多面体及其链环对偶性质的研究
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摘要
凸多面体是几何学的一个古老而重要的研究对象,它与人们的生活密切相关。多面体链环是将古老的凸多面体结构和重要的纽结理论相结合,在解决近年来实验室中发现的病毒结构和合成的超分子结构的理论研究中,得到的理论成果。多面体链环是由多个环相互嵌套成的具有多面体形状的一种拓扑几何结构。本论文基于凸多面体的理论成果,从不同问题入手进行了两大方面的研究:
一:对偶是几何学的一个重要概念和操作,多面体链环作为在多面体结构上构造的新型结构,是否存在对偶性,如何定义,如何操作,是关于多面体链环的一个重要的基础性研究。更进一步,多面体链环是链环结构的特例,纽结与链环一直是数学研究的热点,面对庞大而复杂的纽结表,如何分类,它们之间存在怎样的联系?我们将对偶理论引入纽结理论,给出对偶链环的定义和构造方法,为纽结的分类提供依据,同时这一构造方法的提出,为纽结超分子的合成提供新的思路,对实验的指导也具有重要的意义。
1.在图论中间图理论和拓扑理论的基础上,提出对偶多面体链环的定义。以五个柏拉图多面体为例,运用“三交叉——双线覆盖”的方法构造四面体链环,六面体链环和十二面体链环,利用“球面游走”的拓扑学方法,得到相应的对偶多面体链环。结果显示,四面体链环是自对偶结构,六面体链环和八面体链环,十二面体和二十面体链环互为对偶。从手性角度考虑,对偶变换具有手性保持的特征,十个多面体链环分为六组对偶多面体链环。从构型角度考虑,四面体链环的自对偶是“平凡”的,六面体链环和八面体链环,十二面体和二十面体链环的对偶是“非平凡”的。这一研究说明多面体链环具有对偶性,其对偶变换是拓扑的。
2.基于图论中反转中间图的方法和纽结理论中的缠绕理论,我们提出了构建对偶链环的新方法。这一方法定义了两种有向4-度平面图:G_e和G_o分别用E-tangles和O-tangles覆盖两种有向4-度平面图的顶点。结果得到两种对偶链环:E-dual links和O-dual links,它们具有许多不同的拓扑特征,特别是它们的手性规则。研究表明,通过有向4-度平面图和缠绕可以构建得到对偶链环。这一研究提出了对偶链环的定义及其构建的方法。对偶链环为链环的研究提供了新的思路,这一构建方法可以被用于指导手性分子的合成。
二:病毒是比任何细菌都小的感染性遗传物质,介于生命与非生命之间的一类无细胞结构的生物,绝大多数病毒的衣壳蛋白装配成二十面体对称的结构,“准等价”原理是解决严格遵从二十面体对称性的病毒衣壳几何特征的有效而成熟的方法。但是随着研究手段的不断发展,更多的病毒结构被发现,它们的衣壳结构违背准等价原理。我们以这些奇特结构为研究对象,找到它们的几何模型,补充多面体结构,为病毒研究提供理论指导。
乳多空病毒和多瘤病毒的72个五聚体的衣壳结构违背Caspar-Klug(CK)“准等价”原理。而且,72个五边形的球面包裹问题是一个有待解决的数学难题。我们在十二面体框架的基础上,利用“球面拉伸”的方法,得到新的具有二十面体对称性,72个五边形构造的的多面体结构。新型多面体结构为72个五聚体的病毒衣壳结构的模拟提出新的理论,为五边形排列问题提供了新的思路,丰富了多面体世界。
In geometry, convex polyhedra are old and important researchful objects, which contacts peoples' daily life. Polyhedral links are new theoretical results, some interlocked structures on the basis of old convex polyhedra and knot theory, which derived from the theoretical research for viral capsid structure and supermolecule structure. Polyhedral links are a class of topological links with polyhedral shape which are linked with a collection of finitely separate closed curves..On the basis of theoretical result of convex polyhedra, the thesis includes two parts of research which aim at different problems.
一、Duality is an important conception and manipulation in geometry. As new linked structures, polyhedral links are constructed on the frame of polyhedra. Whether polyhedral links possess of duality, how to define, and how to manipulate are important basis researches of polyhedral links. Furthermore, polyhedral links just a particular family of knots and links, which are attentive in the field of mathematics. Facing to the huge and complicated knot table, how to class, and what kind of relationships there are between them? When duality is applied to knots theory, it puts forward the definition of dual links and the methodology for the construction of dual links. Dual links open a new approach for the research of links, and the methodology may also be used to direct the synthesis of chiral molecules.
1. The novel topology of Platonic polyhedral links is discussed on the basis of the graph theory and topological principles. This interesting problem of the dual polyhedral links has been solved by using our method of the "sphere-surface-movement". There are three classes of dual polyhedral links which can be explored: the tetrahedral link is self-dual, the hexahedral and octahedral link, as well as the dodecahedral and icosahedral link are dual to each other. Our results show that the duality of self-dual tetrahedral link is "trivial", and the duality of hexahedral and octahedral link as well as dodecahedral and icosahedral link are "nontrivial". This study provides further insight into the molecular design and theoretical characterization of the new polyhedral links.
2. A new method for understanding the construction of dual links has been developed on the basis of medial graph in graph theory and tangle in knot theory. The method defines two types of oriented 4-valent plane graph: G_e and G_o, whose vertices are covered by E-tangles and O-tangles, respectively. The result shows that there are two types of dual links: E-dual links and O-dual links, which have many differences in topological properties, especially their chiral rule. In our paper, we show that dual links can be constructed by oriented 4-valent plant graphs and tangles. This research puts forward the definition of dual links and the methodology for the construction of dual links. Dual links open a new approach for the research of links, and the methodology may also be used to direct the synthesis of chiral molecules.
二、A virus is a unit of infectious genetic material smaller than any bacteria and embodying properties placing it on the borderline between life and non-life. The vast majority of the virus capsid protein assembled into icosahedral symmetry of the structure. Caspar-Klug Theory has become a fundamental concept for the classification of icosahedral viral capsids based on the principle of quasi-equivalence. However, recent experiments have shown that there are viruses that do not follow the organisation predicted by this theory. We take these novel viruses as objects of investigation, and constructing the geometry models to explain these novel architectures, which enrich the world of polehedra and give theoretic guidance of viral investigation.
The outer shells of papilloma virions and polyoma virus contain 72 pentamers, the architectures of which do not follow the Caspar-Klug (CK) "quasi-equivalence" theory. Moveover, the spherical pentagon packing problem for 72 pentagons is a mathematical problem. On the basis of the frame of dodecahedron, we apply the method of "spherical stretching" to the frame, we can obtain a novel polyhedron with I_h symmetry, which contains 72 pentagons. The novel polyhedral structure improves new theory for the simulating of viral capsid with 72 pentamers, and gives additional insight into the mathematical problem of the spherical pentagon packing problem for 72 pentagons, and enrichs the world of polehedra.
一:对偶是几何学的一个重要概念和操作,多面体链环作为在多面体结构上构造的新型结构,是否存在对偶性,如何定义,如何操作,是关于多面体链环的一个重要的基础性研究。更进一步,多面体链环是链环结构的特例,纽结与链环一直是数学研究的热点,面对庞大而复杂的纽结表,如何分类,它们之间存在怎样的联系?我们将对偶理论引入纽结理论,给出对偶链环的定义和构造方法,为纽结的分类提供依据,同时这一构造方法的提出,为纽结超分子的合成提供新的思路,对实验的指导也具有重要的意义。
1.在图论中间图理论和拓扑理论的基础上,提出对偶多面体链环的定义。以五个柏拉图多面体为例,运用“三交叉——双线覆盖”的方法构造四面体链环,六面体链环和十二面体链环,利用“球面游走”的拓扑学方法,得到相应的对偶多面体链环。结果显示,四面体链环是自对偶结构,六面体链环和八面体链环,十二面体和二十面体链环互为对偶。从手性角度考虑,对偶变换具有手性保持的特征,十个多面体链环分为六组对偶多面体链环。从构型角度考虑,四面体链环的自对偶是“平凡”的,六面体链环和八面体链环,十二面体和二十面体链环的对偶是“非平凡”的。这一研究说明多面体链环具有对偶性,其对偶变换是拓扑的。
2.基于图论中反转中间图的方法和纽结理论中的缠绕理论,我们提出了构建对偶链环的新方法。这一方法定义了两种有向4-度平面图:G_e和G_o分别用E-tangles和O-tangles覆盖两种有向4-度平面图的顶点。结果得到两种对偶链环:E-dual links和O-dual links,它们具有许多不同的拓扑特征,特别是它们的手性规则。研究表明,通过有向4-度平面图和缠绕可以构建得到对偶链环。这一研究提出了对偶链环的定义及其构建的方法。对偶链环为链环的研究提供了新的思路,这一构建方法可以被用于指导手性分子的合成。
二:病毒是比任何细菌都小的感染性遗传物质,介于生命与非生命之间的一类无细胞结构的生物,绝大多数病毒的衣壳蛋白装配成二十面体对称的结构,“准等价”原理是解决严格遵从二十面体对称性的病毒衣壳几何特征的有效而成熟的方法。但是随着研究手段的不断发展,更多的病毒结构被发现,它们的衣壳结构违背准等价原理。我们以这些奇特结构为研究对象,找到它们的几何模型,补充多面体结构,为病毒研究提供理论指导。
乳多空病毒和多瘤病毒的72个五聚体的衣壳结构违背Caspar-Klug(CK)“准等价”原理。而且,72个五边形的球面包裹问题是一个有待解决的数学难题。我们在十二面体框架的基础上,利用“球面拉伸”的方法,得到新的具有二十面体对称性,72个五边形构造的的多面体结构。新型多面体结构为72个五聚体的病毒衣壳结构的模拟提出新的理论,为五边形排列问题提供了新的思路,丰富了多面体世界。
In geometry, convex polyhedra are old and important researchful objects, which contacts peoples' daily life. Polyhedral links are new theoretical results, some interlocked structures on the basis of old convex polyhedra and knot theory, which derived from the theoretical research for viral capsid structure and supermolecule structure. Polyhedral links are a class of topological links with polyhedral shape which are linked with a collection of finitely separate closed curves..On the basis of theoretical result of convex polyhedra, the thesis includes two parts of research which aim at different problems.
