目前解決奈米尺寸結構力學行為之方法主要以分子動力學為主，然此方法其計算時間間隔約為百兆分之ㄧ秒，因電腦硬體容量限制，故可計算之原子數約為數萬顆數目左右，此一因素，將對未來分析較大型之分子結構力學行為造成限制。本研究發展使用一新理論，用以改善前述分子動力學方法之不足。本研究以有限單元法為基礎，建立一原子-連體力學理論，用以計算奈米尺寸結構之動態力學行為。於該理論中，吾人首先將原子轉換為有限單元之節點，而原子間之化學鍵能即可利用原子-連體力學理論表現之。換言之，如已知各原子之位置及其原子間之作用力，即可使用本法對其結構進行力學行為分析，故本研究方法亦可對探討多重尺度(由奈米尺度至微米尺度)結構之力學行為。吾人並利用暫態有限單元方法分析奈米結構之力學行為，亦可更詳細地獲得於外負載下奈米結構中之機械力學的變化。本研究中所提出的原子-連體理論計算最多可同時模擬數百萬個原子的行為，並可將計算時間間隔增加至千萬分之一秒，故本研究之計算理論可較分子動力學理論有更長時間的歷程分析。本研究之理論可以同時考慮多種化學鍵能之共同作用，期許在未來能有更廣泛之應用。
本研究將利用前述之原子-連體力學法，利用雙股螺旋去氧核醣核酸(double-stranded DNA, dsDNA)與病毒外殼蛋白質結構做為分析載具。首先，於合理的假設下，將原本離散的dsDNA分子結構轉換成為連續體模型，並利用有限單元法分析該結構之力學性質。本研究利用勢能函數與全域力場理論，將DNA骨架間相鄰原子之共價鍵的游離能利用摩斯勢能函數(Morse potential function)描述。且本研究將原子間作用力轉換為一連接兩節點的原子-連體轉換元素，此原子-連體轉換元素其材料特性由上述摩斯勢能函數計算求得，而各相鄰鹼基對間之作用力(氫鍵與凡得瓦力)亦可用此原子-連體轉換單元代表之。本研究於原子間之作用力部份，不僅考慮了鍵結拉伸項，亦同時考慮鍵角變形項。在病毒外鞘薄膜之機械強度探討中，本研究亦將使用薄殼理論與有限單元法來進行計算與模擬。於有限單元法中，本研究首先將組成病毒薄殼之蛋白質等效為一有限單元，而後由文獻所提供之薄殼機械參數(楊氏係數與蒲松比)帶入此等效薄殼單元，計算病毒吞噬核酸後其薄殼所承受之應力，並與文獻所量得之實驗值相互驗證。
因此，本研究將利用原子-連體理論建立自由旋動邊界條件下之dsDNA分子數值模型，並且施加外負載於該數值模型，如此一來，則可經由暫態有限單元法軟體求解此模型力學行為。吾人將前述之數值模擬與實驗量測模擬結果比對，可獲得相當的一致性，並且可以賦予dsDNA分子於拉伸負載下結構變形之力學解釋。而在病毒外鞘薄殼應力分析中，利用等效方法所建構之類真實病毒薄膜模型，改善了薄膜理論解之不足處，且模擬結果與實驗值相比亦落在合理範圍內。於未來方面，期許本研究方法所建構之原子-連體力學法不僅可應用於生物奈米結構，更可解決任意奈米尺寸結構(如半導體之90奈米以下結構或發光二極體之發光層結構)之力學行為分析。
關鍵字：原子-連體力學法、奈米力學、雙股螺旋DNA(ds-DNA)、病毒薄殼、有限單元法

REFERENCES
[1] J. Watson, T. Baker, S. Bell, A. Gann, M. Levine and R. Loick, Molecular Biology of the Gene, 5th ed., Benjamin/Cummings, San Francisco, USA, 2004.
[2] B. Lewin, Genes, 7th edition, Oxford University Press, New York, USA, 2000.
[3] W. Wood, J. Wilson, R. Benbow and L. Hood, Biochemistry: a problems approach, 2nd ed., Benjamin/Cummings, 1981, pp. 315.
