随机过程在非平衡态统计物理和系统生物学建模中的应用
|
摘要
本文一方面是把随机过程模型应用到近代非平衡态统计物理中,从定义到性质给出了一套相对完整的数学理论;另一方面是把随机过程模型应用到系统生物学中,详细总结了生物化学系统的随机建模方法,并深入探讨了酵母细胞环布尔网络模型、简单单分子酶动力学模型以及磷酸化去磷酸化生物开关模型的性质。
首先,我们把[91,153,154,155,157]中的概念和结论推广到非时齐马氏链的情形,并引入了瞬时可逆性和瞬时熵产生率的概念,而且讨论了这二者之间的关系。同时,我们还讨论了环上的周期性非时齐生灭马氏链作为布朗马达模型最关心的量—-旋转数。我们发现当该生灭马氏链瞬时可逆或周期可逆时,它的旋转数都等于零。更进一步,我们还给出了瞬时熵产生率的测度论定义。
紧接着,我们以非时齐马氏链和非时齐扩散过程为模型,定义并证明了其中的推广Jarzynski等式,并且还通过把该结论应用到物理学家和化学家的工作上,阐明了它的物理意义,其中包括Jarzynski和Crooks的开创性工作,Hummer和Szabo的工作,Hatano-Sasa等式以及钱纮关于化学计量学系统中Gibbs自由能差的工作。
然后,我们关注的是一般随机过程的Evans-Searles型暂态涨落定理[167],严格证明了样本熵产生的暂态涨落定理对于非常一般的随机过程都成立,并不需要马氏性、平稳性以及时齐性的条件,从而很好地确信了该理论的广泛适用性;然后对于很多随机过程模型检验了该定理的条件,包括时齐,非时齐马氏链和一般的平稳扩散过程。其中,关于非时齐马氏链和一般扩散过程样本熵产生的暂态涨落定理都是新的结论,之前还没有被讨论过。
近些年来,由于人类基因组计划的成功实施和众多模式生物材料如果蝇等100余种生物全基因组序列相继被测定,以及随之而来的各种“组学”的推动,系统生物学迅速崛起,成为21世纪初生命科学领域的大事件。
在生物体内的噪声越来越得到重视和研究的时候,如何定量的描述随机模型中的同步化行为就变得越来越重要,因为极限环以及固定相位差的概念在随机模型中将不再成立。因此,需要研究随机模型中对应于确定性模型极限环概念的合理推广;而在关于开系统非平衡定态的数学理论[91]中,环流恰好可以扮演这样一个角色。于是,我们就把环流理论应用于2004年李等[116]建立的酵母细胞环布尔网络模型中,得到其同步化行为的完整刻画,并且比较了环流理论和功率谱方法的联系。
另一方面,由于单分子实验技术的快速发展,在实验中追踪单个的分子已经成为可能[119,187,188,189]。我们详细讨论了三状态可逆Michaelis-Menten酶动力学模型,定义了环流、环等待时间和步进概率等概念并详细探讨它们之间的相互关系。然后,我们证明了推广Haldane等式,并将所有结论推广到n状态的情形,同时详细对比了前人在理论和实验方面的诸多工作[15,101,106]。蛋白质磷酸化和去磷酸化过程是生物信号传输过程中非常重要的一类生化反应,并有多种酶参与其中。蛋白质的生物活性经常是被磷酸化过程所激活,而被去磷酸化过程所关闭,所以这样的ATP ?ADP环(PdPC)过程就是在传输着生物信号,被称为控制着信息流的“生物开关”。
我们研究了时间合作现象的基本概念和理论,然后把它应用到磷酸化和去磷酸化环的简单开关以及超灵敏度开关上;我们的想法是把PdPC中出现的合作现象通过其完整的化学主方程求出一个近似模型,通过能量参数(γ= 1)和前人的工作相衔接。为说明这种做法的合理性,我们用了一些初等数学运算和计算机模拟,而不去牵涉过多的数学。最后,我们通过分析若干个经典结构合作模型,得到了时间合作现象和结构合作现象数学上的等同性,这也正是我们称这种现象为“时间合作”的原因。
In the present thesis, we apply stochastic processes to modern nonequilibrium statisti-cal physics, for which we construct a completely mathematical theory including bothdefinitions and properties; on the other hand, we also apply stochastic processes to sys-tems biology, summarizing the stochastic modelling methods of biochemical systemsand investigating the Boolean network model of yeast cell-cycle, the single-moleculeenzyme kinetics and the phosphorylation-dephosphorylation biological switches.
At first, we extend the notions and results of [91, 153, 154, 155, 157] to the sit-uation of a general inhomogeneous Markov chain, then introduce the concepts of in-stantaneous reversibility and instantaneous entropy production rate and investigate theirrelationship. Furthermore, for a time-periodic birth-and-death chain, which can be re-garded as a simple version of physical model (Brownian motors), we prove that itsrotation number is zero when it is instantaneously reversible or periodically reversible.In addition, we also give the measure-theoretical definition of the instantaneous entropyproduction rate of inhomogeneous Markov chains.
Consequently, we define and prove the generalized Jarzynski’s equalities of in-homogeneous Markov chains and multidimensional diffusions. Then, we explain itsphysical meaning and applications through several previous work including Jarzynskiand Crooks’original work, Hummer and Szabo’s work, Hatano-Sasa equality and theGibbs free energy differences in stoichiometric chemical systems.
After that, we focus on the derivation of Evans-Searles ?uctuation theorem [167]for general stochastic processes, and rigorously prove that the transient ?uctuation theo-rem (TFT) of sample entropy production holds for general stochastic processes withoutthe assumption of Markovian, homogeneous, or stationary properties, confirming thevalidity of its universality. Then we verify the condition of our main result for variousstochastic processes, including homogeneous, inhomogeneous Markov chains and gen- eral diffusion processes. Among these cases, the applications to inhomogeneous case,discrete time case and general diffusion processes are all new, which have not ever beenpointed out before.
Recently, the field now commonly referred to as systems biology has developedrapidly. With the sequencing of whole genomes and the development of analysis meth-ods to measure many of the cellular components, we have now entered the realm ofcomplete descriptions at a cellular level. It is believed that systems biology will be-come one of the most active fields of science in the 21st century.
As there is a growing awareness and interest in studying the effects of noise in bi-ological networks, it becomes more and more important to quantitatively characterizethe synchronized dynamics mathematically in stochastic models, because the conceptsof limit cycle and fixed phase difference no longer holds in this case. Therefore, alogical generalization of limit cycle in stochastic models needs to be developed, andinterestingly, the concept of circulation in the mathematical theory of nonequilibriumsteady states [91] actually plays the role. We apply the circulation theory to investigatethe synchronized stochastic dynamics of a Boolean network model of yeast cell-cycleregulation, providing a clear picture of the synchronized dynamics. Furthermore, wecompare this circulation theory with the power spectrum method always used by physi-cists.
On the other hand, recent advances in single-molecule spectroscopy and manip-ulation have now made it possible to study enzyme kinetics at the level of singlemolecules [119, 187, 188, 189]. We thoroughly investigate a more realistic reversiblethree-step mechanism of the Michaelis-Menten kinetics in detail. We also prove thegeneralized Haldane equality and extend all the results to the n-step cycle. Finally,experimental and theoretically based evidences are also included [15,101,106].
Protein phosphorylation is one class of the most important biochemical reactionsin signal transduction system of living cells. The biological activity of a protein is often“turned on”by the phosphorylation, and“turned off”by a dephosphorylation reaction.The turning on and off of the biological activity of a protein has been widely recognizedas a switch in controlling information ?ow.
In this thesis, we investigate the basic concepts and theories of tem-poral cooperativity phenomenon, and apply them to the simple and ultrasen- sitive phosphorylation-dephosphorylation switches; our aim is to connect thephosphorylation-dephosphorylation cooperativity phenomenon to the previous worksthrough the energy parameterγin the simple model reduced from the complete stochas-tic model based on chemical master equations. We use some simple mathematical cal-culations and numerical simulations in order to confirm the rationality of our method,and finally we point out the mathematical equivalence between the temporal and struc-tural cooperativity phenomenons. That is just why we call this phenomenon as“tem-poral cooperativity”.
首先,我们把[91,153,154,155,157]中的概念和结论推广到非时齐马氏链的情形,并引入了瞬时可逆性和瞬时熵产生率的概念,而且讨论了这二者之间的关系。同时,我们还讨论了环上的周期性非时齐生灭马氏链作为布朗马达模型最关心的量—-旋转数。我们发现当该生灭马氏链瞬时可逆或周期可逆时,它的旋转数都等于零。更进一步,我们还给出了瞬时熵产生率的测度论定义。
紧接着,我们以非时齐马氏链和非时齐扩散过程为模型,定义并证明了其中的推广Jarzynski等式,并且还通过把该结论应用到物理学家和化学家的工作上,阐明了它的物理意义,其中包括Jarzynski和Crooks的开创性工作,Hummer和Szabo的工作,Hatano-Sasa等式以及钱纮关于化学计量学系统中Gibbs自由能差的工作。
然后,我们关注的是一般随机过程的Evans-Searles型暂态涨落定理[167],严格证明了样本熵产生的暂态涨落定理对于非常一般的随机过程都成立,并不需要马氏性、平稳性以及时齐性的条件,从而很好地确信了该理论的广泛适用性;然后对于很多随机过程模型检验了该定理的条件,包括时齐,非时齐马氏链和一般的平稳扩散过程。其中,关于非时齐马氏链和一般扩散过程样本熵产生的暂态涨落定理都是新的结论,之前还没有被讨论过。
近些年来,由于人类基因组计划的成功实施和众多模式生物材料如果蝇等100余种生物全基因组序列相继被测定,以及随之而来的各种“组学”的推动,系统生物学迅速崛起,成为21世纪初生命科学领域的大事件。
在生物体内的噪声越来越得到重视和研究的时候,如何定量的描述随机模型中的同步化行为就变得越来越重要,因为极限环以及固定相位差的概念在随机模型中将不再成立。因此,需要研究随机模型中对应于确定性模型极限环概念的合理推广;而在关于开系统非平衡定态的数学理论[91]中,环流恰好可以扮演这样一个角色。于是,我们就把环流理论应用于2004年李等[116]建立的酵母细胞环布尔网络模型中,得到其同步化行为的完整刻画,并且比较了环流理论和功率谱方法的联系。
另一方面,由于单分子实验技术的快速发展,在实验中追踪单个的分子已经成为可能[119,187,188,189]。我们详细讨论了三状态可逆Michaelis-Menten酶动力学模型,定义了环流、环等待时间和步进概率等概念并详细探讨它们之间的相互关系。然后,我们证明了推广Haldane等式,并将所有结论推广到n状态的情形,同时详细对比了前人在理论和实验方面的诸多工作[15,101,106]。蛋白质磷酸化和去磷酸化过程是生物信号传输过程中非常重要的一类生化反应,并有多种酶参与其中。蛋白质的生物活性经常是被磷酸化过程所激活,而被去磷酸化过程所关闭,所以这样的ATP ?ADP环(PdPC)过程就是在传输着生物信号,被称为控制着信息流的“生物开关”。
我们研究了时间合作现象的基本概念和理论,然后把它应用到磷酸化和去磷酸化环的简单开关以及超灵敏度开关上;我们的想法是把PdPC中出现的合作现象通过其完整的化学主方程求出一个近似模型,通过能量参数(γ= 1)和前人的工作相衔接。为说明这种做法的合理性,我们用了一些初等数学运算和计算机模拟,而不去牵涉过多的数学。最后,我们通过分析若干个经典结构合作模型,得到了时间合作现象和结构合作现象数学上的等同性,这也正是我们称这种现象为“时间合作”的原因。
In the present thesis, we apply stochastic processes to modern nonequilibrium statisti-cal physics, for which we construct a completely mathematical theory including bothdefinitions and properties; on the other hand, we also apply stochastic processes to sys-tems biology, summarizing the stochastic modelling methods of biochemical systemsand investigating the Boolean network model of yeast cell-cycle, the single-moleculeenzyme kinetics and the phosphorylation-dephosphorylation biological switches.