一、Duality is an important conception and manipulation in geometry. As new linked structures, polyhedral links are constructed on the frame of polyhedra. Whether polyhedral links possess of duality, how to define, and how to manipulate are important basis researches of polyhedral links. Furthermore, polyhedral links just a particular family of knots and links, which are attentive in the field of mathematics. Facing to the huge and complicated knot table, how to class, and what kind of relationships there are between them? When duality is applied to knots theory, it puts forward the definition of dual links and the methodology for the construction of dual links. Dual links open a new approach for the research of links, and the methodology may also be used to direct the synthesis of chiral molecules.
1. The novel topology of Platonic polyhedral links is discussed on the basis of the graph theory and topological principles. This interesting problem of the dual polyhedral links has been solved by using our method of the "sphere-surface-movement". There are three classes of dual polyhedral links which can be explored: the tetrahedral link is self-dual, the hexahedral and octahedral link, as well as the dodecahedral and icosahedral link are dual to each other. Our results show that the duality of self-dual tetrahedral link is "trivial", and the duality of hexahedral and octahedral link as well as dodecahedral and icosahedral link are "nontrivial". This study provides further insight into the molecular design and theoretical characterization of the new polyhedral links.
2. A new method for understanding the construction of dual links has been developed on the basis of medial graph in graph theory and tangle in knot theory. The method defines two types of oriented 4-valent plane graph: G_e and G_o, whose vertices are covered by E-tangles and O-tangles, respectively. The result shows that there are two types of dual links: E-dual links and O-dual links, which have many differences in topological properties, especially their chiral rule. In our paper, we show that dual links can be constructed by oriented 4-valent plant graphs and tangles. This research puts forward the definition of dual links and the methodology for the construction of dual links. Dual links open a new approach for the research of links, and the methodology may also be used to direct the synthesis of chiral molecules.
二、A virus is a unit of infectious genetic material smaller than any bacteria and embodying properties placing it on the borderline between life and non-life. The vast majority of the virus capsid protein assembled into icosahedral symmetry of the structure. Caspar-Klug Theory has become a fundamental concept for the classification of icosahedral viral capsids based on the principle of quasi-equivalence. However, recent experiments have shown that there are viruses that do not follow the organisation predicted by this theory. We take these novel viruses as objects of investigation, and constructing the geometry models to explain these novel architectures, which enrich the world of polehedra and give theoretic guidance of viral investigation.
The outer shells of papilloma virions and polyoma virus contain 72 pentamers, the architectures of which do not follow the Caspar-Klug (CK) "quasi-equivalence" theory. Moveover, the spherical pentagon packing problem for 72 pentagons is a mathematical problem. On the basis of the frame of dodecahedron, we apply the method of "spherical stretching" to the frame, we can obtain a novel polyhedron with I_h symmetry, which contains 72 pentagons. The novel polyhedral structure improves new theory for the simulating of viral capsid with 72 pentamers, and gives additional insight into the mathematical problem of the spherical pentagon packing problem for 72 pentagons, and enrichs the world of polehedra.
引文
[1]R.P.Goodman,R.M.Berry,A.J.Turberfield,The single-step synthesis of a DNA tetrahedron,Chem.Commun.,12(2004) 1372.
[2]J.Chen,N.C.Seeman,Synthesis from DNA of a molecule with the connectivity of a cube,Nature,350(1991) 631.
[3]W.M.Shih,J.D.Quispe,G.F.Joyce,A 1.7-kilobase single-stranded DNA that folds into a nanoscale octahedron,Nature,427(2004) 618.
[4]N.C.Seeman,Nucleic acid nanostructures and topology,Angew.Chem.,110(1998) 3408;Angew.Chem.,Int.Ed.37(1998)3220.
[5]M.Chandra,S.Keller,C.Gloeckner,B.Bornemann,A.Marx,New branched DNA constructs,Chem.Eur.J.,13(2007) 3558.
[6]M.Scheffler,A.Dorenbeck,S.Jordan,M.W(u|¨)stefeld,G.von Kiedrowski,Self-assembly of trisoligonucleotidyls:the case for nano-acetylene and nano-cyclobutadiene,Angew.Chem.,1999,111,3513;Angew.Chem.,Int.Ed.38(1999) 3311.
[7]Y.He,T.Ye,M.Su,C.Zhang,A.E.Ribbe,W.Jiang,C.Mao,Hierarchical self-assembly of DNA into symmetric supramolecular polyhedra,Nature,452(2008) 198.
[8]C.Zhang,M.Su,et al,Conformational flexibility facilitates self-assembly of complex DNA nanostructures,Proc.Natl.Acad.Sci.U.S.A.,105(2008) 10665.
[9]Y.Zhang, N.C.Seeman, Construction of a DNA-truncated octahedron, J.Am.Chem.Soc, 116 (1994)1661.
[10]F.A.Aldaye, H.F.Sleiman, Modular access to structurally switchabie 3D discrete DNA assemblies, J.Am.Chem.Soc, 129 (2007)13376.
[11]F.Rakotondradany, H.F.Sleiman, M.A.Whitehead, Theoretical study of self-assembled hydrogen-bonded azodibenzoic acid tapes and rosettes, J.Mol.Struct.THEOCHEM, 806 (2007)39.
[12]F.A.Aldaye, H.F.Sleiman.Dynamic DNA templates for discrete gold nanoparticle assemblies:control of geometry, modularity, write/erase and structural switching, J.Am.Chem.Soc, 129(2007)4130.
[13]C.M.Erben, R.P.Goodman, A.J.Turberfield, A self-assembled DNA bipyramid, J.Am.Chem.Soc, 129(2007)6992.
[14]S.J.Martin, The Biochemistry of Viruses, Cambridge University Press, Combrige London, New York, Melbourne., 1978,
[15]王继科,曲连东．病毒形态结构与结构参数,中国农业出版社,2000．
[16]R.L.Duda.Protein chainmail:catenated protein in viral capsids, Cell, 94 (1998)55.
[17]W.R.Wikoff, L.Liljas, R.L.Duda, H.Tsuruta, R.W.Hendrix, J.E.Johnson, Topologically linked protein rings in the Bacteriophage HK97 capsid, Science, 289 (2000)2129.
[18]J.F.Conway, W.R.Wikoff, N.Cheng, R.L.Duda, R.W.Hendrix, J.E.Johnson, A.C.Steven,Virus maturation involving large subunit rotations and local refolding, Science, 292 (2001)744.
[19]C.Helgstrand, W.R.Wikoff, R.L.Duda, R.W.Hendrix, J.E.Johnson, L.Liljas, The refined structure of a protein catenane:the HK.97 bacteriophage capsid at 3.44A resolution, J.Mol.Biol.,334 (2003)885.
[20]L.Tang, K.N.Johnson, L.A.Ball, T.Lin, M.Yeager, J.E.Johnson, The structure of pariacoto virus reveals a dodecahedral cage of duplex RNA, Nat.Struct.Biol., 8 (2001)77.
[21]A.Klug, Structure of viruses of the papilloma-polyoma type Ⅱ, J.Mol.Biol., 11 (1965)424.
[22]A.Klug, J.T.Finch, Structure of viruses of the papilloma-polyoma type I , J.Mol.Biol., 11 (1965)403.
[23]J.T.Finch, A.Klug, Structure of viruses of the papillomapolyoma type Ⅲ:structure of rabbit papilloma virus, J.Mol.Biol., 13 (1965)1.
[24]J.T.Finch, The surface structure of polyoma, J.Gen.Virol., 24 (1974)359.
[25]T.S.Baker, W.W.Newcomb, N.H.Olson, L.M.Cowsert, C.Olson, J.C.Brown, Structures of bovine and human papillomaviruses:analysis by cryoelectron microscopy and three-dimensional image reconstruction, Biophysical Journal, 60 (1991)1445.
[26]B.L.Trus, R.B.Roden, H.L.Greenstone, M.Vrhel, J.T.Schiller, F.P.Booy, Novel structural features of bovine papillomavirus capsid revealed by a three-dimensional reconstruction to 9 AI resolution, Nature Structural Biology, 4 (1997)413.
[27]D.M.Belnap, N.H.Olson, N.M.Cladel, W.W.Newcomb, J.C.Brown, J.W.Kreider, N.D.Christensen, T.S.Baker, Conserved features in papillomavirus and polyomavirus capsids, J.Mol.Biol., 259 (1996)249.
[28]R.L.Finnen, K.D.Erickson, X.S.Chen, R.L.Garcea, Interactions between papillomavirus LI and L2 capsid proteins, J.Virol., 77 (2003)4818.
[29]M.Sapp, C.Fligge, I.Petzak, J.R.Harris;R.E.Streeck, Papillomavirus assembly requires trimerization of the major capsid protein by disulfides between two highly conserved cysteines, J.Virol., 72 (1998)6186.
[30]C.Fligge, F.Schafer, H.C.Selinka, C.Sapp, M.Sapp, DNA-induced structural changes in the papillomavirus capsid, J.Virol., 75 (2001)7727.
[31]I.Rayment, T.S.Baker, D.L.D.Caspar, W.T.Murakami., Polyoma virus capsid structure at 22.5 A resolution, Nature, 295 (1982)110.
