分子马达两态模型定向运动机制的研究
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摘要
分子马达是广泛存在于细胞内部的具有马达功能的酶蛋白生物大分子。生命活动中的许多过程都是基于分子马达的运动。分子马达运动所需能量来自于它所催化的三磷酸腺苷分子(ATP)水解所释放的化学能，它将化学能直接转化为机械能。因此，分子马达就起到了能量转换器的作用。当分子马达的运动协调起来就产生了宏观的运动而做功，其中蕴含的动力学机制已成为众多研究者们关注的焦点。
本文主要分三部分来介绍。第一部分介绍了分子马达的生物学研究进展，包括马达的种类、结构特征和沿轨道运动的性质。第二部分介绍了研究马达的随机跃迁方法-福克普朗克方程的特征根解法，着重推导了如何用特征根方法解福克普朗克方程。第三部分根据马达的运动特点建立起简单的跃迁模型—两态模型。具体计算了两态模型的几率流、有效势的倾斜斜率。主要计算有效势的倾斜斜率和几率流同时随跃迁频率和温度的变化关系。计算结果表明：(1)跃迁率很小时几率流和有效势的倾斜随着跃迁率的增加而增加，当跃迁率达到一定值40左右时，两者几乎同时达到最大值，跃迁率大于某一定值时，两者随跃迁率的增加又同时减小。这表明给定温度情况下，几率流和有效势具有相关性。(2)温度很低时，几率流和有效势倾斜的斜率的变化不同步，随着温度的升高几率流增加较为缓慢，有效势倾斜斜率增加较快。只有当温度达到某一值时，几率流和有效势的斜率才同时随温度的增加而减小，这时几率流和有效势倾斜的斜率密切相关。(3)几率流不仅与有效势的整体倾斜的斜率有关，同时还与有效势的势垒高度有关。所以有效势倾斜斜率和其势垒高度对几率流都有共同影响。在低温时，虽然斜率大，但由于温度相对势垒较低，马达难以跨越势垒，从而定向运动的几率流较小，几率流与有效势倾斜的比值(随温度的变化)也就不会是简单的线性相关。在高温时，马达可以跨越势垒，几率流与有效势倾斜的比值接近线性相关。同时还将力-速度的曲线与实验做了比较，结果表明二者定性吻合。
Molecular motors are a kind of enzyme proteins which exist in the cell and play an important role in the process of the muscle contraction, intracellular transport, DNA duplication, mitosis and so on. The energy for the motion of the motors comes from the chemical energy of ATP hydrolyzation. The motors can transduce the chemical energy into mechanical work. So the motors act as a function of the energy transducer. When the motion of the motors is coordinated, the macroscopic directed motion is brought. Especially, kinesin, which has the feature of the directivity and stepping motion, has been focused on its particular dynamical mechanism.
The paper is composed of three parts mainly. In the first part, the biology study evolvement of molecular motor are introduced, including their categories, structure features and motion properties. In the second parts, a method on studying stochastic transition of the Fokker-Planck equation is introduced, how to use the method of the latent root to on the Fokker-Planck equation is mainly deduced. In the third part, according to the motion characteristic of the motor a simple transition model (two-state model) is established. The current, the effective potential and the slope of the effective potential of the motor are calculated. First, it is shown that the slope of the effective potential increase or decrease faster than the current. All of these facts indicate that the current and the slope are not simply linear relevant. Second, the results show that the current and the slope of the effective potential change simultaneously when the temperature is in a certain range. Third, the slope of the effective pote
ntial corresponds to an average force. Because of the existence of the average force, the motor protein can have a directed motion against the load
force. If the load force is zero, the directed motion is mainly dependent on the average force without the effect of the load force. When the temperature T is smaller, even if there is the slope of the effective potential, most Brownian particles (motor proteins) can not span the potential. Therefore the current is small, the ratio of the slope of the effective potential to the current is not linear relevant simply. When the temperature T is larger, the particles can span the potential at the effect of the average force, so the ratio of them is near linear relevant. Besides, the interaction between particles is not considered. This interaction also has an effect on the current.
Our purpose is to discuss the force and the motion of the motor. So we can understand the motion mechanism of the motor deeply. At the same time, we compare the force-velocity theoretical curve with the experiment, the result shows that they are identical basically.