At first, we extend the notions and results of [91, 153, 154, 155, 157] to the sit-uation of a general inhomogeneous Markov chain, then introduce the concepts of in-stantaneous reversibility and instantaneous entropy production rate and investigate theirrelationship. Furthermore, for a time-periodic birth-and-death chain, which can be re-garded as a simple version of physical model (Brownian motors), we prove that itsrotation number is zero when it is instantaneously reversible or periodically reversible.In addition, we also give the measure-theoretical definition of the instantaneous entropyproduction rate of inhomogeneous Markov chains.
Consequently, we define and prove the generalized Jarzynski’s equalities of in-homogeneous Markov chains and multidimensional diffusions. Then, we explain itsphysical meaning and applications through several previous work including Jarzynskiand Crooks’original work, Hummer and Szabo’s work, Hatano-Sasa equality and theGibbs free energy differences in stoichiometric chemical systems.
After that, we focus on the derivation of Evans-Searles ?uctuation theorem [167]for general stochastic processes, and rigorously prove that the transient ?uctuation theo-rem (TFT) of sample entropy production holds for general stochastic processes withoutthe assumption of Markovian, homogeneous, or stationary properties, confirming thevalidity of its universality. Then we verify the condition of our main result for variousstochastic processes, including homogeneous, inhomogeneous Markov chains and gen- eral diffusion processes. Among these cases, the applications to inhomogeneous case,discrete time case and general diffusion processes are all new, which have not ever beenpointed out before.
Recently, the field now commonly referred to as systems biology has developedrapidly. With the sequencing of whole genomes and the development of analysis meth-ods to measure many of the cellular components, we have now entered the realm ofcomplete descriptions at a cellular level. It is believed that systems biology will be-come one of the most active fields of science in the 21st century.
As there is a growing awareness and interest in studying the effects of noise in bi-ological networks, it becomes more and more important to quantitatively characterizethe synchronized dynamics mathematically in stochastic models, because the conceptsof limit cycle and fixed phase difference no longer holds in this case. Therefore, alogical generalization of limit cycle in stochastic models needs to be developed, andinterestingly, the concept of circulation in the mathematical theory of nonequilibriumsteady states [91] actually plays the role. We apply the circulation theory to investigatethe synchronized stochastic dynamics of a Boolean network model of yeast cell-cycleregulation, providing a clear picture of the synchronized dynamics. Furthermore, wecompare this circulation theory with the power spectrum method always used by physi-cists.
On the other hand, recent advances in single-molecule spectroscopy and manip-ulation have now made it possible to study enzyme kinetics at the level of singlemolecules [119, 187, 188, 189]. We thoroughly investigate a more realistic reversiblethree-step mechanism of the Michaelis-Menten kinetics in detail. We also prove thegeneralized Haldane equality and extend all the results to the n-step cycle. Finally,experimental and theoretically based evidences are also included [15,101,106].
Protein phosphorylation is one class of the most important biochemical reactionsin signal transduction system of living cells. The biological activity of a protein is often“turned on”by the phosphorylation, and“turned off”by a dephosphorylation reaction.The turning on and off of the biological activity of a protein has been widely recognizedas a switch in controlling information ?ow.
In this thesis, we investigate the basic concepts and theories of tem-poral cooperativity phenomenon, and apply them to the simple and ultrasen- sitive phosphorylation-dephosphorylation switches; our aim is to connect thephosphorylation-dephosphorylation cooperativity phenomenon to the previous worksthrough the energy parameterγin the simple model reduced from the complete stochas-tic model based on chemical master equations. We use some simple mathematical cal-culations and numerical simulations in order to confirm the rationality of our method,and finally we point out the mathematical equivalence between the temporal and struc-tural cooperativity phenomenons. That is just why we call this phenomenon as“tem-poral cooperativity”.
引文
[1] Adair, G.S.: The hemoglobin system. VI. The oxygen dissociation curve of hemoglobin. J.Biol. Chem. 63, 529-545 (1925)
[2] Amit, D.J.: Modeling brain function: The world of attractor neural networks. CambridgeUniversity Press 1989
[3] Anderson, W.J. : Continuous-time Markov chains: an applications-oriented approach. NewYork: Springer-Verlag 1991
[4] Arizmendi, C.M. and Family, F.: Algorithmic complexity and efficiency of a ratchet. PhysicaA 269, 285–292 (1999)
[5] Astumian, R.D.: Thermodynamics and kinetics of a Brownian motor. Science 276, 917–922(1997)
[6] Astumian, R.D. and Bier, M.: Fluctuation driven ratchets: molecular motors. Phys. Rev. Lett.72, 1766–1769 (1994)
[7] Ayton, G. and Evans, D.J.: On the asymptotic convergence of the transient and steady-state?uctuation theorems. J. Stat. Phys. 97, 811–815 (1999)
[8] Baiesi, M., Jacobs, T., Maes, C. and Skantzos, N.S.: Fluctuation symmetries for work andheat. Phys. Rev. E 74, 021111 (2006)
[9] Bai, C.L., Wang, C., Xie, X.S. and Wolynes, P.G.: Single molecule physics and chemistry.Proc. Natl. Acad. Sci. USA 96 11075-11076 (1999)
[10] Baras, F., Malek Mansour, M. and Pearson, J.E.: Microscopic simulation of chemical bista-bility in homogeneous systems. J. Chem. Phys. 105(18), 8257–8261 (1996)
[11] Beard, D.A. and Qian, H.: Chemical Biophysics: Quantitative Analysis of Cellular Systems.Cambridge Texts in Biomedical Engineering Cambridge University Press 2008.
[12] Berdichevsky, V. and Gitterman, M.: Stochastic resonance and ratchets-new manifestations.Physica A 249, 88–95 (1998)
[13] Berg, O.G., Paulsson, J. and Ehrenberg, M.: Fluctuations and quality of control in biologicalcells: zero-prder ultrasensitivity reinvestigated. Biophys. J. 79 1228-1236 (2000)
[14] Bier, M.: Brownian ratchets in physics and biology. Contemporary Phys. 38(6), 371–379(1997)
[15] Carter, N.J. and Cross, R.A.: Mechanics of the kinesin step. Nature. 435 308-312 (2005)
[16] Chen, K.C., Calzone, L., Csikasz-Nagy, A., Cross, F.R., Novak, B., Tyson, J.J.: Integrativeanalysis of cell cycle control in budding yeast. Mol. Biol. Cell 15 3841-3862 (2004)
[17] Chen, Y., Chen, X. and Qian, M.P.: Green-Kubo formula, autocorrelation function and ?uc-tuation spectrum for finite Markov chains with continuous time. J. Phys. A: Math. Gen. 392539-2550 (2006)
[18] Ciliberto, A., Novak, B., Tyson, J.J.: Mathematical model of the morphogenesis checkpointin budding yeast. J. Cell Biol. 163 1243-1254 (2003)
[19] Cornish-Bowden, A.: Fundamentals of enzyme kinetics. (3nd ed.) London: Portland Press2004
[20] Crooks, G.E.: Nonequilibrium measurements of free energy differences for microscopicallyreversible Markovian systems. J. Stat. Phys. 90, 1481-1487 (1998)
[21] Crooks, G.E.: Entropy production ?uctuation theorem and the nonequilibrium work relationfor free energy differences. Phys. Rev. E 60, 2721-2726 (1999)
[22] Crooks, G.E.: Path-ensemble averages in systems driven far from equilibrium. Phys. Rev. E61(3), 2361-2366 (2000)
[23] Delbru¨ck, M.: Statistical ?uctuations in autocatalytic reactions. J. Chem. Phys. 8, 120–124(1940)
[24] Dollard, J.D. and Friedman, J.N.: Product integration with applications to differential equa-tions. (Encyclopedia of Mathematics and applications 10) Reading [U.S.A.]: Addison-WesleyPublishing Company 1979
[25] Durrett, R.: Probability: Theory and Examples (Second Edition). Pacific Grove, CA: DuxburyPress 1996
[26] Dynkin, E.B.: Markov processes. Vol. 1, 2. Springer-Verlag 1965
[27] Dynkin, E.B.: Die Grundlagen der Theorie der Markoffschen Prozesse. Berlin: Springer-Verlag 1961
[28] Einstein, A.: Uber die von der Molekularkinetischen Theorie der Warme Geforderte Bewe-gung yon in Ruhenden Flüssigkeiten Suspendierten Teilchen. Ann. Phys. (Leipzig) 17, 549–560 (1905)
[29] Elf, J., Paulsson, J., Berg, O.G. and Ehrenberg, M.: Near-critical phenomenon in intercellularmetabolite pools. Biophys. J. 84 154-170 (2003)
[30] Elf, J., Paulsson, J., Berg, O. and Ehrenberg, M.: Mesoscopic kinetics and its applicationsin protein synthesis. In L.Alberghina and H.V.Westerhoff (eds.): Topics in Current Genetics:Systems Biology: Definitions and Perspectives. Springer-Verlag: Berlin 2005
[31] Elston, T.C. and Doering, C.R.: Numerical and analytical studies of nonequilibrium?uctuation-induced transport processes. J. Stat. Phys. 83, 359–383 (1996)
[32] English, B.P., Min, W., van Oijen, A.M., Lee, K.T.; Luo, G., Sun, H., Cherayil, B.J., Kou,S.C. and Xie, X.S.: Ever-?uctuating single enzyme molecules: Michaelis-Menten equationrevisited. Nat. Chem. Bio. 2 87-94 (2006)
[33] Evans, D.J., Cohen, E.G.D. and Morriss, G.P.: Probability of second law violation in steady?ows. Phys. Rev. Lett. 71, 2401–2404 (1993)
[34] Evans, D.J. and Searles, D.J.: Equilibrium microstates which generate second law violatingsteady states. Phys. Rev. E 50(2), 1645–1648 (1994)