[32]R.C.Liddington, Y.Yan, J.Moulai, R.Sahli, T.L.Benjamin, S.C.Harrison, Structure of simian virus 40 at 3-8-A resolution, Nature, 354 (1991)278.
[33]T.S.Baker, J.Drak, M.Bina, Reconstruction of the three-dimensional structure of simian virus 40 and visualization of the chromatin core, Proc.Natl.Acad.Sci., 85 (1988)422.
[34]T.Stehle, Y.Yan, T.L.Benjamin, S.C.Harrison, Structure of murine polyomavirus complexed with an oligosaccharide receptor fragment, Nature, 369 (1994)160.
[35]D.M.Salunke, D.L.D.Caspar, R.L.Garcea, Self-assembly of purified polyomavirus capsid protein VP1, Cell, 46 (1986)895.
[36]Y.Modis, B.L.Trus, S.C.Harrison, Atomic model of the papillomavirus capsid, EMBO J., 21 (2002)4754.
[37]T.Stehle, S.J.Gamblin, Y.Yan, S.C.Harrison, The structure of simian virus 40 refined at 3.1 A resolution, Structure, 4 (1996)165.
[38]J.N.Brady, V.D.Winston, R.A.Consigli, Dissociation of polyomavirus by the chelation of calcium ions found associated with purified virions, J.Virol., 23 (1977)717.
[39]J.N.Brady, C.Lavialle, N.P.Salzman, Efficient transcription of a compact nucleoprotein complex isolated from purified simian virus 40 virions, J.Virol., 35 (1980)371.
[40]P.P.Li, A.Naknanishi et al, Importance of Vpl calcium-binding residues in assembly, cell entry, and nuclear entry of simian virus 40, J.Virol., 77 (2003)75.
[1]叶伟文译,典雅的几何(Sacred Geometry+Platonic and Archimedean Solids)M.Lundy+D.Sutton原著,天下文化“科学天地2023”,2002.
[2]Http://teach.mlc.edu.tw/cel/regularpolygon
[3]翟新东,DNA和蛋白质纽结理论:多面体链环[D].兰州:兰州大学化学化工院,2005.
[4]A.K.van der Vegt,Order of Space,Vereniging voor Studie-en Studentenbelangen te Delft,2001.
[5]D.Rolfsen,Knots and Links,Publish or Perish Inc,Berkely,1976.
[6]姜伯驹,绳圈的数学,湖南教育出版社,1998年.
[7]C.C.Adams,The Knot Book:An Elementary Introduction to the Mathematical Theory of Knots,W.H.Freeman & Company,New York,1994.
[8]R.W.B,Lickorish,An Introduction to Knot Theory,Springer-Verlag,New York,1997.
[9]E.Flapan,When Topology Meets Chemistry-A Topological Look at Molecular Chirality,Cambridge University Press,Cambridge,2000.
[10]J.Baez,Gauge fields,Knots and gravity,World Scientific,4(1994) 310.
[11]M.Thistlewaite,Knot tabulations and related topics,Aspects of Topology,ed.I.M.James and E.H.Kronheimer,Cambridge Univ.Press,1885.
[12]J.B.Listing,Vorstudien zur topologie,Goettinger Studien(Abtheilung 1),1(1847) 811.
[13]http://www.maths.ed.ac.uk/～aar/knots/
[14]J.W.Alexander, Topological invariants of knots and links, Trans.Amer.Math.Soc, 30 (1928)275.
[15]H.Doll, J.Hoste, A tabulation oforiented links, Math.Comput., 57 (1991)747.
[16]V.Jones, A polynomial invariant for knots via von Neumann algebras, Bull.Am.Math.Soc, 12(1985)103.
[17]P.Freyd, D.Yetter, J.Hoste, et al., A new polynomial invariant of knots and links, Bull.Amer.Math.Soc, 12(1985)239.
[18]V.A.Vassiliev, Cohorrjology of knot spaces, Theory of Singularities and Its ApplicatiQns (Ed.V.I.Arnold), Providence, RI:Amer.Math.Soc, 1 (1990)23.
[19]B.J.Jiang, X.S.Lin, et al., Chirality of knots and links, Topology and Its Applications, 2 (2002)185, arXiv:math/9905158.
[20]C.Rourke, B.Sanderson, A new classification of links and some calculations using it, 22 (2002)1, arXiv:math/0006062v2.
[21]M.Azram, Achirality of knots, Acta Math.Hungar., 101 (2003)217.
[22]C.Cerf, Atlas of oriented knots and links, Topology Atlas Invited Contributions 3, 2 (1998)1.http://at.yorku.ca/t/a/i/c/31.htm.
[23]C.Cerf, Nullification writhe and chirality of alternating links, J.Knot Theory Ramif, 6 (1997)621.
[24]L.H.Kauffman, An invariant of regular isotopy, Trans.Amer.Math.Soc, 318 (1990)417.
[25]S.R.Henry, J.R.Weeks, Symmetry groups of hyperbolic knots and links, J.Knot Theory Ramif., 1(1992)185.
[26]C.Liang, K.Mislow, On amphicheiral knots, J.Math.Chem., 15 (1994)1.
[27]C.Liang, K.Mislow, Classification of topologically chiral molecules, J.Math.Chem., 15 (1994)245.
[28]C.Liang, K.Mislow, Topological chirality and achirality of links, J.Math.Chem., 18 (1995)1.
[29]C.Liang, K.Mislow, Specification of chirality for links and knots, J.Math.Chem., 19 (1996)241.
[30]W.Y.Qiu, X.D.Zhai, Molecular design of Goldberg polyhedral link, J.Mol.Struct.THEOCHEM., 756 (2005)163.
[31]W.Y.Qiu, X.D.Zhai, Y.Y.Qiu, Architecture of Platonic and Archimedean polyhedral links, Sci.China Ser.B-Chem., 51 (2008)13.
[32]G Hu, X.D.Zhai, D.Lu, W.Y.Qiu, The architecture of Platonic polyhedral links.J.Math.Chem., (2008), DOI:10.1007/s10910-008-9487-z.
[33]G Hu, W.Y.Qiu, Extended Goldberg polyhedra, MATCH Commun.Math.Comput.Chem.,59(2008)585.
[34]G Hu, W.Y.Qiu, Extended Goldberg polyhedral links with even tangles, MATCH Commun.Math.Comput.Chem., 61 (2009)737.
[35]G Hu, W.Y.Qiu, Extended Goldberg polyhedral links with odd tangles, MATCH Commun.Math.Comput.Chem., 61 (2009)753.
[36]http://www.math.vt.edu/people/gao/duality/duality.html
[37]M.Mukerjee, Explaining everything, Sci.Am, 274 (1996)88.
[38]A.H.Jeffrey, Magnetic monopoles, duality, and supersymmetry, 2 (1996)1/96, arXiv:hepth/9603086
[39]A.N.Mitra, Duality in physical sciences and beyond, Pramana-J.Phys., 27 (1986)73.
[40]H.Goldstein, Classical Mechanics, Addison-Wesley, Reading, MA, USA, 1950.
[41]PA.M.Dirac, In A Lifetime of Physics, IAEA, Vienna, 1968.
[42]Y.Nambu, In Symmetries and Quark Models, (ed)R.Chand, Gordon and Breach, N.Y 1970.
[43]F.J.Dyson, Lecture Notes on Advanced Quantum Mechanics, Cornell Univ, 1952.
[44]N.Bohr and L.Rosenfeld, In Developments in the Theory of the Electron, (ed)A.Pais,Princeton U Press, 1948.
[45]J.von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton U Press,1955.
[46]http://www.hoodong.com/wiki/%E6%B3%A2%E7%B2%92%E4%BA%8C%E8%B 1 %A 1%E6%80%A7.
[47]B.G Englert, M.O.Scully, H.Walther, The duality in matter and light, Sci.Am., 271 (1994)86.
[48]P.K.Tripathy, Self-dual maxwell chern-simons solitons in 1+1 dimensions, Phys.Lett.D, 57 (1998)5166.
[49]陈文俊,李康,电磁对偶的经典电动力学与电荷量子化,浙江大学学报(理学版),6(2001)626.
[50]M.J.Duff,The theory formerly known as strings,Sci.Amer.,278(1998) 64.
[51]李淼,《超弦史话》,北京大学出版社,2005.
[52]J.Polchinski,Strung Theory,Volumn I,An Introduction to The Bosonic String,Cambridge Univercity Press,1998.
[53]J.Polchinski,Strung Theory,Volumn Ⅱ,Superstring Theory and Beyond,Cambridge Univercity Press,1998.,
[54]E.Bergshoeff,M.de Roo,D-branes and T-duality,Phys.Lett.B,380(1996) 265.
[55]http://www.superstringtheory.com/basics/basic6a.html
[56]http://www.superstringtheory.com/basics/basic6al.html
[57]http://en.wikipedia.org/wiki/U-duality
[58]http://en.wikipedia.org/wiki/Dual_graph
[59]D.Archdeacon,J.Siran,M.Skoviera,Self-dual regular maps from medial graphs,Acta Math.Univ.Comenianae.,61(1992) 57.
[1]C.C.Adams,The Knot Book:An Elementary Introduction to the Mathematical Theory of Knots,W.H.Freeman & Company,New York,1994.
[2]P.R.Cromwell,Knots and Links,Cambridge University Press,Cambridge,2004.
[3]S.Jablan,R.Sazdanovic,Linknot- Knot Theory by Computer,World Scientific,Singapore,2007.
[4]K.Mislow,A commentary on the topological chirality and achirality of molecules,Croat.Chem.Acta.,69(1996) 485.
[5]K.Mislow,C.Z.Liang,Knotted structures in chemistry,biochemistry,and molecular biology,Croat.Chem.Acta.,69(1996) 1385.
[6]J.P.Sauvage,C.D.Buchecker(Eds.),Molecular Catenanes,Rotaxanes and Knots,Wiley-VCH,New York,1999.
