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谐振势阱中的布朗运动——磁镊实验与模拟

谐振势阱中的布朗运动——磁镊实验与模拟 ÔnÆ� Acta Phys. Sin. Vol. 61, No. 17 (2012) 170503 ��³³³²²²¥¥¥��ÙÙÙKKK$$$ÄÄÄ ^^^ppp¢¢¢��[[[* .]y (§²ÆÔn>f&Eó§Æ�,§² 325035 ) ( 2012c 1� 29FÂ�; 2012c 2� 16FÂ�?Uv ) l^p¢�Ú�[�ÝïÄ ?u��³²¥�ÙK$Ä.|^¢�Ú�[�(J�y nØ.,�ÏLnØ ¢��éì,é^p¢�¥ DNA©f�±ÈÝ...

ÔnÆ� Acta Phys. Sin. Vol. 61, No. 17 (2012) 170503 ��³³³²²²¥¥¥��ÙÙÙKKK$$$ÄÄÄ ^^^ppp¢¢¢��[[[* .]y (§²ÆÔn>f&Eó§Æ�,§² 325035 ) ( 2012c 1� 29FÂ�; 2012c 2� 16FÂ�?Uv ) l^p¢�Ú�[�ÝïÄ ?u��³²¥�ÙK$Ä.|^¢�Ú�[�(J�y nØ.,�ÏLnØ ¢��éì,é^p¢�¥ DNA©f�±ÈÝ�é�¥ £©Ù�K,±9^p¢�¥�ÿåØ� '©Û.©ÛÑ:±ÈÝ�Czé÷ DNAóþ�ÙK$ÄK;�� å^eå�ÿþ¬Ñy� Ø�. 'c: ÙK$Ä,��³²,^p, DNA PACS: 05.40.Jc, 87.80.Ek, 87.80.Nj, 87.15.A 1 Ú ó ÙK$Ä´]23N½íN¥�þ ?��âf�Ã5K$Ä, u 1827 c=I ÔÆ[NË ¢ÙK¤uy. 1905c,OÏd"JÑ ÙK$Ä�nØ. �d#©¿ÅdÄ (Smolu- chowski), J� (Langevin)uL ¦�nØ. {IÔnÆ[�= (Perrin)u 1908c¢�y¢ OÏd"�nØ,Ï y¢ ©fÚ©fÃ5K 9$Ä�*3. TyÄu©f$ÄØn ): âf±�þ�0©fØä/EÂâf, -EåÅ,þ�åØU��-, ù� �) ÀÞáå¦âfÃ5$Ä [1¡3]. î�d" (Orstein),¿ÔO (Uhlenbeck)Ú� ²�n��üØ©éÙ K$ÄnØ �o(ÚuÐ [4;5]. ùüØ ©ïÄ ��³²¥�ÙK$Ä, �� n Ø(J.Cc5ù«a.�ÙK$Ä31p [6],^ p [7¡11] �¢�Ãã¥�� 2�A^. äN 5`, 1p�)�1²Ò´��³², ÙKâ f¤É�£Eå�'uâf l1²¥%�ål. ÿþ1²¥�ÙK$Ä±^u(½1²�§Ý Xê [6;12]. ^pKduë�3^5ÙKâfþ � DNA©f�å,@?u��³²¥. ÿþ ÙK$Ä^uÿþ pNþ?�å,ù3ü©f) ÔÔnïÄ¥´«~^{ [13;14]. Ï ,ù {ØïÄóä ,,¡�y��³² ¥�ÙK$ÄnØJø ýZ�¢��.. �©l^p¢�Ú�[�ÝïÄù«ÙK$ Ä.Äk|^¢�Ú�[�(J�ynØ.,�Ï LnØ¢��éì,é^p¥�ÙK$Äÿå {UÑy�Ø�, ±9¢�¥ DNA©f�±È Ý�é�¥� £©Ù�K'©Û,å ãÿÐÙK$ÄnØ3^pêâ©Û¥�A^,¿ ?Ú²(^pEâUÑy�¯KÚÛ5. 2 n Ø Ä?u F = ¡kx ³²¥�ÙK$Ä, Ù¥ k ��³²�§ÝXê, x l²ï *úôg,ÆÄ7 (1OÒ: Y4110357)ÚI[g,ÆÄ7 (1OÒ: 20934004)]Ï�K. y E-mail: syran@mails.gucas.ac.cn c° 2012 ¥¥¥IIIÔÔÔnnnÆÆÆ¬¬¬ Chinese Physical Society http://wulixb.iphy.ac.cn 170503-1 ÔnÆ� Acta Phys. Sin. Vol. 61, No. 17 (2012) 170503 � £. Ù £VÇ©Ù¼ê [4] P (x0; x; t) = 0@ k 2¼kBT (1¡ e ¡2k¢t f ) 1A1=2 £ exp 8>>><>>>:¡ k ¢ µ x¡ x0 ¢ e¡ k¢t f ¶2 2kBT µ 1¡ e ¡2k¢t f ¶ 9>>>=>>>; ; (1) Ù¥ kB Å�[ù~ê, T 9åÆ§Ý, x0 Ð© , t m, f {¢Xê, éu¥Gâ f, kd÷d'X: f = 6¼´r, ùp ´ 6N� Å¢Xê, r ´�¥». XJÐ© x0 = 0, k ¢ t f À 1, K (1) ª{zmÃ'�© Ù¼ê P (x) = µ k 2¼kBT ¶1=2 exp µ ¡k ¢ x 2 2kBT ¶ : (2) 3 ¢ � ¢�3�wº (Nikon-TE2000U) þ? 1. �¬³S¿÷ pH 7.5 ��íÀ. |^)zÃãò» 1.4 ¹m �^^¥ë �þ ¸-DNA©f, DNA©f�,à�½u ý9þ,/¤Xã 1(a)¤«�(�. �¬³ýS C ÃÄöp¤�[^c, ÏLáÚ^^ ¥é DNA\.