[35] Fall, C.P., Marland, E.S., Wagner, J.M. and Tyson, J.J.: Computational cell biology. NewYork: Springer-Verlag 2002
[36] Ferry, R.M., and Green, A.A.: Studies in the chemistry of hemoglobin III: The equilibriumbetween oxygen and hemoglobin and its relation to changing hydrogen ion activity. J. Biol.Chem. 81, 175 (1929)
[37] Fischer, E.: Ein?uss der configuration auf die Wirkung der enzyme. Berichter der DeutschenChemischen Gesellschaft 27, 2985-2993 (1894)
[38] Fischer, E.H. and Krebs, E.G.: Conversion of phosphorylase b to phosphorylase a in muscleextracts. J. Mol. Chem. 216, 121-133 (1955)
[39] Fischer, E.H., Heilmeyer, L.M.G. and Haschke, R.H.: Phosphorylation and the control ofglycogen degradation. Curr. Top. Cell. Regul. 4, 211-251 (1971)
[40] Fox, R.F.: Stochastic versions of the Hodgkin-Huxley equations. Biophys. J. 72 2068-2074(1997)
[41] Fox, R.F. and Lu, Y.: Emergent collective behavior in large numbers of globally coupledindependently stochastic ion channels. Phys. Rev. E 49(4) 3421-3431 (1994)
[42] A. Friedman: Partial differential equations of parabolic type. Englewood Cliffs 1964
[43] Gallavotti, G.: Reversible Anosov diffeomorphisms and large deviations. Math. Phys. Elec-tronic J. 1, 1–12 (1995)
[44] Gallavotti, G.: Chaotic hypothesis: Onsager reciprocity and ?uctuation-dissipation theorem.J. Stat. Phys. 84, 899–926 (1996)
[45] Gallavotti, G. and Cohen, E.G.D.: Dynamical ensembles in stationary states. J. Stat. Phys. 80,931–970 (1995)
[46] Gammaitoni, L., Ha¨nggi, P., Jung, P. and Marchesoni, F.: Stochastic resonance. Rev. Mod.Phys. 70(1), 223–287 (1998)
[47] Gardiner, C.W.: Handbook of Stochastic Methods for Physics, Chemistry and the NaturalSciences. Sringer-Verlag Berlin 1985
[48] Ge, H., Jiang, D.Q. and Qian, M.: A simple discrete model of Brownian motors: time-periodicMarkov chains. J. Stat. Phys. 123(4), 831–859 (2006)
[49] Ge, H., Jiang, D.Q. and Qian, M.: Reversibility and entropy production of inhomogeneousMarkov chains. J. Appl. Prob. 43(4) 1028–1043 (2006)
[50] Ge, H. and Qian, M.: Ge, H., and Qian, M.: Generalized Jarzynski’s equality in inhomoge-neous Markov chains. J. Math. Phys. 48 053302 (2007)
[51] Ge, H., Qian, H. and Qian, M.: Synchronized dynamics and nonequilibrium steady states in ayeast cell-cycle network. Math. Biosci. 211 132-152 (2008)
[52] Ge, H. and Qian, M.: Reversibility and potentiality of exclusion processes on countable dis-crete groups. Appl. Math. Lett. 20 1110-1114 (2007)
[53] Ge, H. and Jiang, D.Q.: Transient ?uctuation theorem of sample entropy production for gen-eral stochastic processes. J. Phys. A 40 F713-F723 (2007)
[54] Ge, H.: Waiting cycle times and generalized Haldane equation in the steady-state cycle kinet-ics of single enzymes. J. Phys. Chem. B 112 61-70 (2008)
[55] Ge, H. and Jiang, D.Q.: Generalized Jarzynski’s equality of multidimensional inhomogeneousdiffusion processes. J. Stat. Phys. 131 675–689 (2008)
[56] Ge, H. and Qian, M.: Steady-state cycle kinetics of single enzymes: competing substrates andmulti-conformations. J. Theor. Comput. Chem. (2008) (accepted)
[57] Ge, H. and Qian, M.: Sensitivity amplification in the phosphorylation-dephosphorylation cy-cle: nonequilibrium steady states, chemical master equation and temporal cooperativity. J.Chem. Phys. (2008) (accepted)
[58] Gillespie, D.T.: A general method for numerically simulating the stochastic time evolution ofcoupled chemical reactions. J. Comput. Phys. 22, 403 (1976)
[59] Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem.81(25), 2340–2361 (1977)
[60] Gillespie, D.T.: Approximating the master equation by Fokker-Planck-type equations forsingle-variable chemical systems J. Chem. Phys. 72(10), 5363–5370 (1980)
[61] Gillespie, D.T.: A rigorous derivation of the chemical master equation. Physica A 188, 404–425 (1992)
[62] Gillespie, D.T.: The chemical Langevin equations. J. Chem. Phys. 113(1), 297–306 (2000)
[63] Gillespie, D.T.: The chemical Langevin and Fokker-Planck equations for the reversible iso-merization reaction. J. Phys. Chem. A 106, 5063–5071 (2002)
[64] Glansdorff, P. and Prigogine, I.: Thermodynamic theory of structure, stability and ?uctuations.London: Wiley-Interscience 1971
[65] Goldbeter, A.: Biochemical Oscillations and Cellular Rhythms: the Molecular Bases of Peri-odic and Chaotic Behaviour, New York: Cambridge Univ. Press 1996
[66] Goldbeter, A. and Koshland, Jr. D.E.: An amplified sensitivity arising from covalent modifi-cation in biological systems. Proc. Natl. Acad. Sci. USA 78 6840-6844 (1981)
[67] Haken, H.: Synergetics: an introduction: nonequilibrium phase transitions and self-organization in physics, chemistry, and biology. Berlin, New York: Springer-Verlag 1977
[68] Haken, H.: Advanced synergetics: instability hierarchies of self-organizing systems and de-vices. Berlin, New York: Springer-Verlag 1983
[69] Hanggi, P., Grabert, H., Talkner, P. and Thomas, H.: Bistable systems: Master equation versusFokker-Planck modeling. Phys. Rev. A 29(1), 371–378 (1984)
[70] Hasegawa, H.: On the construction of a time-reversed Markoff process, Prog. Theor. Phys. 55,90–105 (1976);Variational principle for non-equilibrium states and the Onsager-Machlup formula, ibid. 56,44–60 (1976);Thermodynamic properties of non-equilibrium states subject to Fokker-Planck equations, ibid.57, 1523–1537 (1977);Variational approach in studies with Fokker-Planck equations, ibid. 58, 128–146 (1977)
[71] Haber, J.E. and Koshland, D.E.: Relation of protein subunit interactions to the molecularspecies observed during cooperative binding of ligands. Proc. Natl. Acad. Sci. USA 58, 2087-2093 (1967)
[72] Has’minskii, R.Z.: Stochastic stability of differential equations. Alphen aan den Rijn-Germantown: Sijthoff and Noorrdhoff 1980
[73] Hatano, T. and Sasa, S.: Steady-states thermodynamics of Langevin systems. Phys. Rev. Lett.E. 86, 3463–3466 (2001)
[74] Hill, A.V.: The possible effects of the aggregation of the molecules of haemoglobin on itsdissociation curves. Journal of Physiology 40, iv-vii (1910)
[75] Hill, T.L. and Plesner, I.W.: Studies in irreversible thermodynamics II. A simple class of latticemodels for open systems. J. Chem. Phys. 43, 267-285 (1965)
[76] Hill, T.L.: Studies in irreversible thermodynamics IV. diagrammatic representation of steadystate ?uxes for unimolecular systems. J. Theoret. Biol. 10, 442-459 (1966)
[77] Hill, T.L.: Approach for certain systems, including membranes, to steady state. J. Chem. Phys.54, 34-35 (1971)
[78] Hill, T.L.: Free energy transduction in biology. New York: Academic Press 1977
[79] Hill, T.L.: Free energy transduction and biochemical cycle kinetics. New York: Springer-Verlag 1995
[80] Hill, T. and Chen, Y.: Stochastics of cycle completions (?uxes) in biochemical kinetic dia-grams. Proc. Natl. Acad. Sci. USA 72, 1291–1295 (1975)
[81] Hopfield, J.J.: Neural networks and physical systems with emergent collective computationalabilities. Proc. Natl. Acad. Sci. USA 79 2554-2558 (1982)
[82] Hopfield, J.J.: Neurons with graded response have collective computational properties likethose of two-state neurons. Proc. Natl. Acad. Sci. USA 81 3088-3092 (1984)
[83] Hopfield, J.J. and Herz, Andreas V.M.: Rapid local synchronization of action potentials: To-ward computation with coupled integrate-and-fire neurons. Proc. Natl. Acad. Sci. USA 926655-6662 (1995)
[84] 胡迪鹤: 可数状态的马尔可夫过程论. 武汉大学出版社 1983
[85] Hummer, G. and Szabo, A.: Free energy reconstruction from nonequilibrium single-moleculepulling experiments. Proc. Natl. Acad. Sci. USA 98(7), 3658–3661 (2001)
[86] Jarzynski, C.: Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78, 2690–2693 (1997)
[87] Jarzynski, C.: Equilibrium free-energy differences from nonequilibrium measurements: Amaster-equation approach. Phys. Rev. E 56, 5018–5035 (1997)
[88] Jarzynski, C.: Microscopic analysis of Clausius–Duhem processes. J. Stat. Phys. 96, 415-427 (1999)
[89] Jarzynski, C.: Hamiltonian derivation of a detailed ?uctuation theorem. J. Stat. Phys. 98,77–102 (2000)
[90] Jarzynski, C.: Rare events and the convergence of exponentially averaged work values. Phys.Rev. E 73, 046105 (2006)
[91] Jiang, D.Q., Qian, M. and Qian, M.P.: Mathematical theory of nonequilibrium steady states- On the frontier of probability and dynamical systems. (Lect. Notes Math. 1833) Berlin:Springer-Verlag 2004
[92] Jiang, D.Q., Qian, M. and Zhang, F.X.: Entropy production ?uctuations of finite Markovchains. J. Math. Phys. 44(9), 4176–4188 (2003)
[93] Jiang, D.Q. and Zhang, F.X.: The Green-Kubo formula and power spectrum of reversibleMarkoc processes. J. Math. Phys. 44(10), 4681–4689 (2003)
[94] Ju¨licher, F., Ajdari, A. and Prost, J.: Modeling molecular motors. Rev. Mod. Phys. 69(4),1269–1281 (1997)
[95] Karatzas, I. and Shreve, E.S.: Brownian Motion and Stochastic Calculus. New York: Springer-Verlag 1988
[96] Karlin, S.I. and Taylor, H.M.: A First Course in Stochastic Processes. Academic Press 1975
[97] Kawai, R., Parrondo, J.M.R. and Van den Broeck, C.: Dissipation: The phase-space perspec-tive. Phys. Rev. Lett. 98, 0701397 (2007)
[98] Keizer, J.: Master equations, Langevin equations and the effect of diffusion on concentration?uctuations. J. Chem. Phys. 67(4), 1473–1476 (1977)
[99] Keizer, J.: Statistical thermodynamics of nonequilibrium processes. New York: Springer-Verlag 1987
