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The external environment affects the structural form of biological system. Many biological systems are surrounded by cell solutions, such as DNA and bacteria. The solution will offer a viscous resistance as the biological system moves in the viscous fluid. How does the viscous resistance affect the stability of biological system and what mode will be selected after instability? In this paper, we establish a super-long elastic rod model which contains the viscous resistance to model this phenomenon. The stability and instability of the super-long elastic rod in the viscous fluid are studied. The dynamic equations of motion of the super-long elastic rod in viscous fluid are given based on the Kirchhoff dynamic analogy. Then a coordinate basis vector perturbation scheme is reviewed. According to the new perturbation method, we obtain the first order perturbation representation of super-long elastic rod dynamic equation in the viscous fluid, which is a group of the second order linear partial differential equations. The stability of the super-long elastic rod can be determined by analyzing the solutions of the second order linear partial differential equations. The results are applied to a twisted planar DNA ring. The stability criterion of the twisted planar DNA ring and its critical region are obtained. The results show that the viscous resistance has no effect on the stability of super-long elastic rod dynamics, but affects its instability. The mode selection and the influence of the viscous resistance on the instability of DNA ring are discussed. The amplitude of the elastic loop becomes smaller under the influence of the viscous resistance, and a bifurcation occurs. The mode number of instability of DNA loop becomes bigger with the increase of viscous resistance.
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Keywords:
- super-long elastic rod /
- perturbation method /
- viscous fluid /
- instability
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[2] Le Bret M 1978 Biopolymers 17 1939
[3] Travers A A, Thompson J M T 2004 Phil. Trans. R. Soc. Lond. A 362 1265
[4] Benham C J, Mielke S P 2005 Annu. Rev. Biomed. Eng. 7 21
[5] Shi Y M, Hearst J E 1994 J. Chem. Phys. 101 5186
[6] Zhou H J, Ouyang Z C 1999 J. Chem. Phys. 110 1247
[7] Xue Y, Liu Y Z, Chen L Q 2004 Chin. Phys. 13 794
[8] Wang P, Xue Y 2016 Nonlinear Dyn. 83 1815
[9] Liu Y Z 2006 Nonlinear Mechanics of Thin Elastic Rod-Theoretical Basis of Mechanical Model of DNA (Beijing: Tsinghua Press Springer) p85 (in Chinese) [刘延柱 2006 弹性细杆非线性力学-DNA力学模型的理论基础 (北京: 清华大学出版社 Springer) 第85页]
[10] Bustamante C, Bryant Z 2003 Nature 421 423
[11] Tobias I, Swigon D, Coleman B D 2000 Phys. Rev. E 61 747
[12] Manning R S, Bluman G B 2005 Proc. R. Soc. Lond. A 461 2423
[13] Liu Y Z, Zu J W 2004 Acta Mech. 164 29
[14] Liu Y Z, Sheng L W 2007 Acta Phys. Sin. 56 2305 (in Chinese) [刘延柱, 盛立伟 2007 56 2305]