[7]W.R.Taylor,K.Lin,Protein knots:a tangled problem,Nature,421(2003) 25.
[8]C.Helgstrand,W.R.Wikoff,R.L.Duda,R.W.Hendrix,J.E.Johnson,L.Liljas,The refined structure of a protein catenane:the HK97 bacteriophage capsid at 3.44 A resolution,J.Mol.Biol.,334(2003) 885.
[9]R.P.Goodman,R.M.Berry,A.J.Turberfield,The single-step synthesis of a DNA tetrahedron,Chem.Commun.,12(2004) 1372.
[10]R.P.Goodman,I.A.T.Schaap,C.F.Tardin,C.M.Erben,R.M.Berry,C.F.Schmidt,A.J.Turberfield,Rapid chiral assembly of rigid DNA building blocks for molecular nanofabrication,Science,310(2005) 1661.
[11]J.H.Chen, N.C.Seeman, Synthesis from DNA of a molecule with the connectivity of a cube, Nature, 350(1991)631.
[12]N.C.Seeman, DNA in a material world, Nature, 421 (2003)427.
[13]Y.Zhang, N.C.Seeman, Construction of a DNA-truncated octahedron, J.Am.Chem.Soc,116(1994)1661.
[14]F.F.Andersen et.al, Assembly and structural analysis of a covalently closed nano-scale DNA cage, Nucl.Acids Res., 36 (2008)1113.
[15]C.M.Erben, R.P.Goodman, A.J.Turberfield, A self-assembled DNA bipyramid, J.Am, Chem.Soc, 129 (2007)6992.
[16]P.G.Mezey, Tying knots around chiral centres:chirality polynomials and conformational invariants for molecules, J.Am.Chem.Soc, 108 (1986)3976.
[17]E.Flapan, When Topology Meets Chemistry-a Topological Look at Molecular Chirality, Cambridge University Press, Cambridge, 2000.
[18]W.Y.Qiu, H.W.Xin, Molecular design and topological chirality of the Tq-Mobius ladders, J.Mol.Struct.(THEOCHEM), 401 (1997)151.
[19]W.Y.Qiu, H.W.Xin, Molecular design and tailor of the doubled knots, J.Mol.Struct.(THEOCHEM), 397 (1997)33.
[20]W.Y.Qiu, H.W.Xin, Topological structure of closed circular DNA, J.Mol.Struct.(THEOCHEM), 428 (1998)35.
[21]W.Y.Qiu, H.W.Xin, Topological chirality and achirality of DNA knots, J.Mol.Struct.(THEOCHEM), 429 (1998)81.
[22]W.Y.Qiu, Knot theory, DNA topology, and molecular symmetry breaking, in:D.Bonchev,D.H.Rouvray (Eds.), Chemical Topology-Applications and Techniques, Mathematical Chemistry Series, Vol.6, Gordon and Breach Science Publishers, Amsterdam, 2000.
[23]J.S.Siegel, Chemical topology and interlocking molecules, Science, 304 (2004)1256.
[24]C.Liang, K.Mislow, Topological chirality of proteins, J.Am.Chem.Soc, 116 (1994)3588.
[25]J.Rudnick, R.Bruinsma, Icosahedral packing of RNA viral genomes, Phy.Rev.Lett., 94(2005)038101.
[26]W.Y.Qiu, X.D.Zhai, Molecular design of Goldberg polyhedral links, J.Mol.Struct.(THEOCHEM), 756 (2005)163.
[27]G.Hu, X.D.Zhai, D.Lu, W.Y.Qiu, The architecture of platonic polyhedral links, J.Math.Chem., 2008, (DOI 10.1007/sl0910-008-9487-z).
[28]Y.M.Yang, W.Y.Qiu, Molecular design and mathematical analysis of carbon nanotube links, MATCH Commun.Math.Comput.Chem., 58 (2007)635.
[29]G.Hu, W.Y.Qiu, Extended Goldberg polyhedral links with even tangles, MATCH Commun.Math.Comput.Chem., 61 (2009)737.
[30]G.Hu, W.Y.Qiu, Extended Goldberg polyhedral links with odd tangles, MATCH Commun., Math.Comput.Chem., 61 (2009)753.
[31]A.K..van der Vegt, Order of Space, Vereniging voor Studie-en Studentenbelangen te Delft, 2001.
[32]W.Y.Qiu, X.D.Zhai, Y.Y Qiu, Architecture of platonic and archimedean polyhedral links,Sci.China Sen B-Chem., 51 (2007)13.
[33]M.J.Duff, The theory formerly known as strings, Sci.Amer., 278 (1998)64.
[34]D.Archdeacon, J.Siran, M.Skoviera, Self-dual regular maps from medial graphs, Acta.Math.Univ.Comenianae., 1 (1992)57.
[35]C.C.Adams, The Knot Book, W.H.Freeman and Company, New York, 1994.
[36]M.Mukerjee, Explaining everying, Sci.Am, 274 (1996)88.
[37]B.G.Englert, M.O.Scully, H.Walther, The duality in matter and light, Sci.Am, 271 (1994)86.
[38]E.Bergshoeff, M.de Roo, D-branes and T-duality, Phys.Lett.B., 380 (1996)265.
[1]M.Mukerjee,Explaining everything,Sci.Am.,274(1996) 88.
[2]B.G.Englert,M.O.Scully,H.Walther,The duality in matter and light,Sci.Am.,271(1994)86.
[3]M.J.Duff,The theory formerly known as strings,Sci.Am.,278(1998) 64.
[4]E.Bergshoeff,M.de Roo,D-branes and T-duality,Phys.Lett.B,380(1996) 265.
[5]A.K.van der Vegt,Order of Space,Vereniging voor Studie-en Studentenbelangen te Delft,2001.
[6]D.Archdeacon,J.Siran,M.Skoviera,Self-dual regular maps from medial graphs,Acta Math.Univ.Comenianae,61(1992) 57.
[7]C.C.Adams,The Knot Book:An Elementary Introduction to the Mathematical Theory of Knots,Freeman,New York,1994.
[8]P.R.Cromwell,Knots and Links,Cambridge University Press,Cambridge,2004.
[9]S.Jablan, R.Sazdanovic, Linknot:Knot Theory by Computer, World Scientific, Singapore,2007.
[10]E.Flapan, When Topology Meets Chemistry —a Topological Look at Molecular Chirality.Cambridge University Press, Cambridge, 2000.
[11]R.Kerner, Classification and evolutionary trends of icosahedral viral capsids, Comput.Math.Meth.Med., 9 (2008)175.
[12]K.Mislow, A commentary on the topological chirality and achirality of molecules, Croatica Chemical Acta, 69 (1996)485.
[13]K.Mislow, C.Z.Liang, Knotted structures in chemistry, biochemistry, and molecular biology, Croatica Chemical Acta, 69 (1996)1385.
[14]W.Y.Qiu, X.D.Zhai, Molecular design of Goldberg polyhedral links, J.Mol.Struct.(THEOCHEM), 756 (2005)163.
[15]G.Hu, X.D.Zhai, D.Lu, W.Y.Qiu, The architecture of platonic polyhedral links, J.Math.Chem., 2008, (DOI 10.1007/sl0910-008-9487-z).
[16]Y.M.Yang, W.Y.Qiu, Molecular design and mathematical analysis of carbon nanotube links, MATCH Commun.Math.Comput.Chem., 58 (2007)635.
[17]G.Hu, W.Y.Qiu, Extended Goldberg polyhedral links with even tangles, MATCH Commun.Math.Comput.Chem., 61 (2009)737.
[18]G.Hu, W.Y.Qiu, Extended Goldberg polyhedral links with odd tangles, MATCH Commun.Math.Comput.Chem., 61 (2009)753.
[19]P.G.Mezey, Tying knots around chiral centres:chirality polynomials and conformational invariants for molecules, J.Am.Chem.Soc, 108 (1986)3976.
[20]W.Y.Qiu, H.W.Xin, Molecular design and topological chirality of the Tq-Mobius ladders, J.Mol.Struct.(THEOCHEM), 401 (1997)151.
[21]W.Y.Qiu, H.W.Xin, Molecular design and tailor of the doubled knots, J.Mol.Struct.(THEOCHEM), 397 (1997)33.
[22]W.Y.Qiu, H.W.Xin, Topological structure of closed circular DNA, J.Mol.Struct.(THEOCHEM), 428 (1998)35.
[23]W.Y.Qiu, H.W.Xin, Topological chirality and achirality of DNA knots, J.Mol.Struct.(THEOCHEM), 429 (1998)81.
[24]W.Y.Qiu,Knot theory,DNA topology,and molecular symmetry breaking,in:D.Bonchev,D.H.Rouvray(Eds.),Chemical Topology-Applications and Techniques,Mathematical Chemistry Series,Vol.6,Gordon and Breach Science Publishers,Amsterdam,2000.
[25]J.S.Siegel,Chemical topology and interlocking molecules,Science,304(2004) 1256.
[26]C.Liang,K.Mislow,Topological chirality of proteins,J.Am.Chem.Soc.,116(1994) 3588.
[27]J.Rudnick,R.Bruinsma,Icosahedral packing of RNA viral genomes,Phy.Rev.Lett.,94(2005) 038101
[28]C.Z..Liang,C.Cerf,K.Mislow,Specfication of chirality for links and knots,J.Math.Chem.,19(1996) 241.
[1]F.H.C.Crick,J.D.Watson,The structure of small viruses,Nature,177(1956),473.
[2]C.B.Goodhart,Who goes to Oxbridge? Nature,303(1983) 278.
[3]I.Rayment,T.S.Baker,D.L.Caspar,W.T.Murakami,Polyoma virus capsid structure at 22.5A resolution,Nature,295(1982) 110.