[100] Kitano, H.(北野宏明): Foundations of Systems Biology(系统生物学基础). (刘笔锋,周艳红等译) 化学工业出版社 2007
[101] Kolomeisky, A.B., Stukalin, E.B. and Popov, A.A.: Understanding mechanochemical cou-pling in kinesins using first-passage-time processes. Phys. Rev. E 71 031902(8) (2005)
[102] Koshland, D.E., Jr.: Application of a theory of enzyme specificity to protein synthesis. Proc.Natl. Acad. Sci. USA 44, 98-104 (1958)
[103] Koshland, D.E., Jr.: Mechanisms of transfer enzymes. 305-306 in The Enzymes. 2nded.(Edited by Boyer, P.D., Lardy, H. and Myrback, K.) volume 1, Academic Press, New York1959
[104] Koshland, D.E., Jr.: Enzyme ?exibility an enzyme action. Journal of Cellular and Compar-ative Physiology 54, supplement 1, 245-258 (1959)
[105] Koshland, D.E.,Jr., Nemethy, G. and Filmer, D.: Comparison of experimental binding dataand theoretical models in proteins containing subunits. Biochmistry 5, 365-385 (1966)
[106] Kou, S.C., Cherayil, B.J., Min W., English, B.P. and Xie, X.S.: Single-molecule michaelis-menten equations. J. Phys. Chem. B 109 19068-19081 (2005)
[107] Kramers, H.A.: Brownian motion in a field of force and the diffusion model of chemicalreactions. Physica(Utrecht). 7), 284–304 (1940)
[108] Kurchan, J.: Fluctuation theorem for stochastic dynamics. J. Phys. A: Math. Gen. 31, 3719–3729 (1998)
[109] Kurtz, T.G.: The relationship between stochastic and deterministic midels for chemical reac-tions. J. Chem. Phys. (57), 2976–2978 (1972)
[110] Kurtz, T.G.: Solutions of ordinary differential equations as limits of pure jump Markov pro-cesses. J. Appl. Prob. 7, 49–58 (1970)
[111] Kurtz, T.G.: Limit theorems for sequences of jump Markov processes approximating ordi-nary differential processes. J. Appl. Prob., 8 344–356 (1971)
[112] Kurtz, T.G.: Limit theorems and diffusion approximations for density dependent Markovchains. Math. Progr. Stud., 67(5) 67–78 (1976)
[113] Kurtz, T.G.: Strong approximation theorems for density dependent Markov chains. Stoc.Proc. Appl. 6, 223–240 (1978)
[114] Lasota, A. and Mackey, M.C.: Chaos, fractals, and noise: stochastic aspects of dynamics.New York: Springer-Verlag 1994
[115] Lebowitz, J.L. and Spohn, H.: A Gallavotti-Cohen-type symmetry in the large deviationfunctional for stochastic dynamics. J. Stat. Phys. 95(1–2), 333–365(1999)
[116] Li, F., Long, T., Lu, Y., et al.: The yeast cell-cycle network is robustly designed. Proc. Natl.Acad. Sci. USA 101 4781-4786 (2004)
[117] 林祥, 张汉君和侯振挺: 非时齐马氏链转移矩阵的性质。长沙铁道学院学报 18(3),86–90 (2000)
[118] Liphardt J, Dumont S, Smith S B, Tinoco I and Bustamante C: Equilibrium Information fromNonequilibrium Measurements in an Experimental Test of Jarzynski’s Equality. Science. 296,1832–1835 (2002)
[119] Lu, H.P., Xun, L. and Xie, X.S.: Single-molecule enzymatic dynamics. Science 282 1877-1881 (1998)
[120] Machlup, S. and Onsarger, L.: Fluctuations and irreversible processes.II.systems with kineticenergy. Phys. Rev. 91, 1512–1515 (1953)
[121] Magnasco, M.O.: Forced thermal ratchets. Phys. Rev. Lett. 71, 1477–1481 (1993)
[122] Magnasco, M.O.: Molecular combustion motors. Phys. Rev. Lett. 72, 2656–2659 (1994)
[123] Matkowsky, B.J., Schuss, Z., Knessel, C., Tier, C. and Mangel, M.: in Fluctuations andsensitivity in nonequilibrium systems. edited by Horsthemke, W. and Kondepudi, D.K. NewYork: Springer 1984
[124] McQuarrie, D.A.: Stochastic approach to chemical kinetics. J. Appl. Prob. 4(3), 413-478(1967)
[125] Min, W., English, B. P., Luo, G., Cherayil, B.J., Kou, S.C. and Xie, X. S.: FluctuatingEnzymes: Lessons from Single-Molecule Studies. Acc. Chem. Res. 38 923-931 (2005)
[126] Min, W., Gopich, I.V., English, B.P., Kou, S.C., Xie, X. S. and Szabo, A.: When Does theMichaelis-Menten Equation Hold for Fluctuating Enzymes. J. Phys. Chem. B 110 20093-7(2006)
[127] Monod, J., Changeux, J.P. and Jacob, F.: Allosteric proteins and cellular control systems. J.Mol. Biol. 6, 306-329 (1963)
[128] Monod, J., Wyman, J. and Changeux, J.P.: On the nature of allosteric transitions: a plausiblemodel. J. Mol. Biol. 12, 88-118 (1965)
[129] Moyal, J.E.: Stochastic processes and statistical physics. J. R. Stat. Soc. London, Ser.B.11(2), 150–210 (1949)
[130] Murray, J.D: Mathematical biology, 3rd Ed. New York: Springer 2002
[131] Nicolis, G. and Lefever, R.: Comment on the kinetic potential and the maxwell constructionin non-equilibrium chemical phase transitions. Phys. Lett. A 62, 469 (1977)
[132] Nicolis, G. and Prigogine, I.: Self-organization in nonequilibrium systems: from dissipativestructures to order through ?uctuations. New York: Wiley 1977
[133] Nicolis, G. and Turner, J.W.: . Ann. New York Acad. Sci. 316, 251 (1979)
[134] Onsarger, L. and Machlup, S.: Fluctuations and irreversible processes. Phys. Rev. 91, 1505–1512 (1953)
[135] Oono, Y. and Paniconi, M.: Steady state thermodynamics. Prog. Theor. Phys. Suppl. 130,29–44 (1998)
[136] Oppenheim, I., Schuler, K.E. and Weiss, G.H.: Stochastic and Deterministic Formulation ofChemical Rate Equations. J. Chem. Phys. 50, 460 (1969)
[137] Pauling, L.: The oxygen equilibrium of hemoglobin and its structural interpretation. Proc.Natl. Acad. Sci. USA 21,186-191 (1935)
[138] Perutz, M.F., Rossmann, M.G., Cullis, A.F., Muirhead, H., Will, G. and North, A.C.T.: Struc-ture of haemoglobin: a three-dimensional Fourier synthesis at 5.5 A˙ resolution, obtained byX-ray analysis. Nature 185,416-422 (1960)
[139] Puglisi, A., Rondoni, L. and Vulpiani, A.: Relevance of initial and final conditions for the?uctuation relation in Markov processes. J. Stat. Mech. P08010 (2006)
[140] Pury, P.A. and Caceres, M.O.: Mean first-passage and residence times of random walks onasymetric disordered chains. J. Phys. A: Math. Gen. 36, 2695-2706 (2003)
[141] Qian, H.: Mathemataical formalism for isothermal linear reversibility. Proc. Roy. Soc. Lon-don Ser. A. 457, 1645–1655 (2001)
[142] Qian, H.: Cycle kinetics, steady-state thermodynamics and motors–a paradigm for livingmatter physics. J. Phys. Cond. Matt. 17, S3783-S3794 (2005)
[143] Qian, H.: Open-system nonequilibrium steady-state: statistical thermodynamics, ?uctuationsand chemical oscillations. J. Phys. Chem. B. 110 15063-15074 (2006)
[144] Qian, H. and Elson E.L.: Single-molecule enzymology: stochastic Michaelis-Menten kinet-ics. Biophysical chemistry 101-102 565-576 (2002)
[145] Qian, H., Qian, M. and Tang, X.: Thermodynamics of the general diffusion process: time-reversibility and entropy production. J. Stat. Phys. 107, 1129-1141 (2002)
[146] Qian, H. and Reluga, T.C.: Nonequilibrium thermodynamics and nonlinear kinetics in acellular signaling switch. Phys. Rev. Lett. 94 028101 (2005)
[147] Qian, H., Saffarian S. and Elson E.L.: Concentration ?uctuation in a mesoscopic ocillatingchemical reaction system. Proc. Natl. Acad. Sci. USA 99 10376–10381 (2002)
[148] Qian, H. and Wang H.: Continuous time random walks in closed and open single-moleculesystems with microscopic reversibility. Europhys. Lett. 76(1) 15-21 (2006)
[149] Qian, H. and Xie, X.S.: Generalized Haldane equation and ?uctuation theorem in the steady-state cycle kinetics of single enzymes. Phys. Rev. E 74 010902(4) (2006)
[150] Qian, H.: Thermodynamic and kinetic analysis of sensitivity amplication in biological signaltransduction. Biophys. Chem. 105 585-593 (2003)
[151] Qian, H.: Phosphorylation energy hypothesis: Open chemical systems and their biologicalfunctions. Annu. Rev. Phys. Chem. 58 113-142 (2007)
[152] 钱敏平,龚光鲁: 随机过程(第二版). 北京大学出版社 1997
[153] Qian, M.P., Qian, C. and Qian, M.: Circulations of Markov chains with continuous timeand the probability interpretation of some determinants. Sci. Sinica (Series A) 27(5), 470–481(1984)
[154] Qian, M.P. and Qian, M.: Circulation for recurrent Markov chains. Z. Wahrsch. Verw. Gebiete59, 203–210 (1982)
[155] Qian, M.P. and Qian, M.: The entropy production and reversibility of Markov processes.Science Bulletin 30(3), 165–167 (1985)
[156] Qian, M.P. and Qian, M.: The entropy production and reversibility of Markov processes.In: Prohorov, Yu.A. and Sazonov, V.V.(eds.) Probability theory and applications (volume 1).Proceedings of the 1st World Congress of the Bernoulli Society, Tashkent, USSR 1986, pp.307–316. Utrecht, The Netherlands: VNU Science Press 1987
[157] Qian, M.P., Qian, M. and Gong, G.L.: The reversibility and the entropy production of Markovprocesses. Contemp. Math. 118, 255–261 (1991)
[158] Qian, M, Qian, M.P. and Zhang, X.J.: Fundamental facts concerning reversible master equa-tions. Phys. Lett 309, 371–376 (2003)
[159] 钱敏,张雪娟: 非平衡定态、随机共振和分子马达. 2008 (preprint)
[160] Reid, J.C., Sevick, E.M. and Evans, D.J.: A unified description of two theorems in non-equilibrium statistical mechanics: The ?uctuation theorem and the work relation. Europhys.Lett. 72(5), 726–732 (2005)
[161] Reimann, P.: Brownian motors: noisy transport far from equilibrium. Physics Reports 361,57–265 (2002)
[162] Ricard, J. and Cornish-Bowden, A.: Co-operativity and alosteric enzymes: 20 years on. Eur.J. Biochem. 166,255-272 (1987)
[163] Risken, H.: The Fokker-Planck equation: methods of solution and applications. Berlin:Springer 1989
[164] Schnakenberg, J.: Network theory of microscopic and macroscopic behaviour of masterequation systems. Rev.Mod.Phys. 48, 571–585, (1976)
[165] Schnakenberg, J.: Thermodynamic Network Analysis of Biological Systems. Springer-Verlag,Berlin Heidelberg 1977
[166] Scott, A.C.: Neuroscience: A Mathematical Primer. New York: Springer-Verlag 2002.