本文主要分三部分来介绍。第一部分介绍了分子马达的生物学研究进展，包括马达的种类、结构特征和沿轨道运动的性质。第二部分介绍了研究马达的随机跃迁方法-福克普朗克方程的特征根解法，着重推导了如何用特征根方法解福克普朗克方程。第三部分根据马达的运动特点建立起简单的跃迁模型—两态模型。具体计算了两态模型的几率流、有效势的倾斜斜率。主要计算有效势的倾斜斜率和几率流同时随跃迁频率和温度的变化关系。计算结果表明：(1)跃迁率很小时几率流和有效势的倾斜随着跃迁率的增加而增加，当跃迁率达到一定值40左右时，两者几乎同时达到最大值，跃迁率大于某一定值时，两者随跃迁率的增加又同时减小。这表明给定温度情况下，几率流和有效势具有相关性。(2)温度很低时，几率流和有效势倾斜的斜率的变化不同步，随着温度的升高几率流增加较为缓慢，有效势倾斜斜率增加较快。只有当温度达到某一值时，几率流和有效势的斜率才同时随温度的增加而减小，这时几率流和有效势倾斜的斜率密切相关。(3)几率流不仅与有效势的整体倾斜的斜率有关，同时还与有效势的势垒高度有关。所以有效势倾斜斜率和其势垒高度对几率流都有共同影响。在低温时，虽然斜率大，但由于温度相对势垒较低，马达难以跨越势垒，从而定向运动的几率流较小，几率流与有效势倾斜的比值(随温度的变化)也就不会是简单的线性相关。在高温时，马达可以跨越势垒，几率流与有效势倾斜的比值接近线性相关。同时还将力-速度的曲线与实验做了比较，结果表明二者定性吻合。
Molecular motors are a kind of enzyme proteins which exist in the cell and play an important role in the process of the muscle contraction, intracellular transport, DNA duplication, mitosis and so on. The energy for the motion of the motors comes from the chemical energy of ATP hydrolyzation. The motors can transduce the chemical energy into mechanical work. So the motors act as a function of the energy transducer. When the motion of the motors is coordinated, the macroscopic directed motion is brought. Especially, kinesin, which has the feature of the directivity and stepping motion, has been focused on its particular dynamical mechanism.
The paper is composed of three parts mainly. In the first part, the biology study evolvement of molecular motor are introduced, including their categories, structure features and motion properties. In the second parts, a method on studying stochastic transition of the Fokker-Planck equation is introduced, how to use the method of the latent root to on the Fokker-Planck equation is mainly deduced. In the third part, according to the motion characteristic of the motor a simple transition model (two-state model) is established. The current, the effective potential and the slope of the effective potential of the motor are calculated. First, it is shown that the slope of the effective potential increase or decrease faster than the current. All of these facts indicate that the current and the slope are not simply linear relevant. Second, the results show that the current and the slope of the effective potential change simultaneously when the temperature is in a certain range. Third, the slope of the effective pote
ntial corresponds to an average force. Because of the existence of the average force, the motor protein can have a directed motion against the load
force. If the load force is zero, the directed motion is mainly dependent on the average force without the effect of the load force. When the temperature T is smaller, even if there is the slope of the effective potential, most Brownian particles (motor proteins) can not span the potential. Therefore the current is small, the ratio of the slope of the effective potential to the current is not linear relevant simply. When the temperature T is larger, the particles can span the potential at the effect of the average force, so the ratio of them is near linear relevant. Besides, the interaction between particles is not considered. This interaction also has an effect on the current.
Our purpose is to discuss the force and the motion of the motor. So we can understand the motion mechanism of the motor deeply. At the same time, we compare the force-velocity theoretical curve with the experiment, the result shows that they are identical basically.
引文
[1] Hirokawa N. Kinesin and Dynein Superfamily Proteins and the Mechanism of Organelle Transport[J]. Science, 1998, 279: 519-526.
[2] Vale R D, Milligan R A. The Way Things Move: Looking Under the Hood of Molecular Motor Protein[J]. Science, 2000, 288: 88-95.
[3] Schief W R, Howard J. Conformational Changes during Kinesin Motility[J]. Current Opinion In Cell Biology, 2001, 13: 19-28.
[4] Qian H. A Simple Theory of Motor Protein Kinesin and Energetics [J]. Biophys. Chem., 1997, 67: 263-267.
[5] Kolomeisky A B, Widom B. A Simplified "Ratchet" Model of Molecular Motors[J]. J. Stat. Phys., 1998, 93: 633-645.
