The Stochastic Simulations For Biochemical Reaction Kinetics

Stochastic problems arise extensively in a variety of fields of natural science,such as physics,chemistry,biology and engineering([1,2,13,54]).Stochastic differential equations are one of the frequently used tools modeling stochastic phenomena([23,27,29,33]).How-ever,most stochastic differential equations are difficult or impossible to solve analytically due to their complexity.Therefore the numerical solution has become an indispensable approach.The stochastic simulation algorithm(SSA)initiated by Gillespie([4,9,12,14,18])is an exact and effective numerical method for the simulation of chemical reactions.Unfortunately,for complex chemical reaction systems,such as systems with a largely various reactions and a largely various reaction rates,SSA is of low efficiency,which calls for improvements.This thesis investigates pros and cons of SSA as well as some derivative algorithms,the composite algorithm of the stochastic simulation algorithm and traditional implicit methods and the stochastic Nystrom method.This thesis is divided into four chapters:Chapter 1 gives the research background of this thesis and introduces the basic elements and development of stochastic differential equations.In Chapter 2,we mainly study the SSA,the descendable r-leap algorithm,the chemical Langevin equation(CLE),the slow-scale SSA(ssSSA).This chapter indicates in detail the process of the time to the next reaction and index to the next reaction,so do the propensity function in general and calculation of the slow-scale propensity function and then some nu-merical experiments are done.We also study the ?-leap algorithm and ?-leap EM method deduced by the chemical Langevin equation.And then we put forward and analyze the r-leap Heun method inspired by the Euler-Heun method([4,5]).The SSA is an exact solution.Biochemical reactions can be transferred to differential equations(not always)and the process is reversible.Besides,we need to know ahead the next value when we use traditional implicit methods in a computer.So in chapter 3 we combine the SSA and implicit methods(implicit Euler method,midpoint method,trapezoid method),denoted as the implicit SSA.And the implicit SSA are more suitable for the long-time and large-step calculation than traditional implicit methods after a lot of numerical experiments.At last,we tell why the midpoint method and trapezoid method are superior to the implicit Euler method from the perspective of the propensity function.For biochemical reactions,the SSA tells the intrinsic noise,inspired by which the chapter 4 mainly studies the external noise.And the method we use is the stochastic Nystrom method with additive noise,denoted as SN.We compare it with the stochastic Runge-Kutta(SRK)method from the aspect of moments after the definition of SN has been given to illustrate that SN is more accurate than SRK.Then the strong and weak order,stability and convergence are discussed.Finally numerical experiments are given from a simple unimolecular decomposition model to a genetic regulatory network model to state that SN can better perform the cell behavior.In the last chapter,the thesis summarizes main contributions and some problems to be solved.And then we look ahead into the future direction.