[167] Searles, D.J. and Evans, D.J.: Fluctuation theorem for stochastic systems. Phys. Rev. E 60(1),159–164 (1999)
[168] Searles, D.J. and Evans, D.J.: The ?uctuation theorem and Green-Kubo relations. J. Chem.Phys. 112(22), 9727–9735 (2000)
[169] Searles, D.J. and Evans, D.J.: Ensemble dependence of the transient ?uctuation theorem. J.Chem. Phys. 113(9), 3503–3509 (2000)
[170] Searles, D.J. and Evans, D.J.: Fluctuation theorem for heat ?ow. Int. J. Thermophys. 22(1),123–134 (2001)
[171] Sekimoto, K.: Kinetic characterization of heat bath and the energetics of thermal ratchetmodels. J. Phys. Soc. Japan 66, 1234–1237 (1997)
[172] Sel’kov, E.E.: On the mechanism of single-frequency self-oscillations in glycolysis. I. asimple kinetic model. Eur. J. Biochem 4, 79–86 (1968)
[173] Sophia L.Kalpazidou: Cycle representations of Markov processes. (Applications of Mathe-matics. 28) Berlin: Springer-Verlag 1995
[174] Strook, D.W. and Varadhan, S.R.S.: Multidimentional diffusion processes. Berlin, New York:Springer-Verlag 1979
[175] Strogatz, S.: Sync: The Emerging Science of Spontaneous Order. New York:Hyperion 2003
[176] 谭景莹、董志伟主编,李玉瑞审校: 英汉生物化学及分子生物学词典. 科学出版社2000
[177] Van Kampen, N.G.: A power series expansion of the master equations. Can. J. Phys. 39,551–567 (1961)
[178] Van Kampen N.G.: Stochastic processes in physics and chemistry. Amsterdam: North-Holland 1981
[179] Varadhan, S.R.S.: Large deviations and applications. Philadelphia: Society for Industrialand Applied Mathematics 1984
[180] Vellela, M. and Qian, H.: A quasistationary analysis of a stochastic chemical reaction:Keizer’s paradox revisited. Bull. Math. Biol. 69(5), 1727-1746 (2007)
[181] Walters, P.: An introduction to ergodic theory. New York: Springer-Verlag 1982
[182] Wang, H. and Qian, H.: On detailed balance and reversibility of semi-Markov processes andsingle-molecule enzyme kinetics. J. Math. Phys. 48 013303 (2007)
[183] Wang, H., Peskin, C.S. and Elston, T.C.: A robust numerical algorithm for studyingbiomolecular transport processes. J. Theor. Biol. 221, 491–511 (2003)
[184] Wang, S., Zhang, Y. and Ouyang, Q.: Stochastic model of coliphage lambda regulatorynetwork. Phys. Rev. E 73, 041922 (2006)
[185] Wilkinson, D.J.: Stochastic Modelling for Systems Biology. Chapman and Hall/CRC 2006
[186] Winfree, A.T.: The Geometry of Biological Time, 2nd. Ed. Berline: Springer-Verlag 2000
[187] Xie, X.S. and Lu, H.P.: Single-molecule enzymology. J. Biol. Chem. 274(23), 15967–15970(1999)
[188] Xie, X.S.: Single-molecule approach to enzymology. Single Mol. 2(4), 229–236 (2001)
[189] Xie, X.S.: Single-molecule approach to dispersed kinetics and dynamic disorder: Probingconformational ?uctuation and enzymatic dynamics. J. Chem. Phys. 117(24), 11024–11032(2002)
[190] Xing, J., Wang, H.Y. and Oster, G.: From continuum Fokker-Planck models to discrete ki-netic models. Biophys. J. 89, 1551–1563 (2005)
[191] 阎平凡,张长水: 人工神经网络与模拟进化计算. 清华大学出版社2005
[192] Yin, G.G. and Zhang, Q.: Continuous-time Markov chains and applications. (Applicationsof Mathematics 37) Berlin: Springer 1998
[193] Zhang, F.X.: Exclusion processes on groups: entropy production density and reversibility.Phys. A 348, 131–139 (2005)
[194] Zhang, Y., Qian, M.P., Ouyang, Q., et al.: Stochastic model of yeast cell-cycle network.Physica D 219, 35–39 (2006)
[195] Zhang, Y., Yu, H., Deng, M. and Qian, M.P.: Nonequilibrium model for yeast cell cycle.Computational Intelligence and Bioinformatics. International Conference on Intelligent Com-puting, ICIC 2006, Kunming, China, August 16-19, 2006. Proceedings, Part III, 786-791(2006)
[196] 张筑生: 数学分析。共3册 北京大学出版社 1991
[197] Zhou, T.S., Chen, L.N., and Wang, R.Q.: A mechanism of synchronization in interactingmulti-cell genetic systems. Physica D 211, 107–127 (2005)
[198] van Zon, R. and Cohen, E.G.D.: Stationary and transient work-?uctuation theorems for adragged Brownian particle. Phys. Rev. E 67, 046102 (2003)
[199] van Zon, R. and Cohen, E.G.D.: Extension of the Fluctuation Theorem. Phys. Rev. Lett. 91,110601 (2003)
[200] van Zon, R. and Cohen, E.G.D.: Extended heat-?uctuation theorems for a system with deter-ministic and stochastic forces. Phys. Rev. E 69, 056121 (2004)
[2] Amit, D.J.: Modeling brain function: The world of attractor neural networks. CambridgeUniversity Press 1989
[3] Anderson, W.J. : Continuous-time Markov chains: an applications-oriented approach. NewYork: Springer-Verlag 1991
[4] Arizmendi, C.M. and Family, F.: Algorithmic complexity and efficiency of a ratchet. PhysicaA 269, 285–292 (1999)
[5] Astumian, R.D.: Thermodynamics and kinetics of a Brownian motor. Science 276, 917–922(1997)
[6] Astumian, R.D. and Bier, M.: Fluctuation driven ratchets: molecular motors. Phys. Rev. Lett.72, 1766–1769 (1994)
[7] Ayton, G. and Evans, D.J.: On the asymptotic convergence of the transient and steady-state?uctuation theorems. J. Stat. Phys. 97, 811–815 (1999)
[8] Baiesi, M., Jacobs, T., Maes, C. and Skantzos, N.S.: Fluctuation symmetries for work andheat. Phys. Rev. E 74, 021111 (2006)
[9] Bai, C.L., Wang, C., Xie, X.S. and Wolynes, P.G.: Single molecule physics and chemistry.Proc. Natl. Acad. Sci. USA 96 11075-11076 (1999)
[10] Baras, F., Malek Mansour, M. and Pearson, J.E.: Microscopic simulation of chemical bista-bility in homogeneous systems. J. Chem. Phys. 105(18), 8257–8261 (1996)
[11] Beard, D.A. and Qian, H.: Chemical Biophysics: Quantitative Analysis of Cellular Systems.Cambridge Texts in Biomedical Engineering Cambridge University Press 2008.
