分子马达高斯跃迁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。分子马达通过催化三磷酸腺苷分子(ATP)水解，将化学能转化为机械能产生自身的定向运动。
当对分子马达的动力学进行初步的理论分析时，构造简明合理的物理模型至关重要。在现有的理论中，有人提出马达在不同状态之间跃迁发生在某些固定的位置，这就是所谓的定点跃迁理论，这种理论的假定过强，在物理上难以被人们接受。也有人提出均匀跃迁理论，认为马达在不同状态之间的跃迁在各个位置都是等几率的。但这种模型通常只对相互作用势取分段线性函数形式才有解析解，所处理的问题具有较大的局限性。
针对上述两种模型的局限性，在布朗马达理论框架中，我们提出了分子马达的高斯跃迁模型。我们认为马达在不同状态之间的跃迁不再局限于某些固定的跃迁点，也不是在各个位置都是等几率的，而是发生在某些点附近的一定宽度范围内，用跃迁宽度这一特征量表征跃迁范围的大小，能够较好地说明分子马达的动力学行为。在理论分析中，假定跃迁速率与位置有关且在跃迁点附近具有高斯函数形式。将布朗粒子的几率密度在跃迁点附近展开，可以进行任意阶的计算。
我们选取简单的两态模型，讨论了布朗粒子定向运动产生的几率流随温度、跃迁速率和跃迁宽度的变化关系。计算结果表明，在温度很低或很高时，定向运动的几率流都趋于零；在适当的温度范围内，对于某一确定的跃迁速率，总存在一个几率流的最大值，而且随着速率的增加相应于最大几率流的温度值升高。也就是说只有在适当的温度和跃迁速率下，才会有较大的流，说明几率流对温度和跃迁速率都是有选择的。在此基础上，我们着重讨论了跃迁宽度对几率流的影响。当温度很低或很高时，跃迁宽度对流的影响并不明显。然而在适当的温度范围内，允许粒子在一定宽度内跃迁，粒子在这一范围内出现几率增加，就使得更多的粒子跃迁几率增加，产生了较大的几率流。所以说跃迁宽度对流的影响是较为明显的。此外，我们选取的相互作用势更接近于定点跃迁模型中的简单锯齿势的形式，计算外力对几率流的影响。几率流随外力变化的总体趋势与定点跃迁模型的结果是一致的。这一新的理论可以涵盖早期的定点跃迁模型和均匀跃迁模型，是讨论多态动力学体系相关问题的一个更一般的理论框架。
Molecular motors are multi-component molecular structures. They do mechanical work through the hydrolization of ATP: the energy source of the cell. During hydrolization of ATP, conformational changes occur within the motor protein (a mechanochemical process), and mechanical work is done. They are specialized molecules which convert chemical energy to mechanical work, thereby generating directional motion.
The complex behavior of these systems consequently offers a formidable challenge for theoretical descriptions and numerical approaches that aim at a fundamental analysis of the underlying interaction mechanisms as well as interpretations. The methods of non-equilibrium statistical mechanics permit a general analysis of how to generate the movement of an individual motor. Up to now, some people represent certain-position transition model in which molecular motors transition are supposed to occur at some fixed positions. The molecular motors are described by M internal states and undergo transitions at K spatial locations within the period of the molecular force potentials. But their assumption is too narrow to be accepted. Other people advance equal- probability transition model in which molecular motors transition are supposed to occur at any positions. But this model is usually applied to discuss the case of the simplest indented sawtooth potential.
So a nonuniform ratchet model with Gauss-transition rates is proposed to discuss the directed motion of Brownian particles in asymmetrical periodic potentials. It is assumed that the particles experience several internal states in a single mechanical-chemical circle. In this model, the transition rates between different states are position-dependent which have the form of
Gaussian function. For any internal states, the probability distribution as a function of the time and position may be expanded near the transition points to any rank if necessary.
