| Because of the discreteness of molecular dynamics and thermodynamic fluctuations,chemical kinetics in cells is,by its very nature,stochastic.Thus,it is very important to study the stochastic dynamics of chemical reactions.The chemical master equation(CME)is a popular tool to study this kind of problems.However,it is very difficult,if not impossible,to solve the chemical master equation exactly for complex systems involving many reactants.Fortunately,chemical reaction process can be accurately simulated by stochastic simulation algorithm(SSA)proposed by Gillespie.The commonly used numerical methods for approximating CME are the chemical Langevin equation(CLE),the linear noise approximation(LNA)and the moment-closure approximation(MA),the moment expansion approximation and the approximate effective mesoscopic rate equations(EMREs).The main purpose of this paper is to apply these methods to the simple systems consisting of zero-order and first-order reactions,the Lotka-Volterra(LV)model,the toggle switch model,the Goodwin oscillator.Their stochastic and deterministic dynamic behaviors are analyzed and compared.In investigating stochastic reaction systems,this paper mainly displays single sample orbit of each system and the average behavior and stationary distribution of each system is analysed.The moment equations of systems with second-order and higher-order reactions are not closed.Therefore,the moment approximation method is given,and the effectiveness of different moment approximation methods is analyzed in this paper,that is to say,the moment closure approximation method can effectively approximate the system of moment equations under what system volumes.This article is divided into four chapters:Chapter 1 presents the background of this research and introduces the basic knowledge of SSA,CLE,LNA,approximate EMREs,moment expansion methods and moment closure approximations(normal MA,Possion MA,CMN-MA).In the second chapter,the dynamics of the simple system consisting of zero-order and first-order reactions and the damped oscillated LV system is studied.for a reaction system containing only zero-order and first-order reactions,the stochastic mean is equal to the deterministic concentration and moment equations are closed.In addition,this chapter also studies the random behavior of the LV system with damped oscillation.The steady-state distribution of the LV system with damped oscillation shows its random behavior is not consistent with its deterministic behavior.Because the moment equations of the LV system with damped oscillation are not closed,LNA,approximate EMREs,second-order normal MA and second-order Poisson MA are used to approximate the moment equations derived from CME.The validity of the approximation methods is discussed.The second-order moment obtained by LNA and approximate EMREs is negative,and the first-order moment obtained by the second-order normal MA and the second-order Poisson MA is quite different from the stochastic mean obtained by SSA,thus they are invalid.In the third chapter,the LV system and toggle switch model are investigated.The deterministic LV system oscillates periodically.The stochastic simulation results of LV system show that either the extinction of one reactant leads to the extinction of another reactant or the extinction of one reactant leads to the infinite growth of another reactant.The relationship between the average extinction time and the volume of the system is given.The deterministic toggle switch is bistable,and the steady state of the stochastic toggle switch system has a bimodal distribution,so the stochastic toggle switch is also bistable.In addition,the numerical simulation results of the second-order normal MA and Poisson MA are compared with the stochastic mean,and the validity of the two moment closure methods is examined.In the last chapter,the stochastic behaviors of the classical Goodwin oscillator and the Goodwin oscillator with enzyme of M-M degradation are investigated.The two reaction systems are oscillatory(sustained oscillation or damped oscillation)regardless of the Hill coefficient.A large number of experimental results show that the classical Goodwin oscillator oscillates continuously when the Hill coefficient is greater than 6 in the presence of noise and the Goodwin oscillator with enzyme of M-M degradation oscillates continuously when the Hill coefficient is greater than or equal to 1 in the presence of noise.Although each stochastic orbit of the two reaction systems is oscillatory,the stochastic mean converges to a stable point.The second-order normal MA and the second-order Poisson MA are also used in this chapter to approximate the moment equations of two reaction systems and the validity intervals of system volume for the two methods are obtained under different Hill coefficients.Finally,the main contributions of this paper are summarized and some problems to be solved are put forward. |
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