[12] Berdichevsky, V. and Gitterman, M.: Stochastic resonance and ratchets-new manifestations.Physica A 249, 88–95 (1998)
[13] Berg, O.G., Paulsson, J. and Ehrenberg, M.: Fluctuations and quality of control in biologicalcells: zero-prder ultrasensitivity reinvestigated. Biophys. J. 79 1228-1236 (2000)
[14] Bier, M.: Brownian ratchets in physics and biology. Contemporary Phys. 38(6), 371–379(1997)
[15] Carter, N.J. and Cross, R.A.: Mechanics of the kinesin step. Nature. 435 308-312 (2005)
[16] Chen, K.C., Calzone, L., Csikasz-Nagy, A., Cross, F.R., Novak, B., Tyson, J.J.: Integrativeanalysis of cell cycle control in budding yeast. Mol. Biol. Cell 15 3841-3862 (2004)
[17] Chen, Y., Chen, X. and Qian, M.P.: Green-Kubo formula, autocorrelation function and ?uc-tuation spectrum for finite Markov chains with continuous time. J. Phys. A: Math. Gen. 392539-2550 (2006)
[18] Ciliberto, A., Novak, B., Tyson, J.J.: Mathematical model of the morphogenesis checkpointin budding yeast. J. Cell Biol. 163 1243-1254 (2003)
[19] Cornish-Bowden, A.: Fundamentals of enzyme kinetics. (3nd ed.) London: Portland Press2004
[20] Crooks, G.E.: Nonequilibrium measurements of free energy differences for microscopicallyreversible Markovian systems. J. Stat. Phys. 90, 1481-1487 (1998)
[21] Crooks, G.E.: Entropy production ?uctuation theorem and the nonequilibrium work relationfor free energy differences. Phys. Rev. E 60, 2721-2726 (1999)
[22] Crooks, G.E.: Path-ensemble averages in systems driven far from equilibrium. Phys. Rev. E61(3), 2361-2366 (2000)
[23] Delbru¨ck, M.: Statistical ?uctuations in autocatalytic reactions. J. Chem. Phys. 8, 120–124(1940)
[24] Dollard, J.D. and Friedman, J.N.: Product integration with applications to differential equa-tions. (Encyclopedia of Mathematics and applications 10) Reading [U.S.A.]: Addison-WesleyPublishing Company 1979
[25] Durrett, R.: Probability: Theory and Examples (Second Edition). Pacific Grove, CA: DuxburyPress 1996
[26] Dynkin, E.B.: Markov processes. Vol. 1, 2. Springer-Verlag 1965
[27] Dynkin, E.B.: Die Grundlagen der Theorie der Markoffschen Prozesse. Berlin: Springer-Verlag 1961
[28] Einstein, A.: Uber die von der Molekularkinetischen Theorie der Warme Geforderte Bewe-gung yon in Ruhenden Flüssigkeiten Suspendierten Teilchen. Ann. Phys. (Leipzig) 17, 549–560 (1905)
[29] Elf, J., Paulsson, J., Berg, O.G. and Ehrenberg, M.: Near-critical phenomenon in intercellularmetabolite pools. Biophys. J. 84 154-170 (2003)
[30] Elf, J., Paulsson, J., Berg, O. and Ehrenberg, M.: Mesoscopic kinetics and its applicationsin protein synthesis. In L.Alberghina and H.V.Westerhoff (eds.): Topics in Current Genetics:Systems Biology: Definitions and Perspectives. Springer-Verlag: Berlin 2005
[31] Elston, T.C. and Doering, C.R.: Numerical and analytical studies of nonequilibrium?uctuation-induced transport processes. J. Stat. Phys. 83, 359–383 (1996)
[32] English, B.P., Min, W., van Oijen, A.M., Lee, K.T.; Luo, G., Sun, H., Cherayil, B.J., Kou,S.C. and Xie, X.S.: Ever-?uctuating single enzyme molecules: Michaelis-Menten equationrevisited. Nat. Chem. Bio. 2 87-94 (2006)
[33] Evans, D.J., Cohen, E.G.D. and Morriss, G.P.: Probability of second law violation in steady?ows. Phys. Rev. Lett. 71, 2401–2404 (1993)
[34] Evans, D.J. and Searles, D.J.: Equilibrium microstates which generate second law violatingsteady states. Phys. Rev. E 50(2), 1645–1648 (1994)
[35] Fall, C.P., Marland, E.S., Wagner, J.M. and Tyson, J.J.: Computational cell biology. NewYork: Springer-Verlag 2002
[36] Ferry, R.M., and Green, A.A.: Studies in the chemistry of hemoglobin III: The equilibriumbetween oxygen and hemoglobin and its relation to changing hydrogen ion activity. J. Biol.Chem. 81, 175 (1929)
[37] Fischer, E.: Ein?uss der configuration auf die Wirkung der enzyme. Berichter der DeutschenChemischen Gesellschaft 27, 2985-2993 (1894)
[38] Fischer, E.H. and Krebs, E.G.: Conversion of phosphorylase b to phosphorylase a in muscleextracts. J. Mol. Chem. 216, 121-133 (1955)
[39] Fischer, E.H., Heilmeyer, L.M.G. and Haschke, R.H.: Phosphorylation and the control ofglycogen degradation. Curr. Top. Cell. Regul. 4, 211-251 (1971)
[40] Fox, R.F.: Stochastic versions of the Hodgkin-Huxley equations. Biophys. J. 72 2068-2074(1997)
[41] Fox, R.F. and Lu, Y.: Emergent collective behavior in large numbers of globally coupledindependently stochastic ion channels. Phys. Rev. E 49(4) 3421-3431 (1994)
[42] A. Friedman: Partial differential equations of parabolic type. Englewood Cliffs 1964
[43] Gallavotti, G.: Reversible Anosov diffeomorphisms and large deviations. Math. Phys. Elec-tronic J. 1, 1–12 (1995)
[44] Gallavotti, G.: Chaotic hypothesis: Onsager reciprocity and ?uctuation-dissipation theorem.J. Stat. Phys. 84, 899–926 (1996)
[45] Gallavotti, G. and Cohen, E.G.D.: Dynamical ensembles in stationary states. J. Stat. Phys. 80,931–970 (1995)
[46] Gammaitoni, L., Ha¨nggi, P., Jung, P. and Marchesoni, F.: Stochastic resonance. Rev. Mod.Phys. 70(1), 223–287 (1998)
[47] Gardiner, C.W.: Handbook of Stochastic Methods for Physics, Chemistry and the NaturalSciences. Sringer-Verlag Berlin 1985
[48] Ge, H., Jiang, D.Q. and Qian, M.: A simple discrete model of Brownian motors: time-periodicMarkov chains. J. Stat. Phys. 123(4), 831–859 (2006)
[49] Ge, H., Jiang, D.Q. and Qian, M.: Reversibility and entropy production of inhomogeneousMarkov chains. J. Appl. Prob. 43(4) 1028–1043 (2006)
[50] Ge, H. and Qian, M.: Ge, H., and Qian, M.: Generalized Jarzynski’s equality in inhomoge-neous Markov chains. J. Math. Phys. 48 053302 (2007)
[51] Ge, H., Qian, H. and Qian, M.: Synchronized dynamics and nonequilibrium steady states in ayeast cell-cycle network. Math. Biosci. 211 132-152 (2008)
[52] Ge, H. and Qian, M.: Reversibility and potentiality of exclusion processes on countable dis-crete groups. Appl. Math. Lett. 20 1110-1114 (2007)
[53] Ge, H. and Jiang, D.Q.: Transient ?uctuation theorem of sample entropy production for gen-eral stochastic processes. J. Phys. A 40 F713-F723 (2007)
[54] Ge, H.: Waiting cycle times and generalized Haldane equation in the steady-state cycle kinet-ics of single enzymes. J. Phys. Chem. B 112 61-70 (2008)
[55] Ge, H. and Jiang, D.Q.: Generalized Jarzynski’s equality of multidimensional inhomogeneousdiffusion processes. J. Stat. Phys. 131 675–689 (2008)
[56] Ge, H. and Qian, M.: Steady-state cycle kinetics of single enzymes: competing substrates andmulti-conformations. J. Theor. Comput. Chem. (2008) (accepted)
[57] Ge, H. and Qian, M.: Sensitivity amplification in the phosphorylation-dephosphorylation cy-cle: nonequilibrium steady states, chemical master equation and temporal cooperativity. J.Chem. Phys. (2008) (accepted)
[58] Gillespie, D.T.: A general method for numerically simulating the stochastic time evolution ofcoupled chemical reactions. J. Comput. Phys. 22, 403 (1976)
[59] Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem.81(25), 2340–2361 (1977)
[60] Gillespie, D.T.: Approximating the master equation by Fokker-Planck-type equations forsingle-variable chemical systems J. Chem. Phys. 72(10), 5363–5370 (1980)
[61] Gillespie, D.T.: A rigorous derivation of the chemical master equation. Physica A 188, 404–425 (1992)
[62] Gillespie, D.T.: The chemical Langevin equations. J. Chem. Phys. 113(1), 297–306 (2000)
[63] Gillespie, D.T.: The chemical Langevin and Fokker-Planck equations for the reversible iso-merization reaction. J. Phys. Chem. A 106, 5063–5071 (2002)
[64] Glansdorff, P. and Prigogine, I.: Thermodynamic theory of structure, stability and ?uctuations.London: Wiley-Interscience 1971
[65] Goldbeter, A.: Biochemical Oscillations and Cellular Rhythms: the Molecular Bases of Peri-odic and Chaotic Behaviour, New York: Cambridge Univ. Press 1996
[66] Goldbeter, A. and Koshland, Jr. D.E.: An amplified sensitivity arising from covalent modifi-cation in biological systems. Proc. Natl. Acad. Sci. USA 78 6840-6844 (1981)
[67] Haken, H.: Synergetics: an introduction: nonequilibrium phase transitions and self-organization in physics, chemistry, and biology. Berlin, New York: Springer-Verlag 1977
[68] Haken, H.: Advanced synergetics: instability hierarchies of self-organizing systems and de-vices. Berlin, New York: Springer-Verlag 1983
[69] Hanggi, P., Grabert, H., Talkner, P. and Thomas, H.: Bistable systems: Master equation versusFokker-Planck modeling. Phys. Rev. A 29(1), 371–378 (1984)
[70] Hasegawa, H.: On the construction of a time-reversed Markoff process, Prog. Theor. Phys. 55,90–105 (1976);Variational principle for non-equilibrium states and the Onsager-Machlup formula, ibid. 56,44–60 (1976);Thermodynamic properties of non-equilibrium states subject to Fokker-Planck equations, ibid.57, 1523–1537 (1977);Variational approach in studies with Fokker-Planck equations, ibid. 58, 128–146 (1977)