[4] J. Watson and F. Crick, “A structure for deoxyribose nucleic acid,” Nature, vol. 171, pp. 737-739, 1953.
[5] C. Bustamante, Z. Bryant and S. B. Smith, “Ten years of tension: single-molucule DNA mechanics,” Nature, 421, 423-427, 2003.
[6] A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett., vol. 24, pp. 156, 1970.
[7] K. Visscher, S. P. Gross and S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Quantum Electronics, vol. 2, no. 4, pp. 1066-pp.1079, 1996.
[8] K. C. Neman, E. H. Chadd, G. F. Liou, K. Bergman, and S. M. Block, “Characterization of photodamage to Escherichia coli in optical traps,” Biophys. J., vol. 77, pp. 2856-pp. 2863, 1999.
[9] G. J. L. Wuite, R. J. Davenport, A. Rappaport, C. Bustamante, “An integrated laser trap/flow control video microscope for the study of single biomolecules,” Biophys. J., vol. 79, pp. 1155-pp.1167, 2000.
[10] S. B. Smith, L. Finzi, and C. Bustamate, “Direct mechanical measurements of the elasticity of single DNA molecules by using magnetic beads,” Science, vol. 258, p. 1122-pp.1126, 1992.
[11] B. Brown, and R. R. Browan, Optical tweezers: theory and current applications, American Laboratory, 2001.
[12] A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci. USA, vol. 94, pp. 4853-pp.4860, 1997.
[13] A. Ashkin, , J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett., vol. 11, pp. 286, 1986.
[14] S. Smith, Y. Cui and C. Bustamante, “Overstretching B-DNA: The elastic response of individual double-stranded and single-stranded DNA molecules,” Science, vol. 271, pp. 795-799, 1996
[15] Y. Cui, The elasticity of single DNA molecules and chromatin fibers determined by force measuring laser tweezers, PhD dissertation, University of Oregon , 1998, pp. 51.
[16] D. E. Smith, S. J. Tans, S. B. Smith, S. Grimes, D. L. Anderson and C. Bustamante, “The bacteriophage ψ29 portal motor can package DNA against a large internal force,” Nature, vol. 413, pp. 748-752, 2001.
[17] K. N. Chiang, C. A. Yuan, C. N. Han and C. Y. Chou, “Mechanical characteristic of ssDNA/dsDNA molecule under external loading,” Applied Physics Letters, vol. 88, pp. 023902-1-023902-3, 2006..
[18] M. C. Morias, K. H. Choi, J. S. Koti, P. R. Chipman, D. L. Anderison and M. G. Rossmann, “Conservation of capsid structure in tailed dsDNA bacteriophage: the pseudoatomic structure of phi29,” Molecular Cell, vol. 18, pp.149-159, 2005.
[19] N. V. Hud and K. H. Downing, “Cryoelectron microscopy of λ phage DNA condensates in vitreous ice: the fine structure of DNA toroids,” PNAS, vol. 98, pp. 14925-14930, 2001.
[20] P. K. Purohit, J. Kondev and R. Phillips, “Mechanics of DNA packaging in viruses,” PNAS, vol. 100, pp. 3173-3178, 2003.
[21] I. L. Ivanovska, P. J. de Pablo, B. Lbarra, G. Sgalari, F. C. Mackintosh, J. L. Carrsacosa, C. F. Schmidt and G. J. L. Wuite, “Bacteriophage capsids: tough nanoshells with complex elastic properties,” PNAS, vol. 101, pp. 7600-7605, 2004.
[22] Y. R. Jeng, and C. M. Tan, “Computer simulation of tension experiments of a thin film using an atomic model,” Physical Review B, vol. 65, pp.174107-1-174107-7, 2002.
[23] S. Izumi, T. Kawakami, and S. Sakai, “Study of combined FEM-MD method,” JSME International Journal A, vol. 44, pp. 152-159, 2001.
