Helfrich手征弹性理论的推广及高分子螺旋双稳态的理论解释
摘要
在高分子中，螺旋是一种很普遍的高分子组态。近来实验发现，一些高分子，
如细菌鞭毛丝，在离子溶液中能够形成双稳态螺旋，即左螺旋与右螺旋在相同的
条件下能够同时存在。对于这种现象的形成机制还没有一个令人满意的解释。
Helfrich提出一种四阶手征弹性链理论，并解释了高分子易形成螺旋而不是直
线。我们对这个问题从稳定性的角度进行了考察，结果支持了Helfrich的做法。由
于四阶理论不能解释螺旋双稳态的现象，我们将Helfrich理论进行推广到八阶，发
现五阶理论能够很好的解释螺旋双稳态的现象。
In Polymer, helix is a quite universal configuration Recent experiments show that
some polymers such as bacterial flagellar filaments can form bistable helices in ion
solutions, which means that left handed helix and right handed helix can coexist in the
same environment The mechanism of this phenomenon has not been explained
satisfactorily.
Helfrich proposed a fourth order chiral elastic string theory which explained the
preference of helix over the straight line, we examine the same problem by means of
stability and results support Helfrich's treatment .Because of the difliculty in interpretation
of bistable helices in fourth order theory, we generalized this theory to eighth order.
Fortunately, we found that the fifth order theory suffices to explain the existence of the
bistable helices.
如细菌鞭毛丝，在离子溶液中能够形成双稳态螺旋，即左螺旋与右螺旋在相同的
条件下能够同时存在。对于这种现象的形成机制还没有一个令人满意的解释。
Helfrich提出一种四阶手征弹性链理论，并解释了高分子易形成螺旋而不是直
线。我们对这个问题从稳定性的角度进行了考察，结果支持了Helfrich的做法。由
于四阶理论不能解释螺旋双稳态的现象，我们将Helfrich理论进行推广到八阶，发
现五阶理论能够很好的解释螺旋双稳态的现象。
In Polymer, helix is a quite universal configuration Recent experiments show that
some polymers such as bacterial flagellar filaments can form bistable helices in ion
solutions, which means that left handed helix and right handed helix can coexist in the
same environment The mechanism of this phenomenon has not been explained
satisfactorily.
Helfrich proposed a fourth order chiral elastic string theory which explained the
preference of helix over the straight line, we examine the same problem by means of
stability and results support Helfrich's treatment .Because of the difliculty in interpretation
of bistable helices in fourth order theory, we generalized this theory to eighth order.
Fortunately, we found that the fifth order theory suffices to explain the existence of the
bistable helices.
引文
[1] 程传煊，《生物物理化学》，科学技术文献出版社，北京，1998。
[2] M. V. Volkenstein, Physics and Biology , Academic Press, New York, 1982．
[3] R . J . Nossal and H . Lecar, Molecular and Cell Biophysics, Addison-Wesley Publishing Company, Redwood City, 1991．
[4] 唐有祺，《生命的化学》，湖南科技出版社，长沙，1999。
[5] W. Helfrich, Langmuir, 7 (1991) ,567．
[6] O.Giraldo, S.L.Brock, etal, Nature , vol 405, 4 (2000) ,38．
[7] V.Berl, I.Huc, R.G.Khoury, Nature, vol 407, 12 (2000) ,720．
[8] J. D. Watson and F. H .C. Crick, Nature, 171 (1953) ,737．
[9] F . H .C. Crick and J . D . Watson, Proc. Roy.Soc. (London), Ser . A, 80 (1954) ,223．