[71] Haber, J.E. and Koshland, D.E.: Relation of protein subunit interactions to the molecularspecies observed during cooperative binding of ligands. Proc. Natl. Acad. Sci. USA 58, 2087-2093 (1967)
[72] Has’minskii, R.Z.: Stochastic stability of differential equations. Alphen aan den Rijn-Germantown: Sijthoff and Noorrdhoff 1980
[73] Hatano, T. and Sasa, S.: Steady-states thermodynamics of Langevin systems. Phys. Rev. Lett.E. 86, 3463–3466 (2001)
[74] Hill, A.V.: The possible effects of the aggregation of the molecules of haemoglobin on itsdissociation curves. Journal of Physiology 40, iv-vii (1910)
[75] Hill, T.L. and Plesner, I.W.: Studies in irreversible thermodynamics II. A simple class of latticemodels for open systems. J. Chem. Phys. 43, 267-285 (1965)
[76] Hill, T.L.: Studies in irreversible thermodynamics IV. diagrammatic representation of steadystate ?uxes for unimolecular systems. J. Theoret. Biol. 10, 442-459 (1966)
[77] Hill, T.L.: Approach for certain systems, including membranes, to steady state. J. Chem. Phys.54, 34-35 (1971)
[78] Hill, T.L.: Free energy transduction in biology. New York: Academic Press 1977
[79] Hill, T.L.: Free energy transduction and biochemical cycle kinetics. New York: Springer-Verlag 1995
[80] Hill, T. and Chen, Y.: Stochastics of cycle completions (?uxes) in biochemical kinetic dia-grams. Proc. Natl. Acad. Sci. USA 72, 1291–1295 (1975)
[81] Hopfield, J.J.: Neural networks and physical systems with emergent collective computationalabilities. Proc. Natl. Acad. Sci. USA 79 2554-2558 (1982)
[82] Hopfield, J.J.: Neurons with graded response have collective computational properties likethose of two-state neurons. Proc. Natl. Acad. Sci. USA 81 3088-3092 (1984)
[83] Hopfield, J.J. and Herz, Andreas V.M.: Rapid local synchronization of action potentials: To-ward computation with coupled integrate-and-fire neurons. Proc. Natl. Acad. Sci. USA 926655-6662 (1995)
[84] 胡迪鹤: 可数状态的马尔可夫过程论. 武汉大学出版社 1983
[85] Hummer, G. and Szabo, A.: Free energy reconstruction from nonequilibrium single-moleculepulling experiments. Proc. Natl. Acad. Sci. USA 98(7), 3658–3661 (2001)
[86] Jarzynski, C.: Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78, 2690–2693 (1997)
[87] Jarzynski, C.: Equilibrium free-energy differences from nonequilibrium measurements: Amaster-equation approach. Phys. Rev. E 56, 5018–5035 (1997)
[88] Jarzynski, C.: Microscopic analysis of Clausius–Duhem processes. J. Stat. Phys. 96, 415-427 (1999)
[89] Jarzynski, C.: Hamiltonian derivation of a detailed ?uctuation theorem. J. Stat. Phys. 98,77–102 (2000)
[90] Jarzynski, C.: Rare events and the convergence of exponentially averaged work values. Phys.Rev. E 73, 046105 (2006)
[91] Jiang, D.Q., Qian, M. and Qian, M.P.: Mathematical theory of nonequilibrium steady states- On the frontier of probability and dynamical systems. (Lect. Notes Math. 1833) Berlin:Springer-Verlag 2004
[92] Jiang, D.Q., Qian, M. and Zhang, F.X.: Entropy production ?uctuations of finite Markovchains. J. Math. Phys. 44(9), 4176–4188 (2003)
[93] Jiang, D.Q. and Zhang, F.X.: The Green-Kubo formula and power spectrum of reversibleMarkoc processes. J. Math. Phys. 44(10), 4681–4689 (2003)
[94] Ju¨licher, F., Ajdari, A. and Prost, J.: Modeling molecular motors. Rev. Mod. Phys. 69(4),1269–1281 (1997)
[95] Karatzas, I. and Shreve, E.S.: Brownian Motion and Stochastic Calculus. New York: Springer-Verlag 1988
[96] Karlin, S.I. and Taylor, H.M.: A First Course in Stochastic Processes. Academic Press 1975
[97] Kawai, R., Parrondo, J.M.R. and Van den Broeck, C.: Dissipation: The phase-space perspec-tive. Phys. Rev. Lett. 98, 0701397 (2007)
[98] Keizer, J.: Master equations, Langevin equations and the effect of diffusion on concentration?uctuations. J. Chem. Phys. 67(4), 1473–1476 (1977)
[99] Keizer, J.: Statistical thermodynamics of nonequilibrium processes. New York: Springer-Verlag 1987
[100] Kitano, H.(北野宏明): Foundations of Systems Biology(系统生物学基础). (刘笔锋,周艳红等译) 化学工业出版社 2007
[101] Kolomeisky, A.B., Stukalin, E.B. and Popov, A.A.: Understanding mechanochemical cou-pling in kinesins using first-passage-time processes. Phys. Rev. E 71 031902(8) (2005)
[102] Koshland, D.E., Jr.: Application of a theory of enzyme specificity to protein synthesis. Proc.Natl. Acad. Sci. USA 44, 98-104 (1958)
[103] Koshland, D.E., Jr.: Mechanisms of transfer enzymes. 305-306 in The Enzymes. 2nded.(Edited by Boyer, P.D., Lardy, H. and Myrback, K.) volume 1, Academic Press, New York1959
[104] Koshland, D.E., Jr.: Enzyme ?exibility an enzyme action. Journal of Cellular and Compar-ative Physiology 54, supplement 1, 245-258 (1959)
[105] Koshland, D.E.,Jr., Nemethy, G. and Filmer, D.: Comparison of experimental binding dataand theoretical models in proteins containing subunits. Biochmistry 5, 365-385 (1966)
[106] Kou, S.C., Cherayil, B.J., Min W., English, B.P. and Xie, X.S.: Single-molecule michaelis-menten equations. J. Phys. Chem. B 109 19068-19081 (2005)
[107] Kramers, H.A.: Brownian motion in a field of force and the diffusion model of chemicalreactions. Physica(Utrecht). 7), 284–304 (1940)
[108] Kurchan, J.: Fluctuation theorem for stochastic dynamics. J. Phys. A: Math. Gen. 31, 3719–3729 (1998)
[109] Kurtz, T.G.: The relationship between stochastic and deterministic midels for chemical reac-tions. J. Chem. Phys. (57), 2976–2978 (1972)
[110] Kurtz, T.G.: Solutions of ordinary differential equations as limits of pure jump Markov pro-cesses. J. Appl. Prob. 7, 49–58 (1970)
[111] Kurtz, T.G.: Limit theorems for sequences of jump Markov processes approximating ordi-nary differential processes. J. Appl. Prob., 8 344–356 (1971)
[112] Kurtz, T.G.: Limit theorems and diffusion approximations for density dependent Markovchains. Math. Progr. Stud., 67(5) 67–78 (1976)
[113] Kurtz, T.G.: Strong approximation theorems for density dependent Markov chains. Stoc.Proc. Appl. 6, 223–240 (1978)
[114] Lasota, A. and Mackey, M.C.: Chaos, fractals, and noise: stochastic aspects of dynamics.New York: Springer-Verlag 1994
[115] Lebowitz, J.L. and Spohn, H.: A Gallavotti-Cohen-type symmetry in the large deviationfunctional for stochastic dynamics. J. Stat. Phys. 95(1–2), 333–365(1999)
[116] Li, F., Long, T., Lu, Y., et al.: The yeast cell-cycle network is robustly designed. Proc. Natl.Acad. Sci. USA 101 4781-4786 (2004)
[117] 林祥, 张汉君和侯振挺: 非时齐马氏链转移矩阵的性质。长沙铁道学院学报 18(3),86–90 (2000)
[118] Liphardt J, Dumont S, Smith S B, Tinoco I and Bustamante C: Equilibrium Information fromNonequilibrium Measurements in an Experimental Test of Jarzynski’s Equality. Science. 296,1832–1835 (2002)
[119] Lu, H.P., Xun, L. and Xie, X.S.: Single-molecule enzymatic dynamics. Science 282 1877-1881 (1998)
[120] Machlup, S. and Onsarger, L.: Fluctuations and irreversible processes.II.systems with kineticenergy. Phys. Rev. 91, 1512–1515 (1953)
[121] Magnasco, M.O.: Forced thermal ratchets. Phys. Rev. Lett. 71, 1477–1481 (1993)
[122] Magnasco, M.O.: Molecular combustion motors. Phys. Rev. Lett. 72, 2656–2659 (1994)
[123] Matkowsky, B.J., Schuss, Z., Knessel, C., Tier, C. and Mangel, M.: in Fluctuations andsensitivity in nonequilibrium systems. edited by Horsthemke, W. and Kondepudi, D.K. NewYork: Springer 1984
[124] McQuarrie, D.A.: Stochastic approach to chemical kinetics. J. Appl. Prob. 4(3), 413-478(1967)
[125] Min, W., English, B. P., Luo, G., Cherayil, B.J., Kou, S.C. and Xie, X. S.: FluctuatingEnzymes: Lessons from Single-Molecule Studies. Acc. Chem. Res. 38 923-931 (2005)
[126] Min, W., Gopich, I.V., English, B.P., Kou, S.C., Xie, X. S. and Szabo, A.: When Does theMichaelis-Menten Equation Hold for Fluctuating Enzymes. J. Phys. Chem. B 110 20093-7(2006)
[127] Monod, J., Changeux, J.P. and Jacob, F.: Allosteric proteins and cellular control systems. J.Mol. Biol. 6, 306-329 (1963)
[128] Monod, J., Wyman, J. and Changeux, J.P.: On the nature of allosteric transitions: a plausiblemodel. J. Mol. Biol. 12, 88-118 (1965)
[129] Moyal, J.E.: Stochastic processes and statistical physics. J. R. Stat. Soc. London, Ser.B.11(2), 150–210 (1949)
[130] Murray, J.D: Mathematical biology, 3rd Ed. New York: Springer 2002
[131] Nicolis, G. and Lefever, R.: Comment on the kinetic potential and the maxwell constructionin non-equilibrium chemical phase transitions. Phys. Lett. A 62, 469 (1977)
[132] Nicolis, G. and Prigogine, I.: Self-organization in nonequilibrium systems: from dissipativestructures to order through ?uctuations. New York: Wiley 1977
[133] Nicolis, G. and Turner, J.W.: . Ann. New York Acad. Sci. 316, 251 (1979)
[134] Onsarger, L. and Machlup, S.: Fluctuations and irreversible processes. Phys. Rev. 91, 1505–1512 (1953)
[135] Oono, Y. and Paniconi, M.: Steady state thermodynamics. Prog. Theor. Phys. Suppl. 130,29–44 (1998)