å. ±ÏLUC[^c� 5UC^å�. �¥�$ÄÏL � CCD±z ¦ 25v¤¢Dx�OÅÌÅþSC�ã æ8k, ¿¹�ø©Û.äN�¢�[!ë ©z [11]. é^¥ XY ²¡þ�Éå©ÛÙÉ� ^å F , DNA ©f��å T , Å¢{å Fv ±9Þ áå Fr �^ (ã 1(b)). �¥3 DNA©f�åe �$Ä±w¤´{Z{$Ä.Þáåu 4�¥ l²ï ±x, ^å F Ïd�)£E åÝ.Cq@ x�£Eå Fx = ¡F sin µ, ùp µ { l²ï �Ý, F Ru x �^å. XJ µ ��, sin µ ¼ µ = ±x= hLi, u´ k Fx ¼ ¡F±x= hLi = ¡kx±x,Ù¥ hLi DNA� "àåÝ=^¥�ý9�ål�²þ (Cq@ T{Z{�Ýð½), kx = F hLi ,´T��³² �§ÝXê. ù��¥ x�$Ä±w ��f$Ä,ÙUþ�u 1 2 kxh±x2i, âUþþ ©½n [15;16],�¥3 xgdÝþ�Uþ 1 2 kBT , ù� 1 2 kBT = 1 2 kxh±x2i = 12 F hLih±x 2i; �� F = kBT hLi h±x2i ; (3) Ù¥ ±x2 ® ^¥ÙK$Ä £ x �þ�. (3) ª=^p|^ÙK$Äÿå��n. ¢�¥^¥ �$Ä;,|^Äu¯FpC�g' {§S©Ûæ8�Àª��. ù� ±xÚ hLi± ÏLã©Û��¿ÏLþª¦� å�. 3 0.110 pNS, DNA�åÆ5±^ UÁGó (Worm-Like-Chain, WLC)�.5£ã [17]. âT�. F ¢ Lp kBT = L L0 + 1 4(1¡ L=L0)2 ¡ 1 4 ; (4) Ù¥ L0´ DNA��ÓÐ�Ý,�¢�¤^ ¸- DNA� L0 16.4 ¹m, F å, L´ DNA�" àåÝ, Lp DNA �±ÈÝ, ´p©fÔn Æ¥^5L�p©ff5�ëê. 3)nßÝ e DNA�±ÈÝ� 50 nm,�©��[æ^ù ê. ã 1 (a)¢�«¿ã; (b)�¥�Éå©Û (Ù¥ Fv � �¥Ý, Fr �Å) ÷ DNAó� yþ^¥�$ÄÓ�À ��f/ª,Ù§ÝXê ky = @F @L = kBT 2L0Lp " 2 + µ 1¡ L L0 ¶¡3# : (5) 170503-2 ÔnÆ� Acta Phys. Sin. Vol. 61, No. 17 (2012) 170503 Tªí�A^ (4)ª. 4 � [ l¹��³²�J�§Ñu [4] d2x dt2 + f m dx dt + k m x = F (t) m ; (6) Ù¥ F (t) ´±6N©f-EÚå�ÅÞá å, m´ÙKâfþ, k ´³²�§ÝXê,{¢ Xê f = 6¼´r. du{¢å��ÝCzu) u t.5 = m f �A�mºÝS,éÙ?1� �3 10¡6 sþ?. ¢��ÿþmm 0.04 s� uTmm,Ïd (6)ª�.5 d2x dt2 ±� Ñ.±@3 ¢tmS,�¥� £d��³² E¤� £ ¡kx f ¢tÚÅÞáå F (t)E¤� Å1rÚå ±xrand(¢t)üÜ©�z. éu�ö,d ÙK$ÄnØ ±xrand(¢t)÷vpd©Ù P (±xrand;¢t) = r 1 2¼¾2 e¡ ±x2rand 2¾2 ; Ù¥IO� ¾ = p 2D ¢¢t, D´*ÑXê. du¢ �¥ý9�C^¥,¬^K*ÑXê� �, ÏdIé?1�� [18], 3 x (²1ý9 )Ú y (÷ DNA)ü©Ok Dx ¼ D0 µ 1¡ 9 16 r L ¶ ; (7) Dy ¼ D0 µ 1¡ 9 8 r L ¶ ; (8) Ù¥ D0 = kBT f , gdÙK$Ä�*ÑXê, r ^¥», L ^¥ålý9�Ý, @ ´ DNA�"àåÝ.