[4]R.C.Liddington,Y.Yah,J.Moulai,R.Sahli,T.L.Benjamin,S.C.Harrison,Structure of simian virus 40 at 3.8-(?) resolution,Nature,354(1991) 278.
[5]T.S.Baker,W.W.Newcomb,N.H.Olson,L.M.Cowsert,C.Olson,J.C.Brown,Structures of bovine and human papillomaviruses:analysis by cryoelectron microscopy and three-dimensional image reconstruction,Biophys J.,60(1991) 1445.
[6]D.A.Kottwitz,The densest packing of equal circles on a sphere,Acta Cryst.Section A,47(1991) 158.
[7]T.Tarnai,Z.Gaspar,L.Szalait,pentagon packing models for "All-Pentamer" pirus structures,Biophysical Journal,69(1995) 612.
[8]R.Twarock, A tiling approach to virus capsid assembly explaining a structural puzzle in virology, Journal of Theoretical Biology, 226 (2004)477.
[9]G.Hu, W.Y.Qiu, Extended goldberg polyhedra, MATCH Commun.Math.Comput.Chem.59 (2008)585.
[10]Y.Modis, B.Trus, S.C.Harrison, Atomic model of the papillomavirus capsid, EMBO J.21 (2002)4754.
[11]D.M.Salunke , D.L.Caspar, R.L.Garcea, Polymorphism in the assembly of polyomavirus capsid protein VP1, Biophys J.56 (1989)887.,
[12]Z.Xie, R.W.Hendrix, Assembly in vitro of bacteriophage HK97 proheads, J Mol Biol, 253 (1995)74.
[13]T.Tarnai, Z.Gaspar, Packing of equal regular pentagons on a sphere, PROCEEDINGS OF THE ROYAL SOCIETY-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, 457 (2001)1043.
[14]A.Klug, Structure of viruses of the papilloma-polyoma type Ⅱ, J Mol Biol, 11 (1965)424.
[15]A.Klug, J.T.Finch, Structure of viruses of the papilloma-polyoma type Ⅰ, J Mol Biol, 11 (1965)403.
[16]J.T.Finch, A.Klug, Structure of viruses of the papillomapolyoma type Ⅲ, J.Mol.Biol., 13 (1965)1.
[17]J.T.Finch, The surface structure of polyoma, J.Gen.Virol., 24 (1974)359.
[18]T.S.Baker, W.W.Newcomb, N.H.Olson, L.M.Cowsert, C.Olson, J.C.Brown,Structures of bovine and human papillomaviruses:Analysis by cryoelectron microscopy and three-dimensional image reconstruction, Biophysical Journal, 60 (1991)1445.
[19]B.L.Trus, R.B.Roden, H.L.Greenstone, M.Vrhel, J.T.Schiller, F.P.Booy, Novel structural features of bovine papillomavirus capsid revealed by a three-dimensional reconstruction to 9 AI resolution, Nature Structural Biology, 4 (1997)413.
[20]D.M.Belnap, N.H.Olson, N.M.Cladel, W.W.Newcomb, J.C.Brown, J.W.Kreider, N.D.Christensen, T.S.Baker, Conserved features in papillomavirus and polyomavirus capsids, J.Mol.Biol., 259 (1996)249.
[21]R.L.Finnen, K.D.Erickson, X.S.Chen, R.L.Garcea, Interactions between papillomavirus L1 and L2 capsid proteins, J.Virol., 77 (2003)4818.
[22]M.Sapp, C.Fligge, I.Petzak, J.R.Harris, R.E.Streeck, Papillomavirus assembly requires trimerization of the major capsid protein by disulfides between two hghly conserved,cysteines, J.Virol., 72 (1998)6186.
[23]C.Fligge, F.Schafer, H.C.Selinka, C.Sapp, M.Sapp, DNA-induced structural changes in the papillomavirus capsid, J.Virol., 75 (2001)7727.
[24]http://hpv-web.lanl.gov
[25]X.S.Chen, R.L.Garcea, I.Goldberg, G Casini, S.C.Harrison, Structure of small virus-like particles assembled from the L1 protein of human papillomavirus 16, Mol.,Cell, 5 (2000)557.
[26]I.Rayment, T.S.Baker, D.L.D.Caspar, W.T.Murakami, Polyoma virus capsid structure at 22.5 A resolution, Nature (Lond), 295 (1982)110.
[27]R.C.Liddington, Y Yan, J.Moulai, R.Sahli, T.L.Benjamin, S.C.Harrison, Structure of simian virus 40 at 3.8-A resolution, Nature, 354 (1991)278.
[28]R.C.Liddington, Y Yan, J.Moulai, Sahli, T.L.R.Benjamin, S.C.Harrison, Structure of simian virus 40 at 3-8-.4 resolution, Nature, 354 (1991)278.
[29]Y Modis, B.L.Trus, S.C.Harrison, Atomic model of the papillomavirus capsid, EMBO J., 21(2002)4754.
[30]T.Stehle, S.J.Gamblin, Y Yan, S.C.Harrison, The structure of simian virus 40 refined at 3.1 A resolution, Structure, 4 (1996)165.
[31]J.N.Brady, V.D.Winston, R.A.Consigli, Dissociation of polyomavirus by the chelation of calcium ions found associated with purified virions, J.Virol., 23 (1977)717.
[32]J.N.Brady, C.Lavialle, N.P.Salzman, Efficient transcription of a compact nucleoprotein complex isolated from purified simian virus 40 virions, J.Virol., 35 (1980)371.
[33]J.Nilsson, et al., Structure and assembly of a T=l virus-like particle in BK polyomavirus, JOURNAL OF VIROLOGY, 79 (2005)5337.
[34]S.Kanesashi, et al., Simian virus 40 VP1 capsid protein forms polymorphic assemblies in vitro, Journal of General Virology, 84 (2003)1899.
[35]M.Goldberg, A class of multi-symmetric polyhedra, Tohoku Math.J., 43 (1937)104.
[36]A.K.van der Vegt, Order of Space, Vereniging voor Studie-en Studentenbelangen te Delft,2001.
[37]T.Tarnai, et al., Packing of equal regular pentagons on a sphere, Proc.R.Soc.Lond.A, 457 (2001)1043.
[38]P.Guyot, News on five-fold symmetry, Nature, 326 (1987)640.
[39]H.Reichert, et al., Observation of five-fold local symmetry in liquid lead, Nature, 408 (2000)839.
[40]F.Spaepen, Five-fold symmetry in liquids, Nature, 408 (2000)781.
[41]T.Schilling, S.Pronk, B.Mulder, D.Frenkel, Monte Carlo study of hard pentagons, Physice review E, 71 (2005)036138(6).
[42]V.G.Gryaznov, et al., Pentagonal symmety and disclinations in small particles cryst, Res.Technol., 34(1999)1091.
[2]J.Chen,N.C.Seeman,Synthesis from DNA of a molecule with the connectivity of a cube,Nature,350(1991) 631.
[3]W.M.Shih,J.D.Quispe,G.F.Joyce,A 1.7-kilobase single-stranded DNA that folds into a nanoscale octahedron,Nature,427(2004) 618.
[4]N.C.Seeman,Nucleic acid nanostructures and topology,Angew.Chem.,110(1998) 3408;Angew.Chem.,Int.Ed.37(1998)3220.
[5]M.Chandra,S.Keller,C.Gloeckner,B.Bornemann,A.Marx,New branched DNA constructs,Chem.Eur.J.,13(2007) 3558.
[6]M.Scheffler,A.Dorenbeck,S.Jordan,M.W(u|¨)stefeld,G.von Kiedrowski,Self-assembly of trisoligonucleotidyls:the case for nano-acetylene and nano-cyclobutadiene,Angew.Chem.,1999,111,3513;Angew.Chem.,Int.Ed.38(1999) 3311.
[7]Y.He,T.Ye,M.Su,C.Zhang,A.E.Ribbe,W.Jiang,C.Mao,Hierarchical self-assembly of DNA into symmetric supramolecular polyhedra,Nature,452(2008) 198.
[8]C.Zhang,M.Su,et al,Conformational flexibility facilitates self-assembly of complex DNA nanostructures,Proc.Natl.Acad.Sci.U.S.A.,105(2008) 10665.
[9]Y.Zhang, N.C.Seeman, Construction of a DNA-truncated octahedron, J.Am.Chem.Soc, 116 (1994)1661.
[10]F.A.Aldaye, H.F.Sleiman, Modular access to structurally switchabie 3D discrete DNA assemblies, J.Am.Chem.Soc, 129 (2007)13376.
[11]F.Rakotondradany, H.F.Sleiman, M.A.Whitehead, Theoretical study of self-assembled hydrogen-bonded azodibenzoic acid tapes and rosettes, J.Mol.Struct.THEOCHEM, 806 (2007)39.
[12]F.A.Aldaye, H.F.Sleiman.Dynamic DNA templates for discrete gold nanoparticle assemblies:control of geometry, modularity, write/erase and structural switching, J.Am.Chem.Soc, 129(2007)4130.
[13]C.M.Erben, R.P.Goodman, A.J.Turberfield, A self-assembled DNA bipyramid, J.Am.Chem.Soc, 129(2007)6992.
[14]S.J.Martin, The Biochemistry of Viruses, Cambridge University Press, Combrige London, New York, Melbourne., 1978,
[15]王继科,曲连东．病毒形态结构与结构参数,中国农业出版社,2000．
[16]R.L.Duda.Protein chainmail:catenated protein in viral capsids, Cell, 94 (1998)55.
[17]W.R.Wikoff, L.Liljas, R.L.Duda, H.Tsuruta, R.W.Hendrix, J.E.Johnson, Topologically linked protein rings in the Bacteriophage HK97 capsid, Science, 289 (2000)2129.