[6] Derrida B, Velocity and Diffusion Constant of a Periodic One-Dimensional Hopping Model[J]. J. Stat. Phys., 1983, 31: 433-450.
[7] Fisher M E, Kolomeriky A B. The Force Exerted by a Molecular Motor[J]. Proc. Natl. Acad. Sci. USA,1999, 96: 6597-6602.
[8] Qian H. A Simple Theory of Motor Protein Kinetics and Energetics.Ⅱ[J]. Biophys. Chem., 2000, 83:35-43.
[9] Fisher M E, Kolomeriky A B. Molecular Motor and the Force they Exert[J]. Physica A, 1999, 274: 241-266.
[10] Ahmet Yildiz, Michio Tomishige, Ronald D.Vale, Paul R.Selvin. Kinesin Walks Hand-Over-Hand, Science, 2004,303:676-678
[11] William R.Shief, Rutilio H. Clark, Alvaro H.Crevenna and Jonathon Howard. Inhibition of kinesin mobility by ADP and phosphate supports a hand-over-hand mechanism, Pans, 2004, 101: 1183-1188
[12] Charles L. Asbury, Adrian N. Fehr, Steven M. Block. Kinesin Moves by an Asymmetric Hand-Over-Hand Mechanism, Science, 2003, 302:2130-2134
[13] Lisa M. Klumpp, Adnreas Hoenger, Susan P. Gilbert. Kinesin's second step, Pans, 2004, 101:3444-3449
[14] 卓益忠，赵同军，展永，分子马达与布朗马达，物理，2000，29(12)：712-718．
[15] Walter Steffen, David Smith, John Sleep. The working stroke upon myosin-nucleotide complexes binding to actin, Pans2003, 100:6434-6439
[16] Thomas J. Purcell, Carl Morris, James A. Spudich, H. LeeSweeney, Role of the lever arm in the processive stepping of myosin v, Pans, 2002, 99:14159-14164
[17] K Kazuo, T Makio, H Atsuko, et al, A Single Myosin Head Moves Along an Actin Filament with Regular Steps of 5.3 Nanometres. Nature, 1999,397:129-134.
[18] Ahmet Yildiz, Joseph N.Forkey, Scan A. Mckinney, Taekjip Ha, Yale E. Goldman, Paul R. Selvin Myosin V Walks Hand-Over-Hand: Single Fluorophore Imaging with 1.5-nm Localization, Science, 2003, 300:2061-2065
[19] K Svoboda, C F Schmidt, B J Schnapp et al, Direct Observation of Kinesin Stepping by Optical Trapping Interferometry. Nature, 1993,365:721-727.
[20] S Rice, W L Abel, S Daniel, et al, A Structural Change in the Kinesin Motor Protein That Drives Motility. Nature, 1999, 402: 778-784.
[21] M J Schnitzer, K Visscher and S M Block, Force Production by Single Kinesin Motors. Nature Cell Biology 2000, 2:718-723.
[22] S. P. Gilbert, M. l. Moyer, K. A. Johonson. Pathway of ATP Hydrolysis by Monomeric and Dimeric Kinesin Biochem, 1998, 37: 800-813.
[23] M. E. Fisher, A. B. Kolomeriky. Simple Mechanoehemistry Describes the Dynamics of Kinesin Molecules. Proc. Natl. Acad. Sci., 2001, 98: 7748-7753.