��Ñ�[�§ ¢x = ¡kx f ¢t+ ±xrand(¢t): (9) �[ ±xrand(¢t)^IO� 1�pdÅ© Ù)¤¼ê¦± p 2Dx ¢¢t½ p 2Dy ¢¢t�� |ÅÚå. éu�¢�¥� xÚ y ,\ g�§ÝXê k�Lª,=��üþ� �[§¿?1�[,��ãmS�¥� £m�Cz. 5 (J?Ø 5.1 ¢¢¢��ÚÚÚ��[[[(((JJJnnnØØØ��éééììì ã 2(a) ´¢�¤ÿ� 1 pN åe^¥3 m 7 min S�I©Ùã. âTêâé x Ú y� £ÚO�Ñã 2(b)Ú (c)¤«�Î ã 2 (a) ¢�ÿþ��¥� ©Ùã (±²ï I":); (b) 1 pN å^e�¥ l²ï £ ±x� ÚO©Ù,¢L«nØ©Ù; (c) 1 pN å^e�¥ l² ï £ ±y �ÚO©Ù, ¢L«nØ©Ù. ¢�§Ý T 298 K,T§Ýe6N�Å¢Xê ´Cq@�uY�Å¢X ê 825.9 ¹Pa¢s;nØ©Ù¥ ky Lª�±ÈÝëê Lp �½ 45 nm 170503-3 ÔnÆ� Acta Phys. Sin. Vol. 61, No. 17 (2012) 170503 ã 3 (a)(d)OÅ�[�� 1 pN åe�¥� £ mCz,±9 £©ÙnØ©Ù�éì G©Ùã. òA�ëê\ kx Ú ky Lª, ,�òT§ÝXê\ (2) ª�ÑnØ©Ù ,=ã¥�¢.±w�¢�êâ�ÚO©Ù ÚnØ©ÙÄ�ÎÜ.�e5·ÁX�[T^p ¢�. â (9)ª,ò x��[mm ¢t� ½¢�æ8mm 0.04 s. y þKI Ä�Ù§ÝXê,¦ ¡kyx f ¢t�ýéØ ugdÙK$Ä�ÚåÌÝ,ò ¢t�½ 0.004 s. �[Iêâêþ 10500. ã 3(a)Ú (c)= �[�� 1 pN^e^¥3üþ�;, ã. ©OéÙIêâ?1ÚO¿nØ©Ùéì, Xã 3(b)Ú (d)¤«. �[êâ�©ÙnØ©Ù Ó�©ÎÜ. o(5`,^p¢��³²¥�Ù K$Ä¢��.,ÃØ´l¢�´�[�ÝÑy ² ��³²¥�ÙK$ÄnØ^uêâ©Û� 5. 5.2 ±±±ÈÈÈÝÝÝééé £££©©©ÙÙÙ��KKK (@nØ^u^p¢�êâ©Û�k�5�, ±ò|^5½þ©Û±ÈÝé©Ù�K. ±ÈÝ´£ãp©fåÆ5��Ô nëê, L��´p©f�f5. Ù�, p© fó�Ø´�ò. 3¢�¥,duM¸�ØÓ, DNA a�p©f�±ÈÝÏ~¬Cz. â¢�êâO(¦�±ÈÝÑp©f� f5M¸�'X. lcã¢��n± ÈÝ�UCéü�§ÝXêÑ¬kK. äN ó, Ó��å^e, ±ÈÝ�UC ¬E¤ DNA©f"àåÝ�Cz,l K kx ��. 3 y , l (5) ª±wÑÙ´ ky �'.ã 4(a)Ú (b)Ñ ð½ 1 pN å^e,±ÈÝ�Czéü £©Ù� Kª³. 3 x,±ÈÝl 50 nmü� 25 nm ,©Ù¸l 1.66 ¹m¡1 O\� 1.73 ¹m¡1,O '~ 4.2%. ±wÑ,¦+±ÈÝCzé x �ÙK$ÄkK, �¿Ø²w. �, ± ÈÝ±ÈÝl 50 nmü� 25 nm,y � ©Ù¸Kl 5.54 ¹m¡1 ~�� 4.54 ¹m¡1, eü '~ 18%. dd±�Ñ(Ø:±ÈÝ�Cz é÷ DNAó� y �ÙK$ÄK²w. Ó ,ùJø , «ÿþ±ÈÝ�{ |^nØ £VÇ©Ù¼ê[Ü¢�êâ� £ 170503-4 ÔnÆ� Acta Phys. Sin. Vol. 61, No. 17 (2012) 170503 ©Ù. ¿, dþã©Û[Ü÷ DNA ó �ÙK$Ä°(Ýp,ÏÙé±ÈÝCzA (¯. ã 4 (a) 1 pN å^e±ÈÝCzé x £©Ù¼ ê�K,â¸�,lþ�e±ÈÝg 25, 30, 35, 40, 45, 50 nm; (b)Ó� åe±ÈÝé y £©Ù¼ê �K,â¸�,lþ�e±ÈÝg 50, 45, 40, 35, 30, 25 nm 5.3 ^^^ppp¢¢¢��ÿÿÿåååØØØ��©©©ÛÛÛ å�ÿþ3^pEâ¥´�'�Ú, (3) ª�¤áÄ±9¢�¥Ù¦Ï�KÑ¬K ÿå�O(5. ©z [19]ÒQÑâå�� ØÓ,A�ÿåØ�ØÓ.