[18]J.F.Conway, W.R.Wikoff, N.Cheng, R.L.Duda, R.W.Hendrix, J.E.Johnson, A.C.Steven,Virus maturation involving large subunit rotations and local refolding, Science, 292 (2001)744.
[19]C.Helgstrand, W.R.Wikoff, R.L.Duda, R.W.Hendrix, J.E.Johnson, L.Liljas, The refined structure of a protein catenane:the HK.97 bacteriophage capsid at 3.44A resolution, J.Mol.Biol.,334 (2003)885.
[20]L.Tang, K.N.Johnson, L.A.Ball, T.Lin, M.Yeager, J.E.Johnson, The structure of pariacoto virus reveals a dodecahedral cage of duplex RNA, Nat.Struct.Biol., 8 (2001)77.
[21]A.Klug, Structure of viruses of the papilloma-polyoma type Ⅱ, J.Mol.Biol., 11 (1965)424.
[22]A.Klug, J.T.Finch, Structure of viruses of the papilloma-polyoma type I , J.Mol.Biol., 11 (1965)403.
[23]J.T.Finch, A.Klug, Structure of viruses of the papillomapolyoma type Ⅲ:structure of rabbit papilloma virus, J.Mol.Biol., 13 (1965)1.
[24]J.T.Finch, The surface structure of polyoma, J.Gen.Virol., 24 (1974)359.
[25]T.S.Baker, W.W.Newcomb, N.H.Olson, L.M.Cowsert, C.Olson, J.C.Brown, Structures of bovine and human papillomaviruses:analysis by cryoelectron microscopy and three-dimensional image reconstruction, Biophysical Journal, 60 (1991)1445.
[26]B.L.Trus, R.B.Roden, H.L.Greenstone, M.Vrhel, J.T.Schiller, F.P.Booy, Novel structural features of bovine papillomavirus capsid revealed by a three-dimensional reconstruction to 9 AI resolution, Nature Structural Biology, 4 (1997)413.
[27]D.M.Belnap, N.H.Olson, N.M.Cladel, W.W.Newcomb, J.C.Brown, J.W.Kreider, N.D.Christensen, T.S.Baker, Conserved features in papillomavirus and polyomavirus capsids, J.Mol.Biol., 259 (1996)249.
[28]R.L.Finnen, K.D.Erickson, X.S.Chen, R.L.Garcea, Interactions between papillomavirus LI and L2 capsid proteins, J.Virol., 77 (2003)4818.
[29]M.Sapp, C.Fligge, I.Petzak, J.R.Harris;R.E.Streeck, Papillomavirus assembly requires trimerization of the major capsid protein by disulfides between two highly conserved cysteines, J.Virol., 72 (1998)6186.
[30]C.Fligge, F.Schafer, H.C.Selinka, C.Sapp, M.Sapp, DNA-induced structural changes in the papillomavirus capsid, J.Virol., 75 (2001)7727.
[31]I.Rayment, T.S.Baker, D.L.D.Caspar, W.T.Murakami., Polyoma virus capsid structure at 22.5 A resolution, Nature, 295 (1982)110.
[32]R.C.Liddington, Y.Yan, J.Moulai, R.Sahli, T.L.Benjamin, S.C.Harrison, Structure of simian virus 40 at 3-8-A resolution, Nature, 354 (1991)278.
[33]T.S.Baker, J.Drak, M.Bina, Reconstruction of the three-dimensional structure of simian virus 40 and visualization of the chromatin core, Proc.Natl.Acad.Sci., 85 (1988)422.
[34]T.Stehle, Y.Yan, T.L.Benjamin, S.C.Harrison, Structure of murine polyomavirus complexed with an oligosaccharide receptor fragment, Nature, 369 (1994)160.
[35]D.M.Salunke, D.L.D.Caspar, R.L.Garcea, Self-assembly of purified polyomavirus capsid protein VP1, Cell, 46 (1986)895.
[36]Y.Modis, B.L.Trus, S.C.Harrison, Atomic model of the papillomavirus capsid, EMBO J., 21 (2002)4754.
[37]T.Stehle, S.J.Gamblin, Y.Yan, S.C.Harrison, The structure of simian virus 40 refined at 3.1 A resolution, Structure, 4 (1996)165.
[38]J.N.Brady, V.D.Winston, R.A.Consigli, Dissociation of polyomavirus by the chelation of calcium ions found associated with purified virions, J.Virol., 23 (1977)717.
[39]J.N.Brady, C.Lavialle, N.P.Salzman, Efficient transcription of a compact nucleoprotein complex isolated from purified simian virus 40 virions, J.Virol., 35 (1980)371.
[40]P.P.Li, A.Naknanishi et al, Importance of Vpl calcium-binding residues in assembly, cell entry, and nuclear entry of simian virus 40, J.Virol., 77 (2003)75.
[1]叶伟文译,典雅的几何(Sacred Geometry+Platonic and Archimedean Solids)M.Lundy+D.Sutton原著,天下文化“科学天地2023”,2002.
[2]Http://teach.mlc.edu.tw/cel/regularpolygon
[3]翟新东,DNA和蛋白质纽结理论:多面体链环[D].兰州:兰州大学化学化工院,2005.
[4]A.K.van der Vegt,Order of Space,Vereniging voor Studie-en Studentenbelangen te Delft,2001.
[5]D.Rolfsen,Knots and Links,Publish or Perish Inc,Berkely,1976.
[6]姜伯驹,绳圈的数学,湖南教育出版社,1998年.
[7]C.C.Adams,The Knot Book:An Elementary Introduction to the Mathematical Theory of Knots,W.H.Freeman & Company,New York,1994.
[8]R.W.B,Lickorish,An Introduction to Knot Theory,Springer-Verlag,New York,1997.
[9]E.Flapan,When Topology Meets Chemistry-A Topological Look at Molecular Chirality,Cambridge University Press,Cambridge,2000.
[10]J.Baez,Gauge fields,Knots and gravity,World Scientific,4(1994) 310.
[11]M.Thistlewaite,Knot tabulations and related topics,Aspects of Topology,ed.I.M.James and E.H.Kronheimer,Cambridge Univ.Press,1885.
[12]J.B.Listing,Vorstudien zur topologie,Goettinger Studien(Abtheilung 1),1(1847) 811.
[13]http://www.maths.ed.ac.uk/～aar/knots/
[14]J.W.Alexander, Topological invariants of knots and links, Trans.Amer.Math.Soc, 30 (1928)275.
[15]H.Doll, J.Hoste, A tabulation oforiented links, Math.Comput., 57 (1991)747.
[16]V.Jones, A polynomial invariant for knots via von Neumann algebras, Bull.Am.Math.Soc, 12(1985)103.
[17]P.Freyd, D.Yetter, J.Hoste, et al., A new polynomial invariant of knots and links, Bull.Amer.Math.Soc, 12(1985)239.
[18]V.A.Vassiliev, Cohorrjology of knot spaces, Theory of Singularities and Its ApplicatiQns (Ed.V.I.Arnold), Providence, RI:Amer.Math.Soc, 1 (1990)23.
[19]B.J.Jiang, X.S.Lin, et al., Chirality of knots and links, Topology and Its Applications, 2 (2002)185, arXiv:math/9905158.
[20]C.Rourke, B.Sanderson, A new classification of links and some calculations using it, 22 (2002)1, arXiv:math/0006062v2.
[21]M.Azram, Achirality of knots, Acta Math.Hungar., 101 (2003)217.
[22]C.Cerf, Atlas of oriented knots and links, Topology Atlas Invited Contributions 3, 2 (1998)1.http://at.yorku.ca/t/a/i/c/31.htm.
[23]C.Cerf, Nullification writhe and chirality of alternating links, J.Knot Theory Ramif, 6 (1997)621.
[24]L.H.Kauffman, An invariant of regular isotopy, Trans.Amer.Math.Soc, 318 (1990)417.
[25]S.R.Henry, J.R.Weeks, Symmetry groups of hyperbolic knots and links, J.Knot Theory Ramif., 1(1992)185.
[26]C.Liang, K.Mislow, On amphicheiral knots, J.Math.Chem., 15 (1994)1.
[27]C.Liang, K.Mislow, Classification of topologically chiral molecules, J.Math.Chem., 15 (1994)245.
[28]C.Liang, K.Mislow, Topological chirality and achirality of links, J.Math.Chem., 18 (1995)1.
[29]C.Liang, K.Mislow, Specification of chirality for links and knots, J.Math.Chem., 19 (1996)241.
[30]W.Y.Qiu, X.D.Zhai, Molecular design of Goldberg polyhedral link, J.Mol.Struct.THEOCHEM., 756 (2005)163.
[31]W.Y.Qiu, X.D.Zhai, Y.Y.Qiu, Architecture of Platonic and Archimedean polyhedral links, Sci.China Ser.B-Chem., 51 (2008)13.
[32]G Hu, X.D.Zhai, D.Lu, W.Y.Qiu, The architecture of Platonic polyhedral links.J.Math.Chem., (2008), DOI:10.1007/s10910-008-9487-z.
[33]G Hu, W.Y.Qiu, Extended Goldberg polyhedra, MATCH Commun.Math.Comput.Chem.,59(2008)585.
[34]G Hu, W.Y.Qiu, Extended Goldberg polyhedral links with even tangles, MATCH Commun.Math.Comput.Chem., 61 (2009)737.
[35]G Hu, W.Y.Qiu, Extended Goldberg polyhedral links with odd tangles, MATCH Commun.Math.Comput.Chem., 61 (2009)753.
[36]http://www.math.vt.edu/people/gao/duality/duality.html
[37]M.Mukerjee, Explaining everything, Sci.Am, 274 (1996)88.
[38]A.H.Jeffrey, Magnetic monopoles, duality, and supersymmetry, 2 (1996)1/96, arXiv:hepth/9603086
[39]A.N.Mitra, Duality in physical sciences and beyond, Pramana-J.Phys., 27 (1986)73.