[10] J .H .Fuhrhop , P . Schnieder , J . Rosenberg and E . Bookema, J. Am . Chem.Soc. 109 (1987) ,3387．
[11] P.Bual and S.Fujita,J. Chem.Phys. , 50(1969) 3137．
[12] J .H .Fuhrhop , C . Demoulin, J. Rosenberg and C . Bottcher ,J.Am. Chem. Soc.,112(1990) ,2827．
[13] 罗辽复，科学，51卷3期(1998) ，22。
[14] H. Hotani ,J. Mol. Biol. ,156(1982) ,791．
[15] N.H.Mendelson,Proc.Natl.Acad.Sci. U.S.A. ,73(1976) ,1740．
[16] M.J.Tilby, Nature(London),266( 1977) ,450．
[17] S.Wolfl,B.Witting, and A.Rich, Biochim.Biophys.Acta, 1264 (1995) , 294．
[18] V.Muller et al ,Proc.Natl.AcadSci. U.S.A. ,93 (1996) ,780．
[19] A.Rich and A.Herbert, J.Biol.Chem, 271(1996) , 11595．
[20] A.Rich, Gene, 135(1993) , 99．
[21] A.Wang et al, Science ,211(1981) , 171．
[22] P.Nelson,Proc.Natl.Acad.Sci. U.S.A. ,96(1999) , 14342．
[23] I.Yamashita , K.Hasegawa, H.Suzuki , F.Vonderviszt et al, Nature Struct. Biol. ,5(1998) ,125．
[24] A.E.Love, A Treatise on the Mathematical Theory of Elasticity, 4th ed, Cambridge University Press,London, UK, 1927．
[25] L.D.Landau and E.M.Lifshitz, Theory of Elasticity, Pergamon Press, New York, 1959．
[26] J.Langer and D.A.Singer, SLAM Review, 38(1996) ,605 .
[27] 梅向明，黄敬之，《微分几何》，高等教育出版社，北京，1988。
[28] M.P.do Carmo , Differential Geometry of Curves and Surface , Printice Hall,Englewood, Cliffs, 1976．
[29] Zhao Wei ,Zhou Haijun and Ou-Yang Zhong-Can, Phys.Rew.E, 58(1998) ,8040．
[30] V.S.Kulkarni, J.M.Boggs and R.E.Brown, Biophys. J. ,77(1999) , 319．
[31] M.Nizette,and A.Goriely,Phys. Rew. E, 61(1999) ,319．
[32] D.D.Archibald ,and S.Mann, Chem.Phys.Lipids, 69(1994) ,51．
[33] J.H.Fuhrhop, and W.Helfrich ,Chem. Rew. ,93(1993) , 1565．
[34] B.Hu, Q.H.Liu, X.Wang and O.YZhong-Can, (unpublished).
[35] 徐建平，桂子鹏，《变分方法》，同济大学出版社，上海，1999。
[36] 齐卫宏，刘全慧，湖南大学学报(自然版)(即将发表)。
[37] Weihong Qi and Quanhui Liu, Chinese Physics letter (submitted).
[38] T.A.Ivey and D.A.Singer, Proc. London Mat. Soc., 79(1999) , 429．
[39] A.Goriely and M .Tabor Physica D,105(1997) ,45 ;Physica D, 105(1997) ,20．
[40] A.Goriely and M.Tabor, Proc.Roy.Soc. ,453(1997) , 2583 .
[41] A.Goriely and M.Tabor, Proc.Roy.Soc. ,455(1998) ,3183 .
[42] J.lega and A.Goriely, (unpublished work).
[43] E.H.Dill, Arch. Hist. Exact. Sci., 44(1992) , 2．
[44] B.D.Coleman, E.h.Dill, M.Lembo et al, Arch. Rational Mech. Anal. ,121(1993) , 339．
[45] R.E.Goldstein, T.R.Powers, and C.H.Wiggins, Phys.Rew.Lett., 80(1998) , 5232．
[46] F.Juicher, Phys.Rew.E, 60(1999) ,7239．
[47] B.Fain and J.Rudnick, Phys.Rew. E, 60(1999) , 7239．
[48] I.Tobias, D.Swigon ,and B.D.Coleman ,Phys.Rew.E, 61(2000) ,747．
[49] O.Kratky and G.Porod, Rec. Trav. Chim., 68(1949) , 1106．
[50] P.Bugl and S.Fujita, J. Chem.Phys., 50 (1969) , 3173 .