[24] Noji H. Nture,1998,282:1844
[25] George Oster, Hongyun Wang. Biochimica et Biophysica Acta 2000,1458:482-510
[26] Abrahams, J, Leslie, A, Lutter R. & Walker. J. Nature, 1994,370: 621-628
[27] A. Senior, J. Bioenerg. Biomembr, 1992, 24 : 479-483
[28] J. Weber, A. E. Senior, Biochim. Biophys, Acta, 1997, 1319:19-58
[29] George Oster, Hongyun Wang and Michael Grabe, Phil Trans. R. Soc. Lond. B, 2000,355:523-528
[30] R.Yasuda, H. Noji, K. Kinosita, M. Yushida, Cell, 1998,93:1117-1124
[31] 翟中和，王喜忠，丁明孝．细胞生物学，2000，219
[32] P. D.Boyer, W. E. Kohlbrenner. B. Selman, S. Selman-Reiner (Eds.),1981 ,PP, 231-240
[33] Kazuhiko Kinosita Jr, Ryohei Yasuda, Hiroyuki Noji and Kengo Adachi. Phil. Trans. R. Soc. Lond. B, 2000,355:473-489
[34] Senior A.E. Annu Rev Biophys Cleim, 1990,19: 7-41
[35] Peter Dimroth, Hongyun Wang, Michael Grabe and George Oster. Proc. Natl. Acad. Sci. USA,
1999,96:4924-4929
[36] K. Sekimoto and S. Sawasa, Phys. Soc. Jpn, 1997,66:3326
[37] Reimann P. Brownian motors: noisy transport far from equilibrium. Phys. Rep, 2002, 361(2-4):57～265
[38] Schief WR, Howard J. Conformational changes during kinesin motility. Current Opinion in Cell Biology 2001, 3:19～28
[39] Vale RD, Milligan RA. The way things move: Looking under the hood of molecular motor proteins. Science, 2000, 288:88～95
[40] Ryohei Yasuda, Hiroyuki Noji, Masasuke Yoshida. Kinosita Jr and Hiroyasu. Kinetic analysis of F_1-ATPase. Nature, 2001, 410(19):898～904
[41] Ma JP, Flynn TC, Cui Q, Andrea L, Walker JE. and Karplus M. A Dynamic Analysis of the Rotation Mechanism in F_1-ATPase. Structure, 2002, 10:921～931,
[42] Parmeggiani A, Julicher F, Ajdari A, Prost J. Energy transduction of isothermal rachets: Generic aspects and specific examples close to and far from equilibrium. Phys. Rev. E, 1999, 60(2):2127～2140
[43] Derényi I, Vicsek T. Cooperative transport of Brownian particles. Phys. Rev. Lett, 1995, 75:374～377
[44] Jung P, Kissner JG, Hnggi P. Regular and Chaotic Transport in Asymmetric Periodic Potentials: Inertia Ratchets. Phys. Rev. Lett, 1996, 76:3436～3439
[45] Bartussek R, Hnggi P, and Kissner JG.. Periodically rocked thermal rachets. Europhys. Lett, 1994, 28:459～464
[46] Doering CR, Horsthemke W, Riordan J. Nonequilibrium Fluctuation-Induced Transport. Phys. Rev. Lett, 1994, 72:984～2987
[47] Reimann P, Bartussek R, Haussler R, Hnggi R Brownian motors driven by temperature oscillations. Phys. Lett. A, 1996, 215:26～31
[48] Ajdari A, Prost J. Current reversal in asymmetric pumping. Europhys. Lett, 1995, 32:373～378
[49] Van den Broeck C, Reimann P, Kawai R, and Hnggi P. Coupled Brownian Motors, Lecture Notes in Physics 1999, 527:99～111
[50] Zhao TJ, Zhan Y, Yu H and Ji Q. Motion and Efficiency of a Two-State Model for a Single Molecular Motor. Commun. Theor. Phys, 2003, 39(1):121～124
[51] Zhao TJ, Zhuo YZ, Zhan Y, Ji Q and Cao TG. Two-Dimentional Rachets with Non-Conservative Impulsive Force. Mode. Phys. Lett. B, 2002,16(26):999～1006
[52] Zhao TJ, Cao TG, Zhan Y, Zhuo YZ. Rocking Rachets with Stochastic Potentials. Physica A, 2002,
312:109～118
[53] Zhao TJ, Zhan Y, Wu JH, Wang YH. A Flashing Model for Transport of Brownian Motors. Chin.Phys.Lett, 2002, 19(9):1248～1250
[54] Mielke A. Transport in a fluctuating potential. Ann. Physik 1995, 4:721～738
[55] Lattanzi G, Maritan A. Force Dependence of the Michaelis Constant in a Two-State Ratchet Model for Molecular Motors. Phys. Rev. Lett, 2001, 86:1134～1137
[56] Julicher F, Ajdari A, Prost J. Modeling molecular motors. Rev. Mod. Phys, 1997, 69(4): 1269～1281
[57] Gilbert SP, Moyer ML, Johonson KA. Pathway of ATP Hydrolysis by Monomeric and Dimeric Kinesin. Biochem, 1998, 37(3):800～813
[2] Vale R D, Milligan R A. The Way Things Move: Looking Under the Hood of Molecular Motor Protein[J]. Science, 2000, 288: 88-95.
[3] Schief W R, Howard J. Conformational Changes during Kinesin Motility[J]. Current Opinion In Cell Biology, 2001, 13: 19-28.