The focus of our study is concentrated on a two-state model, in which we choose (M, K) = (2, 2) and calculate the average current as a function of the transition width, temperature and transition rates. The results are summarized as follows: for small temperature, as well as for very high temperature the current vanishes. For the appropriate temperature and each given transition rates, the current has a maximum. With the increasing of the amplitude of transition rates the maximal current appears at the position of higher temperature. So we known the maximal current appear at the appropriate temperature and the appropriate flashing frequency. Then we focus on the relationship between the current and the transition width. For small temperature, as well as for very high temperature the current does not inflect with the change of the transition width. At the appropriate temperature, the current increases with the transition width increasing. It is revealed that the transitions width influences the current greatl
y.
当对分子马达的动力学进行初步的理论分析时，构造简明合理的物理模型至关重要。在现有的理论中，有人提出马达在不同状态之间跃迁发生在某些固定的位置，这就是所谓的定点跃迁理论，这种理论的假定过强，在物理上难以被人们接受。也有人提出均匀跃迁理论，认为马达在不同状态之间的跃迁在各个位置都是等几率的。但这种模型通常只对相互作用势取分段线性函数形式才有解析解，所处理的问题具有较大的局限性。
针对上述两种模型的局限性，在布朗马达理论框架中，我们提出了分子马达的高斯跃迁模型。我们认为马达在不同状态之间的跃迁不再局限于某些固定的跃迁点，也不是在各个位置都是等几率的，而是发生在某些点附近的一定宽度范围内，用跃迁宽度这一特征量表征跃迁范围的大小，能够较好地说明分子马达的动力学行为。在理论分析中，假定跃迁速率与位置有关且在跃迁点附近具有高斯函数形式。将布朗粒子的几率密度在跃迁点附近展开，可以进行任意阶的计算。
我们选取简单的两态模型，讨论了布朗粒子定向运动产生的几率流随温度、跃迁速率和跃迁宽度的变化关系。计算结果表明，在温度很低或很高时，定向运动的几率流都趋于零；在适当的温度范围内，对于某一确定的跃迁速率，总存在一个几率流的最大值，而且随着速率的增加相应于最大几率流的温度值升高。也就是说只有在适当的温度和跃迁速率下，才会有较大的流，说明几率流对温度和跃迁速率都是有选择的。在此基础上，我们着重讨论了跃迁宽度对几率流的影响。当温度很低或很高时，跃迁宽度对流的影响并不明显。然而在适当的温度范围内，允许粒子在一定宽度内跃迁，粒子在这一范围内出现几率增加，就使得更多的粒子跃迁几率增加，产生了较大的几率流。所以说跃迁宽度对流的影响是较为明显的。此外，我们选取的相互作用势更接近于定点跃迁模型中的简单锯齿势的形式，计算外力对几率流的影响。几率流随外力变化的总体趋势与定点跃迁模型的结果是一致的。这一新的理论可以涵盖早期的定点跃迁模型和均匀跃迁模型，是讨论多态动力学体系相关问题的一个更一般的理论框架。
Molecular motors are multi-component molecular structures. They do mechanical work through the hydrolization of ATP: the energy source of the cell. During hydrolization of ATP, conformational changes occur within the motor protein (a mechanochemical process), and mechanical work is done. They are specialized molecules which convert chemical energy to mechanical work, thereby generating directional motion.
The complex behavior of these systems consequently offers a formidable challenge for theoretical descriptions and numerical approaches that aim at a fundamental analysis of the underlying interaction mechanisms as well as interpretations. The methods of non-equilibrium statistical mechanics permit a general analysis of how to generate the movement of an individual motor. Up to now, some people represent certain-position transition model in which molecular motors transition are supposed to occur at some fixed positions. The molecular motors are described by M internal states and undergo transitions at K spatial locations within the period of the molecular force potentials. But their assumption is too narrow to be accepted. Other people advance equal- probability transition model in which molecular motors transition are supposed to occur at any positions. But this model is usually applied to discuss the case of the simplest indented sawtooth potential.