[51] B.Pfannemuller and W.Welte ,Chem.Phys.Lipids, 37(1985) ,227．
[52] N.Nakishima, S.Asakuma and T.Kunitake J.Am.Chem.Soc. ,107(1985) ,509．
[53] W.Helfrich and J.Prost,Phys.Rev. A, 38(1988) ,3065．
[54] Ou-Yang Zhong-Can and Liu Ji-Xin ,Phys.Rev.Lett. 65(1990) , 1679．
[55] Ou-Yang Zhong-Can and Liu Ji-Xin ,Phys.Rev.A ,43(1991) ,6826．
[56] J.V.Selinger and J.M.Schnur ,Phys.Rev.Lett. ,71(1993) ,4091．
[57] J.H.Fuhrhop and W.Helfrich ,Chem.Rev, 93(1993) ,1565．
[58] Ou-Yang Zhong-Can and W.Helfrich, (unpublished work).
[59] 赵伟，从四阶手征弹性模型的观点研究DNA微坏的Kink现象(学位论文)， 中国科学院理论物理研究所，北京，1998。
[60] J.Langer and D.A.Singer, J.Diff.Geom., 20(1984) ,1．
[61] Quan-hui Liu ,(unpublished work).
[2] M. V. Volkenstein, Physics and Biology , Academic Press, New York, 1982．
[3] R . J . Nossal and H . Lecar, Molecular and Cell Biophysics, Addison-Wesley Publishing Company, Redwood City, 1991．
[4] 唐有祺，《生命的化学》，湖南科技出版社，长沙，1999。
[5] W. Helfrich, Langmuir, 7 (1991) ,567．
[6] O.Giraldo, S.L.Brock, etal, Nature , vol 405, 4 (2000) ,38．
[7] V.Berl, I.Huc, R.G.Khoury, Nature, vol 407, 12 (2000) ,720．
[8] J. D. Watson and F. H .C. Crick, Nature, 171 (1953) ,737．
[9] F . H .C. Crick and J . D . Watson, Proc. Roy.Soc. (London), Ser . A, 80 (1954) ,223．
[10] J .H .Fuhrhop , P . Schnieder , J . Rosenberg and E . Bookema, J. Am . Chem.Soc. 109 (1987) ,3387．
[11] P.Bual and S.Fujita,J. Chem.Phys. , 50(1969) 3137．
[12] J .H .Fuhrhop , C . Demoulin, J. Rosenberg and C . Bottcher ,J.Am. Chem. Soc.,112(1990) ,2827．
[13] 罗辽复，科学，51卷3期(1998) ，22。
[14] H. Hotani ,J. Mol. Biol. ,156(1982) ,791．
[15] N.H.Mendelson,Proc.Natl.Acad.Sci. U.S.A. ,73(1976) ,1740．
[16] M.J.Tilby, Nature(London),266( 1977) ,450．
[17] S.Wolfl,B.Witting, and A.Rich, Biochim.Biophys.Acta, 1264 (1995) , 294．
[18] V.Muller et al ,Proc.Natl.AcadSci. U.S.A. ,93 (1996) ,780．
[19] A.Rich and A.Herbert, J.Biol.Chem, 271(1996) , 11595．
[20] A.Rich, Gene, 135(1993) , 99．
[21] A.Wang et al, Science ,211(1981) , 171．
[22] P.Nelson,Proc.Natl.Acad.Sci. U.S.A. ,96(1999) , 14342．
[23] I.Yamashita , K.Hasegawa, H.Suzuki , F.Vonderviszt et al, Nature Struct. Biol. ,5(1998) ,125．
[24] A.E.Love, A Treatise on the Mathematical Theory of Elasticity, 4th ed, Cambridge University Press,London, UK, 1927．
[25] L.D.Landau and E.M.Lifshitz, Theory of Elasticity, Pergamon Press, New York, 1959．
[26] J.Langer and D.A.Singer, SLAM Review, 38(1996) ,605 .