[24] B. Liu, H. Jiang, Y. Huang, S. Qu, M. F. Yu, and K. C. Hwang, “Atomic-scale finite element method in multiscale computation with applications to carbon nanotubes,” Physical Review B, vol. 72, pp.035435-1-035435-8, 2005.
[25] G. M. Odegard, T. A. Gates, L. M. Nicholson, and K. E. Wise, “Equivalent-Continuum modeling of nano-structure materials,” Composites Science and Technology, vol. 62, no. 14, pp. 1869-1880, 2002.
[26] H. C. Cheng, K. N. Chiang, and M. H. Lee, “An alternative local/global finite element approach or ball grid array type packages,” ASME Journal of Electronic Packaging, vol. 120, pp. 129-134, 1998.
[27] K, N. Chiang and W. L. Chen, “Package reflow shapes prediction for solder mask defined ball grid array,” ASME Journal of Electronic Packaging, vol. 120, pp. 175-178, 1998.
[28] C. T. Lin, and K. N. Chiang, “Investigation of nano-scale single crystal silicon using atomistic-continuum mechanics with Stillinger-Weber potential function,” IEEE conference on Emerging Technologies-Nanoelectronics: NanoSingapore 2006, pp. 5-9, Singapore, 10-13 Jan. 2006.
[29] C. N. Han, C. Y. Chou, C. J. Wu, and K. N. Chiang, “Investigation of ssDNA Backbone Molecule Mechanical Behavior Using Atomistic-Continuum Mechanics Method,” 2006 NSTI Nanotechnology Conference and Trade Show, 7-11 May, Boston, 2006.
[30] C. J. Wu, C. Y. Chou, C. N. Han, and K. N. Chiang, “Investigation of Carbon Nanotube Mechanical Properties Using The Atomistic-Continuum Mechanics Method,” 2006 NSTI Nanotechnology Conference and Trade Show, 7-11 May, Boston, 2006.
[31] A. Y. T. Leung, X. Guo, X. Q. He, and S. Kitipornchai, “A continuum model for zigzag single-wall carbon nanotubes,” Applied Physics Letters, vol. 86, pp. 083110-1-083110-3, 2005.
[32] P. Zhang, Y. Huang, P. H. Geubelle, P. A. Klein, and K. C. Hwang, ”The elastic modulus of single-wall carbon nanotubes: a continuum analysis incorporating interatomic potentials,” International Journal of Solid and Structure, vol. 39, pp. 3893-3906, 2002.
[33] Y. L. Liu, “On equivalent continuum mechanics analysis for the material properties of carbon nanotube,” Master Thesis, National Tsing Hua University, Taiwan, 2004.
[34] F. Treves, Basic linear partial differential equations, Academic press, 1975.
[35] W. Bangerth and R. Rannacher, Adaptive finite element methods for differential equations, Birkhauser Verlag, Boston, 2003.
[36] R. Serway, C. Moses and C. Moyer, Modern physics, 2nd ed., Saunders College Publishing, Philadephia, USA, 1997.
[37] M. Fayer, Elements of quantum mechanics, 1st ed., Oxford University Press, New York, USA, 2001.
[38] D. Griffiths, Introduction to quantum mechanics, 2nd ed., Pearson Education International, London, UK, 2005.
[39] P. Dirac, The principles of quantum mechanics, 4th ed., Oxford Science Publications, 1958.
[40] R. Eisberg and R. Resnick, Quantum physics of atoms, molecules, solids nuclei and particles, 2nd ed., John Wiley and Sons, 1985.
[41] W. Heisenberg, The physical principles of quantum theory, 1st ed., Dover , New York, 1930.
[42] R. Feynman, R. Leighton and M. Sands, The Feynman lecture on physics, Vol III, Addison-Wesley Publishing Company, 1963.
[43] I. Tinoco, Jr., K. Sauer, J. Wang and J. Pugisi, Physical chemistry principles and applications in biological sciences, 4th ed., Prentice Hall, New Jersey, 2002.
[44] D. Bensimon, A. J. Simon, V. Croquette and A. Bensimon, “Stretching DNA with a receding meniscus: experiments and models,” Phys. Rev. Lett., vol. 74, no 23, pp. 4754-4757, 1995.