So a nonuniform ratchet model with Gauss-transition rates is proposed to discuss the directed motion of Brownian particles in asymmetrical periodic potentials. It is assumed that the particles experience several internal states in a single mechanical-chemical circle. In this model, the transition rates between different states are position-dependent which have the form of
Gaussian function. For any internal states, the probability distribution as a function of the time and position may be expanded near the transition points to any rank if necessary.
The focus of our study is concentrated on a two-state model, in which we choose (M, K) = (2, 2) and calculate the average current as a function of the transition width, temperature and transition rates. The results are summarized as follows: for small temperature, as well as for very high temperature the current vanishes. For the appropriate temperature and each given transition rates, the current has a maximum. With the increasing of the amplitude of transition rates the maximal current appears at the position of higher temperature. So we known the maximal current appear at the appropriate temperature and the appropriate flashing frequency. Then we focus on the relationship between the current and the transition width. For small temperature, as well as for very high temperature the current does not inflect with the change of the transition width. At the appropriate temperature, the current increases with the transition width increasing. It is revealed that the transitions width influences the current greatl
y.
引文
[1] Svoboda K, Schmidt C F, Schnapp B J, et al, Direct Observation of Kinesin Stepping by Optical Trapping Interferometry. Nature, 1993, 365:721.
[2] T. Uyeda, et al, The neck region of the myosin motor domain acts as a lever arm to generate movement. Proc. Natl. Acad. Sci. 1996, 93:4459.
[3] Hirokawa N. Kinesin and Dynein Superfamily Proteins and the Mechanism of Organelle Transport. Science, 1998, 279:519-526.
[4] Bhattacharjee S M, et al, Helicase on DNA: A Phase coexistence based mechanism, arXiv: cond-mat/0205254.
[5] Schief W R, Howard J. Conformational Changes during Kinesin Motility. Current Opinion In Cell Biology, 2001, 13: 19-28.
[6] Vale R D, Milligan R A, The Way Things Move: Looking Under the Hood of Molecular Motor Protein. Science, 2000, 288: 88.
[7] Nishiyama M, et al, Substeps within the 8-nm step of the ATPase cycle of single kinesin molecules. Nature cell Biology, 2002, 3:425.
[8] Oster G, Wang H Y, et al, Reverse engineering a protein: the mechanochemistry of ATP synthase. Biochimica Biophysica Acta, 2000, 1458:482.
[9] Noji H, The Rotary Enzyme of the Cell: The Rotation of Fl-ATPase Science. 1998, 282:1844.
[10] Fisher M E, Kolomeriky A B. The Force Exerted by a Molecular Motor. Proc. Natl. Acad. Sci. USA, 1999, 96: 6597-6602.
[11] Fisher M E, Kolomeriky A B. Molecular Motor and the Force they Exert. Physica A, 1999, 274:241-266.
[12] Magnasco M O, Szilard's heat engine. Europhys. Lett., 1996, 33:583.
[13] Julicher F, Ajdari A and Prost J, Modeling Molecular Motors. Rev. Mod. Phys., 1997, 69:1269.
[14] Zhao Tongjun, Cao Tianguang, Zhan Yong, et al, Rocking Ratchets with Stochastic Potentials. Physica A, 2002, 313:109.
[15] Qian H., A Simple Theory of Motor Protein Kinetics and Energetics Ⅱ. Biophys. Chem., 2000, 83:35-43.
[16] Reinhard Lipowsky and Thomas Harms, Molecular motor and nonuniform ratchets. Eur Biophys J., 2000, 29: 542.
[17] Zhao Tongjun, Zhan Yong, Zhuo Yizhong, et al, Brownian Ratchets Driven by Flashing Noise Strength. Chin. Scien. Bull., 1999, 21: 1956.
