| Acquired immunodeficiency syndrome (AIDS) caused by human immun-odeficiency virus (HIV)infection, is one of the world’s ten largest fatal disease, seriously threatened human life and health.With highspeed, widespread, and high mortality rate, AIDS hindered the economic development of the world, and has become a very urgent social problem.Group specific antigen (Gag) protein is the main component of HIV,which has significance of HIV’s lifecycle.Late stage of HIV’s replication includes as-sembly,release and maturation. This paper studies the two preparation stages before assembly which the Gag protein participated in:Gag trafficking insid-e the cytoplasm, Gag trimerization at the plasma.The trafficking process is a basic step of successful infection,as well as the first step of gene transfer.The trimerization process is the key before HIV assembly,as it produces multi-mers.These multimers concentrate at the plasma membrane,which speed up virion assembly.After that,an immature new virus particles will be formed and separated from the cells.By studying these two reactions,we can enlarge the understanding of Gag’s role in HIV lifecycle.These two reactions can be used as a reference for the control of HIV infection and relieve AIDS disease.This paper researches the above two processes from the perspective of mathematics in four chapters. Chapter one is the research background,introducing the formation of human cell,the lifecycle of HIV,the action of Gag protein on the late stage of HIV’s replication,and the research status of Gag protein.Then introduces some basic mathematical definition and methods,and gives the main points and innovation of this paper.In chapter two,we research the model of Gag proteins trafficking toward-s the cell membrane.Firstly we use the perturbation method to get the ini-tial approximate analytical solution.Then,by analyzing the differences between perturbation solution and numerical solution,we estimate the protein concen-tration in the neibourhood of cell membrane with the law of conservation of mass,and calculate the analytical expression of the time for the first virions to appear (written as Tv).Finally we discuss the relation between generation time of virus and some factors in detail.such as cell size,active trafficking speed of Gag protein (written as s),generate speed(written as g1),and diffusion coef-ficient (written as D).In chapter three,we research a trimerization model of Gag monomeric pro-tein on the inner wall of the cell membrance.First,we use singular perturbation theory to get an approximate analytical solution of the model which is uni-formly valid. Then, we discuss the stability of Gag protein in the equilibrium concentration.From the trajectory map,we identify two stages of monomer and trimer reaction,estimate the dimensionless expression of the reaction duration on the first stage, and get a conclusion that the reaction duration on the first stage is quite short.Finally we diagram the results of (ε= 0.1,0.5,1) and dif-ferent initial conditions,getting the numerical solution of trimer reaction and external perturbation solution,compare the change trend of the solutions with the initial conditions over time, and analyze the two main reasons of error.The fourth chapter is the summary of the thesis,summarizes the theoreti-cal and practical significance of this paper,points out the deficiency and some problems to be solved in the future. |
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