ùpÁXé Ñ©Û. £� (3) ª�í�L§, 5¿�¦^ sin µ ¼ µ �Cq, TCq3�Ý �u 5± â¤á. ± 5± ÿå{��. ��Ý, ÁX¦� DNA �"àåÝ�uõ� , ÿåªØ2·^. d sin µ ¼ µ = ±x= hLi � h±x2i = hLi2 ¢ sin2 5±,\ (3)ª� F hLi = kBT sin2 5± : (10) þª(Ü (4) ª)� L=L0 = 0:427, Ò´`n Øþ DNA ©f�éÝ�uT, (3) ª^ uÿåØ2·^. éu�¢�� ¸-DNA, �. �Ý Lc = 7 ¹m. þãØ�©Û´lnØí� �CqcJÑu�Ñ. ¢SþØ�kÙ¦5 , X DNA"àåÝCá�, ý9^¥^ U¬�^¥$ÄØ2´üX�²Ä, ÍÜk =Ä (�� å^eT$Ä�³,�ÑØ O),��f�Uþ\þ=ÄU,Ä ²ÄU � (3)ªÏ �. äN/,±òâ�[�� ¥$Ä;,¦Ñ� ånØéì,ã 5 éØ� DNAéÝ�Czª³ã, Ù¥z êâ:d 15gÕá�Óëê�[�êâÚO� Ñ. ±wÑ, cã©Û, ÝCá, ÿå Ø�¬²wO\,$U¬pu 10%. o( ó, 3|^ÙK$Äÿþå��,5¿ÙUÑ y�Ø�,±Kêâ�O(�ä. ã 5 |^�[êâOÑ5� å�éØ�éÝ� Czª³ 6 ( Ø �©l^p¢�Ú�[�Ý�y ��³² ¥�ÙK$ÄnØ, ¿ònØÿÐA^u^p¢ ��êâ©Û.¢�Ú�[�(JL²?u��³ ²¥��¥� £÷vpd©Ù,nØýÿ�© Ù¼êÎÜûÐ.ÏLnØ¢��éì, é¢� ¥ DNA©f�±ÈÝ�é�¥3 xÚ y þ� £©Ù�K,±9^p¢��ÿåØ� '©Û.©ÛÑ:±ÈÝ�Czé÷ DNA óþ�ÙK$ÄK,|^nØ[ÜT þ�ÙK$Ä¤��±ÈÝÏdO(;� � å^eÿå¬Ñy�Ø�. 170503-5 ÔnÆ� Acta Phys. Sin. Vol. 61, No. 17 (2012) 170503 [1] Einstein A 1956 Investigations on the Theory of the Brownian Movement (Dover Pubns) p1 [2] Robert MM 2009 Brownian Motion: Fluctuations, Dynamics, and Applications (NY: Oxford University Press) p46 [3] Hao B L 2011 Physics 40 1 (in Chinese) [Ïy� 2011Ôn 40 1] [4] Uhlenbeck G E, Ornstein L S 1930 Phys. Rev. 36 823 [5] Wang M C, Uhlenbeck G E 1945 Rev. Mod. Phys. 17 323 [6] Berg-Sørensen K, Flyvbjerg H 2004 Rev. Sci. Instrum. 75 594 [7] Strick T R, Allemand J F, Bensimon D, Croquette V 1998 Biophys. J. 74 2016 [8] Ran S Y, Wang X L, Fu W B, Lai Z H, Wang W C, Liu X Q, Mai Z H, Li M 2006 Chin. Phys. Lett. 23 504 [9] Ran S Y, Wang X L, Fu W B, Wang W C, Li M 2008 Chin. Sci. Bull. 53 836 [10] Wang X L, Zhang X H, Wei K J, Sun B, Li M 2008 Acta Phys. Sin. 57 3905 (in Chinese) [�¡ ,Ü,u,3,Æ,o² 2008ÔnÆ� 57 3905] [11] Ran S Y, Wang Y W, Yang G C 2011 Physics Experimentation 31(7) 1 (in Chinese) [.], �ý, 1ë 2011 Ôn¢� 31(7) 1] [12] Jiang Y Q, Guo H L, Liu C X, Li Z L, Cheng B Y, Zhang D Z, Jia S T 2004 Acta Phys. Sin. 53 1721 (in Chinese) [ür,Hù ê,4S,oî�,§Z=,Ü�¥,_£, 2004ÔnÆ� 53 1721] [13] Ran S Y, Sun