[15] Xue Y, Liu Y Z 2009 Acta Phys. Sin. 58 6737 (in Chinese) [薛纭, 刘延柱 2009 58 6737]
[16] Xue Y, Chen L Q, Liu Y Z 2004 Acta Phys. Sin. 53 4029 (in Chinese) [薛纭, 陈立群, 刘延柱 2004 53 4029]
[17] Goriely A, Tabor M 1997 Physica D 105 20
[18] Goriely A, Tabor M 1996 Phys. Rev. Lett. 77 3537
[19] Moulton D E, Lessinnes T, Goriely A 2013 J. Mech. Phys. Solids 61 398
[20] Klapper I 1996 J. Comput. Phys. 125 325
[21] Goldstein R E, Powers T R, Wiggins C H 1998 Phys. Rev. Lett. 80 5232
[22] Wolgemuth C W, Powers T R, Goldstein R E 2000 Phys. Rev. Lett. 84 1623
[23] Liu Y Z, Sheng L W 2007 Chin. Phys. 16 0891
[24] Keller J B, Rubinow S I 1976 J. Fluid Mech. 75 705
[25] Manning R S, Maddocks J H, Kahn J D 1996 J. Chem. Phys. 105 5626
[26] Kehrbaum S 1997 Ph. D. Dissertation (Maryland: University of Maryland, College Park, USA)
[27] Hagerman P 1988 Rev. Biophys. Chem. 17 265
[28] Schlick T 1995 Curr. Opinion Struct. Biol. 5 245
[29] Zajac E E 1962 Trans. ASME. J. Appl. Mech. 29 136
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[1] Beham C J 1977 Proc. Natl. Acad. Sci. USA 74 2397
[2] Le Bret M 1978 Biopolymers 17 1939
[3] Travers A A, Thompson J M T 2004 Phil. Trans. R. Soc. Lond. A 362 1265
[4] Benham C J, Mielke S P 2005 Annu. Rev. Biomed. Eng. 7 21
[5] Shi Y M, Hearst J E 1994 J. Chem. Phys. 101 5186
[6] Zhou H J, Ouyang Z C 1999 J. Chem. Phys. 110 1247
[7] Xue Y, Liu Y Z, Chen L Q 2004 Chin. Phys. 13 794
[8] Wang P, Xue Y 2016 Nonlinear Dyn. 83 1815
[9] Liu Y Z 2006 Nonlinear Mechanics of Thin Elastic Rod-Theoretical Basis of Mechanical Model of DNA (Beijing: Tsinghua Press Springer) p85 (in Chinese) [刘延柱 2006 弹性细杆非线性力学-DNA力学模型的理论基础 (北京: 清华大学出版社 Springer) 第85页]
[10] Bustamante C, Bryant Z 2003 Nature 421 423
[11] Tobias I, Swigon D, Coleman B D 2000 Phys. Rev. E 61 747
[12] Manning R S, Bluman G B 2005 Proc. R. Soc. Lond. A 461 2423
[13] Liu Y Z, Zu J W 2004 Acta Mech. 164 29
[14] Liu Y Z, Sheng L W 2007 Acta Phys. Sin. 56 2305 (in Chinese) [刘延柱, 盛立伟 2007 56 2305]
[15] Xue Y, Liu Y Z 2009 Acta Phys. Sin. 58 6737 (in Chinese) [薛纭, 刘延柱 2009 58 6737]
[16] Xue Y, Chen L Q, Liu Y Z 2004 Acta Phys. Sin. 53 4029 (in Chinese) [薛纭, 陈立群, 刘延柱 2004 53 4029]
[17] Goriely A, Tabor M 1997 Physica D 105 20
[18] Goriely A, Tabor M 1996 Phys. Rev. Lett. 77 3537
[19] Moulton D E, Lessinnes T, Goriely A 2013 J. Mech. Phys. Solids 61 398
[20] Klapper I 1996 J. Comput. Phys. 125 325
[21] Goldstein R E, Powers T R, Wiggins C H 1998 Phys. Rev. Lett. 80 5232
[22] Wolgemuth C W, Powers T R, Goldstein R E 2000 Phys. Rev. Lett. 84 1623
[23] Liu Y Z, Sheng L W 2007 Chin. Phys. 16 0891
[24] Keller J B, Rubinow S I 1976 J. Fluid Mech. 75 705
[25] Manning R S, Maddocks J H, Kahn J D 1996 J. Chem. Phys. 105 5626
[26] Kehrbaum S 1997 Ph. D. Dissertation (Maryland: University of Maryland, College Park, USA)
[27] Hagerman P 1988 Rev. Biophys. Chem. 17 265
[28] Schlick T 1995 Curr. Opinion Struct. Biol. 5 245
[29] Zajac E E 1962 Trans. ASME. J. Appl. Mech. 29 136
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