[45] G. A. Jeffery, W. Saneger, Hydrogen Bonding in Biological Structures, 1st ed., Springer-Verlag, Germay, 1994
[46] J. Conery, W. Peticolas and T. Rush III, “A parallel Algorithm for Calculating the potential energy in DNA,” Proc. of 28th Annual Hawaii International Conf. on System sciences, pp. 123-pp. 131, Hawaii, USA, 1995
[47] Y. Zhang, H. Zhou and Z. Ou-Yang, “Stretching single-stranded DNA: interplay of electrostatic, base-pairing, and base-pair stacking interactions,” Biophy. J., vol. 81, pp. 1133- 1143, 2001.
[48] H. Zhou, Y. Zhang and Z. Ou-Yang, “Elastic property of single double-stranded DNA molecules: theoretical study and comparison with experiments,” Phys. Rev. E, vol. 62, no. 1, pp. 1045-1058, 2000.
[49] H. R. Tabar, and G. A. Mansoori, “Interatomic potential models for nanostructures,” Encyclopedia of Nanoscience and Nanotechnology, vol.x, pp.1-17, 2003.
[50] J. E. Lennard-Jones, “Forces between atoms and ions,” Proc. R. Soc. (Lond.) A, vol. 109, pp. 584, 1925.
[51] P. M. Morse, “Diatomic molecules according to the wave mechanics II vibrational levels,” Physical Review, vol. 34, pp.57-64, 1929.
[52] L. A. Girifalco and V. G. Weizer, “Application of the Morse Potential Function to Cubic Metals,” Physical Review, Vol.114, No. 3, pp.687-690, 1959.
[53] T. Namazu, Y. Isono, and T. Tanaka, “Evaluation of size effect on mechanical properties of single crystal silicon by nanoscale bending test using AFM,” Journal of Microelectromechanical Systems, vol. 9, pp. 450-459, 2000.
[54] W. Suwito, M. L. Dunn, S. J. Cunningham, and D. T. Read, “Elastic moduli, strength, and fracture initiation at sharp notches in etched single crystal silicon microstructures,” Journal Applied Physics, vol. 85, pp. 3519-3534, 1999
[55] C. A. Yuan, G. Q. Zhang, C. N. Han, K. N. Chiang and Yujia Cui “Numerical simulation on the mechanical characteristics of double-stranded DNA under stretching and lateral unzipping,” Journal Applied Physics, vol. 101, pp.074702-1-074702-4, 2007
[56] C. Benham, “Onset of writhing in circular elastic polymers,” Phys. Rev. A, 39, 2582-2586, 1989.
[57] J. F. Marko and E. D. Siggia, “Statistical mechanics of supercoiled DNA,” Phys. Rev. E, 52, 2912-2938, 1995.
[58] A. K. Rappe, C. J. Casewit, K. S. Colwell, W. A. Goddard Ⅲ, and W. M. Skiff, “UFF, a full periodic table force field for molecular mechanics and molecular dynamics simulations,” J. Am. Chem. Soc., vol. 114, pp. 10024-10035, 1992.
[59] W. D. Cornell, P. Cieplak, C. I. Bayly, I. R. Gould, K. M. Merz, Jr. D. M. Ferguson, D. C. Spellmeyer, T. Fox, J. W. Caldwell, and P. A. Kollman, “A second generation force filed for the simulation of proteins, nucleic, acids, and organic molecules,” Journal of American Chemical Society, vol. 117, pp. 5179-5197, 1995.
[60] M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, S. M. Block, “Force and velocity measured for single molecules of RNA polymerase,” Science, vol. 282, pp.902-907, 1998.
[61] A. A. Simpson, Y. Z. Tao, P. G. Leiman, M. O. Badasso, Y. N. He, P. J. Jardine, N. H. Olson, M. C. Morais, S. Grimes, L. D. Anderson, “Structure of the bacteriphage Phi-29 DNA packaging motor,” Nature, vol. 408, pp. 745-750, 2000.