[18] F. Julicher, A. Ajdari and J. Prost., Modeling Molecular Motors. Rev. Mod. Phys., 1997, 69: 1269.
[19] A. Parmeggiani, E Julicher, A. Ajdari and J. Prost., Energy Transduction of Isothermal Ratchets: Generic Aspects and Specific Examples Close to and Far from Equilibrium. Phys. Rev. E., 1999, 60: 2127.
[20] Erwin J. G. Peterman, Hernando Sosa, Lawrence S. B. Goldstein, et al, Polarized Fluorescence Microscopy of Individual and Many Kinesin Motors Bound to Axonemal Microtubules. Biophysical journal, 2001, 81: 2851-2863.
[21] S. M. Block, Trends Cell Biol., Nanometres and piconewtons: the macromolecular mechanics of kinesin. 1995, 5: 169.
[22] K Svoboda, C F Schmidt, B J Schnapp, et al, Direct observation of kinesin stepping by optical trapping interferometry. Nature, 1993, 365:721-727.
[23] Mandelkow E. and Hoenger A. A. Structures of Kinesin and Kinesin-Microtubule Interactions. Curr. Opin. Cell Biol., 1999, 11:34-44
[24] Holmes K. C., The swinging lever-arm hypothesis of muscle contraction. Curr. Biol., 1997, 7:112-118.
[25] Highsmith S., Lever arm model of force generation by actin-myosin-AYP. Biochemistry, 1999, 38: 9791-9797.
[26] Geeves M. A. and Holmes K. C., Structural mechanism of muscle contraction. Annu. Rev. Biochem., 1999, 68: 687-728.
[27] Walter Steffen, David Smith, John Sleep. The working stroke upon myosin-nucleotide complexes binding to actin. Pans, 2003, 100:6434-6439
[28] 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
[29] 卓益忠，赵同军，展永，分子马达与布朗马达，物理，2000，29：712-718．
[30] J 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.
[31] George Oster, Hongyun Wang,. Reverse engineering a protein: the mechanochemistryof ATP synthase. Biochimica et Biophysica Acta, 2000, 1458:482-510.
[32] R. Yasuda, H. Noji, K. Kinosita, M. Yushida, A Rotary Motor Made of a Single Molecule. Cell, 1998,93:1117-1124
[33] 郭维生，罗辽复，肌球蛋白工作循环的一个新模型，生物化学于生物物理进展，2003，30：216-220．
[34] A. Houdusse and H. L. Sweeney, Myosin motors: missing structures and hidden springs. Curr. Opin. Stru. Biol., 2001, 11: 182-194.
[35] Y. Suzuki, T. Yasunaga and R. Ohkura, Swing of the lever arm of a myosin motor at the isomerization and phosphate-release steps. Nature, 1998, 396: 380-383.
[36] Wu Wei-Xia, Zhan Yong, Zhao Tong-jun, Directed Motion of a Molecular Motor Based on the Four-state Model with Unequal Substeps. Comm. Theor. Phys., 2003, 40:9-14
[37] S. P. Gilbert, M. I. Moyer, K. A. Johonson, Pathway of ATP Hydrolysis by Monomeric and Dimeric Kinesin. Biochem, 1998, 37: 800-813.
[38] M. E. Fisher, A. B. Kolomeriky. Simple mechanochemistry describes the dynamics of kinesin molecules. Proc. Natl. Acad. Sci., 2001, 98: 7748-7753.
[39] Mark J. Schnitzer, Koen Visscher and Steven M. Block, Force production by single kinesin motors. Nature Cell Biol., 2000, 2: 718-723.
[40] R. A. Cross, I. Crevel, N. J. Carter, et al, The conformational cycle of kinesin. The Royal Society, 2000, 355:459-464.
[41] Justin E. Molloy and Claudia Veigel, Myosin Motor Walk the Walk. Science, 2003, 300: 2045-2046.
