symmetry reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.3 Exhausted enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.4 The final two monomers . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.5 Index of the end-to-end distances . . . . . . . . . . . . . . . . . . . . 12
2.2 Efficiency of the Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 16
3.1 Partition functions of the ISAWs . . . . . . . . . . . . . . . . . . . . . 16
3.1.1 Partition functions of ISAWs in the square lattice . . . . . . . . . 17
3.1.2 Partition functions of ISAWs in the simple cubic lattice . . . . . 18
3.2 Decomposition of physical quantities by zeros of the exact partition function 21
3.2.1 Collapse and freezing transitions . . . . . . . . . . . . . . . . . . 26
3.3 Partition functions of the ISAWs with end-to-end distance . . . . . . . . 28
3.3.1 Partition functions in the square lattice with end-to-end distance . 29
3.3.2 Partition functions in the simple cubic lattice with end-to-end distance
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.3 Magic numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Two chain system in the charged HP model . . . . . . . . . . . . . . . . 32
4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.1 Efficiency of the algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Decomposition of physical quantities by zeros of the exact partition function 36
4.3 Magic number in three-dimensional lattice . . . . . . . . . . . . . . . . . 37
4.4 Two chain systems in the charged HP model . . . . . . . . . . . . . . . . 38
5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
A Partition functions of the ISAWs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
A.1 Partition functions in the square lattice . . . . . . . . . . . . . . . . . . 40
A.2 Partition functions in the simple cubic lattice . . . . . . . . . . . . . . . 45
B Partition functions of the ISAWs with end-to-end distance 49
B.1 Partition functions of ISAWs in the square lattice with end-to-end distance 49
B.2 Partition functions of ISAWs in the simple cubic lattice with end-to-end
distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
C Publication list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

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