[62] A. Guasch, J. Pous, B. Ibarra, X. F. Gomis-Ruth, J. M. Valpuesta, N. Sousa, J. L. Carrascosa and M.Coll, “Architecture of a DNA Translocating Machine: The High-resolution Structure of the Bacteriophage Phi-29 Connector Particle,” J. Mol. Biol., vol. 315, pp. 663-676, 2002.
[63] Furio Ercolessi, “A molecular dynamics primer,” website: http://www.sissa.it/furio/.
[64] C. A .Yuan, C. N. Han and K. N. Chiang, “Investigation of the Sequence- Dependent dsDNA Mechanical Behavior using Clustered Atomistic-Continuum Method,” NSTI Nanotechnology Conference, Anaheim, California, U.S.A., May 8-12, 2005.
[65] C. A. Yuan, C. N. Han and K. N. Chiang, “Atomistic to Continuum Mechanical Investigation of ssDNA and dsDNA using Transient Finite Element Method,” Inter-Pacific Workshop on Nanoscience and Nanotechnology, City University of Hong Kong, Hong Kong SAR., Nov. 22-Nov. 24, 2004.
[66] C. A. Yuan and K. N. Chiang, “Investigation of dsDNA stretching meso-mechanics using finite Element Method,” 2004 Nanotechnology Conference, March 7-11, 2004, Boston, Massachusetts, U.S.A.
[67] C. A. Yuan and K. N. Chiang, “Numerical Simulation for B-S Structural Transition of Nicked dsDNA Using Enriched Finite Element Method,” Taiwan International Conference on Nano Science and Technology, HsinChu, Taiwan, June 30-July 3, 2004.
[68] C. A. Yuan and K. N. Chiang, “Investigation of dsDNA stretching meso-mechanics using LS-DYNA,” FEA information worldwide news(e-journal), May, 2004.
[69] J. Marko and S. Cocco, “The micromechanics of DNA,” Physics World, pp. 37-41, March, 2003
[70] G. Bao and S. Suresh, “Cell and molecular mechanics of biological materials,” Nature Materials, vol. 2, pp. 715-725, Nov, 2003.
[71] http://www.ocms.ox.ac.uk/ocms/EMBOworkshop.html
[72] H. M. Berman, W. K. Olson, D. L. Beveridge, J. Westbrook, A. Gelbin, T. Demeny, S. H. Hsieh, A. R. Srinivasan, and B. Schneider, “The Nucleic Acid Database: A Comprehensive Relational Database of Three-Dimensional Structures of Nucleic Acids,” Biophys. J., 63, 751-759,1992.
[73] W. D. Cornell, P. Cieplak, C. I. Bayly, I. R. Gould, K. M. Merz, Jr. D. M. Ferguson, D. C. Spellmeyer, T. Fox, J. W. Caldwell, and P. A. Kollman, “A second generation force filed for the simulation of proteins, nucleic, acids, and organic molecules,” Journal of American Chemical Society, vol. 117, pp. 5179-5197, 1995.
[74] N. J. Dimmock, A. J. Easton and K. N. Leppard, “Introduction to Modern Virology,” Fifth edition, Blackwell Science, 2001.
[75] J. M. Valpuesta, J. J. Fernandez, J. M. Carazo and J. L. Carrascosa, "The three-dimensional structure of a DNA translocating machine at 10 Å resolution," Struct. Fold. Des. 7, pp. 289-296, 1999.
[76] M. Rief, H. Clausen-Schaumann and H. Gaub, “Sequence-dependent mechanics of single DNA molecules,” Nature, vol. 6, no. 4, 1999.
[77] S. Kanamaru, P. G. Leiman, V. A. Kostyuchenko, P. R. Chipman, V. V. Mesyanzhinov, F. Arisaka, and M. G. Rossmann, “Structure of the cell-puncturing device of bacteriophage T4,” Nature, vol. 415, pp. 553-557, 2002.
[78] W. R. Wikoff and J. E. Johnson, “Virus assembly: Imaging a molecular machine,” Curr. Biol., vol. 9, pp. 296-300, 1999.