[42] Ahmet Yildiz, Joseph N. Forkey, Sean A. McKinney, et al, Myosin V Walks Hand-over-hand: Single Fluorophore Imaging with 1.5-nm Localization. Science, 2003, 300:2061-2065.
[43] Imre Derenyi, Tamas Vicsek, Realistic models of biological motion. Physica A, 1998,249:397-406.
[44] Qian H. A Simple Theory of Motor Protein Kinesin and Energetics. Biophys. Chem., 1997, 67: 263-267.
[45] Anatoly B. Kolomeisky and Michael E. Fisher, A Simple Kinetic Model Describes the Processivity of Myosin-V. 2003, 84: 1642-1650.
[46] Risken, The Fokker Pianck Equation (Springer Verlag ,Berlin). 1984.
[2] T. Uyeda, et al, The neck region of the myosin motor domain acts as a lever arm to generate movement. Proc. Natl. Acad. Sci. 1996, 93:4459.
[3] Hirokawa N. Kinesin and Dynein Superfamily Proteins and the Mechanism of Organelle Transport. Science, 1998, 279:519-526.
[4] Bhattacharjee S M, et al, Helicase on DNA: A Phase coexistence based mechanism, arXiv: cond-mat/0205254.
[5] Schief W R, Howard J. Conformational Changes during Kinesin Motility. Current Opinion In Cell Biology, 2001, 13: 19-28.
[6] Vale R D, Milligan R A, The Way Things Move: Looking Under the Hood of Molecular Motor Protein. Science, 2000, 288: 88.
[7] Nishiyama M, et al, Substeps within the 8-nm step of the ATPase cycle of single kinesin molecules. Nature cell Biology, 2002, 3:425.
[8] Oster G, Wang H Y, et al, Reverse engineering a protein: the mechanochemistry of ATP synthase. Biochimica Biophysica Acta, 2000, 1458:482.
[9] Noji H, The Rotary Enzyme of the Cell: The Rotation of Fl-ATPase Science. 1998, 282:1844.
[10] Fisher M E, Kolomeriky A B. The Force Exerted by a Molecular Motor. Proc. Natl. Acad. Sci. USA, 1999, 96: 6597-6602.
[11] Fisher M E, Kolomeriky A B. Molecular Motor and the Force they Exert. Physica A, 1999, 274:241-266.
[12] Magnasco M O, Szilard's heat engine. Europhys. Lett., 1996, 33:583.
[13] Julicher F, Ajdari A and Prost J, Modeling Molecular Motors. Rev. Mod. Phys., 1997, 69:1269.
[14] Zhao Tongjun, Cao Tianguang, Zhan Yong, et al, Rocking Ratchets with Stochastic Potentials. Physica A, 2002, 313:109.
[15] Qian H., A Simple Theory of Motor Protein Kinetics and Energetics Ⅱ. Biophys. Chem., 2000, 83:35-43.
[16] Reinhard Lipowsky and Thomas Harms, Molecular motor and nonuniform ratchets. Eur Biophys J., 2000, 29: 542.
[17] Zhao Tongjun, Zhan Yong, Zhuo Yizhong, et al, Brownian Ratchets Driven by Flashing Noise Strength. Chin. Scien. Bull., 1999, 21: 1956.
[18] F. Julicher, A. Ajdari and J. Prost., Modeling Molecular Motors. Rev. Mod. Phys., 1997, 69: 1269.
[19] A. Parmeggiani, E Julicher, A. Ajdari and J. Prost., Energy Transduction of Isothermal Ratchets: Generic Aspects and Specific Examples Close to and Far from Equilibrium. Phys. Rev. E., 1999, 60: 2127.
[20] Erwin J. G. Peterman, Hernando Sosa, Lawrence S. B. Goldstein, et al, Polarized Fluorescence Microscopy of Individual and Many Kinesin Motors Bound to Axonemal Microtubules. Biophysical journal, 2001, 81: 2851-2863.