[4] Qian H. A Simple Theory of Motor Protein Kinesin and Energetics [J]. Biophys. Chem., 1997, 67: 263-267.
[5] Kolomeisky A B, Widom B. A Simplified "Ratchet" Model of Molecular Motors[J]. J. Stat. Phys., 1998, 93: 633-645.
[6] Derrida B, Velocity and Diffusion Constant of a Periodic One-Dimensional Hopping Model[J]. J. Stat. Phys., 1983, 31: 433-450.
[7] Fisher M E, Kolomeriky A B. The Force Exerted by a Molecular Motor[J]. Proc. Natl. Acad. Sci. USA,1999, 96: 6597-6602.
[8] Qian H. A Simple Theory of Motor Protein Kinetics and Energetics.Ⅱ[J]. Biophys. Chem., 2000, 83:35-43.
[9] Fisher M E, Kolomeriky A B. Molecular Motor and the Force they Exert[J]. Physica A, 1999, 274: 241-266.
[10] Ahmet Yildiz, Michio Tomishige, Ronald D.Vale, Paul R.Selvin. Kinesin Walks Hand-Over-Hand, Science, 2004,303:676-678
[11] William R.Shief, Rutilio H. Clark, Alvaro H.Crevenna and Jonathon Howard. Inhibition of kinesin mobility by ADP and phosphate supports a hand-over-hand mechanism, Pans, 2004, 101: 1183-1188
[12] Charles L. Asbury, Adrian N. Fehr, Steven M. Block. Kinesin Moves by an Asymmetric Hand-Over-Hand Mechanism, Science, 2003, 302:2130-2134
[13] Lisa M. Klumpp, Adnreas Hoenger, Susan P. Gilbert. Kinesin's second step, Pans, 2004, 101:3444-3449
[14] 卓益忠，赵同军，展永，分子马达与布朗马达，物理，2000，29(12)：712-718．
[15] Walter Steffen, David Smith, John Sleep. The working stroke upon myosin-nucleotide complexes binding to actin, Pans2003, 100:6434-6439
[16] Thomas J. Purcell, Carl Morris, James A. Spudich, H. LeeSweeney, Role of the lever arm in the processive stepping of myosin v, Pans, 2002, 99:14159-14164
[17] K Kazuo, T Makio, H Atsuko, et al, A Single Myosin Head Moves Along an Actin Filament with Regular Steps of 5.3 Nanometres. Nature, 1999,397:129-134.
[18] Ahmet Yildiz, Joseph N.Forkey, Scan A. Mckinney, Taekjip Ha, Yale E. Goldman, Paul R. Selvin Myosin V Walks Hand-Over-Hand: Single Fluorophore Imaging with 1.5-nm Localization, Science, 2003, 300:2061-2065
[19] K Svoboda, C F Schmidt, B J Schnapp et al, Direct Observation of Kinesin Stepping by Optical Trapping Interferometry. Nature, 1993,365:721-727.
[20] S Rice, W L Abel, S Daniel, et al, A Structural Change in the Kinesin Motor Protein That Drives Motility. Nature, 1999, 402: 778-784.
[21] M J Schnitzer, K Visscher and S M Block, Force Production by Single Kinesin Motors. Nature Cell Biology 2000, 2:718-723.
[22] S. P. Gilbert, M. l. Moyer, K. A. Johonson. Pathway of ATP Hydrolysis by Monomeric and Dimeric Kinesin Biochem, 1998, 37: 800-813.
[23] M. E. Fisher, A. B. Kolomeriky. Simple Mechanoehemistry Describes the Dynamics of Kinesin Molecules. Proc. Natl. Acad. Sci., 2001, 98: 7748-7753.