[79] M. E. Cerritelli, N. Cheng, A. H. Rosenberg, C. E. McPherson, F. P. Booy and A. C. Steven, “Encapsidated Conformation of Bacteriophage T7 DNA,” Cell, vol. 91, pp. 271-280, 1997.
[80] N. H. Olson, M. Gingery, F. A. Eiserling and T. S. Baker, “The Structure of Isometric Capsids of Bacteriophage T4,” Virology, vol. 279, pp. 385-391, 2001.
[81] J. Kindt, S. Tzlil, A. Ben-Shaul and W. M. Gelbart, “DNA packaging and ejection forces in bacteriophage,” PNAS, vol. 98, pp. 13671-13674, 2001.
[82] W. C. Earnshaw, B. M. Honda, R. A. Laskey and J. O. Thomas, “Assembly of nucleosomes: the reaction involving X. laevis nucleoplasmin,” Cell, Vol. 21, Issue 2, pp. 373-383, 1980.
[83] Y. Tao, N. H. Olson, W. Xu, D. L. Anderson, M. G. Rossmann and T. S. Baker, “Assembly of a Tailed Bacterial Virus and Its Genome Release Studied in Three Dimensions,” Cell, Vol. 95, Issue 3, pp. 431-437, 1998.
[84] B. Ibarra, J. R. Castón , O. Llorca , M. Valle , J. M. Valpuesta and J. L. Carrascosa, “Topology of the components of the DNA packaging machinery in the phage phi29 prohead,” J. Mol. Biol., Vol. 298, Issue 5, pp. 807-815, 2000.
[85] G. M. Rossmann, R. Bernal, and S. V. Pletnev, “Combining Electron Microscopic with X-Ray Crystallographic Structures,” J. Struct. Biol., Vol. 136, Issue 3, pp. 190-200, 2001.
[86] A. Guasch, J. Pous, B. Ibarra, F. X. Gomis-Rüth, J. M. Valpuesta, N. Sousa, J. L. Carrascosa and M. Coll, “Detailed architecture of a DNA translocating machine: the high-resolution structure of the bacteriophage phi29 connector particle,” J. Mol. Biol., Vol. 315, Issue 4, pp. 663-676, 2001.
[87] J. E. Johnson, J. A. Speir, “Quasi-equivalent viruses: a paradigm for protein assemblies,” J. Mol. Biol., Vol. 269, Issue 5, pp. 665-675, 1997.
[88] A. C. Ugural, Stresses in plates and shells, 2nd edition, McGRAW-Hill international editions, 1999.
[89] T. Belytschko, W. K. Liu, and B. Moran, Nonlinear finite elements for continua and structures, 1st ed., John Wiley & Sons, NY, USA, 2000.
[90] J. O. Hallquist, LS-DYNA theory manual, Livermore Software Technology Corp., Livermore, CA, USA, 1998.
[91] C. T. Lin, Investigation of nanoscale single crystal group IV-A mechanical properties using atomistic-continuum mechanics (ACM), PhD dissertation, of National Tsing Hua University, 2006.
[92] T. Belytschko, H. R. Yen, and R. Muller, “Mixed methods for time integration,” Computer Methods in Applied Mechanics and Engineering, 17, pp. 259-275, 1979.
[93] T. Belytschko, “Partitioned and adaptive algorithms for explicit time integration,” Nonlinear Finite Element Analysis in Structural Mechanics, pp. 572-584, 1980.
[94] G. M. Hulbert, and T. J. R. Hughes, “Numerical evaluation and comparison of subcycling algorithms for structural dynamics,” Technical Report, DNA-TR-88-8, 1988.
[95] T. Belytschko, and C. S. Tsay, “WHAMSE: A program for three-dimensional nonlinear structural dynamics,” Report EPRI NP-2250, Project 1065-3, Electric Power Research Institute, Palo Alto, CA. 1982.
[96] C. S. Chen, System study and construction of high-resolution optical tweezers system, Master thesis, University of National Tsing-Hua University, 1992.