Applications Of Stochastic Processes Into Nonequilibrium Statistical Physics And Systems Biology

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".