[24] Noji H. Nture,1998,282:1844
[25] George Oster, Hongyun Wang. Biochimica et Biophysica Acta 2000,1458:482-510
[26] Abrahams, J, Leslie, A, Lutter R. & Walker. J. Nature, 1994,370: 621-628
[27] A. Senior, J. Bioenerg. Biomembr, 1992, 24 : 479-483
[28] J. Weber, A. E. Senior, Biochim. Biophys, Acta, 1997, 1319:19-58
[29] George Oster, Hongyun Wang and Michael Grabe, Phil Trans. R. Soc. Lond. B, 2000,355:523-528
[30] R.Yasuda, H. Noji, K. Kinosita, M. Yushida, Cell, 1998,93:1117-1124
[31] 翟中和，王喜忠，丁明孝．细胞生物学，2000，219
[32] P. D.Boyer, W. E. Kohlbrenner. B. Selman, S. Selman-Reiner (Eds.),1981 ,PP, 231-240
[33] Kazuhiko Kinosita Jr, Ryohei Yasuda, Hiroyuki Noji and Kengo Adachi. Phil. Trans. R. Soc. Lond. B, 2000,355:473-489
[34] Senior A.E. Annu Rev Biophys Cleim, 1990,19: 7-41
[35] Peter Dimroth, Hongyun Wang, Michael Grabe and George Oster. Proc. Natl. Acad. Sci. USA,
1999,96:4924-4929
[36] K. Sekimoto and S. Sawasa, Phys. Soc. Jpn, 1997,66:3326
[37] Reimann P. Brownian motors: noisy transport far from equilibrium. Phys. Rep, 2002, 361(2-4):57～265
[38] Schief WR, Howard J. Conformational changes during kinesin motility. Current Opinion in Cell Biology 2001, 3:19～28
[39] Vale RD, Milligan RA. The way things move: Looking under the hood of molecular motor proteins. Science, 2000, 288:88～95
[40] Ryohei Yasuda, Hiroyuki Noji, Masasuke Yoshida. Kinosita Jr and Hiroyasu. Kinetic analysis of F_1-ATPase. Nature, 2001, 410(19):898～904
[41] Ma JP, Flynn TC, Cui Q, Andrea L, Walker JE. and Karplus M. A Dynamic Analysis of the Rotation Mechanism in F_1-ATPase. Structure, 2002, 10:921～931,
[42] Parmeggiani A, Julicher F, Ajdari A, Prost J. Energy transduction of isothermal rachets: Generic aspects and specific examples close to and far from equilibrium. Phys. Rev. E, 1999, 60(2):2127～2140
[43] Derényi I, Vicsek T. Cooperative transport of Brownian particles. Phys. Rev. Lett, 1995, 75:374～377
[44] Jung P, Kissner JG, Hnggi P. Regular and Chaotic Transport in Asymmetric Periodic Potentials: Inertia Ratchets. Phys. Rev. Lett, 1996, 76:3436～3439
[45] Bartussek R, Hnggi P, and Kissner JG.. Periodically rocked thermal rachets. Europhys. Lett, 1994, 28:459～464
[46] Doering CR, Horsthemke W, Riordan J. Nonequilibrium Fluctuation-Induced Transport. Phys. Rev. Lett, 1994, 72:984～2987
[47] Reimann P, Bartussek R, Haussler R, Hnggi R Brownian motors driven by temperature oscillations. Phys. Lett. A, 1996, 215:26～31
[48] Ajdari A, Prost J. Current reversal in asymmetric pumping. Europhys. Lett, 1995, 32:373～378
[49] Van den Broeck C, Reimann P, Kawai R, and Hnggi P. Coupled Brownian Motors, Lecture Notes in Physics 1999, 527:99～111
[50] Zhao TJ, Zhan Y, Yu H and Ji Q. Motion and Efficiency of a Two-State Model for a Single Molecular Motor. Commun. Theor. Phys, 2003, 39(1):121～124
[51] Zhao TJ, Zhuo YZ, Zhan Y, Ji Q and Cao TG. Two-Dimentional Rachets with Non-Conservative Impulsive Force. Mode. Phys. Lett. B, 2002,16(26):999～1006
[52] Zhao TJ, Cao TG, Zhan Y, Zhuo YZ. Rocking Rachets with Stochastic Potentials. Physica A, 2002,
312:109～118
[53] Zhao TJ, Zhan Y, Wu JH, Wang YH. A Flashing Model for Transport of Brownian Motors. Chin.Phys.Lett, 2002, 19(9):1248～1250
[54] Mielke A. Transport in a fluctuating potential. Ann. Physik 1995, 4:721～738
[55] Lattanzi G, Maritan A. Force Dependence of the Michaelis Constant in a Two-State Ratchet Model for Molecular Motors. Phys. Rev. Lett, 2001, 86:1134～1137
[56] Julicher F, Ajdari A, Prost J. Modeling molecular motors. Rev. Mod. Phys, 1997, 69(4): 1269～1281
[57] Gilbert SP, Moyer ML, Johonson KA. Pathway of ATP Hydrolysis by Monomeric and Dimeric Kinesin. Biochem, 1998, 37(3):800～813