B, Li M 2007 Physics 36 228 (in Chinese) [.], Æ,o² 2007Ôn 36 228] [14] Hinterdorfer P, Van Oijen A 2009 Handbook of single-molecule biophysics (Springer Verlag) p1 [15] Reif F 1965 Fundamentals of Statistical and Thermal Physics (New York: McGraw-Hill) p249 [16] Wang C T 1991 Statistical Physics p343 (in Chinese) [�ª� 1991ÚOÔnÆ (uÆÑ�)1 343] [17] Marko J F, Siggia E D 1995Macromolecules 28 8759 [18] Lin B, Yu J, Rice S A 2000 Phys. Rev. E 62 3909 [19] Leuba S H, Karymov M A, Tomschik M, Ramjit R, Smith P, Zla- tanova J 2003 Proc. Natl. Acad. Sci. 100 495 170503-6 ÔnÆ� Acta Phys. Sin. Vol. 61, No. 17 (2012) 170503 Brownian motion in a harmonic trap: magnetic tweezers experiment and its simulation¤ Ran Shi-Yongy ( School of Physics and Electronic Information Engineering, Wenzhou University, Wenzhou 325035, China ) ( Received 29 January 2012; revised manuscript received 16 February 2012 ) Abstract The Brownian motion in a harmonic trap is studied by magnetic tweezers experiment and computer simulation. The results of the experiment and simulation validate the theory. Then the theory is used to analyze the experimental results including the effect of persistent length of DNA on the displacement distribution of the bead and the error in force measurements. It can be concluded that the variation of the persistent length affects more the Brownian motion along the DNA chain than in the other direction; under a small force, a considerable error of the force measurement will occur. Keywords: Brownian motion, hamonic trap, magnetic tweezers, DNA PACS: 05.40.Jc, 87.80.Ek, 87.80.Nj, 87.15.A * Project supported by the Natural Science Foundation of Zhejiang Province (Grant No. Y4110357), and the National Natural Science Foundation of China (Grant No. 20934004). y E-mail: syran@mails.gucas.ac.cn 170503-7

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谐振势阱中的布朗运动——磁镊实验与模拟

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