Numerical Methods For Genetic Regulatory Networks

The exploration on gene regulation is one of the key problems in molecular biology and cell biology. The investigation of genetic regulatory networks allows us to understand the cell mechanisms more clearly, and also has important theoretical and practical significance in biopharmaceuticals and clinical medicine. The principal methods for the simulation of genetic regulatory networks (GRNs) have been Rung-Kutta (RK) methods, which often fail to produce satisfactory results or even lose effect in long-term integration with large step sizes. The purpose of this paper is to develop structure-preserving and highly effective algorithms for the simulation of genetic regulatory networks based on some of their characteristic structures.This paper is divided into four chapters:Chapter 1 gives the background of this paper and presents basic principles of genetic regulations, mathematical models of GRNs, especially ordinary differential equation models. We also discussed fundamental research methods.In chapter 2, splitting methods based on Runge-Kutta (RK) methods are developed to solve the gene regulation systems with a structure of globally stable equlibrium. The vector field of the ODEs of GRNs into two parts, the linear principal part and the higher order nonlinear part. A special type Lie-Trotter splitting method is obtained by using the exact flow for the linear principal part and the numerical flow of a classical Runge-Kutta method for the nonlinear part. Four practical splitting methods, Split(exact:Euler), Split(exact: Heun), Split(exact:RK4) and Split(exact:RK3/8) are derived. When applied to the one-gene self-regulatory network, the two-gene cross-regulatory network and tumor-repressing proteins p53-Mdm2 regulatory network, splitting methods are shown to be more accurate and more efficient than Euler, Heun, RK4 and RK3/8 methods. Moreover, splitting methods are more suitable for the long-time integration with large step sizes. By analysis, a splitting method has a larger stability region than the corresponding RK method.In chapter 3, we consider two-derivative Runge-Kutta (TDRK) methods. These methods have the property that they integrate exactly the first order and second order structures of the true solution. When applied to solve the two models with phosphorylation, Drosophila period protein and mammalian circadian clock, the two-stage fourth-order method TDRK2s4 is shown to be more accurate and more efficient than the classical four-stage fourth-order method RK4.In Chapter 4, we consider genetic regulatory networks with delays. By means of simu-lating the period protein PER in Drosophila and the circadian clock in mammals, the effect of delays on the dynamic behavior of genetic regulatory networks is explored.