[40]H.Goldstein, Classical Mechanics, Addison-Wesley, Reading, MA, USA, 1950.
[41]PA.M.Dirac, In A Lifetime of Physics, IAEA, Vienna, 1968.
[42]Y.Nambu, In Symmetries and Quark Models, (ed)R.Chand, Gordon and Breach, N.Y 1970.
[43]F.J.Dyson, Lecture Notes on Advanced Quantum Mechanics, Cornell Univ, 1952.
[44]N.Bohr and L.Rosenfeld, In Developments in the Theory of the Electron, (ed)A.Pais,Princeton U Press, 1948.
[45]J.von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton U Press,1955.
[46]http://www.hoodong.com/wiki/%E6%B3%A2%E7%B2%92%E4%BA%8C%E8%B 1 %A 1%E6%80%A7.
[47]B.G Englert, M.O.Scully, H.Walther, The duality in matter and light, Sci.Am., 271 (1994)86.
[48]P.K.Tripathy, Self-dual maxwell chern-simons solitons in 1+1 dimensions, Phys.Lett.D, 57 (1998)5166.
[49]陈文俊,李康,电磁对偶的经典电动力学与电荷量子化,浙江大学学报(理学版),6(2001)626.
[50]M.J.Duff,The theory formerly known as strings,Sci.Amer.,278(1998) 64.
[51]李淼,《超弦史话》,北京大学出版社,2005.
[52]J.Polchinski,Strung Theory,Volumn I,An Introduction to The Bosonic String,Cambridge Univercity Press,1998.
[53]J.Polchinski,Strung Theory,Volumn Ⅱ,Superstring Theory and Beyond,Cambridge Univercity Press,1998.,
[54]E.Bergshoeff,M.de Roo,D-branes and T-duality,Phys.Lett.B,380(1996) 265.
[55]http://www.superstringtheory.com/basics/basic6a.html
[56]http://www.superstringtheory.com/basics/basic6al.html
[57]http://en.wikipedia.org/wiki/U-duality
[58]http://en.wikipedia.org/wiki/Dual_graph
[59]D.Archdeacon,J.Siran,M.Skoviera,Self-dual regular maps from medial graphs,Acta Math.Univ.Comenianae.,61(1992) 57.
[1]C.C.Adams,The Knot Book:An Elementary Introduction to the Mathematical Theory of Knots,W.H.Freeman & Company,New York,1994.
[2]P.R.Cromwell,Knots and Links,Cambridge University Press,Cambridge,2004.
[3]S.Jablan,R.Sazdanovic,Linknot- Knot Theory by Computer,World Scientific,Singapore,2007.
[4]K.Mislow,A commentary on the topological chirality and achirality of molecules,Croat.Chem.Acta.,69(1996) 485.
[5]K.Mislow,C.Z.Liang,Knotted structures in chemistry,biochemistry,and molecular biology,Croat.Chem.Acta.,69(1996) 1385.
[6]J.P.Sauvage,C.D.Buchecker(Eds.),Molecular Catenanes,Rotaxanes and Knots,Wiley-VCH,New York,1999.
[7]W.R.Taylor,K.Lin,Protein knots:a tangled problem,Nature,421(2003) 25.
[8]C.Helgstrand,W.R.Wikoff,R.L.Duda,R.W.Hendrix,J.E.Johnson,L.Liljas,The refined structure of a protein catenane:the HK97 bacteriophage capsid at 3.44 A resolution,J.Mol.Biol.,334(2003) 885.
[9]R.P.Goodman,R.M.Berry,A.J.Turberfield,The single-step synthesis of a DNA tetrahedron,Chem.Commun.,12(2004) 1372.
[10]R.P.Goodman,I.A.T.Schaap,C.F.Tardin,C.M.Erben,R.M.Berry,C.F.Schmidt,A.J.Turberfield,Rapid chiral assembly of rigid DNA building blocks for molecular nanofabrication,Science,310(2005) 1661.
[11]J.H.Chen, N.C.Seeman, Synthesis from DNA of a molecule with the connectivity of a cube, Nature, 350(1991)631.
[12]N.C.Seeman, DNA in a material world, Nature, 421 (2003)427.
[13]Y.Zhang, N.C.Seeman, Construction of a DNA-truncated octahedron, J.Am.Chem.Soc,116(1994)1661.
[14]F.F.Andersen et.al, Assembly and structural analysis of a covalently closed nano-scale DNA cage, Nucl.Acids Res., 36 (2008)1113.
[15]C.M.Erben, R.P.Goodman, A.J.Turberfield, A self-assembled DNA bipyramid, J.Am, Chem.Soc, 129 (2007)6992.
[16]P.G.Mezey, Tying knots around chiral centres:chirality polynomials and conformational invariants for molecules, J.Am.Chem.Soc, 108 (1986)3976.
[17]E.Flapan, When Topology Meets Chemistry-a Topological Look at Molecular Chirality, Cambridge University Press, Cambridge, 2000.
[18]W.Y.Qiu, H.W.Xin, Molecular design and topological chirality of the Tq-Mobius ladders, J.Mol.Struct.(THEOCHEM), 401 (1997)151.
[19]W.Y.Qiu, H.W.Xin, Molecular design and tailor of the doubled knots, J.Mol.Struct.(THEOCHEM), 397 (1997)33.
[20]W.Y.Qiu, H.W.Xin, Topological structure of closed circular DNA, J.Mol.Struct.(THEOCHEM), 428 (1998)35.
[21]W.Y.Qiu, H.W.Xin, Topological chirality and achirality of DNA knots, J.Mol.Struct.(THEOCHEM), 429 (1998)81.
[22]W.Y.Qiu, Knot theory, DNA topology, and molecular symmetry breaking, in:D.Bonchev,D.H.Rouvray (Eds.), Chemical Topology-Applications and Techniques, Mathematical Chemistry Series, Vol.6, Gordon and Breach Science Publishers, Amsterdam, 2000.
[23]J.S.Siegel, Chemical topology and interlocking molecules, Science, 304 (2004)1256.
[24]C.Liang, K.Mislow, Topological chirality of proteins, J.Am.Chem.Soc, 116 (1994)3588.
[25]J.Rudnick, R.Bruinsma, Icosahedral packing of RNA viral genomes, Phy.Rev.Lett., 94(2005)038101.
[26]W.Y.Qiu, X.D.Zhai, Molecular design of Goldberg polyhedral links, J.Mol.Struct.(THEOCHEM), 756 (2005)163.
[27]G.Hu, X.D.Zhai, D.Lu, W.Y.Qiu, The architecture of platonic polyhedral links, J.Math.Chem., 2008, (DOI 10.1007/sl0910-008-9487-z).
[28]Y.M.Yang, W.Y.Qiu, Molecular design and mathematical analysis of carbon nanotube links, MATCH Commun.Math.Comput.Chem., 58 (2007)635.
[29]G.Hu, W.Y.Qiu, Extended Goldberg polyhedral links with even tangles, MATCH Commun.Math.Comput.Chem., 61 (2009)737.
[30]G.Hu, W.Y.Qiu, Extended Goldberg polyhedral links with odd tangles, MATCH Commun., Math.Comput.Chem., 61 (2009)753.
[31]A.K..van der Vegt, Order of Space, Vereniging voor Studie-en Studentenbelangen te Delft, 2001.
[32]W.Y.Qiu, X.D.Zhai, Y.Y Qiu, Architecture of platonic and archimedean polyhedral links,Sci.China Sen B-Chem., 51 (2007)13.
[33]M.J.Duff, The theory formerly known as strings, Sci.Amer., 278 (1998)64.
[34]D.Archdeacon, J.Siran, M.Skoviera, Self-dual regular maps from medial graphs, Acta.Math.Univ.Comenianae., 1 (1992)57.
[35]C.C.Adams, The Knot Book, W.H.Freeman and Company, New York, 1994.
[36]M.Mukerjee, Explaining everying, Sci.Am, 274 (1996)88.
[37]B.G.Englert, M.O.Scully, H.Walther, The duality in matter and light, Sci.Am, 271 (1994)86.
[38]E.Bergshoeff, M.de Roo, D-branes and T-duality, Phys.Lett.B., 380 (1996)265.
[1]M.Mukerjee,Explaining everything,Sci.Am.,274(1996) 88.
[2]B.G.Englert,M.O.Scully,H.Walther,The duality in matter and light,Sci.Am.,271(1994)86.
[3]M.J.Duff,The theory formerly known as strings,Sci.Am.,278(1998) 64.
[4]E.Bergshoeff,M.de Roo,D-branes and T-duality,Phys.Lett.B,380(1996) 265.
[5]A.K.van der Vegt,Order of Space,Vereniging voor Studie-en Studentenbelangen te Delft,2001.
[6]D.Archdeacon,J.Siran,M.Skoviera,Self-dual regular maps from medial graphs,Acta Math.Univ.Comenianae,61(1992) 57.
[7]C.C.Adams,The Knot Book:An Elementary Introduction to the Mathematical Theory of Knots,Freeman,New York,1994.
[8]P.R.Cromwell,Knots and Links,Cambridge University Press,Cambridge,2004.
[9]S.Jablan, R.Sazdanovic, Linknot:Knot Theory by Computer, World Scientific, Singapore,2007.
[10]E.Flapan, When Topology Meets Chemistry —a Topological Look at Molecular Chirality.Cambridge University Press, Cambridge, 2000.
[11]R.Kerner, Classification and evolutionary trends of icosahedral viral capsids, Comput.Math.Meth.Med., 9 (2008)175.
[12]K.Mislow, A commentary on the topological chirality and achirality of molecules, Croatica Chemical Acta, 69 (1996)485.