[27] 梅向明，黄敬之，《微分几何》，高等教育出版社，北京，1988。
[28] M.P.do Carmo , Differential Geometry of Curves and Surface , Printice Hall,Englewood, Cliffs, 1976．
[29] Zhao Wei ,Zhou Haijun and Ou-Yang Zhong-Can, Phys.Rew.E, 58(1998) ,8040．
[30] V.S.Kulkarni, J.M.Boggs and R.E.Brown, Biophys. J. ,77(1999) , 319．
[31] M.Nizette,and A.Goriely,Phys. Rew. E, 61(1999) ,319．
[32] D.D.Archibald ,and S.Mann, Chem.Phys.Lipids, 69(1994) ,51．
[33] J.H.Fuhrhop, and W.Helfrich ,Chem. Rew. ,93(1993) , 1565．
[34] B.Hu, Q.H.Liu, X.Wang and O.YZhong-Can, (unpublished).
[35] 徐建平，桂子鹏，《变分方法》，同济大学出版社，上海，1999。
[36] 齐卫宏，刘全慧，湖南大学学报(自然版)(即将发表)。
[37] Weihong Qi and Quanhui Liu, Chinese Physics letter (submitted).
[38] T.A.Ivey and D.A.Singer, Proc. London Mat. Soc., 79(1999) , 429．
[39] A.Goriely and M .Tabor Physica D,105(1997) ,45 ;Physica D, 105(1997) ,20．
[40] A.Goriely and M.Tabor, Proc.Roy.Soc. ,453(1997) , 2583 .
[41] A.Goriely and M.Tabor, Proc.Roy.Soc. ,455(1998) ,3183 .
[42] J.lega and A.Goriely, (unpublished work).
[43] E.H.Dill, Arch. Hist. Exact. Sci., 44(1992) , 2．
[44] B.D.Coleman, E.h.Dill, M.Lembo et al, Arch. Rational Mech. Anal. ,121(1993) , 339．
[45] R.E.Goldstein, T.R.Powers, and C.H.Wiggins, Phys.Rew.Lett., 80(1998) , 5232．
[46] F.Juicher, Phys.Rew.E, 60(1999) ,7239．
[47] B.Fain and J.Rudnick, Phys.Rew. E, 60(1999) , 7239．
[48] I.Tobias, D.Swigon ,and B.D.Coleman ,Phys.Rew.E, 61(2000) ,747．
[49] O.Kratky and G.Porod, Rec. Trav. Chim., 68(1949) , 1106．
[50] P.Bugl and S.Fujita, J. Chem.Phys., 50 (1969) , 3173 .
[51] B.Pfannemuller and W.Welte ,Chem.Phys.Lipids, 37(1985) ,227．
[52] N.Nakishima, S.Asakuma and T.Kunitake J.Am.Chem.Soc. ,107(1985) ,509．
[53] W.Helfrich and J.Prost,Phys.Rev. A, 38(1988) ,3065．
[54] Ou-Yang Zhong-Can and Liu Ji-Xin ,Phys.Rev.Lett. 65(1990) , 1679．
[55] Ou-Yang Zhong-Can and Liu Ji-Xin ,Phys.Rev.A ,43(1991) ,6826．
[56] J.V.Selinger and J.M.Schnur ,Phys.Rev.Lett. ,71(1993) ,4091．
[57] J.H.Fuhrhop and W.Helfrich ,Chem.Rev, 93(1993) ,1565．
[58] Ou-Yang Zhong-Can and W.Helfrich, (unpublished work).
[59] 赵伟，从四阶手征弹性模型的观点研究DNA微坏的Kink现象(学位论文)， 中国科学院理论物理研究所，北京，1998。
[60] J.Langer and D.A.Singer, J.Diff.Geom., 20(1984) ,1．
[61] Quan-hui Liu ,(unpublished work).