【作者】 葛颢；

【导师】 钱敏； 陈大岳； 蒋达权；

【作者基本信息】 北京大学 ， 概率论与数理统计， 2008， 博士

【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。更多还原

【Abstract】 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”.更多还原