[136] Oppenheim, I., Schuler, K.E. and Weiss, G.H.: Stochastic and Deterministic Formulation ofChemical Rate Equations. J. Chem. Phys. 50, 460 (1969)
[137] Pauling, L.: The oxygen equilibrium of hemoglobin and its structural interpretation. Proc.Natl. Acad. Sci. USA 21,186-191 (1935)
[138] Perutz, M.F., Rossmann, M.G., Cullis, A.F., Muirhead, H., Will, G. and North, A.C.T.: Struc-ture of haemoglobin: a three-dimensional Fourier synthesis at 5.5 A˙ resolution, obtained byX-ray analysis. Nature 185,416-422 (1960)
[139] Puglisi, A., Rondoni, L. and Vulpiani, A.: Relevance of initial and final conditions for the?uctuation relation in Markov processes. J. Stat. Mech. P08010 (2006)
[140] Pury, P.A. and Caceres, M.O.: Mean first-passage and residence times of random walks onasymetric disordered chains. J. Phys. A: Math. Gen. 36, 2695-2706 (2003)
[141] Qian, H.: Mathemataical formalism for isothermal linear reversibility. Proc. Roy. Soc. Lon-don Ser. A. 457, 1645–1655 (2001)
[142] Qian, H.: Cycle kinetics, steady-state thermodynamics and motors–a paradigm for livingmatter physics. J. Phys. Cond. Matt. 17, S3783-S3794 (2005)
[143] Qian, H.: Open-system nonequilibrium steady-state: statistical thermodynamics, ?uctuationsand chemical oscillations. J. Phys. Chem. B. 110 15063-15074 (2006)
[144] Qian, H. and Elson E.L.: Single-molecule enzymology: stochastic Michaelis-Menten kinet-ics. Biophysical chemistry 101-102 565-576 (2002)
[145] Qian, H., Qian, M. and Tang, X.: Thermodynamics of the general diffusion process: time-reversibility and entropy production. J. Stat. Phys. 107, 1129-1141 (2002)
[146] Qian, H. and Reluga, T.C.: Nonequilibrium thermodynamics and nonlinear kinetics in acellular signaling switch. Phys. Rev. Lett. 94 028101 (2005)
[147] Qian, H., Saffarian S. and Elson E.L.: Concentration ?uctuation in a mesoscopic ocillatingchemical reaction system. Proc. Natl. Acad. Sci. USA 99 10376–10381 (2002)
[148] Qian, H. and Wang H.: Continuous time random walks in closed and open single-moleculesystems with microscopic reversibility. Europhys. Lett. 76(1) 15-21 (2006)
[149] Qian, H. and Xie, X.S.: Generalized Haldane equation and ?uctuation theorem in the steady-state cycle kinetics of single enzymes. Phys. Rev. E 74 010902(4) (2006)
[150] Qian, H.: Thermodynamic and kinetic analysis of sensitivity amplication in biological signaltransduction. Biophys. Chem. 105 585-593 (2003)
[151] Qian, H.: Phosphorylation energy hypothesis: Open chemical systems and their biologicalfunctions. Annu. Rev. Phys. Chem. 58 113-142 (2007)
[152] 钱敏平,龚光鲁: 随机过程(第二版). 北京大学出版社 1997
[153] Qian, M.P., Qian, C. and Qian, M.: Circulations of Markov chains with continuous timeand the probability interpretation of some determinants. Sci. Sinica (Series A) 27(5), 470–481(1984)
[154] Qian, M.P. and Qian, M.: Circulation for recurrent Markov chains. Z. Wahrsch. Verw. Gebiete59, 203–210 (1982)
[155] Qian, M.P. and Qian, M.: The entropy production and reversibility of Markov processes.Science Bulletin 30(3), 165–167 (1985)
[156] Qian, M.P. and Qian, M.: The entropy production and reversibility of Markov processes.In: Prohorov, Yu.A. and Sazonov, V.V.(eds.) Probability theory and applications (volume 1).Proceedings of the 1st World Congress of the Bernoulli Society, Tashkent, USSR 1986, pp.307–316. Utrecht, The Netherlands: VNU Science Press 1987
[157] Qian, M.P., Qian, M. and Gong, G.L.: The reversibility and the entropy production of Markovprocesses. Contemp. Math. 118, 255–261 (1991)
[158] Qian, M, Qian, M.P. and Zhang, X.J.: Fundamental facts concerning reversible master equa-tions. Phys. Lett 309, 371–376 (2003)
[159] 钱敏,张雪娟: 非平衡定态、随机共振和分子马达. 2008 (preprint)
[160] Reid, J.C., Sevick, E.M. and Evans, D.J.: A unified description of two theorems in non-equilibrium statistical mechanics: The ?uctuation theorem and the work relation. Europhys.Lett. 72(5), 726–732 (2005)
[161] Reimann, P.: Brownian motors: noisy transport far from equilibrium. Physics Reports 361,57–265 (2002)
[162] Ricard, J. and Cornish-Bowden, A.: Co-operativity and alosteric enzymes: 20 years on. Eur.J. Biochem. 166,255-272 (1987)
[163] Risken, H.: The Fokker-Planck equation: methods of solution and applications. Berlin:Springer 1989
[164] Schnakenberg, J.: Network theory of microscopic and macroscopic behaviour of masterequation systems. Rev.Mod.Phys. 48, 571–585, (1976)
[165] Schnakenberg, J.: Thermodynamic Network Analysis of Biological Systems. Springer-Verlag,Berlin Heidelberg 1977
[166] Scott, A.C.: Neuroscience: A Mathematical Primer. New York: Springer-Verlag 2002.
[167] Searles, D.J. and Evans, D.J.: Fluctuation theorem for stochastic systems. Phys. Rev. E 60(1),159–164 (1999)
[168] Searles, D.J. and Evans, D.J.: The ?uctuation theorem and Green-Kubo relations. J. Chem.Phys. 112(22), 9727–9735 (2000)
[169] Searles, D.J. and Evans, D.J.: Ensemble dependence of the transient ?uctuation theorem. J.Chem. Phys. 113(9), 3503–3509 (2000)
[170] Searles, D.J. and Evans, D.J.: Fluctuation theorem for heat ?ow. Int. J. Thermophys. 22(1),123–134 (2001)
[171] Sekimoto, K.: Kinetic characterization of heat bath and the energetics of thermal ratchetmodels. J. Phys. Soc. Japan 66, 1234–1237 (1997)
[172] Sel’kov, E.E.: On the mechanism of single-frequency self-oscillations in glycolysis. I. asimple kinetic model. Eur. J. Biochem 4, 79–86 (1968)
[173] Sophia L.Kalpazidou: Cycle representations of Markov processes. (Applications of Mathe-matics. 28) Berlin: Springer-Verlag 1995
[174] Strook, D.W. and Varadhan, S.R.S.: Multidimentional diffusion processes. Berlin, New York:Springer-Verlag 1979
[175] Strogatz, S.: Sync: The Emerging Science of Spontaneous Order. New York:Hyperion 2003
[176] 谭景莹、董志伟主编,李玉瑞审校: 英汉生物化学及分子生物学词典. 科学出版社2000
[177] Van Kampen, N.G.: A power series expansion of the master equations. Can. J. Phys. 39,551–567 (1961)
[178] Van Kampen N.G.: Stochastic processes in physics and chemistry. Amsterdam: North-Holland 1981
[179] Varadhan, S.R.S.: Large deviations and applications. Philadelphia: Society for Industrialand Applied Mathematics 1984
[180] Vellela, M. and Qian, H.: A quasistationary analysis of a stochastic chemical reaction:Keizer’s paradox revisited. Bull. Math. Biol. 69(5), 1727-1746 (2007)
[181] Walters, P.: An introduction to ergodic theory. New York: Springer-Verlag 1982
[182] Wang, H. and Qian, H.: On detailed balance and reversibility of semi-Markov processes andsingle-molecule enzyme kinetics. J. Math. Phys. 48 013303 (2007)
[183] Wang, H., Peskin, C.S. and Elston, T.C.: A robust numerical algorithm for studyingbiomolecular transport processes. J. Theor. Biol. 221, 491–511 (2003)
[184] Wang, S., Zhang, Y. and Ouyang, Q.: Stochastic model of coliphage lambda regulatorynetwork. Phys. Rev. E 73, 041922 (2006)
[185] Wilkinson, D.J.: Stochastic Modelling for Systems Biology. Chapman and Hall/CRC 2006
[186] Winfree, A.T.: The Geometry of Biological Time, 2nd. Ed. Berline: Springer-Verlag 2000
[187] Xie, X.S. and Lu, H.P.: Single-molecule enzymology. J. Biol. Chem. 274(23), 15967–15970(1999)
[188] Xie, X.S.: Single-molecule approach to enzymology. Single Mol. 2(4), 229–236 (2001)
[189] Xie, X.S.: Single-molecule approach to dispersed kinetics and dynamic disorder: Probingconformational ?uctuation and enzymatic dynamics. J. Chem. Phys. 117(24), 11024–11032(2002)
[190] Xing, J., Wang, H.Y. and Oster, G.: From continuum Fokker-Planck models to discrete ki-netic models. Biophys. J. 89, 1551–1563 (2005)
[191] 阎平凡,张长水: 人工神经网络与模拟进化计算. 清华大学出版社2005
[192] Yin, G.G. and Zhang, Q.: Continuous-time Markov chains and applications. (Applicationsof Mathematics 37) Berlin: Springer 1998
[193] Zhang, F.X.: Exclusion processes on groups: entropy production density and reversibility.Phys. A 348, 131–139 (2005)
[194] Zhang, Y., Qian, M.P., Ouyang, Q., et al.: Stochastic model of yeast cell-cycle network.Physica D 219, 35–39 (2006)
[195] Zhang, Y., Yu, H., Deng, M. and Qian, M.P.: Nonequilibrium model for yeast cell cycle.Computational Intelligence and Bioinformatics. International Conference on Intelligent Com-puting, ICIC 2006, Kunming, China, August 16-19, 2006. Proceedings, Part III, 786-791(2006)
[196] 张筑生: 数学分析。共3册 北京大学出版社 1991
[197] Zhou, T.S., Chen, L.N., and Wang, R.Q.: A mechanism of synchronization in interactingmulti-cell genetic systems. Physica D 211, 107–127 (2005)
[198] van Zon, R. and Cohen, E.G.D.: Stationary and transient work-?uctuation theorems for adragged Brownian particle. Phys. Rev. E 67, 046102 (2003)
[199] van Zon, R. and Cohen, E.G.D.: Extension of the Fluctuation Theorem. Phys. Rev. Lett. 91,110601 (2003)
[200] van Zon, R. and Cohen, E.G.D.: Extended heat-?uctuation theorems for a system with deter-ministic and stochastic forces. Phys. Rev. E 69, 056121 (2004)