[21] S. M. Block, Trends Cell Biol., Nanometres and piconewtons: the macromolecular mechanics of kinesin. 1995, 5: 169.
[22] K Svoboda, C F Schmidt, B J Schnapp, et al, Direct observation of kinesin stepping by optical trapping interferometry. Nature, 1993, 365:721-727.
[23] Mandelkow E. and Hoenger A. A. Structures of Kinesin and Kinesin-Microtubule Interactions. Curr. Opin. Cell Biol., 1999, 11:34-44
[24] Holmes K. C., The swinging lever-arm hypothesis of muscle contraction. Curr. Biol., 1997, 7:112-118.
[25] Highsmith S., Lever arm model of force generation by actin-myosin-AYP. Biochemistry, 1999, 38: 9791-9797.
[26] Geeves M. A. and Holmes K. C., Structural mechanism of muscle contraction. Annu. Rev. Biochem., 1999, 68: 687-728.
[27] Walter Steffen, David Smith, John Sleep. The working stroke upon myosin-nucleotide complexes binding to actin. Pans, 2003, 100:6434-6439
[28] 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
[29] 卓益忠，赵同军，展永，分子马达与布朗马达，物理，2000，29：712-718．
[30] J 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.
[31] George Oster, Hongyun Wang,. Reverse engineering a protein: the mechanochemistryof ATP synthase. Biochimica et Biophysica Acta, 2000, 1458:482-510.
[32] R. Yasuda, H. Noji, K. Kinosita, M. Yushida, A Rotary Motor Made of a Single Molecule. Cell, 1998,93:1117-1124
[33] 郭维生，罗辽复，肌球蛋白工作循环的一个新模型，生物化学于生物物理进展，2003，30：216-220．
[34] A. Houdusse and H. L. Sweeney, Myosin motors: missing structures and hidden springs. Curr. Opin. Stru. Biol., 2001, 11: 182-194.
[35] Y. Suzuki, T. Yasunaga and R. Ohkura, Swing of the lever arm of a myosin motor at the isomerization and phosphate-release steps. Nature, 1998, 396: 380-383.
[36] Wu Wei-Xia, Zhan Yong, Zhao Tong-jun, Directed Motion of a Molecular Motor Based on the Four-state Model with Unequal Substeps. Comm. Theor. Phys., 2003, 40:9-14
[37] S. P. Gilbert, M. I. Moyer, K. A. Johonson, Pathway of ATP Hydrolysis by Monomeric and Dimeric Kinesin. Biochem, 1998, 37: 800-813.
[38] M. E. Fisher, A. B. Kolomeriky. Simple mechanochemistry describes the dynamics of kinesin molecules. Proc. Natl. Acad. Sci., 2001, 98: 7748-7753.
[39] Mark J. Schnitzer, Koen Visscher and Steven M. Block, Force production by single kinesin motors. Nature Cell Biol., 2000, 2: 718-723.
[40] R. A. Cross, I. Crevel, N. J. Carter, et al, The conformational cycle of kinesin. The Royal Society, 2000, 355:459-464.
[41] Justin E. Molloy and Claudia Veigel, Myosin Motor Walk the Walk. Science, 2003, 300: 2045-2046.
[42] Ahmet Yildiz, Joseph N. Forkey, Sean A. McKinney, et al, Myosin V Walks Hand-over-hand: Single Fluorophore Imaging with 1.5-nm Localization. Science, 2003, 300:2061-2065.
[43] Imre Derenyi, Tamas Vicsek, Realistic models of biological motion. Physica A, 1998,249:397-406.
[44] Qian H. A Simple Theory of Motor Protein Kinesin and Energetics. Biophys. Chem., 1997, 67: 263-267.
[45] Anatoly B. Kolomeisky and Michael E. Fisher, A Simple Kinetic Model Describes the Processivity of Myosin-V. 2003, 84: 1642-1650.
[46] Risken, The Fokker Pianck Equation (Springer Verlag ,Berlin). 1984.