[13]K.Mislow, C.Z.Liang, Knotted structures in chemistry, biochemistry, and molecular biology, Croatica Chemical Acta, 69 (1996)1385.
[14]W.Y.Qiu, X.D.Zhai, Molecular design of Goldberg polyhedral links, J.Mol.Struct.(THEOCHEM), 756 (2005)163.
[15]G.Hu, X.D.Zhai, D.Lu, W.Y.Qiu, The architecture of platonic polyhedral links, J.Math.Chem., 2008, (DOI 10.1007/sl0910-008-9487-z).
[16]Y.M.Yang, W.Y.Qiu, Molecular design and mathematical analysis of carbon nanotube links, MATCH Commun.Math.Comput.Chem., 58 (2007)635.
[17]G.Hu, W.Y.Qiu, Extended Goldberg polyhedral links with even tangles, MATCH Commun.Math.Comput.Chem., 61 (2009)737.
[18]G.Hu, W.Y.Qiu, Extended Goldberg polyhedral links with odd tangles, MATCH Commun.Math.Comput.Chem., 61 (2009)753.
[19]P.G.Mezey, Tying knots around chiral centres:chirality polynomials and conformational invariants for molecules, J.Am.Chem.Soc, 108 (1986)3976.
[20]W.Y.Qiu, H.W.Xin, Molecular design and topological chirality of the Tq-Mobius ladders, J.Mol.Struct.(THEOCHEM), 401 (1997)151.
[21]W.Y.Qiu, H.W.Xin, Molecular design and tailor of the doubled knots, J.Mol.Struct.(THEOCHEM), 397 (1997)33.
[22]W.Y.Qiu, H.W.Xin, Topological structure of closed circular DNA, J.Mol.Struct.(THEOCHEM), 428 (1998)35.
[23]W.Y.Qiu, H.W.Xin, Topological chirality and achirality of DNA knots, J.Mol.Struct.(THEOCHEM), 429 (1998)81.
[24]W.Y.Qiu,Knot theory,DNA topology,and molecular symmetry breaking,in:D.Bonchev,D.H.Rouvray(Eds.),Chemical Topology-Applications and Techniques,Mathematical Chemistry Series,Vol.6,Gordon and Breach Science Publishers,Amsterdam,2000.
[25]J.S.Siegel,Chemical topology and interlocking molecules,Science,304(2004) 1256.
[26]C.Liang,K.Mislow,Topological chirality of proteins,J.Am.Chem.Soc.,116(1994) 3588.
[27]J.Rudnick,R.Bruinsma,Icosahedral packing of RNA viral genomes,Phy.Rev.Lett.,94(2005) 038101
[28]C.Z..Liang,C.Cerf,K.Mislow,Specfication of chirality for links and knots,J.Math.Chem.,19(1996) 241.
[1]F.H.C.Crick,J.D.Watson,The structure of small viruses,Nature,177(1956),473.
[2]C.B.Goodhart,Who goes to Oxbridge? Nature,303(1983) 278.
[3]I.Rayment,T.S.Baker,D.L.Caspar,W.T.Murakami,Polyoma virus capsid structure at 22.5A resolution,Nature,295(1982) 110.
[4]R.C.Liddington,Y.Yah,J.Moulai,R.Sahli,T.L.Benjamin,S.C.Harrison,Structure of simian virus 40 at 3.8-(?) resolution,Nature,354(1991) 278.
[5]T.S.Baker,W.W.Newcomb,N.H.Olson,L.M.Cowsert,C.Olson,J.C.Brown,Structures of bovine and human papillomaviruses:analysis by cryoelectron microscopy and three-dimensional image reconstruction,Biophys J.,60(1991) 1445.
[6]D.A.Kottwitz,The densest packing of equal circles on a sphere,Acta Cryst.Section A,47(1991) 158.
[7]T.Tarnai,Z.Gaspar,L.Szalait,pentagon packing models for "All-Pentamer" pirus structures,Biophysical Journal,69(1995) 612.
[8]R.Twarock, A tiling approach to virus capsid assembly explaining a structural puzzle in virology, Journal of Theoretical Biology, 226 (2004)477.
[9]G.Hu, W.Y.Qiu, Extended goldberg polyhedra, MATCH Commun.Math.Comput.Chem.59 (2008)585.
[10]Y.Modis, B.Trus, S.C.Harrison, Atomic model of the papillomavirus capsid, EMBO J.21 (2002)4754.
[11]D.M.Salunke , D.L.Caspar, R.L.Garcea, Polymorphism in the assembly of polyomavirus capsid protein VP1, Biophys J.56 (1989)887.,
[12]Z.Xie, R.W.Hendrix, Assembly in vitro of bacteriophage HK97 proheads, J Mol Biol, 253 (1995)74.
[13]T.Tarnai, Z.Gaspar, Packing of equal regular pentagons on a sphere, PROCEEDINGS OF THE ROYAL SOCIETY-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, 457 (2001)1043.
[14]A.Klug, Structure of viruses of the papilloma-polyoma type Ⅱ, J Mol Biol, 11 (1965)424.
[15]A.Klug, J.T.Finch, Structure of viruses of the papilloma-polyoma type Ⅰ, J Mol Biol, 11 (1965)403.
[16]J.T.Finch, A.Klug, Structure of viruses of the papillomapolyoma type Ⅲ, J.Mol.Biol., 13 (1965)1.
[17]J.T.Finch, The surface structure of polyoma, J.Gen.Virol., 24 (1974)359.
[18]T.S.Baker, W.W.Newcomb, N.H.Olson, L.M.Cowsert, C.Olson, J.C.Brown,Structures of bovine and human papillomaviruses:Analysis by cryoelectron microscopy and three-dimensional image reconstruction, Biophysical Journal, 60 (1991)1445.
[19]B.L.Trus, R.B.Roden, H.L.Greenstone, M.Vrhel, J.T.Schiller, F.P.Booy, Novel structural features of bovine papillomavirus capsid revealed by a three-dimensional reconstruction to 9 AI resolution, Nature Structural Biology, 4 (1997)413.
[20]D.M.Belnap, N.H.Olson, N.M.Cladel, W.W.Newcomb, J.C.Brown, J.W.Kreider, N.D.Christensen, T.S.Baker, Conserved features in papillomavirus and polyomavirus capsids, J.Mol.Biol., 259 (1996)249.
[21]R.L.Finnen, K.D.Erickson, X.S.Chen, R.L.Garcea, Interactions between papillomavirus L1 and L2 capsid proteins, J.Virol., 77 (2003)4818.
[22]M.Sapp, C.Fligge, I.Petzak, J.R.Harris, R.E.Streeck, Papillomavirus assembly requires trimerization of the major capsid protein by disulfides between two hghly conserved,cysteines, J.Virol., 72 (1998)6186.
[23]C.Fligge, F.Schafer, H.C.Selinka, C.Sapp, M.Sapp, DNA-induced structural changes in the papillomavirus capsid, J.Virol., 75 (2001)7727.
[24]http://hpv-web.lanl.gov
[25]X.S.Chen, R.L.Garcea, I.Goldberg, G Casini, S.C.Harrison, Structure of small virus-like particles assembled from the L1 protein of human papillomavirus 16, Mol.,Cell, 5 (2000)557.
[26]I.Rayment, T.S.Baker, D.L.D.Caspar, W.T.Murakami, Polyoma virus capsid structure at 22.5 A resolution, Nature (Lond), 295 (1982)110.
[27]R.C.Liddington, Y Yan, J.Moulai, R.Sahli, T.L.Benjamin, S.C.Harrison, Structure of simian virus 40 at 3.8-A resolution, Nature, 354 (1991)278.
[28]R.C.Liddington, Y Yan, J.Moulai, Sahli, T.L.R.Benjamin, S.C.Harrison, Structure of simian virus 40 at 3-8-.4 resolution, Nature, 354 (1991)278.
[29]Y Modis, B.L.Trus, S.C.Harrison, Atomic model of the papillomavirus capsid, EMBO J., 21(2002)4754.
[30]T.Stehle, S.J.Gamblin, Y Yan, S.C.Harrison, The structure of simian virus 40 refined at 3.1 A resolution, Structure, 4 (1996)165.
[31]J.N.Brady, V.D.Winston, R.A.Consigli, Dissociation of polyomavirus by the chelation of calcium ions found associated with purified virions, J.Virol., 23 (1977)717.
[32]J.N.Brady, C.Lavialle, N.P.Salzman, Efficient transcription of a compact nucleoprotein complex isolated from purified simian virus 40 virions, J.Virol., 35 (1980)371.
[33]J.Nilsson, et al., Structure and assembly of a T=l virus-like particle in BK polyomavirus, JOURNAL OF VIROLOGY, 79 (2005)5337.
[34]S.Kanesashi, et al., Simian virus 40 VP1 capsid protein forms polymorphic assemblies in vitro, Journal of General Virology, 84 (2003)1899.
[35]M.Goldberg, A class of multi-symmetric polyhedra, Tohoku Math.J., 43 (1937)104.
[36]A.K.van der Vegt, Order of Space, Vereniging voor Studie-en Studentenbelangen te Delft,2001.
[37]T.Tarnai, et al., Packing of equal regular pentagons on a sphere, Proc.R.Soc.Lond.A, 457 (2001)1043.
[38]P.Guyot, News on five-fold symmetry, Nature, 326 (1987)640.
[39]H.Reichert, et al., Observation of five-fold local symmetry in liquid lead, Nature, 408 (2000)839.
[40]F.Spaepen, Five-fold symmetry in liquids, Nature, 408 (2000)781.
[41]T.Schilling, S.Pronk, B.Mulder, D.Frenkel, Monte Carlo study of hard pentagons, Physice review E, 71 (2005)036138(6).
[42]V.G.Gryaznov, et al., Pentagonal symmety and disclinations in small particles cryst, Res.Technol., 34(1999)1091.